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ROCK PHYSICS AND SEISMIC SIGNATURES OF SUB-RESOLUTION

SAND-SHALE SYSTEMS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF GEOPHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE

OF DOCTOR OF PHILOSOPHY

Piyapa Dejtrakulwong

December 2012

http://creativecommons.org/licenses/by-nc-sa/3.0/us/

This dissertation is online at: http://purl.stanford.edu/jz636wn5735

© 2012 by Piyapa Dejtrakulwong. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.

ii



http://purl.stanford.edu/jz636wn5735

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Gerald Mavko, Primary Adviser


Tapan Mukerji, Co-Adviser


Stephan Graham

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

v

Abstract

This dissertation aims to improve the interpretation of thinly bedded sand-shale

systems that are below resolution of conventional well-log and seismic data by using

rock physics and quantitative seismic analysis. The key contributions of this

dissertation are (1) incorporation of parameter uncertainties into existing models that

are used for estimating petrophysical properties of sub-resolution sand-shale systems,

(2) a new method for approximating fluid substitution in thinly bedded sand-shale

reservoirs that is applicable at the measurement scale without the need to downscale

the measurements, and (3) an application of rock physics, spatial statistics, and

feature-extraction based attributes to quantitatively interpret seismic data for sub-

resolution reservoir properties such as net-to-gross ratios, saturations, and stacking

patterns.

Most of rock physics relations are derived for rocks that are considered

homogeneous at particular scales. If these relations are applied to measurements at

other scales, the relations often fail when the measurements represent average

properties of heterogeneous rocks (for example, a stack of interbedded sand and shale

layers). First, we investigate the Thomas-Stieber model, a model commonly used for

estimating volume fraction of sand and its porosity in thinly bedded sand-shale

sequences. We present sensitivity and uncertainty analyses of this model under various

scenarios, especially when the model assumptions are violated. We also extend the

model by incorporating uncertainties into the model parameters using Monte Carlo

simulations in a Bayesian framework.

vi

Next, we propose a simple graphical mesh interpretation and accompanying

equations for approximating fluid substitution in sub-resolution interbedded sand-

shale sequences. The advantages of our method are as follows. Even when it is applied

to the measurements at their original scales, our method appropriately changes fluid in

the sands only, without the need to downscale the measurements. The interbedded

sand layers can be either clean or shaly (i.e., sand with dispersed clay). We illustrate

the performance of the model using both synthetic and real well log data and present

sensitivity analysis of the model parameters.

Estimating reservoir properties of sub-resolution sand-shale reservoirs from

seismic data is not straightforward because the relation between seismic signatures and

rock properties are not unique. This relation is even further complicated by the spatial

arrangement of the sub-resolution layers. This dissertation presents a workflow for

seismic interpretation of such thin reservoirs. The workflow consists of four main

steps: (1) estimate transition matrices (Markov chain model) at the well location from

log data, (2) use the matrices to create various sand-shale sequences with varying

reservoir properties such as net-to-gross ratios, saturations, and stacking patterns, (3)

generate synthetic seismograms corresponding to the sequences and from these

seismograms extract attributes which will be used as a training set, (4) and finally use

the training set to estimate reservoir properties of the area away from the well. Most of

seismic attributes discussed here are obtained using feature-extraction techniques,

which compare amplitudes of seismogram segments and find new representations of

these seismograms in a new, smaller set of features. We apply the workflow to both

synthetic data and real data from channelized turbidite deposits in West Africa.

vii

Acknowledgement

My PhD journey at Stanford University is one of the most meaningful experiences

in my life. I would not have come this far on my journey without help and support

from several people. First of all, I am deeply grateful to Gary Mavko, my adviser, for

his teaching and guidance throughout my time at Stanford. His insightful ideas have

tremendous impact on my research. I would like to thank Tapan Mukerji, my co-

adviser, for his advice and inspirational ideas. I would like to thank both Gary and

Tapan for helping me improve my presentation skills.

I am grateful to Jack Dvorkin for his continuous support. I am very impressed by

his kindness and willingness to help students. I am grateful to Steve Graham for his

wonderful teaching and advice on geology concepts. I am indebted to Claude Reichard

for his teaching on technical writing. His advice helps improve the writing of this

dissertation tremendously. I would like to thank Tiziana Vanorio for her support

throughout these years.

I would like to thank Fuad Nijim, Tara Illich, and all administrative staffs for their

help. I also would like to thank the SRB program and its affiliates and thank the SEG

scholarship for financial support.

I would like to thank all SRB colleagues for their support, advice, and friendship:

Kyle Spikes, Kevin Wolf, Tanima Dutta, Kaushik Bandyopadhyay, Richa, Carmen

Gomez, Franklin Ruiz, Ratnanabha Sain, Ramil Ahmadov, Danica Dralus, Nishank

Saxena, Adam Tew, Yu Xia, Kenichi Akama, Dario Grana, Adam Allan, Amrita Sen,

Sabrina Aliyeva, Yuki Kobayashi, Ammar El Husseiny, Priyanka Dutta, Humberto

viii

Samuel Arevalo-Lopez Sr, Chisato Konishi, Huyen Bui, Cinzia Scotellaro, Stephanie

Vialle, and Fabian Krzikalla. I am also grateful to Ezequiel Gonzalez and Juan-

Mauricio Florez for useful discussion and suggestion on my research.

I would like to thank Peerapong Dhangwatnotai for always being by my side.

Lastly, I dedicate this dissertation to my family in Thailand. Their unconditional love

and support always encourage me to keep on walking.

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Contents

Abstract……………………………………………………………………………….v

Acknowledgements…………………………………………………………………vii

Contents……………………………………………………………………………...ix

List of Tables……………………………………………………………………….xiii

List of Figures……………………………………………………………………….xv

Chapter 1 Introduction ................................................................................................ 1

1.1 Objective ........................................................................................................ 1

1.2 Background and motivation ........................................................................... 2

1.2.1 Rock physics relations and sub-resolution heterogeneity ...................... 3

1.2.2 Seismic property estimation of sub-resolution heterogeneous systems . 4

1.3 Definitions of terms, symbols, and abbreviations .......................................... 5

1.4 Chapter description ........................................................................................ 6

1.5 References ...................................................................................................... 7

Chapter 2 Sensitivity and uncertainty analysis of the Thomas-Stieber model for

property estimation of thin sand-shale reservoirs ..................................................... 9 2.1 Abstract .......................................................................................................... 9

2.2 Introduction .................................................................................................. 10

2.3 Volumetric properties of sand-shale mixtures ............................................. 12

2.3.1 Total porosity and volume of shale ...................................................... 12

2.3.2 Important assumptions of the Thomas-Stieber model .......................... 16

2.4 Sensitivity and uncertainty analysis ............................................................. 19

2.4.1 Model with correct input parameters .................................................... 19

2.4.2 Model with uncertain input parameters ................................................ 24

x

2.4.3 Example with well log data .................................................................. 29

2.5 The Thomas-Stieber model on rock-physics cross-plots ............................. 31

2.5.1 Density and volume fraction of shale ................................................... 32

2.5.2 Neutron porosity and density ................................................................ 33

2.5.3 Velocity and total porosity ................................................................... 34

2.5.4 Vp/Vs ratio and acoustic impedance ................................................... 36

2.5.5 Application to real data ........................................................................ 37

2.6 Discussion .................................................................................................... 38

2.7 Conclusions .................................................................................................. 42

2.8 Acknowledgements ...................................................................................... 42

2.9 References .................................................................................................... 42

Chapter 3 Fluid substitution for sub-resolution interbedded sand-shale sequences

using the mesh method ............................................................................................... 47 3.1 Abstract ........................................................................................................ 47

3.2 Introduction .................................................................................................. 49

3.3 Elastic properties of interbedded sands ........................................................ 52

3.3.1 Models .................................................................................................. 52

3.3.2 Modeling the V-point using the Voigt-Reuss-Hill average .................. 56

3.4 Fluid substitution for interbedded sands ...................................................... 60

3.4.1 Approximate fluid substitution in shaly sands ..................................... 60

3.4.2 Graphical interpretation and equation derivations for fluid substitution

in interbedded sands ............................................................................. 62

3.4.3 Important note ...................................................................................... 66

3.5 Synthetic examples ...................................................................................... 66

3.5.1 Case 1: Shaly sand (sand with dispersed clay) with fully-oil-saturated

effective porosity (𝑆𝑊𝑒=0) .................................................................. 67

3.5.2 Case 2: Clean sand interbedded with shale, varying 𝑉𝑠𝑎𝑛𝑑, and fully-

oil-saturated effective porosity (𝑆𝑊𝑒=0) ............................................. 69

3.5.3 Case 3: Shaly sand (𝑉𝑑𝑖𝑠𝑝= 0.15) interbedded with shale, varying

𝑉𝑠𝑎𝑛𝑑, and fully-oil-saturated effective porosity (𝑆𝑊𝑒=0) ................ 70

3.5.4 Case 4: Shaly sand (normal distribution of 𝑉𝑑𝑖𝑠𝑝 with a mean of 0.1

and a standard deviation of 0.05) interbedded with shale, varying

𝑉𝑠𝑎𝑛𝑑, and fully-oil-saturated effective porosity (𝑆𝑊𝑒=0) ................ 71

3.5.5 Case 5: Interbedded sand-shale sequences with systematic changes of

both the 𝑉𝑑𝑖𝑠𝑝 in the sand layers and the 𝑉𝑠𝑎𝑛𝑑 ................................ 73

3.6 Pitfalls in interpretation ................................................................................ 74

3.7 Sensitivity analysis ....................................................................................... 75

3.8 Real data example ........................................................................................ 78

3.9 Discussion .................................................................................................... 84

3.9.1 The mesh method using both bulk and shear moduli ........................... 84

3.9.2 Possible modification of the mesh method when key assumptions are

relaxed .................................................................................................. 85

3.9.3 Using rock-physics trends to constrain clean-sand properties ............. 86

xi

3.9.4 Limitation of the mesh method ............................................................. 87

3.9.5 Comparison with alternative methods .................................................. 87

3.9.6 Upscaled Gassmann’s equations .......................................................... 89

3.10 Conclusions ................................................................................................ 102

3.11 Acknowledgements .................................................................................... 103

3.12 References .................................................................................................. 103

Chapter 4 Seismic signature and uncertainty in petrophysical property

estimation of thin sand-shale reservoirs ................................................................. 107

4.1 Abstract ...................................................................................................... 107

4.2 Introduction ................................................................................................ 109

4.3 Forward modeling for seismic response and attributes .............................. 111

4.3.1 Markov chain models in stratigraphic sequences ............................... 112

4.3.2 Rock-physics models for sand-shale mixtures ................................... 116

4.3.3 Seismic attributes ................................................................................ 118

4.4 Seismic signatures for 1-D Synthetic example .......................................... 124

4.4.1 Model setup ........................................................................................ 124

4.4.2 Scenario 1: Effect of net-to-gross ratios ............................................. 125

4.4.3 Scenario 2: Effect of saturation .......................................................... 138

4.4.4 Scenario 3: Effect of stacking patterns ............................................... 150

4.4.5 Discussions ......................................................................................... 157

4.5 Net-to-gross estimation from 2-D sections ................................................ 160

4.5.1 Model setup ........................................................................................ 161

4.5.2 Results ................................................................................................ 162

4.5.3 Net-to-gross estimation using a Bayesian framework ........................ 163

4.6 Local net-to-gross estimation in non-stationary sequences ....................... 164

4.6.1 Model setup ........................................................................................ 165

4.6.2 Results and discussion ........................................................................ 167

4.7 Discussion .................................................................................................. 170

4.7.1 Comparisons with amplitude attributes .............................................. 170

4.7.2 Notes on the feature-extraction based attributes ................................ 171

4.8 Conclusions ................................................................................................ 173


4.10 References .................................................................................................. 174

Chapter 5 Seismic signature and uncertainty in petrophysical property

estimation of thin sand-shale reservoirs: Case studies .......................................... 179

5.1 Abstract ...................................................................................................... 179

5.2 Introduction ................................................................................................ 181

5.3 Geological background .............................................................................. 181

5.4 Case study 1: Effect of reservoir properties and stacking pattern on seismic

signatures at well location .......................................................................... 183

5.4.1 Net-to-gross estimation from well log data ........................................ 183

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5.4.2 Seismograms at well location ............................................................. 186

5.4.3 Transition matrices and rock property calibration .............................. 187

5.4.4 Synthetic seismograms ....................................................................... 191

5.4.5 Effect of reservoir properties and stacking pattern on seismic signatures

............................................................................................................ 192

5.4.6 Discussion ........................................................................................... 193

5.5 Case study 2: Estimating sub-resolution reservoir properties from a 2-D

seismic section ................................................................................................ 203

5.5.1 2-D seismic section ............................................................................. 203

5.5.2 Rock property calibration at well location ......................................... 203

5.5.3 Non-stationarity when moving away from the well ........................... 205

5.5.4 Synthetic seismogram generation and seismic attribute extraction .... 207

5.5.5 Sensitivity analysis ............................................................................. 211

5.5.6 Property estimation of the 2-D seismic section .................................. 214

5.5.7 Discussion ........................................................................................... 220

5.6 Conclusions ................................................................................................ 221


5.8 References .................................................................................................. 222

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List of Tables

Table 1.1: Definitions of terms, symbols, and abbreviations used throughout this

dissertation. Note that Vi is the volume of the ith

layer illustrated in Figure 1.1, and

i = 1, 2,…, 6. ........................................................................................................... 5

Table 3.1: Models used to generate elastic moduli of brine-saturated shaly-sand. ...... 57

Table 3.2: Models used to estimate effective solid moduli for fluid substitution. ....... 57

Table 3.3: Summary of methods usually used in fluid substitution and comments for

when these methods are applied to sub-resolution interbedded sand-shale

sequences. ............................................................................................................. 88

Table 4.1: Kernel functions used for extracting seismic attributes. Note that xi is the ith

seismogram. ........................................................................................................ 122

Table 4.2: Form of transition matrices for generating sequences used in investigating

net-to-gross effects on seismic signatures. Values of parameter k range from 0.45

to 0.95. The four lithologic states are sand (s), shaly-sand (sh-s), sandy-shale (s-

sh), and shale (sh). ............................................................................................... 126

Table 4.3: Summary of the methods used to compute seismic attributes for net-to-

gross estimation. Choices of parameters for each method are also included.

Performance of each method is shown as a success rate in classifying a data point

into three net-to-gross classes: <0.38, 0.38-0.465, and >0.465 .......................... 137

Table 4.4: Summary of the methods used to compute seismic attributes for water-

saturation (in sand layers) effect. Choices of parameters for each method are

similar to those listed in Table 4.2, unless otherwise specified. Performance of

each method is shown as a success rate in classifying a data point into three

water-saturation categories: Sw=0.1, 0.5, and 1. Results are shown in the columns

corresponding to the transition matrices used in simulations. The actual transition

matrices and sample sequences are shown in Figure 4.23. ................................. 149

xiv

Table 4.5: Summary of the methods used to compute seismic attributes for stacking-

pattern effects using sequences generated from the fixed-sampling transition

matrices. Choices of parameters for each method are also included. Performance

of each method is shown as a success rate in classifying a data point into three

stacking patterns: retrogradational, progradational, and aggradational patterns. 152

Table 4.6: for stacking-pattern effects using sequences generated from the fixed-

sampling transition matrices. Choices of parameters for each method are also

included. Performance of each method is shown as a success rate in classifying a

data point into three stacking patterns: retrogradational, progradational, and

aggradational patterns. ........................................................................................ 156

Table 5.1: Specifications for synthetic earth models with various net-to-gross ratios,

stacking patterns, saturating fluids, and layer thicknesses. ................................. 191

xv

List of Figures

Figure 1.1: Different types of clay/shale distribution in sand. The volume of the ith

layer and the volume of dispersed clay are denoted by Vi and Vdisp, respectively. . 6

Figure 2.1: Shale distribution in sand (Modified after Mavko et al., 2009). ................ 12

Figure 2.2: Relation between total porosity and shale volume. Point A and E represent

clean sand and pure shale point, respectively. Points B and C represent sand with

dispersed clay, and point D is where the original sand pore space is completely

filled with dispersed clay. ..................................................................................... 13

Figure 2.3: Graphical determination of the laminated shale and the dispersed clay

volumes (𝑉𝑙𝑎𝑚and 𝑉𝑑𝑖𝑠𝑝, respectively) for a measurement at point F. In each

case, the shale volume is determined by drawing a line from the shale point (E),

passing through point F, and intersecting line AD. The intersection point (C)

represents shaly sand that is the sandy end-member of the lamination, and this

shaly sand point has 𝑉𝑑𝑖𝑠𝑝equal to its x-coordinate (𝑉𝑑𝑖𝑠𝑝= 0.2). Then, 𝑉𝑙𝑎𝑚 is

simply the ratio between the lengths of line CF and CE, which is equal to 0.4. .. 15

Figure 2.4: Distorting total porosity diagram (blue) for computing effective porosity

(red). ...................................................................................................................... 16

Figure 2.5: Total porosity and shale volume after cement is added into the remaining

sand pore space of the sands on the dispersed sand line. The blue line is the

resulting dispersed sand line after adding cement with a volume of 25% of the

porosity of the sand end-point into the pore space (Modified after Juhasz, 1986).

............................................................................................................................... 17

Figure 2.6: Thomas-Stieber diagram (left) with plotted properties from the

corresponding earth model which is represented by an interbedded sand-shale

sequence (right). The properties of the individual sand types (A, B, and C) and

shale are shown in circles. The average total porosity and shale volume fraction

of the earth model is marked with an “X.” The volume fractions of laminated

shale and dispersed clay can be determined graphically by first drawing a line that

xvi

originates from the shale point, passes through point X, and intersects the

dispersed line. The volume fraction of dispersed clay is simply equal to the x-

coordinate of the intersection point, and the volume fraction of laminated shale

(𝑉𝑙𝑎𝑚 = 1 – 𝑉𝑠𝑎𝑛𝑑) is the ratio between the distance from point X to the

intersection point and the distance from the shale point to the intersection point.20

Figure 2.7: Estimated sand fraction and sand property when applying Thomas-Stieber

diagram to point “X” in Figure 2.6 (left). The estimated values are compared with

the true values from the earth model (Figure 2.6, right). ...................................... 20

Figure 2.8: Thomas-Stieber diagram (left) with plotted properties from the

corresponding earth model which is represented by an interbedded (shaly) sand-

(sandy) shale sequence (right). The properties of sand and shale layers within the

earth model are shown in circles. The average total porosity and shale volume

fraction of the earth model is marked with an “X.” The volume fractions of

laminated shale and dispersed clay can be determined graphically by first drawing

a line that originates from the shale point, passes through point X, and intersects

the dispersed line. The volume fraction of dispersed clay is simply equal to the x-

coordinate of the intersection point, and the volume fraction of laminated shale

(𝑉𝑙𝑎𝑚 = 1 – 𝑉𝑠𝑎𝑛𝑑) is the ratio between the distance from point X to the

intersection point and the distance from the shale point to the intersection point.21


diagram to point “X” in Figure 2.8 (left). The estimated values are compared with

the true values from the earth model (Figure 2.8, right). ...................................... 22

Figure 2.10: Relations between total porosity and shale volume fraction of bimodal

mixtures using the fractional packing model by Kolterman and Gorelick (1995)

and the Thomas-Stieber model (i.e., ideal mixing model). Data points

representing sand/shaly-sand interbedded with shale (solid circles) are generated

using the fractional packing model with varying sand fractions and volume

fractions of dispersed clay..................................................................................... 24

Figure 2.11: Percentage differences between the estimated sand fractions (left), the

estimated volume fractions of dispersed clay in the sand (right) and the true

values. The ideal mixing model and the fractional packing model (i.e., non-ideal

mixing) are outlined. ............................................................................................. 24

Figure 2.12: Multiple realizations of Thomas-Stieber diagrams generated from a set of

sand and shale end-points. Three data points are labeled. ................................... 26

Figure 2.13: Posterior distributions for estimated sand fractions of the three data points

shown in Figure 2.12. ............................................................................................ 27

Figure 2.14: Posterior distributions of volume fraction of dispersed clay in sand (left)

and sand fraction (right) for the three data points shown in Figure 2.12. ............. 27

Figure 2.15: Standard deviations of estimated properties at different locations on the

Thomas-Stieber diagram. The two estimated values are (left) volume fraction of

dispersed clay and (right) sand fraction. The three data points are the same points

as in Figure 2.12. ................................................................................................... 29

Figure 2.16: Three lithofacies from a detailed core analysis by Lowe (2004).

Lithofacies 1 represents thick-bedded to massive sandstone. Lithofacies 2 and 3

represent interbedded, thin-bedded sandstone and mudstone. In lithofacies 2, the

xvii

sandstone beds are 2 – 20 cm thick and in lithofacies 3 the beds are less than 2 cm

thick (Modified from Dutta, 2009). ...................................................................... 30

Figure 2.17: Petrophysical analysis of the selected well-log interval using the Thomas-

Stieber model. (Left) Total porosity and gamma ray values for three lithofacies in

the selected interval. The median total porosity and gamma ray values for each

lithofacies are shown in solid circles. From these median points, the up-down or

left-right bars indicate the interquartile ranges (i.e., from 1st to 3

rd quartiles) of

each property. A Thomas-Stieber diagram is also superimposed on the data.

(Right) Variation of gamma ray values with depth. Data points are color-coded by

lithofacies similar to the left panel. The gamma ray log shows an upward-fining

trend. ..................................................................................................................... 30

Figure 2.18: Probability density functions of properties of the sand end-point and the

shale end-point. (Left) total porosity and (right) gamma ray value. ..................... 31

Figure 2.19: Estimated sand fractions for the selected well-log interval. (Left) gamma

ray log of the selected interval and (right) posterior distributions of estimated

sand fractions. ....................................................................................................... 31

Figure 2.20: Relationship between density and volume fraction of shale in shaly sand

lamination. Point A and E represent clean sand point and pure shale point,

respectively. Points B and C represent sand with dispersed clay, and point D is

where the original sand pore space is completely filled with dispersed clay

(Modified after Mavko et al., 2009). ..................................................................... 32

Figure 2.21: Density-neutron plot for the dispersed and laminated sand-shale systems.

............................................................................................................................... 34

Figure 2.22: Velocity to total porosity curves for the dispersed sand-shale system. The

model used here follows Dvorkin and Gutierrez, 2002. ....................................... 35

Figure 2.23: Velocity to total porosity curves for the dispersed sand-shale system. The

model used here follows Dvorkin and Gutierrez, 2002. Each black line represents

lamination between sand (or shaly sand) and shale with volume fraction of

laminated shale ranging from 0 to 1. Each blue line represents lamination between

sand with a volume fraction of dispersed clay ranging from 0 to 𝜙𝑐𝑙𝑒𝑎𝑛 𝑠𝑎𝑛𝑑

and shale with a constant volume fraction of laminated shale. ............................. 35

Figure 2.24: Rock-physics template shown as a cross-plot between Vp/Vs and AI with

superimposed rock-physics trends. The green and magenta curves represent the

shale and wet-sand lines, respectively. Along these lines, the change in porosity is

due to packing or grain sorting. At each porosity value, the red curve which is

connected to the wet-sand line represents the corresponding gas-saturated sand

with varying saturations. The dispersed and laminated sand-shale system is

constructed using the sand and shale points shown in red circles. Each blue line

represents lamination between sand with a volume fraction of dispersed clay

ranging from 0 to 𝜙𝑐𝑙𝑒𝑎𝑛 𝑠𝑎𝑛𝑑 and shale with a constant volume fraction of

laminated shale. ..................................................................................................... 37

Figure 2.25: Selected well-log data on different cross-plots for laminated and

dispersed sand-shale systems. The data points are color-coded by their

corresponding gamma ray values. ......................................................................... 38

xviii

Figure 2.26: The effect of vertical resolution on cross-plots between total porosity and

shale volume of a synthetic earth model between shaly sand and shale. The left

panel shows a cross-plot between the two measurements with two different

resolutions; whereas, the middle panel shows a cross-plot when both

measurements share the same resolution. The right panel is a short section of the

corresponding synthetic earth model. Sand and shale layers are shown in white

and black, respectively. ......................................................................................... 40


diagram to points in Figure 2.26 to investigate the effect of vertical resolution on

interpretation using cross-plots between total porosity and shale volume............ 40

Figure 2.28: The effect of vertical resolution on cross-plots between total porosity and

shale volume of a synthetic (non-stationary) earth model between shaly sand and

shale. The left panel shows a cross-plot between the two measurements with two

different resolutions; whereas, the middle panel shows a cross-plot when both

measurements share the same resolution. The right panel is a short section of the

corresponding synthetic earth model. Sand and shale layers are shown in white

and black, respectively. ......................................................................................... 41


diagram to points in Figure 2.28 to investigate the effect of vertical resolution on

interpretation using cross-plots between total porosity and shale volume............ 41

Figure 3.1: Inverted-V relation between P-wave velocity and total porosity for a

dispersed sand-shale system, following the model of Dvorkin and Gutierrez

(2002). In this case, curves are computed using the Hashin-Shtrikman lower

bound (HSLB). The solid magenta and blue curves represent the sandy-shale and

shaly-sand legs of the dispersed sand-shale system. ............................................. 53

Figure 3.2: Relation between velocity and total porosity for a dispersed sand-shale

system, following the model of Dvorkin and Gutierrez (2002) with a slight

modification. Each red line represents interbedding of sand (or shaly sand) and

shale with volume fraction of interbedded shale ranging from 0 to 1. Each blue

line represents interbedding of sand with a volume fraction of dispersed clay

ranging from 𝑉𝑑𝑖𝑠𝑝 = 0 to 𝑉𝑑𝑖𝑠𝑝 = 𝜙𝑆 and shale with a constant volume

fraction of interbedded shale. ................................................................................ 54

Figure 3.3: P-wave compliance (C=1/M) versus effective porosity curves for a

interbedded sand-shale system. Each red line represents interbedding of sand (or

shaly sand) and shale with volume fraction of laminated (interbedded) shale

ranging from 𝑉𝑙𝑎𝑚 = 0 to 𝑉𝑙𝑎𝑚 = 1. Each blue line represents interbedding of

sand with a volume fraction of dispersed clay ranging from 𝑉𝑑𝑖𝑠𝑝 = 0 to

𝑉𝑑𝑖𝑠𝑝 = 𝜙𝑆 and shale with a constant volume fraction of laminated (interbedded)

shale. ..................................................................................................................... 55

Figure 3.4: Shaly sand lines before and after fluid substitution when only P-wave

moduli are used in calculations. The model used to generate wet shaly sand is

listed in the lower right corner ((a) – (d) and Table 3.1). Starting with the brine-

saturated shaly sand (black dash-line), oil is substituted using the Gassmann

xix

shaly-sand equation (Dvorkin et al., 2007) and three different models for effective

solid moduli (Table 3.2). ....................................................................................... 59

Figure 3.5: Shaly sand lines before and after fluid substitution when both bulk and

shear moduli are used in calculations. The model used to generate wet shaly sand

is listed in the lower right corner ((a) – (d) and Table 3.1). Starting with the brine-

saturated shaly sand (black dash-line), oil is substituted using the Gassmann

shaly-sand equation and three different models for effective solid moduli (Table

3.2). ....................................................................................................................... 60

Figure 3.6: Approximate fluid substitution for a dispersed shaly-sand line. Applying

fluid substitution to this shaly-sand line is approximately equivalent to moving

the clean sand point up or down, following the usual Gassmann’s equation, while

the V-point at the other end of this line is fixed. Then, the new clean sand point is

connected to the fixed V-point by another straight line. Here, we show an

example when the clean sand point is moved down after fluid substitution. ........ 62

Figure 3.7: Graphical interpretation of fluid substitution by our mesh method.

Applying approximate fluid substitution to any clean sand (or sand with dispersed

clay) interbedded with shale is simply equivalent to distorting the mesh. Here the

distortion is shown in the P-compliance (C=1/M) versus effective porosity plane.

The distortion moves the mesh accordingly with the change in the clean sand

compliance after fluid substitution. The blue arrow shows how a data point inside

the triangular diagram moves after the distortion. ................................................ 63

Figure 3.8: Schematic diagram and terminology for the laminated (interbedded) sand-

shale system. ......................................................................................................... 64

Figure 3.9: Fluid substitution results from four different procedures for shaly sands

with no interbedding. The procedures are Gassmann’s equation applied to sand

layers only + upscaling, Gassmann shaly-sand equation (Dvorkin et al., 2007)

applied to each sand layers only + upscaling, Gassmann’s equation applied at the

measurement scale, and our mesh method. For this synthetic case, volume

fractions of dispersed clay in the sand range from 𝑉𝑑𝑖𝑠𝑝= 0 to 𝑉𝑑𝑖𝑠𝑝 = 𝜙𝑠. ..... 68

Figure 3.10: Fluid substitution results from four different procedures for interbedded

clean sand-shale sequences. The procedures are Gassmann’s equation applied to

sand layers only + upscaling, Gassmann shaly-sand equation (Dvorkin et al., 2007)

applied to each sand layers only + upscaling, Gassmann’s equation applied at the

measurement scale, and our mesh method. Note that results of Gassmann’s

equation + upscaling, Gassmann shaly-sand equation + upscaling, and the mesh

are on top of each other. The x-axis represents pseudo-depth. ............................. 69

Figure 3.11: Sand fractions of the synthetic model for interbedded clean sand-shale

sequences. The x-axis represents pseudo-depth. ................................................... 69

Figure 3.12: Percentage differences between velocities after fluid substitution by

Gassmann’s equation and Gassmann’s equation applied to only sand layers

followed by upscaling. These differences are plotted against sand fraction and

volume fraction of shale (𝑉𝑠ℎ). In this clean sand case, sand fraction is simply

equivalent to 1 – 𝑉𝑠ℎ. ........................................................................................... 70


shaly sand-shale sequences, with a fixed volume fraction of dispersed clay. The

xx

procedures are Gassmann’s equation applied to sand layers only + upscaling,

Gassmann shaly-sand equation (Dvorkin et al., 2007) applied to each sand layers

only + upscaling, Gassmann’s equation applied at the measurement scale, and our

mesh method. The x-axis represents pseudo-depth. ............................................. 70

Figure 3.14: Sand fractions of the synthetic model for interbedded shaly sand-shale

sequences, with a fixed volume fraction of dispersed clay. The x-axis represents

pseudo- depth. ....................................................................................................... 71


shaly sand-shale sequences, with varying volume fractions of dispersed clay. The

procedures are Gassmann’s equation applied to sand layers only + upscaling,

Gassmann shaly-sand equation (Dvorkin et al., 2007) applied to each sand layers

only + upscaling, Gassmann’s equation applied at the measurement scale, and our

mesh method. The x-axis represents pseudo-depth. ............................................. 71

Figure 3.16: Sand fractions of the synthetic model for interbedded shaly sand-shale

sequences, with varying volume fractions of dispersed clay. The x-axis represents

pseudo-depth. ........................................................................................................ 72

Figure 3.17: Changes in P-wave velocity after fluid substitution from four different

procedures: Gassmann’s equation applied to sand layers only + upscaling,

Gassmann shaly-sand equation (Dvorkin et al., 2007) applied to sand layers only

+ upscaling, Gassmann’s equation applied at the measurement scale, and our

mesh method. Each location on the triangular diagram represents a interbedded

sand-shale sequence with a unique pair of sand fraction and volume fraction of

dispersed clay values. ............................................................................................ 73

Figure 3.18: Pitfalls in interpretation of fluid substitution results when thin

interbedding exists. (Left) Synthetic gas-saturated data with sand fractions greater

than or equal to 0.5. (Middle) Results after fluid substitution from gas to brine

using our mesh approach. (Right) Results after fluid substitution from gas to brine

using Gassmann’s equation. Data points are color-coded by water saturation

values before fluid substitution. ............................................................................ 75

Figure 3.19: Sensitivity analysis results of the mesh method for five input parameters:

P-wave velocity and total porosity of clean sand, effective porosity and effective

water saturation of the interbedded package, and elastic modulus of fluid. The

thick lines on the top of each subplot are the oil-saturated velocity and the

reference velocities after fluid substitution from oil to brine using four procedures

with all correct input parameters: Gassmann ignoring interbedding, Gassmann

applied to each sand layer + upscaling, Gassmann shaly-sand equation (Dvorkin

et al., 2007) applied to each sand layer + upscaling, and our mesh method. Results

of sensitivity analysis for each parameter are shown as a velocity distribution,

which is normalized to one. Sand fractions and volume fractions of dispersed clay

for each synthetic model are shown on above each subplot. ................................ 78

Figure 3.20: Selected dataset represents an interbedded sand-shale sequence. The data

is color-coded by Gamma ray values. The Thomas-Stieber-Yin-Marion model is

superimposed onto the data. The set of parallel lines labeled as 𝑉𝑠𝑎𝑛𝑑 represent

volume fractions of laminated (interbedded) sand according to the Thomas-

Stieber-Yin-Marion model. ................................................................................... 81

xxi

Figure 3.21: Comparisons of fluid substitution results which are color-coded by sand

fractions estimated from the Thomas-Stieber-Yin-Marion model. Four fluid

substitution procedures are used: Gassmann’s equation (i.e., ignoring the effect of

sand-shale interbedding), Gassmann shaly-sand equation (i.e., assuming that shale

is all dispersed; Dvorkin et al., 2007), the downscaling-upscaling procedure, and

the mesh method. Note that points with very low sand fractions are excluded from

the plot. Lines X = Y are super-imposed for comparison purpose. ...................... 81


fractions estimated from the P-compliance versus effective porosity plane. Four

fluid substitution procedures are used: Gassmann’s equation (i.e., ignoring the

effect of sand-shale interbedding), Gassmann shaly-sand equation (i.e., assuming

that shale is all dispersed; Dvorkin et al., 2007), the downscaling-upscaling

procedure, and the mesh method. Note that points with very low sand fractions

are excluded from the plot. Lines X = Y are super-imposed for comparison

purpose. ................................................................................................................. 83

Figure 3.23: Instability of the inversion step in the downscaling-upscaling procedure

leading to spikes in P-wave velocity after fluid substitution. ............................... 84

Figure 3.24: Fluid substitution results by four different procedures for shaly sands,

with no interbedding. The procedures are Gassmann using total porosity and

ignoring interbedding, Gassmann using total porosity (applied to sand layers only)

+ upscaling, Gassmann using effective porosity by Dvorkin et al., 2007 (applied

to each sand layers only) + upscaling and our method (i.e., the mesh). For this

synthetic case, volume fractions of dispersed clay in the sand range from

𝑉𝑑𝑖𝑠𝑝 =0 to 𝑉𝑑𝑖𝑠𝑝 = 𝜙𝑠. Both bulk and shear moduli are used for this synthetic

example. ................................................................................................................ 85

Figure 3.25: Modified upper bounds for modeling the clean-sand diagenetic trend.

This example represents a quartz-water system. ................................................... 87

Figure 3.26: Percentage differences between the baselines and the predicted velocities

by the upscaled Gassmann’s equation (left) and the Gassmann shaly-sand

equation (Dvorkin et al., 2007) (right) for the first scenario, where P-wave moduli

and approximate Gassmann’s equation are used for the upscaled Gassmann. Each

location on the triangular diagram represents an interbedded sand-shale sequence

with a unique pair of sand fraction and volume fraction of dispersed clay values.

Note that the color scales of the two panels are different. .................................... 96



equation (Dvorkin et al., 2007) (right) for the second scenario, where both bulk

and shear moduli, and approximate Gassmann’s equation are used in calculations.

Each location on the triangular diagram represents an interbedded sand-shale

sequence with a unique pair of sand fraction and volume fraction of dispersed

clay values. Note that the color scales of the two panels are different. ................ 96



equation (Dvorkin et al., 2007) (right) for the second scenario, where both bulk

and shear moduli, and the actual Gassmann’s equation are used for the upscaled

xxii

Gassmann. Each location on the triangular diagram represents an interbedded

sand-shale sequence with a unique pair of sand fraction and volume fraction of

dispersed clay values. Note that the color scales of the two panels are different.

The color scale on the left panel does not cover either the actual maximum or

minimum values of the percentage differences. The color scale is adjusted to

show values of the majority of the results. Those results indicated by a magenta

ellipse are unreliable due to instability of the method at low effective porosity. . 97

Figure 3.29: Comparisons of five sets of fluid substitution results which are color-

coded by sand fractions estimated from the Thomas-Stieber-Yin-Marion model.

Five fluid substitution procedures are used: Gassmann’s equation (i.e., ignoring

the effect of sand-shale interbedding), Gassmann shaly-sand equation (i.e.,

assuming that shale is all dispersed; Dvorkin et al., 2007), the downscaling-

upscaling procedure, the mesh method, and the upscaled Gassmann’s equation.

Note: points with very low sand fractions are excluded from the plot. Lines X = Y

are superimposed for comparison purpose. ......................................................... 100

Figure 3.30: Sensitivity analysis results of the upscaled Gassmann for six input

parameters: P-wave velocity of wet porous shale, volume fraction of dispersed

clay, sand fraction, effective water saturation, bulk modulus of fluid, and effective

porosity of the interbedded package. The thick lines on the top of each subplot are

the oil-saturated velocity and the reference velocities after fluid substitution from

oil to brine using the downscaling-upscaling procedure and the upscaled

Gassmann with all correct input parameters. Results of sensitivity analysis for

each parameter are shown as a velocity distribution, which is normalized to one.

Sand fractions and volume fractions of dispersed clay for each synthetic model

are shown on the top of each subplot. ................................................................. 102

Figure 4.1: Overall workflow for seismic-signature study and property estimation.

First, thin sand-shale sequences are generated using Markov-chain models and

rock physics relations. Then, seismic responses for these sequences are modeled,

and seismic attributes are extracted. Finally, the attributes are related to reservoir

properties, which can be used for reservoir characterization of target areas. ..... 112

Figure 4.2: Three examples of transition matrices with fixed sampling intervals:

retrogradational, progradational, and aggradational sequences. The lithologic

states in the transition matrices are sand (s), shaly sand (sh-s) sandy shale (s-sh)

and shale (sh). The off diagonal elements marked by arrows control the

directionality of the sequences. ........................................................................... 114

Figure 4.3: Examples of an embedded-form transition matrix with realizations of

sequences. The lithologic states in the transition matrix are sand (s), shaly sand

(sh-s) sandy shale (s-sh) and shale (sh). An example of thickness distributions

used is shown in the lower left corner................................................................. 115

Figure 4.4: Examples of SP log responses showing stacking patterns in parasequence

sets. (Left) Retrogradational, (middle) progradational and (right) aggradational

patterns (Modified after Van Wagoner et al., 1990). .......................................... 116

Figure 4.5: Illustrations of sand-shale mixtures, with their porosity and velocity values

related to clay content (Modified after Marion et al., 1992). Porosity versus clay

xxiii

content shows a V-shaped trend, where the two end points are the pure sand and

pure shale porosity. Selected clay fractions corresponding to the four lithologic

states are marked. The states are sand (s), shaly-sand (sh-s), sandy-shale (s-sh),

and shale (sh). ..................................................................................................... 117

Figure 4.6: Wavelet-transform analysis for extracting attributes from a seismogram.

(Left) modulus of wavelet-transform coefficients for the seismogram. The red

arrow indicates the modulus along a particular scale. (Right) a log-log plot of

scale versus variance of modulus of wavelet-transform coefficients. The plot is

shown in open circles which are fit by a straight line. The slope and intercept of

this line are used as seismic attributes. ............................................................... 119

Figure 4.7: Modulus of wavelet coefficients (shown in the middle two boxes) of two

seismogram segments. Dissimilarity between these seismograms is defined as

distance between their modulus maps. ................................................................ 120

Figure 4.8: Workflow for generating seismic responses of multiple realizations of thin

sand-shale sequences. The four lithologic states in the transition matrix are sand

(s), shaly-sand (sh-s), sandy-shale (s-sh), and shale (sh). ................................... 125

Figure 4.9: Distributions of net-to-gross ratios from sequence realizations generated

using the transition matrices in Table 4.2. Red lines indicate values of limiting

sand distributions. From top left to bottom right, values of the parameter k defined

in the transition matrices are 0.45, 0.55, 0.65, 0.75, 0.85 and 0.95, respectively.

............................................................................................................................. 126

Figure 4.10: Slope and intercept of wavelet transforms of seismic responses for 2

different standard deviations of velocity distributions: (left) v = 0.1 km/s and

(right) v = 0.2 km/s. Points correspond to 6 sets of 200 realizations generated

from different transition matrices with various net-to-gross ratios (𝝅sand). Sw is

0.1 for sand layers and 1 for the others. .............................................................. 127

Figure 4.11: Probability distributions of slope and intercept values corresponding to

data points in Figure 4.10. (Left column) v = 0.1 km/s and (right column) v =

0.2 km/s. The direction of increasing net-to-gross ratios (𝝅sand) is to the left for all

plots. .................................................................................................................... 127

Figure 4.12: Classical MDS results for studying effect of net-to-gross ratios on seismic

signature. (Top left) Dissimilarity matrix showing pairwise Euclidean distances

between any two seismograms from realizations with standard deviations of

velocities equal to 0.1 km/s. (Bottom left) Correlation coefficient between

dissimilarity and distance (between points in new coordinates resulting from

classical MDS) versus numbers of included coordinates. (Top right) Classical

MDS results with the first two coordinates. Each point represents one seismogram,

color-coded by its true net-to-gross value from its corresponding sand-shale

sequence. (Bottom right) Classical MDS results with the first two coordinates

converted into polar coordinates. ........................................................................ 131

Figure 4.13: Metric MDS results for studying effect of net-to-gross ratios on seismic

signature. (Top left) Dissimilarity matrix showing pairwise (Euclidean) distances



dissimilarity and distance (between points in new coordinates resulting from

xxiv

metric MDS) versus numbers of included coordinates. (Top right) Metric MDS

results with the first two coordinates. Each point represents one seismogram,


sequence. (Bottom right) Metric MDS results with the first two coordinates


Figure 4.14: Non-metric MDS results for studying effect of net-to-gross ratios on

seismic signature. (Top left) Dissimilarity matrix: pairwise (Euclidean) distances



dissimilarity and distance (between points in new coordinates resulting from non-

metric MDS) versus numbers of included coordinates. (Top right) Non-metric-

MDS results with the first two coordinates. Each point represents one seismogram,


sequence. (Bottom right) Non-metric MDS results with the first two coordinates


Figure 4.15: Gaussian KPCA results for studying effect of net-to-gross ratios on

seismic signature. (Left) Kernel matrix using a Gaussian kernel. Each element in

the matrix (K(xi,xj)) corresponds to the Gaussian kernel function evaluated using

a pair of seismograms (xi,xj) from realizations with standard deviations of

velocities equal to 0.1 km/s. (Right) Projections of seismograms, which

correspond to sequences with different net-to-gross ratios, onto the first two

principal components from the Gaussian KPCA. Each point is color-coded by the

net-to-gross ratio from its corresponding sand-shale sequence. ......................... 134

Figure 4.16: Dynamic similarity KPCA results for studying effect of net-to-gross

ratios on seismic signature. (Left) Kernel matrix using a dynamic similarity

kernel. Each element in the matrix (K(xi,xj)) corresponds to the dynamic

similarity kernel function evaluated using a pair of seismograms (xi,xj) from

realizations with standard deviations of velocities equal to 0.1 km/s. (Right)

Projections of seismograms, which correspond to sequences with different net-to-

gross ratios, onto the first two principal components from the dynamic similarity

KPCA. Each point is color-coded by the net-to-gross ratio from its corresponding

sand-shale sequence. ........................................................................................... 134

Figure 4.17: Inverse multi-quadric KPCA results for studying effect of net-to-gross

ratios on seismic signature. (Left) Kernel matrix using an inverse multi-quadric

kernel. Each element in the matrix (K(xi,xj)) corresponds to the inverse multi-

quadric kernel function evaluated using a pair of seismograms (xi,xj) from

realizations with standard deviations of velocities equal to 0.1 km/s. (Right)

Projections of seismograms, which correspond to sequences with different net-to-

gross ratios, onto the first two principal components from the inverse multi-

quadric KPCA. Each point is color-coded by the net-to-gross ratio from its

corresponding sand-shale sequence .................................................................... 135

Figure 4.18: Polynomial KPCA results for studying effect of net-to-gross ratios on

seismic signature. (Left) Kernel matrix using a polynomial kernel. Each element

in the matrix (K(xi,xj)) corresponds to the polynomial kernel function evaluated

using a pair of seismograms (xi,xj) from realizations with standard deviations of

xxv

velocities equal to 0.1 km/s. (Right) Projections of seismograms, which

correspond to sequences with different net-to-gross ratios, onto the first two

principal components from the polynomial KPCA. Each point is color-coded by

the net-to-gross ratio from its corresponding sand-shale sequence. ................... 135

Figure 4.19: Linear KPCA results for studying effect of net-to-gross ratios on seismic

signature. (Left) Kernel matrix using a linear kernel. Each element in the matrix

(K(xi,xj)) corresponds to the linear kernel function evaluated using a pair of

seismograms (xi,xj) from realizations with standard deviations of velocities equal

to 0.1 km/s. (Right) Projections of seismograms, which correspond to sequences

with different net-to-gross ratios, onto the first two principal components from the

linear KPCA. Each point is color-coded by the net-to-gross ratio from its

corresponding sand-shale sequence. ................................................................... 136

Figure 4.20: Change in MDS classification success rate when the number of

coordinates included as net-to-gross attributes increases. Results from three MDS

methods are shown. ............................................................................................. 136

Figure 4.21: Change in KPCA classification success rate when the number of principal

components included as net-to-gross attributes increases. .................................. 138

Figure 4.22: Parallel coordinate plot. (Left) Polylines of the first five principal

components from the Gaussian KPCA. Each line is color-coded by its

corresponding net-to-gross class. Three classes are <0.38, 0.38-0.465, and >0.465.

(Right) The solid lines are the median (i.e., the 0.5-quantile) of the component

values. The dash lines surrounding the median are the 0.45- and 0.55- quantiles.

............................................................................................................................. 138

Figure 4.23: Three selected matrices with the same limiting distribution and their

sample sequences. From left to right, sand layers become more blocky (i.e.,

groups of sand layers become thicker). The lithologic states are sand (s), shaly

sand (sh-s) sandy shale (s-sh) and shale (sh). Note that each row of matrix B is

equal to a fixed probability vector, and thus the lithologic states generated using

this matrix are considered as independent random events (i.e., the current state

has no dependency on the previous states). ........................................................ 139

Figure 4.24: Slope and intercept attributes for varying water saturation within the sand

layers. Results in each plot are from sequences generated by using the transition

matrix shown in the lower-right corner of each plot. From left to right, the

transition matrices correspond to A, B, and C in Figure 4.23............................ 143

Figure 4.25: Projections of seismograms, which correspond to sequences with

different water-saturation values, onto the first two coordinates from classical

MDS. Each point is color-coded by the water-saturation value of the sand layers.

From left to right, subplots correspond to transition matrices A, B, and C,

respectively. ........................................................................................................ 143


different water-saturation values, onto the two coordinates from metric MDS.

Each point is color-coded by the water-saturation value of the sand layers. From

left to right, subplots correspond to transition matrices A, B, and C, respectively.

............................................................................................................................. 143

xxvi


different water-saturation values, onto the two coordinates from non-metric MDS.

Each point is color-coded by the water-saturation value of the sand layers. From

left to right, subplots correspond to transition matrices A, B, and C, respectively.

............................................................................................................................. 144


different water-saturation values, onto the first two principal components from the

Gaussian KPCA. Each point is color-coded by the water-saturation value of the

sand layers. From left to right, subplots correspond to transition matrices A, B,

and C, respectively. ............................................................................................. 144



dynamic similarity KPCA. Each point is color-coded by the water-saturation

value of the sand layers. From left to right, subplots correspond to transition

matrices A, B, and C, respectively. ..................................................................... 144



inverse multi-quadric KPCA. Each point is color-coded by the water-saturation

value of the sand layers. From left to right, subplots correspond to transition

matrices A, B, and C, respectively. ..................................................................... 145



polynomial KPCA. Each point is color-coded by the water-saturation value of the

sand layers. From left to right, subplots correspond to transition matrices A, B,

and C, respectively. ............................................................................................. 145

Figure 4.32: First 21 eigenvalues from classical MDS, color-coded by the transition

matrices used in generating sand-shale sequences for investigating the effect of

saturations. .......................................................................................................... 145

Figure 4.33: Correlation coefficient between dissimilarity and distance (between

points in new coordinates resulting from an MDS algorithm) versus numbers of

included coordinates from MDS results for investigating the effect of saturations.

(Left) metric MDS and (right) non-metric MDS. ............................................... 146


coordinates included as saturation attributes increases. Three MDS algorithms are

compared. Results when using sequences generated by transition matrices A (left),

B (middle) and C (right) are shown. ................................................................... 146

Figure 4.35: Change in KPCA classification success rate as the number of principal

components included as attributes increases, when using sequences generated by

transition matrices A. .......................................................................................... 146



transition matrices B. .......................................................................................... 147



transition matrices C. .......................................................................................... 147

xxvii

Figure 4.38: Component-wise analysis for the Gaussian KPCA. (Left) Parallel

coordinate plot of the first ten principal components of the Gaussian KPCA for

matrix B. The solid lines are the median (the 0.5 quantile). The dash lines

surrounding the median are the 0.45 and 0.55 quantiles. (Right) Projections of

seismograms onto the first and sixth principal components of the Gaussian KPCA.

............................................................................................................................. 148

Figure 4.39: Component-wise analysis for the inverse multi-quadric KPCA. (Left)

Parallel coordinate plot of the first ten principal components of the inverse multi-

quadric KPCA for matrix C. The solid lines are the median (the 0.5 quantile). The

dash lines surrounding the median are the 0.45 and 0.55 quantiles. (Right)

Projections of seismograms onto the first and sixth principal components of the

inverse multi-quadric KPCA. .............................................................................. 148

Figure 4.40: Sample sequences from three selected transition matrices with the same

limiting distribution: [.25 .25 .25 .25]. From left to right, columns represent

retrogradational, progradational, and aggradational stacking patterns, respectively.

Red arrows schematically indicate transitions from coarse to fine grains, and vice

versa. Note that other interpretations of transitional patterns are possible. The

lithologic states are sand (s), shaly sand (sh-s) sandy shale (s-sh) and shale (sh).

............................................................................................................................. 150

Figure 4.41: Parallel coordinate plot of the first ten components of the Gaussian

KPCA result for investigating the effect of stacking pattern on seismic signatures.

The solid lines are the median (i.e., the 0.5-quantile) of the component values.

The dash lines surrounding the median are the 0.45- and 0.55- quantiles. All lines

are color-coded by the types of stacking patterns. .............................................. 151

Figure 4.42: Sample sequences using the embedded-form transition matrices shown on

the left with varying averages of exponentially-distributed lithologic thicknesses

along the sequences. From (A) to (C), the three stacking patterns are

retrogradational (overall thinning- and fining-upward), progradational (overall

thickening- and coarsening-upward), and aggradational stacking patterns in

parasequence sets, respectively. Red arrows schematically show a series of

progradational parasequences within retrogradational, progradational, and

aggradational parasequence sets. Note that other interpretations of such

parasequence patterns are possible The lithologic states are sand (s), shaly sand

(sh-s) sandy shale (s-sh) and shale (sh). .............................................................. 154

Figure 4.43: Change in KPCA classification success rate when the number of principal

components included as attributes increases. ...................................................... 157

Figure 4.44: Parallel coordinate plot of the first ten coordinates of the classical MDS

result. The solid lines are the median (i.e., the 0.5-quantile) of the component . 157

Figure 4.45: Projections of seismograms with added noise onto the first two principal

components from the dynamic similarity KPCA. Each projected point is color-

coded by the water-saturation value of the sand layers within the corresponding

sequence. Percentages of noise added are specified in each panel. The success

rate of classifying a projected seismogram into its corresponding saturation class

is shown in the lower left corner of each panel................................................... 159

xxviii

Figure 4.46: A realization of one 2-D geologic section used in the numerical example.

The area at the left end marked with the red box corresponds to the well location.

