hierarchical simulation of multiple-facies reservoirs
TRANSCRIPT
HIERARCHICAL SIMULATION OF
MULTIPLE-FACIES RESERVOIRS USING
MULTIPLE-POINT GEOSTATISTICS
A REPORT
SUBMITTED TO THE DEPARTMENT OF PETROLEUM
ENGINEERING
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE
By
Amisha Maharaja
June 2004
I certify that I have read this report and that in my
opinion it is fully adequate, in scope and in quality, as
partial fulfillment of the degree of Master of Science in
Petroleum Engineering.
Andre Journel(Principal advisor)
ii
Abstract
Joint simulation with presently available multiple-point geostatistical simulation al-
gorithms leads to poor shape reproduction as the number of facies increases. This is
because the training image (Ti) cannot depict with enough replicates all alternative
patterns that can be found. Moreover, the size of the Ti itself and that of the tem-
plate required to capture large-scale structures in the Ti is limited due to memory
restriction, especially in 3D.
Hierarchical simulation of facies with distinct shapes and spatial continuity is
proposed to overcome the disadvantages of joint simulation. The idea is to iden-
tify a hierarchy of deposition within the reservoir using geologic rules of deposition
and simulate the facies accordingly. This amounts to first simulating the large-scale
structures and then simulating smaller structures conditional to the pre-simulated
large-scale structures. The hierarchical approach is demonstrated using synthetic 2D
and 3D meandering fluvial reservoirs and the results are compared with that of joint
simulation. Finally, the hierarchical simulation techinique is applied to a real life
dense data set from the Rhine-Meuse delta.
The advantage of hierarchical approach is better reproduction of large-scale struc-
tures by reducing the size of the Ti and the number of facies being simulated si-
multaneously. Most of the memory demand during mp-simulation comes from the
size of the search-tree, which in turn depends on the size of the Ti, the size of the
data template, and the number of facies in the Ti. Large-scale structures require a
bigger template to capture them, however that increases the size of the search tree.
In hierarchical simulation, the increase in the size of the search tree due to a larger
template is compensated by the reduction in the number of facies being simulated
simultaneously. Once the large-scale structures are simulated the smaller structures
iii
are simulated conditioned to the former. The Ti for simulating the small-scale fea-
tures need not be very large and rich, thereby further reducing the size of search tree.
The hierarchical approach is geologically sound because the sequence of simulations
follows the natural sequence of deposition.
iv
Contents
Abstract iii
Acknowledgements v
Table of Contents vi
List of Tables viii
List of Figures ix
1 Introduction 1
2 Joint Simulation 5
2.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Hierarchical Simulation 12
3.1 Hierarchical simulation of three facies . . . . . . . . . . . . . . . . . . 12
3.1.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Hierarchical simulation of four facies . . . . . . . . . . . . . . . . . . 13
3.2.1 Two-step cookie-cut simulation method . . . . . . . . . . . . . 13
3.2.2 Three-step hierarchical simulation method . . . . . . . . . . . 15
3.2.3 Two-step hierarchical simulation method . . . . . . . . . . . 15
3.3 3D Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Implementation of hierarchical simulation 27
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5 Rhine-Meuse Delta Case Study 29
5.1 Introduction to the Data Set . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6 Conclusions and Future Work 30
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
References 32
vii
List of Tables
2.1 Input parameters and simulation statistics for the joint simulation of
2, 3 and 4 facies, see Figures 2.2, 2.4, 2.6, and 2.8. The servo-system
parameter is in [0,1], with 1 corresponding to maximum control. The
parameters ‘Nodes in data template’ and ‘Radii of search ellipsoid’
determine the size and geometry of the data template. ‘Average nodes
retained in template’ indicates the average number of nodes that were
retained in the data template. . . . . . . . . . . . . . . . . . . . . . . 8
3.1 Input parameters and simulation statistics for the Two-step cookie-cut
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Input parameters and simulation statistics for the Three-step hierar-
chical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Input parameters and simulation statistics for the Two-step hierarchi-
cal approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
viii
List of Figures
1.1 The morphological elements of a meandering-river system. . . . . . . 4
2.1 Binary training image of channels (100 x 300). Green: channel 22%;
grey: floodbasin 78% . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Unconditional realization using two facies training image (100 x 100).
