interwell connectivity evaluation using injection and

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2014-01-23

Interwell Connectivity Evaluation Using Injection and

Production Fluctuation Data

Soroush, Mohammad

Soroush, M. (2014). Interwell Connectivity Evaluation Using Injection and Production Fluctuation

Data (Unpublished doctoral thesis). University of Calgary, Calgary, AB.

doi:10.11575/PRISM/26620

http://hdl.handle.net/11023/1285

doctoral thesis

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UNIVERSITY OF CALGARY

Interwell Connectivity Evaluation Using Injection and Production

Fluctuation Data

By

Mohammad Soroush

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF CHEMICAL AND PETROLEUM ENGINEERING

CALGARY, ALBERTA

JANUARY, 2014

© Mohammad Soroush 2014

II

ABSTRACT

Evaluating interwell connectivities can provide important information for reservoir

management by identifying flow conduits, barriers, and injection imbalances. Injection

and production rates contain connectivity information and a number of methods have

been proposed to predict connectivity based on this data. The capacitance model (CM)

has recently been applied successfully in several real field cases for this purpose. For

non-ideal conditions occurring in the field, however, further investigation on the CM is

needed to have a better understanding of reservoir heterogeneity and to facilitate more

informed comparisons between the CM results and geological information and other

available data.

The CM is based on a linear productivity model assuming a pseudo steady state flow

regime for slightly compressible fluids. Therefore, we expect within a specific range of

fluid and reservoir properties that the results are reliable. The first aim of this work is to

determine the range of applicability of the CM before applying it to field data by a

sensitivity analysis on accuracy of results. We also briefly address how to extend the

model for transient flow regime effects.

Secondly, the CM equation is derived from a productivity model assuming radial flow in

the drainage area of each producer and most of the pressure drop will occur within a few

feet of the wellbore. In this work, we show that heterogeneities close to the wellbore have

more effect on production and the CM parameters than interwell heterogeneities between

injector-producer pairs. We demonstrate that the CM is able to assess these near producer

heterogeneities. Also, we suggest methods to decouple the effects of well geometry from

near well heterogeneities.

Thirdly, we illustrate the application of the CM in heavy oil reservoirs and wormhole

assessment. We propose a modification to the CM to make it perform better in real fields

when we have producer shut-ins and mini shut-ins or skin changes. The results of one

conventional and one heavy oil field cases are analyzed at the end of this work.

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Applying earlier methods in real field cases may give misleading connectivity results and

inaccurate rate predictions. Adopting the approaches described in this work helps

geoscientists and engineers have a better understanding of reservoir heterogeneity and its

effects on fluid flow in the reservoir.

IV

ACKNOWLEDGMENTS

I would like to express my deepest gratitude to my supervisor professor Jerry L. Jensen

(University of Calgary) and Dr. Danial Kaviani (ConocoPhillips, Canada) for their help

during this work. For their helpful guidance to select this topic and their assistance

throughout this project I am thankful.

My thanks also extend to Dr. Hassanzadeh and Dr. Maini for being the members of

committee and the examiners Dr. Laurence R. Lines and Dr. Eduardo Gildin for giving

me their valuable time to review my thesis.

The financial support for this study was provided by PTRC (Petroleum Technology

Research Center, Regina). This support is gratefully acknowledged.

V

Dedicated to My Family

VI

TABLE OF CONTENTS

ABSTRACT ...................................................................................................................... II

ACKNOWLEDGMENTS .............................................................................................. IV

TABLE OF CONTENTS ............................................................................................... VI

LIST OF FIGURES ........................................................................................................ XI

LIST OF TABLES ...................................................................................................... XXII

CHAPTER 1 INTRODUCTION ......................................................................................1

1.1 Connectivity Evaluation.............................................................................................1

1.2 Importance of Topic ...................................................................................................1

1.3 General Achievements ...............................................................................................2

CHAPTER 2 LITERATURE REVIEW ..........................................................................5

2.1 Introduction ................................................................................................................5

2.2 Statistically-Based Methods.......................................................................................5

2.2.1 Spearman Rank Correlation ........................................................................................... 5

2.2.2 Artificial Neural Network .............................................................................................. 7

2.2.3 Injector-Producer Relationships Using an Extended Kalman Filter .............................. 8

2.2.4 Interwell Relation Based on Wavelet Analysis .............................................................. 9

2.3 Material (fluid) Propagation-Based Methods ..........................................................12

2.3.1 Streamline-based Method ............................................................................................. 12

2.3.2 Non-reactive Tracer Test .............................................................................................. 13

2.4 Potential (pressure) Change Propagation-Based Methods .......................................13

2.4.1 BHP-based Connectivity .............................................................................................. 14

2.4.2 MLR Model .................................................................................................................. 15

2.4.3 MPI-based Connectivity ............................................................................................... 17

2.4.4 Capacitance Model (CM) ............................................................................................. 18

2.5 Conclusions ..............................................................................................................24

CHAPTER 3 CM SENSITIVITY ANALYSIS .............................................................29

3.1 Introduction ..............................................................................................................29

VII

3.2 Sensitivity to the Diffusivity Constant .....................................................................30

3.3 Sensitivity to the Sampling Time .............................................................................38

3.4 Sensitivity to the Reservoir Area .............................................................................40

3.5 Sensitivity to the Number of Producers ...................................................................40

3.6 CM Number .............................................................................................................43

3.7 Sensitivity to the Number of Data ...........................................................................45

3.8 Sensitivity to Noise ..................................................................................................48

3.9 Error Assessment Using the Bootstrap ....................................................................51

3.10 Field Examples.......................................................................................................56

3.11 Conclusions ............................................................................................................59

CHAPTER 4 THE CM AND HORIZONTAL WELLS ..............................................61

4.1 Introduction ..............................................................................................................61

4.2 Horizontal Well Effect on the CM Parameters ........................................................61

4.3 Well Trajectory Effect ..............................................................................................66

4.3.1 One-branch Horizontal Well ........................................................................................ 66

4.3.2 Horizontal Well Direction ............................................................................................ 67

4.3.3 Deviated Wells ............................................................................................................. 68

4.4 Well Length Effect ...................................................................................................69

4.5 Analytical Method ...................................................................................................71

4.6 Applying the Reverse CM .......................................................................................74

4.7 Heterogeneous Reservoir .........................................................................................76

4.8 Conclusions ..............................................................................................................78

CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY ....................79

5.1 Introduction ..............................................................................................................79

5.2 Near Wellbore Effect ...............................................................................................79

5.3 Interwell Connectivity Assessment..........................................................................80

5.4 Near well Connectivity Assessment ........................................................................84

5.4.1 Median and Interquartile Range ................................................................................... 85

5.4.2 Equivalent Skin Factor ................................................................................................. 86

5.5 -skin Relationship ..................................................................................................89

VIII

5.6 Conclusions ..............................................................................................................90