Sand and shale are colored in yellow and blue respectively. The thicknesses of

sand layers decrease linearly away from the well. The total thickness of reservoir

is 150 m. An example of the thickness distribution used to simulate the sequence

at the well location is also shown........................................................................ 161

Figure 4.47: Results from realizations of 1,000 2-D sections show how slope and

intercept vary with varying net-to-gross ratios. .................................................. 162

Figure 4.48: Contour plots and marginal distributions for the high (i.e., equal to or

greater than 0.6) and low (i.e., equal to or less than 0.35) net-to-gross values. .. 162

Figure 4.49: (Lower left corner) Posterior distributions of net-to-gross ratios for three

selected locations on the unknown seismic section labeled as (1), (2) and (3). The

true values for each location are shown in the table at the lower right corner. ... 164

Figure 4.50: Two transition matrices used to generate non-stationary sequences and

examples of sequence realizations. For simulation steps in the top half of the

reservoir interval, the probability of using matrix A is 0.8. The same is true for

the bottom half of the reservoir with matrix B. The lithologic states considered

are sand (S) and shale (Sh). ................................................................................. 166

Figure 4.51: Segmentation of seismograms and sequences. (Left) application of an 80-

ms moving window to a seismogram (Right) seismograms segments and their

corresponding sequence segments. ..................................................................... 166

Figure 4.52: The first two components of KPCA with Gaussian kernel. A total of 1200

seismogram segments are represented as points which are color-coded by the net-

to-gross values of the corresponding sequence segments. .................................. 167

Figure 4.53: The first two components of KPCA with Gaussian kernel. A total of 2070

seismogram segments generated from the same forms of transition matrices as

described in the text are represented as points which are color-coded by the net-

to-gross values of the corresponding sequence segments. Nine seismogram

segments are plotted in black and treated as unknowns. ..................................... 169

Figure 4.54: Probability distributions of local net-to-gross estimation for the three

(unknown) seismograms. Each row represents the results for each unknown

seismogram. Each column represents local net-to-gross estimations at a specified

position of the sequences which correspond to the unknown seismograms. The

red dash lines represent the true net-to-gross values. .......................................... 169

Figure 4.55: Amplitude attributes extracted from seismograms which correspond to

sequences with different net-to-gross ratios. ....................................................... 171

Figure 4.56: Success rates in classifying a data point into three net-to-gross classes

(<0.38, 0.38-0.465, and >0.465) by using the attributes specified on the horizontal

axis. The rates are shown on top of each bar. Att1: RMS and absolute amplitude at

the top of the reservoir, Att 2: the first two principal components of Gaussain

KPCA, Att3: the first two principal components of dynamic similarity KPCA, and

Att4: the first 20 principal components of linear KPCA. .................................... 171

Figure 5.1: Six identified lithofacies from the study area offshore Equatorial Guinea,

West Africa (Lowe, 2004; Dutta, 2009). ............................................................ 182

xxix

Figure 5.2: Well A from deep-water turbidite deposits, offshore Equatorial Guinea,

West Africa. The zone of interest is highlighted. From left to right, the curves are

gamma ray, bulk density, density-derived porosity, P-wave velocity, and water

saturation (SW), respectively. .............................................................................. 184


Stieber model. (Left) Total porosity and gamma ray values for three lithofacies in

the selected interval. The median total porosity and gamma ray values for each

lithofacies are shown in solid circles. From these median points, the up-down or

left-right bars indicate the interquartile ranges (i.e., from 1st to 3

rd quartiles) of

each property. A Thomas-Stieber diagram is also superimposed on the data.

(Right) Variation of gamma ray values with depth. Data points are color-coded by

lithofacies similar to the left panel. The gamma ray log shows an upward-fining

trend. ................................................................................................................... 185

Figure 5.4: Estimated net-to-gross ratios for the selected well-log interval. (Left)

gamma ray log of the selected interval and (right) posterior distributions of

estimated net-to-gross ratios. .............................................................................. 185

Figure 5.5: Acoustic impedance and the corresponding synthetic seismogram of the

target zone. (Left) Acoustic impedance of the target zone shown in magenta,

together with 200 data points above and 200 data points below the target zone.

(Right) Synthetic seismogram generated using a 30-Hz zero-phase Ricker wavelet.

............................................................................................................................. 187

Figure 5.6: Synthetic-seismogram test set. Note that each panel shows four repetitions

of one seismogram. From left to right, the panels show seismograms that are

generated from the original well log (i.e., fining-upward), the inverted log (i.e.,

coarsening-upward), the shuffled log (i.e., serrated), the brine-saturated log, and

the log with smaller layer thicknesses within the target zone. ............................ 187

Figure 5.7: Sample realizations of synthetic earth models which cover a range of net-

to-gross ratios and various stacking patterns. The sequence realizations are

generated from the three categories of transition matrices: with an upward-

increasing trend (left), with an upward-decreasing trend (middle), and with no

trend in net-to-gross ratios (right). ...................................................................... 190

Figure 5.8: Probability density functions of acoustic impedance for all lithologic states:

oil sand (blue), shale (red), and wet sand (black). .............................................. 190

Figure 5.9: Synthetic seismograms corresponding to five sets of sand-shale sequences.

Model I to V correspond to sequence realizations generated from various

transition matrices, sand properties, and layer thicknesses. Refer to Table 5.1 for

detailed specifications. ........................................................................................ 192

Figure 5.10: Projections of the noise-free seismograms from the training set and the

test set onto the first two principal components after the application of the linear

KPCA. Each point from the training set is color-coded by the net-to-gross ratio

from its corresponding sand-shale sequence. The test set, shown in black symbols,

is plotted on top of the projections of the training set. ........................................ 193

Figure 5.11: Cross-plot between the RMS and the maximum amplitude of noise-free

seismograms from both the training set and the test set. The points from the

training set are color-coded by the net-to-gross ratios from their corresponding

xxx

sand-shale sequences. The points from the test set are shown in black symbols.

Two distinct trends corresponding to changes in stacking pattern and net-to-gross

ratio are marked by the blue and the red arrows, respectively. ........................... 194

Figure 5.12: Projections of the noisy seismograms from the test set onto the first two

principal components from the linear KPCA. The test set shown in black symbols

is plotted on top of the projections of the training set, which are color-coded by

the net-to-gross ratios from their corresponding sand-shale sequences. The

percentages of noise added to the seismograms are shown in the lower left corner

of each plot. ......................................................................................................... 196

Figure 5.13: Cross-plot between the RMS and the maximum amplitude of noisy

seismograms from both the training set and the test set. The points from the

training set are color-coded by the net-to-gross ratios from their corresponding

sand-shale sequences, and the points from the test set are shown in black symbols.

The percentages of noise added to the seismograms are shown in the lower left

corner of each plot. .............................................................................................. 197

Figure 5.14: Linear KPCA results (0% noise) with net-to-gross (NTG)/stacking-

pattern class labels. NTG1, NTG2, and NTG3 represent NTG≤0.46, NTG>0.46

and NTG≤0.54, and NTG>0.54, respectively. The three stacking patterns are

fining-upward (F), coarsening-upward (C), and serrated (S). ............................. 197

Figure 5.15: Classification success rate when using linear KPCA attributes, amplitude

attributes, and dynamic similarity KPCA attributes at various noise levels. ...... 198

Figure 5.16: Effect of layer thickness on linear KPCA results. Each plot represents the

projections of seismograms onto the first two principal components of linear

KPCA when layer thickness is 1.5 cm (left) or 2.5 cm (right). Points are color-

coded by the net-to-gross ratios from their corresponding sand-shale sequences.

............................................................................................................................. 199

Figure 5.17: Effect of layer thickness on amplitude results (RMS versus maximum

amplitude). Layer thickness used for modeling sequences is 1.5 cm for the left

plot and 2.5 cm for the right plot. Points are color-coded by the net-to-gross ratios

from their corresponding sand-shale sequences. ................................................. 199

Figure 5.18: Probability density functions of two sets of acoustic impedance showing

small and large contrast (shown in red and green, respectively) between the sand

and the shale (shown in solid and dash lines, respectively). ............................... 200

Figure 5.19: Effect of impedance contrast on linear KPCA results. Each plot

represents the projections of seismograms onto the first two principal components

of linear KPCA when the overlap between the sand and the shale impedances is

small (left) or large (right). Points are color-coded by the net-to-gross ratios from

their corresponding sand-shale sequences. ......................................................... 201

Figure 5.20: Effect of impedance contrast on amplitude results (RMS versus

maximum amplitude). The overlap between the sand and the shale impedances is

small in the left panel and large in the right panel. Points are color-coded by the

net-to-gross ratios from their corresponding sand-shale sequences. ................... 201

Figure 5.21: Effect of phase rotation on linear KPCA results. Points represent

projections of the seismograms from the Model I training set and the well location

onto the first two principal components from the linear KPCA. Each point is

xxxi

color-coded by the net-to-gross ratio from its corresponding sand-shale sequence.

The three seismograms at the well with three different phases shown in black

symbols are plotted on top of the projections of the training set. The projection of

the zero trace is used as a reference point. .......................................................... 202

Figure 5.22: A 2-D seismic section extracted along the turbidite channel from

proximal (left) to distal (right) directions. The location of well A is marked. .... 203

Figure 5.23: Prior distributions for thickness (left) and net-to-gross ratio (right),

assigned to sequence models when generating a training set. ............................ 204

Figure 5.24: Relative locations of two existing wells (well A and well B). Well A

(black symbol) is located in the distal direction along the channel, and well B (red

symbol) is located in the proximal direction along the channel. Distance is

measured from a reference point. The selected 2-D seismic line is shown in blue.

............................................................................................................................. 205

Figure 5.25: Average properties of the overburden and underburden of the interval of

interest from proximal and distal wells. Three properties are shown: P-wave

velocity (Vp), bulk density (RHOB), and acoustic impedance (AI). .................. 206

Figure 5.26: Probability density functions of acoustic impedance for the sand (blue)

and the shale (red) lithologic states used in generating a training set. ................ 207

Figure 5.27: Projections of the training set and the test seismograms onto the first two

principal components of KPCA with a linear kernel. The training points are color-

coded by the reservoir thicknesses of their corresponding sand-shale sequences.

The test points are labeled by their trace numbers. ............................................. 208

Figure 5.28: Projections of the training set onto the first two principal components of

KPCA with a linear kernel. In each panel, the training points are color-coded by

the reservoir parameter specified at the top of each panel. ................................. 209


KPCA with a Gaussian kernel. In each panel, the training points are color-coded

by the reservoir parameter specified at the top of each panel. ............................ 209


KPCA with a linear kernel, when the values of underburden multiplier,

overburden multiplier, and reservoir thickness are fixed. Points are color-coded

by net-to-gross ratio. ........................................................................................... 210

Figure 5.31: The first 20 eigenvalues of the linear KPCA results. ............................. 211

Figure 5.32: Comparison of two empirical cumulative distributions for the overburden

multiplier values in class 3. The prior distribution and the class-conditional

distribution are shown in blue and red, respectively. The shaded area represents

area between the two curves. .............................................................................. 213

Figure 5.33: Normalized difference measure between a class-conditional empirical

cumulative distribution and a prior empirical distribution for a parameter. The red

line represents a threshold used in determining whether a parameter has a

significant impact on the output response (i.e., seismic signature). .................... 214

Figure 5.34: Pareto plot of estimated sensitivity values. Parameters are ranked

according to their sensitivity values. ................................................................... 214

Figure 5.35: Estimated probability density functions for reservoir thickness for all

seismic traces in the test set. The last three traces (#55-57) correspond to the

xxxii

additional three traces which are added for evaluating the performance of our

property estimation. Each density function is shown as a column-wise color scale.

............................................................................................................................. 216

Figure 5.36: The mean values of reservoir thickness for all seismic traces in the test set.

The last three points are added for evaluating the performance of our property

estimation. ........................................................................................................... 216

Figure 5.37: Estimated probability density functions for the underburden multiplier for

all seismic traces in the test set. The last three traces (#55-57) correspond to the



............................................................................................................................. 216

Figure 5.38: The mean values of underburden multiplier for all seismic traces in the

test set. The last three points are added for evaluating the performance of our

property estimation. ............................................................................................ 216

Figure 5.39: Estimated probability density functions for the overburden multiplier for

all seismic traces in the test set. The last three traces (#55-57) correspond to the



............................................................................................................................. 217

Figure 5.40: The mean values of overburden multiplier for all seismic traces in the test

set. The last three points are added for evaluating the performance of our property

estimation. ........................................................................................................... 217

Figure 5.41: Estimated probability density functions for net-to-gross ratio for all

seismic traces in the test set. The last three traces (#55-57) correspond to the



............................................................................................................................. 217

Figure 5.42: The mean values of net-to-gross ratio for all seismic traces in the test set.

The last three traces (#55-57) correspond to the additional three traces which are

added for evaluating the performance of our property estimation. ..................... 217

Figure 5.43: The estimated probability densities of reservoir thickness for seismic

traces at the well and trace #55-57, which are added for evaluating the

performance of our property estimation. The true thicknesses for all four test

traces are marked by the magenta line. ............................................................... 218

Figure 5.44: The estimated probability densities of underburden multiplier for seismic


performance of our property estimation. The true multipliers for three test traces

are marked by the lines shown in the same colors as their densities. Note that

since trace #56 is associated with well B, the “true” multiplier of well A is

irrelevant and thus not shown here. .................................................................... 218

Figure 5.45: The estimated probability densities of overburden multiplier for seismic


performance of our property estimation. The true multipliers for three test traces

are marked by the lines shown in the same colors as their densities. Note that

xxxiii

since trace #56 is associated with well B, the “true” multiplier of well A is

irrelevant and thus not shown here. .................................................................... 219

Figure 5.46: The estimated probability densities of net-to-gross ratio for seismic traces

at the well and trace #55-57, which are added for evaluating the performance of

our property estimation. The true net-to-gross ratios for all four test traces are

marked by the magenta line. ............................................................................... 219

Figure 5.47: The estimated probability densities of net-to-gross ratio for seismic trace

# 57 for both when the non-stationarity is taken into account (blue curve) and

when the non-stationarity is not taken into account (red curve). The true net-to-

gross ratio for trace #57 is marked by the blue line. ........................................... 220

1

Chapter 1

Introduction

1.1 Objective

The overall goal of this dissertation is to improve the interpretation and property

estimation of thinly bedded sand-shale systems that cannot be resolved with

conventional well log and seismic data by modifying existing rock physics relations

and establishing new ones to account for heterogeneity of rocks at the measurement

scales, and by using quantitative seismic analysis which combines rock physics,

spatial statistics of the geology, and seismic attributes to infer rock properties of the

sub-resolution sand-shale systems. The specific objectives are as follows:

To incorporate natural variations and uncertainties in real observations into

existing models for estimating petrophysical properties of thinly bedded

sand-shale reservoirs;

CHAPTER 1: Introduction 2

To establish rock physics relations that can be directly applied to

heterogeneous rocks at the measurement scale, without the need to

downscale the measurements;

To investigate the effects of rock properties and stacking patterns on

seismic signatures of thinly bedded sand-shale reservoirs;

To estimate reservoir properties (e.g., volume fraction of sand, saturation,

and stacking patterns) of thinly bedded, sub-resolution sand-shale

reservoirs from seismic data.

1.2 Background and motivation

Definitions of thin beds vary depending on the context. In both formation

evaluation analysis and seismic study, thin beds generally refer to beds with

thicknesses below the vertical resolution of measuring tools. This is the definition we

follow in this dissertation. The resolution varies depending on measuring tools. While

conventional logging tools have typical vertical resolutions of a few feet, the vertical

resolution of seismic data, generally defined as a quarter of the dominant wavelength,

is typically in the range of tens of meters. Geologically, thin beds vary in thickness

from 3 to 10 cm. Layers with thicknesses less than 1 cm are referred to as laminae

(Campbell, 1967; Boggs, 2001; Passey et al., 2006).

The problem of interpreting thin beds has received much study. For example,

thinly interbedded sand-shale sequences can cause low-resistivity pay, a well-known

phenomenon in formation evaluation. The thin hydrocarbon-bearing sand layers have

high resistivity, while the thin wet shale layers have low resistivity. When resistivity

logging devices cannot resolve each individual layer, the resistivity measurements are

averages of the thin sand and shale layers (arithmetic or harmonic average, depending

on the direction of the current flow). Due to this averaging, the resistivity

measurements of the hydrocarbon interval can appear to be lower than they actually

are (Boyd et al., 1995; Passey et al., 2006). Core samples provide direct indicators of

thin beds (Passey et al., 2006), but their use is often limited because of their high

operating cost and limited availability. Other logs, such as high-resolution fullbore


formation microimager (FMI) and nuclear magnetic resonance log (NMR), provide

indirect indicators of thin beds. However, these logs are still subject to environmental

effects, inaccurate calibration, and improper processing (Passey et al., 2006).

Moreover, even though well log data provide high-resolution information about

reservoirs, they lack spatial coverage. Therefore, the need remains for better

interpretation of conventional well log and seismic data in thin sub-resolution bedded

reservoirs. This dissertation attempts to improve such interpretation by applying rock

physics relations and quantitative seismic interpretation.

1.2.1 Rock physics relations and sub-resolution heterogeneity

Rock physics relations are often obtained from controlled laboratory experiments

or derived for rocks that are homogeneous at particular scales. However, in practice,

these relations are often applied to measurements at scales where the rocks are

heterogeneous. When the scale of heterogeneity is smaller than what geophysical tools

can resolve, the measurements represent average values of the heterogeneous rocks

(for example, a stack of interbedded sand-shale layers that cannot be individually

resolved by well logging tools). Applying rock physics relations to such measurements

directly at the measurement scale can result in erroneous predictions. For example,

Dvorkin and Uden (2006) demonstrated that when the rock physics transform between

well-log-scale impedance and hydrate saturation is applied to seismic-scale

impedances of a 1-D synthetic earth model which contains three sand layers with

methane hydrate, the predicted hydrate saturations can significantly differ from the

true saturations. Another case of when rock physic relations fail to account for

heterogeneity is the application of the isotropic Gassmann’s fluid substitution equation

(Gassmann, 1951) to the measurements at their original scales, without accounting for

the sub-resolution sand-shale interbedding. In this case, the predicted changes in

elastic properties can be erroneous because Gassmann’s equation is not appropriate for

fluid substitution in the low-permeability shale layers (e.g., Katahara, 2004; Skelt,

2004a; Skelt, 2004b; Chopra, 2005). This dissertation aims to investigate existing rock

physics relations to account for heterogeneity, and establish new relations that can be


directly applied to heterogeneous rocks without the need to downscale the

measurements.

1.2.2 Seismic property estimation of sub-resolution heterogeneous systems

Estimating reservoir properties from seismic response can be challenging when

thicknesses of sedimentary beds are below seismic resolution. Examples of previous

studies on seismic property estimation of thin layered reservoirs are as follows. Vernik

et al. (2002) estimated seismic-scale sand volume of deepwater turbidite deposits by

cross-plotting P- and S-impedance inverted from seismic data. Dvorkin (2005)

introduced cumulative attributes (CATTS), which can be computed by integrating the

seismic trace repeatedly, for estimating total pore volume of the sub-resolution

reservoir. These studies provide useful attributes for estimating properties of sub-

resolution reservoirs, based on rock physics transforms (i.e., from elastic to reservoir

properties). However, because these transforms are applied to seismic-scale attributes,

for example inverted P-impedance, which is the upscale value of the fine-scale

heterogeneity, these attributes cannot be used to infer the fine-scale patterns of the

reservoir. Stright (2011) proposed a methodology for predicting sub-seismic

lithofacies using P-impedance and Vp/Vs. The strategy behind this method is to

generate a seismic-scale attribute template with lithofacies information by

incorporating fine-scale heterogeneity from well log and core data. This template can

then be applied to the seismic volume to infer sub-seismic lithofacies proportions.

Because this method transforms the attributes into lithofacies on a point-by-point basis,

the predicted lithofacies of all adjacent points along the vertical direction may not

reflect a spatial pattern of the actual lithofacies arrangement. Xie et al. (2004)

predicted seismic facies of thin reservoirs by comparing seismic traces away from the

well with those traces at the wells whose facies are known. The correlation coefficient

is used as a similarity measure among the seismic traces. This method is based solely

on pattern recognition of seismic traces without accounting for the facts that different

sets of rock properties and geometry can result in similar seismic signatures and that

similar reservoir properties but different geometry can result in different seismic

signatures (Takahashi, 2000; Spikes, 2008). This dissertation aims to investigate both


the effects of rock properties (e.g., sand volume, saturation) and the stacking patterns

of thinly bedded sand-shale reservoirs on seismic signatures and ultimately to estimate

these properties from seismic data.

1.3 Definitions of terms, symbols, and abbreviations

Important terms, symbols, and definitions which will be used throughout this

dissertation are defined in Table 1.1. Several volumetric parameters such as volume

fraction of laminated shale and dispersed clay are illustrated by the clay/shale

distribution diagram in Figure 1.1.

Table 1.1: Definitions of terms, symbols, and abbreviations used throughout this dissertation. Note that Vi is the volume of the ith layer illustrated in Figure 1.1, and i = 1, 2,…, 6.

Term Symbol or

abbreviation

Definition

Effective porosity 𝜙𝐸𝑓𝑓 volume fraction of effective pore space which excludes

volume of clay-bound water

Effective

saturation 𝑆𝑊𝑒 volume fraction of water in the effective pore space

Net-to-gross

(reservoir) ratio

NTG volume fraction of reservoir sand layers

𝑁𝑇𝐺 =𝑉2 + 𝑉4 𝑉𝑖6𝑖=1

Sand fraction 𝑉𝑠𝑎𝑛𝑑 volume fraction of all sandy layers including shaly sand

𝑉𝑠𝑎𝑛𝑑 =𝑉2 + 𝑉4 + 𝑉6 𝑉𝑖6𝑖=1

Shale volume 𝑉𝑠ℎ volume fraction of shale including dispersed clay and

laminated shale

𝑉𝑠ℎ =𝑉1 + 𝑉3 + 𝑉5 + 𝑉4 ∗ 𝑉𝑑𝑖𝑠𝑝 1 + 𝑉6 ∗ 𝑉𝑑𝑖𝑠𝑝 2

𝑉𝑖6𝑖=1

Total porosity 𝜙𝑇 volume fraction of void space in the rock

Volume of

dispersed clay 𝑉𝑑𝑖𝑠𝑝 volume fraction of clay dispersed in the sand pore space

𝑉𝑑𝑖𝑠𝑝 =𝑉4 ∗ 𝑉𝑑𝑖𝑠𝑝 1 + 𝑉6 ∗ 𝑉𝑑𝑖𝑠𝑝 2

𝑉𝑖6𝑖=1

Volume of

laminated shale 𝑉𝑙𝑎𝑚 volume fraction of total shale layers

𝑉𝑙𝑎𝑚 = 1 − 𝑉𝑠𝑎𝑛𝑑 =𝑉1 + 𝑉3 + 𝑉5 𝑉𝑖6𝑖=1


Figure 1.1: Different types of clay/shale distribution in sand. The volume of the ith layer and the volume of dispersed clay are denoted by Vi and Vdisp, respectively.

In rock physics, a laminated sand-shale sequence generally refers to a sequence of

alternating sand and shale units. However, in sedimentology this same alternating

sequence is generally described as interbedded sandstone and shale. This dissertation

uses the terms laminated and interbedded interchangeably.

1.4 Chapter description

The remainder of this dissertation is organized into four chapters, each of which

discusses about either existing or new tools for interpreting sub-resolution sand-shale

reservoirs.

Chapter 2 presents the sensitivity and uncertainty analyses of the Thomas-Stieber

model (Thomas and Stieber, 1975). This model is commonly used in thinly-bedded

sand-shale reservoirs to estimate volumetric fractions of various types of clay/shale

distribution within the sand-shale mixtures. We investigate various scenarios when

certain model assumptions are not met and show their effects on the estimated

properties from the model. We also show how uncertainty in the input parameters can

be incorporated into the model.

Chapter 3 proposes a simple graphical mesh interpretation and accompanying

equations for fluid substitution in sub-resolution interbedded sand-shale sequences.

The sand layers can be either clean or shaly (i.e., sand with dispersed clay). Our mesh

method can be applied to measurements directly at their original scale, without the

need to downscale the measurements, while still changing fluid in the sand layers only.

Laminated shale

Clean sand

Sand with dispersed clay ( V disp1 )

Sand with pore space completely filled with clay ( V disp2 )


Chapter 4 investigates seismic signatures of thin sand-shale reservoirs using

wavelet-transform based attributes and feature-extraction based attributes. Note that

feature extraction techniques transform input data into new representations in lower-

dimensional space (Cunningham, 2008). We use the standard statistical rock-physics

approach to explore what-if scenarios of how seismic signatures of the thin reservoirs

change with varying reservoir properties, such as net-to-gross ratio, saturation, and

stacking pattern. We present synthetic examples of how to apply these seismic

attributes to estimate reservoir properties in real-data applications.

Chapter 5 applies the seismic attributes introduced in Chapter 4 to real well log

and seismic data from deep-water deposits in West Africa.

1.5 References

Boggs, S., 2001, Principles of sedimentology and Stratigraphy, 3rd

ed.

Boyd, A., Darling, H., Tabanou, J., Davis, B., Lyon, B., Flaum, C., Klein, J., Sneider,

R.M., Sibbit, A., and Singer, J., 1995, The lowdown on low-resistivity pay:

Schlumberger Oilfield Review, 7(3), 4-18.

Campbell, C.V., 1967, Lamina, laminaset, bed and bedset: Sedimentology, 8(1), 7-26.

Chopra, S., 2005, Expert Answers: Gassmann’s equation: CSEG Recorder in May, 8-

12.

Cunningham, P., 2008, Dimension reduction: Machine learning techniques for

multimedia: case studies on organization and retrieval, Springer, Eds. M. Cord and

P. Cunningham.

Dvorkin, J., 2005, Cumulative seismic attribute for ϕh determination: SEG Expanded

Abstracts, 24, 1550-1553.

Dvorkin, J. and Uden, R., 2006, The challenge of scale in seismic mapping of hydrate

and solutions: The leading edge, 25(5), 637-642.

Gassmann, F., 1951, Uber die elastizitat poroser medien: Vier Natur Gesellschaft, 96,

1-23.

Katahara, K., 2004, Fluid Substitution in Laminated Shaly Sands: SEG Expanded

Abstracts, 23, 1718-1721.

Passey, Q. R., Dahlberg, K. E., Sullivan, K. B., Yin, H., Brackett, R. A., Xiao, Y. H.,

and Guzmán-Garcia, A. G., 2006, Petrophysical evaluation of hydrocarbon pore-


thickness in thinly bedded clastic reservoirs: AAPG Archie series, 1, American

Association of Petroleum Geologists, Tulsa, Oklahoma, U.S.A.

Skelt, C., 2004a, Fluid substitution in laminated sands: The Leading Edge, 23, 485–

488.

Skelt, C., 2004b, The influence of shale distribution on the sensitivity of

compressional slowness to reservoir fluid changes: SPWLA 45th

Annual Logging

Symposium, June 6-9.

Spikes, K.T., 2008, Probabilistic seismic inversion based on rock-physics models for

reservoir characterization, Ph.D. Thesis, Stanford University.

Stright, L.E, 2011, Multiscale modeling of deep-water channel deposits: An

interdisciplinary study integrating geostatistics, geology and geophysics The

cretaceous Cerro Toro formation, southern Chile, the Eocene Ardath & Scripps

formations, southern California, and the Oligocene Puchkirchen formation, upper

Austria, Ph.D. Thesis, Stanford University.

Takahashi, I., 2000, Quantifying Information and Uncertainty of Rock Property

Estimation from Seismic Data, Ph.D. Thesis, Stanford University.

Thomas, E. C. and Stieber, S. J., 1975, The distribution of shale in sandstones and its

effect upon porosity: 16th Annual Logging Symposium, SPWLA, Paper T.

Vernik, L., Fisher, D., and Bahret, S., 2002, Estimation of net-to-gross from P and S

impedance in deepwater turbidites: The leading edge, 21(4), 380-387.

Xie, D., Wood, J.R., and Pennington, W.D., 2004, Quantitative seismic facies analysis

for thin-bed reservoirs: a case study of the central Boonsville field, Fort Worth

Basin, north-central Texas: SEG Expanded Abstracts, 23, 1484-1487.

9

Chapter 2

Sensitivity and uncertainty analysis

of the Thomas-Stieber model for

property estimation of thin sand-

shale reservoirs

2.1 Abstract

This chapter presents sensitivity and uncertainty analyses of the Thomas-Stieber

model and its application for estimating petrophysical properties in thinly bedded

sand-shale reservoirs. This model describes how total porosity relates to different

types of clay/shale distribution within the sand-shale mixture. By inputting the clean

sand and shale properties, this model can be used deterministically to estimate both the

volume fraction of the sandy layers within the lamination and the volume fraction of

CHAPTER 2: Sensitivity and uncertainty analysis of the Thomas-Stieber model 10

dispersed clay within the sandy layers. Even though applying this model to a dataset is

straightforward, several factors including simplistic assumptions and uncertainties of

estimated input parameters can affect the data interpretation.

The objective of this study is to incorporate natural variations and uncertainties

which can be found in real observations into the model and to point out some pitfalls

in the interpretation when these variations are neglected. For a case of bimodal mixing

(i.e., sand and pure shale), this model can be used to infer volume fraction of sand and

its porosity. However, when the sand beds within the interval of interest have a range

of properties (e.g., varying clay content), the model cannot deterministically estimate

property of each individual sand layer. Instead, it gives an average property of all the

sand beds within the stack. The uncertainties of the input parameters of the model can

be used in stochastic methods to quantify uncertainty of our interpretation. We use

Monte Carlo simulations propagated through the Thomas-Stieber model to estimate

the posterior distributions of interpreted volume fractions of sand in a Bayesian

framework. We apply this stochastic approach to well log data from deep-water

turbidite deposits in West Africa. We also present the analogs of the Thomas-Stieber

model on other cross-plots.

2.2 Introduction

Geophysical measurements at various scales are used to infer the properties and

structure of the subsurface. These data often represent average properties of multiple

thin layers when the heterogeneities are below the resolution of the measuring tools.

The lack of knowledge about the existence and properties of thin lamination could

mislead the interpretation and thus affect the evaluation of the thin reservoirs. For

example, a direct application of isotropic Gassmann’s equation (Gassmann, 1951)

without accounting for the sub-resolution, interbedded sand-shale layers would be

erroneous because fluid substitution is likely to be applicable only in permeable sand

beds (e.g., Katahara, 2004; Skelt, 2004a; Skelt, 2004b; Chopra, 2005). Thus, a more

accurate description of the thin reservoirs is needed for better reservoir

characterization.


The Thomas-Stieber model (Thomas and Stieber, 1975) is a useful tool for

evaluating thinly bedded sand-shale reservoirs because it provides not only the

evidence of fine-scale lamination but also the petrophysical properties of rocks within

the lamination (e.g., Dutta, 2009; Yadav et al., 2010; Voleti et al., 2012). This model

describes the relations between total porosity and shale volume of the sand-shale

mixtures. These relations are not unique, but dependent on how clay/shale is

distributed in the mixtures. Since any rocks that can be described using the Thomas-

Stieber model are assumed to be mixtures between the two end-member rocks (i.e.,

sand and shale), by inputting the properties of the sand and the shale end-points the

model can be constructed and used to infer volume fractions of different types of

clay/shale distributions. These inferred shale volumes are crucial for improving

reservoir evaluation. Even though the Thomas-Stieber model can be constructed and

applied easily, several assumptions of this model can be too simplistic for real data.

For example, by using this model the interpretation of any data point is limited to

lamination between sandy layers and shale layers, and all the sandy layers share the

same petrophysical properties. This interpretation may not always be true. Therefore,

it is important to be aware of the limitations of the model and to explore how data

interpretation is affected under more realistic scenarios or under uncertainty in the

model’s input parameters.

In this chapter, we generate synthetic rocks to test the model under various

scenarios which do not fit the model assumptions. Then, we incorporate uncertainties

in input parameters into the interpretation using Monte Carlo simulations. The

remainder of this chapter is organized as follows. Section 2.3 briefly reviews the

Thomas-Stieber volumetric model for a sand-shale mixture system, including

dispersed and laminar mixes, and discusses important assumptions of the model. In

Section 2.4, we perform sensitivity and uncertainty analyses of the model by

incorporating natural variations into the model in order to quantify uncertainty in our

interpretation. We present results of the analyses through both synthetic examples and

real well log data. Section 2.5 shows analogs of the Thomas-Stieber model on other

cross-plots, and finally Section 2.6 discusses other sources of uncertainties that can be


found in real data application and provides possible constraints on choices of input

parameters.

2.3 Volumetric properties of sand-shale mixtures

Here a volumetric property refers to a petrophysical property whose representative

value for a rock sample depends on the volume fractions and the individual properties

of the rock’s constituents, but not how they are arranged or distributed. The volumetric

properties covered in this section are total porosity, effective porosity, and shale

volume.

2.3.1 Total porosity and volume of shale

Thomas and Stieber1 (1975) provided a simple model for how total porosity of a

sand-shale mixture varies with shale volume, depending on the distribution of

clay/shale in the mixture. There are three main types of distribution: dispersed clay,

laminated shale, and structural clay (Figure 2.2). Here we focus on the dispersed and

laminar types. As noted in Chapter 1, in rock physics, a laminated sand-shale sequence

generally refers to a sequence of alternating sand and shale units. However, in

sedimentology this same alternating sequence is generally described as interbedded

sandstone and shale. We use the terms laminated and interbedded interchangeably.

Figure 2.1: Shale distribution in sand (Modified after Mavko et al., 2009).

1 Thomas-Stieber model was originally derived using gamma ray response.


Figure 2.2: Relation between total porosity and shale volume. Point A and E represent clean sand and pure shale point, respectively. Points B and C represent sand with dispersed clay, and point D is where the original sand pore space is completely filled with dispersed clay.

The variation of total porosity with shale volume in a sand-shale mixture is

schematically shown in Figure 2.2. The dispersed lines AD and DE describe the

topology of bimodal mixtures of sand grains and clay, and a V-shaped relation

between the shale volume and the mixture porosity (Marion, 1990). When clay is

dispersed into the original sand pore space without disturbing the sand pack, total

porosity linearly decreases with volume of shale (the shaly-sand line AD) as

(0 ≤ 𝑉𝑠ℎ ≤ 𝜙𝑠): 𝜙𝑇 = 𝜙𝑠 − 1 − 𝜙𝑠ℎ 𝑉𝑠ℎ .

(2.1)

In equation 2.1, 𝜙𝑇, 𝜙𝑠, and 𝜙𝑠ℎ are the total, sand, and shale porosity respectively,

and 𝑉𝑠ℎ is the shale volume.

At this stage, sand grains provide the load-bearing matrix of the mixture. When

clay is continuously added until its volume is equal to sand porosity (D), the original

sand pore space is completely filled with clay:

𝑉𝑠ℎ = 𝜙𝑠 : 𝜙𝑇 = 𝜙𝑠𝜙𝑠ℎ .

(2.2)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Shale volume

Tota

l poro

sity

Dispersed line

AV

lam = 0.2

Vlam

= 0.4

Vlam

= 0.6

Vlam

= 0.8

EF

B

C

D


When shale volume is greater than sand porosity, sand grains are displaced and

disconnected; as a result, the mixture changes from grain-supported to clay-supported

sediments. At this stage, porosity increases linearly with increasing shale volume (the

sandy-shale line DE) because the solid sand grains are replaced by porous clay (Yin,

1992; Avseth et al., 2005):

𝜙𝑠 < 𝑉𝑠ℎ ≤ 1 : 𝜙𝑇 = 𝜙𝑠ℎ𝑉𝑠ℎ .

(2.3)

We refer to the volume of dispersed clay as 𝑉𝑑𝑖𝑠𝑝 .

In the laminar distribution, the total porosity is simply a weighted average of the

porosities of all the end-members. For example, in the case of clean sand at point A

laminated (interbedded) with pure shale at point E, the total porosity of the mixture

can be expressed as

𝜙𝑇 = 1 − 𝑉𝑙𝑎𝑚 𝜙𝑠 + 𝑉𝑙𝑎𝑚 𝜙𝑠ℎ ,

𝑉𝑙𝑎𝑚 = 1 − 𝑉𝑠𝑎𝑛𝑑 ,

(2.4)

where 𝑉𝑙𝑎𝑚 is the volume of laminated shale and 𝑉𝑠𝑎𝑛𝑑 is the volume fraction of

laminated sand which can be clean or shaly. We refer to this volume as the sand

fraction.

The other line segments (BE, CE and DE) correspond to lamination between shaly

sands (i.e., sand with dispersed clay) and pure shale. Note that these line segments

radiate from the shale point (E), because the Thomas-Stieber model assumes that one

of end-members in a lamination is pure shale. Different volumes of laminated shale

within the mixture are marked by the iso-laminar lines, represented by the set of blue

lines in Figure 2.2 (Thomas and Stieber, 1975).

In order to construct the diagram in Figure 2.2, we need a minimum of two input

parameters: porosities of the clean sand end-point (at 𝑉𝑠ℎ = 0) and the pure shale end-

point (at 𝑉𝑠ℎ = 1). If volume of shale is derived from gamma ray logs, we need two


additional parameters: the gamma ray values for clean sand and shale. Assuming that

there are only two types of rocks alternating within the lamination—clean sand (or

shaly sand) and pure shale—the diagram, which combines both dispersed and

laminated models, can be used graphically or algebraically to determine the existence

of thin lamination even when measurements cannot resolve each layer, and to estimate

the amount and properties of sand in the lamination. For example in Figure 2.2, point

F (𝑉𝑠ℎ = 0.52 and total porosity = 0.138) represents lamination between shaly sand (C)

and shale (E), with 𝑉𝑙𝑎𝑚 = 0.4. The shaly sand has a total porosity equal to 0.138, and

thus this shaly sand has 𝑉𝑑𝑖𝑠𝑝 = 0.2 (Equations 2.1 and 2.4). Alternatively, both

𝑉𝑙𝑎𝑚 and 𝑉𝑑𝑖𝑠𝑝 of point F can be obtained graphically (Figure 2.3). Since pure shale is

assumed to be one of the end-members in the lamination, a line is drawn by always

starting from the shale point (E) and then passing through point F until the line

intersects the dispersed line (AD). This intersection point (C) represents the other end-

member of the lamination. Being on the dispersed line, point C corresponds to shaly

sand whose 𝑉𝑑𝑖𝑠𝑝 is simply obtained by reading off the x-coordinate of the point. 𝑉𝑙𝑎𝑚

is then equivalent to the ratio between the lengths of line CF and line CE.

Figure 2.3: Graphical determination of the laminated shale and the dispersed clay volumes (𝑉𝑙𝑎𝑚 and 𝑉𝑑𝑖𝑠𝑝 , respectively) for a measurement at point F. In each case, the shale volume is determined by drawing a line from the shale point (E), passing through point F, and intersecting line AD. The intersection point (C) represents shaly sand that is the sandy end-member of the lamination, and this shaly sand point has 𝑉𝑑𝑖𝑠𝑝 equal to its x-coordinate (𝑉𝑑𝑖𝑠𝑝 = 0.2). Then, 𝑉𝑙𝑎𝑚 is simply the ratio between the lengths of line CF and CE, which is equal to 0.4.

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Shale volume

Tota

l poro

sity

Dispersed line

AV

lam = 0.2

Vlam

= 0.4

Vlam

= 0.6

Vlam

= 0.8

EF

B

C

D


2.3.1.1 Computing effective porosity by distorting grids

From total porosity (Equations 2.1 and 2.3), we can easily write an expression for

effective porosity:

0 ≤ 𝑉𝑠ℎ ≤ 1 𝜙𝐸𝑓𝑓 = 𝜙𝑇 − 𝜙𝑠ℎ𝑉𝑠ℎ ,

(2.5)

where 𝜙𝑠 < 𝑉𝑠ℎ ≤ 1. Note that Equation 2.5 is applicable under the assumption that

both dispersed clay and laminated shale have the same mineralogy and properties (e.g.,

porosity).

According to Equation 2.5, the transformation from total porosity to effective

porosity is equivalent to moving the pure shale corner and the sandy-shale line DE in

Figure 2.2 down to zero; any data point inside the triangular diagram is distorted

proportionally (Figure 2.4, Mavko et al., 2009).

Figure 2.4: Distorting total porosity diagram (blue) for computing effective porosity (red).

2.3.2 Important assumptions of the Thomas-Stieber model

The Thomas-Stieber model assumes that there are only two types of rocks

alternating within the lamination. Consequently, the interpretation of any data point in

the grid in Figure 2.2 is limited to mixing between sand and shale or between shaly

sand (sand with dispersed clay) and shale. However, laminations consisting of only

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Shale volume

Tota

l poro

sity


two alternating elements may not always be present. For example, both the sand and

shale properties within the interval of interest can vary due to variations in amount of

clay present. Dawson et al. (2008) characterized shale samples from deep-water

depositional settings and reported different types of shale due to differences in

compositions, fabric and sealing capacity. Since this assumption of two rocks

alternating in the lamination is only valid for very simple rocks, in the next section we

will study synthetic examples to understand the limitations in interpretation of more

complex rocks when using this model.

The model assumes that any porosity reduction is caused only by clay filling the

pore space, not by cementation or grain sorting (Thomas and Stieber, 1975; Ball et al.,

2004; Mavko et al., 2009). For the cross-plot of total porosity versus shale volume,

adding cement to the remaining sand pore space reduces total porosity but does not

affect shale volume. For example if cement (25% of the sand porosity) is added into

the sands on the dispersed sand line, the resulting total porosity and shale volume can

be plotted as in Figure 2.5. The blue line is the line of 25%-cement volume, which is

expressed as a percentage of the porosity of the sand end-point.

Figure 2.5: Total porosity and shale volume after cement is added into the remaining sand pore space of the sands on the dispersed sand line. The blue line is the resulting dispersed sand line after adding cement with a volume of 25% of the porosity of the sand end-point into the pore space (Modified after Juhasz, 1986).

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Shale volume

Tota

l poro

sity Shale

Sand

cement volume added = 25% of the

sand porosity


The dispersed distribution in the Thomas-Stieber model represents ideal mixing

and assumes that one component does not disturb the packing of the other. This

assumption holds under a necessary but not sufficient condition that there is a large

difference between the mean sizes of two components mixed together (Cumberland

and Crawford, 1987; Koltermann and Gorelick, 1995; Dvorkin and Gutierrez, 2002;

Revil et al., 2002). Even though the size of clay particles is much smaller than the

mean size of sand grains, other factors that affect real sedimentary rocks (e.g., range of

grain size, grain shape, geological process) can easily lead to deviation from the ideal

packing (Koltermann and Gorelick, 1995; Boggs, 2001; Revil et al., 2002). Similar to

the Thomas-Stieber model, a fractional packing model developed by Koltermann and

Gorelick (1995) describes mixtures, where clay and sand are dispersedly mixed, in

terms of relations between total porosity and shale volume fraction. However, this

fractional packing model relaxes the assumption that one component does not disturb

the other by allowing the shaly-sand domain and the sandy-shale domain to coexist in

the dispersed mixing. This model is shown to fit laboratory measurements of sand-clay

mixtures better than the ideal mixing model (Koltermann and Gorelick, 1995). In the

next section using synthetic examples, we will explore how interpretation is affected

by applying the ideal mixing model to the non-ideal mixtures.

The Thomas-Stieber model assumes that both the dispersed clay in the sand and

the laminated clay/shale have the same properties. This assumption is not always true

because the compositions and properties of clay/shale can vary depending on the

origin of the clay/shale within the sand-shale mixture. For example, while detrital clay

is transported and deposited at the deposition site, authigenic clay is diagenetically

formed after deposition (Thomas and Stieber, 1975; Asquith, 1990; Worden and

Morad, 2003; Worthington, 2003).

Even assuming that all of the model assumptions hold, in real applications of this

model, the input parameters (i.e., properties of the end-points used to construct the

diagram) have to be estimated. Uncertainties in this parameter estimation can affect

the interpretation of data. To understand the effects, we perform sensitivity and

uncertainty analysis of the Thomas-Stieber model in the next section.


2.4 Sensitivity and uncertainty analysis

In the previous section, we have introduced the Thomas-Stieber model and related

cross-plots, and we showed that once the cross-plots are constructed, they can be used

to deterministically infer the volume fraction and property of sand within thin sand-

shale lamination. In this section, we perform sensitivity and uncertainty analyses on

the model, particularly the cross-plot between total porosity and volume fraction of

shale (or gamma ray), and study how uncertainties in various aspects of this model

affect the estimated sand fractions and sand properties. Two sources of uncertainties

are considered: uncertainties from violations of the model assumptions and

uncertainties from the model parameters. In section 2.4.1, we investigate three

scenarios, where the model parameters are correct but the earth models do not follow

the model assumptions, and analyze the effects on data interpretation. In section 2.4.2,

we investigate how uncertainties in the model parameters propagate through the model.

2.4.1 Model with correct input parameters

2.4.1.1 Mixing various sands with shale

In Figure 2.6 (right), the earth model is composed of shale layers interbedded with

sand types A, B, and C, each of which has a different amount of dispersed clay in its

pore space and thus a different total porosity. Assuming that the layers in this earth

model are too thin to be resolved by logging tools, the measured properties of the earth

model therefore represent average properties of the layers. The average total porosity

and volume fraction of shale of the model is shown in Figure 2.6 (left) and marked

with an ―X,‖ with the properties of the individual sand layers being shown in circles.

By constructing the Thomas-Stieber diagram with the correct clean sand and pure

shale end-points, the sand fraction and property of the sand can be estimated. Figure

2.7 shows a comparison of the estimated sand fraction and the estimated sand property

with the true values. While the sand fraction is estimated correctly, the volume

fraction of dispersed clay in sand (i.e., the sand property) is either over- or

underestimated, compared to the actual properties of sand A, B, or C. The estimated

sand property is only the volumetric average of all the sand layers. Because the

Thomas-Stieber model assumes that only one sand type is laminated (interbedded)


with shale, given only the average (i.e., upscaled) properties at point ―X,‖ it is

impossible to estimate (i.e., downscale) properties of individual sand types, or even to

know how many sand types are present within the interval.

Figure 2.6: Thomas-Stieber diagram (left) with plotted properties from the corresponding earth model which is represented by an interbedded sand-shale sequence (right). The properties of the individual sand types (A, B, and C) and shale are shown in circles. The average total porosity and shale volume fraction of the earth model is marked with an ―X.‖ The volume fractions of laminated shale and dispersed clay can be determined graphically by first drawing a line that originates from the shale point, passes through point X, and intersects the dispersed line. The volume fraction of dispersed clay is simply equal to the x-coordinate of the intersection point, and the volume fraction of laminated shale (𝑉𝑙𝑎𝑚 = 1 – 𝑉𝑠𝑎𝑛𝑑 ) is the ratio between the distance from point X to the intersection point and the distance from the shale point to the intersection point.

Figure 2.7: Estimated sand fraction and sand property when applying Thomas-Stieber diagram to point ―X‖ in Figure 2.6 (left). The estimated values are compared with the true values from the earth model (Figure 2.6, right).

2.4.1.2 Mixing shaly sand and sandy shale

The earth model composed of shaly sand interbedded with sandy shale is shown in

Figure 2.8, right. The measured properties of this earth model, which are the average

True V sand True V disp in sand

Estimated Estimated

V disp

in sand

A B C

E

stim

ate

d V

dis

p in

sa

nd

Total

porosity

Vshale

B

C

B

A

C

A

A

estimated volume of laminated shale

estimated volume of dispersed clay

X

To

tal p

oro

sity

E

stim

ate

d V

sa

nd


total porosity and volume fraction of shale, are shown in Figure 2.8 (left) and marked

with an ―X,‖ with the actual shaly sand and sandy shale properties being shown in

circles. By constructing Thomas-Stieber diagram with the correct clean sand and pure

shale end-points, the sand fraction and sand property can be estimated. Figure 2.9

shows a comparison of the estimated sand fraction and the estimated sand property

with the true values. Even though both end-points used for the Thomas-Stieber

diagram are correct, the sand fraction and sand property are not estimated correctly

because the diagram always assumes that the shale end-point (or pure shale) is one of

two end-members in the sand-shale lamination, whereas in this example the

lamination does not contain pure shale, but the sandy shale. Figure 2.9 shows the

possible errors which can result when one of the end-members is chosen incorrectly.

Figure 2.8: Thomas-Stieber diagram (left) with plotted properties from the corresponding earth model which is represented by an interbedded (shaly) sand-(sandy) shale sequence (right). The properties of sand and shale layers within the earth model are shown in circles. The average total porosity and shale volume fraction of the earth model is marked with an ―X.‖ The volume fractions of laminated shale and dispersed clay can be determined graphically by first drawing a line that originates from the shale point, passes through point X, and intersects the dispersed line. The volume fraction of dispersed clay is simply equal to the x-coordinate of the intersection point, and the volume fraction of laminated shale (𝑉𝑙𝑎𝑚 = 1 – 𝑉𝑠𝑎𝑛𝑑 ) is the ratio between the distance from point X to the intersection point and the distance from the shale point to the intersection point.

Total

X porosity

Vshale

estimated volume of laminated shale

estimated volume of dispersed clay

To

tal p

oro

sity


Figure 2.9: Estimated sand fraction and sand property when applying Thomas-Stieber diagram to point ―X‖ in Figure 2.8 (left). The estimated values are compared with the true values from the earth model (Figure 2.8, right).

2.4.1.3 Deviation from ideal mixing

Assuming that the shaly sand layers in the earth model represent non-ideal

mixtures, we generate total porosities of the shaly sand layers using the fractional

packing model of Koltermann and Gorelick (1995). For shale volume fractions

ranging from 0 to 1, the total porosities predicted by the fractional packing model are

represented as points on the blue curve in Figure 2.10. The model’s input parameters

used in this example are 𝜙𝑠= 0.34, 𝜙𝑠ℎ= 0.3, and 𝜙𝑚𝑖𝑛 =0.18. These parameters are

estimated by Kolterman and Gorelick (1995) to fit laboratory measurements for sand-

clay mixtures with diameter ratios greater than 100 and under 30 MPa confining

pressure. Note that 𝜙𝑚𝑖𝑛 is the minimum porosity observed when the shale volume

fraction is equal to the sand porosity. At this point, the Thomas-Stieber model assumes

an ideal mixture where the sand pore space is completely filled with clay and the

model predict that 𝜙𝑚𝑖𝑛 is equal to 𝜙𝑠𝜙𝑠ℎ (Equation 2.2). In contrast, the fractional

packing model assumes a non-ideal mixture where clay disturbs the packing of the

sand, prohibiting the complete filling of the sand pore space by clay. Therefore, in the

fractional packing model 𝜙𝑚𝑖𝑛 is greater than the corresponding value for the ideal

packing model, 𝜙𝑠𝜙𝑠ℎ (Koltermann and Gorelick, 1995).

Using the fractional packing model, we generate a set of data points (solid circles

in Figure 2.10) by varying sand fractions and volume fraction of dispersed clay to

create various sand-shale sequences. Then we estimate sand fractions and sand

properties of these points using the Thomas-Stieber model (the red curve in Figure

True V disp in sand Estimated

V disp

in sand

True V sand Estimated

sand fraction

E

stim

ate

d V

sa

nd

E

stim

ate

d V

dis

p in

sa

nd


2.10) to show how our data interpretation is affected when applying the ideal mixing

model to the non-ideal mixtures. Results are shown in Figure 2.11 as percentage

differences between the estimated and true value using the relation

% 𝑑𝑖𝑓𝑓 = 100 ∗𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 − 𝑡𝑟𝑢𝑒

𝑡𝑟𝑢𝑒 .

(2.6)

In this analysis of the ideal-mixing assumption, both the sand fractions and the

volume fractions of dispersed clay are underestimated (i.e., negative percentage

differences). Consequently, the sand in the reservoir is interpreted to have less volume

fraction but be cleaner than the actual properties. The largest underestimations for the

sand fraction and the volume fraction of dispersed clay are approximately 12% and

26%, respectively. Therefore, the estimated volume fractions of dispersed clay are

affected more severely from using the ideal mixing model than the estimated sand

fractions are. Applying the ideal mixing model to the non-ideal mixtures also affects

the interpretation of sand-shale sequences at different locations on the triangular

diagram differently. Fortunately, the better reservoir quality sequences (i.e., those

located near the upper left corner of the diagram, where the sand fractions within the

sequences are greater and the volume fractions of dispersed clay in the sand are

smaller) are least affected by the assumption of ideal mixing. The overall magnitude

of the effect depends on the value of 𝜙𝑚𝑖𝑛 . In general, if 𝜙𝑚𝑖𝑛 increases, the mixtures

deviate more and more from the ideal mixing, and thus the errors in the interpretation

resulting from using the ideal mixing model on the non-ideal mixtures will get larger.


Figure 2.10: Relations between total porosity and shale volume fraction of bimodal mixtures using the fractional packing model by Kolterman and Gorelick (1995) and the Thomas-Stieber model (i.e., ideal mixing model). Data points representing sand/shaly-sand interbedded with shale (solid circles) are generated using the fractional packing model with varying sand fractions and volume fractions of dispersed clay.

Figure 2.11: Percentage differences between the estimated sand fractions (left), the estimated volume fractions of dispersed clay in the sand (right) and the true values. The ideal mixing model and the fractional packing model (i.e., non-ideal mixing) are outlined.

2.4.2 Model with uncertain input parameters

We have previously shown that even when the correct sand and shale end-points

are used in the Thomas-Stieber model, other complications (e.g., multiple sand types)

0 0.2 0.4 0.6 0.8 1

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Shale volume

Tota

l poro

sity

Fractional packing model

Ideal mixing model

Shale

Sand

0.2 0.4 0.6 0.8

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Shale volume

Tota

l poro

sity

0.2 0.4 0.6 0.8

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Shale volume

Tota

l poro

sity

-10

-8

-6

-4

-2

-25

-20

-15

-10

-5

Fractional

packing

model

Ideal mixing

model

Color-coded by % difference between

the estimated

and the true sand fractions

Color-coded by % difference

between the estimated and the

true volumes of dispersed clay


can lead to errors in the estimation of sand fraction and sand properties. Moreover, in

practice it is almost impossible to know what the correct end-points are, without

detailed core analyses. Therefore, there are always uncertainties associated with the

model’s input parameters (i.e., the properties of the sand and the shale end-points). In

this section, we perform sensitivity analyses of how uncertainties in the input

parameters affect the estimated sand fractions and sand properties using Monte Carlo

simulations. Note that there is also uncertainty in the data. We will not present this

issue here; however, the same analysis shown in this section can be extended to

incorporate data uncertainties into the model.