Green: channel 29.9%; grey: floodbasin 70.1% . . . . . . . . . . . . . 7
2.3 Three facies training image (100 x 300). Green: channel 22%; blue:
levee 7%; grey: floodbasin 71% . . . . . . . . . . . . . . . . . . . . . 9
2.4 Joint simulation of three facies using a 3-facies training image (100 x
100). Green: channel 28.9%; blue: levee 7.6%; grey: floodbasin 63.5% 9
2.5 Three facies training image (100 x 300). Green: channel 20%; red:
crevasse 7%; grey: floodbasin 73% . . . . . . . . . . . . . . . . . . . . 10
2.6 Joint simulation of three facies using a 3-facies training image (100 x
100). Green: channel 29.8%; red: crevasse 7.1%; grey: floodbasin 63.1% 10
2.7 Four facies training image (100 x 300). Green: channel 21%; blue:
levee 6%; red: crevasse 5%; grey: floodbasin 68% . . . . . . . . . . . 11
2.8 Joint simulation of four facies using a 4-facies training image (100 x
100). Green: channel 25%; blue: levee 8%; red: crevasse 5%; grey:
floodbasin 61.8% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 Hierarchical simulation of only levee and floodbasin using a 3-facies Ti
(100 x 100). Channels were previously simulated (Figure 2.2) and set
as hard data. Green: channel 30%; red: levee 7%; grey: floodbasin 63% 17
ix
3.2 Hierarchical simulation of only crevasse and floodbasin using a 3-facies
Ti (100 x 100). Channels are previously simulated (Figure 2.2). Green:
channel 30%; red: crevasse 8%; grey: floodbasin 62% . . . . . . . . . 18
3.3 Combining Figures 3.1 and 3.2 using cookie-cut. Crevasse erodes levees
and floodbasin. Green: channel 30%; blue: levee 7%; red: crevasse 8%;
grey: floodbasin 55% . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Flowchart for two-step cookie-cut simulation method . . . . . . . . . 21
3.5 Step 3 of three-step method: Simulation of only crevasse and floodbasin
using a 4-facies Ti. The channels were simulated in step 1 (Figure 2.2)
and levees were simulated in step 2 (Figure 3.1) and set as hard data.
Simulated facies proportions:- Green: channel 30%; blue: levee 7%;
red: crevasse 6%; grey: floodbasin 57% . . . . . . . . . . . . . . . . . 22
3.6 Flowchart for three-step hierarchical simulation method . . . . . . . . 22
3.7 Step 2 of two-step hierarchical simulation method. Simulation of crevasse
and floodbasin using 4-facies Ti. The channel and levee have been pre-
viously simulated in step 1 and set as hard data (Figure 2.4). Simu-
lated facies proportions: Green: channel 28.9%; blue: levee 7.6%; red:
crevasse 6.4%; white: floodbasin 57% . . . . . . . . . . . . . . . . . . 23
3.8 Flowchart for two-step hierarchical simulation method . . . . . . . . . 24
3.9 Four facies training image (100 x 100 x 100). Green: channel 25%;
blue: pointbar 4%; red: levee 2%; grey: floodbasin 69% . . . . . . . . 25
3.10 Hierarchical simulation of channel, pointbar, levee and floodbasin (100
x 100 x 100). Green: channel 28%; blue: pointbar 5%; red: levee 4%;
grey: floodbasin 63% . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
x
Chapter 1
Introduction
Stochastic simulation is widely used for generating heterogeneous reservoir models.
Traditional variogram-based simulation methods utilize the correlation between only
two points at a time and therefore cannot simulate curvilinear structures. Exam-
ples are SGS and SIS methods. Multiple-point geostatistical simulation techniques
consider the relation between three or more points taken together and are able to
reproduce curvilinear structures and account for complex patterns between the vari-
ables being simulated.
The pros and cons of various methods that utilize multiple-point geostatistics is
discussed in Strebelle (2000). Boolean object-based methods can reproduce the curvi-
linear geometries but are difficult to condition to dense data. Pixel-based methods are
easier to condition to dense data from different sources. Simulated annealing, MCMC
simulation and similar algorithms are pixel-based but iterative techniques and their
rate of convergence is not known a priori.
The first non-iterative multiple-point algorithm was suggested by Guardiano and
Srivastava (1993) in which mp-statistics were obtained directly by scanning a train-
ing image (Ti). However, the algorithm was extremely CPU demanding because
the entire Ti had to be scanned completely at each unsampled node to obtain con-
ditional probability distribution for that node. Strebelle (2000, 2002) proposed an
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CHAPTER 1. INTRODUCTION 2
algorithm snesim which was based on the idea of Guardiano and Srivastava of deriv-
ing probabilities directly from a Ti, however, it required scanning the Ti only once
and cataloguing the conditional probabilities using a dynamic data structure called
search-tree. The snesim algorithm is pixel-based, non-iterative, and general so that
any random geometry can be accomodated.