CHAPTER 6 THE CM IN HEAVY OIL RESERVOIRS - WORMHOLE

ASSESSMENT .................................................................................................................91

6.1 Introduction ..............................................................................................................91

6.2 Connectivity Evaluation in Heavy Oil Reservoirs ...................................................91

6.3 Wormhole assessment ............................................................................................101

6.3.1 Wormhole detection ................................................................................................... 101

6.3.2 Equivalent skin associated with the wormhole .......................................................... 103

6.3.3 Rate of wormhole growth ........................................................................................... 105

6.4 Conclusions ............................................................................................................108

CHAPTER 7 THE CM IN TIGHT FORMATIONS ..................................................109

7.1 Introduction ............................................................................................................109

7.2 Transient MPI ........................................................................................................109

7.3 Connectivity Parameters in Transient Regime ......................................................112

7.4 Transient CM .........................................................................................................116

7.5 Conclusions ............................................................................................................117

CHAPTER 8 MULTIWELL COMPENSATED CM .................................................118

8.1 Introduction ............................................................................................................118

8.2 CM and Compensated CM (CCM) ........................................................................118

8.3 Skin and the CCM ..................................................................................................119

8.4 Multiwell CCM (MCCM) ......................................................................................120

8.5 Application of MCCM for Mini Shut-ins ..............................................................127

8.6 Conclusions ............................................................................................................133

CHAPTER 9 FIELD APPLICATION .........................................................................134

9.1 Introduction ............................................................................................................134

9.2 Marsden South Field ..............................................................................................134

9.2.1 Field Description ........................................................................................................ 134

9.2.2 Applications of analytical connectivity values ........................................................... 136

9.2.3 Window selection to apply the model ........................................................................ 137

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9.2.4 Selecting the number of producers in each window ................................................... 138

9.2.5 Including production hours ........................................................................................ 138

9.2.6 Using bootstrap technique .......................................................................................... 139

9.2.7 Comparing ′ values to the sand body map ............................................................... 139

9.2.8 Comparing median of ′ to the sand body map .......................................................... 143

9.2.9 Analysis of dye test results ......................................................................................... 144

9.3 Storthoaks Field .....................................................................................................148

9.3.1 Field Description ........................................................................................................ 148

9.3.2 MCCM Results ........................................................................................................... 148

9.4 Conclusions ............................................................................................................153

CHAPTER 10 CONCLUSIONS AND RECOMMENDATIONS .............................154

10.1 Conclusions ..........................................................................................................154

10.2 Recommendations ................................................................................................156

NOMENCLATURE .......................................................................................................158

REFERENCES ...............................................................................................................162

APPENDIX 1 ..................................................................................................................166

Derivation of MPI Formulas ............................................................................................... 166

APPENDIX 2 ..................................................................................................................168

Derivation of the CM .......................................................................................................... 168

APPENDIX 3 ..................................................................................................................170

Derivation of the Analytical Formula for ’s Using MPI ................................................... 170

APPENDIX 4 ..................................................................................................................172

Derivation of the Reverse CM ............................................................................................ 172

APPENDIX 5 ..................................................................................................................175

Derivation of the Relationship between and Skin ............................................................ 175

APPENDIX 6 ..................................................................................................................176

Derivation of the Analytical Formula for ’s in Transient Regime .................................... 176

X

APPENDIX 7 ..................................................................................................................178

Derivation of Transient CM ................................................................................................ 178

APPENDIX 8 ..................................................................................................................181

Derivation of the MCCM for Skin Changes ................................................................181

APPENDIX 9 ..................................................................................................................183

Calculating Average Production Rate Using the CM ..................................................183

XI

LIST OF FIGURES

Figure 2-1 The principle of correlation analysis; the injector is highly connected to the

producer. From Fedenczuk et al. (1998). ............................................................................ 6

Figure 2-2 Schematic of a typical artificial neural network; the input layer reads the input

data, the output layer determines the required results, and the hidden layer processes the

intermediate results. From Panda and Chopra (1998). ....................................................... 8

Figure 2-3 Extended Kalman filter; each injection production pair is assumed to be a

subsystem in which the injection rate is transformed to the production rate by means of a

moving average model. Function “f” decouples the effect of distance from noise free

injection rates, “n” indicates noise function and “m” is the abbreviation of measured

parameters. From Liu and Mendel (2009). ......................................................................... 9

Figure 2-4 Variance of high frequency component of injection rate and water production;

a similar trend is a result of good communication between well pairs. After Jansen et al.

(1997). ............................................................................................................................... 10

Figure 2-5 Effective flow units estimated by the streamline model are indicated in red and

those estimated by this model are indicated in blue for a 5×4 pattern (5 injectors and 4

producers). From Lee et al. (2011). .................................................................................. 11

Figure 2-6 Weight factors estimated using the CM are shown in red, and estimates by this

model are outlined in blue for a specific field. From Lee et al. (2011). ........................... 11

Figure 2-7 Injection efficiency of I5 is the oil produced at the offset producers (P3, P4, P5

and P7) divided by the injection rate of I5. The offset oil produced is the sum of oil

produced by the red, green, orange and yellow streamline bundles. From Thiele and

Batycky (2006).................................................................................................................. 13

Figure 2-8 The map on the left shows weight factors that were estimated using this

method in a 5×4 homogeneous reservoir. The length of the arrow is proportional to the

value of the coefficient. The map on the right shows interwell permeability calculated

using this method as well. The length of the arrow is proportional to the value of the

relative interwell permeability. From Dinh and Tiab (2013). ........................................... 15

Figure 2-9 The map on the left shows the ’s estimated for a 5×4 homogeneous synthetic

field. The length of the arrow is proportional to the value of the coefficient. The map on

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the right shows the ’s estimated for a 5×4 synthetic field with a sealing fault. From

Albertoni and Lake (2003). ............................................................................................... 16

Figure 2-10 Normalized connectivity map (map of heterogeneity matrix, Aoptimized-

Ahomogeneous); the gray features are barriers and the white feature is a conduit. For a

homogeneous case, the connectivity values are zero. From Kaviani and Valkó (2010). . 18

Figure 2-11 Map of the ’s (left) and the ’s (right) for a homogeneous 5×4 system; the

length of the arrow is proportional to the or values. From Yousef et al. (2006). ...... 20

Figure 2-12 The left plot shows a log-log plot of the ’s versus ’s. I04 and I05 are

located in a higher permeability layer. The right plot shows a schematic of a different

trend of the F-C curve estimated from the CM parameters, according to the corresponding

geological features around a producer. From Yousef et al. (2009). ................................. 20

Figure 2-13 Homogeneous 2×2 reservoir; P01 and P02 are in equal distance (x) from I01.