Using the decomposition of conditional probability, we have the following

relations:

𝑃 𝐴 𝐷 = 𝑃 𝐴,𝐸 𝐷 𝑑𝐸 = 𝑃 𝐴 𝐸,𝐷 𝑃 𝐸 𝐷 𝑑𝐸 ,

𝑃 𝐴 𝐷 = 1

𝑃(𝐷) 𝑃 𝐴 𝐸,𝐷 𝑃 𝐷 𝐸 𝑃 𝐸 𝑑𝐸 ,

(2.7)

where A is the property of interest (e.g., sand fraction), D is a data point, and E

represents the two input properties (e.g., clean sand and pure shale total porosities).

P(D) is a normalization constant.

For simplicity, P(E) is assumed to be known. For a set of acquired data points,

information such as the location, geological setting, and depth of the collection site

can be used to constrain and estimate the distributions of the properties of the

endpoints. In our examples, the gamma ray and total porosity values of both the clean

sand and pure shale points are assumed to be normally distributed, and all parameters

are assumed to be independent. Note that the distributions of the properties of the

endpoints need not be normal. Also note that we compute shale volume fraction (to be

used in Equations 2.1 – 2.4) from gamma ray values by using a linear transformation.

We also assume that for a pair of endpoints any point inside the diagram that is

constructed using these endpoints is equally likely to occur. Thus the probability


density function is equal to a constant value (i.e., the inverse of the area of the

triangular diagram) that is defined over the entire the triangle. The probability that a

small area surrounding a data point will be chosen or P(a small area surrounding D|E)

is equal to that small area divided by the area of the triangle.

In our example, both the gamma ray and porosity distributions of the shale

endpoint have larger variances than those of the sand endpoint. To run a Monte Carlo

simulation, we first separately draw the properties of the two endpoints, each of which

is a pair of gamma ray and total porosity values, from the selected normal distributions

and input the two end-points into the Thomas-Stieber model. Then we use the model

to infer the values of sand fractions and volume fractions of dispersed clay in the sand

for three selected data points. By drawing multiple realizations of the endpoints

(Figure 2.12), we finally obtain both the joint posterior distributions between sand

fraction and volume fraction of dispersed clay (Figure 2.13) and the marginal

distributions of each variable (Figure 2.14) for the three data points.

Figure 2.12: Multiple realizations of Thomas-Stieber diagrams generated from a set of sand and shale end-points. Three data points are labeled.


Figure 2.13: Posterior distributions for estimated sand fractions of the three data points shown in Figure 2.12.

Figure 2.14: Posterior distributions of volume fraction of dispersed clay in sand (left) and sand fraction (right) for the three data points shown in Figure 2.12.

All three data points have quite similar ranges of uncertainty in the estimated

volume fraction of dispersed clay, with the range of point 1 being slightly smaller

point 2 and point 3. Furthermore, point 1, which has the highest sand fraction, has the

smallest uncertainty in the estimated sand fraction partially due to its location on the

triangular diagram; this point is closer to the sand end-point that is assumed to be less

uncertain than the other endpoint. Due to shale-distribution-dependent relations

between total porosity and gamma ray of the Thomas-Stieber diagram (i.e.,

geometrical interpretation), the variations in the input parameters can influence

estimated properties of points with high and low sand fractions differently.

Sand fraction

Volu

me o

f dis

pers

ed c

lay in s

and

0.4 0.5 0.6 0.7 0.8 0.9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

(1)

(2)

(3)

0 0.1 0.2 0.3 0.40

5

10

15

20

25

30

Volume of dispersed clay in sand

(1)

(2)

(3)

0.4 0.6 0.8 10

5

10

15

20

Sand fraction

(1)

(2)

(3)


To calculate standard deviations of estimated properties at different locations on

the diagram, we use a simple error propagation technique:

𝜎𝑦2 =

𝜕𝑦

𝜕𝑥𝑖

2𝑛𝑖=1 ∙ 𝜎𝑥𝑖

2 ,

(2.8)

where y is a function of x1, x2, …, xn and 𝜎𝑘 is the standard deviation of the variable k.

Note that all parameters are assumed to be independent. Since this equation is based

on a linear approximation of the actual model, its result is reliable only for small

parameter uncertainties (Gujer, 2008). We take variable y to be either the estimated

sand fraction or the estimated volume fraction of dispersed clay, each of which is a

function of six parameters: gamma ray measurement, total porosity measurement,

gamma ray and total porosity values of the sand end-point, and gamma ray and total

porosity values of the shale end-point. We use the mean gamma ray and total porosity

values of the end-points from Figure 2.12 to construct the diagrams in Figure 2.15.

Then, we assign non-zero standard deviations to both the gamma ray and the total

porosity values of both endpoints, while assigning zero standard deviations to the

other parameters. Finally, by using Equation 2.8 we propagate uncertainties in the

properties of the endpoints through the Thomas-Stieber model.

Figure 2.15 shows how uncertainty in the estimated properties varies for different

locations on the Thomas-Stieber diagram. Laminated sequences with cleaner sand and

higher sand fractions, which are represented in the upper left corner of the diagram,

are least affected by the uncertainties of the endpoints. However, the resulting patterns

of uncertainty shown in Figure 2.15 are not universal, but they depend on the setup of

the analysis.


Figure 2.15: Standard deviations of estimated properties at different locations on the Thomas-Stieber diagram. The two estimated values are (left) volume fraction of dispersed clay and (right) sand fraction. The three data points are the same points as in Figure 2.12.

2.4.3 Example with well log data

In this section, we show examples of estimating sand fractions and volume fraction

of dispersed clay with well log data from deep-water turbidite deposits, offshore

Equatorial Guinea, West Africa. An evidence of fine-scale lamination is provided by a

detailed core analysis of this study area (Lowe, 2004). Six lithofacies are identified,

but in this example we focus on only three lithofacies (Figure 2.16). The first

lithofacies is characterized by thick-bedded to massive sandstone, whose bed

thicknesses are greater than 20 cm. Both the second and the third lithofacies are

characterized by interbedded, thin-bedded sandstone and mudstone. In the second

lithofacies, the sandstone bed thicknesses range from 2 to 20 cm and the rock units in

this lithofacies have volume fractions of sandstone greater than 20%. In the third

lithofacies, the sandstone thicknesses are less than 2 cm (Lowe, 2004; Dutta, 2009).

For the selected interval, the gamma log shows an upward-fining trend (Figure

2.17, right). In Figure 2.17 (left), total porosity values are plotted against

corresponding gamma ray values. Suppose that some geological information provides

us the estimates of total porosity and gamma ray distributions for both the sand and the

shale end-points (Figure 2.18). Then Equation 2.7 can be applied to each data point in

Figure 2.17 (left) to estimate its corresponding posterior distributions of sand fraction

(Figure 2.19).

20 40 60 80

0.1

0.15

0.2

0.25

0.3

Gamma ray

Tota

l poro

sity

Color-coded by standard deviation

of estimated sand fraction

20 40 60 80

0.1

0.15

0.2

0.25

0.3

Gamma ray

Tota

l poro

sity

Color-coded by standard deviation

of volume of dispersed clay

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.02

0.03

0.04

0.05

0.06

0.07

0.08

>0.084

(1)

(2)

(3)

(1)

(2)

(3)


Figure 2.16: Three lithofacies from a detailed core analysis by Lowe (2004).

Lithofacies 1 represents thick-bedded to massive sandstone. Lithofacies 2 and 3 represent interbedded, thin-bedded sandstone and mudstone. In lithofacies 2, the sandstone beds are 2 – 20 cm thick and in lithofacies 3 the beds are less than 2 cm thick (Modified from Dutta, 2009).

Figure 2.17: Petrophysical analysis of the selected well-log interval using the Thomas-Stieber model. (Left) Total porosity and gamma ray values for three lithofacies in the selected interval. The median total porosity and gamma ray values for each lithofacies are shown in solid circles. From these median points, the up-down or left-right bars indicate the interquartile ranges (i.e., from 1st to 3rd quartiles) of each property. A Thomas-Stieber diagram is also superimposed on the data. (Right) Variation of gamma ray values with depth. Data points are color-coded by lithofacies similar to the left panel. The gamma ray log shows an upward-fining trend.

20 30 40 50 60 70 80 900.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Gamma ray

Tota

l poro

sity

Lithofacies 1

Lithofacies 2

Lithofacies 3

40 60 80

1070

1075

1080

1085

1090

1095

1100

Gamma ray

Depth

(m

)

0.25 m

Lithofacies 1 Lithofacies 2 Lithofacies 3


Figure 2.18: Probability density functions of properties of the sand end-point and the shale end-point. (Left) total porosity and (right) gamma ray value.

Figure 2.19: Estimated sand fractions for the selected well-log interval. (Left) gamma ray log of the selected interval and (right) posterior distributions of estimated sand fractions.

2.5 The Thomas-Stieber model on rock-physics cross-plots

In sections 2.3 and 2.4, we have discussed two types of clay/shale distribution (i.e.,

dispersed and laminar) in a sand-shale mixture and how the volume fraction of each

type can be estimated using the Thomas-Stieber model, which can be represented as a

plot between total porosity and shale volume fraction (or gamma ray). In this section,

we present the extension of this model to other rock properties on various examples of

0 0.2 0.4 0.60

2

4

6

8

10

Total porosity

0 50 1000

0.02

0.04

0.06

0.08

0.1

Gamma ray

Sand

Shale

40 60 80

1075

1080

1085

1090

1095

Gamma ray

Depth

(m

)

Lithof acies 1

Lithof acies 2

Lithof acies 3

0 0.5 1

1075

1080

1085

1090

1095

Sand fraction

Depth

(m

)


rock-physics cross-plots (Sections 2.5.1 – 2.5.4). Then in Section 2.5.5, we apply all

the cross-plots to the same dataset previously described in Section 2.4.3.

2.5.1 Density and volume fraction of shale

Although the Thomas-Stieber model is developed for porosity and shale volume

fraction, a similar model can be derived analogously for other volumetric properties

(e.g., density versus shale volume fraction). Density of any point on the diagram in

Figure 2.20 can be written as follows:

𝜌 = 1 − 𝑉𝑙𝑎𝑚 𝜌𝑠ℎ𝑎𝑙𝑦 𝑠𝑎𝑛𝑑 + 𝑉𝑙𝑎𝑚 𝜌𝑠ℎ ,

𝜌𝑠ℎ𝑎𝑙𝑦 𝑠𝑎𝑛𝑑 = 𝜌𝑠 + 𝑉𝑑𝑖𝑠𝑝 𝜌𝑠ℎ − 𝜌𝑓𝑙 ,

(2.9)

where 𝜌𝑠 ,𝜌𝑠ℎ and 𝜌𝑓𝑙 are the densities of clean sand, shale, and saturating fluid in the

effective pore space, respectively. Note that 0 ≤ 𝑉𝑑𝑖𝑠𝑝 ≤ 𝜙𝑠 . Thus, to construct the

diagram, we need not only the two end-points ( 𝜌𝑠 and 𝜌𝑠ℎ ) but also 𝜙𝑠.

Figure 2.20: Relationship between density and volume fraction of shale in shaly sand lamination. Point A and E represent clean sand point and pure shale point, respectively. Points B and C represent sand with dispersed clay, and point D is where the original sand pore space is completely filled with dispersed clay (Modified after Mavko et al., 2009).

If we assume that the mineral compositions in clean sand and pure shale are

known, an alternative parameterization can be derived. For example, when sand and

0 0.2 0.4 0.6 0.8 11.9

2

2.1

2.2

2.3

2.4

2.5

Vshale

De

nsity

Vlam = 0.2

Vlam = 0.6

Vlam = 0.8

Vlam = 0.4

B

C

D

A

E


shale are only composed of quartz and clay respectively, we can replace 𝜌𝑠 and 𝜌𝑠ℎ in

Equation 2.9 using the following expressions:

𝜌𝑠 = 𝜌𝑞𝑡𝑧 1 − 𝜙𝑠 + 𝜌𝑐𝑙𝑎𝑦 𝑉𝑑𝑖𝑠𝑝 1 − 𝜙𝑠ℎ + 𝜌𝑤𝑉𝑑𝑖𝑠𝑝 𝜙𝑠ℎ + 𝜌𝑓𝑙 𝜙𝑠 − 𝑉𝑑𝑖𝑠𝑝 ,

𝜌𝑠ℎ = 𝜌𝑐𝑙𝑎𝑦 + 𝜌𝑐𝑙𝑎𝑦 𝜙𝑠ℎ + 𝜌𝑤𝜙𝑠ℎ ,

(2.10)

where 𝜌𝑞𝑡𝑧 ,𝜌𝑐𝑙𝑎𝑦 ,𝜌𝑤𝑎𝑛𝑑 𝜌𝑓𝑙 are the densities of quartz, clay, water, and saturating

fluid in the effective pore space, respectively. In this alternative parameterization, we

write any density on the diagram in Figure 2.20 in terms of the two ―hidden‖

endpoints: 𝜙𝑠 and 𝜙𝑠ℎ . Note that while the Thomas-Stieber model, which relates total

porosity and shale volume fraction (Figure 2.2), is totally independent of saturating

fluids, the relation between density and shale volume fraction (Figure 2.20) does

depend on properties of the saturating fluids. Consequently when superimposing a

density model onto any dataset of sand-shale sequences, the fluid inside the effective

pore space of all the rock data must be the same.

2.5.2 Neutron porosity and density

Neutron-density cross-plots are commonly used in the shaly-sand analysis to

determine volume fraction of clay minerals (Asquith, 1990). Katahara (2008)

suggested that the contrast between neutron porosity and density porosity is often

directly related to clay content more than gamma ray values because radioactivity

measured by the gamma ray log may originate from other non-clay minerals (e.g.,

feldspar), or some clay minerals may be considered non-radioactive (e.g., kaolinite,

chlorite) (Guest, 1990). Figure 2.21 illustrates a neutron-density cross-plot for both the

dispersed and the laminated sand-shale systems.


Figure 2.21: Density-neutron plot for the dispersed and laminated sand-shale systems.

2.5.3 Velocity and total porosity

Analogous to the Thomas-Stieber volumetric model, there are several rock-physics

models for elastic properties of the dispersed and laminated sand-shale systems

(Avseth et al., 2005). The dispersed system consists of two different domains: sandy

shale and shaly sand. The sandy shale describes a mixture where sand grains are

suspended in clay-rich matrix, whereas the shaly sand describes a mixture where

porous clay is added into the sand pore space. The transition between the two domains

occurs when the original sand pore space is completely filled with clay (Marion, 1990;

Yin, 1992; Dvorkin and Gutierrez, 2002). We refer to this transition point as the V-

point.

Dvorkin and Gutierrez (2002) used the Hashin-Shtrikman lower bound (HSLB) to

model sandy shale and the V-point, since the bound is realized by a mixture where

elastically stiffer material (i.e., quartz grains) is enveloped by softer material (i.e.,

porous shale). At the same time, they treated shaly sand as a HSLB mixture between

clean sand and the V-point. In this case, HSLB is also used; the bound is realized by a

mixture where stiffer material (i.e., sand filled with clay) is enveloped by softer

material (i.e., clean sand). When plotting elastic properties against total porosity for

the dispersed sand-shale system, an inverted-V trend is observed (Figure 2.22) and has

been shown by several authors to be good approximation for data from this type of

0 0.2 0.4 0.6 0.8 1

1

1.5

2

2.5

3

water

wet clay

clean sand

Neutron porosity

Density (

g/c

c)

direction of increasing

sand fractions


sand-shale system (e.g., Dvorkin and Gutierrez, 2002; Avseth et al., 2005; Flórez,

2005).

Figure 2.22: Velocity to total porosity curves for the dispersed sand-shale system. The model used here follows Dvorkin and Gutierrez, 2002.

Figure 2.23: Velocity to total porosity curves for the dispersed sand-shale system. The model used here follows Dvorkin and Gutierrez, 2002. Each black line represents lamination between sand (or shaly sand) and shale with volume fraction of laminated shale ranging from 0 to 1. Each blue line represents lamination between sand with a volume fraction of dispersed clay ranging from 0 to 𝜙𝑐𝑙𝑒𝑎𝑛 𝑠𝑎𝑛𝑑 and shale with a constant volume fraction of laminated shale.

For a wave traveling perpendicular to the layers in thin laminations, the effective

P-wave modulus (M) is computed using the Backus average (Mavko et al., 1998):

0 0.1 0.2 0.3 0.4 0.5 0.6

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

Total porosity

Vp

Shaly-sand line

Sandy-shale line

Shale

Sand

Quartz

V-point: Sand completelyfilled with clay

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.42

2.5

3

3.5

Shale

Sand

Total porosity

Vp (

km

/s)

V-point

Shaly-sand line

increasing

sand fractions


𝑀𝑙𝑎𝑚𝑖𝑛𝑎𝑟−1 = (1 − 𝑉𝑙𝑎𝑚 )𝑀𝑠𝑎𝑛𝑑 (𝑜𝑟 𝑠ℎ𝑎𝑙𝑦 𝑠𝑎𝑛𝑑 )

−1 + 𝑉𝑙𝑎𝑚𝑀𝑠ℎ−1 ,

𝑉𝑙𝑎𝑚 = 1 − 𝑉𝑠𝑎𝑛𝑑 .

(2.11)

In Figure 2.23, each black line represents lamination between sand (or shaly sand)

and shale with a volume fraction of laminated shale ranging from 0 to 1. Each blue

line represents lamination between sand with a volume fraction of dispersed clay

ranging from 0 to 𝜙𝑐𝑙𝑒𝑎𝑛 𝑠𝑎𝑛𝑑 and shale with a constant volume fraction of laminated

shale.

2.5.4 Vp/Vs ratio and acoustic impedance

A cross-plot between Vp/Vs ratio and acoustic impedance (AI) is a useful rock-

physics template (RPT) for lithology and pore fluid interpretations. The cross-plot is

constructed by first computing the dry rock moduli at the high-porosity end member

using Hertz-Mindlin contact theory. Then this high-porosity point is connected to the

zero-porosity mineral point with either the modified lower or upper Hashin-Shtrikman

bounds, depending on the choice of sedimentological trends (e.g., using the lower

bound for the sorting trend). The corresponding saturated rock moduli are computed

using Gassmann’s equations (Avseth et al., 2005). Figure 2.24 illustrates an example

of Vp/Vs versus AI cross-plots with incorporated dispersed and laminated sand-shale

models.


Figure 2.24: Rock-physics template shown as a cross-plot between Vp/Vs and AI with superimposed rock-physics trends. The green and magenta curves represent the shale and wet-sand lines, respectively. Along these lines, the change in porosity is due to packing or grain sorting. At each porosity value, the red curve which is connected to the wet-sand line represents the corresponding gas-saturated sand with varying saturations. The dispersed and laminated sand-shale system is constructed using the sand and shale points shown in red circles. Each blue line represents lamination between sand with a volume fraction of dispersed clay ranging from 0 to 𝜙𝑐𝑙𝑒𝑎𝑛 𝑠𝑎𝑛𝑑 and shale with a constant volume fraction of laminated shale.

2.5.5 Application to real data

In Sections 2.5.1 – 2.5.4, we explored the analogs of the Thomas-Stieber model on

various cross-plots. In this section, we integrate different cross-plots and apply them to

the same dataset used in Section 2.4.3; before cross-plotting the data, we perform fluid

substitution on the data using our mesh method2

(Chapter 3) so that they are

completely saturated with brine. Fluid substitution results are shown on various cross-

plots in Figure 2.25. The laminated trend fits the data points on all cross-plots quite

well.

2 The mesh method refers to our proposed fluid-substitution method for interbedded sand-shale

sequences. Refer to Chapter 3 for more details about this method.

2 3 4 5 6 7 8 9

1.5

2

2.5

3

3.5

4

10% porosity

20%

30%

40%

50%

60%

10% porosity

20%

30%

0% gas

100%

AI (g/cm3 x km/s)

Vp/V

s

Shale

Sand

70%

40%

V-pointincreasing

sand fractions

shaly-sand line


Figure 2.25: Selected well-log data on different cross-plots for laminated and dispersed sand-shale systems. The data points are color-coded by their corresponding gamma ray values.

2.6 Discussion

2.6.1 Extension of the uncertainty analysis

We have shown the sensitivity and uncertainty analyses on the Thomas-Stieber

model represented as cross-plots between total porosity and gamma ray (or shale

volume fraction). The same analyses (e.g., uncertainty in endpoints) can be applied to

cross-plots of other volumetric or elastic properties. Note that this chapter does not

incorporate all available data for evaluating thinly bedded sand-shale reservoirs. Other

useful data such as resistivity, elastic anisotropy, NMR-T2 distribution can also be

incorporated into the analyses (e.g., Georgi and Schӧn, 2005; Passey et al., 2006).

Furthermore, as noted earlier, even though we do not account for uncertainty in the

measurements in this chapter, this uncertainty can be incorporated into the analyses

easily by following similar procedures as described in Section 2.4.2.

0 0.1 0.2 0.3 0.42

2.5

3

3.5

Shale

Sand

Total porosity

Vp (

km

/s)

0 0.5 10

0.1

0.2

0.3

0.4

Shale

Sand

Shale volume

Tota

l poro

sity

0 0.5 12

2.2

2.4

2.6

2.8

ShaleSand

Shale volume

Density (

g/c

c)

0.1 0.2 0.3 0.4

2.1

2.2

2.3

2.4

2.5

water

Shale

Sand

Neutron porosity

Density (

g/c

c)

30

40

50

60

70

gamma ray


2.6.2 Constraint on properties of the end-points

By integrating elastic properties into the analysis of thinly bedded sand-shale

reservoirs, rock-physics models can help constrain the choices of the end-point

properties and reduce uncertainty in these input parameters into the Thomas-Stieber

model. For example, elastic properties of well-sorted, clean sands can be computed

using Hertz-Mindlin contact theory and Gassmann’s fluid substitution equations.

Since elastic properties are combined with volumetric properties, the cross-plot

analyses are applicable not only to well log data for investigating the sub-resolution

properties, but also to data at other scales, for example, to seismic inversion results

(e.g., Vp/Vs, AI).

2.6.3 Effect of resolution on consistency in interpretation of different cross-plots

Because geophysical or well logging tools have their own vertical resolutions,

their measurements may represent properties of different volumes of rocks. Therefore,

combining various measurements can create inconsistency in the interpretation. For

example, when cross-plotting measurements from a lower-resolution gamma ray log

against those from a higher-resolution density log, if the rocks are very heterogeneous

at a short scale, the cross-plot may illustrate a misleading pattern or trend, which

affects the interpretation. Resolution of well logging tools can be affected by many

factors including logging speed, tool geometry (Serra, 1984).

We generate two synthetic examples to illustrate the effect of resolution on

interpretation from cross-plots. The first example is represented by an earth model

composed of shaly sand interbedded with shale (Figure 2.26, right). All layers have

the same thickness equal to one unit. The shaly sand and shale layers have fixed

volume fractions of dispersed clay equal to 0.1 and 1, respectively. Their

corresponding total porosity values are 0.225 and 0.25, respectively. We compute

arithmetic averages of both the shale volume and total porosity using window lengths

of 45 and 15 units to simulate a lower-resolution gamma ray log (i.e., shale volume)

and a higher-resolution density log (i.e., density porosity), respectively. The cross-plot

between total porosity and shale volume with different resolutions is shown in Figure


2.26 (left). For comparison, we also compute and cross-plot total porosity and shale

volume when both measurements have the same resolutions (Figure 2.26, middle).

When cross-plotting the two properties that have different vertical resolutions, the data

points are slightly misplaced. As a result, the interpretation of sand fractions (V𝑠𝑎𝑛𝑑 )

and volume fractions of dispersed clay ( V𝑑𝑖𝑠𝑝 ) using this cross-plot is slightly

erroneous (Figure 2.27). Note that the magnitude of the error also depends on the

contrast in layer properties that are averaged and the geometric arrangement of these

layers.

Figure 2.26: The effect of vertical resolution on cross-plots between total porosity and shale volume of a synthetic earth model between shaly sand and shale. The left panel shows a cross-plot between the two measurements with two different resolutions; whereas, the middle panel shows a cross-plot when both measurements share the same resolution. The right panel is a short section of the corresponding synthetic earth model. Sand and shale layers are shown in white and black, respectively.

Figure 2.27: Estimated sand fraction and sand property when applying Thomas-Stieber diagram to points in Figure 2.26 to investigate the effect of vertical resolution on interpretation using cross-plots between total porosity and shale volume.

0 0.5 10.05

0.1

0.15

0.2

0.25

0.3

Shale volume

Tota

l poro

sity

Different vertical resolutions

0 0.5 10.05

0.1

0.15

0.2

0.25

0.3

Shale volume

Tota

l poro

sity

Same vertical resolution

sand point sand point

shale pointshale point

0 0.5 10

0.2

0.4

0.6

0.8

1

True Vsand

Estim

ate

d V

sand

0 0.2 0.40

0.1

0.2

0.3

0.4

True Vdisp

Estim

ate

d V

dis

p


The second example of the effect of resolution has a setup similar to the first

example, but the volume fractions of dispersed clay in the shaly-sand layers are not

fixed and the layer arrangement is non-stationary along the sequence. When cross-

plotting the two properties that have different vertical resolutions, the data points are

clearly misplaced (Figure 2.28, left). Interestingly, the interpretation of sand fractions

( V𝑠𝑎𝑛𝑑 ) is only slightly affected by the difference in resolutions; whereas, the

interpretation of volume fractions of dispersed clay (V𝑑𝑖𝑠𝑝 ) is more affected (Figure

2.29).

Figure 2.28: The effect of vertical resolution on cross-plots between total porosity and shale volume of a synthetic (non-stationary) earth model between shaly sand and shale. The left panel shows a cross-plot between the two measurements with two different resolutions; whereas, the middle panel shows a cross-plot when both measurements share the same resolution. The right panel is a short section of the corresponding synthetic earth model. Sand and shale layers are shown in white and black, respectively.

Figure 2.29: Estimated sand fraction and sand property when applying Thomas-Stieber diagram to points in Figure 2.28 to investigate the effect of vertical resolution on interpretation using cross-plots between total porosity and shale volume.

0 0.5 10.05

0.1

0.15

0.2

0.25

0.3

0.35

Shale volume

Tota

l poro

sity

Different vertical resolutions

0 0.5 10.05

0.1

0.15

0.2

0.25

0.3

0.35

Shale volume

Tota

l poro

sity

Same vertical resolution

shale point

sand pointsand point

shale point

0 0.5 10

0.2

0.4

0.6

0.8

1

True Vsand

Estim

ate

d V

sand

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

True Vdisp

Estim

ate

d V

dis

p


2.7 Conclusions

The Thomas-Stieber model can be used for evaluating thinly bedded sand-shale

reservoirs as it can help determine both the sand fractions within the lamination and

the property of the sand. We present the sensitivity and uncertainty analyses of the

model, when natural variations are integrated into the model in order to quantify

uncertainty associated with reservoir evaluations. Using synthetic examples, we

illustrate that even when the correct end-points are used in the Thomas-Stieber model,

rocks in nature can be so complex that they violate several assumptions of the model,

resulting in erroneous interpretation of reservoir properties. When the properties of the

end-points are uncertain, this uncertainty can be incorporated into the model using

Monte Carlo simulations in a Bayesian framework. The Thomas-Stieber model is not

only limited to a cross-plot between total porosity and gamma ray (or shale volume),

but also other rock properties including elastic properties. Rock-physics models can

then be used to constrain the properties of the end-points. If measurements come from

tools with different vertical resolutions, there can be inconsistency among the cross-

plots. Through synthetic examples, we show that this resolution problem can lead to

erroneous interpretation and that the estimated sand fractions seem to be affected by

the resolution less severe than the estimated volume fractions of dispersed clay.

2.8 Acknowledgements

This work was supported by the SRB Project, the American Chemical Society

grant number 46350-AC8 and the Stanford Center for Reservoir Forecasting. We

would like to thank Hess Corporation for providing the data.

2.9 References

Asquith, G. B., 1990, Log evaluation of shaly sandstones: A practical guide: American

Association of Petroleum Geologists, Course note series # 31.

Avseth, P., Mukerji, T. and Mavko, G., 2005. Quantitative Seismic Interpretation,

Cambridge.


Ball, V., Erickson, S., Brown, L., 2004. A Model-centric Approach to Seismic

Petrophysics: SEG Expanded Abstracts, 23, 1730.

Boggs, S., 2001, Principles of sedimentology and Stratigraphy, 3rd

ed.


12.

Cumberland, D. J. and Crawford, R. J., 1987, The packing of particles: Handbook of

powder technology, v.6.

Dawson, W. C., W. R. Almon, K. Dempster, and Sutton, S. J., 2008, Shale variability

in deep-marine depositional systems: implications for seal character – subsurface

and outcrop examples: American Association of Petroleum Geologists, Search and

Discovery Article #50128, accessed 3 March 2009;

http://www.searchanddiscovery.com/documents/2008/08144dawson/ndx_dawson.

pdf

Dutta, T., 2009. Integrating Sequence Stratigraphy and Rock-physics to Interpret

Seismic Amplitudes and Predict Reservoir Quality, Ph.D. Thesis, Stanford

University.

Dvorkin, J. and Gutierrez, M. A., 2002, Grain sorting, porosity and elasticity:

Petrophysics, 43(3), 185-196.

Flórez, J., 2005. Integrating Geology, Rock physics, and Seismology for Reservoir-

quality Prediction, Ph.D. Thesis, Stanford University.

Gassmann, F., 1951. Uber die elastizitat poroser medien: Vier Natur Gesellschaft, 96,

1-23.

Georgi, D. and Schӧn, J., 2005, Elastic wave anisotropy and shale distribution:

SPWLA 46th

Annual Logging Symposium, June 26-29, Paper Q.

Guest, K., 1990, The use of core derived quantitative mineralogical data to improve

formation evaluation: Advances in core evaluation: Accuracy and precision in

reserves estimation, Ed. Paul F. Worthington, Pennsylvania: Gordon and Breach

Science Publishers, 187-210.

Gujer, W., 2008, Systems analysis for water technology, Springer.

Juhasz, I., 1986, Assessment of the distribution of shale, porosity and hydrocarbon

saturation in shaly sands: Transactions of the SPWLA 10th European Formation

Evaluation Symposium, Aberdeen, 22 April, AA1-15.

Katahara, K., 2004, Fluid substitution in laminated shaly sands: 74th Annual Meeting,

SEG Expanded Abstracts, 1718-1721.


Katahara, K., 2008, What is Shale to a Petrophysicist?: The Leading Edge, 27, 738-

741.

Koltermann, C. E. and Gorelick, S. V., 1995, Fractional packing model for hydraulic

conductivity derived from sediment mixtures: Water Resources Research, 31(12),

3283 – 3297.

Lowe, D. R., 2004, Report on core logging, lithofacies, and basic sedimentology of

Equatorial Guinea: Hess internal report.

Marion, D., 1990, Acoustical, mechanical and transport properties of sediments and

granular materials: Ph.D. Thesis, Stanford University.

Mavko, G., Mukerji, T., and Dvorkin, J., 1998. The Rock Physics Handbook, 1st

edition, Cambridge.

Mavko, G., Mukerji, T., and Dvorkin, J., 2009, The Rock Physics Handbook, 2nd

edition, Cambridge.

Passey, Q. R., Dahlberg, K. E., Sullivan, K. B., Yin, H., Brackett, R. A., Xiao, Y. H.,

and Guzmán-Garcia, A. G., 2006, Petrophysical evaluation of hydrocarbon pore-

thickness in thinly bedded clastic reservoirs: AAPG Archie series, 1, American

Association of Petroleum Geologists, Tulsa, Oklahoma, U.S.A.

Revil, A., Grauls, D., and Brévart, O., 2002, Mechanical compaction of sand/clay

mixtures: Journal of Geophysical Research, 107(B11).

Serra, O., 1984, Fundamentals of well-log interpretation: The acquisition of logging

data, 1st edition, Elsevier.


488.



Annual Logging



effect upon porosity: 16th Annual Logging Symposium, SPWLA, paper T.

Voleti, D., Tyagi, A., Singh, A., Mundayat, V., Pandey, V., and Saxena, K., 2012, A

new petrophysical interpretation approach to characterize the thinly laminated

reservoir using conventional tools: SPE Oil and Gas India conference and

exhibition, SPE 152912.

Worden, R. H., and Morad, S., 2003, Clay minerals in sandstones: controls on

formation distribution and evolution: Int. Assoc. Sedimentol. Spec. Publ., 34, 3 –

41.


Worthington, P. F., 2003, Effect of clay content upon some physical properties of

sandstone reservoirs: Int. Assoc. Sedimentol. Spec. Publ., 34, 191 – 211.

Yadav, L., Dutta, T., Kundu, A., and Sinha, N., 2010, A new approach for the realistic

evaluation of very thin reservoirs of Krishna Godavari basin, East coast, India:

SPE Asia Pacific Oil & Gas conference and exhibition, SPE 132970.

Yin, H., 1992, Acoustic Velocity and Attenuation of Rocks: Isotropy, Intrinsic

Anisotropy, and Stress Induced Anisotropy, Ph.D. Thesis, Stanford University.

47

Chapter 3

Fluid substitution for sub-resolution

interbedded sand-shale sequences

using the mesh method

3.1 Abstract

This chapter provides a simple graphical mesh interpretation and accompanying

equations for approximating fluid substitution in sub-resolution interbedded sand-

shale sequences. The sand layers can be either clean or shaly (i.e., sand with dispersed

clay). Because geophysical logging tools have limited vertical resolution, their

measurements often represent average properties of multiple sedimentary layers. If

this sub-resolution sand-shale interbedding is not properly accounted for, applying

fluid substitution at the measurement scale can lead to erroneous predictions. Given

that all the properties and input parameters are known, physically the most accurate

CHAPTER 3: Fluid substitution for interbedded sand-shale sequences 48

way to perform fluid substitution in these interbedded sequences is to first downscale

(i.e., invert) the measurements for the properties of the two end-members, which are

sand and shale. Gassmann’s equation is then applied only to the sand end-member,

and finally the properties of the sand with a new fluid and the shale are upscaled (i.e.,

averaged) back to the measurement scale. However, in practice the required input

parameters are only estimates. Therefore, fluid substitution by the downscaling-

upscaling procedure can be unstable and non-robust.

We propose a mesh method which combines rock-physics models for dispersed

and interbedded sand-shale systems, the Thomas-Stieber model, and Gassmann’s fluid

substitution equation so that our method can be used for fluid substitution in

interbedded sand-shale sequences directly at the measurement scale, without the need

to downscale the measurements, while still changing fluid in the sand layers only. We

apply our method to both synthetic and real well log data, and compare the results

with those predicted by simply using Gassmann’s equation (which ignores the effect

of thin sand-shale interbedding) and by the downscaling-upscaling procedure. Since

ideally the downscaling-upscaling procedure is the appropriate way to perform fluid

substitution in interbedded sand-shale sequences, we use fluid substitution results from

this procedure as baselines for both synthetic and real data. These baselines are

considered correct in synthetic data, because all the parameters to be estimated are

known. However, as previously noted the downscaling-upscaling procedure may yield

unreliable results for real data, especially in data with very low sand fractions.

Therefore, we select only a portion of the results from the downscaling-upscaling

procedure as the baselines for real data.

The results of our comparison are the following. Since Gassmann’s equations do

not account for the effect of sub-resolution sand-shale interbedding, the changes in

elastic moduli after fluid substitution are overpredicted when compared with the

baselines. In contrast, the results from the mesh method agree well with the baselines.

Thus even when our mesh method is applied at the measurement scale, the method

appropriately accounts for the effect of sub-resolution sand-shale interbedding. We

also perform a sensitivity analysis of five input parameters needed for the mesh


method which are in decreasing order of sensitivity clean sand velocity, clean sand

porosity, effective porosity, effective water saturation and fluid elastic modulus.

3.2 Introduction

Fluid substitution is an important rock physics technique used to quantitatively

predict elastic properties of rocks when one saturating pore fluid replaces another. One

tool commonly used for fluid substitution is the isotropic Gassmann’s equation

(Gassmann, 1951). Fluid substitution using this equation involves two steps. First, the

rock’s dry-frame bulk modulus (𝐾𝑑𝑟𝑦 ) is estimated from the initial measurements of

rocks saturated with the original “fluid 1” (e.g., P- and S-wave velocities, density)

using the following relation (Mavko et al., 2009):

𝐾𝑑𝑟𝑦 =𝐾𝑠𝑎𝑡 ∗ 𝜙𝑇𝐾𝑚𝑖𝑛 𝐾𝑓𝑙∗ + 1 − 𝜙𝑇 − 𝐾𝑚𝑖𝑛𝜙𝑇𝐾𝑚𝑖𝑛 𝐾𝑓𝑙∗ + 𝐾𝑠𝑎𝑡 ∗ 𝐾𝑚𝑖𝑛 − 1 − 𝜙𝑇

.

(3.1)

In equation 3.1, 𝐾𝑠𝑎𝑡 ∗ and 𝐾𝑓𝑙∗ are the bulk modulus of the saturated rock and the bulk

modulus of the original saturating fluid (fluid 1). 𝐾𝑚𝑖𝑛 is the bulk modulus of the solid

mineral, and 𝜙𝑇 is the total porosity. Second, this dry-frame modulus is used to predict

the rock’s bulk modulus when pore fluid 1 is replaced by “fluid 2” using the following

relation:

𝐾𝑠𝑎𝑡 =𝜙𝑇 1 𝐾𝑚𝑖𝑛 − 1 𝐾𝑓𝑙 + 1 𝐾𝑚𝑖𝑛 − 1 𝐾𝑑𝑟𝑦

𝜙𝑇 𝐾𝑑𝑟𝑦 1 𝐾𝑚𝑖𝑛 − 1 𝐾𝑓𝑙 + 1 𝐾𝑚𝑖𝑛 1 𝐾𝑚𝑖𝑛 − 1 𝐾𝑑𝑟𝑦 ,

(3.2)

where 𝐾𝑓𝑙 is the bulk modulus of fluid 2. Although the bulk modulus of the saturated

rock depends on the pore fluid, Gassmann’s equations predict that the shear modulus

remains unchanged under fluid substitution (e.g., Berryman, 1999; Smith et al., 2003).

Even though Gassmann’s fluid substitution equation seems simple and easy to

implement, applying it to real rocks often yields erroneous results, because real rocks


can be so complex that several assumptions in Gassmann’s theory are violated (e.g.,

Smith et al., 2003; Han and Batzle, 2004; Chopra, 2005). For example, Gassmann’s

theory assumes that rocks are homogeneous. However, since geophysical tools have

limited vertical resolution, their measurements always represent averages of the

intrinsically heterogeneous rocks. Consider, for example, a stack of sub-resolution

interbedded sand and shale layers. Shaly portions of the stack have low permeability

and contain bound water associated with their clay-mineral component; these

characteristics violate Gassmann’s assumption regarding pore connectivity and perfect

pore-fluid communication. As a result, the traditional Gassmann’s equation may not

be appropriate for fluid substitution in the shale layers (Chopra, 2005; Dvorkin et al.,

2007; Katahara, 2008). Therefore, if the isotropic Gassmann’s fluid substitution

equation is applied to the measurements at their original scales, without accounting for

the sub-resolution sand-shale interbedding, the resulting predictions may be erroneous

(e.g., Katahara, 2004; Skelt, 2004a; Skelt, 2004b; Chopra, 2005).

The physically correct way to perform fluid substitution in the thinly-layered

system is to first downscale the measurements for the separate sand and shale end-

members’ properties, then apply Gassmann’s equation to the sand layers only, and

finally upscale the layers back to the measurement scale by using the Backus average

(Katahara, 2004; Skelt, 2004a; Chopra, 2005; Singleton and Keirstead, 2011). This

downscaling-upscaling procedure is an ideal solution when the shale properties and

sand fraction are known. However, in practice uncertainties in estimated parameters

and uncertainties in measurements can affect fluid substitution results, especially

during the downscaling step. Several authors (e.g., Katahara, 2004; Skelt, 2004a;

Singleton and Keirstead, 2011) have proposed methods for fluid substitution in

interbedded sand-shale sequences in which the sand is assumed to be clean and its

volume fraction is equivalent to simply (1 − 𝑉𝑠𝑕), where 𝑉𝑠𝑕 is shale volume fraction.

Thus these methods are not intended for the case where the sandy layers are shaly (i.e.,

sand having dispersed clay inside its pore space).

The presence of clay dispersed in the pore space of sand complicates the fluid

substitution problem, because as previously noted, immobile bound-water in clay may


block communication of the pore fluids, which violates a basic assumption of

Gassmann’s equation (Smith et al., 2003; Mavko et al., 2006; Dvorkin et al., 2007;

Simm, 2007). Several proposed methods for fluid substitution in shaly sand are simply

modifications of the traditional Gassmann’s equation. Simm (2007) discussed

workflows for fluid substitution in shaly sand and suggested an alternative approach

which adjusts the dry-rock modulus parameters in Gassmann’s equation using

empirical trends. Dvorkin et al (2007) proposed a method for fluid substitution in

shaly sand which uses effective porosity instead of total porosity in the traditional

Gassmann’s equation, and also assumes a new composite mineral phase by combining

wet porous shale and solid minerals together. Both of these proposed methods are

intended for fluid substitution in shaly sand, but not for interbedded sequences. Skelt

(2004b) showed that when laminated (interbedded) shale is modeled as dispersed,

large errors may be introduced into the predicted compressional velocities. Thus, none

of the above methods provides a complete solution for the problem of fluid

substitution in interbedded (shaly) sand-shale sequences.

In this chapter, we propose a simple graphical “mesh” method for fluid

substitution in sub-resolution interbedded sand-shale sequences, which appropriately

changes fluid in the sands only, without the need to downscale the measurements.

Unlike previous methods which are limited to fluid substitution for interbedded clean

sand and shale or for simply shaly sand, our mesh method is applicable also to

interbedded sand-shale sequences, in which the sand layers can be either clean or

shaly. The remainder of this chapter is organized as follows. Section 3.3 discusses

rock-physics models for a sand-shale mixture system, including dispersed and laminar

(interbedded) mixes. In Section 3.4, we integrate fluid substitution into the sand-shale

models, leading to the graphical mesh interpretation and accompanying equations for

fluid substitution in interbedded sand-shale sequences. Section 3.5 shows a

comparison between fluid substitution results by the mesh approach and by other

methods using synthetic data. Section 3.6 discusses some pitfalls in interpretation

when sub-resolution is not accounted for. Then, in Section 3.7, we run a sensitivity

analysis of input parameters needed for the mesh method. In Section 3.8, we test our


method on real data, and finally in Section 3.9 we discuss limitations and possible

extensions of the mesh approach.

3.3 Elastic properties of interbedded sands

3.3.1 Models

While the Thomas-Stieber volumetric model1 (Thomas and Stieber, 1975) relates

total porosity to shale distribution in the sand-shale mixtures, several rock-physics

models have explored elastic properties of the dispersed and interbedded sand-shale

systems (Avseth et al., 2005). The dispersed portion of the sand-shale system consists

of two different domains: “sandy shale” and “shaly sand” (Figure 3.1). The sandy

shale describes a mixture where sand grains are suspended in clay-rich matrix,

whereas the shaly sand describes a mixture where porous clay is added into the sand

pore space. The transition between the two domains occurs when the original sand

pore space is completely filled with clay (Marion, 1990; Yin, 1992; Dvorkin and

Gutierrez, 2002). We call this transition point the V-point.

Dvorkin and Gutierrez (2002) used the Hashin-Shtrikman lower bound (HSLB) to

model sandy shale and the V-point, since the bound is realized by a mixture where

elastically stiffer material (i.e., quartz grains) is enveloped by softer material (i.e.,

porous shale). At the same time, they treated shaly sand as a HSLB mixture between

clean sand and the V-point. In this case, the HSLB is also used; the bound is realized

by a mixture where stiffer material (i.e., sand filled with clay) is enveloped by softer

material (i.e., clean sand). When plotting elastic properties against total porosity, the

shaly-sand and sandy-shale legs of the dispersed sand-shale system form an “inverted-

V” trend (Figure 3.1), and this trend has been shown by several authors to be good

approximation for data from this type of sand-shale system (e.g., Dvorkin and

Gutierrez, 2002; Avseth et al., 2005; Flórez, 2005).

1 Refer to Chapter 2 for more details.


Figure 3.1: Inverted-V relation between P-wave velocity and total porosity for a

dispersed sand-shale system, following the model of Dvorkin and Gutierrez (2002). In this case, curves are computed using the Hashin-Shtrikman lower bound (HSLB). The solid magenta and blue curves represent the sandy-shale and shaly-sand legs of the dispersed sand-shale system.

We follow an approach similar to that of Dvorkin and Guiterrez (2002) to model

elastic properties of shaly sand with a slight modification in the calculation of the V-

point. Instead of using HSLB for the V-point, we use the Voigt-Reuss-Hill average

(VRH) between properties of quartz at zero total porosity (𝜙𝑇= 0) and porous shale at

𝜙𝑇= 𝜙𝑠𝑕 . This achieves better consistency in fluid substitution results for synthetic

data. The elastic modulus at the V-point corresponds to the value of the VRH average,

evaluated at the point where 𝜙𝑇= 𝜙𝑠𝜙𝑠𝑕 . The shaly-sand line can then be modeled by

a lower elastic bound that connects the V-point and the clean sand point (𝜙𝑇= 𝜙𝑠).

When the elastic contrast between the two materials is relatively small, using either

HSLB or the Reuss lower bound (with P-modulus only) gives similar results (Dvorkin

and Gutierrez, 2002). In this study, we model the shaly-sand line using the Reuss

average of P-wave moduli rather than the bulk and shear modulus, since S-wave

information is often unavailable or unreliable. Using only the P-wave moduli,

equations for elastic properties of the V-point and the shaly-sand line are as follows:

𝑉𝑠𝑕 = 𝜙𝑠 : 𝑀𝑉𝑜𝑖𝑔𝑡 ≈ (1 − 𝜙𝑠)𝑀𝑞𝑡𝑧 + 𝜙𝑠𝑀𝑠𝑕 ,

𝑉𝑠𝑕 = 𝜙𝑠 : 𝑀𝑅𝑒𝑢𝑠𝑠−1 ≈ (1 − 𝜙𝑠)𝑀𝑞𝑡𝑧

−1 + 𝜙𝑠𝑀𝑠𝑕−1 ,

𝑉𝑠𝑕 = 𝜙𝑠 : 𝑀𝑠𝑎𝑛𝑑 −𝑓𝑖𝑙𝑙𝑒𝑑 −𝑤𝑖𝑡 𝑕−𝑠𝑕𝑎𝑙𝑒 ≈ 0.5 ∗ 𝑀𝑉𝑜𝑖𝑔𝑡 + 𝑀𝑅𝑒𝑢𝑠𝑠 |𝑣𝑠𝑕= 𝜙𝑠 ,

0 ≤ 𝑉𝑠𝑕 ≤ 𝜙𝑠 : 𝑀𝑠𝑕𝑎𝑙𝑦 −𝑠𝑎𝑛𝑑−1 ≈ (1 − 𝐹)𝑀𝑠

−1 + 𝐹𝑀𝑠𝑎𝑛𝑑 −𝑓𝑖𝑙𝑙𝑒𝑑 −𝑤𝑖𝑡 𝑕−𝑠𝑕𝑎𝑙𝑒−1 ,

0 0.1 0.2 0.3 0.4 0.5 0.6

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

Total porosity

Vp

Shaly-sand line

Sandy-shale line

Shale

Sand

Quartz

V-point: Sand completelyfilled with clay


𝑤𝑖𝑡𝑕 𝐹 =𝑉𝑠𝑕𝜙𝑠

.

(3.3)

In equation 3.3, 𝑀 is the P-wave modulus, and subscripts s, sh, and qtz stand for clean

sand, pure shale and quartz, respectively.

Figure 3.2: Relation between velocity and total porosity for a dispersed sand-shale system, following the model of Dvorkin and Gutierrez (2002) with a slight modification. Each red line represents interbedding of sand (or shaly sand) and shale with volume fraction of interbedded shale ranging from 0 to 1. Each blue line represents interbedding of sand with a volume fraction of dispersed clay ranging from 𝑉𝑑𝑖𝑠𝑝 = 0 to 𝑉𝑑𝑖𝑠𝑝 = 𝜙𝑆 and shale with a constant volume fraction of interbedded shale.

For a wave traveling perpendicular to the layers in thin sand-shale sequences, the

effective P-wave modulus is computed using the Backus average (Backus, 1962;

Mavko et al., 1998):

𝑀𝑙𝑎𝑚𝑖𝑛𝑎𝑟−1 = (1 − 𝑉𝑙𝑎𝑚 )𝑀𝑠𝑎𝑛𝑑 (𝑜𝑟 𝑠𝑕𝑎𝑙𝑦 𝑠𝑎𝑛𝑑 )

−1 + 𝑉𝑙𝑎𝑚𝑀𝑠𝑕−1 ,

𝑉𝑙𝑎𝑚 = 1 − 𝑉𝑠𝑎𝑛𝑑 ,

(3.4)

where 𝑉𝑙𝑎𝑚 is the volume fraction of laminated (interbedded) shale and 𝑉𝑠𝑎𝑛𝑑 is the

volume fraction of laminated (interbedded) sand which can be clean or shaly. We refer

to this volume as the sand fraction. As noted in Chapter 1, in this dissertation we use

0 0.1 0.2 0.3 0.4

2

2.5

3

3.5

4

4.5

5

Total porosity

Vp (

km

/s)

Shale

Sand

V-point

(HSLB)

V-point

(VRH)

increasing

sand fractions

Shaly-sand line


the terms laminated and interbedded interchangeably. Both the laminated sand-shale

sequence and the interbedded sand-shale sequence here refer to a sequence of

alternating sand and shale units.

Both Equations 3.3 and 3.4 are simply harmonic averages of the P-wave moduli of

all components. These equations are equivalent to the volume-weighted arithmetic

averages of the P-wave compliances or compressional compliances (C = 1/M). Figure

3.3 shows a cross-plot between the compressional compliance and effective porosity.

In this domain, all the curves become straight lines. In both Figure 3.2 and Figure 3.3,

each red line represents interbedding of sand (or shaly sand) and shale. Along the line,

the volume fraction of laminated (interbedded) shale (𝑉𝑙𝑎𝑚 ) changes from 𝑉𝑙𝑎𝑚 = 0 to

𝑉𝑙𝑎𝑚 = 1 but 𝑉𝑑𝑖𝑠𝑝 is constant. Each blue line represents interbedding of sand and shale,

where 𝑉𝑙𝑎𝑚 is held constant and 𝑉𝑑𝑖𝑠𝑝 changes from 𝑉𝑑𝑖𝑠𝑝 = 0 to 𝑉𝑑𝑖𝑠𝑝 = 𝜙𝑠. We refer to

the set of red and blue lines as the mesh. (Note that non-linear parameterization of the

shaly-sand curves can be chosen by using alternative models for the curves instead of

using the Reuss average.)

Figure 3.3: P-wave compliance (C=1/M) versus effective porosity curves for a interbedded sand-shale system. Each red line represents interbedding of sand (or shaly sand) and shale with volume fraction of laminated (interbedded) shale ranging from 𝑉𝑙𝑎𝑚 = 0 to 𝑉𝑙𝑎𝑚 = 1. Each blue line represents interbedding of sand with a volume fraction of dispersed clay ranging from 𝑉𝑑𝑖𝑠𝑝 = 0 to 𝑉𝑑𝑖𝑠𝑝 =𝜙𝑆 and shale with a constant volume fraction of laminated (interbedded) shale.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.02

0.04

0.06

0.08

0.1

Effective porosity

1/M

(1/G

Pa)

Increasing sand fractions

Clean sand

Pure shale

V point: sand completely

filled with clay


In this work, we assume that fluid substitution only occurs in the effective porosity.

Since the shale point and the points along the sandy-shale line have zero effective

porosity, their properties remain unchanged under fluid substitution.

3.3.2 Modeling the V-point using the Voigt-Reuss-Hill average

Even though the P-wave velocity of the V-point from VRH is much larger than

that estimated from the HSLB (Figure 3.2), the VRH average is chosen here to ensure

consistency with the fluid substitution calculations. In the case of non-monomineralic

rocks, the effective mineral bulk modulus is needed as an input in Gassmann’s

equation. Since we will follow common practice and use the VRH average to estimate

this effective modulus (Dvorkin and Guiterrez, 2002; Smith et al., 2003; Kumar, 2006),

it is essential to also model shaly-sand in our synthetic data using the VRH to calculate

the V-point.

To illustrate possibly inconsistent results during fluid substitution, we test various

scenarios where existing models are mixed and matched to generate three elements

needed in fluid substitution for synthetic data: the shaly-sand line, V-point, and

effective mineral modulus. First, we model the brine-saturated shaly sand line and the

V-point with the specifications listed in Table 3.1. In every case, brine will be replaced

by oil. The method of Dvorkin et al (2007) for fluid substitution in shaly sand uses

effective porosity instead of total porosity in Gassmann’s equation (Equations 3.1 and

3.2) and also assumes a composite mineral phase consisting of wet porous shale and

solid minerals together. Hereafter we refer to this method as the Gassmann shaly-sand

model. The effective porosity (𝜙𝐸𝑓𝑓 ) can be calculated by following the dispersed

clay-porosity relation in the Thomas-Stieber model (Equation 3.5; refer to Chapter 2

for more details).