The size of the search-tree depends on three factors
• Size of the training image
• Size of the data template used to scan the training image
• Total number of facies in the training image
Consider a categorical variable Z(u) valued in 1, . . . , K. Let NTI be the total
number of locations in the Ti. Since for a given data event size j, there can not be
more than NTI different data events in the Ti, a (crude) upper-bound of the memory
demand of the search tree is:
Memory Demand ≤J∑
j=1
min(Kj , NTI)
Hydrocarbon reservoirs often consist of multiple facies with different shapes and
sizes. For example, a meandering fluvial system typically contains channels, point-
bars, levees, crevasse splays and floodbasin (Figure 1.1). Note that the geologic
definition of facies is “Accumulation of deposits that exhibits specific characteristics
and grades laterally into other sedimentary accumulations that were formed at the
same time but exhibit different characteristics (Leet, 1982).” In this paper, the term
‘facies’ refers to the morphological elements of a depositional system that have a dis-
tinct shape.
The shapes and proportions of these different facies vary greatly from one fluvial
system to another. Moreover, the facies are related to each other by some definite
geologic rules, for example, crevasse splays must be attached to channel belt. In mp-
simulation, the shapes, relative proportions and the spatial dependence between the
CHAPTER 1. INTRODUCTION 3
different facies is conveyed through a Ti. Fluvial systems tend to be very complex,
hence a large and rich Ti is required if all the facies are simulated jointly, which can
be very memory demanding. Instead of simulating all the facies jointly, they can
be simulated sequentially. Strebelle (2000) proposed a hierarchical method for sim-
ulating four fluvial facies namely, channels, levees, crevasse splays, and floodbasin.
However, the specific hierarchy used to simulate the facies and the manner in which
the hierarchy was implemented is different from the implementation presented in this
report, see Chapter 4 for details. The hierarchy used here is based on geologic rules
of deposition of the facies, see Chapter 3
Once the shapes of the facies are simulated with the correct proportion and spatial
arrangement, sand and mud with varying net-to-gross can be simulated within each
facies using traditional variogram based algorithms such as SGS. This is similar to
first simulating the different containers and then their specific contents. The net-to-
gross ratio, defined as the percentage of sand, is generally highest in the channels,
followed by the crevasse splays and levees. Floodbasin contains the largest volume of
fine sediments in the fluvial system.
Because this is a pixel-based stochastic simulation, conditioning to dense data of
various support sizes is easier. Moreover, a measure of uncertainty can be attained by
simulating multiple realizations using the same Ti, or by using multiple realizations
of several Tis depicting alternative plausible geological scenarios.
Chapter 2 shows results of joint simulation of two, three and four facies reservoirs.
Hierarchical simulation of three and four facies reservoirs is introduced in Chapter
3. Three alternative approaches are discussed for the 2D reservoir and the results
are compared with that of joint simulation. Finally, a 3D example is presented. The
implementation of hierarchical simulation in the snesim algorithm is discussed in
Chapter 4. The Rhine-Meuse delta case-study is presented in Chapter 5. Conclusions
and recommendations for future work are presented in Chapter 6.
Chapter 2
Joint Simulation
To perform joint simulation of multiple facies, all the facies must be provided in the Ti
exhibiting the proper relationship between these facies. The proportion of the facies
in the Ti need not be equal to the desired simulated proportions as the latter propor-
tions can be controlled by a servo-system. Fluvial systems can be quite complicated
because facies of different shapes and sizes are involved, hence it is desirable to use a
Ti that is larger than the size of the simulation grid to provide enough replicates of any
particular pattern or structure. This comes at a cost of larger RAM memory demand.
A binary training image consisting of channel and floodbasin (Figure 2.1) is used
to simulate the realization shown in Figure 2.2. All the training images in this paper
have been generated using the fluvsim algorithm developed by Deutsch and Tran
(2002). It is important that the Ti provides the correct channel attributes such as,
width, thickness, and sinuosity, to get the desired results. These attributes can be
obtained directly from seismic amplitude maps, outcrop analogues or inferred from
well data (Bridge and Tye, 2000). Moreover, in order to simulate long, thin, contin-
uous channels, it is important to provide a large Ti. In this case, the Ti is 100 x 300
pixels, while the simulation grid is 100 x 100 pixels.
The 3 facies channel, levee, and floodbasin are jointly simulated (Figure 2.4) us-
ing a 3-facies channel-levee-floodbasin Ti (Figure 2.3). Similarly, channel, crevasse,
and floodbasin are jointly simulated (Figure 2.6) using the corresponding 3-facies
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CHAPTER 2. JOINT SIMULATION 6
channel-crevasse-floodbasin Ti (Figure 2.5). Finally, all four facies are jointly simu-
lated (Figure 2.8) using a 4-facies Ti (Figure 2.7).