........................................................................................................................................... 25

Figure 3-1 The injection rates are selected from the above injection rate profiles. Three

sets of rates were generated based on the left figure, and 2 sets were generated based on

the right figure. To investigate the probable effect of higher rates on the results, we

generated 5 more sets of rates by multiplying the first 3 sets of rates by 5, and the other 2

rates by 1.7. ....................................................................................................................... 32

Figure 3-2 Location of the vertical wells for Case 3.1; 5 injectors and 4 producers are

located in a homogenous reservoir. .................................................................................. 32

Figure 3-3 The algorithm shows the summary of the procedure to calculate the CV of CM

parameters and for each case. ..................................................................................... 33

Figure 3-4 The median of CV of the estimated ’s and ’s is shown versus permeability

for a homogeneous system (Case 3.1). ............................................................................. 34

Figure 3-5 By decreasing the permeability, the prediction error of the estimated rates

increases to 1% (Case 3.1). ............................................................................................... 34

Figure 3-6 At large compressibilities, both ’s and ’s have large CV’s for different

injection rates. At small compressibilities, however, only values are unstable (Case 3.1).

........................................................................................................................................... 36

Figure 3-7 Combining the results of Figure 3-4 and 3-6 shows that the effects of

compressibility and permeability are almost the same (Case 3.1). ................................... 37

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Figure 3-8 In Case 3.2, three barriers and one channel exist in the reservoir. .................. 38

Figure 3-9 Similar to the homogeneous case, in Case 3.2 ’s are unstable at both high and

low permeabilities and the ’s are unstable at small permeability values. ....................... 38

Figure 3-10 At 5-day sampling, the range of stable ’s and ’s shifts to higher

permeabilities (Case 3.1). ................................................................................................. 39

Figure 3-11 The median of CV of the estimated ’s and ’s is shown versus sampling

time for a homogeneous system (Case 3.1). ..................................................................... 39

Figure 3-12 Trend of CV’s for both ’s and ’s versus the reservoir area is similar to the

trend of CV’s versus total compressibility (Case 3.1). ...................................................... 40

Figure 3-13 Permeability map of Case 3.4; four barriers and one channel exist in the

system. .............................................................................................................................. 42

Figure 3-14 For Case 3.4; since both the area and number of producers are twice those of

Case 3.1, we expect to have the same range of stable parameters as Case 3.1. However,

this range is slightly narrower than Case 3.1. ................................................................... 42

Figure 3-15 For well pairs with smaller values, the uncertainty in the estimated ’s

increases. Here, the CV of the estimated ’s for Case 3.4 at k = 1 md is plotted. ............ 43

Figure 3-16 By calculating the CM number (C) for different reservoir conditions,

sampling time, reservoir area, and well numbers, the CM results are stable and repeatable

for 0.3<C<10. .................................................................................................................... 45

Figure 3-17 By decreasing the length of analysis window, the range of stable ’s (left)

and ’s (right) will be shorter. .......................................................................................... 46

Figure 3-18 Including more data leads to a more accurate estimation of ’s (Case 3.1). At

small CM numbers, errors increase as L decreases........................................................... 47

Figure 3-19 By calculating the CM number by changing parameters except permeability

(Mix), we observed that the trend of increasing the AAD of by decreasing the CM

number is similar to what we had by changing permeability. .......................................... 48

Figure 3-20 By adding noise, CM parameter errors increase. In (a), (c), and (e) the

median of CV of ’s is shown at L = 4, 8, and 32 respectively. (b), (d), and (f) show the

median of CV of ’s at L = 4, 8, and 32 respectively. By “ideal” we mean the noise-free

case with a large number of samples. By introducing noise to the data, estimates

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become more variable for small and medium numbers of samples. However, values are

stable at small number of samples with moderate noise (10%). ....................................... 51

Figure 3-21 This figure shows AAD of ’s versus different amounts of noise, number of

data, and CM number. ....................................................................................................... 51

Figure 3-22 In the bootstrap technique, based on selected subsamples of data, we assign

the appropriate weight to each time step. Then we apply the CM for each scenario and

finally we calculate the standard deviation of the estimated ’s. ..................................... 53

Figure 3-23 Standard deviation of the estimated ’s using the bootstrap correlates well

with the AADs. .............................................................................................................. 54

Figure 3-24 Independent of the number of wells, by applying bootstrap and estimating the

standard deviation, we can estimate the error in estimated ’s. ....................................... 54

Figure 3-25 The error in the estimated ’s from averaging results of several bootstraps

(vertical axis) is, in general, more accurate than ones obtained from a single run of CM

(horizontal axis). ............................................................................................................... 55

Figure 3-26 Applying 10 to 20 resamplings, we can get a good estimation of the standard

deviation of the bootstrap. For larger number of samples, we need a smaller number of

resamplings. ...................................................................................................................... 56

Figure 3-27 Contour plot of median CV’s of ’s at different CM numbers and L; for the

source of each point, see Table 3-2. Most of the cases have stable ’s. ........................... 58

Figure 3-28 Contour plot of median CV’s of ’s at different CM numbers and L; for

description of each point see Table 3-2. Compared to the ’s (Figure 3-28), a smaller

number of cases has stable ’s. ......................................................................................... 59

Figure 4-1 Schematic of increasing production rate of horizontal well and its effect on the

CM parameters; subscript H and V stand for a horizontal and vertical well respectively.

C(t) stands for non-waterflood terms. High amounts of production from horizontal well

increase the ’s of that horizontal well and decreases the ’s of vertical wells. .............. 62

Figure 4-2 Map of ’s (left) and ’s (right) for Case 4.1; all the wells are fully penetrating

vertical wells. .................................................................................................................... 63

Figure 4-3 W-E cross section of the simulation model for the Case 4.2; P01 is a two-

branch horizontal well with a length of 550ft which is drilled in the bottom layer. ......... 64

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Figure 4-4 Map of ’s (left) and ’s (right) for the Case 4.2; producer P01 is a W-E two-

branch horizontal well with a length of 550 ft. .................................................................. 64

Figure 4-5 Cross plot of ’s (left) and ’s (right) for Case 4.2 versus Case 4.1; horizontal

well ’s are enhanced and vertical well ’s are decreased. On the other hand, all ’s

decrease. ............................................................................................................................ 66

Figure 4-6 W-E cross section of the simulation model for Case 4.3; P01 is a one-branch

horizontal well with a length of 550 ft which is drilled in the bottom layer. .................... 67

Figure 4-7 Map of ’s (left) and ’s (right) for Case 4.3; producer P01 is a one-branch

horizontal well with a length of 550 ft. .............................................................................. 67

Figure 4-8 Effect of well direction; P01 is a two-branch horizontal well: four different

orientations were considered, including W-E, SW-NE direction, S-N direction and SE-

NW directions (Case 4.4). ................................................................................................. 68

Figure 4-9 W-E cross section of the simulation model for Case 4.5; P01 is a 76 deviated

well. ................................................................................................................................... 69

Figure 4-10 Map of ’s (left) and ’s (right) for Case 4.5; producer P01 is a 76 deviated

well. ................................................................................................................................... 69