0 ≤ 𝑉𝑠𝑕 ≤ 1 𝜙𝐸𝑓𝑓 = 𝜙𝑇 − 𝜙𝑠𝑕𝑉𝑠𝑕 .

(3.5)

Various methods of computing the effective mineral modulus are listed in Table 3.2.

Results of fluid substitution are shown in Figure 3.4 and Figure 3.5. Note that in both

figures all the lines labeled as “oil-model…” are fluid substitution predictions. The


“oil- shaly-sand” lines are constructed by connecting the V-points and the clean sand

points that are saturated with oil.

Table 3.1: Models used to generate elastic moduli of brine-saturated shaly-sand.

Case V-point

(𝝓𝑻 = 𝝓𝒔𝝓𝒔𝒉)

Models for shaly-sand lines

(mixing between V-point and clean

sand)

MHH VRH*

VRH

MHR VRH* Reuss

MRR Reuss**

Reuss

MLR HSLB***

Reuss *Voigt-Reuss-Hill average of quartz and wet-shale moduli **Reuss average of quartz and wet-shale moduli ***Hashin-Shtrikman lower bound of quartz and wet-shale moduli

Table 3.2: Models used to estimate effective solid moduli for fluid substitution.

Model # Models for effective mineral (mixing between quartz and wet

shale)

1 VRH

2 Reuss

3 HSLB

Figure 3.4 shows results when only P-wave moduli are used in all calculations

(assuming S-wave information is unavailable or unreliable). When the models used to

compute the V-point modulus and the effective mineral modulus for the Gassmann

shaly-sand equation (Dvorkin et al., 2007) are not the same, Dvorkin’s fluid

substitution (from brine to oil) near zero effective porosity can result in either

inconsistent or negative moduli.

First, we discuss cases where inconsistent results are observed. Since only fluids in

the effective porosity are replaced, at zero effective porosity we expect no fluid-

substitution effect. Thus, as effective porosity decreases and approaches zero, the

fluid-substituted P-wave compliance values should approach the P-wave compliance

of the wet-shaly sand at zero effective porosity. However, this is not the case when the

V-point modulus is larger than the effective mineral modulus used in the Gassmann

shaly-sand equation (Note that V-point modulus from VRH > HSLB > Reuss). For


example, in Figure 3.4(a) and (b), the V-point modulus is computed using the VRH

average which is larger than the modulus computed using either HSLB or the Reuss

average; as a result, after fluid substitution is applied, there are non-zero changes in

compliance at zero effective porosity (i.e., the cyan and red curves), creating

inconsistent results. This change in compliance at zero effective porosity after fluid

substitution is also observed in Figure 3.4(d) when HSLB is used to compute the V-

point modulus, and the Reuss average is used to compute the effective mineral

modulus. However, the changes in compliance at zero effective porosity in Figure

3.4(d) are smaller than those in Figure 3.4(a) and (b).

Second, we discuss cases where the fluid-substituted P-wave compliance near zero

effective porosity becomes negative and exhibits a mathematical singularity. This

implausible value is observed when the V-point modulus is smaller than the effective

mineral modulus used in the Gassmann shaly-sand equation. For example, in Figure

3.4(c), the V-point modulus is computed using the Reuss average which is the smallest

modulus among those computed using all the considered models. When either VRH or

HSLB is used to compute the effective mineral modulus, both negative compliance

and singularity can be observed (the green and cyan curves in Figure 3.4(c)).

When the same model is used to compute the V-point modulus and the effective

mineral modulus for the Gassmann shaly-sand equation, the resulting compliances

after fluid substitution are very similar to the compliances along the oil- shaly sand

lines.

All of the above observations are the results of the inconsistency test when only P-

wave moduli are used in the calculations (Figure 3.4). When both bulk and shear

moduli are used (Figure 3.5), the results are similar, except that the magnitude of the

inconsistency becomes smaller (Figure 3.5(a) and (b)).

In summary, choices of models used for constructing the shaly-sand lines, the V-

point, and for computing effective mineral moduli used in fluid substitution equations

can vary; however, consistency among models need to be taken into consideration to

ensure that fluid substitution results are consistent with all the assumptions (e.g., no


change in compliance at zero effective porosity) and that the results fall within a range

of plausible values (e.g., no singularity). Based on our inconsistency test results, when

applying fluid substitution using the Gassmann shaly-sand equation, the model for

computing the V-point modulus and the model for computing the effective mineral

modulus should be the same. In the next section, we introduce fluid substitution for

sub-resolution interbedded sand-shale sequences with the mesh, and we will see that

our mesh method is much less sensitive to the modeling choices for computing the V-

point.

Figure 3.4: Shaly sand lines before and after fluid substitution when only P-wave moduli are used in calculations. The model used to generate wet shaly sand is listed in the lower right corner ((a) – (d) and Table 3.1). Starting with the brine-saturated shaly sand (black dash-line), oil is substituted using the Gassmann shaly-sand equation (Dvorkin et al., 2007) and three different models for effective solid moduli (Table 3.2).

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

Effective porosity

1/M

(1/G

Pa)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

Effective porosity

1/M

(1/G

Pa)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

1/M

(1/G

Pa)

Effective porosity

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

Effective porosity

1/M

(1/G

Pa)

wet-shaly sand

oil-model1 (VRH)

oil-model2 (Reuss)

oil-model3 (HSLB)

oil-shaly sand

MHH MHR

MRR MLR

(a) (b)

(c) (d)


Figure 3.5: Shaly sand lines before and after fluid substitution when both bulk and shear moduli are used in calculations. The model used to generate wet shaly sand is listed in the lower right corner ((a) – (d) and Table 3.1). Starting with the brine-saturated shaly sand (black dash-line), oil is substituted using the Gassmann shaly-sand equation and three different models for effective solid moduli (Table 3.2).

3.4 Fluid substitution for interbedded sands

In Section 3.3.1, we have shown the diagram between P-compliance and effective

porosity for the dispersed and interbedded sand-shale system. In this section, we use

this diagram to show how to do approximate fluid substitution in shaly sands,

introduce a simple graphical interpretation, and derive equations for fluid substitution

in interbedded sands including sand with dispersed clay.

3.4.1 Approximate fluid substitution in shaly sands

Based on the results of the inconsistency test in Section 3.3.2, from this point on

we will only discuss cases where the same model is used for computing both the V-

point moduli and the effective mineral moduli. When the shaly-sand lines are modeled

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

1/M

(1/G

Pa)

Effective porosity

0 0.1 0.2 0.30

0.05

0.1

0.15

Effective porosity

1/M

(1/G

Pa)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

Effective porosity

1/M

(1/G

Pa)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

Effective porosity

1/M

(1/G

Pa)

wet-shaly sand

oil-model1 (VRH)

oil-model2 (Reuss

oil-model3 (HSLB)

oil-shaly sand

MHR

MRR MLR

(d)

(a) (b)

(c)

MHH


by the Reuss average, these lines become straight in the plot of P-compliance versus

effective porosity (Figure 3.4(b)-(d) and Figure 3.5(b)-(d)). If we apply the Gassmann

shaly-sand equation (Dvorkin et al., 2007) to each point along these (dispersed) shaly-

sand lines, the points with a new fluid form curves that are approximately straight

lines. In our synthetic example, we consistently use VRH to compute both the V-point

moduli and the effective mineral moduli for the Gassmann shaly-sand equation, and

we approximate that the shaly-sand line remains straight under fluid substitution.

Because of this linear approximation, applying fluid substitution to the entire shaly-

sand line can be done by following a few simple steps. First, fluid substitution (using

the traditional Gassmann’s equation) is applied to the clean sand point. Then this clean

sand point is moved to the new fluid modulus. Finally, the new shaly-sand line (with

the new fluid) is constructed by drawing a straight line connecting the new clean sand

point and the V-point (Figure 3.6). As a result of this linearity in fluid substitution of

the shaly-sand line, the changes in P-compliance (after fluid substitution) for sands

with different amounts of dispersed clay are directly proportional to the change in P-

compliance of the clean sand point.

When the linearity of the shaly-sand line is approximately preserved under fluid

substitution (from fluid 1 to fluid 2), it is equivalent to say that the Reuss average

(used for the shaly-sand line) is approximately consistent with the fluid substitution

equation. In other words, if we use the Reuss average to model the shaly-sand line

with an original fluid and apply fluid substitution to this line, we would obtain the

same results as if we were to model the shaly-line with the new fluid directly. The

same consistency with fluid substitution equations can be seen when using the

Raymer-Hunt-Garder models (Spike and Dvorkin, 2005).


Figure 3.6: Approximate fluid substitution for a dispersed shaly-sand line. Applying fluid substitution to this shaly-sand line is approximately equivalent to moving the clean sand point up or down, following the usual Gassmann’s equation, while the V-point at the other end of this line is fixed. Then, the new clean sand point is connected to the fixed V-point by another straight line. Here, we show an example when the clean sand point is moved down after fluid substitution.

3.4.2 Graphical interpretation and equation derivations for fluid substitution in

interbedded sands

Using results in Figure 3.6, here we introduce a simple graphical interpretation for

fluid substitution in interbedded sands. A sedimentary package of sand (or shaly sand)

interbedded with shale can be represented as a point inside the triangular diagram in

Figure 3.3. The effective porosity of the package is fully-saturated with an original

fluid. Instead of estimating the effect of fluid substitution on a point inside the diagram

one by one, we investigate how the whole triangular diagram is changed when the

original fluid in the effective porosity is replaced by a new fluid.

The triangular diagram in Figure 3.3 is composed of two main elements: the end-

points (i.e., the clean sand, V-point, and pure shale), and the dispersed lines (i.e., the

shaly-sand line connecting the clean sand to the V-point, and the sandy-shale line

connecting the V-point to the pure shale). The mesh is simply filled in by modeling

the interbedding of points along the shaly-sand line and the pure-shale point. Under

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.02

0.04

0.06

0.08

0.1

Effective porosity

1/M

(1/G

Pa)

Pure shaleClean sand

(fluid 1)

Clean sand

(fluid 2)

Increasing sand fractions

Shaly-sand

line

V point: sand completely

filled with clay


fluid substitution, the sandy-shale line, which extends from the pure shale and V-point,

remains unchanged due to their zero effective porosity. Therefore, both the pure shale

and V-point corners are fixed. In Section 3.4.1, we approximate that the shaly-sand

line remains straight and moves accordingly with the clean sand point under fluid

substitution. Since we know how the two main elements of the triangular diagram

change under fluid substitution to a new fluid, we can draw a new triangular diagram

and fill in the mesh. In summary, the change of the whole triangular diagram under

fluid substitution simply depends on how the clean sand point moves. Therefore,

applying approximate fluid substitution to any data point (e.g., either sand or shaly

sand interbedded with shale) inside the triangular diagram can be done by simply

distorting the mesh (Figure 3.7).

Figure 3.7: Graphical interpretation of fluid substitution by our mesh method. Applying approximate fluid substitution to any clean sand (or sand with dispersed clay) interbedded with shale is simply equivalent to distorting the mesh. Here the distortion is shown in the P-compliance (C=1/M) versus effective porosity plane. The distortion moves the mesh accordingly with the change in the clean sand compliance after fluid substitution. The blue arrow shows how a data point inside the triangular diagram moves after the distortion.

To derive the corresponding equations, we let an arbitrary point X in Figure 3.8

represent a sedimentary sequence where shaly sand is interbedded with shale. The

volume fraction of dispersed clay in the sand layers is equal to 𝑉𝑑𝑖𝑠𝑝 . We assume that

the starting effective porosity in shaly sand is fully saturated with fluid 1. In the

derivations, we use a set of variables defined as follows:


𝐶𝑠𝑎𝑡1 is the P-wave compliance of point X with its effective porosity

saturated with fluid 1

𝐶𝑠𝑕 is the P-wave compliance of shale

𝐶𝑑𝑖𝑠𝑝 𝑓𝑙𝑢𝑖𝑑 1 is the P-wave compliance of the shaly-sand layer whose

effective porosity is saturated with fluid 1

∆𝐶𝑠 and ∆𝐶𝑠𝑎𝑡 are the changes in the P-wave compliance for clean sand and

point X when fluid 1 is substituted by fluid 2, respectively.

Figure 3.8: Schematic diagram and terminology for the laminated (interbedded) sand-shale system.

With these definitions, we can write P-wave compliances of point X when

saturated with fluid 1 and 2 as follows:

𝐶𝑠𝑎𝑡1 = 𝑉𝑠𝑎𝑛𝑑 ∗ 𝐶𝑑𝑖𝑠𝑝 𝑓𝑙𝑢𝑖𝑑 1 − 1 − 𝑉𝑠𝑎𝑛𝑑 ∗ 𝐶𝑠𝑕 ,

(3.6)

𝐶𝑠𝑎𝑡2 = 𝑉𝑠𝑎𝑛𝑑 ∗ 𝐶𝑑𝑖𝑠𝑝 𝑓𝑙𝑢𝑖𝑑 2 − 1 − 𝑉𝑠𝑎𝑛𝑑 ∗ 𝐶𝑠𝑕 .

(3.7)

0Effective porosity

1/M

(1/G

Pa)

}

shale clean sand point

Csat1

Csat2

{ Point X

Cdisp

f luid1

}

Cdisp

Csand

Csat

Cdisp

fluid2

sandsand - vdisp


We are interested in the change in the compliances of point X when fluid 1 is

substituted fluid 2. This change can be expressed as

∆𝐶𝑠𝑎𝑡 = 𝐶𝑠𝑎𝑡1 − 𝐶𝑠𝑎𝑡2,

= 𝑉𝑠𝑎𝑛𝑑 ∗ ∆𝐶𝑑𝑖𝑠𝑝 (by substituting Equations 3.6 and 3.7).

(3.8)

From Figure 3.8, we have

∆𝐶𝑠/ ∆𝐶𝑑𝑖𝑠𝑝 = 𝜙𝑠/(𝜙𝑠 − 𝑉𝑑𝑖𝑠𝑝 ) ,

or ∆𝐶𝑑𝑖𝑠𝑝 = (𝜙𝑒𝑓𝑓 𝑠𝑎𝑛𝑑 /𝜙𝑠)∆𝐶𝑠 (by substituting 𝜙𝑒𝑓𝑓 𝑠𝑎𝑛𝑑 = 𝜙𝑠 − 𝑉𝑑𝑖𝑠𝑝 ),

(3.9)

where 𝜙𝑒𝑓𝑓 𝑠𝑎𝑛𝑑 is the effective porosity in the sand layers.

We substitute Equation 3.9 into Equation 3.8 to yield the approximate equation for

the change in the P-compliance after performing fluid substitution at point X:

∆𝐶𝑠𝑎𝑡 = 𝑉𝑠𝑎𝑛𝑑 ∗ (𝜙𝑒𝑓𝑓 𝑠𝑎𝑛𝑑 /𝜙𝑠)∆𝐶𝑠 ,

or ∆𝐶𝑠𝑎𝑡 = (𝜙𝐸𝑓𝑓 /𝜙𝑠)∆𝐶𝑠 ,

(3.10)

where 𝜙𝐸𝑓𝑓 is the effective porosity of the interbedded sand-shale sequence at point X.

Finally, we obtain an expression for compliance of the interbedded sand-shale

sequence at point X with new fluid (𝐶𝑠𝑎𝑡2):

𝐶𝑠𝑎𝑡2 = 𝐶𝑠𝑎𝑡1 − (𝜙𝐸𝑓𝑓 /𝜙𝑠)(𝐶𝑠𝑎𝑛𝑑 𝑓𝑙𝑢𝑖𝑑 1 − 𝐶𝑠𝑎𝑛𝑑 𝑓𝑙𝑢𝑖𝑑 2) .

(3.11)

In summary, the change in P-compliance after fluid substitution for an interbedded

sand-shale sequence is directly proportional to the change in P-compliance of the

clean sand point. The magnitude of this change is approximately equal to the ratio of

the effective porosity of the interbedded sand-shale sequence to the porosity of clean

sand. It is important to note that the starting fluid in the clean sand must be consistent


with the fluid in the effective pore space of the laminated (interbedded) sequence. For

example, let’s assume that a data point represents interbedding of 50% shale and 50%

shaly sand, whose effective pore space is only half saturated with oil. In this case, the

effective water saturation in the shaly sand layer is 0.5. If we want to apply fluid

substitution to this interbedded sequence by replacing oil with brine, we need to

compute the change in P-compliance of the clean sand point by going from (effective)

water saturation of 0.5 to water saturation of 1.

3.4.3 Important note

Our mesh method for fluid substitution in interbedded sand-shale sequences is

guaranteed to be robust for any rock constructed from the sand and shale end-members.

The method approximates changes in P-compliance for interbedded sand-shale

sequences to be proportional to the change in P-compliance of the clean sand point

after fluid substitution. The change of compliance at the clean sand point can be

estimated using the traditional Gassmann’s equation. Since clean (quartz-) sandstone

is a type of rock where Gassmann’s assumptions are most valid, applying fluid

substitution to this rock should result in a reasonable value. Based on our method,

applying fluid substitution to any interbedded sand-shale sequence is equivalent to

distorting the triangular diagram by moving only the clean sand’s corner, whose

change resulting from the traditional Gassmann’s equation is in a reasonable range

(Figure 3.8). Therefore, if a point X representing an interbedded sand-shale sequence

lies inside the triangular diagram, it is guaranteed that the fluid substitution result of

this point X is robust. Interestingly, the assurance for robust results holds regardless

of model choices for the V-point.

3.5 Synthetic examples

In this section we use synthetic examples to compare fluid substitution results

from our mesh method with three other procedures. We create synthetic examples by

first generating interbedded sand-shale sequences according to the specifications in

each case (e.g., clean sand interbedded with shale). Properties such as density, porosity


and velocity are assigned to each individual sedimentary layer. Then we upscale

volumetric and elastic properties using volumetric average and Backus average,

respectively. These upscaled values (i.e., the average properties of multiple layers)

simulate traditional well-logging measurements when the logging tools cannot resolve

individual layers.

We then apply fluid substitution (from oil to brine) to the interbedded sand-shale

sequences. First our mesh method is applied to the upscaled properties (i.e., the

equation is applied at the measurement scale). Then we apply three other fluid-

substitution procedures:

Gassmann’s equation applied to each sand layer, followed by upscaling

Gassmann shaly-sand equation (Dvorkin et al., 2007) applied to each sand

layer, followed by upscaling

Gassmann’s equation applied to upscaled properties (i.e., the equation is

applied at the measurement scale)

Note that the accurate way to perform fluid substitution in sub-resolution interbedded

sand-shale sequences is to first downscale the measurements for properties of sand and

shale end-members, apply Gassmann’s equation to the sand layers, and then upscale

the layers back to the measurement scale by Backus averaging. However, in real

applications of this downscaling-upscaling procedure, a number of parameters need to

be estimated, which can lead to non-robust and unreliable results. From the above list,

the first two procedures which apply fluid substitution in the sand layers only followed

by upscaling are equivalent to performing the downscaling-upscaling procedure, when

we know the actual properties of sand and shale end-members. These fluid substitution

results from the downscaling-upscaling procedure are considered our baselines.

3.5.1 Case 1: Shaly sand (sand with dispersed clay) with fully-oil-saturated

effective porosity (𝑺𝑾𝒆=0)

In case 1, we generate synthetic shaly sands with volume fractions of dispersed

clay in the sands ranging from 𝑉𝑑𝑖𝑠𝑝 = 0 to 𝑉𝑑𝑖𝑠𝑝 = 𝜙𝑠 . The starting effective pore


space in the shaly sands is fully-saturated with oil (𝑆𝑊𝑒=0). In this example, all clay is

dispersed, and thus there is no laminated (interbedded) shale. Oil is replaced by brine

using the four selected procedures. Results are shown in Figure 3.9.

Since the rocks in this example are simply shaly sands with no interbedding of

sand and shale, either applying Gassmann’s equation to the upscaled properties or

applying the equation to the sand layers followed by upscaling will give the same

results (the green dashed line and the black solid line in Figure 3.9). After fluid

substitution, the mesh method yields slightly higher velocities than those of the

Gassmann shaly-sand equation. However, results from both methods are similar.

Figure 3.9: Fluid substitution results from four different procedures for shaly sands with no interbedding. The procedures are Gassmann’s equation applied to sand layers only + upscaling, Gassmann shaly-sand equation (Dvorkin et al., 2007) applied to each sand layers only + upscaling, Gassmann’s equation applied at the measurement scale, and our mesh method. For this synthetic case, volume fractions of dispersed clay in the sand range from 𝑉𝑑𝑖𝑠𝑝 = 0 to 𝑉𝑑𝑖𝑠𝑝 = 𝜙𝑠.

0.1 0.15 0.2 0.25 0.3 0.35 0.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

Vp (

km

/s)

Total porosity

G. + upscaling

G. shaly-sand + upscaling

G. ignoring lamination

Mesh

Oil-saturated model

V-point (Vdisp

= clean sand porosity)

Clean sand (Vdisp

= 0)


3.5.2 Case 2: Clean sand interbedded with shale, varying 𝑽𝒔𝒂𝒏𝒅, and fully-oil-

saturated effective porosity (𝑺𝑾𝒆=0)

Figure 3.10: Fluid substitution results from four different procedures for interbedded clean sand-shale sequences. The procedures are Gassmann’s equation applied to sand layers only + upscaling, Gassmann shaly-sand equation (Dvorkin et al., 2007) applied to each sand layers only + upscaling, Gassmann’s equation applied at the measurement scale, and our mesh method. Note that results of Gassmann’s equation + upscaling, Gassmann shaly-sand equation + upscaling, and the mesh are on top of each other. The x-axis represents pseudo-depth.

Figure 3.11: Sand fractions of the synthetic model for interbedded clean sand-shale sequences. The x-axis represents pseudo-depth.

In case 2, we generate an interbedded (clean) sand-shale sequence. Sand fractions

vary with pseudo-depth (Figure 3.11). The starting effective pore space in the clean

sand is fully-saturated with oil. Oil is replaced by brine using the four selected

procedures. Results are shown in Figure 3.10. When both Gassmann’s equation and

the mesh method are applied at the measurement scale, Gassmann’s equation yields

higher velocities than the baselines, while the mesh yields exactly the same results as

the baselines. Thus, the mesh correctly accounts for the effect of sub-resolution sand-

shale interbedding, but Gassmann’s equation applied to the upscaled values

2.2

2.4

2.6

2.8V

p (

km

/s)

Depth

G. + upscaling

G. shaly-sand + upscaling


Mesh

Oil-saturated model

0

0.2

0.4

0.6

0.8

1

Volu

me f

raction o

f

sand in lam

ination

Depth


overpredicts the changes in velocities after fluid substitution. This overprediction

becomes smaller when sand fractions are either very small or very large (Figure 3.12).

For this example of clean sand interbedded with shale, there is no difference between

the traditional Gassmann and the Gassmann shaly-sand equations, because the sand

layers are clean (i.e., no dispersed clay).

Figure 3.12: Percentage differences between velocities after fluid substitution by Gassmann’s equation and Gassmann’s equation applied to only sand layers followed by upscaling. These differences are plotted against sand fraction and volume fraction of shale (𝑉𝑠𝑕 ). In this clean sand case, sand fraction is simply equivalent to 1 – 𝑉𝑠𝑕 .

3.5.3 Case 3: Shaly sand (𝑽𝒅𝒊𝒔𝒑= 0.15) interbedded with shale, varying 𝑽𝒔𝒂𝒏𝒅, and

fully-oil-saturated effective porosity (𝑺𝑾𝒆=0)

Figure 3.13: Fluid substitution results from four different procedures for interbedded shaly sand-shale sequences, with a fixed volume fraction of dispersed clay. The procedures are Gassmann’s equation applied to sand layers only + upscaling, Gassmann shaly-sand equation (Dvorkin et al., 2007) applied to each sand layers only + upscaling, Gassmann’s equation applied at the measurement scale, and our mesh method. The x-axis represents pseudo-depth.

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

9

10

11

Vshale, Sand fraction

%D

iff

:= 1

00*(

Vp G

. no lam

ination -

Vp G

.&up)/

(Vp G

.&up)

Vshale

Sand fraction

2.2

2.4

2.6

2.8

3

3.2

Vp (

km

/s)

Depth

G. + upscaling

G. shaly-sand+ upscaling


Mesh

Oil-saturated model


Figure 3.14: Sand fractions of the synthetic model for interbedded shaly sand-shale sequences, with a fixed volume fraction of dispersed clay. The x-axis represents pseudo- depth.

In case 3, we generate an interbedded (shaly) sand-shale sequence. The volume

fraction of dispersed clay in shaly sand is kept constant at 𝑉𝑑𝑖𝑠𝑝 =0.15. Sand fractions

vary with pseudo-depth (Figure 3.14). The starting effective pore space in the shaly

sand layers is fully-saturated with oil. Oil is replaced by brine using the four selected

procedures. Results are shown in Figure 3.13. Since our mesh method yields velocity

results very similar to the baselines, it accounts for both the interbedding part and the

shaly-sand (sand with dispersed clay) part during fluid substitution. Note that even

when effective water saturation changes to other values, we observe that results of the

mesh method agree well with baselines.

3.5.4 Case 4: Shaly sand (normal distribution of 𝑽𝒅𝒊𝒔𝒑 with a mean of 0.1 and a

standard deviation of 0.05) interbedded with shale, varying 𝑽𝒔𝒂𝒏𝒅, and fully-oil-

saturated effective porosity (𝑺𝑾𝒆=0)

Figure 3.15: Fluid substitution results from four different procedures for interbedded shaly sand-shale sequences, with varying volume fractions of dispersed clay. The procedures are Gassmann’s equation applied to sand layers only + upscaling, Gassmann shaly-sand equation (Dvorkin et al., 2007) applied to each sand layers only + upscaling, Gassmann’s equation applied at the measurement scale, and our mesh method. The x-axis represents pseudo-depth.

0

0.2

0.4

0.6

0.8

1

Volu

me f

raction o

f

sand in lam

ination

depth

2.2

2.4

2.6

2.8

3

3.2

Vp (

km

/s)

Depth

G. + upscaling

G.shaly-sand + upscaling


Mesh

Oil-saturated model


Figure 3.16: Sand fractions of the synthetic model for interbedded shaly sand-shale sequences, with varying volume fractions of dispersed clay. The x-axis represents pseudo-depth.

In case 4, we generate interbedding of sedimentary layers consisting of shaly sand

and shale. The volume fractions of dispersed clay assigned to the shaly-sand layers are

drawn from a normal distribution with a mean of 0.1 and a standard deviation of 0.05.

Sand fractions vary with pseudo-depth (Figure 3.16). The starting effective pore space

in the shaly sand layers is fully-saturated with oil. Oil is replaced by brine using the

four selected procedures. Results are shown in Figure 3.15.

Note that this case is more complicated than the previous cases because each

shaly-sand layer has different volume fractions of dispersed clay. If we follow the

normal routine of downscaling the measurements to the end-members’ properties

before applying fluid substitution, it is impossible in this case to extract all the end-

members from their average values alone. However, this is not a problem for the mesh

because we can apply our method directly to the measurements even when they

represent averages of multiple layers.

Although there are variations in properties of the interbedded sand layers (i.e.,

volume fractions of dispersed clay), the mesh still yields fluid substitution results

which agree well with those of the Gassmann shaly-sand equation (or Gassmann’s

equation) applied to each sand layers only followed by upscaling (Figure 3.15).

0

0.2

0.4

0.6

0.8

1

Volu

me f

raction o

f

sand in lam

ination

depth


3.5.5 Case 5: Interbedded sand-shale sequences with systematic changes of both

the 𝑽𝒅𝒊𝒔𝒑 in the sand layers and the 𝑽𝒔𝒂𝒏𝒅

In case 5, we generate interbedding of sedimentary layers between sand and shale.

We systematically vary volume fractions of dispersed clay in the sand layers (i.e.,

from 𝑉𝑑𝑖𝑠𝑝 = 0 to 𝑉𝑑𝑖𝑠𝑝 = 𝜙𝑆) for a constant sand fraction. Then we repeat the same

process for other sand fractions. As a result, we roughly cover all possible

interbedding scenarios allowed by the triangular diagram (Figure 3.3). The starting

effective pore space in the sand or shaly sand layers is fully-saturated with oil (𝑆𝑊𝑒=0).

Oil is then replaced by brine using the four selected procedures. Results are shown in

Figure 3.17.

Figure 3.17: Changes in P-wave velocity after fluid substitution from four different procedures: Gassmann’s equation applied to sand layers only + upscaling, Gassmann shaly-sand equation (Dvorkin et al., 2007) applied to sand layers only + upscaling, Gassmann’s equation applied at the measurement scale, and our mesh method. Each location on the triangular diagram represents a interbedded sand-shale sequence with a unique pair of sand fraction and volume fraction of dispersed clay values.

0 0.1 0.2 0.3

0.04

0.06

0.08

Effective porosity

1/M

(1/G

Pa)

Gassmann ignoring lamination

0

50

100

150

200

250

0 0.1 0.2 0.3

0.04

0.06

0.08

Effective porosity

1/M

(1/G

Pa)

Gassmann + upscaling

0

50

100

150

200

250

0 0.1 0.2 0.3

0.04

0.06

0.08

Effective porosity

1/M

(1/G

Pa)

Gassmann shaly-sand equation

+ upscaling

0

50

100

150

200

250

0 0.1 0.2 0.3

0.04

0.06

0.08

Effective porosity

1/M

(1/G

Pa)

Mesh

0

50

100

150

200

250

change in

velocity (m/s)

change in

velocity (m/s)

change in

velocity (m/s)change in

velocity (m/s)

Velocity changes after fluid substitution (from oil to brine) using


By applying Gassmann’s equation at the measurement scale and thus ignoring the

effect of thin sand-shale interbedding, the changes in velocity after fluid substitution

(from oil to brine) are much larger than the changes from the other three procedures.

These large velocity changes are reflected in a distinct color pattern in the triangular

diagram (Figure 3.17, upper left corner). Our mesh method yields fluid substitution

results similar to those obtained by applying fluid substitution to the shaly-sand layers

only, followed by upscaling.

3.6 Pitfalls in interpretation

We show two examples of pitfalls in interpretation of fluid substitution results

when sub-resolution interbedding exists. First, we generate a synthetic dataset that

represents interbedding of shaly-sand and shale, with sand fractions ranging between

𝑉𝑠𝑎𝑛𝑑 = 0.5 and 𝑉𝑠𝑎𝑛𝑑 = 1, and volume fractions of dispersed clay ranging between

𝑉𝑑𝑖𝑠𝑝 =0 and 𝑉𝑑𝑖𝑠𝑝 = 𝜙𝑆. Effective porosity is fully-saturated with gas. We perform

fluid substitution (gas to brine) using the mesh and Gassmann’s equation (ignoring

interbedding).

In the first example, suppose that thin interbedding is not accounted for. Then

changes in velocities after fluid substitution are overpredicted. Because of this

overprediction, data with small sand fractions can look as if they have larger sand

fractions. Figure 3.18 (Right) shows fluid substitution results when the effect of thin

sand-shale interbedding is ignored. In this panel, all data points are located between

the lines of 𝑉𝑠𝑎𝑛𝑑 = 0.75 and 1. However, the actual sand fractions range from 𝑉𝑠𝑎𝑛𝑑 =

0.5 to 𝑉𝑠𝑎𝑛𝑑 = 1. Thus, due to inappropriate fluid substitution the overpredicted

velocities move the data points to higher positions, where they can be misinterpreted

as having sand fractions higher than the actual values. In contrast, our mesh approach

preserves the sand-fraction properties in the process of fluid substitution (Figure 3.18,

middle).

In the second example, suppose that an interbedded sand-shale sequence with a

small sand fraction is saturated with gas. The change in velocities after fluid


substitution from gas to brine is generally expected to be high. However, because the

sequence has low sand fraction, its velocity change after fluid substitution is small,

which can be misinterpreted as if the fluid changes from oil to brine, as illustrated in

Figure 3.18 (middle).

Figure 3.18: Pitfalls in interpretation of fluid substitution results when thin interbedding exists. (Left) Synthetic gas-saturated data with sand fractions greater than or equal to 0.5. (Middle) Results after fluid substitution from gas to brine using our mesh approach. (Right) Results after fluid substitution from gas to brine using Gassmann’s equation. Data points are color-coded by water saturation values before fluid substitution.

3.7 Sensitivity analysis

In the synthetic examples, when performing fluid substitution using the

downscaling-upscaling procedures, we use all correct parameters. However, in

practice the downscaling process involves an inversion or estimations for volume

fractions and properties of the end-members (e.g., shale) in the interbedding, possibly

leading to non-robust results. In contrast, our mesh method guarantees robustness. The

method also does not depend on the shale elastic property. It only requires the

estimations of effective porosity, effective saturation, fluid properties, and the clean

sand porosity and velocity.

In this section, we show how uncertainties in input parameters affect fluid

substitution results of our mesh method (i.e., using Equation 3.11). The five input

parameters considered are clean sand’s P-wave velocity (𝑉𝑝𝑠𝑎𝑛𝑑 ), clean sand porosity

0.1 0.2 0.3 0.41.5

2

2.5

3

3.5

4

4.5

Shale Sand

Total porosity

Vp (

km

/s)

Sand

original water saturation

before fluid substitution

0.2

0.4

0.6

0.8

0.1 0.2 0.3 0.41.5

2

2.5

3

3.5

4

4.5

Shale Sand

Total porosity

Vp (

km

/s)

Sand

0.2

0.4

0.6

0.8

0.1 0.2 0.3 0.41.5

2

2.5

3

3.5

4

4.5

Shale Sand

Total porosity

Vp (

km

/s)

Sand

original water saturation

before fluid substitution

0.2

0.4

0.6

0.8

GAS BRINE - MESH BRINE –GASSMANN

Sand fraction = 0.5 Sand fraction = 0.75

Color-coded by original Sw before fluid substitution

Vsand = 0.5 Vsand = 0.75


(𝜙𝑠), effective porosity of the laminate package (𝜙𝐸𝑓𝑓 ), effective water saturation in

the sand layers (𝑆𝑊𝑒 ), and fluid properties (i.e., bulk modulus, 𝐾𝑓𝑙 ) .

To run this analysis, we generate synthetic stacks of interbedded sand-shale

sequences, where the sand can be either clean or dirty (i.e., dispersed clay exists in the

sand pore space). We explore cases where the sand fractions range from 𝑉𝑠𝑎𝑛𝑑 = 0 to

𝑉𝑠𝑎𝑛𝑑 = 1, and the volume fractions of dispersed clay range from 𝑉𝑑𝑖𝑠𝑝 = 0 to 𝑉𝑑𝑖𝑠𝑝 =

𝜙𝑆. The starting effective pore space is half filled with oil. Then, we upscale properties

of multiple layers to mimic well logging measurements. All the four procedures

described in Section 3.5 are applied to the synthetic measurements to substitute oil in

the effective porosity with brine, using all correct parameters. The results from these

procedures are used as references.

In real data application, we would not know the true parameters; instead, we would

only have distributions of possible values. We analyze the effect of these uncertainties

in the input parameters on fluid-substitution results, one parameter at a time. For each

input parameter, we randomly draw a value from its assigned distribution while

keeping all the other parameters at their true values, use these parameters in our

method, and keep the result. The process is repeated multiple times. Finally, we have a

distribution of outcomes resulting from uncertainties in that one particular parameter.

We repeat the same process for all parameters on the list.

We assume that all the input parameters previously listed are independent and

normally distributed with means (𝜇) and variances 𝜎2 . The means of the distributions

are set to be the true values used in the forward-modeling process. The standard

deviations 𝜎 are set to be 10% of the means. We test the effect of uncertainties in the

input parameters one by one. Note that a similar Monte Carlo simulation was used in

Artola and Alvarado (2006) for sensitivity analyses of Gassmann’s equations.

Results of the sensitivity analysis for the mesh method are shown in Figure 3.19 as

distributions of velocities after fluid substitution. Each subplot is indicated by a

volume fraction of laminated (interbedded) sand and a volume fraction of dispersed

clay in the sand layers. The thick lines on the top of each subplot are the oil-saturated


velocity and the reference velocities after fluid substitution from oil to brine using the

four procedures with all correct parameters.

For specific values of sand fraction and volume fraction of dispersed clay, all

distributions of velocities after fluid substitution using our method are generally

centered around the thick magenta-dashed line, which is the reference result using our

method with all correct input parameters. This centering is expected since we set the

correct values as the means of distributions for the input parameters that we want to

test.

For specific values of sand fraction and volume fraction of dispersed clay,

uncertainties in the clean sand velocity result in the largest spread of velocities after

fluid substitution using our method. Thus, our method is most sensitive to

uncertainties in the clean sand velocity. The subsequent parameters, in decreasing

order of sensitivity, are the clean sand porosity, effective porosity, effective water

saturation and fluid elastic modulus.

For the results of each tested parameter, when a volume fraction of laminated

(interbedded) sand increases with a fixed volume fraction of dispersed clay, the spread

of velocities after fluid substitution using our method becomes larger.

For the results of each tested parameter, when a volume fraction of dispersed clay

in the sand layers increases with a fixed volume fraction of laminated (interbedded)

sand, the spread of velocities after fluid substitution using our method becomes

smaller.


Figure 3.19: Sensitivity analysis results of the mesh method for five input parameters: P-wave velocity and total porosity of clean sand, effective porosity and effective water saturation of the interbedded package, and elastic modulus of fluid. The thick lines on the top of each subplot are the oil-saturated velocity and the reference velocities after fluid substitution from oil to brine using four procedures with all correct input parameters: Gassmann ignoring interbedding, Gassmann applied to each sand layer + upscaling, Gassmann shaly-sand equation (Dvorkin et al., 2007) applied to each sand layer + upscaling, and our mesh method. Results of sensitivity analysis for each parameter are shown as a velocity distribution, which is normalized to one. Sand fractions and volume fractions of dispersed clay for each synthetic model are shown on above each subplot.

3.8 Real data example

In this section, we show fluid substitution results from real log data. We compare

four procedures: Gassmann’s equation (i.e., ignoring the effect of thin sand-shale

interbedding), Gassmann shaly-sand equation (i.e., assuming that shale is all dispersed;

Dvorkin et al., 2007), the downscaling-upscaling procedure, and our mesh method.

The downscaling-upscaling procedure here refers to the most appropriate

procedure for fluid substitution in thinly interbedded sand-shale sequences, given that

all parameters are correct. The procedure includes steps as follows:

2.2 2.4 2.6 2.8 3 3.2 3.40

1

Sand fraction = 0.33 Vdis = 0

2.2 2.4 2.6 2.8 3 3.2 3.40

1

Sand fraction = 0.33 Vdis = 0.12

2.2 2.4 2.6 2.8 3 3.2 3.40

1


2.2 2.4 2.6 2.8 3 3.2 3.40

1

Sand fraction = 0.67 Vdis = 0

Norm

aliz

ed d

ensity f

unction

2.2 2.4 2.6 2.8 3 3.2 3.40

1


2.2 2.4 2.6 2.8 3 3.2 3.40

1


2.2 2.4 2.6 2.8 3 3.2 3.40

1

Sand fraction = 1 Vdis = 0

P-velocity (km/s)2.2 2.4 2.6 2.8 3 3.2 3.4

0

1

Sand fraction = 1 Vdis = 0.12

P-velocity (km/s)

2.2 2.4 2.6 2.8 3 3.2 3.40

1

Sand fraction = 1 Vdis = 0.24

P-velocity (km/s)

V

p sand

Sand porosity

Effective porosity

Sw e

Koil


G.+upscaling

G. shaly-sand +upscaling

Mesh

Oil-saturated model


1. Determine parameters including P-wave velocity, density, and total porosity of

the shale end-member, volume fraction of laminated (interbedded) shale, and

volume fraction of dispersed clay present in the sand layers. These end-

member parameters can be estimated by cross-plotting measurements with

superimposed theoretical model curves. For example, the Thomas-Stieber

triangular diagram (Chapter 2) in the total-porosity and shale-volume-fraction

plane helps break down volume fraction of laminated (interbedded) shale from

dispersed clay. Note that the Thomas-Stieber model also requires some

additional input parameters (e.g., clean sand porosity).

2. Downscale for the elastic properties of the sandy end-member by inverting the

Backus average in Equation 3.4 for the inverse P-wave modulus of the sandy

layers.

From 𝑀𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑−1 = (1 − 𝑉𝑙𝑎𝑚 )𝑀𝑠𝑎𝑛𝑑 (𝑜𝑟 𝑠𝑕𝑎𝑙𝑦 𝑠𝑎𝑛𝑑 )

−1 + 𝑉𝑙𝑎𝑚𝑀𝑠𝑕−1 , we can write

𝑀𝑠𝑎𝑛𝑑 (𝑜𝑟 𝑠𝑕𝑎𝑙𝑦 𝑠𝑎𝑛𝑑 )−1 = (𝑀𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

−1 − 𝑉𝑙𝑎𝑚𝑀𝑠𝑕−1)/(1 − 𝑉𝑙𝑎𝑚 ),

(3.12)

where 𝑀𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑−1 is the inverse P-wave modulus of the measurements. Using

equation 3.12 requires estimates of 𝑀𝑠𝑕 .

3. Apply fluid substitution to the sandy end-member using Gassmann shaly-sand

equation (Dvorkin et al., 2007).

4. Upscale the sandy and shale layers back using Equation 3.4. At the end of this

step, only the sandy layers are saturated with the new fluid.

Even though all steps in the downscaling-upscaling procedure are straightforward,

fluid substitution results from this procedure can be unstable and non-robust in

practice due to uncertainties in real measurements and errors in parameter estimation.

When these uncertainties and errors are passed on through the inversion and the fluid

substitution steps, unreliable results may be observed (e.g., negative values, velocity

spikes).


In contrast, even though the mesh method also requires parameter estimation steps,

it skips the actual Backus downscaling-upscaling steps. Since the mesh method is

based on the Thomas-Stieber-Yin-Marion model which describes sand-shale mixtures,

the mesh implicitly performs downscaling and upscaling without any actual

computation. For the same reason, we do not have to actually compute volume

fraction of laminated (interbedded) shale and dispersed clay as this shale distribution

information is embedded into the mesh method. As a result, the mesh is more robust

than the downscaling-upscaling process in practice. One could say that the mesh

method is an upscaled version of Gassmann for their type of heterogeneous system.

The well log data comes from a sand-shale system. A section representing wet

sand interbedded with shale is selected and shown in Figure 3.20. The Thomas-

Stieber-Yin-Marion model is superimposed onto the cross-plot to estimate the volume

fractions of laminated (interbedded) shale and dispersed clay, which will be used later

in the downscaling-upscaling procedure.

Starting with all brine-saturated data, we substitute oil for brine using four

procedures: Gassmann’s equation (i.e., ignoring the effect of sub-resolution sand-shale

interbedding), Gassmann shaly-sand equation (i.e., assuming that shale is all

dispersed), the downscaling-upscaling procedure, and our mesh method. Note that

here both bulk and shear moduli are used in all calculations. Since neither the

Gassmann’s equation nor the Gassmann shaly-sand equation correctly account for

sub-resolution interbedding, we expect that using these procedures the changes in P-

wave velocities after fluid substitution are overpredicted with respect to the true

velocity change. For example, if fluid substitution from brine to oil decreases P-wave

velocities, an overpredicted result will give a lower velocity value. Since the

downscaling-upscaling procedure is the appropriate way to perform fluid substitution

in interbedded sand-shale sequences, we use results from this procedure as baselines.

However, these results show some velocity spikes, especially at low sand-fraction

points. Thus, to avoid any erratic behavior, the baselines from the downscaling-

upscaling procedure are chosen to include only those points with gamma ray values

less than 85.


Figure 3.20: Selected dataset represents an interbedded sand-shale sequence. The data

is color-coded by Gamma ray values. The Thomas-Stieber-Yin-Marion model is superimposed onto the data. The set of parallel lines labeled as 𝑉𝑠𝑎𝑛𝑑 represent volume fractions of laminated (interbedded) sand according to the Thomas-Stieber-Yin-Marion model.


fractions estimated from the Thomas-Stieber-Yin-Marion model. Four fluid substitution procedures are used: Gassmann’s equation (i.e., ignoring the effect of sand-shale interbedding), Gassmann shaly-sand equation (i.e., assuming that shale is all dispersed; Dvorkin et al., 2007), the downscaling-upscaling procedure, and the mesh method. Note that points with very low sand fractions are excluded from the plot. Lines X = Y are super-imposed for comparison purpose.

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Shale

Sand

Vshale

Tota

l poro

sity

Gamma ray

20

30

40

50

60

70

80

90

100

Vsand

= 1

Vsand

= 0.75

Vsand

= 0.5

Vsand

= 0.25

Vsand

= 0

1.8 2 2.2 2.4 2.61.8

2

2.2

2.4

2.6

2.8

Vp downscaling-upscaling

(km/s)

Vp m

esh (

km

/s)

1.8 2 2.2 2.4 2.61.8

2

2.2

2.4

2.6

2.8


(km/s)

Vp G

assm

ann s

haly

-sand

(k

m/s

)

1.8 2 2.2 2.4 2.61.8

2

2.2

2.4

2.6

2.8


(km/s)

Vp G

assm

ann (

km

/s)

1.8 2 2.2 2.4 2.61.8

2

2.2

2.4

2.6

2.8


(km/s)

Vp d

ow

nscalin

g-u

pscalin

g

(k

m/s

)

sand fraction

0.2

0.4

0.6

0.8

1

Line X = Y

X > Y region


P-wave velocities after fluid substitution from the downscaling-upscaling

procedure (i.e., baselines) are compared with results from other procedures (Figure

3.21). The data points in Figure 3.21 are color-coded by sand fractions (𝑉𝑠𝑎𝑛𝑑 )

estimated from the Thomas-Stieber-Yin-Marion model in Figure 3.20. Black lines (X

= Y) are super-imposed onto the plots for comparison purpose. For clean sand or very

high sand-fractions data, we see similar P-wave velocities after fluid substitution

regardless of procedures used, because these data represent no or almost no sub-

resolution interbedding. For data with intermediate sand fractions, P-wave velocities

after fluid substitution using both the Gassmann’s equation and the Gassmann shaly-

sand equation are lower than velocity results from the downscaling-upscaling

procedure, because Gassmann’s equation do not account for sub-resolution

interbedding. These lower velocity values (i.e., larger changes in velocities during

fluid substitution) are clearly seen in the top row of Figure 3.21 as data points deviate

from the lines X = Y and stay in the X > Y region.

P-wave velocities after fluid substitution using the mesh method are much closer

to those of the downscaling-upscaling procedure; however, their values are not

perfectly the same (Figure 3.21, lower left corner). One possible explanation for their

small mismatches is inconsistency of sand fractions used in the downscaling-upscaling

process. We use the Thomas-Stieber-Yin-Marion model, specifically in the volumetric

(i.e., total porosity VS volume fraction of shale) space, to estimate sand fractions.

Then, we input these values into Equation 3.12 to invert for the P-compliance of sandy

layers. This inversion step operates in the elastic (i.e., velocity) space. The sand

fractions estimated from the volumetric space may not be consistent with those from

the elastic space. Since the mesh method internally operates in the elastic space, the

downscaling-upscaling procedure, that uses (potentially) different sand fractions

estimated from the volumetric space, can cause a mismatch between fluid-substitution

results of the two methods. To solve this problem, we should select a set of end-points

(e.g., clean-sand porosity, velocity) such that sand-fraction values are as consistent as

possible across all spaces. An example of fluid-substitution results when sand-

fractions values are estimated in the elastic space are shown in Figure 3.22. Note that


the color scale is the same in both Figure 3.21 and Figure 3.22. Using sand fractions

estimated from the elastic space, we achieve a slightly better match of P-wave

velocities after fluid substitution using the mesh method and the downscaling-

upscaling procedure.


fractions estimated from the P-compliance versus effective porosity plane. Four fluid substitution procedures are used: Gassmann’s equation (i.e., ignoring the effect of sand-shale interbedding), Gassmann shaly-sand equation (i.e., assuming that shale is all dispersed; Dvorkin et al., 2007), the downscaling-upscaling procedure, and the mesh method. Note that points with very low sand fractions are excluded from the plot. Lines X = Y are super-imposed for comparison purpose.

As noted earlier in Section 3.8, one drawback of the downscaling-upscaling

procedure is non-robustness during the Backus-inversion step, potentially resulting in

velocity spikes and/or negative velocity. Using the downscaling-upscaling procedure,

we observe P-wave velocity spikes after applying fluid substitution to the well log data

(Figure 3.23). In contrast, velocity spikes are not observed in the fluid substitution

results from the mesh method.

1.8 2 2.2 2.4 2.61.8

2

2.2

2.4

2.6

2.8


(km/s)

Vp m

esh (

km

/s)

1.8 2 2.2 2.4 2.61.8

2

2.2

2.4

2.6

2.8


(km/s) V

p G

assm

ann s

haly

-sand

(k

m/s

)

1.8 2 2.2 2.4 2.61.8

2

2.2

2.4

2.6

2.8


(km/s)

Vp G

assm

ann (

km

/s)

1.8 2 2.2 2.4 2.61.8

2

2.2

2.4

2.6

2.8


(km/s)

Vp d

ow

nscalin

g-u

pscalin

g

(km

/s)

Sand fraction

0.2

0.4

0.6

0.8

1

Line X = Y

X > Y region


Figure 3.23: Instability of the inversion step in the downscaling-upscaling procedure leading to spikes in P-wave velocity after fluid substitution.

3.9 Discussion

3.9.1 The mesh method using both bulk and shear moduli

Most of results shown in this chapter are computed using P-wave moduli only.

However, the method is also applicable if both bulk and shear moduli are used. This

applicability is supported by results and discussion from the consistency test in

Section 3.3.2 (Figure 3.5). For example, we rerun the synthetic shaly-sand case

(Section 3.5.1) using both bulk and shear moduli in all fluid-substitution procedures.

The mesh results are still in good agreement with other procedures (Figure 3.24).

1 1.5 2 2.5 3

P-wave velocity after fluid substitution (km/s)

Depth

Downscaling-upscaling

process

the mesh


Figure 3.24: Fluid substitution results by four different procedures for shaly sands, with no interbedding. The procedures are Gassmann using total porosity and ignoring interbedding, Gassmann using total porosity (applied to sand layers only) + upscaling, Gassmann using effective porosity by Dvorkin et al., 2007 (applied to each sand layers only) + upscaling and our method (i.e., the mesh). For this synthetic case, volume fractions of dispersed clay in the sand range from 𝑉𝑑𝑖𝑠𝑝 =0 to 𝑉𝑑𝑖𝑠𝑝 = 𝜙𝑠. Both bulk and shear moduli are used for this synthetic example.

3.9.2 Possible modification of the mesh method when key assumptions are

relaxed

Our mesh method is based on two key assumptions. First, the shaly-sand line is

assumed to remain straight under fluid substitution. This assumption is only an

approximation of the actual fluid substitution results, which show that the shaly-sand

(straight) line undergoing fluid substitution turns into a non-linear curve (Section

3.4.1). If a non-linear functional form is chosen, we need to also adjust equation

derivations for fluid substitution in interbedded sand-shale sequences. Since all the

derivations are done in the compliance-effective porosity space, interbedding of any

two end-members in this space are simply equivalent to weighted averages of their

compliance and effective porosity, regardless of whether the shaly-sand line is straight

or not. This computational advantage in the compliance-effective porosity space

makes equation derivations easy even for non-linear shaly-sand curves.

The second key assumption of our mesh method is that fluid substitution is

assumed to occur only in the sand, but not in the shale (i.e., fluid substitution occurs in

0.1 0.15 0.2 0.25 0.3 0.35 0.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

Vp (

km

/s)

Total porosity

G. + upscaling

G. (eff) + upscaling


Mesh

Oil-saturated model

V-point (Vdisp = clean sand porosity)

Clean sand (Vdisp = 0)


effective porosity). As a result, performing fluid substitution using our method is

equivalent to distorting the mesh according to how the clean sand end-point moves,

while the shale end-point is fixed (i.e., no change in elastic property). However, it is

possible to modify our method to account for fluid substitution in shale by simply

moving the shale end-point according to the change in its elastic property under fluid

substitution, which may be computed using Gassmann’s equation (Lucier et al., 2011)

or using effective medium models (Ruiz and Azizov, 2011). If fluid substitution is

allowed in the shale layers, it is necessary to model fluid substitution also in the

dispersed clay (i.e., the shaly-sand line) for consistency. Once we are able to

approximate the elastic change of the shaly-sand line, we can modify the mesh model

accordingly by following a procedure similar to that shown in Section 3.4.2. As

previously mentioned, since all the derivations are done in the compliance-effective

porosity space, the modification of our mesh method to account for fluid substitution

in shale should not be difficult.