2.1 Comments
Table 2.1 summarizes the input parameters and simulation statistics for the joint
simulation approach. The poor quality of results from joint simulation of more than
two facies is evident from the examples in Figures 2.4, 2.6, and 2.8. The complexity
of the Ti in Figure 2.5 is greater than that in Figure 2.3 because the crevasse splays
have a distinctly different shape than the channels. The crevasse splays of Figure 2.5
are discontinuous, fan-like bodies, while the levees of Figure 2.3 are fairly continuous
facies that border the channels with the same elongated rectangular shape. Hence,
the joint simulation of a channel-levee-floodbasin system (Figure 2.4) gives better
results than that of a channel-crevasse-floodbasin system (Figure 2.6).
Since the Ti is three times as long as the simulation grid, the simulated channels
and levees of Figure 2.4 are reasonably continuous. Moreover, the levees and crevasse
splays are simulated close to the channels as specified by the training images. The
servo-system has been enforced, hence the target facies proportions are correctly re-
produced, see Table 2.1. The shape of the crevasse splays is adequately reproduced
in both Figures 2.6 and 2.8.
When the four-facies are simulated jointly (Figure 2.8), the results are much poorer
as the complexity of the Ti has increased considerably. The quality of results can be
improved by using a much larger Ti as the variety of patterns found in the Ti as well
as their number of replicates would increase, however for the same reason the size
of the search tree and hence the RAM demand will increase. Moreover, to ensure a
good reproduction of the large scale features a large data template is required, which
in case of 3 or more facies quickly leads to a large RAM demand, especially in 3D.
Multiple-grid simulation approach (Tran, 1994) has been proposed as a work-around
for simulating large-scale structure, however, a larger template size is still desirable.
CHAPTER 2. JOINT SIMULATION 7
Figure 2.1: Binary training image of channels (100 x 300). Green: channel 22%; grey:floodbasin 78%
Figure 2.2: Unconditional realization using two facies training image (100 x 100).Green: channel 29.9%; grey: floodbasin 70.1%
CHAPTER 2. JOINT SIMULATION 8
Parameters 2 facies 3 facies 3 facies 4 faciesFigure 2.2 Figure 2.4 Figure 2.6 Figure 2.8
Training Image 2 facies 3 facies 3 facies 4 faciesFigure 2.1 Figure 2.3 Figure 2.5 Figure 2.7
Ti facies proportion(%) Channel:22 Channel:22 Channel:20 Channel:21Levee:7 Crevasse:7 Levee:6
Crevasse:5Cond. data proportion(%) None None None NoneTarget proportion (%) Channel:30 Channel:30 Channel:30 Channel:23
Levee:7 Crevasse:7 Levee:7Crevasse:7
Simulated proportion(%) Channel:29.9 Channel:28.9 Channel:29.8 Channel:24.7Levee:7.6 Crevasse:7.1 Levee:8.4
Crevasse:5.1Servosystem parameter 0.95 0.95 0.95 0.95# of multiple grids 3 3 3 3Nodes in data template 50 36 36 50Radii of search ellipsoid 50,50,1 50,50,1 50,50,1 50,50,1Average nodesretained in template 27.4 19.8 19.3 21.9simulation time (sec) 7.2 8.5 7.2 19.5on 3 GB RAM machinemax. RAM used (GB) 0.45 0.46 0.45 0.58
Table 2.1: Input parameters and simulation statistics for the joint simulation of 2, 3and 4 facies, see Figures 2.2, 2.4, 2.6, and 2.8. The servo-system parameter is in [0,1],with 1 corresponding to maximum control. The parameters ‘Nodes in data template’and ‘Radii of search ellipsoid’ determine the size and geometry of the data template.‘Average nodes retained in template’ indicates the average number of nodes that wereretained in the data template.
CHAPTER 2. JOINT SIMULATION 9
Figure 2.3: Three facies training image (100 x 300). Green: channel 22%; blue: levee7%; grey: floodbasin 71%
Figure 2.4: Joint simulation of three facies using a 3-facies training image (100 x 100).Green: channel 28.9%; blue: levee 7.6%; grey: floodbasin 63.5%
CHAPTER 2. JOINT SIMULATION 10
Figure 2.5: Three facies training image (100 x 300). Green: channel 20%; red:crevasse 7%; grey: floodbasin 73%
Figure 2.6: Joint simulation of three facies using a 3-facies training image (100 x 100).Green: channel 29.8%; red: crevasse 7.1%; grey: floodbasin 63.1%
CHAPTER 2. JOINT SIMULATION 11
Figure 2.7: Four facies training image (100 x 300). Green: channel 21%; blue: levee6%; red: crevasse 5%; grey: floodbasin 68%
Figure 2.8: Joint simulation of four facies using a 4-facies training image (100 x 100).Green: channel 25%; blue: levee 8%; red: crevasse 5%; grey: floodbasin 61.8%
Chapter 3
Hierarchical Simulation
3.1 Hierarchical simulation of three facies
The hierarchical simulation of 3 facies is done in two steps. The first step is joint
simulation of channel and floodbasin facies using the binary Ti shown in Figure 2.1.