Figure 4-11 (left) and (right) versus length of horizontal well (two-branch W-E

horizontal well); As P01’s length increases, its ’s are increasing and the vertical well ’s

are decreasing. However, by increasing the horizontal well length, all the ’s are

decreasing (Case 4.6). ....................................................................................................... 70

Figure 4-12 (left) and (right) versus length of horizontal well (one-branch horizontal

well toward I01); the trend is similar to the Figure 4-11, except the rate of change of ’s

and ’s between P01 and I01 (closest injector) which is higher (Case 4.6)...................... 71

Figure 4-13 Horizontal well P01 is divided into 11 vertical producers (each vertical well

in one grid block). ............................................................................................................. 72

Figure 4-14 Cross plot of the optimized ’s using the CM (Case 4.2) versus the analytical

’s using the MPI; P01 has a length of 550 ft. .................................................................. 73

Figure 4-15 By allowing the number of producer elements to approach infinity and

increasing the number of grid blocks in the simulation model, the difference between

analytical and optimized ’s is minimized. ...................................................................... 74

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Figure 4-16 Map of the ’s (left) and the ’s (right) for Case 4.1 using the reverse CM;

the length of the arrow is proportional to the or values. ......................................... 75

Figure 4-17 Map of the ’s (left) and the ’s (right) for the Case 4.2 using the reverse

CM; the horizontal well does not impact these values. .................................................... 76

Figure 4-18 The ′’s for the system of vertical wells (left) and the ′’s for the system with

horizontal well(s) (right); ′’s in black color have a positive value and ′’s in red color

have a negative value. The blue rectangle shows a barrier with permeability close to zero

and the green rectangle shows a fracture with permeability of about 100 times the system

permeability (Case 4.7). .................................................................................................... 77

Figure 4-19 The ′’s for the system of vertical wells (left) and the ′’s for the system

with horizontal well(s) (right) using the reverse CM; the blue rectangle shows a barrier

with permeability close to zero and the green rectangle shows a fracture with

permeability of about 100 times the system permeability (Case 4.7). ............................. 78

Figure 5-1 Schematic subdivision of drainage area into a rapidly drained area and total

drained area; a large number of streamlines traverse a segment which is located in the

rapidly drained area (right). The largest pressure drop occurs in the rapidly drained area

(left)................................................................................................................................... 80

Figure 5-2 Schematic subdivision of the area between one injector-producer pair; the near

producer area has more effect on production and connectivity parameters. ..................... 80

Figure 5-3 Base permeability is 100 md and near wellbore altered permeabilities are 1000

md for P01 and P02 and 10 md for P03 and P04 (left). Applying the apparent skin

diminishes near producer heterogeneity effect (Case 5.1). ............................................... 82

Figure 5-4 The figure on the left shows ′’s are less affected by the interwell features. If

we apply apparent skin the ′’s could be better representative of interwell heterogeneity

(right). Permeability distribution is generated by SGeMS (Case 5.2). ............................. 83

Figure 5-5 Figures show the cross plot of ′’s using pseudo skin versus normalized

interwell permeability for Cases 5.1 (left) and 5.2 (right). ............................................... 84

Figure 5-6 The median and interquartile range (IQR) of the′ values for each producer

(right); the red circle signifies a positive median and the blue circle indicates a negative

median (Case 5.1). ............................................................................................................ 86

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Figure 5-7 The median and interquartile range (IQR) of the′ values for each producer

(right); the red circle signifies a positive median and the blue circle indicates a negative

median (Case 5.2). ............................................................................................................ 86

Figure 5-8 Flowchart used in the CM-MPI code; equivalent skin will be optimized via

Matlab. .............................................................................................................................. 87

Figure 5-9 Equivalent skin obtained using the CM-MPI algorithm; the red circle signifies

negative skin and the blue circle indicates positive skin (Case 5.1) ................................. 88

Figure 5-10 Equivalent skin obtained using the CM-MPI algorithm; the red circle

signifies negative skin and the blue circle indicates positive skin (Case 5.2). ................. 89

Figure 5-11 Plot of skin versus 1/for P01; calculated R2’s

equal one confirm that the

relationship is linear. ......................................................................................................... 89

Figure 6-1 ’s variations over time for Case 6.1 (mobility ratio=10). As time increases,

the ’s converge to the ideal ones. Each solid line shows the between a well pair. The

dashed lines show the ideal ’s (from unit-mobility ratio). .............................................. 93

Figure 6-2 ’s map when all the producers have a skin of +2 (left) and +1 (right). ......... 93

Figure 6-3 ’s variations over time for the Case 6.2 (mobility ratio=1000); as time

increases, the ’s becomes stable. Unlike Case 6.1., the ’s at the last time step do not

converge to the ideal ones. Each solid line shows the between a well pair. The dashed

lines show the ideal ’s (from the unit-mobility ratio). .................................................... 94

Figure 6-4 The AAD variations over time for the Cases 6.1 (mobility ratio=10) and 6.2

(mobility ratio=1000) based on the data for each time step; the AAD for these cases never

exceeds 0.02. ..................................................................................................................... 95

Figure 6-5 ’s variations by moving the analysis window for Case 6.1 (mobility

ratio=10); applying the CM to early data will lead to unstable results. Each solid line

shows the between a well pair. The dashed lines show the ideal ’s (from the unit-

mobility ratio). .................................................................................................................. 96

Figure 6-6 ’s variations by moving the analysis window for the Case 6.2 (mobility

ratio=1000); applying the CM only on very early data will lead to unstable results. Each

solid line shows the between a well pair. The dashed lines show the ideal ’s (from the

unit-mobility ratio). ........................................................................................................... 97

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Figure 6-7 The AAD variations by moving the analysis window for Case 6.1 (mobility

ratio=10) and 6.2 (mobility ratio = 1000); at a very late time the lower mobility ratio

provides less variable ’s. ................................................................................................. 97

Figure 6-8 ’s variations by moving the analysis window for the case where I03 was

shut-in for the first 100 months (Case 6.2.1); the results are not significantly different

from Case 6.2. Each solid line shows the between a well pair. The dashed lines show

the ideal ’s (from unit-mobility ratio). ............................................................................ 98

Figure 6-9 ’s variations by moving the analysis window for the case where I02 and I04

were shut-in for the first 100 months (Case 6.2.2); in comparison to previous cases, we

observed a slightly different trend in the ’s. Each solid line shows the between a well

pair. The dashed lines show the ideal ’s (from the unit-mobility ratio). ........................ 99

Figure 6-10 Plot of average absolute error in for M=1000 relative to M=1 for both a

system with and without horizontal well; x axis is the starting time of every 50 month

time interval in which the CM is applied, the ’s in a vertical well system approach stable

values after 50 months, whereas the ’s in horizontal well system approach stable values

after 150 months. The average absolute error in for horizontal well system is larger than

that of vertical well system. ............................................................................................ 101

Figure 6-11 CM is a robust tool to detect the presence of a wormhole (Case 6.4);

however, the wormhole geometry has a subtler effect (left vs. right). ........................... 102

Figure 6-12 Simple wormhole model; P01 is a 4 - branch horizontal well where the

length of branches is growing evenly in the reservoir (Case 6.5). .................................. 104

Figure 6-13 Type curves to evaluate equivalent skin associated with the wormhole for

any specific time; negative skin values in x-axis is associated to the wormhole (Case 6.5).