3.9.3 Using rock-physics trends to constrain clean-sand properties

The key parameters needed for the mesh method are the velocity and porosity of

the clean sand point. Since the results from our sensitivity analysis showed that the

mesh method is most sensitive to the clean-sand velocity (relative to the other

parameters tested), clean-sand properties must be chosen carefully. We can estimate

such properties from data with small gamma ray values as they represent cleaner sand

sections. Additionally, we can use the clean-sand diagenetic trend to constrain the

estimation of the clean-sand properties. The diagenetic trend describes the relationship

between the clean-sand velocity and porosity while clean sandstones undergoing

diagenetic processes (Figure 3.25). These processes (e.g., compaction, cementation)

cause porosity reduction, creating steep velocity-porosity curves often seen in

sandstone data which covers a wide depth range. These trends can be modeled using

the modified Hashin-Shtrikman upper bound (or the modified Voigt bound) which

connects the newly deposited sediments at their critical porosities with the

corresponding mineral points (Nur et al., 1998; Avseth et al., 2005). The clean sand

trend line also establishes a guideline for selecting clean-sand properties along depth.


Figure 3.25: Modified upper bounds for modeling the clean-sand diagenetic trend. This example represents a quartz-water system.

3.9.4 Limitation of the mesh method

Since our mesh method is the elastic analog of the Thomas-Stieber-Yin-Marion

model, our method also carries over one important assumption from that model, which

states that all the sand-shale mixtures (e.g., sand interbedded with shale, or sand with

dispersed clay) are generated by mixing two rock types: the clean sand and pure shale.

When shale is dispersedly added into the pore space of sand, the sand becomes shaly

and its total porosity decreases. The Thomas-Stieber-Yin-Marion model assumes that

the only cause of any porosity reduction is the filling of shale in the pore space. The

model does not account for porosity reduction caused by cementation or changes in

grain sorting (Thomas and Stieber, 1975; Ball et al., 2004; Mavko et al., 2009). This

assumption is therefore passed to our mesh method. As a result, the mesh method does

not properly account for the effects of cementation or grain sorting.

3.9.5 Comparison with alternative methods

The problem of fluid substitution for thin interbedded sand-shale sequences is not

new. In practice, this sub-resolution interbedding problem in fluid substitution is either

completely ignored, or is handled by one of the other fluid substitution methods. When

sub-resolution interbedding is not accounted for, fluid substitution results are often

mispredicted (e.g., overestimation when going from oil- to brine-saturated

measurements) as shown in our synthetic examples (Section 3.5) and literature (e.g.,

Skelt, 2004a). Each fluid substitution method has specific assumptions, and it requires

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

total porosityV

p (

km

/s)

critical porosity

clean

sand

trend

Modified

Voigt bound

Reuss bound

Modified upper

Hashin-Shtrikman

bound

Voigt bound


a set of parameters and possibly other evidence of sub-resolution interbedding. We

want to point out that this chapter provides the mesh method as another alternative to

deal with this fluid substitution problem in interbedded sand-shale sequence. The

mesh method is a new invention that integrates existing models to handle both

interbedded and dispersed sand-shale systems at the same time. The method is based

on assumptions and requires a set of estimated parameters. However, it reduces some

additional procedures needed to be done in other methods. We summarize different

methods used in fluid substitution and comment on the performance of each method

when it is applied to thinly interbedded sand-shale sequences (Table 3.3).

Table 3.3: Summary of methods usually used in fluid substitution and comments for when these methods are applied to sub-resolution interbedded sand-shale sequences.

Method Inputs/ parameters2 Comments

Gassmann’s equation

(using total porosity) 𝜙𝑇 ,𝑉𝑝 ,𝑉𝑠 ,𝜌, 𝑆𝑊 ,

𝐾𝑚𝑖𝑛 ,𝐾𝑓𝑙 , 𝜌𝑓𝑙

Ignoring interbedding and shale

effect

(inappropriate procedure)

Gassmann shaly-sand

equation by Dovrkin et

al., 2007 (using

effective porosity and

newly- defined

composite mineral)

𝜙𝐸𝑓𝑓 ,𝜙𝑇 ,𝑉𝑝 ,𝑉𝑠 ,

𝜌, 𝑆𝑊𝑒 ,𝑉𝑠𝑕

𝐾𝑠𝑕𝑎𝑙𝑒 ,𝜙𝑠𝑕 ,𝐾𝑚𝑖𝑛 , 𝐾𝑓𝑙 , 𝜌𝑓𝑙

Ignoring interbedding

Assuming all shale is dispersed

Downscaling-

upscaling procedure 𝜙𝐸𝑓𝑓 ,𝑉𝑝 ,𝑉𝑠 , 𝜌, 𝑆𝑊𝑒 ,

𝑉𝑙𝑎𝑚 ,𝑉𝑑𝑖𝑠𝑝 ,𝑀𝑠𝑕𝑎𝑙𝑒 ,

𝐾𝑓𝑙 , 𝜌𝑓𝑙

Correct if inputs are correct

Non-robust procedure

Mesh method 𝜙𝐸𝑓𝑓 ,𝑉𝑝 ,𝑉𝑠 , 𝜌, 𝑆𝑊𝑒 ,

𝑀𝑠𝑎𝑛𝑑 ,𝜙𝑠 ,𝐾𝑓𝑙 , 𝜌𝑓𝑙

Correct if inputs are correct

2 𝜙𝑇 ,𝜙𝐸𝑓𝑓 ,𝑉𝑝 ,𝑉𝑠 ,𝜌, 𝑆𝑊 , 𝑆𝑊𝑒 ,𝑉𝑠𝑕 : Total porosity, effective porosity, P-wave velocity, S-wave velocity,

density, water saturation, effective water saturation, and shale volume, respectively. These inputs are

measured data, or they can be derived from other measurements.

𝐾𝑚𝑖𝑛 ,𝐾𝑓𝑙 ,𝜌𝑓𝑙 : Mineral bulk modulus, and bulk moduli and densities of fluids, and respectively.

𝑀𝑠𝑕𝑎𝑙𝑒 ,𝐾𝑠𝑕𝑎𝑙𝑒 ,𝜙𝑠𝑕 : P-wave modulus, bulk modulus, and total porosity of the shale end point,

respectively

𝑀𝑠𝑎𝑛𝑑 ,𝜙𝑠: P-wave modulus and total porosity of the clean sand end point, respectively.

𝑉𝑙𝑎𝑚 ,𝑉𝑑𝑖𝑠𝑝 : Volume of laminated (interbedded) shale and volume of dispersed clay, respectively.


3.9.6 Upscaled Gassmann’s equations

From both synthetic and real data examples in Sections 3.5 and 3.8, when applying

Gassmann’s equation (Table 3.3) directly to sub-resolution interbedded sand-shale

sequences at the measurement scale, the effect of interbedding is not correctly

accounted for, resulting in erroneous predictions. In this section, we show that it is

possible to apply Gassmann’s equation at the measurement scale and account for the

effect of sand-shale interbedding at the same time by simply adjusting several input

parameters in Gassmann’s equation. We refer to this adjustment as an upscaled

Gassmann’s equation for interbedded sand-shale sequences.

We algebraically determine the required input adjustment for the case where clean

sand is interbedded with shale. Note that all the derivations will be done using

approximate Gassmann’s equation, which can be expressed in terms of P-wave moduli

(Mavko et al., 1995):

𝑀𝑠𝑎𝑡2

𝑀𝑚𝑖𝑛 −𝑀𝑠𝑎𝑡2−

𝑀𝑓𝑙2

𝜙 𝑀𝑚𝑖𝑛 −𝑀𝑓𝑙2 =

𝑀𝑠𝑎𝑡1

𝑀𝑚𝑖𝑛 −𝑀𝑠𝑎𝑡1−

𝑀𝑓𝑙1

𝜙 𝑀𝑚𝑖𝑛 −𝑀𝑓𝑙1 ,

(3.13)

where 𝑀 is the P-wave modulus, and subscripts min, fl1, and sat1 are mineral, fluid 1,

and state of being saturated with fluid 1, respectively. 𝜙 is the total porosity. First, we

apply fluid substitution (i.e., replace fluid 1 by fluid 2) to the clean sand layers, whose

P-wave moduli can be written as

𝑀𝑆 𝑠𝑎𝑡2 =

𝑀𝑆 𝑠𝑎𝑡1

𝑀𝑞𝑡𝑧 −𝑀𝑆 𝑠𝑎𝑡1−

𝑀𝑓𝑙1

𝜙𝑠 𝑀𝑞𝑡𝑧 −𝑀𝑓𝑙1 +

𝑀𝑓𝑙2

𝜙𝑠 𝑀𝑞𝑡𝑧 −𝑀𝑓𝑙2 𝑀𝑞𝑡𝑧



𝑀𝑓𝑙1


𝑀𝑓𝑙2

𝜙𝑠 𝑀𝑞𝑡𝑧 −𝑀𝑓𝑙2 + 1

,

(3.14)

where 𝑀𝑆 and 𝜙𝑠 are the P-wave modulus and total porosity of clean sand, respectively.

The mineral used here is quartz. If we let


𝐴 =𝑀𝑆 𝑠𝑎𝑡1


𝑀𝑓𝑙1


𝑀𝑓𝑙2

𝜙𝑠 𝑀𝑞𝑡𝑧 −𝑀𝑓𝑙2 ,

(3.15)

then Equation 3.14 is equivalent to simply

𝑀𝑆 𝑠𝑎𝑡2 =𝐴 ∗ 𝑀𝑞𝑡𝑧

𝐴 + 1 .

(3.16)

For the interbedded sand-shale sequence, in which the clean sand and shale layers are

saturated with fluid 2 and brine respectively, we obtain the P-wave modulus of the

sequence by Backus averaging:

𝑀𝑠𝑎𝑡2 = 𝑉𝑠𝑎𝑛𝑑𝑀𝑆 𝑠𝑎𝑡2

+1 − 𝑉𝑠𝑎𝑛𝑑

𝑀𝑆𝑕 −1

,

(3.17)

where 𝑀𝑆𝑕 is the P-wave modulus of wet shale, and 𝑉𝑠𝑎𝑛𝑑 is the sand fraction of the

interbedded sequence. By replacing 𝑀𝑆 𝑠𝑎𝑡2 in Equation 3.17 by Equation 3.16, we

have

𝑀𝑠𝑎𝑡2 = 𝑉𝑠𝑎𝑛𝑑

𝐴 ∗ 𝑀𝑞𝑡𝑧 +𝑉𝑠𝑎𝑛𝑑𝑀𝑞𝑡𝑧


𝑀𝑆𝑕

−1

,

(3.18)

and if we let X = 𝑉𝑠𝑎𝑛𝑑

𝑀𝑞𝑡𝑧 +

1−𝑉𝑠𝑎𝑛𝑑

𝑀𝑆𝑕, then we obtain

𝑀𝑠𝑎𝑡2 = 𝑉𝑠𝑎𝑛𝑑

𝐴 ∗ 𝑀𝑞𝑡𝑧 + 𝑋

−1

𝑀𝑠𝑎𝑡2 =𝐴𝑀𝑞𝑡𝑧

𝑉𝑠𝑎𝑛𝑑

𝐴 ∗ 𝑋𝑀𝑞𝑡𝑧

𝑉𝑠𝑎𝑛𝑑+ 1

.

(3.19)


Now, if we directly apply Equation 3.13 to measurements which represent average

properties of clean sand and shale layers, we obtain

𝑀𝑠𝑎𝑡2∗ =

𝑀𝑠𝑎𝑡1

∗

𝑀𝑚𝑖𝑛∗ −𝑀𝑠𝑎𝑡1

∗ −𝑀𝑓𝑙1

∗

𝜙𝐸𝑓𝑓 𝑀𝑚𝑖𝑛∗ −𝑀𝑓𝑙1

∗ +

𝑀𝑓𝑙2∗


∗ 𝑀𝑚𝑖𝑛

∗

𝑀𝑠𝑎𝑡1

∗



∗


∗ +

𝑀𝑓𝑙2∗


∗ + 1

.

(3.20)

Note that effective porosity is used instead of total porosity. If we let

𝐵 =𝑀𝑠𝑎𝑡1

∗



∗


∗ +

𝑀𝑓𝑙2∗


∗ ,

(3.21)

then Equation 3.20 is equivalent to simply

𝑀𝑠𝑎𝑡2∗ =

𝐵 ∗ 𝑀𝑚𝑖𝑛∗

𝐵 + 1 .

(3.22)

When Gassmann’s equation is applied at the measurement scale, in order to

correctly account for sub-resolution interbedding, we let

𝑀𝑠𝑎𝑡2∗ = 𝑀𝑠𝑎𝑡2

𝐵 ∗ 𝑀𝑚𝑖𝑛∗

𝐵 + 1=

𝐴𝑀𝑞𝑡𝑧

𝑉𝑠𝑎𝑛𝑑

𝐴 ∗ 𝑋𝑀𝑞𝑡𝑧

𝑉𝑠𝑎𝑛𝑑+ 1

.

(3.23)

One way to make Equation 3.23 hold is to equate the left side and the right side on a

term-by-term basis, which results in the following six equations:

𝑀𝑠𝑎𝑡1

∗


∗ 𝑀𝑚𝑖𝑛∗ =


𝑀𝑞𝑡𝑧 −𝑀𝑆 𝑠𝑎𝑡1 𝑀𝑞𝑡𝑧

𝑉𝑠𝑎𝑛𝑑 ,

(3.24)


𝑀𝑠𝑎𝑡1

∗


∗ = 𝑀𝑆 𝑠𝑎𝑡1

𝑀𝑞𝑡𝑧 −𝑀𝑆 𝑠𝑎𝑡1 𝑋 ∗ 𝑀𝑞𝑡𝑧


(3.25)

𝑀𝑓𝑙1∗



∗ =𝑀𝑓𝑙1

𝜙𝑠 𝑀𝑞𝑡𝑧 −𝑀𝑓𝑙1

𝑀𝑞𝑡𝑧


(3.26)

𝑀𝑓𝑙1∗


∗ =

𝑀𝑓𝑙1


𝑋 ∗ 𝑀𝑞𝑡𝑧


(3.27)

𝑀𝑓𝑙2

∗



∗ =𝑀𝑓𝑙2


𝑀𝑞𝑡𝑧


(3.28)

and 𝑀𝑓𝑙2

∗


∗ =

𝑀𝑓𝑙2


𝑋 ∗ 𝑀𝑞𝑡𝑧

𝑉𝑠𝑎𝑛𝑑.

(3.29)

From the above set of equations, we can solve for

𝑀𝑚𝑖𝑛∗ =

1

𝑋=

𝑉𝑠𝑎𝑛𝑑𝑀𝑞𝑡𝑧


𝑀𝑆𝑕

−1

,

𝑀𝑓𝑙1∗ =

1

𝑀𝑓𝑙1 −

1

𝑀𝑞𝑡𝑧 +

1

𝑀𝑚𝑖𝑛∗

−1

= 1

𝑀𝑓𝑙1 −

1

𝑀𝑞𝑡𝑧 +𝑉𝑠𝑎𝑛𝑑𝑀𝑞𝑡𝑧

+1 − 𝑉𝑠𝑎𝑛𝑑𝑀𝑆𝑕

−1

,

and 𝑀𝑓𝑙2∗ =

1

𝑀𝑓𝑙2 −

1

𝑀𝑞𝑡𝑧 +

1

𝑀𝑚𝑖𝑛∗

−1

= 1

𝑀𝑓𝑙2 −

1

𝑀𝑞𝑡𝑧 +𝑉𝑠𝑎𝑛𝑑𝑀𝑞𝑡𝑧

+1 − 𝑉𝑠𝑎𝑛𝑑𝑀𝑆𝑕

−1

.

(3.30)

The expressions shown in Equation 3.30 are the input adjustment required so that

when Gassmann’s equation is applied at the measurement scale, it also accounts for

the effect of sand-shale interbedding at the same time. In summary, the upscaled


Gassmann’s equation for a interbedded (clean) sand-shale sequence can be applied by

simply using Gassmann’s equation (Equation 3.13) with the following data/ input

parameters: P-wave modulus of a data point which represents interbedding of clean

sand saturated with fluid 1 and wet shale 𝑀𝑠𝑎𝑡1∗ , effective porosity at the

measurement scale (𝜙𝐸𝑓𝑓 ) , P-wave modulus of the effective mineral (𝑀𝑚𝑖𝑛∗ ) in

Equation 3.30, and P-wave moduli of the original and final saturating fluids

(𝑀𝑓𝑙1∗ 𝑎𝑛𝑑 𝑀𝑓𝑙2

∗ ) in Equation 3.30.

The exact input adjustments (Equation 3.30) are derived only for the case where

clean sand is interbedded with shale. However, we heuristically extend the derivation

results to the case where shaly-sand is interbedded with shale. P-wave moduli of the

effective mineral, the original, and the final saturating fluids required for the upscaled

Gassmann’s equation for shaly-sand interbedded with shale are as follows:

𝑚 =1

2 1 − 𝑉𝑑𝑖𝑠𝑝 𝑀𝑞𝑡𝑧 + 𝑉𝑑𝑖𝑠𝑝 𝑀𝑆𝑕 +

1

2

1 − 𝑉𝑑𝑖𝑠𝑝

𝑀𝑞𝑡𝑧 +𝑉𝑑𝑖𝑠𝑝

𝑀𝑆𝑕

−1

,

𝑀𝑚𝑖𝑛∗ =

𝑉𝑠𝑎𝑛𝑑𝑚

+1 − 𝑉𝑠𝑎𝑛 𝑑𝑀𝑆𝑕

−1

𝑀𝑓𝑙1∗ =

1

𝑀𝑓𝑙1 −

1 − 𝑉𝑠𝑎𝑛𝑑𝑀𝑚𝑖𝑛

∗ +1 − 𝑉𝑠𝑎𝑛𝑑𝑀𝑆𝑕

−1

,

and 𝑀𝑓𝑙2∗ =

1

𝑀𝑓𝑙2 −

1 − 𝑉𝑠𝑎𝑛𝑑𝑀𝑚𝑖𝑛

∗ +1 − 𝑉𝑠𝑎𝑛𝑑𝑀𝑆𝑕

−1

.

(3.31)

Note that we use the Voigt-Reuss-Hill average to compute the effective mineral (𝑚) of

the composite between quartz and wet shale (Dvorkin et al., 2007) to represent the

solid part of the shaly-sand layers as an additional step to the clean sand case. When

the volume fraction of dispersed clay (𝑉𝑑𝑖𝑠𝑝 ) is equal to zero (i.e., clean sand), 𝑀𝑚𝑖𝑛∗

in Equation 3.31 is reduced to 𝑀𝑞𝑡𝑧 , which is consistent with Equation 3.30. Also note

that 1 − 𝑉𝑑𝑖𝑠𝑝 and 𝑉𝑑𝑖𝑠𝑝 are not the actual volumetric fractions of quartz and wet shale

in the solid part of the shaly-sand layers, but rather a heuristic approximation.


3.9.6.1 Synthetic data examples

We test our upscaled Gassmann’s equation on synthetic data. Similar to Section

3.5.5, we generate interbedded sequences between sand and shale by systematically

varying both the volume fractions of dispersed clay in the sand layers (i.e., from

𝑉𝑑𝑖𝑠𝑝 = 0 to 𝑉𝑑𝑖𝑠𝑝 = 𝜙𝑆 ) and the sand fractions (i.e., from 𝑉𝑠𝑎𝑛𝑑 = 0 to 𝑉𝑠𝑎𝑛𝑑 = 1).

Thus, we provide a rough coverage for all possible interbedding scenarios allowed by

the triangular diagram (Figure 3.3). We assign a starting effective water-saturation of

0.5 (i.e., 𝑆𝑊𝑒=0.5). Then, we apply both the upscaled Gassmann and the Gassmann

shaly-sand equation (Dvorkin et al., 2007) to the synthetic data directly at the

measurement scale, and compare the results with the baselines, which are the results of

a two-step process –applying the Gassmann shaly-sand equation to the sand layers

only, and then upscaling these sand layers and shale by Backus averaging. This two-

step process is equivalent to the downscaling-upscaling procedure, and these baselines

are considered correct in synthetic data because all the parameters needed are known

(more detailed discussion in Section 3.5). Results are shown as percentage differences

between P-wave velocities after fluid substitution by the selected methods and the

baselines:

% 𝑑𝑖𝑓𝑓 = 100 ∗𝑉𝑝𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑 𝑚𝑒𝑡 𝑕𝑜𝑑 − 𝑉𝑝𝑏𝑎𝑠𝑒𝑙𝑖𝑛𝑒

𝑉𝑝𝑏𝑎𝑠𝑒𝑙𝑖𝑛𝑒 ,

(3.32)

where Vp is the P-wave velocity after fluid substitution.

We test the upscaled Gassmann in three different scenarios. The first scenario is to

compute parameter adjustments by following Equation 3.31, and then all the adjusted

parameters are input into approximate Gassmann’s equation (Mavko et al., 1995). The

next two scenarios are to test the performance of the upscaled Gassmann in the case

where both bulk and shear moduli are used. The second scenario is to compute

parameter adjustments by following steps similar to Equation 3.31, except that the

effective mineral (𝑚) of the composite between quartz and wet shale is now computed

by


𝑚𝐾 =1

2 1 − 𝑉𝑑𝑖𝑠𝑝 𝐾𝑞𝑡𝑧 + 𝑉𝑑𝑖𝑠𝑝 𝐾𝑆𝑕 +

1

2


𝐾𝑞𝑡𝑧 +𝑉𝑑𝑖𝑠𝑝

𝐾𝑆𝑕

−1

,

𝑚𝜇 =1

2 1 − 𝑉𝑑𝑖𝑠𝑝 𝜇𝑞𝑡𝑧 + 𝑉𝑑𝑖𝑠𝑝 𝜇𝑆𝑕 +

1

2


𝜇𝑞𝑡𝑧 +𝑉𝑑𝑖𝑠𝑝

𝜇𝑆𝑕

−1

,

and 𝑚 = 𝑚𝐾 +4

3𝑚𝜇 .

(3.33)

Then, all the adjusted parameters are input into approximate Gassmann’s equation

(Equation 3.13; Mavko et al., 1995). Finally, the third scenario is to compute

parameter adjustments by following steps similar to Equation 3.31, except that all the

P-wave moduli in the equation are replaced by bulk moduli

𝑘 =1

2 1 − 𝑉𝑑𝑖𝑠𝑝 𝐾𝑞𝑡𝑧 + 𝑉𝑑𝑖𝑠𝑝 𝐾𝑆𝑕 +

1

2


𝐾𝑞𝑡𝑧 +𝑉𝑑𝑖𝑠𝑝

𝐾𝑆𝑕

−1

,

𝐾𝑚𝑖𝑛∗ =

𝑉𝑠𝑎𝑛𝑑𝑘


𝐾𝑆𝑕 −1

𝐾𝑓𝑙1∗ =

1

𝐾𝑓𝑙1 −

1 − 𝑉𝑠𝑎𝑛𝑑𝐾𝑚𝑖𝑛∗ +

1 − 𝑉𝑠𝑎𝑛𝑑𝐾𝑆𝑕

−1

,

and 𝐾𝑓𝑙2∗ =

1

𝐾𝑓𝑙2 −

1 − 𝑉𝑠𝑎𝑛𝑑𝐾𝑚𝑖𝑛∗ +

1 − 𝑉𝑠𝑎𝑛𝑑𝐾𝑆𝑕

−1

.

(3.34)

Then, all adjusted parameters are input into the actual Gassmann’s equation. We

compare results of the upscaled Gassmann for all three scenarios with those of the

Gassmann shaly-sand equation and the baselines. To assure consistency of fluid

substitution results among the selected methods, only P-wave moduli are used in all

calculations in the Gassmann shaly-sand equation and the baselines for the first

scenario, but both bulk and shear moduli are used in these two methods for the second

and the third scenarios. Results of the first, second, and third scenarios are shown in

Figure 3.26, Figure 3.27 and Figure 3.28, respectively.



by the upscaled Gassmann’s equation (left) and the Gassmann shaly-sand equation (Dvorkin et al., 2007) (right) for the first scenario, where P-wave moduli and approximate Gassmann’s equation are used for the upscaled Gassmann. Each location on the triangular diagram represents an interbedded sand-shale sequence with a unique pair of sand fraction and volume fraction of dispersed clay values. Note that the color scales of the two panels are different.

Figure 3.27: Percentage differences between the baselines and the predicted velocities by the upscaled Gassmann’s equation (left) and the Gassmann shaly-sand equation (Dvorkin et al., 2007) (right) for the second scenario, where both bulk and shear moduli, and approximate Gassmann’s equation are used in calculations. Each location on the triangular diagram represents an interbedded sand-shale sequence with a unique pair of sand fraction and volume fraction of dispersed clay values. Note that the color scales of the two panels are different.

0 0.1 0.2 0.3

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Effective porosity

1/M

(1/G

Pa)

Upscaled Gassmann

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Effective porosity1/M

(1/G

Pa)


0

5

10

15

20

% diff

Sand with

dispersed

shale

% diff

Clean

sand

Shale

Increasing

sand fractions

0 0.1 0.2 0.3

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Effective porosity

1/M

(1/G

Pa)

Upscaled Gassmann

0

0.5

1

1.5

2

0 0.1 0.2 0.3

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Effective porosity

1/M

(1/G

Pa)


0

5

10

15

20

Shale

Clean

sand

Sand with

dispersed

shale

% diff% diff

Increasing

sand fractions


Figure 3.28: Percentage differences between the baselines and the predicted velocities by the upscaled Gassmann’s equation (left) and the Gassmann shaly-sand equation (Dvorkin et al., 2007) (right) for the second scenario, where both bulk and shear moduli, and the actual Gassmann’s equation are used for the upscaled Gassmann. Each location on the triangular diagram represents an interbedded sand-shale sequence with a unique pair of sand fraction and volume fraction of dispersed clay values. Note that the color scales of the two panels are different. The color scale on the left panel does not cover either the actual maximum or minimum values of the percentage differences. The color scale is adjusted to show values of the majority of the results. Those results indicated by a magenta ellipse are unreliable due to instability of the method at low effective porosity.

In the first scenario (Equation 3.31 & approximate Gassmann), both the upscaled

Gassmann and the Gassmann shaly-sand equation (Dvorkin et al., 2007) show only

positive percentage differences in P-wave velocities relative to the baselines.

Therefore, both fluid substitution methods overpredict velocities. However, while the

prediction errors of the upscaled Gassmann are less than 0.6%, the errors of the

Gassmann shaly-sand equation can reach 20%. This very large difference in errors is

because our method accounts for the effect of sand-shale interbedding, whereas the

Gassmann shaly-sand equation ignores the effect of sand-shale interbedding by

assuming that all shale is dispersed.

Since in the upscaled Gassmann the parameter adjustment for the case where

shaly-sand interbedded with shale is only a heuristic approximation, the prediction

errors of this method are higher in the region near the shaly-sand line (i.e., sand with

dispersed clay line in Figure 3.26). In contrast, since the Gassmann shaly-sand

equation is most valid for shaly sands, the errors of this method are lower in the region

0 0.1 0.2 0.3

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Effective porosity

1/M

(1/G

Pa)

Upscaled Gassmann

-1.5

-1

-0.5

0

0.5

1

0 0.1 0.2 0.3

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Effective porosity

1/M

(1/G

Pa)


0

5

10

15

20

Shale

Clean

sand

Sand with

dispersed

shale

% diff% diff

Increasing

sand fractions


near the shaly-sand line. However, from this shaly-sand line, when moving inward in

the diagram, the errors of the Gassmann shaly-sand equation grow larger, especially in

the middle of the diagram where the volume fractions of laminated (interbedded) shale

are high, because all this laminated (interbedded) shale is misinterpreted as being

dispersed (Figure 3.26, right).

In the second scenario (Equation 3.33; approximate Gassmann), while the

prediction errors by the Gassmann shaly-sand equation reach 20%, the errors by the

upscaled Gassmann are less than 3% (Figure 3.27). The overall distributions of

prediction errors of both methods are similar to the first scenario (Equation 3.31;

approximate Gassmann) in Figure 3.26. While the high errors of the upscaled

Gassmann are more concentrated in the region near the shaly-sand line, the errors of

the Gassmann shaly-sand in the same region are relatively low, but they grow larger

toward the middle of the diagram (Figure 3.27). However, in this scenario even though

we use both bulk and shear moduli in computing the adjustment for the upscaled

Gassmann, we input these adjusted parameters into the approximate Gassmann’s

equation (Equation 3.13), which involves only P-wave moduli. However, we use the

actual Gassmann’s equation, which involves both bulk and shear moduli, to compute

the baselines. The difference in the approximate and the actual Gassmann’s equations

increases prediction errors of the upscaled Gassmann. Therefore, results of the

upscaled Gassmann show that there are non-zero errors near the clean sand point

(Figure 3.27, left), and that the error range in Figure 3.27(left) is larger than that of

Figure 3.26(left).

Instead of comparing results from the approximate and actual Gassmann’s

equations as in the second scenario, in the third scenario (Equation 3.34; actual

Gassmann) the actual Gassmann’s equation is used in all calculations. Therefore, the

overall absolute errors of the upscaled Gassmann in this scenario are smaller than in

the second scenario (Figure 3.27 and Figure 3.28, left). However, because of the

overgeneralization of the parameter adjustment in this scenario (i.e., simple

replacement of P-wave moduli by bulk moduli), a portion of the results corresponding

to low effective-porosity sequences becomes unreliable (those marked by a magenta


ellipse in Figure 3.28, left). Note that Gassmann’s equation is also reported to often

yield unreliable results for low porosity rocks (e.g., Smith et al., 2003).

3.9.6.2 Real data example

We test our upscaled Gassmann’s equation on the same dataset used in Section 3.8.

Starting with all brine-saturated data, we substitute oil for brine using five procedures:

Gassmann’s equation (i.e., ignoring the effect of sand-shale interbedding), the

Gassmann shaly-sand equation (i.e., assuming that shale is all dispersed; Dvorkin et al.,

2007), downscaling-upscaling, our mesh method, and our upscaled Gassmann

(Equation 3.34). Note that here both bulk and shear moduli are used in all calculations.

Sand fractions and volume fractions of dispersed clay, which are needed for the

downscaling-upscaling procedure and the upscaled Gassmann, are estimated using the

Thomas-Stieber-Yin-Marion model in the P-compliance vs. effective porosity space.

Results are shown in Figure 3.29 as P-wave velocities after fluid substitution by the

five selected procedures. As in the previous performance test on this dataset (Figure

3.22), the results of the downscaling-upscaling procedure are used as baselines.

However, to avoid any erratic behavior at very low porosity, the baselines from the

downscaling-upscaling procedure are chosen to include only those points with gamma

ray values less than 85.

In Figure 3.29, results of Gassmann’s equation, the Gassmann shaly-sand

equation, the baselines, and the mesh are exactly the same as the results for this dataset

in the previous performance test (Figure 3.22). Both Gassmann’s equation and the

Gassmann shaly-sand equation overestimate the fluid effect because sub-resolution

interbedding is not correctly accounted for. In contrast, the results from the mesh

method agree well with the baselines, and the results from the upscaled Gassmann

show the best agreement with the baselines. However, when using the upscaled

Gassmann, unreliable results (e.g., velocity spikes) are observed, especially in low

porosity data. Analogous to the synthetic case in Figure 3.28 (left), this instability of

the upscaled Gassmann may come from the overgeneralization in parameter

adjustment, where P-wave moduli are simply replaced by bulk moduli (Equation 3.34).


Figure 3.29: Comparisons of five sets of fluid substitution results which are color-coded by sand fractions estimated from the Thomas-Stieber-Yin-Marion model. Five fluid substitution procedures are used: Gassmann’s equation (i.e., ignoring the effect of sand-shale interbedding), Gassmann shaly-sand equation (i.e., assuming that shale is all dispersed; Dvorkin et al., 2007), the downscaling-upscaling procedure, the mesh method, and the upscaled Gassmann’s equation. Note: points with very low sand fractions are excluded from the plot. Lines X = Y are superimposed for comparison purpose.

3.9.6.3 Sensitivity analysis

In this section, we show how uncertainties in input parameters affect fluid

substitution results of the upscaled Gassmann. For this analysis, we use only P-wave

moduli information (Equation 3.31). The six input parameters are P-wave velocity of

wet porous shale (𝑉𝑝𝑠𝑕), volume fraction of dispersed clay (𝑉𝑑𝑖𝑠𝑝 ), sand fraction

(𝑁𝑇𝐺), effective water saturation in the sand layers (𝑆𝑊𝑒 ), fluid properties (i.e., bulk

modulus, 𝐾𝑓𝑙 ), and effective porosity of the interbedded sequence (𝜙𝐸𝑓𝑓 ).

The procedure and setting used here are similar to those of Section 3.7. The

starting effective pore space is half filled with oil. Then, we apply the upscaled

Gassmann and the downscaling-upscaling procedure to substitute brine for oil using

1.8 2 2.2 2.4 2.61.8

2

2.2

2.4

2.6


(km/s)

Vp m

esh (

km

/s)

1.8 2 2.2 2.4 2.61.8

2

2.2

2.4

2.6


(km/s)

Vp G

assm

ann (

km

/s)

(Dvork

in e

t al.,

2007)

1.8 2 2.2 2.4 2.61.8

2

2.2

2.4

2.6


(km/s)

Vp G

assm

ann (

Tota

l) (

km

/s)

1.8 2 2.2 2.4 2.61.8

2

2.2

2.4

2.6


(km/s)

Vp u

pscale

d G

assm

ann (

km

/s)

0.2

0.4

0.6

0.8

1Sand fraction

X > Y region

Line X = Y


all correct parameters, and these results are used as references. We assume that all five

input parameters are independent and normally distributed with means ( 𝜇 ) and

variances 𝜎2 . The means of the distributions are set to be the true values used in the

forward-modeling process. The standard deviations 𝜎 are set to be 10% of the means.

We test the effect of uncertainties in the input parameters one by one.

Results of the sensitivity analysis for the upscaled Gassmann are shown in Figure

3.30. Each subplot is indicated by a sand fraction (𝑉𝑠𝑎𝑛𝑑 ) and a volume fraction of

dispersed clay in the sand layers (𝑉𝑑𝑖𝑠𝑝 ). The thick lines on the top of each subplot are

the oil-saturated velocity and the reference velocities after fluid substitution from oil

to brine using the downscaling-upscaling procedure and the upscaled Gassmann with

all correct parameters. During the sensitivity analysis, one by one each input

parameter is assumed to be uncertain while the other parameters are assumed to be

correct, and the sensitivity analysis results are shown as a distribution of velocities

after fluid substitution.

When applying the upscaled Gassmann to the data, fluid substitution results are

only slightly affected by volume fraction of dispersed clay, effective water saturation,

and fluid bulk modulus. Uncertainty in effective porosity has a greater effect on the

results when the sand fraction in an interbedded sequence increases. For low sand-

fraction sequences, while the uncertainty in P-wave velocity of shale leads to a large

uncertainty in fluid substitution results, the uncertainty in sand fraction has only a

moderate effect on the results. However, when sand fraction increases, fluid

substitution results become less sensitive to shale velocity and become more sensitive

to the sand fraction. Overall, results show that the upscaled Gassmann is most

sensitive to P-wave velocity of wet porous shale and sand fraction, but this sensitivity

also depends on properties of interbedded sequences (e.g., sand fractions).

It is interesting to note that for our mesh method, fluid substitution results are most

sensitive to P-wave velocity of clean sand (Figure 3.19). The effect of uncertainty in

clean sand velocity on the results is more pronounced for high sand-fraction sequences

than for low sand-fraction sequences. Therefore, an alternative procedure for fluid


substitution in interbedded sand-shale sequences is to use the mesh method in low

sand-fraction sequences and to use the upscaled Gassmann in high sand-fraction

sequences. By doing so, fluid substitution results should be less sensitive to

uncertainty in input parameters.

Figure 3.30: Sensitivity analysis results of the upscaled Gassmann for six input parameters: P-wave velocity of wet porous shale, volume fraction of dispersed clay, sand fraction, effective water saturation, bulk modulus of fluid, and effective porosity of the interbedded package. The thick lines on the top of each subplot are the oil-saturated velocity and the reference velocities after fluid substitution from oil to brine using the downscaling-upscaling procedure and the upscaled Gassmann with all correct input parameters. Results of sensitivity analysis for each parameter are shown as a velocity distribution, which is normalized to one. Sand fractions and volume fractions of dispersed clay for each synthetic model are shown on the top of each subplot.

3.10 Conclusions

We provide a simple mesh method for approximating fluid substitution in sub-

resolution interbedded sand-shale sequences. The method is done by simply applying

Gassmann’s fluid substitution to clean sand and distorting the mesh. As a result, for

any sequence representing sand (or shaly-sand) interbedded with shale, the elastic

property after fluid substitution is scaled to the behavior of the clean sand that has

2.5 3 3.50

1

Sand fraction = 0.33 Vdisp = 0

2.5 3 3.50

1

Sand fraction = 0.33 Vdisp = 0.12

2.5 3 3.50

1


2.5 3 3.50

1

Sand fraction = 0.67 Vdisp = 0

Norm

aliz

ed d

ensity f

unction

2.5 3 3.50

1


2.5 3 3.50

1


2.5 3 3.50

1

Sand fraction = 1 Vdisp = 0

P-velocity (km/s)

2.5 3 3.50

1

Sand fraction = 1 Vdisp = 0.12

P-velocity (km/s)

2.5 3 3.50

1

Sand fraction = 1 Vdisp = 0.24

P-velocity (km/s)

V

p shale

Vdisp

Sand fraction

Sw e

Koil

Effective porosity

Downscaling-upscaling

Upscaled Gassmann

Oil-saturated model


undergone fluid substitution. The mesh method is directly applicable at the

measurement scale, without the need to downscale the measurements, but its fluid

substitution results still agree well with those predicted using the actual downscaling-

upscaling procedure. The sensitivity analysis of five input parameters needed for the

mesh method shows that this method is most sensitive to the clean-sand P-wave

velocities, whose values can be constrained by rock physics trends. The method is

only robust for rocks that can be represented as sand-shale mixtures, in which any

porosity reduction is caused only by filling of shale in the pore space. Therefore, the

method does not account for effects of sorting and cementation. Because the mesh

method does not cover all possible variations in rocks, this method should not be

universally applied to a whole dataset without checking for its applicability. To

alleviate the problem of non-universality of the mesh method, we recommend

applying both the mesh method and the traditional Gassmann’s equation to the data.

The traditional Gassmann’s equation tends to overestimate the change of elastic

moduli after fluid substitution. The more accurate fluid substitution result is the one

that has a smaller change in elastic moduli, but how accurate is this observation should

be addressed in future analyses.


This work was supported by the Stanford Rock Physics and Borehole Geophysics

project.

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inversion using rock physics: The leading edge, 30(1), 70-78.


488.



Annual Logging


Smith, T. M., Condergeld, C. H., and Rai, C. S., 2003, Gassmann Fluid Substitutions:

A Tutorial: Geophysics, 68(2), 430-440.

Spike K. T. and J. P. Dvorkin, 2005, Gassmann-consistency of velocity-porosity

transforms: The Leading Edge, 24, 581-583.


effect upon porosity: 16th Annual Logging Symposium, SPWLA, Paper T.



107

Chapter 4

Seismic signature and uncertainty in

petrophysical property estimation of

thin sand-shale reservoirs

4.1 Abstract

Property estimation of thin sand-shale reservoirs using seismic response is often

challenging due to limited seismic resolvability. This chapter investigates seismic

signatures of sub-resolution sand-shale sequences and shows numerical examples of

using wavelet-transform based attributes and feature-extraction1 based attributes for

estimating reservoir properties which are net-to-gross ratio, saturation, and stacking

1 Feature extraction transforms the original features in high-dimensional data to create new features

which provide new representations of the data in a lower-dimensional space, while preserving as much

information in the data as possible. An example of linear feature extraction techniques is principal

component analysis (PCA) (Cunningham, 2008).

CHAPTER 4: Seismic signatures of thin sand-shale reservoirs 108

patterns. To investigate seismic signature, we generate thin sand-shale sequences (i.e.,

vertical arrangements of lithology) using 1-D Markov chain models. Layer properties

assigned to the sequences come from established binary-mixture models from rock

physics relations, in particular porosity and velocity versus sand-shale ratios. Forward

computation of the seismic response of the sequences is then used to extract attributes

and relate them to the spatial patterns and properties of thin sand-shale reservoirs. The

wavelet-transform based attributes are the slope and intercept of the log-log plot

between wavelet-coefficient modulus and scale. The feature-extraction techniques

compare the entire seismograms, capture variations in the amplitudes, and provide

new representations of the seismograms in a new feature space, which are used as

attributes. The two techniques considered here are multidimensional scaling

techniques (MDS) and kernel principal component analysis (KPCA). To quantify

attribute performance, we use the attributes for classifying seismograms with different

reservoir properties into classes and measure the accuracy of this classification by

computing the classification success rate. 1-D synthetic examples show that some of

the attributes, in particular KPCA, effectively discriminate different net-to-gross,

saturation, or stacking-pattern classes. A workflow similar to our synthetic study can

be applied to real seismic data to characterize thin sand-shale reservoirs. To illustrate

this application, we present two additional numerical examples. In the first example,

we assume that the transition matrix or thickness distributions at the well location are

known. In reality, this information can be directly extracted from the well data. Then,

we create synthetic 2-D spatial models describing geology away from the well point,

explore statistically how attributes vary with a change of net-to-gross ratios, and apply

these statistics to obtain the posterior distributions of net-to-gross ratios at three

selected locations from the unknown seismic section. In the second example, we

investigate seismic signatures of non-stationary thin sand-shale sequences, whose

layering patterns change vertically. The sequences are modeled as a first-order,

discrete, 1-D Markov chain. We use two different transition matrices to generate a

sequence in order to create non-stationarity. We analyze seismograms using moving


windows to obtain seismic signatures locally along the reservoir and estimate posterior

distributions of net-to-gross ratios along the reservoir following a Bayesian framework.

4.2 Introduction

Seismic estimation of reservoir properties is common in petroleum exploration.

However, estimating petrophysical properties of thinly layered reservoirs with layers

below seismic resolvability is often challenging. Previous studies (e.g., Khattri and Gir,

1976; Marion et al., 1994; Mukerji, 1995; Mukerji et al., 1995; Takahashi, 2000; Hart

and Chen, 2004; Hart, 2008) have shown that both the scales of heterogeneity (e.g.,

layer thickness, d) relative to the scale of seismic measurement (i.e., wavelength, λ)

and the difference in sediment stacking patterns affect seismic signatures and their

relations with rock properties. When λ/d is small, waves traveling perpendicular to the

layered medium can be described by the ray theory or short-wavelength limit in which

the total travel time through the media is simply the sum of the travel time of each

layer. The average velocity for the short-wavelength limit is the weighted average of

slowness. In contrast, when λ/d is large, waves travel through the layered medium

which behaves as a homogeneous effective medium with an effective velocity in the

effective medium theory or long-wavelength limit equal to the square root of the ratio

of the effective modulus and the average density. The effective modulus is calculated

using the Backus average (Backus, 1962). In the long-wavelength limit, when the

layered medium is perfectly periodic (e.g., alternating sand-shale system), normally

incident reflection seismograms exhibit no internal reflection between the top and the

base reflections, and the reflected amplitudes directly relate to both the properties and

the proportions of the periodic materials (Stovas et al., 2006). For a non-periodic

layered medium, internal reflections can be observed, and the relation between the top

reflectivity and the proportions of the layered materials become non-deterministic and

non-unique due to not only the natural variation in material properties but also how the

materials are arranged (Takahashi, 2000; Mukerji and Mavko, 2008).


Most stratigraphic sequences in nature reflect non-random stacking of sedimentary

patterns. Depending on depositional environments, the main characteristics of such

patterns include lateral extents, vertical arrangements of lithologies, and layer

thickness distributions (Harms and Tackenberg, 1972). Markov chains have been used

as a tool to simulate bedded sequence to capture these preferred directionality and

asymmetric facies associations signaling depositional process (e.g., Krumbein and

Dacey, 1969; Harbaugh and Bonham-Carter, 1970; Schwarzacher, 1975; Xu and

MacCarthy, 1996; Parks et al., 2000). Velzeboer (1981) modeled sequences by a first-

order Markov chain with distributions of physical properties and theoretically derived

an expression for power spectrum of the seismic reflection response. This expression

is related to sedimentary parameters such as sand-shale ratio. Sinvhal and Sinvhal

(1992) simulated multiple realizations of lithologic models from well log information

using first-order Markov chains and generate the corresponding synthetic seismograms.

Then, several features were extracted from the autocorrelation and the power spectrum

of these seismograms, and these features were used in linear discriminant analysis for

identifying and predicting lithology from seismic data. In this chapter, we build on this

work with the goal of extracting attributes from seismograms statistically and use

them for quantitative seismic interpretation of thin sand-shale reservoirs. Even though

our general workflow is similar to that from Sinvhal and Sinvhal (1992), there are

several important differences. We extend the analysis to investigate the effects of other

properties such as saturation and stacking patterns on seismic signatures of thin

reservoirs. We use rock-physics models to create what-if scenarios with varying

reservoir properties. Instead of using fixed properties for each lithology (Sinvhal and

Sinvhal, 1992), we also incorporate property variations into the models. Most of the

attributes in our study are extracted directly by comparing the amplitudes of the entire

seismogram segments.

In this chapter, we investigate seismic signatures of thin (sub-seismic resolution)

sand-shale sequences using wavelet-transform based attributes and feature-extraction

based attributes. This chapter also presents a practical workflow for seismic

interpretation of reservoir properties of thin sand-shale sequences. The rest of this


chapter is organized as follows. Section 4.3 describes the forward-modeling process

for generating multiple realizations of thin sand-shale sequences and their

corresponding seismic responses. The section also discusses both types of seismic

attributes. Section 4.4 illustrates the effects of net-to-gross ratio, saturation, and

stacking patterns on seismic signatures through 1-D synthetic examples which

demonstrate the performance of various seismic attributes. Section 4.5 provides a

practical workflow through a synthetic example for estimating net-to-gross ratios from

2-D seismic sections using a Bayesian framework. Section 4.6 shows another synthetic

example for estimating net-to-gross ratios along non-stationary sequences whose

layering patterns vary from top to bottom, and finally Section 4.7 discusses the

benefits, limitations, and possible extension of the workflow and the attributes

presented in this chapter.

4.3 Forward modeling for seismic response and attributes

In this section, we briefly review the methodology and tools used in forward

modeling to generate seismic responses and their attributes for thin sand-shale

sequences. Our forward-modeling process has three steps: (1) using Markov chain

models to generate sequences of sedimentary layers, (2) using rock-physics models to

assign physical properties to the layers and simulating seismic responses of the

sequences, and (3) extracting seismic attributes which will be later used in estimating

reservoir properties. An overall workflow of the forward-modeling steps is shown in

Figure 4.1.


Figure 4.1: Overall workflow for seismic-signature study and property estimation.

First, thin sand-shale sequences are generated using Markov-chain models and rock physics relations. Then, seismic responses for these sequences are modeled, and seismic attributes are extracted. Finally, the attributes are related to reservoir properties, which can be used for reservoir characterization of target areas.

4.3.1 Markov chain models in stratigraphic sequences

We generate vertical arrangements of sand and shale layers in thin sand-shale

sequences using Markov chain models. In stratigraphic analysis, a column of

sediments can be described as a chain or spatial arrangement of a finite number of

discrete states (i.e., lithology). Markov chains use conditional probabilities to describe

the dependency of the current state on the previous states. If the transition from one

state to the next depends only on the immediately preceding state, the chain is said to

be first-order (Harbaugh and Bonham-Carter, 1970; Sinvhal and Khattri, 1983).

A Markov chain model is commonly represented by a transition matrix, whose

element pij (at the ith

row and jth

column) represents the probability of a transition from

state i to state j, or the probability of going to state j, given that i is the current state. In

a stratigraphic study, the transition matrix is usually obtained from real geological

observations and is typically constructed in one of two ways: either by counting states


using a fixed sampling interval, or by counting states only when a transition occurs (an

embedded form).

When counting states using a fixed sampling interval, the lithologic state is

determined and considered only at discrete points equally spaced along a stratigraphic

column. This allows successive points to have similar lithology, which implies that the

diagonal element (i.e., the probability that a state has a transition to itself) can be non-

zero (Krumbein and Dacey, 1969). In practice, however, selecting a proper sampling

interval for this method can be problematic. Choosing an interval that is too small

relative to the overall average bed thickness can increase the counts of transitions of a

state to itself. Consequently, the diagonal elements become very large, and

probabilities of the state transiting into the others become unreasonably small. In

contrast, using a sampling interval that is too large can miss very fine-layered

characteristics of the sequences (Sinvhal and Sinvhal, 1992). A sequence simulated by

this fixed-sampling type always yields lithologic states with thicknesses that are

geometrically distributed (Krumbein and Dacey, 1969).

In contrast, in an embedded-form transition matrix, all diagonal elements are zero,

since transitions are considered only when lithologic states change. In this case, the

step size between two consecutive states is not a fixed interval, but the actual observed

bed thicknesses (Parks et al., 2000). Thus, a sequence of states generated from an

embedded transition matrix does not contain information about thicknesses of each

layer. Thicknesses are simulated separately assuming that the layer thickness of each

lithologic state is distributed according to some distributions (i.e., semi Markov

process). A case when the layer thickness is exponentially distributed is called a

continuous-time Markov chain model. Examples of transition matrices and sequences

obtained from both the fixed-sampling and the embedded-form transition matrices are

shown in Figure 4.2 and Figure 4.3.

The probability of transition from state i to state j in n steps (denoted as p(n)

ij) can

be determined by raising the transition matrix to the power n. If a limiting distribution

exists, a successive multiplication leads to a row vector of fixed probabilities

representing proportions of each state in the long-term behavior. In terms of


stratigraphic application, this limiting distribution (or stationary distribution π) implies

proportions of each lithology in the entire sequence provided that sufficient numbers

of transition steps occur (Harbaugh and Bonham-Carter, 1970).

Figure 4.2: Three examples of transition matrices with fixed sampling intervals:

retrogradational, progradational, and aggradational sequences. The lithologic states in the transition matrices are sand (s), shaly sand (sh-s) sandy shale (s-sh) and shale (sh). The off diagonal elements marked by arrows control the directionality of the sequences.

S Sh S ShS Sh

Fixed

Retrograding Prograding Aggrading

(fining-upwards) (coarsening-upwards)

S Sh S ShS Sh

Fixed

S Sh S ShS Sh

FixedFixed

Retrograding Prograding Aggrading

(fining-upwards) (coarsening-upwards)

Fixed

Sh SSh S Sh S

Retrogradational Progradational Aggradational

(fining-upwards) (coarsening -upwards)


Figure 4.3: Examples of an embedded-form transition matrix with realizations of

sequences. The lithologic states in the transition matrix are sand (s), shaly sand (sh-s) sandy shale (s-sh) and shale (sh). An example of thickness distributions used is shown in the lower left corner.

Using either the fixed-sampling or the embedded-form transition matrices, it is

possible to generate sequences that mimic stratigraphic stacking patterns, for example,

by manipulating the fixed-sampling matrices with appropriate off-diagonal patterns, as

shown in Figure 4.2.

Parasequences and parasequence sets are two fundamental stratal units in sequence

stratigraphy. Parasequences contains a series of beds or bedsets that are genetically

related, in which younger beds are deposited and built up basinward (Van Wagoner, et

al., 1988; Van Wagoner, et al., 1990). Most siliciclastic parasequences show

progradational stacking pattern and all exhibit shoaling-upward trends in their vertical

sections with sediments which are generally coarsening-upward (e.g., beach and

deltaic parasequences) but sometimes fining-upward (e.g., tidal-flat parasequences).

Some characteristics of coarsening-upward parasequences include upward increases in

grain size, thickness of sandstone beds, and ratio of sandstone to mudstone. In contrast,

upward decreases in similar elements are observed in fining-upward parasequences

(Van Wagoner, et al., 1988; Van Wagoner, et al., 1990; Kamola and Van Wagoner,

1995; Mulholland, 1998). In response to the relation between depositional rates and

accommodation rates, lateral shifts of successive parasequences result in parasequence

02.02.06.0

2.001.07.0

3.03.004.0

5.03.02.00

S Sh-S S-Sh Sh

S

Sh-S

S-Sh

Sh

0 1 2 3 4 5 60

50

100

150

200

250

Thickness (m)S Sh S Sh S Sh

02.02.06.0

2.001.07.0

3.03.004.0

5.03.02.00

S Sh-S S-Sh Sh

S

Sh-S

S-Sh

Sh

0 1 2 3 4 5 60

50

100

150

200

250


02.02.06.0

2.001.07.0

3.03.004.0

5.03.02.00

S Sh-S S-Sh Sh

S

Sh-S

S-Sh

Sh

02.02.06.0

2.001.07.0

3.03.004.0

5.03.02.00

S Sh-S S-Sh Sh

S

Sh-S

S-Sh

Sh

0 1 2 3 4 5 60

50

100

150

200

250

Thickness (m)0 1 2 3 4 5 6

0

50

100

150

200

250


Sh SSh SSh SSh S

Sh SSh S

0 1 2 3 4 50

50

100

150

Thickness (m)


sets with predictable stacking patterns: retrogradational, progradational or

aggradational. When the depositional rate exceeds the accommodation rate, younger

parasequences are progressively deposited basinward, yielding a progradational

parasequence set. When the depositional rate is smaller than the accommodation rate,

younger parasequences are deposited landward, yielding a backstepping,

retrogradational parasequence set whose individual parasequences are progradational.