The simulated channel pixels (Figure 2.2) are frozen as hard data. In the second
step, the channel-levee-floodbasin Ti (Figure 2.3) and channel-crevasse-floodbasin Ti
(Figure 2.6) are used to generate Figures 3.1 and 3.2 respectively. During the second
step, the Ti-derived probability of simulating a channel pixel is set to zero so that no
more channel is simulated. The servo-system correction is applied at both steps to
ensure reproduction of the target proportions.
3.1.1 Comments
Figure 3.1 indicates a poor reproduction of the elongated shape of the levees, while
Figure 3.2 shows a good reproduction of the crevasse shape. The channels are continu-
ous as they were simulated with a large binary Ti and a large template independently
of the crevasse and levees. Both crevasse and levees are attached to the channels in
spite of having been simulated independently of the channels, however, some isolated
levee pixels are found in Figure 3.1. It is recommended that sequential simulation
proceeds over a single grid for the hierarchical simulation of levee so that isolated
levee pixels are minimized. Indeed, in the multiple-grid approach, the hard data are
12
CHAPTER 3. HIERARCHICAL SIMULATION 13
relocated to the nearest grid node, which combined with the dropping of nodes in the
data template can cause levees to be simulated in the floodbasin detached from the
channels.
3.2 Hierarchical simulation of four facies
In the case of 4 facies, the hierarchical simulation can be done in three ways. In
all three hierarchical approaches the natural sequence of development of the fluvial
facies is adhered to, which is as follows: The main channel belt forms prior to levees
and crevasse splays and all sediments are deposited inside the channel belt, hence in
the first two hierarchical approaches the channels are simulated prior to levees and
crevasse. When excessive sediments are supplied, they cannot be contained within the
channel belt and they spill over to form levees and floodbasin deposits. Consequently,
in the first two approaches levee is simulated after channel, while in the last approach
it is simulated jointly with channel. If the channel has a high sinuosity, the levees can
be breached and crevasse splays are formed adjacent to the channel belt, hence in all
three approaches, crevasse is simulated last.
3.2.1 Two-step cookie-cut simulation method
In this method, a 4-facies training image is not needed. The results from hierarchical
simulation of channel-levee-floodbasin (Figure 3.1) and channel-crevasse-floodbasin
(Figure 3.2) are cookie-cut one onto the other such that the crevasse erodes the levee:
this results in a simulated 4-facies image (Figure 3.3). This approach is geologically
correct because the crevasse splays form by breaching the natural levees. After cookie-
cut the proportion of the levee will be lower because some of the levee pixels are
replaced by crevasse pixels. The flowchart in Figure 3.4 summarizes the simulation
procedure and Table 3.1 summarizes the input parameters and simulation statistics
for this method.
CHAPTER 3. HIERARCHICAL SIMULATION 14
Parameters step 1 step2 step2 step3Figure 2.2 Figure 3.1 Figure 3.2 Figure 3.3
Training Image 2 facies 3 facies 3 facies 4 faciesFigure 2.1 Figure 2.3 Figure 2.5 N/A
Ti facies proportion(%) Channel:22 Channel:22 Channel:20 N/ALevee:7 Crevasse:7
Cond. data prop.(%) None Channel:100 Channel:100 N/ALevee:0 Crevasse:0
Target proportions (%) Channel:30 Channel:30 Channel:30 N/ALevee:7 Crevasse:7
Simulated proportions (%) Channel:29.9 Channel: 29.9 Channel:29.9 Channel:29.9Levee:7.1 Crevasse:8.2 Levee:7
Crevasse:8.2Servosystem parameter 1 0.95 0.95 N/A# of multiple grids 3 1 3 N/ANodes in data template 50 36 36 N/ARadii of search ellipsoid 50,50,1 50,50,1 50,50,1 N/AAverage nodesretained in template 27.4 17.6 19.5 N/Asimulation time (sec) 7.2 2.1 3.9 N/Aon 3 GB RAM machinemax. RAM used (GB) 0.45 0.44 0.48 N/A
Table 3.1: Input parameters and simulation statistics for the Two-step cookie-cutapproach
Comments
In Figure 3.3 the channels are continuous and both levee and crevasse are simulated
close to the channels. Since a 4-facies Ti is not used, the RAM required is smaller than
that for joint simulation of the four facies. This allows the use of a larger template
to capture the large-scale features. The shape of the crevasse is well reproduced.
Since crevasse is simulated independently of levee using a 3-facies channel-crevasse-
floodbasin Ti, it is not affected by the isolated levee pixels. Simulation of channel-
floodbasin took 7.2 seconds while that of levee-floodbasin took 2.1 seconds and that
of crevasse-floodbasin took 3.9 seconds on a 2.0 GHz desktop with 3 GB RAM.