......................................................................................................................................... 104

Figure 6-14 Rate change of ’s with respect to length of wormhole stabilizes after the

wormhole grows some distance away from the borehole. In this plot, only the ’s of I01

are shown. Other injector ’s have the same trend (Case 6.5). ...................................... 106

Figure 6-15 Type curve generated to evaluate equivalent wormhole growth for any

specific time for a homogenous 5-injector 4-producer system; wormhole length

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(summation of all branches length) is divided by the length of the reservoir (Case 6.5).

......................................................................................................................................... 106

Figure 6-16 Liu and Zhao (2005) calculated maximum wormhole length versus time

during the fast growth period for 2 wells. ....................................................................... 107

Figure 6-17 Comparison of the CM results and the simulation model (Case 6.5). ........ 108

Figure 7-1 Average reservoir pressure using MPI (left) and transient MPI (right) with

permeability of 100 md for the Case 7.1; the blue dots indicate simulated average

pressure, while the red shows predicted average pressure. ............................................. 112

Figure 7-2 Average reservoir pressure using MPI (left) and transient MPI (right) with

permeability of 0.1 md for the Case 7.1; the blue line shows simulated average pressure

and the red line shows predicted average pressure. ........................................................ 112

Figure 7-3 At early time is a function of time then it approaches to a constant value

when reservoir boundary is reached. .............................................................................. 113

Figure 7-4 Transient and pseudo steady state is calculated for the Case 7.2. The left

figure is between I01 and P01; the middle figure is between I01 and P03; and the right

figure is between I03 and P01. ........................................................................................ 114

Figure 7-5 Transient is calculated versus permeability after 300 days. The left figure is

between I01 and P01; the middle figure is between I01 and P03; and the right figure is

between I03 and P01. ...................................................................................................... 114

Figure 7-6 Transient is calculated versus reservoir area after 300 days and a

permeability of 100 md. The left figure is between I01 and P01; the middle figure is

between I01 and P03; and the right figure is between I03 and P01. ............................... 115

Figure 7-7 Transient is calculated versus CM Number. The left figure is between I01

and P01; the middle figure is between I01 and P03; and the right figure is between I03

and P01............................................................................................................................ 115

Figure 8-1 In Case 8.1, three barriers and one channel exist in the reservoir. ................ 125

Figure 8-2 Producers’ conditions change for Case 8.1. .................................................. 126

Figure 8-3 Applying the MCCM provides the most accurate ’s for Case 8.1. Estimated

’s using the segmented/compensated CM is also good. The simple CM, however, gives

poor estimates. ................................................................................................................ 126

XX

Figure 8-4 Applying MCCM for the Case 8.1, provides accurate estimation of ’s. The

estimated ’s using the segmented/compensated CM are far from the correct ones. ..... 127

Figure 8-5 If a producer is shut-in temporarily within a sampling interval, it will lead to

an increase in the production rates of its connected producers. ...................................... 128

Figure 8-6 In general, the average rate is different from the instantaneous rate at the end

of the time step and, at smaller diffusivity constants, this difference is larger. The left

figure is for k=40 md and the right figure is for k=500 md. ........................................... 129

Figure 8-7 Number of shut-in days within sampling intervals for Case 8.2. .................. 130

Figure 8-8 Applying the MCCM provides accurate estimates of ’s for Case 8.2. ....... 131

Figure 8-9 Applying the MCCM, the estimated production rate is much more accurate

than the other estimators for Case 8.2. ............................................................................ 131

Figure 8-10 The estimated for the Case 8.2 are relatively inaccurate. ........................ 132

Figure 8-11 In Case 8.3, the estimated ’s using the MCCM are very close to the correct

values. ............................................................................................................................. 133

Figure 9-1 Overlain maps of sand bodies and well locations; red triangles indicate

injectors and black circles represent producers. The names are not actual names of the

wells; I = injector, P = producer, S = suspended and A = abandoned at the time. ......... 136

Figure 9-2 values calculated from analytical model for equivalent homogenous system;

values are between 0 and 0.2. ...................................................................................... 137

Figure 9-3 Analytical and versus well distance to determine window size; we selected

a cut off of 0.05 for at a 3000 ft distance and a cut off of 0.15 for at a 2000 ft

distance. .......................................................................................................................... 138

Figure 9-4 Comparing actual production rate with the model predicted rate ignoring

production hours (left) and including production hours (right). ..................................... 139

Figure 9-5 P35 (blue arrow) is located between two sand bodies. The connectivity values

are only slightly different from what would be obtained for a homogeneous reservoir. The

distance between grid lines is one mile (5280 ft). P signifies a producer, I represents an

injector, and S signifies a well currently shut-in............................................................. 141

Figure 9-6 P50 (blue arrow) is located within a sand body. The connectivity values are

large in absolute value and it could be a sign of wormhole development. ..................... 142

XXI

Figure 9-7 P52 (blue arrow) is mapped as being within a sand body. The connectivity

values are small in absolute value. .................................................................................. 142

Figure 9-8 Comparison of connectivity results and net pay map, Note the change of

scales for ′ for the P35 map (left) and P50/P52 map (right). ........................................ 143

Figure 9-9 Median and interquartile range (IQR) of ′ values for several producers. ... 144

Figure 9-10 Dye test arrival time for some wells in the southeast sand body; injection

started at 9 am. In some wells they did not detect any dye. ........................................... 146

Figure 9-11 Comparison of first arrival time calculated from the analytical model and

actual first arrival time of the dye; ellipses identify times from a common injector. ..... 147

Figure 9-12 Correlations of and with dye travel time. .............................................. 147

Figure 9-13 Seismic impedance-amplitude map and connectivity results for the

Storthoaks field; the yellow color signifies low impedance and the pink color represents

high impedance. .............................................................................................................. 149

Figure 9-14 kh map and connectivity results for the Storthoaks field; the red color

signifies high kh and the blue color represents low kh. .................................................. 150

Figure 9-15 Median of ′’s (′’s) and impedance map for the Storthoaks field. .......... 151

Figure 9-16 Median of ′’s (′’s) and kh map for the Storthoaks field. ........................ 151

Figure 9-17 MCCM predicts the total rate and catches the small fluctuations; 8-17 has a

high average rate). ........................................................................................................... 152

Figure 9-18 MCCM can predict the total rate of producers with a low rate (left) and those

which are shut in during the analysis period (right). ...................................................... 152

XXII

LIST OF TABLES

Table 2-1 Limitations of different methods in the literature ............................................. 26

Table 3-1 Reservoir and simulator parameters used for the Case 3.1 .............................. 33