When the depositional rate is approximately equal to the accommodation rate, there is

no significant lateral shift in deposition of successive parasequences, resulting in an

aggradational parasequence set whose characteristics in terms of facies, thickness, and

sandstone to mudstone ratios stay similar throughout its vertical section (Van Wagoner,

et al., 1990; Mitchum and Van Wagoner, 1991; Kamola and Van Wagoner, 1995;

Boggs, 2001). Examples of SP log responses of parasequence sets with the three

stacking patterns are shown in Figure 4.4.

Figure 4.4: Examples of SP log responses showing stacking patterns in parasequence

sets. (Left) Retrogradational, (middle) progradational and (right) aggradational patterns (Modified after Van Wagoner et al., 1990).

4.3.2 Rock-physics models for sand-shale mixtures

After obtaining multiple sedimentary sequences, we assign petrophysical

properties to those sand-shale layers with rock-physics models and equations. The four

lithologic states (i.e., sand, shaly-sand, sandy-shale and shale) can be represented by

mixtures of sand and clay, where each state is assigned a unique clay fraction. Porosity

values corresponding to the mixtures of sand with the specified clay fractions are

determined by using the Yin-Marion dispersed-mixing model (Marion et al., 1992;


Yin, 1992). This model describes the topology of the bimodal mixtures and a V-

shaped relation between the volume fraction of clay and the mixture porosity.

When a small volume of clay (i.e., less than the sand porosity) is added to an

original packing of sand, clay starts filling the sand pore space without disturbing the

sand packing. Sand grains provide the load-bearing matrix of the mixture. At this stage,

porosity decreases, because clay particles replace some portions of the original sand

pore space. When the clay content is greater than the sand porosity, sand grains are

displaced and disconnected. The mixture changes from grain-supported to clay-

supported sediments. At this stage, porosity increases linearly with increasing clay

content, because the solid sand grains are replaced by porous clay. As a result, the plot

of the volume fraction of clay versus the total porosity of the mixture shows a V-

shaped pattern (Figure 4.5) (Marion et al., 1992; Yin, 1992).

Figure 4.5: Illustrations of sand-shale mixtures, with their porosity and velocity values related to clay content (Modified after Marion et al., 1992). Porosity versus clay content shows a V-shaped trend, where the two end points are the pure sand and pure shale porosity. Selected clay fractions corresponding to the four lithologic states are marked. The states are sand (s), shaly-sand (sh-s), sandy-shale (s-sh), and shale (sh).

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

Volume fraction of clay

Tota

l poro

sity

0.1 0.2 0.3

2

2.5

3

3.5

Total porosity

P-w

ave v

elo

city (

km

/s)

clay

0

0.5

1

S

Sh-sS-sh

Sh

S-sh

Sh-s

S

Sh

sand, shaly sand sandy-shale, shale


Given our sand-clay mixtures, we obtain the corresponding P-wave velocities

using the soft-sand model. This model uses the lower Hashin-Shtrikman bound to

construct velocity-porosity trends for sand mixing with a specified clay volume

(Avseth et al., 2005). The plot of P-wave velocity versus porosity, as clay fractions in

the mixture increase from 0 to 1, shows an inverted-V trend (Figure 4.5). In order to

explore various saturation scenarios, Gassmann’s equation (Gassmann, 1951) is used

to compute properties of rocks saturated with pore-fluid mixtures of water and oil.

Water saturation values (Sw) for each lithologic state are varied depending on case

studies. The density assigned to each layer is simply a weighted average of the

densities of all components in that particular layer. With the rock-physics models,

velocity and density values are now assigned to each sedimentary layer in the

sequences. Layer properties (i.e., velocity, density and thickness) are then input into

the Kennett algorithm (Kennett, 1983) to simulate full-waveform, normally-incident,

reflected seismograms using a zero-phase Ricker wavelet with a central frequency of

30 Hz. The simulations are performed for all reverberations.

4.3.3 Seismic attributes

In our study of seismic signatures for thin sand-shale sequences, we extract two

main types of attributes from seismograms and relate these attributes to the reservoir

properties of interest (i.e., net-to-gross ratios, saturations, and stacking patterns). The

two attributes introduced in this chapter are wavelet-transform based attribute and

feature-extraction based attributes.

4.3.3.1 Wavelet-transform-based analysis

Wavelet transform decomposes a time series signal into a set of scaled and

translated versions of a selected wavelet function. The transform has been used, for

example, to study fractal behavior of seismic data and well logs to characterize

lithofacies (e.g., Álvarez et al., 2003; López and Aldana, 2007). By using well logs,

López and Aldana (2007) showed a possible relation of lithofacies and parameters

including fractal dimension, intercept and slope obtained from linear fits to log-log

plots of variance of wavelet-transform coefficients versus scale.


Using a complex Gaussian wavelet, we perform the transform on simulated

seismograms yielding wavelet-transform coefficients at various scales. We calculate

variance of the modulus of these coefficients for every scale and make a log-log

(base2) plot of the variance versus scale (Figure 4.6). Then, we extract the slope and

intercept of a linear fit as statistical attributes for each realization of the sand-shale

sequences. Wavelet-transform attributes shown in this work are derived using complex

Gaussian wavelets. Note that we have also tried a few other wavelets and found that

behavior of log-log plots between wavelet-coefficient modulus and scale changes

depending on the wavelets and the range of scale. The intercept attribute from this

wavelet-transform analysis relates to the total variance of the seismogram.

Figure 4.6: Wavelet-transform analysis for extracting attributes from a seismogram.

(Left) modulus of wavelet-transform coefficients for the seismogram. The red arrow indicates the modulus along a particular scale. (Right) a log-log plot of scale versus variance of modulus of wavelet-transform coefficients. The plot is shown in open circles which are fit by a straight line. The slope and intercept of this line are used as seismic attributes.

4.3.3.2 Multidimensional scaling and Kernel principal component analysis

Even though the thin sand-shale layers are below seismic resolution, seismograms

from sequences with similar underlying geology and properties potentially share more

statistical similarities than those from sequences with different properties.

Seismograms of length n can be considered as n-dimensional vectors in the space ℝ𝑛 .

To compare the simulated seismograms, we obtain new representations of these

seismograms in a lower dimensional space using two feature extraction techniques:

multidimensional scaling (MDS) and kernel principal component analysis (KPCA).

0 1 2 3 4 5-24

-22

-20

-18

-16

-14

-12

-10

-8

5 10 15 20 25 30

0.05

0.1

0.15

0.2Log

2(v

aria

nce)

Log2(scale)scale-0.1 0 0.1

0

500

1000

1500

2000

2500

Complex Gaussian

Wavelet

0 1 2 3 4 5-24

-22

-20

-18

-16

-14

-12

-10

-8

5 10 15 20 25 30

0.05

0.1

0.15

0.2Log

2(v

aria

nce)


0

500

1000

1500

2000

2500

Complex Gaussian

Wavelet

0 1 2 3 4 5-24

-22

-20

-18

-16

-14

-12

-10

-8

5 10 15 20 25 30

0.05

0.1

0.15

0.2Log

2(v

aria

nce)


0

500

1000

1500

2000

2500

Complex Gaussian

Wavelet


These new representations are used as attributes, which are then related to reservoir

properties.

The first technique used for extracting attributes is multidimensional scaling. To

compare p seismograms of length n using MDS, we first construct a p-by-p

dissimilarity matrix, whose element at the ith

row and jth

column (𝛿ij) is a dissimilarity

measurement between the ith

and jth

seismograms. The dissimilarity values can be

assigned from subjective judgment or objective measure (Cox and Cox, 2001). For

seismogram comparison, dissimilarity need not always be the Euclidean distance

between any two seismograms; we can define dissimilarity in many different ways.

For example, we apply wavelet transform to seismograms and obtain wavelet

coefficients for specified scales. Dissimilarity can be defined as pairwise Euclidean

distance between their modulus of wavelet coefficients at selected scales (Figure 4.7).

Figure 4.7: Modulus of wavelet coefficients (shown in the middle two boxes) of two

seismogram segments. Dissimilarity between these seismograms is defined as distance between their modulus maps.

To obtain attributes, this dissimilarity matrix is then input into an MDS algorithm

which transforms the matrix into points in lower-dimensional Euclidean space. Points

corresponding to their original objects (i.e., seismograms) are configured so that the

inter-point Euclidean distances (dij) match the input dissimilarity values (𝛿ij) as much

as possible. MDS is classified into various categories: classical (metric), metric, and

non-metric MDS. While classical MDS solves an eigendecomposition problem for

coordinates of points in lower-dimensional space, both metric and non-metric MDS

use iterative optimization techniques. Metric MDS minimizes difference between dij

and f(𝛿ij), where f is a continuous monotonic function. Examples of such functions

include identity transformation, logarithmic transformation, etc. (Cox and Cox, 2001;


Zhang, 2008). Non-metric MDS aims to find a configuration of points so that the rank

order of the dissimilarity matrix is preserved (Cox and Cox, 2001; Webb, 2002). The

main result of MDS algorithms is the relative positions of points in lower-dimensional

space; thus, absolute positions do not matter (Scheidt and Caers, 2009). Note that

classical MDS with Euclidean distances is equivalent to principal component analysis

(PCA) (Cox and Cox, 2001; Williams, 2002). We use these new coordinates obtained

by MDS as seismic attributes for reservoir characterization.

The other technique used for extracting seismic attributes is KPCA, which also

provides new representations of seismograms in lower-dimensional space. We first

briefly review the concept of linear principal component analysis (PCA). PCA is

commonly used for dimensionality reduction and linear feature extraction. This

method performs eigenvalue decomposition of a data covariance matrix to compute a

set of orthogonal directions (principal components) which capture maximum

variability of the data. The projection of each data point onto selected principal

components creates a new representation of each point in a low-dimensional space.

While PCA does not work well with non-linear features, kernel principal component

analysis (KPCA), introduced by Schӧlkopf et al. (1997), provides an alternative

nonlinear PCA method. The idea behind KPCA is to map input data nonlinearly into

some high dimensional feature space, in which the data become linearly separable;

then linear PCA is performed (Scheidt and Caers, 2009). Instead of solving an

eigenvalue problem of the covariance matrix, linear PCA in the feature space is done

by solving an equivalent system involving dot products in the feature space. These dot

products can be obtained using kernel functions without an explicit mapping from the

original space to the feature space. Results from KPCA are the projections of input

data onto the principal components.

To obtain attributes, we apply Gaussian, dynamic similarity, inverse multi-quadric,

polynomial and linear kernels (Table 4.1) to the input seismograms and obtain kernel

matrices. Then after centering these matrices, we compute eigenvalues and

eigenvectors of the matrices, and obtain coordinates of points, each of which

corresponds to a seismogram that is projected onto selected principal components.


Hereafter we refer to KPCA with Gaussian, dynamic similarity, inverse multi-quadric,

polynomial, or linear kernels as Gaussian, dynamic similarity, inverse multi-quadric,

polynomial, or linear KPCA, respectively. Note that performing KPCA with a linear

kernel is equivalent to performing linear PCA (e.g., Tenenbaum et al., 2000; Rathi et

al., 2006; van der Maaten, et al., 2009). Likewise, performing KPCA with kernel

functions that depend only on 𝑥𝑖 − 𝑥𝑗 can be interpreted as solving a metric MDS as

an eigenproblem (Williams, 2002). Parameters such as 𝜎, c, or d that are used for our

study will be shown in the note column of result tables for each case scenario.

Table 4.1: Kernel functions used for extracting seismic attributes. Note that xi is the ith seismogram.

Kernel matrix Note

Gaussian kernel

𝑘 𝑥𝑖 , 𝑥𝑗 = 𝑒− 𝑥𝑖−𝑥𝑗

2

2𝜎2

σ > 0; a suggested value of 𝜎 is

about 20% of the range of the

distances (Scheidt and Caers,

2009)

𝑥𝑖 − 𝑥𝑗 is the Euclidean

distance between the ith

and jth

seismograms

Dynamic

similarity kernel

(Yan et al., 2006)

𝑘 𝑥𝑖 , 𝑥𝑗 = 𝑒−𝐷𝑃𝐹2 𝑥𝑖 ,𝑥𝑗

𝜎2 Dynamic partial function (DPF)

by Li et al. (2003)

(see below for more details)

Inverse multi-

quadric kernel 𝑘 𝑥𝑖 , 𝑥𝑗 =

1

𝑐2 + 𝑥𝑖 − 𝑥𝑗 2

𝑥𝑖 − 𝑥𝑗 is the Euclidean

distance between the ith

and jth

seismograms

Polynomial kernel 𝑘 𝑥𝑖 , 𝑥𝑗 = 𝑥𝑖 ∙ 𝑥𝑗 + 𝑐 𝑑

c ≥ 0 and d is the degree of the

polynomial (d ∈ ℕ)

Linear kernel 𝑘 𝑥𝑖 , 𝑥𝑗 = 𝑥𝑖 ∙ 𝑥𝑗 equivalent to performing PCA

The Euclidean distance ( 𝑥𝑖 − 𝑥𝑗 ) used in both the Gaussian kernel and the

inverse multi-quadric kernel is a special case of a more general Minkowski metric,

which is defined as


𝑑(𝐴,𝐵) = ak − bk r

n

k=1

1r

,

(4.1)

where d is the distance or similarity measure between objects A and B, each of which

is represented by a vector of length n (e.g., A = [a1, a2, a3, …, an]). The Euclidean

distance is the Minkowski metric with the parameter r equal to 2. Using the

Minkowski metric to measure similarity between two objects is to compare each and

every element in the objects.

Based on the idea similar to human perceptual similarity in cognitive science, Li et

al. (2003) defined a distance function called dynamic partial function (DPF) as

𝐷𝑃𝐹(𝐴,𝐵) = δkr

δk ϵ∆m

1r

,

(4.2)

where δk = |ak – bk| for k = 1,…, n and Δm is the set of the m smallest values of δ

from {δ1,…, δn}. Thus parameter m ranges from 1 to n. If m = n, Equation 4.2 is

equivalent to the Minkowski metric.

Using DPF, Yan et al. (2006) proposed a new kernel for image classification and

showed that this kernel yields higher classification accuracy than other kernels,

including Gaussian and polynomial kernels. As an extension of the Gaussian kernel,

the new kernel is defined as

𝑘 𝑥𝑖 , 𝑥𝑗 = 𝑒−𝐷𝑃𝐹 2(𝑥𝑖 ,𝑥𝑗 )

𝜎2 .

(4.3)


4.4 Seismic signatures for 1-D Synthetic example

In the previous section, we have described how we generate seismic responses and

extract seismic attributes for thin sand-shale sequences. In this section, we illustrate 1-

D synthetic examples where seismic attributes are related to net-to-gross ratios,

saturations, and various sedimentary stacking patterns.

4.4.1 Model setup

We characterize the lithology into four states: sand, shaly sand, sandy shale, and

shale. Then we use a set of 4x4 fixed-sampling transition matrices, in particular the

aggrading type, to generate multiple realizations of lithology arrangements by keeping

the layer thickness fixed at 0.5 m. Each sand-shale sequence has a total thickness of

100 m, and the sequence is embedded between two 200-m thick shale layers. The four

lithologic states are represented by mixtures of sand and clay with clay fraction values

of 0.1, 0.3, 0.6 and 0.9 in the order of increasing shaliness. Using clay-fraction

information, porosity, velocity and density values are computed following methods

provided in Section 4.3.2. We introduce uncertainties by assuming each lithologic

state having a distribution of velocities with a mean equal to the calculated velocities

and standard deviations (v) of 0.1 and 0.2 km/s. Full-waveform, normally-incident,

reflected seismograms are simulated using the Kennett algorithm (Kennett, 1983) with

a zero-phase Ricker wavelet. The central frequency is 30 Hz and the simulations are

performed for all reverberations. A summary workflow from a transition matrix to

multiple realizations of seismograms is shown in Figure 4.8.


Figure 4.8: Workflow for generating seismic responses of multiple realizations of thin

sand-shale sequences. The four lithologic states in the transition matrix are sand (s), shaly-sand (sh-s), sandy-shale (s-sh), and shale (sh).

We explore three main scenarios to investigate the effect of net-to-gross ratios,

saturations, and stacking patterns on seismic signatures of thin sand-shale sequences.

We define the net-to-gross ratio as the proportion of the first lithologic state (sand) in

the entire sequence.

4.4.2 Scenario 1: Effect of net-to-gross ratios

We study three transition matrices which generate aggrading-type sequences

(Table 4.2). Each transition matrix has a different limiting distribution (𝜋), a row

vector of fixed probabilities representing proportions of each state in the long-term

behavior. Thus in long sequences (i.e., large numbers of layers per sequence) the

proportions of sand in the sequences should converge to a distinct value: 𝜋sand. In our

simulations, all sequences consist of 200 sedimentary layers. We simulate 200

sequence realizations from each transition matrix and compute the true net-to-gross

ratios of these sequences. Figure 4.9 shows that the computed net-to-gross values

scatter around 𝜋sand and that their distributions get narrower when the values of

parameter k in the transition matrices (Table 4.2) approach 1. Since the probabilities of

going from sand (or shaly-sand) to shale and the probabilities of going from sandy-

shale (or shale) to sand are equal to k, as k gets larger the simulations will eventually

turn into sequences with only sand and shale in alternating layers; as a result, the

Sh S

Sh S

Sh S

Transition matrix Thin sand-shale sequenceswith petrophysical properties

Synthetic seismograms


distributions of net-to-gross ratios become very narrow around the value of 0.5. Water

saturation (Sw) is set to be the same in all simulations: Sw=0.1 for sand layers and

Sw=1 for the other lithology states. For the effect of net-to-gross ratios, we run

simulations for velocity distributions with standard deviations (v) of 0.1 and 0.2 km/s.

Table 4.2: Form of transition matrices for generating sequences used in investigating net-to-gross effects on seismic signatures. Values of parameter k range from 0.45 to 0.95. The four lithologic states are sand (s), shaly-sand (sh-s), sandy-shale (s-sh), and shale (sh).

Sand Shaly-sand Sandy-shale Shale

Sand (1−𝑘)

3

(1−𝑘)

3

(1−𝑘)

3 𝑘

Shaly-sand (1−𝑘)

3

(1−𝑘)

3

(1−𝑘)

3 𝑘

Sandy-shale 𝑘 (1−𝑘)

3

(1−𝑘)

3

(1−𝑘)

3

Shale 𝑘 (1−𝑘)

3

(1−𝑘)

3

(1−𝑘)

3

Figure 4.9: Distributions of net-to-gross ratios from sequence realizations generated

using the transition matrices in Table 4.2. Red lines indicate values of limiting sand distributions. From top left to bottom right, values of the parameter k defined in the transition matrices are 0.45, 0.55, 0.65, 0.75, 0.85 and 0.95, respectively.

Using the wavelet transform analysis, the values of slopes and intercepts decrease

when the proportions of sand in the total sequence (𝜋sand) increase. Cross-plots and

probability distributions of both attributes, color-coded by 𝜋sand for each transition

0.2 0.3 0.4 0.50

20

40

60

Net-to-gross ratios

0.2 0.3 0.4 0.50

20

40

60

Net-to-gross ratios

0.2 0.3 0.4 0.50

20

40

60

Net-to-gross ratios

0.2 0.3 0.4 0.50

20

40

60

Net-to-gross ratios0.2 0.3 0.4 0.50

20

40

60

Net-to-gross ratios

0.2 0.3 0.4 0.50

20

40

60

Net-to-gross ratios

True net-

to-gross

ratios

sand


matrix, are shown in Figure 4.10 and Figure 4.11, respectively. For a larger standard

deviation (v) in velocity distributions, both slope and intercept values for different

𝜋sand are less spread out. One explanation for the smaller spread could be that using

larger v results in larger overlaps in velocities drawn for each state. Thus, even when

𝜋sand increases (i.e., when there are more sand layers in the sequences), there is not

much change in the overall velocities drawn for each sequence because all four

lithology states yield velocity values that are very close. Larger overlaps in the slope-

intercept attributes can lead to larger uncertainties in property estimations.

Figure 4.10: Slope and intercept of wavelet transforms of seismic responses for 2

different standard deviations of velocity distributions: (left) v = 0.1 km/s and (right) v = 0.2 km/s. Points correspond to 6 sets of 200 realizations generated from different transition matrices with various net-to-gross ratios (𝝅sand). Sw is 0.1 for sand layers and 1 for the others.

Figure 4.11: Probability distributions of slope and intercept values corresponding to

data points in Figure 4.10. (Left column) v = 0.1 km/s and (right column) v = 0.2 km/s. The direction of increasing net-to-gross ratios (𝝅sand) is to the left for all plots.

2.9 2.95 3-27

-26

-25

-24

-23

-22

-21

-20

Slope

Inte

rce

pt

2.9 2.95 3-27

-26

-25

-24

-23

-22

-21

-20

Slope

Inte

rce

pt

0.32

0.35

0.38

0.42

0.45

0.48

directionof increasing sand fraction

directionof increasing sand fraction

(sand)sand fraction

2.88 2.92 2.96 30

20

40

60

80

Slope

2.88 2.92 2.96 30

20

40

60

80

Slope

-28 -26 -24 -22 -200

0.2

0.4

0.6

0.8

Intercept

-28 -26 -24 -22 -200

0.2

0.4

0.6

0.8

Intercept

0.32

0.35

0.38

0.42

0.45

0.48

Sand fraction (sand)

direction of

increasing

sand fractions


Next, we apply wavelet transform, MDS, and KPCA algorithms to a subset of 600

seismograms corresponding to sequences generated from transition matrices (Table

4.2) with values of k equal to 0.45, 0.75, and 0.95, and v = 0.1 km/s. Then, each

algorithm outputs 600 seismic-attribute vectors as new representations for the

seismograms. For example, using wavelet-transform-based analysis, the attribute

vector consists of two elements, the slope and intercept. When using MDS algorithms,

the attribute vector simply contains coordinates of a point (i.e., seismogram) in the

lower-dimensional space.

To compare how well the attributes characterize net-to-gross ratios, we count the

number of times the linear discriminant analysis algorithm (LDA) successfully sorts

the attribute vectors (i.e., the seismograms) into their corresponding net-to-gross

classes, which are defined to be less than 0.38, between 0.38 and 0.465, and greater

than 0.465. The classification success rate is estimated using a stratified 10-fold cross

validation. The validation process first partitions the attributes into ten approximately

equal-sized subsamples (i.e., folds) such that the proportions of each class in the

subsamples are approximately equal to those in the whole sample (i.e., stratified).

Then, a subsample is held as a validation set, while treating the rest as a training set in

the classification step, where a success rate is obtained. This step is repeated ten times

so that each subsample is used once as a validation set. The overall success rate is

simply an average of the ten classification experiments. To improve the accuracy of

the estimates, this stratified 10-fold cross validation is repeated ten times. The ten

success rates are then averaged to obtain the final success rate (Witten et al., 2011).

Results from MDS and KPCA algorithms are shown in Figure 4.12 – Figure 4.17

as cross-plots between just two attributes, either the first two coordinates or the first

two principal components. The classification results are summarized in Table 4.3.

Note that three numbers are listed under the success rate column, representing the

success rate when classifying samples into classes using only the first coordinate, the

second coordinate, and both coordinates, respectively.

The metric and non-metric MDS yield better classification success rates (74% and

73% respectively) than the classical MDS (56%) when considering results in the polar


coordinate system (i.e., theta and rho values; Figure 4.12 – Figure 4.14, lower right

corner). Note that in the classical MDS, the first two coordinates correspond to the two

largest eigenvalues. Among 2-attribute results, there is not much difference in success

rates among the results from the wavelet transform, metric, and non-metric MDS.

However, the success rate of each individual attribute clearly shows which attribute is

more sensitive to net-to-gross changes. For example, the intercept attribute gives a

success rate of 73% while the slope yields a success rate of 55%. The first principal

components of the Gaussian, dynamic similarity, and inverse multi-quadric KPCA

effectively capture variations in net-to-gross ratios. Using the first component alone,

classification success rates of KPCA can be as high as 85%. When using only the first

two principal components, the dynamic similarity kernel best differentiates the three

net-to-gross classes while both the polynomial and the linear KPCA poorly separate

the net-to-gross classes (Figure 4.15 – Figure 4.18).

As previously noted, since both the classical MDS with Euclidean distances and

the linear KPCA are equivalent to linear PCA, their results are similar (Figure 4.12

(upper left corner) and Figure 4.19 (right)). When plotting only the first two

coordinates or principal components, both the low and high net-to-gross classes are

shown in circular structures which are largely overlapped. As a result, both the

classical MDS and the linear KPCA (i.e., linear PCA) yield only about 50%

classification success rate. These circular structures are also observed in the results

from metric and non-metric MDS (top left corner of Figure 4.13 and Figure 4.14).

Thus, both the linear KPCA (i.e., PCA) and the Euclidean MDS fail to detect the

intrinsic circular structures in the data (Tenenbaum et al., 2000).

The inter-point distances (dij) of MDS results do not reproduce the original

dissimilarity (𝛿ij) of seismograms well if only two coordinates (i.e., two-dimensional

space) are considered, as evidenced by the correlation coefficients between the

distances and dissimilarity for all three MDS algorithms being less than 0.9 (Figure

4.12 – Figure 4.14, lower left corner). However, the correlation increases as the

number of coordinates increases. The classical MDS results have ten large eigenvalues,


which further indicates that more coordinates may need to be included in the results to

better reproduce the dissimilarity matrix and improve classification performance.

The overall classification success rate increases as the number of coordinates or

principal components included as attributes increases (MDS in Figure 4.20 and KPCA

in Figure 4.21). Among the selected kernels for KPCA, the classification success rate

of the dynamic similarity KPCA reaches 90% with just only two principal components

included; whereas, the success rates of the polynomial and the linear KPCA (i.e., PCA)

show a gradual increase over a larger number of components. The linear KPCA yields

98% success rate when using 20 principal components. Note that to achieve this

success rate of 98%, the number of components to be included as attributes may be

less than 20 because we include components in order of decreasing eigenvalues, not

their contributions to the success rate. If only those components that significantly

improve the success rate are selected and used, the rate can rapidly increase and reach

the desired level.

Using more coordinates or components as attributes involves higher-dimensional

vectors; if these attributes are later used in other tools such as neural networks, a larger

number of attributes leads to more computation and more complex models. Moreover,

it is more difficult to visualize the multivariate results. One way to visualize high-

dimensional data is to use a parallel coordinate plot, where an n-dimensional vector is

represented by a polyline connecting the value of each element in the vector. The

horizontal and vertical axes of the plot are the non-negative integer line (e.g., from 1

to n) and the real number line, respectively. The polylines, each of which corresponds

to one attribute vector (i.e., seismogram), can be color-coded by their net-to-gross

classes. Instead of plotting individual polylines, the specified quantiles of each

element can be plotted to better see the distributions of each class at a particular

element. Figure 4.22 is an example of the parallel coordinate plots, using the first five

components of the results from the Gaussian KPCA. While the large overlaps prevent

the polylines from providing much information on how each component is distributed

in each class, the plot using quantile values clearly shows that the three net-to-gross

classes have quite distinct distributions of the first component. This observation


supports the above mentioned KPCA (Gaussian kernel) result, which achieves a

classification success rate of 80% by using only the first component. Note that the

0.45- and 0.55- quantiles are selected only to get a better picture of how each

component may contribute to the classification results. However, using 0.25- and 0.75-

quantiles gives a better statistical summary of each component.

Figure 4.12: Classical MDS results for studying effect of net-to-gross ratios on

seismic signature. (Top left) Dissimilarity matrix showing pairwise Euclidean distances between any two seismograms from realizations with standard deviations of velocities equal to 0.1 km/s. (Bottom left) Correlation coefficient between dissimilarity and distance (between points in new coordinates resulting from classical MDS) versus numbers of included coordinates. (Top right) Classical MDS results with the first two coordinates. Each point represents one seismogram, color-coded by its true net-to-gross value from its corresponding sand-shale sequence. (Bottom right) Classical MDS results with the first two coordinates converted into polar coordinates.

Seismogram #

Seis

mogra

m #

200 400 600

200

400

600

Distance

0

1

2

3

4

-2 -1 0 1 2-3

-2

-1

0

1

2

First coordinate

Second c

oord

inate

0.25

0.3

0.35

0.4

0.45

0.5

Net-to-gross ratios

-4 -2 0 2 40

0.5

1

1.5

2

2.5

Rho

Theta

0.25

0.3

0.35

0.4

0.45

0.5Net-to-gross ratios

5 10 150.5

0.6

0.7

0.8

0.9

1

# of coordinates

Corr

ela

tion c

oeff

icie

nt

betw

een

dis

sim

ilarity

and d

ista

nce

-2 -1 0 1 2

-2

-1

0

1

2

3

First coordinate

Second c

oord

inate

Net-to-gross ratios

0.25

0.3

0.35

0.4

0.45

0.5

-4 -2 0 2 4

0

0.5

1

1.5

2

2.5

Theta

Rho

Net-to-gross ratios

0.25

0.3

0.35

0.4

0.45

0.5


Figure 4.13: Metric MDS results for studying effect of net-to-gross ratios on seismic

signature. (Top left) Dissimilarity matrix showing pairwise (Euclidean) distances between any two seismograms from realizations with standard deviations of velocities equal to 0.1 km/s. (Bottom left) Correlation coefficient between dissimilarity and distance (between points in new coordinates resulting from metric MDS) versus numbers of included coordinates. (Top right) Metric MDS results with the first two coordinates. Each point represents one seismogram, color-coded by its true net-to-gross value from its corresponding sand-shale sequence. (Bottom right) Metric MDS results with the first two coordinates converted into polar coordinates.

Seismogram #

Seis

mogra

m #

200 400 600

200

400

600

Distance

0

1

2

3

4

-2 -1 0 1 2-3

-2

-1

0

1

2

3

Second c

oord

inate

First coordinate

0.25

0.3

0.35

0.4

0.45


2 4 6 80.7

0.8

0.9

1

# of coordinates C

orr

ela

tion c

oeff

icie

nt

betw

een

dis

sim

ilarity

and d

ista

nce

-4 -2 0 2 40

0.5

1

1.5

2

2.5

3

Theta

Rho

0.25

0.3

0.35

0.4

0.45

0.5

Net-to-gross ratios

-2 0 2-3

-2

-1

0

1

2

3

First coordinate

Second c

oord

inate

Net-to-gross ratios

0.25

0.3

0.35

0.4

0.45

0.5

-4 -2 0 2 40

0.5

1

1.5

2

2.5

3

Theta

Rho

Net-to-gross ratios

0.25

0.3

0.35

0.4

0.45

0.5


Figure 4.14: Non-metric MDS results for studying effect of net-to-gross ratios on

seismic signature. (Top left) Dissimilarity matrix: pairwise (Euclidean) distances between any two seismograms from realizations with standard deviations of velocities equal to 0.1 km/s. (Bottom left) Correlation coefficient between dissimilarity and distance (between points in new coordinates resulting from non-metric MDS) versus numbers of included coordinates. (Top right) Non-metric-MDS results with the first two coordinates. Each point represents one seismogram, color-coded by its true net-to-gross value from its corresponding sand-shale sequence. (Bottom right) Non-metric MDS results with the first two coordinates converted into polar coordinates.

Seismogram #

Seis

mogra

m #

200 400 600

200

400

600

Distance

0

1

2

3

4

-2 -1 0 1 2-2

-1

0

1

2

3

Second c

oord

inate

First coordinate

0.25

0.3

0.35

0.4

0.45


2 4 6 80.7

0.8

0.9

1

# of coordinatesCorr

ela

tion c

oeff

icie

nt

betw

een

dis

sim

ilarity

and d

ista

nce

-4 -2 0 2 40

0.5

1

1.5

2

2.5

3

Theta

Rho

0.25

0.3

0.35

0.4

0.45

0.5

Net-to-gross ratios

-2 -1 0 1 2-2

-1

0

1

2

3

First coordinate

Second c

oord

inate

Net-to-gross ratios

0.25

0.3

0.35

0.4

0.45

0.5

-4 -2 0 2 4

0

0.5

1

1.5

2

2.5

3

Theta

Rho

Net-to-gross ratios

0.25

0.3

0.35

0.4

0.45

0.5


Figure 4.15: Gaussian KPCA results for studying effect of net-to-gross ratios on

seismic signature. (Left) Kernel matrix using a Gaussian kernel. Each element in the matrix (K(xi,xj)) corresponds to the Gaussian kernel function evaluated using a pair of seismograms (xi,xj) from realizations with standard deviations of velocities equal to 0.1 km/s. (Right) Projections of seismograms, which correspond to sequences with different net-to-gross ratios, onto the first two principal components from the Gaussian KPCA. Each point is color-coded by the net-to-gross ratio from its corresponding sand-shale sequence.

Figure 4.16: Dynamic similarity KPCA results for studying effect of net-to-gross

ratios on seismic signature. (Left) Kernel matrix using a dynamic similarity kernel. Each element in the matrix (K(xi,xj)) corresponds to the dynamic similarity kernel function evaluated using a pair of seismograms (xi,xj) from realizations with standard deviations of velocities equal to 0.1 km/s. (Right) Projections of seismograms, which correspond to sequences with different net-to-gross ratios, onto the first two principal components from the dynamic similarity KPCA. Each point is color-coded by the net-to-gross ratio from its corresponding sand-shale sequence.

Seismogram #

Seis

mogra

m #

200 400 600

100

200

300

400

500

600

K(xi,xj)

0.2

0.4

0.6

0.8

1

-0.4 -0.2 0 0.2 0.4 0.6-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

First principal component

Second p

rincip

al com

ponent

0.25

0.3

0.35

0.4

0.45


Seismogram #

Seis

mogra

m #

K (xi,xj)

200 400 600

100

200

300

400

500

600

-0.5 0 0.5

-0.4

-0.2

0

0.2

0.4

0.6


Second p

rincip

al com

ponent

0.2

0.4

0.6

0.8

1

0.25

0.3

0.35

0.4

0.45



Figure 4.17: Inverse multi-quadric KPCA results for studying effect of net-to-gross

ratios on seismic signature. (Left) Kernel matrix using an inverse multi-quadric kernel. Each element in the matrix (K(xi,xj)) corresponds to the inverse multi-quadric kernel function evaluated using a pair of seismograms (xi,xj) from realizations with standard deviations of velocities equal to 0.1 km/s. (Right) Projections of seismograms, which correspond to sequences with different net-to-gross ratios, onto the first two principal components from the inverse multi-quadric KPCA. Each point is color-coded by the net-to-gross ratio from its corresponding sand-shale sequence

Figure 4.18: Polynomial KPCA results for studying effect of net-to-gross ratios on

seismic signature. (Left) Kernel matrix using a polynomial kernel. Each element in the matrix (K(xi,xj)) corresponds to the polynomial kernel function evaluated using a pair of seismograms (xi,xj) from realizations with standard deviations of velocities equal to 0.1 km/s. (Right) Projections of seismograms, which correspond to sequences with different net-to-gross ratios, onto the first two principal components from the polynomial KPCA. Each point is color-coded by the net-to-gross ratio from its corresponding sand-shale sequence.

Seismogram #

Seis

mogra

m #

200 400 600

200

400

600

-0.5 0 0.5-0.5

0

0.5


Second p

rincip

al com

ponent

0

0.5

1

0.25

0.3

0.35

0.4

0.45

0.5

K(xi,xj)

Net-to-gross ratios

Seismogram #

Seis

mogra

m #

K (xi,xj)

200 400 600

200

400

600

-2 0 2 4-4

-3

-2

-1

0

1

2


Second p

rincip

al com

ponent

2

4

6

8

10

0.25

0.3

0.35

0.4

0.45

0.5

Net-to-gross ratios


Figure 4.19: Linear KPCA results for studying effect of net-to-gross ratios on seismic

signature. (Left) Kernel matrix using a linear kernel. Each element in the matrix (K(xi,xj)) corresponds to the linear kernel function evaluated using a pair of seismograms (xi,xj) from realizations with standard deviations of velocities equal to 0.1 km/s. (Right) Projections of seismograms, which correspond to sequences with different net-to-gross ratios, onto the first two principal components from the linear KPCA. Each point is color-coded by the net-to-gross ratio from its corresponding sand-shale sequence.


coordinates included as net-to-gross attributes increases. Results from three MDS methods are shown.

seismogram #

seis

mogra

m #

200 400 600

100

200

300

400

500

600-2

0

2

4

6

8

-2 -1 0 1 2-3

-2

-1

0

1

2


Second p

rincip

al com

ponent

0.25

0.3

0.35

0.4

0.45

0.5

K(xi,xj)

Net-to-gross ratios

2 4 6 8 10 12

0.4

0.5

0.6

0.7

0.8

0.9

1

# of coordinates included

Cla

ssific

ation s

uccess r

ate

classical MDS

Metric MDS

Non-metric MDS


Table 4.3: Summary of the methods used to compute seismic attributes for net-to-

gross estimation. Choices of parameters for each method are also included. Performance of each method is shown as a success rate in classifying a data point into three net-to-gross classes: <0.38, 0.38-0.465, and >0.465

Method Compared

objects

Distance/ or

Kernel

Classification

success rate

(using 1st, 2nd,

both

coordinates/com

ponents)

Notes

Wavelet-

transform

– – 0.55, 0.73, 0.72

Slope/intercept/both

MDS

(classical)

Seismograms Euclidean 0.37, 0.49, 0.53

0.50, 0.49, 0.55

(theta/rho/both)

> ten large eigenvalues;

the algorithm cannot

reproduce dissimilarity

well by using only the

first two coordinates

MDS

(metric)

Seismograms Euclidean 0.38,0.43,0.45

0.45, 0.69, 0.73

(theta/rho/both)

Correlation coefficient

between distance and

dissimilarity is 0.83 when

using the first two

coordinates; the value

becomes greater than

0.95 when there are more

than 4 coordinates.

MDS

(non-

metric)

Seismograms Euclidean 0.38,0.43,0.45

0.47, 0.68, 0.72

(theta/rho/both)

Correlation coefficient

between distance and

dissimilarity is 0.88 when

using the first two

coordinates; the value

becomes greater than

0.95 when there are more

than 3 coordinates.

KPCA Seismograms Gaussian 0.81, 0.35, 0.81 σ2 = 0.5

KPCA Seismograms Dynamic

similarity

0.85, 0.58, 0.90 σ = 0.1265, m = 80% of

total seismogram length,

r = 2

KPCA Seismograms Inverse

multi-

quadric

0.79, 0.35, 0.79 c2 = 1

KPCA Seismograms Polynomial 0.50, 0.51,

0.59

c = 1, d = 2

KPCA Seismograms Linear 0.37, 0.49, 0.53 equivalent to PCA


Figure 4.21: Change in KPCA classification success rate when the number of

principal components included as net-to-gross attributes increases.

Figure 4.22: Parallel coordinate plot. (Left) Polylines of the first five principal

components from the Gaussian KPCA. Each line is color-coded by its corresponding net-to-gross class. Three classes are <0.38, 0.38-0.465, and >0.465. (Right) The solid lines are the median (i.e., the 0.5-quantile) of the component values. The dash lines surrounding the median are the 0.45- and 0.55- quantiles.

4.4.3 Scenario 2: Effect of saturation

We use sequences generated from a similar transition matrix, but with varying

saturation values. In this scenario, we show only simulations for velocity distributions

with standard deviation of 0.1 km/s. We use Gassmann’s equation to substitute

mixtures of water and oil with the desired saturations (Sw = 0.1, 0.5, and 1) into only

the sand layers in the sequences. All other lithologies always have Sw equal to 1.

0 10 20 30 40 50

0.4

0.5

0.6

0.7

0.8

0.9

1

# of principal components

Cla

ssific

ation s

uccess r

ate

Gaussian KPCA

dynamic similarity KPCA

inverse multi-quadric KPCA

polynomial KPCA

linear KPCA

1 2 3 4 5

-0.4

-0.2

0

0.2

0.4

0.6

Coordinate

Coord

inate

Valu

e

1 2 3 4 5

-0.4

-0.2

0

0.2

0.4

0.6

Coordinate

Coord

inate

Valu

e

Class 1

Class 2

Class 3


We consider three matrices (A, B, and C) which have the same limiting

distribution: [0.45 0.05 0.05 0.45] (Figure 4.23). Long sequences generated using the

three matrices should have their net-to-gross ratios within the sequence equal to 0.45;

however, in our simulation the values spread around 0.45. Therefore, to separate the

net-to-gross effect from the saturation effect, we select only those sequences with net-

to-gross ratios ranging from 0.4 – 0.5. Thus, the main difference among sequences

generated from the three transition matrices is how the sand layers are distributed. For

example, sand layers in sequences from matrix C are more clustered and blocky than

those from the other two matrices. This blocky pattern is expected since matrix C has

large probabilities of going to the sand and shale states (pss = psh-s,s = ps-sh ,sh = pshsh =

0.85).

Figure 4.23: Three selected matrices with the same limiting distribution and their

sample sequences. From left to right, sand layers become more blocky (i.e., groups of sand layers become thicker). The lithologic states are sand (s), shaly sand (sh-s) sandy shale (s-sh) and shale (sh). Note that each row of matrix B is equal to a fixed probability vector, and thus the lithologic states generated using this matrix are considered as independent random events (i.e., the current state has no dependency on the previous states).

Results from wavelet-transform, MDS, and KPCA algorithms are color-coded by

their corresponding saturation values (i.e., 0.1, 0.5, and 1) and shown in Figure 4.24 –

Figure 4.30 as cross-plots between just two attributes, either the first two coordinates

Sh S Sh S Sh S

(A) .05 .05 .05 .85

.05 .05 .05 .85

.85 .05 .05 .05

.85 .05 .05 .05

(B) .45 .05 .05 .45

.45 .05 .05 .45

.45 .05 .05 .45

.45 .05 .05 .45

(C) .85 .05 .05 .05

.85 .05 .05 .05

.05 .05 .05 .85

.05 .05 .05 .85

(A) (B) (C)


or the first two components. Results are then used in the same cross-validation scheme

described in Section 4.4.2. The classification success rates are summarized in Table

4.4. Note that since the linear KPCA (i.e., PCA) are equivalent to classical MDS with

Euclidean distances. Thereafter, when Euclidean distances are used, we only show the

results from classical MDS.

For wavelet-transform-based attributes, slope and intercept of sequences obtained

from the same transition matrix do not show any clear trend with increasing water

saturation. Their distributions largely overlap, resulting in classification rates less than

65%. However, for the same water saturation slope and intercepts values increase as

the transition matrices change from A to C, i.e., as the sands in the sequences become

more clustered and blocky (Figure 4.24). Thus, spatial statistics at the sub-seismic

scale also have an influence on seismic signatures of sub-resolution systems.

For the same transition matrix, plots of the first two coordinates from classical

MDS and the two coordinates from metric and non-metric MDS in Figure 4.25 –

Figure 4.27 show similar patterns, resulting in similar classification success rates with

the rates of classical MDS being slightly smaller than metric and non-metric MDS. In

matrix A results, all MDS algorithms effectively separate the sequences which have

sand layers fully saturated with water (Sw = 1) from the other sequences relatively well.

The other two saturation classes are almost inseparable, because their distributions

largely overlap. In matrix B results, the water-saturated sequences are still well

separated by all MDS algorithms; however, in this case the saturation class of 0.1

starts to separate out and surround the saturation class of 0.5 with less overlap than the

results in matrix A. In matrix C, both the saturation classes of 0.1 and 1 surround the

saturation class of 0.5. The overall classification success rates using two-coordinate

MDS results are less than 70%.

As with the net-to-gross case, if only two coordinates are considered, the inter-

point distances of MDS results do not reproduce the original dissimilarity of

seismograms well, with the correlation coefficients between the distances and

dissimilarity for metric and non-metric MDS algorithms being less than 0.9 (Figure

4.33). The correlation increases as the number of coordinates increases, with


correlations of matrix C being the lowest for a given number of coordinates. For

classical MDS results, there are more than two large eigenvalues, especially the

eigenvalues in matrix C whose values are almost twice as large as those in matrix A

and B (Figure 4.32). More coordinates can be included as attributes to better reproduce

the dissimilarity matrix and improve classification performance. When the number of

coordinates increases, the overall classification success rate also increases. For the

same transition matrix and same number of coordinates, the success rates from

classical, metric, and non-metric MDS turn out to be very similar (Figure 4.34).

By using the first two principal components, the dynamic similarity KPCA best

differentiates the three saturation classes for all transition matrices (Figure 4.29). Its

classification success rates are greater than 80%, whereas the other kernels have

poorer performance. From Figure 4.28, the first principal components of the Gaussian

KPCA for both the transition matrices A and B separate the sequences which have

sand layers fully saturated with water (Sw = 1) from the other sequences relatively well.

Matrix C shows a similar result, except that the class which is effectively separated by

using the first component is that with Sw equal to 0.5. For all three transition matrices,

the second component does not contribute much in differentiating the three saturation

classes, and the overall classification success rates of the Gaussian KPCA are only in

the 60-70 percent range.

Both the inverse multi-quadric KPCA in Figure 4.30 and the polynomial KPCA in

Figure 4.31 show patterns quite similar to the Gaussian KPCA, in which the water-

saturated sequences (Sw = 1) seem distinguishable from the other classes by just the

first component for both matrices A and B; however, in matrix C this saturation class

(Sw = 1) is less separable from the rest. The classification success rates using the first

two principal components of the inverse multi-quadric KPCA are generally in the 60-

70 percent range, which is slightly higher than the 50-65 percent range of the

polynomial KPCA.

One way to improve the performance of KPCA attributes is to increase the

numbers of components included as attributes. Figure 4.35 – Figure 4.37 illustrate the

changes in classification success rates with increasing numbers of included


components. Among the selected kernels for KPCA, the classification success rate of

the dynamic similarity KPCA reaches 80% with just only two principal components

included; whereas, the success rates of the classical MDS (i.e., linear KPCA or PCA)

show a gradual increase over a larger number of components. The linear KPCA yields

almost 100% success rate when using 20 or more principal components. Note that the

components are included as attributes in order of decreasing eigenvalues, not their

contributions to the success rate. If only those components that can effectively

distinguish different saturation classes are selected, with a smaller number of

components the rate can sometimes be very close to the success rate obtained when

using a larger numbers of components. A small number of components also make it

possible to visualize the results. Selecting those components with higher

discriminating power can be done by using parallel coordinate plots (Figure 4.38 –

Figure 4.39). For example, from the parallel coordinate plots of the first ten

components of Gaussian KPCA results for matrix B, we select the first and sixth

components and they yield a classification success rate of 80%, while the success rate

when using all ten components is 82% (Figure 4.38). As another example, the success

rate when using the first and sixth components of inverse multi-quadric KPCA results

for matrix C is 73%, while the success rate when using the first ten components is 82%

(Figure 4.39). Here our selection process is heuristic-based, selecting any two

components that exhibit large separation among the three classes. However, this

selection scheme does not necessarily work in all cases, especially when several

components behave similarly. Therefore, a systematic comparison of success rates

may be needed in order to select the best possible combination of components.


Figure 4.24: Slope and intercept attributes for varying water saturation within the

sand layers. Results in each plot are from sequences generated by using the transition matrix shown in the lower-right corner of each plot. From left to right, the transition matrices correspond to A, B, and C in Figure 4.23.


different water-saturation values, onto the first two coordinates from classical MDS. Each point is color-coded by the water-saturation value of the sand layers. From left to right, subplots correspond to transition matrices A, B, and C, respectively.


different water-saturation values, onto the two coordinates from metric MDS. Each point is color-coded by the water-saturation value of the sand layers. From left to right, subplots correspond to transition matrices A, B, and C, respectively.

2.9 2.92 2.94 2.96 2.98 3-27

-26

-25

-24

-23

-22

-21

-20

Slope

Inte

rcept

2.9 2.92 2.94 2.96 2.98 3-27

-26

-25

-24

-23

-22

-21

-20

Slope

Inte

rcept

2.9 2.92 2.94 2.96 2.98 3-27

-26

-25

-24

-23

-22

-21

-20

Slope

Inte

rcept

sw=0.1

sw=0.5

sw=1

.05 .05 .05 .85

.05 .05 .05 .85

.85 .05 .05 .05

.85 .05 .05 .05

.45 .05 .05 .45

.45 .05 .05 .45

.45 .05 .05 .45

.45 .05 .05 .45

.85 .05 .05 .05

.85 .05 .05 .05

.05 .05 .05 .85

.05 .05 .05 .85

-2 -1 0 1 2-1.5

-1

-0.5

0

0.5

1

1.5

Second c

oord

inate

First coordinate

-2 -1 0 1 2-1.5

-1

-0.5

0

0.5

1

1.5

Second c

oord

inate

First coordinate-4 -2 0 2 4

-2

-1

0

1

2

Second c

oord

inate

First coordinate

sw = 0.1

sw = 0.5

sw = 1

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Second c

oord

inate

First coordinate-3 -2 -1 0 1 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Second c

oord

inate


-3

-2

-1

0

1

2

3

Second c

oord

inate

First coordinate

sw=0.1

sw=0.5

sw=1



different water-saturation values, onto the two coordinates from non-metric MDS. Each point is color-coded by the water-saturation value of the sand layers. From left to right, subplots correspond to transition matrices A, B, and C, respectively.


different water-saturation values, onto the first two principal components from the Gaussian KPCA. Each point is color-coded by the water-saturation value of the sand layers. From left to right, subplots correspond to transition matrices A, B, and C, respectively.


different water-saturation values, onto the first two principal components from the dynamic similarity KPCA. Each point is color-coded by the water-saturation value of the sand layers. From left to right, subplots correspond to transition matrices A, B, and C, respectively.

-2 -1 0 1 2-1.5

-1

-0.5

0

0.5

1

1.5S

econd c

oord

inate

First coordinate

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Second c

oord

inate


-3

-2

-1

0

1

2

3

Second c

oord

inate

First coordinate

sw=0.1

sw=0.5

sw=1

-0.5 0 0.5-0.5

0

0.5

Second p

rincip

al com

ponent


-0.5 0 0.5-0.5

0

0.5

Second p

rincip

al com

ponent


-0.6 -0.4 -0.2 0 0.2 0.4-0.5

0

0.5

Second p

rincip

al com

ponent


sw=0.1

sw=0.5

sw=1

-1 -0.5 0 0.5 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Second p

rincip

al com

ponent


-1 -0.5 0 0.5 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Second p

rincip

al com

ponent

First principal component-1 -0.5 0 0.5 1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Second p

rincip

al com

ponent


sw=0.1

sw=0.5

sw=1



different water-saturation values, onto the first two principal components from the inverse multi-quadric KPCA. Each point is color-coded by the water-saturation value of the sand layers. From left to right, subplots correspond to transition matrices A, B, and C, respectively.


different water-saturation values, onto the first two principal components from the polynomial KPCA. Each point is color-coded by the water-saturation value of the sand layers. From left to right, subplots correspond to transition matrices A, B, and C, respectively.

Figure 4.32: First 21 eigenvalues from classical MDS, color-coded by the transition

matrices used in generating sand-shale sequences for investigating the effect of saturations.

-0.6 -0.4 -0.2 0 0.2 0.4-0.5

0

0.5

Second p

rincip

al com

ponent

First principal component-0.4 -0.2 0 0.2 0.4 0.6

-0.5

0

0.5

Second p

rincip

al com

ponent

First principal component-0.5 0 0.5

-0.5

0

0.5

Second p

rincip

al com

ponent


sw=0.1

sw=0.5

sw=1

-2 -1 0 1 2 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Second p

rincip

al com

ponent

First principal component-4 -2 0 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Second p

rincip

al com

ponent

First principal component-5 0 5

-5

-4

-3

-2

-1

0

1

2

3

4

Second p

rincip

al com

ponent


sw=0.1

sw=0.5

sw=1

1 5 9 13 17 210

100

200

300

400

Eigenvalue

Matrix A

Matrix B

Matrix C


Figure 4.33: Correlation coefficient between dissimilarity and distance (between

points in new coordinates resulting from an MDS algorithm) versus numbers of included coordinates from MDS results for investigating the effect of saturations. (Left) metric MDS and (right) non-metric MDS.


coordinates included as saturation attributes increases. Three MDS algorithms are compared. Results when using sequences generated by transition matrices A (left), B (middle) and C (right) are shown.


components included as attributes increases, when using sequences generated by transition matrices A.

1 2 3 4 5 6 7

0.7

0.8

0.9

1

# of coordinates

Corr

ela

tion c

oeff

icie

nt

betw

een

dis

sim

ilarity

and d

ista

nce

Matrix A

Matrix B

Matrix C

1 2 3 4 5 6 7

0.7

0.8

0.9

1

# of coordinatesCorr

ela

tion c

oeff

icie

nt

betw

een

dis

sim

ilarity

and d

ista

nce

Matrix A

Matrix B

Matrix C

2 4 6 8 10 12

0.5

0.6

0.7

0.8

0.9

1


Cla

ssific

ation s

uccess r

ate

2 4 6 8 10 12

0.5

0.6

0.7

0.8

0.9

1


Cla

ssific

ation s

uccess r

ate

2 4 6 8 10 12

0.5

0.6

0.7

0.8

0.9

1


Cla

ssific

ation s

uccess r

ate

Classical MDS

Metric MDS

Non-metric MDS

Matrix A Matrix B Matrix C

0 10 20 30 40 50 60 70 80 90

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

# of principal components or coordinates

Cla

ssific

ation s

uccess r

ate

Transition matrix A

Gaussian KPCA



polynomial KPCA

classical MDS (or linear KPCA)



components included as attributes increases, when using sequences generated by transition matrices B.


components included as attributes increases, when using sequences generated by transition matrices C.