CHAPTER 3. HIERARCHICAL SIMULATION 15
3.2.2 Three-step hierarchical simulation method
The 4 facies can also be simulated in three steps as follows: First only the channel
and floodbasin are simulated using a binary Ti and the simulated channels as set as
hard data (Figure 2.2). In the second step, only levees and floodbasin are simulated
(Figure 3.1) using a 3-facies channel-levee-floodbasin Ti (Figure 2.3). The simulated
levees are set as hard data together with the simulated channels from step 1. Finally,
crevasse and floodbasin are simulated (Figure 3.5) using a 4-facies channel-crevasse-
levee-floodbasin Ti (Figure 2.7).
In the second step, the Ti-derived probability of simulating a channel pixel is
set to zero so that no more channel is simulated. Similarly, in step three, the Ti-
derived probabilities of simulating both channel and levee pixels is set to zero so
that no more channel and levees are simulated. Servo-system correction is applied at
all three steps to ensure reproduction of target proportions. The procedure for this
method is summarized in the flowchart in Figure 3.6. Table 3.2 summarizes the input
parameters and simulation statistics for this method.
Comments
In Figure 3.5 the channels are continuous as they were simulated using a large binary
Ti and a large template. Since the crevasse simulation is conditioned to previously
simulated channel as well as levee values, some crevasse pixels are simulated next to
the isolated levee pixels, which is permitted by the Ti. The target proportions of
the four facies are well reproduced because the servo-system is applied. The total
simulation took 14.4 seconds.
3.2.3 Two-step hierarchical simulation method
Comparing the results of joint versus hierarchical simulation of channel-levee-floodbasin
system, it is observed that the levees are better simulated jointly with channels since
they have the same elongated shape as the channels. Hence, in this last approach,
the levees are simulated jointly with channel and floodbasin in the first step using a
3-facies channel-levee-floodbasin Ti. The simulated channel and levee pixels are then
CHAPTER 3. HIERARCHICAL SIMULATION 16
frozen as hard data (Figure 2.4). Finally, only crevasse and floodbasin are simulated
(Figure 3.7) using a 4-facies channel-levee-crevasse-floodbasin Ti (Figure 2.7) by set-
ting to zero the Ti-derived probability of simulating channel and levee pixels. The
servo-system correction is applied at both steps to ensure reproduction of target pro-
portions of the four facies. The flowchart in Figure 3.8 summarizes the procedure and
Table 3.3 summarizes the input parameters and simulation statistics for this method.
Comments
In Figure 3.7 the continuity of the levees is improved and isolated pixels are minimized
by simulating the levees with the channels. The channels are reasonably continuous
in spite of using a 3-facies Ti, because the levee has the same elongated shape as the
channel. The shape of the crevasse is well reproduced and they are attached to the
channels. The target proportions of the four facies are well reproduced because the
servo-system is applied. The first step took 4.9 seconds, the second step took 5.1
seconds.
3.3 3D Example
The hierarchical simulation approach is applied to a synthetic 3D meandering chan-
nel reservoir with four facies namely, channel, pointbar, levee, and floodbasin. The
training image is 100 x 100 x 100 grid blocks (Figure 3.9). In Figure 3.9, the channels
are sinuous and have a U-shaped cross-section. The pointbars are deposited in the
inner bends of the channel and in cross-section they occur along one margin of the
channel. The levees have an elongated shape with a triangular cross-section and are
deposited adjacent to the channel. Some of the channels are amalgamated, hence the
pointbar is found between two channels. Thus, the pointbar needs to be simulated
jointly with the channel so that it can be simulated inside channels as depicted by
the Ti.
The simulation grid is 100 x 100 x 100 grid blocks. The two-step hierarchical
approach is used to generate the final 4-facies realization. In step 1, only the channel,
pointbar and floodbasin are simulated using the corresponding 3-facies Ti. That
CHAPTER 3. HIERARCHICAL SIMULATION 17
Ti was obtained from Figure 3.9 by merging the levee facies with the floodbasin.
Furthermore, only the first 50 layers of the Ti were used for the simulation to reduce
the RAM requirement by reducing the size of the search tree. In step 2, only levees
and floodbasin are simulated conditioned to the previously simulated channel and
pointbar. The first 50 layers of the original 4-facies Ti are used for this simulation.
During the second step, the Ti-derived probability of simulating a channel or point-
bar pixel is set to zero so that no more channel or point-bar is simulated.