Table 3-2 Selected field cases analyzed with the CM; we devoted a number to each field

to show in the contour plot................................................................................................ 57

Table 4-1 Reservoir and simulator parameters used for Case 4.1 .................................... 63

Table 4-2 ’s evaluated for the Case 4.1 (left) and the Case 4.2 (right); right table shows

’s are enhance for P01 and decreased for P02, P03, and P04. ........................................ 65

Table 4-3 Horizontal well P01 is divided into 11 vertical producers and ’s are calculated

between 14 producers and 5 injectors. .............................................................................. 73

Table 4-4 ’s of all 11 vertical wells are summed up to obtain ’s between horizontal well

and each injector. .............................................................................................................. 73

Table 6-1 Reservoir and simulator parameters for Case 6.1 ............................................. 92

Table 7-1 Reservoir and simulator parameters for the Case 7.1 ..................................... 111

Table 8-1 Reservoir and simulator parameters for Case 8.1 ........................................... 125

Table 9-1 Marsden south field properties ....................................................................... 135

Table 9-2 Storthoaks field properties .............................................................................. 148

CHAPTER 1 INTRODUCTION 1

CHAPTER 1 INTRODUCTION

1.1 Connectivity Evaluation

In waterflooding projects, the amount of water injected, location of each injector relative

to the producers, and future oil rate predictions are very important in managing recovery

performance. Evaluating interwell connectivity helps to achieve this information. By

connectivity in the reservoir we mean how the rate change (pressure change) of one well

affects (or relates to) another well rate (pressure). Interwell connectivity is positively

related to the value of permeability between well pairs. For example, a conduit between

two wells increases the connectivity of those wells. On the other hand, there is a very low

connectivity between two wells which are separated by a fault.

To evaluate interwell connectivity, different methods have been suggested, from

Muskat’s (1949, page 572) work until now. Some of these methods, however, are only

based on injection-production rate data such as the capacitance model (CM).

Yousef et al. (2006) developed and applied the CM to predict the total production rate

(oil and water) of each producer as a function of the injection rates of all injectors in the

system and the bottomhole pressures (BHPs) of all producers. The model uses three sets

of parameters; the first quantifies the connectivity between injector-producer pairs, the

second quantifies the amount of fluid storage or time lag between those pairs, and the

third shows the effect of producer BHPs on the production rate. Having the knowledge of

well pair connectivity by means of the CM parameters may result in a better adjustment

of injection rates, infill drilling, or producer shut-in. This model has been tested in several

real fields, and the results have been favourably compared with geological and

geophysical data.

1.2 Importance of Topic

Injection and production rate data are easily accessible and using them does not incur the

costs of running field tests. Unlike simulation-based methods, the CM does not require

geological and geophysical data to generate the initial model. Furthermore, the CM is less

time-consuming than simulation-based methods. Compared to other injection and


production data analysis methods, the CM is more robust in terms of accuracy of

parameters and prediction ability. Although we use nonlinear regression to obtain

connectivity parameters at the final step of our calculation, the CM is derived from

material balance and the productivity model. Therefore, it is based on pressure change

propagation and we can evaluate static connectivity from dynamic data. In Chapter Two

we describe, in detail, mostly rate-based methods of connectivity evaluation in the

literature. In this work we will show how the CM can practically be enhanced or adopted

for non-ideal conditions and how much useful information can be achieved using the

model.

1.3 General Achievements

To evaluate interwell connectivity among all the injection-production rate-based

methods, we use the CM in this work. While the CM is a fast and robust tool for better

understanding of the reservoir heterogeneity in waterflooding projects, some aspects need

improvement. These aspects primarily, concern adapting the CM or interpreting CM

results so that it gives improved results in common but non-ideal situations.

The CM is basically derived from a linear productivity model in the pseudo steady state

regime for slightly compressible fluids. Therefore, we expect the results are stable within

a specific range of fluid and reservoir properties such as permeability, compressibility,

viscosity, porosity, and number of wells per area. By stable results (stability of the

results) we mean that changing the input values (injection rates) does not affect the CM

estimated parameters. Therefore, we should generate the same CM parameters for all the

runs and the CM parameters should be repeated. There is, however, also a statistical side

involving the number of sampling data, sampling interval, and amount of noise which

affects the accuracy of the results. Before applying the CM on field data, a sensitivity

analysis on the accuracy of the expected results is desirable. In Chapter Three, we carry

out a sensitivity analysis on the listed parameters by changing each of them and keeping

the others constant (in ideal conditions) during simulations and defining a dimensionless

number, the CM number (C), to specify a range in which we can apply the CM with

accurate results (0.3 < C < 10). However, we may not have some of this information

while we work on field data. To solve the problem, we apply the bootstrap, which is a


computational method using resampling with replacement to evaluate the uncertainty and

to provide unbiased estimates. Finally, by calculating C and L (the ratio of number of

samples to the number of model parameters) from the available information, we estimate

the uncertainty of evaluation for eleven published field cases. The findings in this chapter

not only help us to estimate the accuracy of the CM before applying it on field data, but

also to assist the CM users to assess repeatability and stability of the results and to

mitigate the effect of noise after applying the model.

The CM equation is derived from a productivity model assuming radial flow in the

drainage area of each producer; most of the pressure drop will occur within a few feet of

the wellbore. Therefore, heterogeneities close to the wellbore have more effect on

production (producers’ productivity) and the CM parameters than interwell

heterogeneities between injector-producer pairs. Depending on the well geometry, we

may have different sizes and shapes of this high pressure drop area in the vicinity of the

producers. For instance, this area will be a circle with radius of a few feet for a vertical

well and a large ellipse around a horizontal well in the pseudo steady state regime. In

Chapter Four, we show the effects of well geometry on the near-wellbore pressure drop

and the implications for the CM. Then we propose methods to decouple the impact of

well geometry from heterogeneities. Afterwards, in Chapter Five, we show how the CM

is robust to assess the near producer heterogeneities and then propose a method to

evaluate equivalent skin around the wells.

In Chapter Six, we illustrate the application of the CM in heavy oil reservoirs. In heavy

oil reservoirs, the non-unit mobility ratio violates the CM assumptions and influences the

CM parameters. A higher mobility ratio results in a larger effect on these parameters,

hence they are less representative of interwell connectivity. We show that at large

mobility contrasts (≈ 1000), analyzing the data after 0.4-0.5 PV of injection leads to

stable CM results. Development of a wormhole around a producer in heavy oil reservoirs

alters the permeability, specifically in the near wellbore region. Depending on the time

interval of investigation, the CM gives us different assessments about the wormhole

development. In this chapter, we suggest an analytical model to predict maximum

wormholes length.


In Chapter Seven, we extend the CM for the case of transient flow, where the

connectivity parameters are a function of time. This approach helps us to apply the model

in tight formations. Finally, in Chapter Eight, we enhance the CM to be applicable in real

fields when we have shut-in and mini shut-in production or, more generally, skin

changes. Results of one conventional and one heavy oil (medium to heavy oil) field cases

will be analyzed in Chapter Nine. As a summary of this work, we close with our

conclusions and recommendations in Chapter Ten.