0 10 20 30 40 50 60 70 80 90

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1


Cla

ssific

ation s

uccess r

ate

Transition matrix B

Gaussian KPCA



polynomial KPCA


0 10 20 30 40 50 60 70 80 900.4

0.5

0.6

0.7

0.8

0.9

1


Cla

ssific

ation s

uccess r

ate

Transition matrix C

Gaussian KPCA



polynomial KPCA



Figure 4.38: Component-wise analysis for the Gaussian KPCA. (Left) Parallel

coordinate plot of the first ten principal components of the Gaussian KPCA for matrix B. The solid lines are the median (the 0.5 quantile). The dash lines surrounding the median are the 0.45 and 0.55 quantiles. (Right) Projections of seismograms onto the first and sixth principal components of the Gaussian KPCA.

Figure 4.39: Component-wise analysis for the inverse multi-quadric KPCA. (Left)

Parallel coordinate plot of the first ten principal components of the inverse multi-quadric KPCA for matrix C. The solid lines are the median (the 0.5 quantile). The dash lines surrounding the median are the 0.45 and 0.55 quantiles. (Right) Projections of seismograms onto the first and sixth principal components of the inverse multi-quadric KPCA.

1 2 3 4 5 6 7 8 9 10-0.6

-0.4

-0.2

0

0.2

0.4

Coordinate

Coord

inate

Valu

e

sw=0.1

sw=0.5

sw=1

-0.5 0 0.5

-0.5

0

0.5


Six

th p

rincip

al com

ponent

Matrix B

1 2 3 4 5 6 7 8 9 10-0.3

-0.2

-0.1

0

0.1

0.2

Coordinate

Coord

inate

Valu

e

-0.4 -0.2 0 0.2 0.4 0.6

-0.4

-0.2

0

0.2

0.4

0.6


Six

th p

rincip

al com

ponent

sw=0.1

sw=0.5

sw=1 Matrix C


Table 4.4: Summary of the methods used to compute seismic attributes for water-saturation (in sand layers) effect. Choices of parameters for each method are similar to those listed in Table 4.2, unless otherwise specified. Performance of each method is shown as a success rate in classifying a data point into three water-saturation categories: Sw=0.1, 0.5, and 1. Results are shown in the columns corresponding to the transition matrices used in simulations. The actual transition matrices and sample sequences are shown in Figure 4.23.

Method Classification success rate

(using 1st, 2nd, both coordinates/components)

Notes

Transition

matrix A

(Aggrading)

Transition

matrix B

Transition

matrix C

(Blocky-sand)

Wavelet-

transform

0.54, 0.49, 0.57 0.59, 0.52,

0.65

0.42, 0.54, 0.53 Slope/intercept/both

MDS

(classical)

0.66, 0.33, 0.65 0.66, 0.33,

0.66

0.49, 0.42, 0.53 See Figure 4.32 for

eigenvalues

equivalent to KPCA

with a linear kernel or

PCA

MDS

(metric)

0.65, 0.36, 0.65 0.67, 0.31,

0.67

0.52, 0.43, 0.57

See Figure 4.33 for

correlation between

dissimilarity and

distance

MDS

(non-metric)

0.66, 0.35, 0.66 0.68, 0.33,

0.69

0.53, 0.44, 0.58 See Figure 4.33 for

correlation between

dissimilarity and

distance

KPCA

(Gaussian)

0.64, 0.32, 0.62

(1&3) 0.73

(1&3&8) 0.80

(1to10) 0.80

0.73, 0.35,

0.73

(1&6) 0.80

(1to10) 0.82

0.60, 0.36, 0.61

(1&5) 0.69

(1to10) 0.74

𝜎2 = 0.5

Trials with other

components

KPCA

(Dynamic

similarity)

0.71, 0.76, 0.88

(1to10) 0.89

0.70, 0.73,

0.87

(1to10) 0.89

0.81, 0.54, 0.84

(1to10) 0.90

𝜎 = 0.0398, m = 80%

of total seismogram

length, r = 4

KPCA

(Inverse multi-

quadric)

0.65, 0.32, 0.64

(1&3) 0.71

(1&3&7) 0.77

(1to10) 0.81

0.68, 0.34,

0.67

(1&7) 0.78

(1to10) 0.82

0.54, 0.42, 0.60

(1&6) 0.73

(1to10) 0.82

c2 = 1

Trials with other

components

KPCA

(Polynomial)

0.68, 0.32, 0.65

(1&7) 0.79

(1&3&7) 0.80

(1to10) 0.80

0.66, 0.35,

0.65

(1&8) 0.76

(1to10) 0.77

0.49, 0.43,

0.52

(1&8) 0.60

(1&10) 0.67

c = 1 and d = 2

Trials with other

components


4.4.4 Scenario 3: Effect of stacking patterns

We select transition matrices to mimic three stacking patterns within

retrogradational, progradational or aggradational parasequence sets while keeping net-

to-gross ratios and saturations the same. The three transition matrices shown in Figure

4.40 have the same limiting distributions: [0.25 0.25 0.25 0.25]. Note that each row of

matrix C is equal to a fixed probability vector. When using this matrix to generate a

sequence, the resulting lithologic states are considered as independent random events

(i.e., the current state has no dependency on the previous states). From the limiting

distribution, long sequences generated using these matrices should have their net-to-

gross ratios within the sequence equal to 0.25; however, in our simulation the values

spread around 0.25. Therefore, to separate the net-to-gross effect from the saturation

effect, we select only sequences with net-to-gross ratios ranging from 0.2 – 0.3. Water

saturation (Sw) is set to be the same in all simulations: Sw=0.1 for sand layers and

Sw=1 for the other lithology states). In this case, we show only simulations for

velocity distributions with standard deviation of 0.1 km/s.

Figure 4.40: Sample sequences from three selected transition matrices with the same

limiting distribution: [.25 .25 .25 .25]. From left to right, columns represent retrogradational, progradational, and aggradational stacking patterns, respectively. Red arrows schematically indicate transitions from coarse to fine grains, and vice versa. Note that other interpretations of transitional patterns are possible. The lithologic states are sand (s), shaly sand (sh-s) sandy shale (s-sh) and shale (sh).

(A) .183 .45 .183 .183

.183 .183 .45 .183

.183 .183 .183 .45

.45 .183 .183 .183

(B) .183 .183 .183 .45

.45 .183 .183 .183

.183 .45 .183 .183

.183 .183 .45 .183

(C) .25 .25 .25 .25

.25 .25 .25 .25

.25 .25 .25 .25

.25 .25 .25 .25

Sh S Sh S Sh S

(A) (B) (C)


To see how well the seismic attributes can differentiate the stacking patterns,

results from wavelet-transform, MDS, and KPCA are used in the cross-validation

scheme described in Section 4.4.2. Three classes considered are retrogradational,

progradational, and aggradational stacking patterns. The classification success rates

are summarized in Table 4.5. When using wavelet-transform attributes (i.e., slope and

intercept), the success rate is only 36%, which is only slightly better than the chance of

guessing the correct class out of three classes (33.33%). The classical, metric, non-

metric MDS, and KPCA with all tested kernels (Gaussian, dynamic similarity, inverse

multi-quadric, and polynomial kernels) yield similar success rates. Even when ten

coordinates or components are included, the success rates are still under 40%. A

parallel coordinate plot of the first ten components of the Gaussian KPCA is shown in

Figure 4.41 as an example to illustrate how each component from the three classes are

largely overlapped each other, causing the low success rate.

Figure 4.41: Parallel coordinate plot of the first ten components of the Gaussian

KPCA result for investigating the effect of stacking pattern on seismic signatures. The solid lines are the median (i.e., the 0.5-quantile) of the component values. The dash lines surrounding the median are the 0.45- and 0.55- quantiles. All lines are color-coded by the types of stacking patterns.

1 2 3 4 5 6 7 8 9 10-0.4

-0.2

0

0.2

0.4

0.6

Coordinate

Coord

inate

Valu

e

(A) Retrograding

(B) Prograding

(C) Aggrading


Table 4.5: Summary of the methods used to compute seismic attributes for stacking-pattern effects using sequences generated from the fixed-sampling transition matrices. Choices of parameters for each method are also included. Performance of each method is shown as a success rate in classifying a data point into three stacking patterns: retrogradational, progradational, and aggradational patterns.

Method Compared

objects

Distance/ or

Kernel

Classification

success rate

(using 1st, 2nd,

both coordinates)

Notes

Wavelet-

transform

– – 0.34, 0.34, 0.36


MDS

(classical)

Seismograms Euclidean 0.37, 0.35, 0.36

0.37,0.36,

0.39,0.37

1,2,1&2

(using 1,2,5,10

coordinates)

equivalent to KPCA with

a linear kernel or PCA

MDS

(metric)

Seismograms Euclidean 0.34,0.36,0.39,

0.37

(using 1,2,5,10

coordinates)

MDS

(non-

metric)

Seismograms Euclidean 0.34,0.36,0.39,

0.37

(using 1,2,5,10

coordinates)

KPCA Seismograms Gaussian 0.35,0.35,0.39,

0.39

(using 1,2,5,10

components)

σ2 = 0.5


similarity

0.34, 0.33, 0.36,

0.39

(using 1,2,5,10

components)

σ = 0.1265, m = 80% of


r = 2


multi-

quadric

0.37,0.36,0.39,

0.39

(using 1,2,5,10

components)

c2 = 1

KPCA Seismograms Polynomial 0.37, 0.37, 0.39,

0.39

c = 1, d = 2

In the above study of stacking-pattern effect on seismic signatures, sequences are

generated following the method of fixed-sampling transition matrices, with a step size

of 0.5 m (Figure 4.40). However, these sequences do not exhibit any thinning- or

thickening-upward trends of the sand layers that are observed in fining- and

coarsening-upward parasequences and parasequence sets (Section 4.3.1). Adjusting

the transition probabilities in the fixed-sampling transition matrices such that the sand

layers become thicker or thinner is possible. However, using only one transition

matrix to generate the entire sequence (i.e., stationary Markov chain model) makes the


thicknesses of sand layers approximately the same throughout the sequences.

Therefore, one way to incorporate thinning- or thickening-upward trends into the

synthetic sequences is to vary the transition matrices along the sequence. If the

embedded-form transition matrix is used, an alternative way to generate these trends is

to vary parameters of thickness distributions along the sequence (e.g., average values

of lithologic thicknesses that are exponentially distributed).

We create a new set of sequences with the three stacking patterns (e.g.,

retrogradational, progradational and aggradational) by using the embedded-form

transition matrices with varying the average thicknesses along the sequences to

capture the thickening- and thinning-upward trends. For example, we create the

retrogradational pattern by using the transition matrix shown in Figure 4.42 (top left)

and varying the average thicknesses for sand from 0.05 m at the top to 4 m at the

bottom of sequences. Examples of sequences for all three stacking patterns are shown

in Figure 4.42. The total thicknesses of all sequences are between 120 and 130 m. To

separate out the net-to-gross and saturation effects, we select only sequences with net-

to-gross ratios ranging from 0.37 to 0.42 and set water saturations (Sw) to be the same

in all simulations: Sw=0.1 for sand layers and Sw=1 for the other lithology states.

Velocity distributions for lithologic states are assumed to be normally distributed with

standard deviations of 0.1 km/s.


Figure 4.42: Sample sequences using the embedded-form transition matrices shown

on the left with varying averages of exponentially-distributed lithologic thicknesses along the sequences. From (A) to (C), the three stacking patterns are retrogradational (overall thinning- and fining-upward), progradational (overall thickening- and coarsening-upward), and aggradational stacking patterns in parasequence sets, respectively. Red arrows schematically show a series of progradational parasequences within retrogradational, progradational, and aggradational parasequence sets. Note that other interpretations of such parasequence patterns are possible The lithologic states are sand (s), shaly sand (sh-s) sandy shale (s-sh) and shale (sh).

Results from wavelet-transform, MDS, and KPCA are used in the cross-validation

scheme with the three stacking-pattern classes: retrogradational, progradational, and

aggradational. The classification success rates are summarized in Table 4.6. Using

wavelet-transform attributes (i.e., slope and intercept) gives the success rate only 34%,

which is approximately the chance of guessing the correct class out of three classes

(33.33%). However, classical, metric, non-metric MDS, and KPCA with both

Gaussian and inverse multi-quadric kernels yield higher success rates, with the

numbers increasing up to 70% when ten coordinates or components are included.

Since the distance used for classical MDS is not Euclidean but city-block distance,

classical MDS is not equivalent to the linear KPCA. Similar to the other scenarios,

Figure 4.43 illustrates that when the numbers of components included as attributes

Sh S

0

121

Sh S

0

124.2Sh S

0

123

Depth

fro

m a

refe

rence p

oin

t (m

)

(B) (C)0 .2 .3 .5

.5 0 .2 .3

.3 .5 0 .2

.2 .3 .5 0

Transition matrix

Stacking patterns of

parasequence sets:

( A) Retrogradational

(B) Progradational

( C) Aggradational

S: sand

Sh: shale

(A)


increase, classification success rates also increase. Among the selected kernels for

KPCA, the classification success rate of the dynamic similarity KPCA reaches a

higher success rate with just only a few principal components included; whereas, the

success rates of the linear KPCA (or PCA) show a gradual increase over a larger

number of components. The linear KPCA yields about 95% success rate when using

20 or more principal components. Note that the components are included as attributes

in order of decreasing eigenvalues, not their contributions to the success rate.

Compared to the previous results of fixed layer thicknesses, when thickening- or

thinning-upward trends are incorporated into sequences, stacking patterns can affect

seismic signatures. Seismograms with the retrogradational parasequence sets are the

most distinguishable among the three stacking patterns. This observation can be seen,

for example, from the parallel coordinate plot of the first ten coordinates of the

classical MDS result. From Figure 4.44, some coordinate values corresponding to the

retrogradational pattern are quite separated out from the other patterns, especially the

1st and 4

th coordinates, while most coordinate values of the progradational and

aggradational patterns are very similar. Note that patterns here refer to the patterns of

parasequence sets, which contain a series of parasequences. The parasequences

themselves are modeled in this study as being progradational (i.e., coarsening- and

thickening-upward). Therefore, it is more difficult to distinguish the progradational

pattern (i.e., both parasequences and their stacks are progradational) and the

aggradational pattern (i.e., parasequences are progradational and their stacks have no

change in the overall trend) because both share similar characteristics throughout the

sequences.


Table 4.6: for stacking-pattern effects using sequences generated from the fixed-sampling transition matrices. Choices of parameters for each method are also included. Performance of each method is shown as a success rate in classifying a data point into three stacking patterns: retrogradational, progradational, and aggradational patterns.

Method Compared

objects

Distance/ or

Kernel

Classification

success rate

(using 1st, 2nd,

both coordinates)

Notes

Wavelet-

transform

– – 0.36, 0.36, 0.34


MDS

(classical)

Seismograms Cityblock 0.49, 0.45, 0.55

0.49,0.55,

0.61,0.71

1,2,1&2

(using 1,2,5,10

coordinates)

MDS

(metric)

Seismograms Cityblock 0.50,0.58,0.63,

0.71

(using 1,2,5,10

coordinates)

MDS

(non-

metric)

Seismograms Cityblock 0.48,0.57,0.63,

0.71

(using 1,2,5,10

coordinates)

KPCA Seismograms Gaussian

0.44,0.47,0.62,

0.68

(using 1,2,5,10

components)

σ2 = 5


similarity

0.67, 0.66, 0.66,

0.72

(using 1,2,5,10

components)

σ = 0.0716, m = 80% of


r = 4


multi-

quadric

0.44,0.44,0.59,

0.68

(using 1,2,5,10

components)

c2 = 100

KPCA Seismograms Polynomial 0.43, 0.44, 0.59,

0.67

(using 1,2,5,10

components)

c = 10, d = 2

KPCA Seismograms Linear 0.44,0.44, 0.58,

0.67

(using 1,2,5,10

components)


Figure 4.43: Change in KPCA classification success rate when the number of

principal components included as attributes increases.

Figure 4.44: Parallel coordinate plot of the first ten coordinates of the classical MDS

result. The solid lines are the median (i.e., the 0.5-quantile) of the component

4.4.5 Discussions

The layer thickness in our simulations has a wavelength-to-mean thickness ratio of

about 100. Seismic waves cannot distinguish the boundaries of these thin layers

0 10 20 30 40 50 60 70 80 900.4

0.5

0.6

0.7

0.8

0.9

1

# of principal components

Cla

ssific

ation s

uccess r

ate

Gaussian KPCA



polynomial KPCA

linear KPCA

1 2 3 4 5 6 7 8 9 10-20

-15

-10

-5

0

5

10

15

20

Coordinate

Coord

inate

Valu

e

(A) Retrograding

(B) Prograding

(C) Aggrading


because of their band-limited characteristics. Hence, interpretations of these sub-

resolution layers can only be based on statistical attributes of the seismograms. In this

study, we generate synthetic seismograms of thin sand-shale sequences and extract

seismic attributes using various techniques such as wavelet transform,

multidimensional scaling (MDS), and kernel principal component analysis (KPCA).

These attributes are then related to properties of interest, which are net-to-gross ratio,

water saturation in the sand layers, and stacking patterns. Our results demonstrate that

these attributes can differentiate sequences with different reservoir properties. For

example, KPCA, especially the dynamic similarity KPCA, captures variations in net-

to-gross ratios and saturations well. However, there is no clear cut answer to which

attributes are the best choice because an attribute that performs really well in

differentiating one reservoir property may not work so well for other properties. For

example, the wavelet-transform attributes are able to capture variations in net-to-gross

ratios within the reservoirs, but their performance deteriorates when dealing with

changes in water saturation and stacking patterns.

Increasing the number of coordinates of MDS or components of KPCA results can

improve the classification of various reservoir properties. Our results show that with

only a small number of components used, KPCA with the dynamic similarity kernel

yields high success rates which are often higher than the other kernels with the same

number of components; however, when the number of components increase, the

success rates from the dynamic similarity KPCA do not vary much, while the success

rates from the linear KPCA gradually increase and eventually reach higher success

rates than those from the dynamic similarity KPCA. Thus, both the dynamic similarity

and the linear KPCA are good candidates for being used as attributes. Note that all

parameters used in the kernel formulas for KPCA (e.g., dynamic similarity,

polynomial KPCA) are estimated to yield good classification success rates; however,

it is not our goal in this work to find the exact parameters that optimize the success

rates.

Using a large number of components as attributes can make visualization more

difficult. Moreover, if these attributes are used in other classification methods or


models such as neural network, a large number of attributes can lead to more complex

models and more computation time. Therefore, instead of just increasing the number, a

subset of coordinates or components which are most relevant to the property of

interest can also be chosen. For example, we heuristically select a few components

that effectively differentiate different property classes using the parallel coordinate

plots. Further analysis is needed for the best possible combination of components.

All seismograms generated in this section (Section 4.4) are noise-free. To

investigate the effect of noise on the seismic attributes (e.g., KPCA), we add noise to

the synthetic seismograms which are generated from transition matrix A with varying

water saturations (Section 4.4.3). Then we run KPCA with a dynamic similarity kernel

and present the results in Figure 4.45. From the first two principal components of the

dynamic similarity KPCA, adding noise does not affect the attributes much. The

success rates of classifying a seismogram into its corresponding saturation class stay

almost the same. Note that the effect of noise on the KPCA attributes may vary

depending on types of kernels and the characteristics of seismograms (e.g., length and

shape).

Figure 4.45: Projections of seismograms with added noise onto the first two principal

components from the dynamic similarity KPCA. Each projected point is color-coded by the water-saturation value of the sand layers within the corresponding sequence. Percentages of noise added are specified in each panel. The success rate of classifying a projected seismogram into its corresponding saturation class is shown in the lower left corner of each panel.

-1 0 1-1

-0.5

0

0.5

1

Second p

rincip

al com

ponent


-1 0 1-1

-0.5

0

0.5

1

Second p

rincip

al com

ponent


-1 0 1-1

-0.5

0

0.5

1

Second p

rincip

al com

ponent


-1 0 1-1

-0.5

0

0.5

1

Second p

rincip

al com

ponent


sw =0.1

sw =0.5

sw =1

0% noise

success rate = 0.87

5% noise

success rate = 0.87

10% noise

success rate = 0.87

20% noise

success rate = 0.85


Results from our study have shown how the seismic signatures can vary with

changes in the properties and spatial arrangements of sub-resolution layers. However,

any patterns in the attribute space associated with changes in net-to-gross ratios, water

saturation, or stacking patterns shown in this study are not universal. In practice, these

patterns can vary depending on rock properties, numbers of lithologic states,

transition-matrix configurations, presence of noise, and other specific parameters.

Even though the study does not investigate all possible scenarios (e.g., all possible

transition-matrix configurations), it shows a promising applicability in characterizing

thin sand-shale reservoirs. In real application, all parameters should be calibrated at

the well locations. By assuming that the stratigraphy in the explored area demonstrates

a lateral continuation within conformable sequences, the inferred transition matrix

from a calibration well could be used to explore statistically how the seismic attributes

(e.g., wavelet-transform attributes, MDS coordinates) would change with varying net-

to-gross ratios and saturations. These statistics of the attributes can then be applied to

observations away from the well to characterize the area and quantify the uncertainties.

We show a numerical example of this application in the next section.

4.5 Net-to-gross estimation from 2-D sections

In the previous section, we investigated seismic signatures of 1-D, thin sand-shale

reservoirs. Various attributes from different methods show patterns relating to changes

in net-to-gross ratios, water saturation, and stacking patterns within the reservoirs.

These patterns can be applied to a real situation by following four main steps: (1)

extract a transition matrix at the well location using well log data, (2) create models

(e.g., 2-D sections) with varying property of interest, (3) extract seismic attributes

corresponding to those models and generate probability distributions of the attributes,

(4) and use the distributions to estimate properties of locations away from the well. In

this section, we demonstrate this four-step procedure through a numerical example.


4.5.1 Model setup

First, we assume a known transition matrix at the well location, which is selected

to be an embedded-form transition matrix with two lithologic states representing sand

and shale. We assign average clay fractions of 0.1 and 0.9 to the sand and shale states,

respectively. Then, we introduce uncertainties by assuming each state has normally-

distributed clay fractions with previously specified average values. Sw of sand and

shale are 0.3 and 1, respectively. The thickness distributions of both states are

exponentially distributed with average thicknesses of 0.3 and 0.5 m, respectively. The

transition matrix and thickness distributions are used to generate 1-D vertical

sequences at the well location only. Then, we create multiple realizations of 2-D

spatial models describing geology away from the well location (i.e., sand layers are

thinning linearly starting from the well location into an area away from the well)

(Figure 4.46). Seismic responses of these realizations are generated at 25 discrete

locations which are equally spaced along the 2-D sections. 5% noise is added to the

seismograms. Then, seismic attributes are extracted and related to the property of

interest, in this case net-to-gross values. For illustration here, we select slope and

intercept attributes from the wavelet-transform analysis. In section 4.5.2 below, we

will explore statistically how the attributes vary with changes in net-to-gross ratios.

These statistics will then be applied to an unknown synthetic seismic section for

estimating net-to-gross ratios of the area away from the well (Section 4.5.3).

Figure 4.46: A realization of one 2-D geologic section used in the numerical example.

The area at the left end marked with the red box corresponds to the well location. Sand and shale are colored in yellow and blue respectively. The thicknesses of sand layers decrease linearly away from the well. The total thickness of reservoir is 150 m. An example of the thickness distribution used to simulate the sequence at the well location is also shown.

0 1 2 3 4 5 6 7 80

20

40

60

80

100

Shale Cap Rock

Shale Cap Rock

decreasing

N/G

Total Thickness

~ 150 m

Thickness (m)

0 1 2 3 4 5 6 7 80

20

40

60

80

100

Shale Cap Rock

Shale Cap Rock

decreasing

N/G

Total Thickness

~ 150 m

Thickness (m)

Shale Cap Rock

Shale Cap Rock

Shale Cap Rock

Shale Cap Rock

decreasing

N/G

Total Thickness

~ 150 m

Thickness (m)


4.5.2 Results

Slope and intercept results from realizations of 1,000 2-D sections is shown in

Figure 4.47. Each point in the plot represents one seismogram. There is a systematic

change in both slope and intercept values as net-to-gross values increase. Contour

plots show distributions of slope and intercept values for the high (i.e., ≥ 0.6) and low

(i.e., ≤ 0.35) net-to-gross groups (Figure 4.48).

Figure 4.47: Results from realizations of 1,000 2-D sections show how slope and

intercept vary with varying net-to-gross ratios.

Figure 4.48: Contour plots and marginal distributions for the high (i.e., equal to or

greater than 0.6) and low (i.e., equal to or less than 0.35) net-to-gross values.

2.95 2.96 2.97 2.98 2.99 3 3.01-29

-28

-27

-26

-25

-24

-23

Slope

Inte

rce

pt

0

0.2

0.4

0 0.2 0.4

net-to-gross<=0.35

net-to-gross>=0.6


4.5.3 Net-to-gross estimation using a Bayesian framework

We estimate posterior distributions of net-to-gross ratios at three selected locations

from the unknown seismic section assumed to be acquired from the area. Posterior

distributions of net-to-gross ratios given a pair of attributes (i.e., slope and intercept)

can be obtained using Bayes’ formula:

𝑃 𝑁𝑇𝐺 𝐴𝑡𝑡𝑟 𝛼𝑃 𝐴𝑡𝑡𝑟 𝑁𝑇𝐺 ∙ 𝑃 𝑁𝑇𝐺 ,

4.1

where NTG is the net-to-gross ratio, Attr represents the seismic attributes (e.g., slope

and intercept from wavelet-transform-based analyses), P(NTG) is the prior probability

of the net-to-gross ratios, P(Attr | NTG) is the likelihood function, and P(NTG | Attr)

is the posterior probability of the net-to-gross ratios.

In this example, we use a simple geological model (i.e., thinning of sand layers) to

create realizations of many 2-D rock sections. From these realizations, we can then

estimate P(NTG) or the prior probability of net-to-gross ratios at a specified location

“X” on the 2-D section and use it together with the likelihood function at that location

“X” to obtain the posterior distribution of the net-to-gross ratios at “X”. Results for

three selected locations are shown in Figure 4.49.


Figure 4.49: (Lower left corner) Posterior distributions of net-to-gross ratios for three

selected locations on the unknown seismic section labeled as (1), (2) and (3). The true values for each location are shown in the table at the lower right corner.

The posterior distributions shown in Figure 4.49 are quite narrow and also capture

the true net-to-gross ratios. This is partly because we assume that the geology is

simple and known in the area, and this information helps reduce uncertainty in the

prediction. However, in reality there may see larger uncertainty due to the more

complicated natural system.

4.6 Local net-to-gross estimation in non-stationary sequences

In the previous numerical examples, we focused mostly on sequences in which an

entire sequence is generated using one transition matrix. However, we often see spatial

statistics of layered media change with depth. Examples of non-stationary sequences

are shown in Section 4.4.4, where thickening- and thinning-upward trends are

incorporated into sequences. In the previous sections, we used seismic attributes

extracted from seismograms to represent the entire reservoir interval, and so we could


only estimate a posterior distribution of net-to-gross ratios for the whole reservoir.

Unlike the previous sections, this section focuses on seismic signatures of thin-bedded

sequences that are generated from two transition matrices and aims to estimate local

distributions of net-to-gross across the reservoir.

4.6.1 Model setup

To generate sequences, we use transition matrices of the fixed sampling type. Only

two lithologic states are considered: sand and shale. The thickness of each individual

layer is 0.5 m, and the total thickness of the reservoir interval is 200 m. To simulate

sequences of lithologic states, we select two fixed-sampling transition matrices (A and

B), as shown in Figure 4.50. At each simulation step, the choice of which matrix to

use for generating the next lithologic state depends on the following rules. At each

step, we draw a random number from a uniform distribution on the interval [0,1]. For

a simulation step within the top half of the reservoir, if the random number is less than

0.8, matrix A is selected (i.e., the probability of using matrix A at each step in the top

half of the reservoir is 0.8). A similar rule is used for every simulation step in the

bottom half of the reservoir, except that matrix A is replaced by matrix B. The

resulting sequence becomes non-stationary. Examples of these sequences are shown in

Figure 4.50.

We assign average clay fractions of 0.1 and 0.9 to the sand and shale states,

respectively. Then we introduce uncertainties by assuming that each state has

normally-distributed clay fractions with the previously specified average values and

standard deviations of 0.01 and 0.03, respectively. Water saturations in sand and shale

layers are equal to 0.1 and 1, respectively. Porosity, velocity and density are then

computed following the methods described in Section 4.3.2. Forward modeling of

these sequence realizations for normal-incident reflected seismograms is performed

using the Kennett algorithm (Kennett, 1983) with a zero-phase Ricker wavelet. The

center frequency is 30 Hz, and the simulations are performed for all reverberations.

The seismograms are noise-free. We run a moving window of 80 ms through each

seismogram, resulting in seismogram segments of equal length. Then the


corresponding sequence segments and their net-to-gross values are obtained (Figure

4.51). These seismogram segments are then used to extract statistical attributes.

Figure 4.50: Two transition matrices used to generate non-stationary sequences and

examples of sequence realizations. For simulation steps in the top half of the reservoir interval, the probability of using matrix A is 0.8. The same is true for the bottom half of the reservoir with matrix B. The lithologic states considered are sand (S) and shale (Sh).

Figure 4.51: Segmentation of seismograms and sequences. (Left) application of an

80-ms moving window to a seismogram (Right) seismograms segments and their corresponding sequence segments.

P(A) = 0.8

P(B) = 0.8

Sh S Sh S

(A) 0.15 0.85

0.10 0.90

(B) 0.90 0.10

0.85 0.15

Transition matrices

Sh SSh S

Sh S

Sh S


4.6.2 Results and discussion

Results from the Gaussian KPCA with σ2 = 0.5 are shown in Figure 4.52. Each

point corresponds to a seismogram segment and is color-coded by its net-to-gross

value. Four net-to-gross classes are defined as follows: [0.16, 0.31), [0.31, 0.43), [0.43,

0.59), and [0.59, 0.77). When the first ten components of the Gaussian KPCA is used,

10-fold cross validation repeated ten times (Section 4.4.2) produces the classification

success rate of 82%.

Figure 4.52: The first two components of KPCA with Gaussian kernel. A total of

1200 seismogram segments are represented as points which are color-coded by the net-to-gross values of the corresponding sequence segments.

In real applications, one of the goals of any seismic signature study is to apply the

results to characterize reservoirs in areas away from well locations. To apply our local

net-to-gross estimation in real situations, first non-stationary transition matrices at the

well location are estimated using well log data. By assuming some prior knowledge of

geology in areas around the well, sequence models with varying net-to-gross values

are generated. Seismic attributes corresponding to these models are then used in local

net-to-gross estimation of locations away from the well.

Here we present a numerical example assuming that the non-stationary transition

matrices at the well location are already obtained and that the geology of areas

spatially around the well is not fully known but it can still be described by the same

-0.4 -0.2 0 0.2 0.4-0.4

-0.2

0

0.2

0.4


Second p

rincip

al com

ponent

Net-to-gross ratios

0.2

0.3

0.4

0.5

0.6

0.7


form of the transition matrices inferred at the well. Sequences are modeled using the

same setup as described in Section 4.6.1 and Figure 4.50, except that the probabilities

of using transition matrix A and B in the top and bottom halves of the reservoir

interval respectively are 1, and that the transition matrices A and B inferred at the well

are as follows: 𝐴𝑤𝑒𝑙𝑙 = 0.12 0.880.37 0.63

and 𝐵𝑤𝑒𝑙𝑙 = 0.67 0.330.78 0.22

.

From the matrices at the well, the same forms of matrices which described areas

around the well are 𝐴 = 1 − 𝑎 𝑎1 − 𝑏 𝑏

and 𝐵 = 𝑐 1− 𝑐𝑑 1 − 𝑑

, where 0.51 ≤ a, b, c, and d

≤ 0.99. These matrices are used to generate sequence realizations with varying net-to-

gross ratios. Then, we generate corresponding synthetic seismograms and run a

moving window of 80 ms through each seismogram, resulting in seismogram

segments of equal length. These seismogram segments are then used to extract the

KPCA attributes. The first two components of the Gaussian KPCA with σ2 = 0.5 are

shown in Figure 4.53. These KPCA attributes are then used to estimate local net-to-

gross distributions for three “unknown” seismograms, each of which corresponds to a

sequence with net-to-gross ratios varying from a low value at the top to a high value at

the bottom of the sequence. Each of the “unknown” seismogram is divided into three

80-ms segments, and the nine “unknown” seismogram segments are represented as

black markers in Figure 4.53. By following the Bayesian framework described in

Section 4.5.3, probability distributions of local net-to-gross for the nine samples are

estimated and compared with the true net-to-gross values (shown in red dash lines)

(Figure 4.54). There is a reasonable match between the most probable and the true net-

to-gross values. Moreover, we are able to extract the trend of net-to-gross ratios (e.g.,

low to high) from the top to the bottom of the reservoir and see where most sand is

located. In this example, our seismic attribute analysis of non-stationary sequences is

used to estimate local net-to-gross distributions along the reservoirs, which give us

more insight into how sand is distributed within the reservoir.


Figure 4.53: The first two components of KPCA with Gaussian kernel. A total of

2070 seismogram segments generated from the same forms of transition matrices as described in the text are represented as points which are color-coded by the net-to-gross values of the corresponding sequence segments. Nine seismogram segments are plotted in black and treated as unknowns.

Figure 4.54: Probability distributions of local net-to-gross estimation for the three

(unknown) seismograms. Each row represents the results for each unknown seismogram. Each column represents local net-to-gross estimations at a specified position of the sequences which correspond to the unknown seismograms. The red dash lines represent the true net-to-gross values.

-0.4 -0.2 0 0.2 0.4-0.4

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Position along the sequence Top Middle Bottom

Unknown 1

Unknown 2

Unknown 3


4.7 Discussion

In Sections 4.4 – 4.6, we showed three different examples where seismic attributes

were extracted from synthetic seismograms and used to infer reservoir properties of

thin sand-shale sequences. In this section, we discuss broader aspects of the attributes

used in this work.

4.7.1 Comparisons with amplitude attributes

Using the synthetic seismograms in Section 4.4.2, we extract two additional

attributes, the root-mean-square (RMS) amplitude and the absolute amplitude at the

top of the reservoir. The cross-plot between the two amplitudes is illustrated in Figure

4.55. Each point is color-coded by the net-to-gross ratios of the corresponding sand-

shale sequences. Then, we use the stratified 10-fold cross validation scheme described

in Section 4.4.2 to compute the success rate of classifying the seismograms

represented by the two amplitude attributes into their corresponding net-to-gross

classes. The success rate is 80%. Figure 4.56 shows the success rates of three selected

sets of KPCA attributes and the rates of these KPCA attributes combined with the

amplitude attributes. By adding the amplitude attributes to the first two principal

components of Gaussian KPCA, the success rate changes from 81% to 82%. Likewise,

for the case of dynamic similarity KPCA, the success rate changes from 90% to 91%.

However, when the amplitude attributes are added to the first 20 principal components

of linear KPCA, the success rate stays almost the same. These observations show that

the amplitude attributes can be combined with the KPCA attributes to improve the

success rate; however, in this example even though the amplitude attributes alone

yield a success rate as high as 80%, the improvement in the success rate is very small.

This implies that the amplitude attribute do not contribute much “new” information to

the KPCA attributes. In other words, these variations and patterns in the amplitudes

are already captured and included in the KPCA attributes.


Figure 4.55: Amplitude attributes extracted from seismograms which correspond to

sequences with different net-to-gross ratios.

Figure 4.56: Success rates in classifying a data point into three net-to-gross classes

(<0.38, 0.38-0.465, and >0.465) by using the attributes specified on the horizontal axis. The rates are shown on top of each bar. Att1: RMS and absolute amplitude at the top of the reservoir, Att 2: the first two principal components of Gaussain KPCA, Att3: the first two principal components of dynamic similarity KPCA, and Att4: the first 20 principal components of linear KPCA.

4.7.2 Notes on the feature-extraction based attributes

In this work, when performing the feature-extraction based techniques, we only

use amplitude data as the input features for the seismograms. However, it is easy to

combine other features (e.g., instantaneous phase, instantaneous frequency) into the

0 0.02 0.04 0.06 0.08-0.15

-0.1

-0.05

0

0.05

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RMS amplitude

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plit

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ate

Att4 Att1+Att4Att1 Att2 Att1+Att2 Att3 Att1+Att3

0.797 0.8090.823

0.8960.905

0.9830.985


analysis by normalizing different features so that each of the features has zero mean

and unit variance before running the usual steps.

Principal component analysis (PCA), kernel principal component analysis (KPCA),

and multidimensional scaling (MDS) used in this work are unsupervised feature-

extraction techniques, which do not account for class labels. As a result, there is no

guarantee that the selected new features (i.e., principal components) optimize class

separation. Instead, by selecting only a number of these new features as new

representations of the input data, the features with more discriminating power may be

discarded (Avseth et al., 2005). An alternative approach is to use supervised feature-

extraction techniques. For example, linear discriminant analysis (LDA) aims to find

new representations of data, while minimizing within class variance and maximizing

class separation. However, LDA has several limitations. For example, when the

dimensionality of data is high, a large number of training samples is required so that

within-class scatter matrix is nonsingular. To avoid this problem, it is common to first

reduce the dimensionality of the data using techniques such as PCA before applying

LDA (also known as PCA plus LDA) (Zhang, et al., 2006; Liew and Wang, 2009;

Thomas and Wilsey, 2011). We also use this workflow by applying PCA, KPCA, or

MDS and followed by LDA.

PCA involves solving an eigenproblem of the data covariance matrix whose size

grows with the dimensionality of the training data, while the computations in both

KPCA and MDS algorithms involve the matrices whose sizes grow with the number

of the training data. As a result, depending on both the dimensionality and the size of

the training dataset, the feasibility of these techniques may be limited (e.g., van der

Maaten, 2009; Shi and Zhang, 2011). The number of training data can be controlled

since that is the number of simulated seismograms.

With the exception of classical MDS, MDS algorithms solve iterative optimization

problems, so they can be very slow. Furthermore, before running the algorithms, the

desired number of dimension needs to be specified.


4.8 Conclusions

This chapter illustrates a workflow which can be applied to thin sand-shale

sequences for quick interpretation of reservoir properties before performing full

stochastic inversion. The workflow consists of four steps: (1) estimate transition

matrices at the well location from log data, (2) create what-if scenarios with varying

reservoir properties, (3) extract seismic attributes from the synthetic models and these

attributes become a training set, and (4) finally use the training set to classify seismic

signatures or estimate properties of locations away from the well. Most of the seismic

attributes shown in this chapter are derived from feature-extraction techniques, which

compare amplitudes (i.e., features) of entire seismogram segments, find new

representations of these seismograms with a new, smaller set of features, and use them

as attributes for estimating reservoir properties. The resulting attributes do not

necessarily relate to specific rock properties (e.g., Vp/Vs, acoustic impedance), nor do

they have physical meaning. However, it should be kept in mind that physical and

geological models are used as inputs to generate the seismograms from which the

attributes are extracted. Our numerical examples demonstrate that the feature-

extraction based attributes are able to capture variations and patterns in the input

amplitudes corresponding to variations in reservoir properties. The decision of how

many features should be selected depends on both the performance of each algorithm

and the goal of the study. By including more informative features as attributes, the

accuracy of predicting reservoir properties may increase; however, if these attributes

are further used in other algorithms, a larger number of attributes increases complexity

and requires more computation time.


This work was supported by the Stanford Rock Physics and Borehole Geophysics

project and the Stanford Center for Reservoir Forecasting.


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179

Chapter 5

Seismic signature and uncertainty in

petrophysical property estimation of

thin sand-shale reservoirs: Case

studies

5.1 Abstract

In Chapter 4, we presented a workflow and provided seismic attributes for

interpreting seismic signatures of thin sand-shale reservoirs. In this chapter, we apply

the workflow and these seismic attributes to real data from channelized turbidite

deposits offshore Equatorial Guinea, West Africa. We focus on a short fining-upward

interval from a distal well. Within this interval, both the well log analyses and the core

images show fine-scale sand-shale lamination. Their bed thicknesses are below the log

CHAPTER 5: Seismic signatures of thin sand-shale reservoirs: Case studies 180

resolution. Here we present two case studies. In the first case study, using rock

properties from the well, we generate synthetic seismograms under various scenarios

and treat these seismograms as our “real” traces in order to remove the effects of noise

and wavelet estimation from seismic signatures. The main seismic attribute used here

is obtained from kernel principal component analysis (KPCA), which projects

seismograms onto principal components. KPCA results of the training set in the first

case study show distinct seismic signatures of thin sand-shale reservoirs for varying

net-to-gross ratio, stacking pattern, water saturation, and reservoir thickness. These

results can be used to characterize the “real” traces successfully. We also present the

link between the KPCA attributes and amplitude attributes and investigate other

factors that can affect seismic signatures. In the second case study, we estimate

reservoir properties of sub-resolution sand-shale sequences from a real 2-D seismic

section. We analyze two existing wells that are located on each side of the seismic

section and find that average impedances of both the overburden and underburden

change spatially. Therefore, we incorporate such non-stationarity by adding two

additional parameters which are referred to as overburden and underburden multipliers.

We generate a training set by varying all parameters (e.g., net-to-gross ratios,

thickness) within specified ranges. The KPCA attributes of the training data on the

first two principal components illustrate distinct trends associated with changes in

some seismic-signature parameters, in particular the reservoir thickness while for other

parameters no clear trend can be observed. Since parameters that have greater impact

on seismic signatures can overshadow subtle effects from the other parameters, we

perform sensitivity analyses to rank all the seismic-signature parameters according to

their influence on seismic signatures. We first estimate probability density functions of

the important parameters for all seismic test traces. Then to reduce the effect of these

parameters and to enhance the effect of the others, for each test trace we create a new

training set (or select a subset from the training set) where these newly estimated

parameters are varied within smaller ranges near their mean values, while keeping the

ranges of the other parameters the same. Then this new training set can be used to

estimate the subsequent parameters. After these iterative steps, we obtain posterior


distributions of reservoir properties from all seismic traces in the 2-D section. To

evaluate estimation results, we add three known traces into the test set. Results show

that for both the well trace and the three additional known traces, their posterior

distributions for all the seismic-signature parameters generally capture the true values.

5.2 Introduction

In chapter 4, we used synthetic examples to illustrate a workflow for petrophysical

property estimation of thin sand-shale sequences. Our workflow has four main steps:

(1) extract transition matrices and calibrate rock properties at well locations, (2) use

the inferred transition matrices and rock-physics models to generate thin sand-shale

sequences with various scenarios (e.g., varying saturations), (3) simulate the

corresponding synthetic seismograms and extract seismic attributes from these

seismograms, and (4) apply the attributes to real seismic data for estimating reservoir

properties. The attributes for our seismic signature study are obtained using wavelet-

transform analysis, multi-dimensional scaling (MDS), and kernel principal component

analysis (KPCA). The results from the synthetic examples show that these attributes

can potentially differentiate thin sand-shale sequences with different net-to-gross

ratios or saturations.

In this chapter, we present two case studies in which we apply our workflow to

real well log and seismic data from deep-water turbidite deposits, offshore Equatorial

Guinea, West Africa. In Section 5.3, we provide geological background for the study

area including depositional environment and preliminary analyses of well log data. In

Sections 5.4, for our first case study we interpret seismic signatures at the well

location by focusing on a short fining-upward interval. In Section 5.5, for our second

case study we interpret seismic signatures within a reservoir zone across the selected

2-D section which extends from proximal to distal directions along a turbidite channel.


5.3 Geological background

The study area is an oil field located in the Rio Muni basin offshore Equatorial

Guinea, West Africa. The reservoirs are parts of a late-Cretaceous submarine canyon

which is characterized as being erosive and sand-rich, as evidenced by seismic, well

log, and core data (Dutta, 2009; Jobe et al., 2011). Using the available cores from five

wells in the study area, Lowe (2004) identified six main lithofacies: (1) thick-bedded

to massive sandstone, (2) interbedded thin-bedded sandstone and mudstone with beds

2-20 cm thick and >20% sandstone, (3) interbedded thin-bedded sandstone and

mudstone with beds <2 cm thick, (4) carbonate-cemented sediments, (5) conglomerate

and breccia, and (6) mudstone with <10% interbedded sandstone (Figure 5.1). Only

lithofacies (4) is a diagenetic lithofacies, while the others are depositional lithofacies.

One of the basic features observed in the cores is the fining-upward trend in which the

coarse-grained, thick-bedded lithofacies grades upward into successively finer-grained,

thinner-bedded lithofacies and/or mudstone (Lowe, 2004). The fining-upward pattern,

core analyses, and interpretation of seismic data suggest that these sediments are

channelized deposits of high-density turbidity currents in the submarine canyon

system (Dutta, 2009).

Figure 5.1: Six identified lithofacies from the study area offshore Equatorial Guinea, West Africa (Lowe, 2004; Dutta, 2009).


5.4 Case study 1: Effect of reservoir properties and stacking pattern

on seismic signatures at well location

In this case study, we select a short interval representing a fining-upward sequence

in a reservoir zone from well A, which locates in the distal direction of the

channelized turbidite deposits (Dutta et al., 2007; Dutta, 2009). Using the well log data,

we generate synthetic seismograms to study the effect of net-to-gross ratio, stacking

pattern, saturating fluid, and reservoir thickness on seismic signatures. In Section 5.4.1,

we apply Thomas-Stieber model to porosity and gamma ray values for estimating net-

to-gross ratio of the fining-upward sequence (Chapter 2). Sections 5.4.2 – 5.4.4 show

synthetic seismogram generation for sequences with varying net-to-gross ratio,

stacking pattern, saturating fluid, and reservoir thickness. In Section 5.4.5, we analyze

signatures of those synthetic seismograms using kernel principal component analysis

(KPCA; Chapter 4). These signatures are then used for interpreting the synthetic

seismograms generated from the log data at the well. Section 5.4.6 discusses other

factors that could affect our interpretations.

5.4.1 Net-to-gross estimation from well log data

Log measurements from well A are shown in Figure 5.2. The gamma-ray log in

the selected interval exhibits a fining-upward pattern of the uncemented lithofacies, in

which the thick-bedded or massive sandstone facies (lithofacies (1)) change into the

thinly interbedded sandstone-mudstone facies (lithofacies (2) and (3)) in the upward

direction (Dutta, 2009). As previously noted, detailed core analyses by Lowe (2004)

showed that the sandstone beds in the thin-bedded lithofacies have thicknesses ranging

from <2 cm to 20 cm. Since the individual thin beds of sandstones and mudstones are

too thin to be resolved by the logging tools, well log measurements represent average

properties of multiple beds. To correct for this averaging effect in order to estimate the

amount and the properties of the sub-resolution sand beds, we use the Thomas-Stieber

model (Thomas and Stieber, 1975) as discussed in Chapter 2.


Figure 5.2: Well A from deep-water turbidite deposits, offshore Equatorial Guinea,

West Africa. The zone of interest is highlighted. From left to right, the curves are gamma ray, bulk density, density-derived porosity, P-wave velocity, and water saturation (SW), respectively.

The Thomas-Stieber model describes how total porosity varies with shale volume

depending on the configuration of shale in the sand-shale mixtures. We focused only

on two shale configurations which are laminar and dispersed. By inputting properties

of the clean sand and shale end-members, we showed how this model can be used

graphically or algebraically to estimate the net-to-gross ratios and volumes of

dispersed shale in the sandy layers. We also demonstrated how to propagate

uncertainties of the input parameters through the model using Monte Carlo simulations

in a Bayesian framework and produced posterior distributions of the estimated

properties. We showed an example of applying the Thomas-Stieber model to the log

data from well A. In that example, the selected well-log interval is exactly the same

fining-upward sequence as shown in Figure 5.2.

In the example involving the log data from well A shown in Chapter 2, the

Thomas-Stieber model was applied to density-derived porosity and gamma ray values

by assuming that the properties of the sand and shale end-members are uncertain and

that they are normally-distributed. The resulting posterior distributions of net-to-gross

ratios along the fining-upward interval are shown in Figure 5.3. Note that The

Thomas-Stieber model for estimating net-to-gross ratios is only applied to the data

from lithofacies 1, 2, and 3. For the rest of the data, their net-to-gross ratios are simply

estimated by 1 – 𝑉𝑠ℎ , where 𝑉𝑠ℎ is the shale volume fraction. The net-to-gross ratio of

0 50 100

1050

1060

1070

1080

1090

1100

1110

Gamma ray

Depth

(m

)

2 2.5

Bulk density

0 0.2

Total porosity

2 3 4

P-wave velocity

0 0.5 1

SW


the entire fining-upward sequence is then equivalent to an arithmetic average of net-

to-gross ratios of all points along the sequence. We estimated this average value to be

0.48. This estimate can vary depending on the properties of the end-members used in

the Thomas-Stieber model.


Stieber model. (Left) Total porosity and gamma ray values for three lithofacies in the selected interval. The median total porosity and gamma ray values for each lithofacies are shown in solid circles. From these median points, the up-down or left-right bars indicate the interquartile ranges (i.e., from 1st to 3rd quartiles) of each property. A Thomas-Stieber diagram is also superimposed on the data. (Right) Variation of gamma ray values with depth. Data points are color-coded by lithofacies similar to the left panel. The gamma ray log shows an upward-fining trend.

Figure 5.4: Estimated net-to-gross ratios for the selected well-log interval. (Left)

gamma ray log of the selected interval and (right) posterior distributions of estimated net-to-gross ratios.

20 30 40 50 60 70 80 900.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Gamma ray

Tota

l poro

sity

Lithofacies 1

Lithofacies 2

Lithofacies 3

40 60 80

1070

1075

1080

1085

1090

1095

1100

Gamma ray

Depth

(m

)

40 60 80

1075

1080

1085

1090

1095

Gamma ray

Depth

(m

)

Lithof acies 1

Lithof acies 2

Lithof acies 3

0 0.5 1

1075

1080

1085

1090

1095

Sand fraction

Depth

(m

)


5.4.2 Seismograms at well location

When using real seismic data, both the noise and the uncertainty from the wavelet

estimation process can affect seismic signatures of thin sand-shale sequences. To

focus on the signatures, in this case study we remove these two sources of

uncertainties by generating a synthetic seismogram at the well location and using it as

our “true” seismic trace. We extract velocity and density values of the target zone

from the logs, together with the upper and the lower 200 points of the target zone

(Figure 5.5, left). Each layer is assumed to be approximately 15-cm thick (i.e., the

spacing in the log measurements). Layer properties (i.e., velocity, density and

thickness) are then input into the Kennett algorithm (Kennett, 1983) to simulate full-

waveform, normally-incident, reflected seismograms using a zero-phase Ricker

wavelet with a central frequency of 30 Hz. The simulations are performed for all

reverberations. Figure 5.5 (right) illustrates the synthetic seismogram at the well

whose gamma ray log exhibits a fining-upward stacking pattern. To expand our

seismogram test set, we generate four additional synthetic seismograms by modifying

the velocity and density logs within the target zone, while both the upper and the

lower parts of the target zone are kept the same.

For the first modification, we turn the velocity and density curves of the target

zone upside down. This new property arrangement corresponds to a “coarsening-

upward” stacking pattern. For the second modification, we randomly shuffle the orders

of the velocity-density pairs within the target zone. This property arrangement

corresponds to a more “serrated” stacking pattern. For the third modification, we apply

fluid substitution to both velocity and density curves to change the saturating fluid

from oil to brine. For the final modification, we decrease the thickness of each layer

within the target zone from 15 cm to 10 cm. Consequently, the total thickness of the

target zone also decreases. The logs produced by the four modifications are then used

to generate corresponding synthetic seismograms. Hereafter, we refer to the

seismograms for the original and the four additional logs as the fining-upward,

coarsening-upward, serrated, brine-saturated, and thinner-reservoir models,

respectively (Figure 5.6).


Figure 5.5: Acoustic impedance and the corresponding synthetic seismogram of the

target zone. (Left) Acoustic impedance of the target zone shown in magenta, together with 200 data points above and 200 data points below the target zone. (Right) Synthetic seismogram generated using a 30-Hz zero-phase Ricker wavelet.

Figure 5.6: Synthetic-seismogram test set. Note that each panel shows four repetitions

of one seismogram. From left to right, the panels show seismograms that are generated from the original well log (i.e., fining-upward), the inverted log (i.e., coarsening-upward), the shuffled log (i.e., serrated), the brine-saturated log, and the log with smaller layer thicknesses within the target zone.

5.4.3 Transition matrices and rock property calibration

This section shows how we create transition matrices and use them to generate

synthetic earth models within the target zone for varying net-to-gross ratio, stacking

5000 10000

1050

1060

1070

1080

1090

1100

1110

1120

1130

Acoustic impedance

(g/cm3 x m/s)

Depth

(m

)

tim

e (

ms)

tim

e (

ms)

Fining-upward

model

tim

e (

ms)

Coarsening-upward

model

tim

e (

ms)

Serrated model

tim

e (

ms)

Brine-saturated

model

tim

e (

ms)

Thinner-reservoir

model


pattern, saturating fluid, and layer thickness. We use Markov chain models to create

synthetic earth models within the target zone by assigning two lithologic states – sand

and shale. Using the fixed-sampling transition matrices (Chapter 4), we consider three

main categories of matrices, each of which generates sequences with distinct spatial

patterns in net-to-gross ratios.