Comments
Figure 3.10 shows the final 4-facies realization. Both channel and pointbar are well-
reproduced in plan view and cross-section. Channels become discontinuous when
they thin out vertically. The sinuosity of the channel is discernable and the pointbar
is simulated in the inner bend of the channel as specified by the Ti. It was possible
to capture this large-scale structure by using a large data template in combination
with the multiple-grid appraoch. By using hierarchy, the problem was reduced to
simulation of 3 facies instead of 4 facies in the first step, which enabled using a large
data template, which might not have been possible if the 4 facies had been jointly
simulated. It was difficult to simulate continuous levees because in the Ti they are
fragmented as they thin upward.
Figure 3.1: Hierarchical simulation of only levee and floodbasin using a 3-facies Ti(100 x 100). Channels were previously simulated (Figure 2.2) and set as hard data.Green: channel 30%; red: levee 7%; grey: floodbasin 63%
CHAPTER 3. HIERARCHICAL SIMULATION 18
Figure 3.2: Hierarchical simulation of only crevasse and floodbasin using a 3-facies Ti(100 x 100). Channels are previously simulated (Figure 2.2). Green: channel 30%;red: crevasse 8%; grey: floodbasin 62%
Figure 3.3: Combining Figures 3.1 and 3.2 using cookie-cut. Crevasse erodes lev-ees and floodbasin. Green: channel 30%; blue: levee 7%; red: crevasse 8%; grey:floodbasin 55%
CHAPTER 3. HIERARCHICAL SIMULATION 19
Parameters step 1 step2 step3Figure 2.2 Figure 3.1 Figure 3.5
Training Image 2 facies 3 facies 4 faciesFigure 2.1 Figure 2.3 Figure 2.7
Ti facies proportion(%) Channel:22 Channel:22 Channel:21Levee:7 Levee:6
Crevasse:5Cond. data prop.(%) None Channel:100 Channel: 76.6
Levee:0 Levee:22.9Crevasse:0
Target proportion(%) Channel:30 Channel:30 Channel:30Levee:7 Levee:7
Crevasse:7Simulated proportion(%) Channel:29.9 Channel:29.9 Channel:29.9
Levee: 7.1 Levee:7.1Crevasse:6.2
Servosystem parameter 1 0.95 0.95# of multiple grids 3 1 3Nodes in data template 50 36 36Radii of search ellipsoid 50,50,1 50,50,1 50,50,1Average nodesretained in template 27.4 17.6 17.8simulation time (sec) 7.2 2.1 5.1on 3 GB RAM machinemax. RAM used (GB) 0.45 0.44 0.58
Table 3.2: Input parameters and simulation statistics for the Three-step hierarchicalapproach
CHAPTER 3. HIERARCHICAL SIMULATION 20
Parameters 3 facies 4 faciesFigure 2.4 Figure 3.7
Training Image 3 facies 4 faciesFigure 2.3 Figure 2.7
Ti facies proportion(%) Channel:22 Channel:21Levee:7 Levee:6
Crevasse:5Cond. data proportion(%) None Channel:79.2
Levee:20.8Crevasse: 0
Target proportion (%) Channel:30 Channel:23Levee:7 Levee:7
Crevasse:7Simulated proportion(%) Channel:28.9 Channel:28.9
Levee:7.6 Levee:7.9Crevasse:6.4
Servosystem parameter 0.95 0.95# of multiple grids 3 3Nodes in data template 36 36Radii of search ellipsoid 50,50,1 50,50,1Average dataretained in template 19.8 18.8simulation time (sec) 8.5 5.1on 3 GB RAM machinemax. RAM used (GB) 0.46 0.6
Table 3.3: Input parameters and simulation statistics for the Two-step hierarchicalapproach
CHAPTER 3. HIERARCHICAL SIMULATION 21
Figure 3.4: Flowchart for two-step cookie-cut simulation method
CHAPTER 3. HIERARCHICAL SIMULATION 22
Figure 3.5: Step 3 of three-step method: Simulation of only crevasse and floodbasinusing a 4-facies Ti. The channels were simulated in step 1 (Figure 2.2) and levees weresimulated in step 2 (Figure 3.1) and set as hard data. Simulated facies proportions:-Green: channel 30%; blue: levee 7%; red: crevasse 6%; grey: floodbasin 57%
Figure 3.6: Flowchart for three-step hierarchical simulation method
CHAPTER 3. HIERARCHICAL SIMULATION 23
Figure 3.7: Step 2 of two-step hierarchical simulation method. Simulation of crevasseand floodbasin using 4-facies Ti. The channel and levee have been previously sim-ulated in step 1 and set as hard data (Figure 2.4). Simulated facies proportions:Green: channel 28.9%; blue: levee 7.6%; red: crevasse 6.4%; white: floodbasin 57%
CHAPTER 3. HIERARCHICAL SIMULATION 24
Figure 3.8: Flowchart for two-step hierarchical simulation method
CHAPTER 3. HIERARCHICAL SIMULATION 25
Figure 3.9: Four facies training image (100 x 100 x 100). Green: channel 25%; blue:pointbar 4%; red: levee 2%; grey: floodbasin 69%
CHAPTER 3. HIERARCHICAL SIMULATION 26
Figure 3.10: Hierarchical simulation of channel, pointbar, levee and floodbasin (100 x100 x 100). Green: channel 28%; blue: pointbar 5%; red: levee 4%; grey: floodbasin63%
Chapter 4
Implementation of hierarchical
simulation
All runs in this paper were done using the snesim algorithm (Strebelle, 2000, 2002).