CHAPTER 2 LITERATURE REVIEW 5

CHAPTER 2 LITERATURE REVIEW

2.1 Introduction

In this work, we focus mostly on interwell connectivity evaluation using dynamic

injection production rate data. To begin, we discuss several methods suggested in the

literature. Generally, we classify these methods into three categories:

1) Statistically-based methods

2) Material (fluid) propagation-based methods

3) Potential (pressure) change propagation-based methods

Methods in the first category are based on statistical evaluations and do not obey physics

of the fluid flow through porous media, while the methods in the second and third

categories are based on the fluid flow equations in porous media. Since the methods in

category two are based on fluid propagation, they should be rate dependent. The methods

in the third category, however, can represent the static connectivity from dynamic data. In

the waterflooding process, pressure change will reach the production well faster than

material will. For example, at the beginning of waterflooding in a system of one injector

and one producer, water breakthrough may not happen in a producer, while any pressure

change from an injection well already affects the production rate of that producer.

2.2 Statistically-Based Methods

Those methods in which the reservoir properties and reservoir fluid flow equations are

not used in the model are included in this category, such as the Spearman rank

correlation, neural networks, wavelet analysis, and extended Kalman filter (EKF). In this

section we explain these methods.

2.2.1 Spearman Rank Correlation

Heffer et al. (1997) applied Spearman rank correlations to relate injector/producer pairs

and associated these relations with geomechanics. By converting the rates to ranks, the

Spearman rank correlation coefficient (a standard non-parametric statistic correlation


coefficient) is calculated for pairs composed of each injection well and its adjacent

production wells. The higher values of rank correlations means the higher the amount of

well pair connectivity.

Refunjol and Lake (1999) used this analysis with a time lag to include the effect of

medium and distance. They chose this time lag in such a way that the correlation

coefficient approaches the largest value. Figure 2-1 depicts the principle of correlation

between injection rate and production rate with a specific time lag (in this figure both

wells are highly connected).

Soeriawanata and Kelkar (1999) defined an arbitrary threshold (they used 0.5 in their

study) for the cross correlation. Afterward, they added the rate of a selected injector to

the injector with the highest cross correlation and, if its cumulative cross correlation is

higher than that threshold, they considered a significant connectivity between those

injectors and the target producer.

Fedenczuk et al. (1998), based on the Spearman rank correlation, generated spider graphs

to depict the communication between injectors and horizontal producers and to visualize

the correlations between all injectors and a specific horizontal producer.

Figure ‎2-1 The principle of correlation analysis; the injector is highly connected to the producer.

From Fedenczuk et al. (1998).

Compared to recent methods, Spearman rank correlation is not a robust tool to infer

interwell connectivity in a system of multiple injectors and producers, as it does not

always result in a correct correlation in good agreement with the geological features. For

example, they obtained some negative values which are not consistent with reality,


however, it is easy to use. Since this is a statistical-based method, and fluid flow through

porous media is not involved in the method derivation, any change in well condition such

as producers’ BHP changes, skin change, and well shut-ins may impact the results. In

addition, they provided spurious correlations for injectors-injectors and producers-

producers pairs. Also, they did not mention fluid phase and flow regime (fluid and

medium properties) for the application of the model.

2.2.2 Artificial Neural Network

Panda and Chopra (1998) determined the interaction between injector/producer pairs by

means of artificial neural networks. There are two steps for developing a neural network:

1) A training phase in which an internal weight matrix is evaluated by iteratively

comparing the output from the neural network with known results (similar to

history matching for flow simulation).

2) Applying the trained net or converged weight matrix to map user-controlled input

data to obtain predicted results.

The network consists of three layers: the input layer reads the input data, the output layer

determines the required results, and a hidden layer processes the intermediate results. In

other words, the values at the input nodes are multiplied by some weights and

transformed to hidden nodes by a transformed function. Afterward, with a similar

treatment, the results of the output nodes are evaluated (Figure 2-2).


Figure ‎2-2 Schematic of a typical artificial neural network; the input layer reads the input data, the

output layer determines the required results, and the hidden layer processes the intermediate results.

From Panda and Chopra (1998).

Demiryurek et al. (2008) quantified the interwell connectivity between injectors and

producers in a reservoir by a sensitivity analysis based on a neural network. They

analyzed the outputs (production rates) by varying the injection rates; i.e. the inputs to the

trained neural network model, and thereby assessed the influence of the candidate

injector on the target producer. Hence, by a specific amount of injection rate change, if

the production rate change is noticeable, that injector and producer are well connected.

They did not explain the time delay in the injection-production relationship.

In these references, they also did not point out whether or how they used fluid and

reservoir properties. Similar to the previous method, any changes in well condition such

as producers BHPs changes, skin change, and well shut-ins may impact the results. They

also did not mention fluid phase and flow regime for the application of the model. To

obtain satisfactory results, a long waterflooding history is presumably necessary in order

to train the internal matrix. There are few examples in the literature on the application of

this method.

2.2.3 Injector-Producer Relationships Using an Extended Kalman Filter

Liu and Mendel (2009) estimated injector-producer relationships between multiple

injectors and a single producer based on measured production and injection rates using an

extended Kalman filter (EKF). In this model, each injector-producer pair is assumed to be


a subsystem in which the injection rate is transformed to the production rate by means of

a moving average model. Hence, the reservoir is considered as a collection of subsystems

that convert injection rates to the production rate (Figure 2-3). They then suggested a

formula to calculate the injector-producer relationship whose parameters are obtained by

the EKF. Ultimately, for N injectors contributing to one producer, 2N parameters will be

estimated to generate N injector-producer relationship values. They also utilized a

modified EKF for processing real data.

Similar to the previous methods, any change in well condition such as producers BHPs

changes, skin change, and well shut-ins may impact the results. They also did not

mention the effects of fluid phase and flow regime (fluid and medium properties) for the

application of the model. Producer-producer relations are not included and there are few

examples in the literature.

Figure ‎2-3 Extended Kalman filter; each injection production pair is assumed to be a subsystem in

which the injection rate is transformed to the production rate by means of a moving average model.

Function “f” decouples the effect of distance from noise free injection rates, “n” indicates noise

function and “m” is the abbreviation of measured parameters. From Liu and Mendel (2009).

2.2.4 Interwell Relation Based on Wavelet Analysis

Jansen et al. (1997) used the wavelet transformation to decompose the production data

into a combination of high frequency (details) and low frequency (smoothed)

components. Consequently, they analyzed the interwell relationship by interpreting those

high frequency components. In other words, they compared the variance of high


frequency components of the injection rate and water production. A similar trend

between the two shows good communication between well pairs (Figure 2-4).

Figure ‎2-4 Variance of high frequency component of injection rate and water production; a similar

trend is a result of good communication between well pairs. After Jansen et al. (1997).