5.4.3.1 Interbedded sand-shale sequences with no clear trend in net-to-gross ratios

along the sequences

To generate simple interbedded sand-shale sequences that span a wide range of

net-to-gross ratios, we set the form of transition matrices as 𝑇1 = 𝑎 1 − 𝑎𝑏 1 − 𝑏

, where a

and b are randomly drawn from a uniform distribution on [0,1]. Since we draw new

values of a and b for each realization, the transition matrix always changes from

realization to realization. When using these 𝑇1 matrices, net-to-gross ratios vary along

the sequences, but there is no one particular trend.

5.4.3.2 Interbedded sand-shale sequences with an upward-decreasing trend in net-to-

gross ratios along the sequences

To incorporate an upward-decreasing trend in net-to-gross ratios into the sand-

shale sequences, we first divide the sequences into two equal subsequences. Then we

set the transition matrices for the top and bottom subsequences as follows:

𝑇2(𝑡𝑜𝑝 ) = 𝑎 1 − 𝑎𝑏 1 − 𝑏

, where a and b are drawn from a uniform distribution on [0,0.5],

𝑇2(𝑏𝑜𝑡 ) = 𝑐 1 − 𝑐𝑑 1 − 𝑑

, where e and f are drawn from a uniform distribution on [0.5,1].

By combining both sets of 𝑇2 transition matrices, we produce the sequences which

exhibit an upward decrease in sand to shale ratios.

5.4.3.3 Interbedded sand-shale sequences with an upward-increasing trend in net-to-

gross ratios along the sequences

To generate an upward-increasing trend in net-to-gross ratios along the sequences,

we first divide the sequences into two equal subsequences and then use both sets of


transition matrices, as shown in Section 5.4.3.2. However, the order of both transition-

matrix sets is reversed:

𝑇3(𝑡𝑜𝑝 ) = 𝑎 1 − 𝑎𝑏 1 − 𝑏

, where a and b are drawn from a uniform distribution on [0.5,1],

𝑇3(𝑏𝑜𝑡 ) = 𝑐 1 − 𝑐𝑑 1 − 𝑑

, where e and f are drawn from a uniform distribution on [0,0.5].

By combining both sets of 𝑇3 transition matrices, we produce the sequences which

exhibit an upward increase in sand to shale ratios.

Figure 5.7 shows sample sequences from the above three categories of transition

matrices which are used to generate arrangements of sand and shale layer within the

target zone. As noted in Section 5.4.2, log measurements are spaced at approximately

15 cm apart, and we assign this spacing as the layer thicknesses when we use well log

data to generate the “true” seismogram at the well location. However, as shown by the

Thomas-Stieber analysis (Section 5.4.1), the target zone contains thin layers below the

vertical resolution of logging tools. Furthermore, core analyses also demonstrate that

sand thicknesses in this zone can be even less than 2 cm (Lowe, 2004; Dutta, 2009).

Therefore, for our synthetic earth models within the target zone, we use the fixed-

sampling transition matrices and select the thickness of the sand or shale layer to be

approximately 1.5 cm.

After generating layer arrangements, we estimate acoustic impedances of both the

sand and the shale lithologic states from the acoustic impedances at the well location.

This estimated sand property represents the oil-saturated sand. Since we also test an

interbedded brine-sand and shale model, we estimate the property of the brine-

saturated sand by applying Gassmann’s fluid substitution to the oil-saturated sand. The

estimated impedances for all three states are used as mean values of normal

distributions with assigned standard deviations (Figure 5.8). We draw values from

these distributions and assign them to layers according to their states.

In summary, we simulate five sets of sand-shale sequences with the properties

shown in Table 5.1. Each set contains 400 realizations.


Figure 5.7: Sample realizations of synthetic earth models which cover a range of net-

to-gross ratios and various stacking patterns. The sequence realizations are generated from the three categories of transition matrices: with an upward-increasing trend (left), with an upward-decreasing trend (middle), and with no trend in net-to-gross ratios (right).

Figure 5.8: Probability density functions of acoustic impedance for all lithologic states: oil sand (blue), shale (red), and wet sand (black).

No t

rend in n

et-

to-g

ross r

atios

Upw

ard

-decre

asin

g t

rend in n

et-

to-g

ross r

atios

Upw

ard

-incre

asin

g t

rend in n

et-

to-g

ross r

atios

white: sand and black: shale

4.5 5 5.5 60

2

4

6

8

10

Acoustic impedance (g/cm3 x km/s)

Pro

babili

ty d

ensity

oil-saturated

sand

shale

brine-saturated

sand


Table 5.1: Specifications for synthetic earth models with various net-to-gross ratios, stacking patterns, saturating fluids, and layer thicknesses.

Model Net-to-gross trend Sand property Reservoir

thickness

I upward-decreasing oil-saturated 17 m

II upward-increasing oil-saturated 17 m

III no trend oil-saturated 17 m

IV upward-decreasing brine-saturated 17 m

V upward-decreasing oil-saturated 11 m

5.4.4 Synthetic seismograms

In the previous section, we have generated five sets of sand-shale sequences for

various net-to-gross ratios, stacking patterns, saturating fluids, and layer thicknesses.

By inputting these sequences into the Kennett algorithm (Kennett, 1983), in this

section we simulate the corresponding synthetic seismograms (Figure 5.9). These

seismograms become our training set for characterizing the seismogram test set from

Section 5.4.2. Note that the Kennett algorithm is used here with a 30-Hz Ricker

wavelet and that all seismograms are noise-free. The seismograms from both the

training set and the test set are then input into kernel principal component analysis

algorithms (KPCA; Chapter 4) to generate seismic attributes.


Figure 5.9: Synthetic seismograms corresponding to five sets of sand-shale sequences.

Model I to V correspond to sequence realizations generated from various transition matrices, sand properties, and layer thicknesses. Refer to Table 5.1 for detailed specifications.

5.4.5 Effect of reservoir properties and stacking pattern on seismic signatures

The seismic attributes generated by KPCA are the projections of the input

seismograms onto the principal components (Figure 5.10). Note that we also run MDS

algorithms (Chapter 4) and KPCA with different kernels (e.g., linear kernel, Gaussian

kernel, and dynamic similarity kernel) and observe similar results. Therefore, here we

show only the results from KPCA with a linear kernel, and we refer to these linear

KPCA attributes as the KPCA attributes, unless otherwise specified. The results from

the training set correspond to points color-coded by net-to-gross ratios from their

corresponding sequences, and the results from the test set (shown in black symbols)

are plotted on top of the results from the training set. Figure 5.10 shows that the

training groups with different stacking pattern, saturating fluid, and reservoir thickness

are nicely separated as distinct clusters. Within these clusters, there are also apparent

trends of net-to-gross ratios. Figure 5.10 also illustrates that all the test points are

placed correctly in their corresponding training groups (e.g., the point corresponds to

the fining-upward log is located on top of the upward-decreasing net-to-gross group).

From the locations of all the test points, we can roughly estimate their net-to-gross

ratios to be between 0.45 and 0.55, which is consistent with what we previously

estimated using the Thomas-Stieber model in Section 5.4.1. Using the training set we

successfully infer the net-to-gross ratio, stacking pattern, saturating fluid, and reservoir

thickness of the test set.

tim

e (

ms)

Model I Model II Model III Model IV Model V


Figure 5.10: Projections of the noise-free seismograms from the training set and the

test set onto the first two principal components after the application of the linear KPCA. Each point from the training set is color-coded by the net-to-gross ratio from its corresponding sand-shale sequence. The test set, shown in black symbols, is plotted on top of the projections of the training set.

5.4.6 Discussion

In the previous section, we demonstrated that KPCA results of the training set

show distinct seismic signatures which correspond to changes in net-to-gross ratio,

stacking pattern, saturating fluid, and reservoir thickness, and that these results can be

used to characterize the seismogram test set successfully. This section discusses

several factors that could also influence seismic signatures.

5.4.6.1 Link between distance-based attributes and amplitude attributes

Using trace amplitudes as inputs, KPCA algorithms output coordinate values

which are projections of the traces onto the principal component axes. These

coordinate values and the principal components may not directly relate to one

particular physical property of the reservoirs (e.g., porosity). Instead, they might be

more connected to the seismic traces themselves (e.g., shape). In order to illustrate a

possible link between KPCA attributes and characteristics of seismic traces, we extract

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08


Second p

rincip

al com

ponent

color-coded by net-to-gross ratios

0.35

0.4

0.45

0.5

0.55

0.6

Model I

Model II

Model III

Model IV

Model V

fining-upward log

coarsening-upward log

serrated log

brine-saturated,

finning-upward log

thinner-layered,

finning-upward log

thickness

effect

fluid effect

increasing

net-to-gross

ratios

changing

stacking

patterns


the following amplitude attributes. We pick the maximum amplitude and compute the

root-mean-square amplitude (RMS) from all the seismograms. A cross-plot between

the two amplitude attributes is shown in Figure 5.11. This plot illustrates trends

similar to those in the KPCA results (Figure 5.10). The first and second principal

components of the KPCA results correlate very well with the maximum amplitude and

the RMS amplitude, respectively. The amplitude attributes can also be used

successfully to infer net-to-gross ratio, stacking pattern, saturating fluid, and reservoir

thickness of the sequences in the test set. However, even though the three stacking-

pattern clusters of the amplitude attributes are separated, the separation is relatively

slight. Therefore, small noise added to seismograms can easily alter the amplitude

attributes so that these clusters may become more overlapped. Consequently, when

using the amplitude attributes, presence of noise can reduce the odds of correctly

inferring a stacking pattern of a sequence from an unknown seismogram.

Figure 5.11: Cross-plot between the RMS and the maximum amplitude of noise-free seismograms from both the training set and the test set. The points from the training set are color-coded by the net-to-gross ratios from their corresponding sand-shale sequences. The points from the test set are shown in black symbols. Two distinct trends corresponding to changes in stacking pattern and net-to-gross ratio are marked by the blue and the red arrows, respectively.

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160.028

0.03

0.032

0.034

0.036

0.038

0.04

Maximum amplitude

RM

S a

mplit

ude


Model I

Model II

Model III

Model IV

Model V

fining-upward log

coarsening-upward log

serrated log

brine-saturated,

fining-upward log

thinner-layered,

finning-upward log

0.35

0.4

0.45

0.5

0.55

0.6

thickness

effect

increasing

net-to-gross

ratios

changing

stacking

patterns

fluid

effect


5.4.6.2 Effect of noise

To investigate the effect of noise on both the feature-extraction based (e.g., KPCA)

and the amplitude attributes, we add random noise to all seismograms from both the

training set and the test set before extracting their attributes. The linear KPCA and the

amplitude attributes are shown in Figure 5.12 and Figure 5.13, respectively. For a

better look at the noise effect, we use only sequences with varying net-to-gross ratio

and different stacking pattern. KPCA attributes for the noise-free case separate the net-

to-gross and the stacking-pattern trends nicely; however, these trends are smeared out

when level of noise increases. Any interpretation using the smeared trends is subject to

more uncertainty. Note that even though the clusters of points in the noisy case

sometimes change their directions from those of the noise-free case (e.g., from

northeast-southwest to northwest-southeast alignments), these changes do not affect

our interpretation. Any rotation or translation of the KPCA results does not matter.

Both the net-to-gross and stacking-pattern trends in amplitude attributes are also

affected and smeared out by noise. To quantify and compare the effect of noise on

KPCA and amplitude attributes, we count the number of times the linear discriminant

analysis algorithm (LDA) successfully sorts the attribute vectors (i.e., the seismograms)

into their corresponding net-to-gross/stacking-pattern classes. Net-to-gross interval is

divided into three subintervals: (1) NTG ≤0.46, (2) NTG >0.46 and NTG ≤0.54, and (3)

NTG >0.54, and the stacking pattern contains three subsets: fining-upward (F),

coarsening-upward (C), and serrated (S). The combination of both the net-to-gross

subintervals and the stacking-pattern subsets results in a total of nine classes: NTG1 +

F, NTG2 + F, NTG3 + F, NTG1 + C, NTG2 + C, NTG3 + C, NTG1 + S, NTG2 + S, and

NTG3 + S. An example of KPCA attributes with class labels is shown in Figure 5.14.

The classification success rate is estimated using a stratified 10-fold cross validation

(Chapter 4). Figure 5.15 illustrates the success rates for both the KPCA and the

amplitude attributes at different noise levels. Both types of attributes are affected by

noise; however, the amplitude attributes are more severely affected, as shown by their

rapidly decreasing classification success rates. Note that in this example, using more

components of KPCA results as attributes does not increase classification success rate.


This is possibly because the well-log interval considered here is relatively short. As a

result, the corresponding seismograms with different properties show difference in

amplitudes only on a small portion of the traces (i.e., about two wavelength).

Therefore, with just a few principal components we are able to capture all the

important features. As previously noted, in this case study we show only results from

the linear KPCA because KPCA with other kernels, including Gaussian kernel and

dynamic similarity kernel, yields similar results. When we investigate the effect of

noise on the dynamic similarity KPCA, we observe that its results do not differ much

from those of the linear KPCA (Figure 5.15).

In summary, even though the amplitude attributes can be useful for property

estimation of thin sand-shale sequences, they can be very sensitive to noise. While the

feature-extraction-based algorithms (e.g., KPCA) also extract important information

from seismograms by comparing their amplitudes, the resulting attributes are less

sensitive to noise.

Figure 5.12: Projections of the noisy seismograms from the test set onto the first two principal components from the linear KPCA. The test set shown in black symbols is plotted on top of the projections of the training set, which are color-coded by the net-to-gross ratios from their corresponding sand-shale sequences. The percentages of noise added to the seismograms are shown in the lower left corner of each plot.

-0.05 0 0.05-0.04

-0.02

0

0.02

0.04


Second p

rincip

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ponent

-0.05 0 0.05-0.04

-0.02

0

0.02

0.04


Second p

rincip

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ponent

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-0.02

0

0.02

0.04


Second p

rincip

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ponent

-0.05 0 0.05-0.04

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0

0.02

0.04


Second p

rincip

al com

ponent

0.35

0.4

0.45

0.5

0.55

0.6

Model I

Model II

Model III

fining-

upward log

coarsening-

upward log

serrated log

net-to-gross ratios

0% noise 5% noise

10% noise 20% noise


Figure 5.13: Cross-plot between the RMS and the maximum amplitude of noisy seismograms from both the training set and the test set. The points from the training set are color-coded by the net-to-gross ratios from their corresponding sand-shale sequences, and the points from the test set are shown in black symbols. The percentages of noise added to the seismograms are shown in the lower left corner of each plot.

Figure 5.14: Linear KPCA results (0% noise) with net-to-gross (NTG)/stacking-pattern class labels. NTG1, NTG2, and NTG3 represent NTG≤0.46, NTG>0.46 and NTG≤0.54, and NTG>0.54, respectively. The three stacking patterns are fining-upward (F), coarsening-upward (C), and serrated (S).

0.12 0.14 0.16

0.036

0.038

0.04

0.042

Maximum amplitude

RM

S a

mplit

ude

0.12 0.14 0.16

0.036

0.038

0.04

0.042

Maximum amplitude

RM

S a

mplit

ude

0.12 0.14 0.16

0.036

0.038

0.04

0.042

Maximum amplitude

RM

S a

mplit

ude

0.12 0.14 0.16

0.036

0.038

0.04

0.042

Maximum amplitude

RM

S a

mplit

ude

0.35

0.4

0.45

0.5

0.55

0.6

Model I

Model II

Model III

fining-

upward log

coarsening-

upward log

serrated log5% noise0% noise

10% noise 20% noise

net-to-gross ratios

-0.04 -0.02 0 0.02 0.04 0.06-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Linear KPCA results (0% noise)

color-coded by nine net-to-gross/stacking-pattern classes


Second p

rincip

al com

ponent

NTG1 + S

NTG2 + S

NTG3 + S

NTG1 + F

NTG2 + F

NTG3 + F

NTG1 + C

NTG2 + C

NTG3 + C


Figure 5.15: Classification success rate when using linear KPCA attributes, amplitude attributes, and dynamic similarity KPCA attributes at various noise levels.

5.4.6.3 Effect of layer thickness used to model the training set

In Section 5.4.3, when modeling the sequences for the training set, using

information from the core analysis we assigned each layer a thickness of 1.5 cm,

equivalent to wavelength to thickness ratio of about 50. In this section, we investigate

the effect on the attribute results of varying layer thickness. We increase the layer

thickness to 2.5 cm, equivalent to wavelength to thickness ratio of about 30 and

decrease the number of layers so that the total thicknesses of the sequences are kept

the same. The linear KPCA results and the amplitude results for the two thickness

models are shown in Figure 5.16 and Figure 5.17, respectively. We show only results

from sequences that are all fining-upward, but with varying net-to-gross ratios. To

quantify and compare the effect of different modeling thicknesses on KPCA and

amplitude attributes, we use each set of attributes and compute their success rates for

classifying the seismograms into three net-to-gross classes: (1) NTG ≤0.46, (2) NTG

>0.46 and NTG ≤0.54, and (3) NTG >0.54. The classification success rates are

displayed in the lower corner of each plot in both Figure 5.16 and Figure 5.17. When

the layer thickness in the sequences changes from 1.5 to 2.5 cm, both the KPCA and

the amplitude attributes show little to no change, and their classification success rates

also stay almost exactly the same.

0 5 15 200.2

0.4

0.6

0.8

1

Cla

ssific

ation s

uccess r

ate

% of noise added

Linear KPCA

(the first tw o components)

Amplitude attributes

(maximum and RMS)

Dynamic similarity KPCA

(the first tw o components)


Figure 5.16: Effect of layer thickness on linear KPCA results. Each plot represents the projections of seismograms onto the first two principal components of linear KPCA when layer thickness is 1.5 cm (left) or 2.5 cm (right). Points are color-coded by the net-to-gross ratios from their corresponding sand-shale sequences.

Figure 5.17: Effect of layer thickness on amplitude results (RMS versus maximum amplitude). Layer thickness used for modeling sequences is 1.5 cm for the left plot and 2.5 cm for the right plot. Points are color-coded by the net-to-gross ratios from their corresponding sand-shale sequences.

5.4.6.4 Effect of impedance contrast

Figure 5.8 illustrates sand and shale impedances used for modeling the sequences

in the training set. In this section, we investigate the effect on the attribute results of

varying impedance contrast between the sand and the shale. We simulate sequences

with two different sets of impedances. The original set represents a larger impedance

contrast between the sand and the shale (i.e., a smaller overlap in impedances), while

the other set represents a smaller contrast (i.e., a larger overlap) (Figure 5.18). . The

-0.04 -0.02 0 0.02 0.04-0.015

-0.01

-0.005

0

0.005

0.01

0.015


Second p

rincip

al com

ponent

1.5-cm thick layers

-0.04 -0.02 0 0.02 0.04-0.015

-0.01

-0.005

0

0.005

0.01

0.015


Second p

rincip

al com

poent

2.5-cm thick layers

0.4

0.45

0.5

0.55

0.6

net-to-gross ratios

classification

success rate = 0.88

classification

success rate = 0.89

0.13 0.135 0.14 0.145 0.150.0365

0.037

0.0375

0.038

0.0385

0.039

0.0395

0.04

0.0405

Maximum amplitude

RM

S a

mplit

ude

1.5-cm thick layers

0.13 0.135 0.14 0.145 0.150.0365

0.037

0.0375

0.038

0.0385

0.039

0.0395

0.04

0.0405

Maximum amplitude

RM

S a

mplit

ude

2.5-cm thick layers

0.4

0.45

0.5

0.55

0.6

net-to-gross ratios

classification

success rate = 0.87classification

success rate = 0.86


linear KPCA results and the amplitude results for the two impedance sets are shown in

Figure 5.19 and Figure 5.20, respectively. We show only results from sequences that

are all fining-upward, but with varying net-to-gross ratios. To quantify and compare

the effect of impedance contrast on KPCA and amplitude attributes, we compute the

success rates of classifying the seismograms into three net-to-gross classes, similar to

Section 5.4.6.3. The classification success rates are displayed in the lower corner of

each plot in both Figure 5.19 and Figure 5.20. When the impedances of the sand and

the shale become more overlapped (i.e., small contrast), the linear KPCA attributes

show little to no change, and their classification success rate stays almost exactly the

same. However, the smaller impedance contrast shows a significant effect on the

amplitude attributes by decreasing their success rate from 86% down to 52%.

Figure 5.18: Probability density functions of two sets of acoustic impedance showing small and large contrast (shown in red and green, respectively) between the sand and the shale (shown in solid and dash lines, respectively).

4 4.5 5 5.5 60

2

4

6

8

10


Pro

babili

ty d

ensity

sand

(large contrast)

shale

sand

(small contrast)


Figure 5.19: Effect of impedance contrast on linear KPCA results. Each plot represents the projections of seismograms onto the first two principal components of linear KPCA when the overlap between the sand and the shale impedances is small (left) or large (right). Points are color-coded by the net-to-gross ratios from their corresponding sand-shale sequences.

Figure 5.20: Effect of impedance contrast on amplitude results (RMS versus maximum amplitude). The overlap between the sand and the shale impedances is small in the left panel and large in the right panel. Points are color-coded by the net-to-gross ratios from their corresponding sand-shale sequences.

5.4.6.5 Effect of wavelet phase rotation

In this section, we investigate how KPCA attributes are affected by wavelet phase.

First, we use the synthetic seismogram generated at the well with a zero-phase Ricker

wavelet. Then we generate two additional seismograms using -20° phase and +40°

phase Ricker wavelets. These three seismograms at the well are our test traces. Linear

KPCA attributes of the three seismograms at the well and the Model I training set

-0.04 -0.02 0 0.02 0.04-0.015

-0.01

-0.005

0

0.005

0.01

0.015


Second p

rincip

al com

ponent

Impedances with a small overlap

-0.04 -0.02 0 0.02 0.04-0.015

-0.01

-0.005

0

0.005

0.01

0.015


Second p

rincip

al com

ponent

Impedances with a large overlap

0.4

0.45

0.5

0.55

0.6

0.65

classification

success rate = 0.88

net-to-gross ratios

classification

success rate = 0.89

0.12 0.125 0.13 0.135 0.14 0.145

0.035

0.036

0.037

0.038

0.039

Maximum amplitude

RM

S a

mplit

ude

Impedances with a small overlap

0.12 0.125 0.13 0.135 0.14 0.145

0.035

0.036

0.037

0.038

0.039

Maximum amplitude

RM

S a

mplit

ude

Impedances with a large overlap

0.4

0.45

0.5

0.55

0.6

0.65

classification

success rate = 0.86

classification

success rate = 0.52

net-to-gross ratios


(Section 5.4.5) are shown in Figure 5.21. Since the training set is generated using a

zero-phase Ricker wavelet, the only test trace that overlaps with this training set is the

one with zero phase. In addition to the training and the test traces, we also add the

KPCA representation of the zero trace (i.e., trace that has all elements equal to zero) as

a reference point. The angle between the vector from the reference point to the -20°

phase trace and the vector from the reference point to the zero-phase trace is equal to

20°. Also, the angle between the vector from the reference point to the +40° phase

trace and the vector from the reference point to the zero-phase trace is equal to 40°.

Note that this is also true when the attribute vectors are replaced by the actual seismic

traces. In this example, phase of the wavelet affect the linear KPCA attributes by

rotating each attribute point around the reference point by an angle of that absolute

phase value. The rotation is clockwise if the phase is positive and counterclockwise if

the phase is negative.

Figure 5.21: Effect of phase rotation on linear KPCA results. Points represent projections of the seismograms from the Model I training set and the well location onto the first two principal components from the linear KPCA. Each point is color-coded by the net-to-gross ratio from its corresponding sand-shale sequence. The three seismograms at the well with three different phases shown in black symbols are plotted on top of the projections of the training set. The projection of the zero trace is used as a reference point.

-0.1 0 0.1 0.2 0.3 0.4-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2


Second p

rincip

al com

ponent


0.4

0.45

0.5

0.55

0.6

Model I

zero trace

0 degree

-20 degree

+40 degree

20o

40o


5.5 Case study 2: Estimating sub-resolution reservoir properties

from a 2-D seismic section

In the previous case study (Section 5.4), we investigated the effect of net-to-gross

ratio, stacking pattern, saturating fluid, and reservoir thickness on the seismic

signatures by using real well log data, but with synthetic seismograms. In this case

study, we use real seismograms from a 2-D section extracted along a turbidite channel

from proximal to distal directions.

5.5.1 2-D seismic section

The 2-D seismic section is shown in Figure 5.22. The location of well A is marked,

and the proximal direction of the channel is on the left side of the plot. In this case

study, we still focus on the same fining-upward interval as discussed in the previous

case study (Section 5.4). This interval at well A corresponds to a seismic segment

shown in the black rectangle (Figure 5.22). By following the same top and bottom

marks of this interval, we extract seismic segments across the 2-D section.

Figure 5.22: A 2-D seismic section extracted along the turbidite channel from proximal (left) to distal (right) directions. The location of well A is marked.

5.5.2 Rock property calibration at well location

In order to generate realizations of sand-shale sequences and their corresponding

synthetic seismograms which will be used as a training set, we first need to estimate

transition matrices and rock properties for each lithologic state from the information at

Tim

e (

ms)

Trace #

0 10 20 30 40 50

Well A


the well. However, since we focus on the same fining-upward interval as discussed in

the previous case study (Section 5.4), we simply use the same matrices and rock

properties which are already estimated in Section 5.4.3 and follow the same steps

described in that section to simulate multiple realizations of sand-shale sequences. The

steps are summarized as follows: (1) select a transition matrix that represents the

desired stacking pattern, (2) use the matrix to generate arrangement of sand and shale

layers, and (3) assign rock properties to each layer by drawing from the distribution

corresponding to its lithologic state. In this case study, all the sand layers are assumed

to be oil-saturated. All three stacking patterns are also included in generating the

training set by following the same step shown in Section 5.4.3.2. However, the

transition-matrix sets for the top and bottom sub-sequences are defined as follows:

𝑇(𝑡𝑜𝑝 ) = 𝑎 1 − 𝑎𝑏 1 − 𝑏

, where a and b are drawn from a uniform distribution on [0,1],

𝑇(𝑏𝑜𝑡 ) = 𝑐 1 − 𝑐𝑑 1 − 𝑑

, where e and f are drawn from a uniform distribution on [0,1].

When we generate the training set, we obtain rock properties of the sediments that are

underlain and overlain the reservoir interval directly from well A.

In order to expand our training set, we also vary reservoir thickness which is

assumed to be uniformly distributed between 8.5m and 25.6 m (Figure 5.23, left

column). Net-to-gross ratios for all sequence models are also recorded, and their

overall distribution is shown in Figure 5.23 (right column).

Figure 5.23: Prior distributions for thickness (left) and net-to-gross ratio (right), assigned to sequence models when generating a training set.

10 15 20 250

1000

2000

3000

Thickness (m)

Counts

0 0.5 10

2000

4000

6000

Net-to-gross ratio

Counts


5.5.3 Non-stationarity when moving away from the well

For the two existing wells, each of which is located on each side of the 2-D

seismic section (Figure 5.24), we observe that the average impedances for both the

overburden and the underburden are different and that these impedances in the

proximal direction are lower than those in the distal direction (Figure 5.25). This

observation shows spatial non-stationarity of rock properties. Another example of the

non-stationarity for this same dataset was provided by Dutta (2009), who showed that

the sand-rich facies becomes better sorted, and its quartz/clay ratio increases spatially

from distal to proximal directions. We observe that the impedance of shale also

increases from the distal to proximal wells, but the increase is very small.

Figure 5.24: Relative locations of two existing wells (well A and well B). Well A (black symbol) is located in the distal direction along the channel, and well B (red symbol) is located in the proximal direction along the channel. Distance is measured from a reference point. The selected 2-D seismic line is shown in blue.

0 0.25 0.5 0.75

0

0.25

0.5

0.75

1

1.25

Distance (km)

Dis

tance (

km

)

well B

(proximal)

well A

(distal)


Figure 5.25: Average properties of the overburden and underburden of the interval of interest from proximal and distal wells. Three properties are shown: P-wave velocity (Vp), bulk density (RHOB), and acoustic impedance (AI).

We incorporate the non-stationarity of rock properties into our modeling steps as

follows. First, to account for spatial variations in the overburden and underburden, we

vary two additional parameters, which are referred to as the overburden and the

underburden multipliers. These multipliers are simply numbers drawn from uniform

distributions over ranges that are estimated from the wells, and these numbers are used

to multiply the overburden and underburden rock properties extracted from well A.

Second, to account for spatial variations of the sand and shale lithologic states within

the target zone, we assign standard deviations to the distributions of impedances for

both the sand state and the shale state so that these distributions cover such variations.

The distributions are shown in Figure 5.26. Note that since the lateral extent of our 2-

D section is not significantly large, the spatial variations in rock properties can be

dealt with by adjusting the distributions. However, when exploring areas significantly

away from the well such that rock properties change vastly, we recommend creating

additional training sets where the rock properties are chosen appropriately (e.g., using

a linear interpolation between two wells to estimate suitable rock properties). Then a

training set can be selected and applied to a group of test points that are located in the

same neighborhood. In the case where only one well is available, spatial trends in

0

1000

2000

3000

4000

5000

6000

distal well

proximal well

Vp RHOB AI Vp RHOB AI

(m/s) (kg/m3) (kg/m2s) (m/s) (kg/m3) (kg/m2s)

Overburden Underburden


sedimentological parameters and rock physics models are key tools for extrapolating

the rock properties of the lithologic states in the area away from the well (e.g., Dutta,

2009).

Figure 5.26: Probability density functions of acoustic impedance for the sand (blue) and the shale (red) lithologic states used in generating a training set.

5.5.4 Synthetic seismogram generation and seismic attribute extraction

After obtaining sequence models with assigned rock properties, we generate the

corresponding synthetic seismograms using the Kennett algorithm (Kennett, 1983) to

simulate full-waveform, normally-incident, reflected seismograms using a zero-phase

Ricker wavelet with a central frequency of 50 Hz. This frequency is selected to match

with the seismic data. The simulations are performed for all reverberations. These

seismograms serve as our training set. Since the wavelet of the real seismograms is not

a Ricker wavelet, we transform the real seismograms with their original wavelet to

match the training set by using

𝑆𝑟𝑒𝑎𝑙∗ 𝜔 = 𝑆𝑟𝑒𝑎𝑙 𝜔 ∙ 𝐻 𝜔 ,

where 𝐻 𝜔 is the Fourier transform of a filter function, and

𝐻 𝜔 =𝑆syn

well A 𝜔

𝑆realwell A 𝜔

.

In the above relations, 𝑆𝑟𝑒𝑎𝑙 and 𝑆𝑟𝑒𝑎𝑙∗ are the Fourier transforms of the seismogram

before and after applying the transfer function, respectively. 𝑆synwell A and 𝑆real

well A are the

4.6 4.8 5 5.2 5.4 5.60

2

4

6

8

10


Pro

babili

ty d

ensity

oil-saturated

sand

shale


Fourier transform of the synthetic seismogram at well A generated using a Ricker

wavelet and the Fourier transform of the real seismogram at well A with its original

wavelet, respectively. After applying the above relations, the Fourier transforms of the

filtered seismograms are transformed back to the time domain.

The filtered seismograms and all the seismograms from the training set are then

input into KPCA algorithms. Here we show only the results from KPCA with a linear

kernel. The projections of all the input seismograms onto the first two principal

components are shown in Figure 5.27. The projections of the training set are color-

coded by reservoir thickness, and the projections of the real seismograms are labeled

by their trace numbers. Each panel in Figure 5.28 shows the results where each

training point is color-coded by the parameter specified at the top of that panel. An

example of results from KPCA with a Gaussian kernel is shown in Figure 5.29, which

shows trends similar to those from the linear KPCA. In this case study, we do not

focus on an extensive comparison of performance of various kernels. Hereafter, we

refer to the linear KPCA attributes as the KPCA attributes, unless otherwise specified.

Figure 5.27: Projections of the training set and the test seismograms onto the first two principal components of KPCA with a linear kernel. The training points are color-coded by the reservoir thicknesses of their corresponding sand-shale sequences. The test points are labeled by their trace numbers.


Figure 5.28: Projections of the training set onto the first two principal components of KPCA with a linear kernel. In each panel, the training points are color-coded by the reservoir parameter specified at the top of each panel.

Figure 5.29: Projections of the training set onto the first two principal components of KPCA with a Gaussian kernel. In each panel, the training points are color-coded by the reservoir parameter specified at the top of each panel.


While there are noticeable trends in KPCA attributes associated with changes in

reservoir thickness and the underburden multiplier, there is almost no trend associated

with varying net-to-gross ratio (Figure 5.28). From this observation, net-to-gross ratio

does not seem to affect seismic signatures (i.e., KPCA attributes). However, when all

other parameters are fixed and only net-to-gross ratio is varying, a net-to-gross trend

becomes visible on the KPCA attribute space (Figure 5.30). The visibility of the net-

to-gross trend implies that each parameter does not affect seismic signatures equally

and that effects of some parameters can be so large that they overshadow effects of

other parameters.

Figure 5.30: Projections of the training set onto the first two principal components of KPCA with a linear kernel, when the values of underburden multiplier, overburden multiplier, and reservoir thickness are fixed. Points are color-coded by net-to-gross ratio.

Even though we show the KPCA results on the first two principal components, we

use the first ten components as our attributes in all later analyses. The number of

components included as attributes is chosen based on the eigenvalues of the KPCA

results (Figure 5.31). The first ten principal components account for 99% of the total

variance.

0 2000 4000 6000 8000 10000 12000-1

-0.5

0

0.5

1x 10

4


Second p

rincip

al com

ponent

Color-coded by net-to-gross ratio

0.2

0.4

0.6

0.8


Figure 5.31: The first 20 eigenvalues of the linear KPCA results.

5.5.5 Sensitivity analysis

In the previous case study (Section 5.4), we generate sequence models where we

vary one reservoir parameter at a time. In this case study, we vary all parameters at the

same time. Since different parameters can have different impact on seismic signatures,

a parameter with a greater impact may overshadow subtle effects from other

parameters (e.g., Figure 5.28). In this section, in order to rank the input parameters

(i.e., reservoir properties) according to their impacts, we perform a sensitivity analysis

by following the method of Fenwick et al (2012), in which the estimated sensitivity for

the ith

parameter 𝑠(𝑝𝑖) using

𝑠 𝑝𝑖 =1

𝐾

𝑑 𝐹,𝑘

𝑑 𝐹,𝑘(95)

𝐾

𝑘=1.

In the above equation, the integer i ranges from 1 to 4, and these numbers correspond

to the following parameters: overburden multiplier, underburden multiplier, reservoir

thickness, and net-to-gross ratio. The training set is divided into 3 classes (i.e., 𝐾 = 3)

based on their Euclidean distances in the KPCA space by using K-mean clustering. 𝑐𝑘

is the kth

class, and the integer k ranges from 1 to 3. For each parameter, 𝑑 𝐹,𝑘 is the L-1

norm measure of difference (or distance) between its class-conditional empirical

cumulative distribution 𝐹 (𝑝𝑖|𝑐𝑘) and its prior empirical distribution 𝐹 (𝑝𝑖), and this

difference can be estimated by integrating the area between the two curves. 𝑑 𝐹,𝑘(95)

is the

95th

-percentile of bootstrapped L-1 norm distances. Note that for a given parameter 𝑝𝑖 ,

5 10 15 200

2

4

6

8

10

12x 10

8

Principal component

Eig

envalu

e


if there exists a class 𝑘 where 𝑑 𝐹,𝑘 ≥ 𝑑 𝐹,𝑘(95)

, then the parameter 𝑝𝑖 is defined as being

sensitive to the classification based on the distances in the KPCA space.

The strategy behind this method is that if a parameter has an impact on seismic

signatures which, in our study, are represented by projections of seismograms into the

principal-component space, then this parameter will separate the projected

seismograms into different classes. Consequently, the statistical distributions of the

sample values of this parameter among all classes will be dissimilar. 𝑑 𝐹,𝑘 is used as a

measure of such dissimilarity. Note that this sensitivity analysis does not account for

any interaction (or co-existence) between two or more parameters that can jointly

affect seismic signatures (Fenwick et al., 2012).

Figure 5.32 illustrates a comparison between a class-conditional empirical

cumulative distribution and a prior empirical distribution of the overburden multiplier,

and the shaded area is the difference measure between the two distributions. Figure

5.33 shows results of our sensitivity analysis. For each parameter, each bar represents

the difference 𝑑 𝐹,𝑘 normalized by 𝑑 𝐹,𝑘(95)

for class 𝑘. The red line represents a threshold.

For a parameter, if there is any class where the normalized difference exceeds this line,

then that parameter is considered to be an impacting parameter. Out of the four

parameters, only the net-to-gross ratio is considered a non-impacting parameter

because none of its classes have the normalized difference exceed the threshold. This

result is consistent with our previous observation that net-to-gross ratio does not seem

to affect seismic signatures (Figure 5.28). However, as previously shown, when all

other parameters are fixed and only net-to-gross ratio is varying, a net-to-gross trend

becomes visible on the KPCA attribute space (Figure 5.30). The pareto plot in Figure

5.34 shows the ranking of the four reservoir parameters according to their estimated

sensitivity values (𝑠). Reservoir parameters in decreasing of the sensitivity values are

reservoir thickness, underburden multiplier, overburden multiplier, and net-to-gross

ratio.

As previously noted, some parameters can overshadow the subtle effects on

seismic signatures from the other parameters. Therefore, for each seismic test trace we


first estimate probability density functions of the three parameters: reservoir thickness,

underburden multiplier, and overburden multiplier. Then for each test trace we create

a new training set by varying the net-to-gross parameter within the same range, but the

thickness and the two multipliers are varied within smaller intervals around their mean

values. Note that a “new” training set may not be needed if the original training set

contains a sufficiently large numbers of points. In this case, we can simply select a

subset of training points that have the values of the thickness and the multipliers fall

within the desired ranges. By limiting the values of an influential parameter within a

smaller range, we can have a better look at the effect of other parameters. Even though

we are not interested in the values of both the underburden multiplier and the

overburden multiplier, the effects of both parameters overshadow the effect from net-

to-gross ratio. Therefore, we also need to estimate posterior distributions for both

multipliers.

In summary, for each seismic test trace we will estimate probability density

functions of the first three parameters which are reservoir thickness, underburden

multiplier, and overburden multiplier, and after doing so we create a new training set

in which these previously estimated parameters will vary uniformly over small ranges

around their mean values. The new training set is then used to estimate net-to-gross

ratio.

Figure 5.32: Comparison of two empirical cumulative distributions for the overburden multiplier values in class 3. The prior distribution and the class-conditional distribution are shown in blue and red, respectively. The shaded area represents area between the two curves.

0.8 0.85 0.9 0.95 1 1.05 1.10

0.2

0.4

0.6

0.8

1

Overburden multiplier

Cum

ula

tive p

robabili

ty

prior empirical distribution

class-conditional

empirical distribution


Figure 5.33: Normalized difference measure between a class-conditional empirical cumulative distribution and a prior empirical distribution for a parameter. The red line represents a threshold used in determining whether a parameter has a significant impact on the output response (i.e., seismic signature).

Figure 5.34: Pareto plot of estimated sensitivity values. Parameters are ranked according to their sensitivity values.

5.5.6 Property estimation of the 2-D seismic section

In the previous sections, we showed the KPCA results of both the training set and

the test set (Figure 5.27 – Figure 5.28), then performed sensitivity analyses, and

presented the order of the reservoir parameters to be estimated which are reservoir

thickness, underburden multiplier, overburden multiplier, and net-to-gross ratio. In this

section, we use these results to estimate reservoir properties of the 2-D seismic section

(Figure 5.22).

To estimate a probability density of a reservoir parameter 𝑝 for a test trace, we use

a k-nearest neighbor (kNN) method which finds k closet points (in a training set) to a

0 4 8 12

Underburden multiplier


Reservoir thickness

Net-to-gross ratio

Normalized difference between distributions

class 1

class 2

class 3

sensitivity threshold

0 2 4 6 8 10 12

Net-to-gross ratio



Reservoir thickness

Estimated sensitivity value


query (or test) point. Assume that the value of 𝑝 is subdivided into equal 𝑛𝑝classes,

each of which is denoted by 𝑝𝑙 for 𝑙 = 1, 2, …, 𝑛𝑝 . We want to find the posterior

probability of parameter 𝑝 given a test point 𝑥 or 𝑃(𝑝𝑙|𝑥). Using the kNN method, this

probability according to Bayes’ theorem is written as

𝑃 𝑝𝑙 𝑥 =𝑃(𝑥, 𝑝𝑙)

𝑃(𝑥, 𝑝𝑗 )𝑛𝑝

𝑗=1

= 𝑘𝑙

𝑘 ,

where 𝑃 𝑥, 𝑝𝑙 is the joint probability estimated as (𝑘𝑙 𝑛 /𝑉. Within a volume 𝑉

surrounding the test point 𝑥, 𝑘𝑙 is the number of the training points that belong to class

𝑝𝑙 , and 𝑛 is the total number of the training points (Tobin et al., 2009).

Using the above equation, we first estimate probability densities for reservoir

thickness, underburden multiplier, and overburden multiplier from the linear KPCA

results (Figure 5.27). Their estimated densities and their mean values are shown in

Figure 5.35 – Figure 5.40. The subsequent estimated probability density and the mean

values for net-to-gross ratio are shown in Figure 5.41 – Figure 5.42. Note that the

estimated underburden multipliers generally increase from left (i.e., proximal) to right

(i.e., distal). This trend is consistent with the observation that the average impedance

of the underburden in the proximal direction is smaller than the impedance in the

distal direction (Figure 5.25). In order to evaluate the performance of our property

estimation, we add three additional seismic traces to the test set, each of which

corresponds to a thin sand-shale sequence with reservoir parameters listed in Table 5.1.

Table 5.1: Summary of reservoir parameters for three seismic traces which are added to the test set for evaluating the performance of property estimation.

Trace # Underburden

multiplier

Overburden

multiplier

Reservoir

Thickness

Water

saturation

Net-to-

gross

ratio

Stacking

pattern

55 (real) 1 1 17 0 ~0.48 Fining-

upward

56

(synthetic)

Well B Well B 17 0 ~0.48 Fining-

upward

57

(synthetic)

0.95 0.87 17 0 ~0.48 Coarsening-

upward


Figure 5.35: Estimated probability density functions for reservoir thickness for all seismic traces in the test set. The last three traces (#55-57) correspond to the additional three traces which are added for evaluating the performance of our property estimation. Each density function is shown as a column-wise color scale.

Figure 5.36: The mean values of reservoir thickness for all seismic traces in the test set. The last three points are added for evaluating the performance of our property estimation.

Figure 5.37: Estimated probability density functions for the underburden multiplier for all seismic traces in the test set. The last three traces (#55-57) correspond to the additional three traces which are added for evaluating the performance of our property estimation. Each density function is shown as a column-wise color scale.

Figure 5.38: The mean values of underburden multiplier for all seismic traces in the test set. The last three points are added for evaluating the performance of our property estimation.

Trace #

Reserv

oir

thic

kness (

m)

Probability

density

5 10 15 20 25 30 35 40 45 50 55

25.6

22.2

18.8

15.4

11.9

8.53 0

0.1

0.2

5 10 15 20 25 30 35 40 45 50 558.5

14.5

20.5

25.6

trace #

Reserv

oir

thic

kness (

m)

Trace #

Underb

urd

en

multip

lier

Probability

density

5 10 15 20 25 30 35 40 45 50 55

1.1

1.04

0.98

0.92

0.86

0.8

2

46

8

10

5 10 15 20 25 30 35 40 45 50 550.85

0.9

0.95

1

1.05

trace #

Underb

urd

en

multip

lier


Figure 5.39: Estimated probability density functions for the overburden multiplier for

all seismic traces in the test set. The last three traces (#55-57) correspond to the additional three traces which are added for evaluating the performance of our property estimation. Each density function is shown as a column-wise color scale.

Figure 5.40: The mean values of overburden multiplier for all seismic traces in the

test set. The last three points are added for evaluating the performance of our property estimation.

Figure 5.41: Estimated probability density functions for net-to-gross ratio for all

seismic traces in the test set. The last three traces (#55-57) correspond to the additional three traces which are added for evaluating the performance of our property estimation. Each density function is shown as a column-wise color scale.

Figure 5.42: The mean values of net-to-gross ratio for all seismic traces in the test set. The last three traces (#55-57) correspond to the additional three traces which are added for evaluating the performance of our property estimation.

Trace #

Overb

urd

en

multip

lier

Probability

density

5 10 15 20 25 30 35 40 45 50 55

1.1

1.06

1.02

0.98

0.94

0.924681012

5 10 15 20 25 30 35 40 45 50 55

0.95

1

1.05

trace #

Overb

urd

en

multip

lier

Trace #

Net-

to-g

ross r

atio

Probability

density

5 10 15 20 25 30 35 40 45 50 55

1

0.8

0.6

0.4

0.2

0

1

2

3

4

5

5 10 15 20 25 30 35 40 45 50 550

0.2

0.4

0.6

0.8

1

trace #

Net-

to-g

ross r

atio


To evaluate the performance of property estimation, we summarize and compare

the result at the well and the results of the three additional points with the true

parameter values (Figure 5.43 – Figure 5.45). Even though the underburden and the

overburden multipliers are not the parameters of interest, we also include their results

here in order to show the performance of the estimation. In general, the true values are

captured within the estimated densities for all the four test traces. Trace #56 shows a

bimodal distribution of the estimated thickness, which is different from the other three

traces. This may be because we generate trace #56 by using rock properties of the

underburden and the overburden sediments from well B, while the rest of the traces

(including the training set) are generated using properties from well A (either with or

without multipliers). Note that since trace #56 is associated with well B, we do not

include the “true” multipliers in Figure 5.44 and Figure 5.45.

Figure 5.43: The estimated probability densities of reservoir thickness for seismic

traces at the well and trace #55-57, which are added for evaluating the performance of our property estimation. The true thicknesses for all four test traces are marked by the magenta line.

Figure 5.44: The estimated probability densities of underburden multiplier for seismic

traces at the well and trace #55-57, which are added for evaluating the performance of our property estimation. The true multipliers for three test traces are marked by the lines shown in the same colors as their densities. Note that since trace #56 is associated with well B, the “true” multiplier of well A is irrelevant and thus not shown here.

10 15 20 250

0.05

0.1

0.15

0.2

Reservoir thickness (m)

Pro

babili

ty d

ensity

well A

Trace 55

Trace 56

Trace 57

0.85 0.9 0.95 1 1.050

2

4

6

8

10

12


Pro

babili

ty d

ensity

well A

Trace 55

Trace 56

Trace 57


Figure 5.45: The estimated probability densities of overburden multiplier for seismic

traces at the well and trace #55-57, which are added for evaluating the performance of our property estimation. The true multipliers for three test traces are marked by the lines shown in the same colors as their densities. Note that since trace #56 is associated with well B, the “true” multiplier of well A is irrelevant and thus not shown here.

Figure 5.46: The estimated probability densities of net-to-gross ratio for seismic

traces at the well and trace #55-57, which are added for evaluating the performance of our property estimation. The true net-to-gross ratios for all four test traces are marked by the magenta line.

When generating the training set, we account for the variation in rock properties of

the underburden and the overburden (Section 5.5.3). If this variation (non-stationarity)

is not accounted for, the estimated net-to-gross ratios can be erroneous. For example,

Figure 5.47 shows the estimated probability densities of net-to-gross ratio for trace #

57 for both when the non-stationarity is taken into account and when the non-

stationarity is not taken into account. For trace #57, when the non-stationarity is

ignored, the estimated probability density of net-to-gross ratio shifts and the density

curve (almost) does not cover the true net-to-gross ratio.

0.95 1 1.050

2

4

6

8

10

12

14

Overburden multiplierP

robabili

ty d

ensity

well A

Trace 55

Trace 56

Trace 57

0.2 0.4 0.6 0.80

1

2

3

4

Net-to-gross ratio

Pro

babili

ty d

ensity

well A

Trace 55

Trace 56

Trace 57


Figure 5.47: The estimated probability densities of net-to-gross ratio for seismic trace

# 57 for both when the non-stationarity is taken into account (blue curve) and when the non-stationarity is not taken into account (red curve). The true net-to-gross ratio for trace #57 is marked by the blue line.

5.5.7 Discussion

In the previous sections, we presented a workflow for reservoir property

estimation in thinly-bedded sand-shale reservoirs and showed results of applying this

workflow to real seismic data. In this section, we provide some additional guidelines

for real application.

In the second case study, in order to match real seismic traces with the training set,

we transform those traces with their original wavelet into the traces with the Ricker

wavelet. The advantage of this transformation is that we can use the same training set

for other sets of real seismic traces that have frequency ranges similar to the range in

the training set. Even though variation in wavelets is not accounted for in this study, it

can be incorporated into the analysis by creating multiple filters for seismograms

(Section 5.5.4). Alternatively, instead of applying a transformation to real seismic data,

a wavelet can be estimated from the data and then used to create a training set.

To reduce the effects of reservoir thickness, overburden multiplier, and

underburden multiplier, we decrease the ranges of these parameters and re-create new

training sets. As a result, seismic signatures become more sensitive to parameters such

as net-to-gross ratio. However, instead of decreasing the ranges of the seismic-

signature parameters around their mean values, an alternative way to appropriately

incorporate their posterior distributions into estimation of the next parameters is to

0.2 0.4 0.6 0.80

1

2

3

4

5

Net-to-gross ratio

Pro

babili

ty d

ensity

not accounting for

non-stationarity

accounting for

non-stationarity


perform Monte Carlo simulations. By doing so, the errors of the previously estimated

parameters are propagated to the estimation of the next parameters.

Prior knowledge of the geology of the area and trends in the sedimentological

parameters (e.g., sorting) can be used to create training sets. For example, when there

is only one well available, if we know that sorting deteriorates in one particular

direction away from the well, then we can use rock physics relations to predict rock

properties and assign these properties to the appropriate lithologic states.

5.6 Conclusions

We apply our workflow for property estimation of thin sand-shale sequences to

real data from channelized turbidite deposits offshore Equatorial Guinea, West Africa.

We focus our analysis on an interbedded sand-shale interval which has bed thickness

below seismic resolution. Within this interval, the average wavelength to thickness

ratio is approximately 35 for massive sandstone, but can be greater than 100 for

interbedded thin-bedded sandstone and mudstone. In the first case study, where real

well log data and synthetic seismograms are used, results show that the effects of net-

to-gross ratio, saturation, thickness, and stacking pattern on seismic signatures can be

distinguished on the KPCA and the amplitude attribute space. However, of the two

types of attributes, the amplitude attribute is more sensitive to noise. In the second

case study, where real well log data and a real 2-D seismic section are used, seismic

signatures are also affected by the properties of the overburden and the underburden

which are non-stationary across the 2-D section. Even though we are not interested in

these two parameters, the sensitivity analysis shows that their impact on seismic

signature is greater than net-to-gross ratio. Therefore, the effect of these two

parameters needs to be reduced before net-to-gross estimation.

The workflow presented here can be extended by combining knowledge of

sedimentological trends For example, if only well B (proximal) is available, we can

incorporate trends such as sorting, quartz/clay ratio (Dutta, 2009) and use rock physics

relations to predict rock properties of lithologic states in the well A area (distal). Then


these properties can be used to generate training sets for estimating reservoir

properties of sub-resolution sequences in the area around well A.

Additional topics for future research include incorporating saturation effect,

finding the optimal values of parameter k (i.e., number of neighbors) in the kNN

method, comparing performance of other kernels for KPCA algorithms, and

minimizing computational time.


We would like to thank Hess Corporation for providing the data. This work was

supported by the Stanford Rock Physics and Borehole Geophysics project and the

Stanford Center for Reservoir Forecasting.

5.8 References

Dutta, T., Mukerji, T., and Mavko, G., 2007, Quantifying spatial trends of sediment

parameters in channelized turbidite, West Africa: SEG Expanded Abstracts, 26,

1674-1678.

Dutta, T., 2009, Integrating sequence Stratigraphy and rock-pshycis to interpret

seismic amplitudes and predict reservoir quality, Ph.D. Thesis, Stanford University.

Fenwick, D., Scheidt, C., and Caers, J., 2012, A distance-based generalized sensitivity

analysis for reservoir modeling: Computational Geosciences.

Jobe, Z.R., Lowe, D.R., and Uchytil, S.J., 2011, Two fundamentally different types of

submarine canyons along the continental margin of Equatorial Guinea: Marine and

Petroleum Geology, 28, 843-860.

Kennett, B., 1983, Seismic wave propagation in stratified media: Cambridge

University Press, Cambridge.

Lowe, D.R., 2004, Report on core logging, lithofacies, and basic sedimentology of

Equitoral Guinea: Hess internal report.

Thomas, E. C., and Stieber, S. J., 1975, The distribution of shale in sandstones and its

effect upon porosity: 16th Annual Logging Symposium, SPWLA, paper T.


Tobin, K.W., Chaum, E., Gregor, J., Karnowski, T.P., Price, J.R., and Wall, J., 2009,

Image informatics for clinical and preclinical biomedical analysis: Computational

Intelligence in Medical Imaging, 1st ed. CRC Press.

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