In order to use hierarchical simulation, modifications to the existing snesim pro-
gram were necessary. To illustrate these changes, consider the two-step hierarchical
approach discussed in the paper.
In the first step, the 3 facies, channel, levee and floodbasin are simulated using the
corresponding 3-facies Ti. The simulated channel and levee pixels should be extracted
from the simulation output file along with their location and saved in a separate file
so that they can be supplied as hard data for the next step. This step requires no
modification in the program. In the second step, a 4-facies channel-levee-crevasse-
floodbasin Ti is used to simulate only two facies, crevasse and floodbasin. In order to
accomplish this, the following must be done at each location to be simulated:
• Get the Ti-derived proportions for a given data event
• Set the local probability of channel and levee to zero
• Re-standardize correspondingly the probabilities of crevasse and floodbasin
• Apply the servo-system correction to crevasse and floodbasin
One last modification is required when obtaining the Ti-derived proportions from
the search tree. When an uninformed node is encountered in the data template, the
27
CHAPTER 4. IMPLEMENTATION OF HIERARCHICAL SIMULATION 28
original snesim program considers the possibility of having any one of the four facies
that exist in the Ti. However, in step 2, since only crevasse and floodbasin can be
simulated, only these two facies should be considered. Indeed channel and levee have
already been simulated in step 1 and set as hard data, hence the uninformed node in
the data template cannot be these two facies.
Chapter 5
Rhine-Meuse Delta Case Study
5.1 Introduction to the Data Set
5.2 Results and Discussion
29
Chapter 6
Conclusions and Future Work
6.1 Conclusions
• Hierarchical simulation gives better results than joint simulation when more
than three facies are involved. This approach is highly recommended for both
algorithmic and geological considerations.
• Three alternative approaches for simulating four facies hierarchically were demon-
strated. As the number of facies increases, the alternatives to simulate them
hierarchically also increases. The actual hierarchy of simulation steps should be
guided by the geological rules of deposition.
• In case of the levees, which have elongation similar to channels, their joint
simulation with channels produces better results than a hierarchical simulation,
in which some isolated pixels can be generated. Thus in similar depositional
contexts, it might be better to simulate similar shaped facies together.
• The shape of crevasse splays is very different from that of the channels and lev-
ees. Hence, simulating crevasse separately from channels and levees gives better
results. Thus, facies with different shapes and continuity should be simulated
separately.
• A 4-facies training image can be used to simulate only two facies as was done in
30
CHAPTER 6. CONCLUSIONS AND FUTURE WORK 31
the two-step hierarchical simulation method, see Figure 3.7. However, the rela-
tionship between the different facies is extracted from the complex four-facies
training image. This concept can be extended to other depositional environ-
ments with multiple facies.
• During hierarchical simulation, facies such as levee and crevasse splays are at-
tached to the channels as specified in the training image, even though the levee
and crevasse are simulated independently of the channels.
• In this particular case-study, the two-step hierarchical simulation method (Fig-
ure 3.7) gives the best results out of the three hierarchical methods, because
channels and levees are simulated jointly, and a 4-facies Ti was used to condition
the relative position of crevasse and floodbasin.
6.2 Future work
Consider the synthetic 2D fluvial reservoir with four facies and assume that condition-
ing data is available for all of them. In the current implementation, during step 1 of
the two-step hierarchical approach the crevasse data are merged with the floodbasin
data, so that only channel, levee and floodbasin data exist. However, crevasse data
are indirect indicator of a channel located nearby. This information is ignored when
the crevasse data are merged with floodbasin data.
To utilize the information carried by the crevasse data, the grid can be pre-
processed to flag the simulation nodes that are within a fixed distance d of the crevasse
data. The distance d depends on the size and shape of the crevasse and should be
provided by the geologist. During simulation if the location to be simulated is flagged,
then the Ti-derived proportion of channel P (A|B) is increased by 10% by multiplying
P (A|B) by a factor 1.1. The 10% percent value increase is an arbitrary choice. The
upper bound is set to 1 so that we do not get proportions greater than 1.
P ∗ = min(1, P (A|B) ∗ 1.1) (6.1)
Step 2 would then proceed as described in section 3.2.3.
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