Lee et al. (2011) considered the production rate of each producer as a linear function of

the filtered injection rates of surrounding injectors. To account for any attenuation and

delay between injection and production rates, they filtered those injection rates.

1

1,2, ,I

j ij ij

i

q t w t other terms j K

...................................................( 2-1)

where jq t is the production rate of producer j, ij is the relative weight of producer j

for injector i, ijw t is a filtered injection rate of injector i for producer j. I is the number

of injectors and K is the number of producers. This model is similar to Albertoni and

Lake 2003 (Section 2.4.2).

They estimated the ij’s using the Haar Wavelet, and compared them with flow units

obtained by a streamline model which is presented in Section 2.3.1 (Figure 2-5) and

weight factors estimated using the CM which is explained in Section 2.4.4 (Figure 2-6).

Fre

qu

en

cy V

aria

nce

Time (months)

Scaled Variance for injection rate

Variance for water production rate


Figure ‎2-5 Effective flow units estimated by the streamline model are indicated in red and those

estimated by this model are indicated in blue for a 5×4 pattern (5 injectors and 4 producers). From

Lee et al. (2011).

Figure ‎2-6 Weight factors estimated using the CM are shown in red, and estimates by this model are

outlined in blue for a specific field. From Lee et al. (2011).


They extended their work (Lee et al., 2011) by plotting a flow-storage capacity diagram

to interpret interwell heterogeneity.

Similarly, any change in well condition such as producers BHPs changes, skin change,

and well shut-ins may impact the results. They also did not mention the effects of fluid

phase and flow regime (fluid and medium properties) on the results for the application of

the model. There are few examples of this in the literature.

2.3 Material (fluid) Propagation-Based Methods

These methods in which the amount of fluid and fluid saturation profile are the basis for

analysis are included in this category, such as streamline based methods and non-reactive

tracer tests. In these methods, the propagation of fluid is the basis of the analysis, rather

than the pressure disturbance. Therefore, the results are changing by rate and are not

strongly representative of static interwell heterogeneity.

2.3.1 Streamline-based Method

Thiele and Batycky (2006) used streamline-based simulation to evaluate interwell

connectivity. They defined a dynamic well allocation factor between injection and

production wells by adding together the rates of all the streamlines associated with a

particular well pair. Afterward, they calculated the efficiency of injection wells as the

ratio of sum of the oil produced at offset producers to the injected water (Figure 2-7).

Water can be reallocated from low efficiency to high efficiency wells. To visualize

reservoir flow in this method they used well positions, well rates, geological description,

fluid properties, relative permeability and reservoir continuity.


Figure ‎2-7 Injection efficiency of I5 is the oil produced at the offset producers (P3, P4, P5 and P7)

divided by the injection rate of I5. The offset oil produced is the sum of oil produced by the red,

green, orange and yellow streamline bundles. From Thiele and Batycky (2006).

Jensen et al. (2004) interpreted streamline-determined connectivity as a steady state

hydraulic connection between well pairs. Therefore, zero values of connectivity for non-

adjacent wells occur because of no direct hydraulic connection. These connectivity values

are injection rate dependent. This model needs information about fluid and reservoir

properties.

2.3.2 Non-reactive Tracer Test

Albertoni and Lake (2003) simulated the injection of a non-reactive tracer in a synthetic

field. They observed that the tracer response is not symmetric, depending on how much

tracer is injected to each injector (or average injection rate). If a tracer is injected as a

pulse input (which is more common in the fields), the tracer response depends not only

on the average injection rate, but also on the period in which the tracer is injected and

produced. Therefore, they concluded that the obtained connectivity values are not unique

compared to the MLR connectivity values (Section 2.4.2).

2.4 Potential (pressure) Change Propagation-Based Methods

Methods in which the pressure disturbance is the basis for the analysis are included in

this category, including the BHP-based method, MLR, MPI-based method, and the CM.


The methods of this group better represent characteristics of the medium such as

permeability changes regardless of the amount of fluid injected. Among these methods,

the CM has been easily tested for a variety of synthetic and real field data.

2.4.1 BHP-based Connectivity

Dinh and Tiab (2008) considered the BHP of each producer as a linear function of the

BHPs of surrounding injectors.

1

1,2, , I

j ij i

i

P t P t j K

.................................................................( 2-2)

where jP t is the estimated BHP of producer j, ij is the relative weight of producer j

for injector i, and iP t is the BHP of injector i.

They used multivariate linear regression to estimate the weight factors and claimed that

these weight factors yield better results in comparison to the flow rate based methods

when there are shut-in periods in the data. Figure 2-8 (left) shows the map of weight

factors estimated using this method for a 5×4 case (5 injectors and 4 producers). Pressure

propagation is faster than fluid propagation in the media thus, roughly, they do not need

to filter injection pressures accounting for attenuation and time lag. This method is

potential based, so pressure perturbations of each injector BHP will be transmitted to the

producer. Larger distances between wells or the existence of smaller permeability regions

between them will result in a smaller effect on producer BHP.

They also coupled their model with a solution for pressure distribution due to fully

penetrated vertical wells in a closed rectangular reservoir, and calculated some reservoir

parameters such as relative interwell permeability (Figure 2-8, right) from the interwell

connectivity coefficients (Dinh and Tiab, 2013).


Figure ‎2-8 The map on the left shows weight factors that were estimated using this method in a 5×4

homogeneous reservoir. The length of the arrow is proportional to the value of the coefficient. The

map on the right shows interwell permeability calculated using this method as well. The length of the

arrow is proportional to the value of the relative interwell permeability. From Dinh and Tiab (2013).

However, a minimum number of data points is required to obtain accurate results; the

model is not practical in real field cases since pressure data is often unavailable or

measured infrequently. They also assumed a constant production rate which limits the

application of this method in the real field. In addition, well productivity, GOR, effective

permeabilities and the total injection rate should be constant. Drilling a new well and/or

shutting in a well may impact the results. The flow rate of producers should be

maintained only by injection rates to reach satisfactory results (for example, no other

external forces should exist in the system, such as active aquifer).

2.4.2 MLR Model

Albertoni and Lake (2003) assumed the production rate of each producer is a linear

function of the filtered injection rates of surrounding injectors.

1

ˆ 1,2, ,I

c

j ij ij

i

q t w t j K

....................................................................( 2-3)

where ˆjq t is the estimated production rate of producer j, ij is the relative weight of

producer j for injector i, and c

ijw t is the filtered injection rate of injector i for

producer j.


They used multiple linear regression (MLR) to estimate the weight factors (’s). In other

words, they considered the reservoir as a system which converts an input signal

(injection) to an output signal (production), and interpreted the ’s as a degree of

interwell connectivity between well pairs.

They introduced a diffusivity filter, accounting for attenuation and time lag. The shape of

the diffusivity filter is a function of the diffusivity constant (which depends on the

medium and the distance between the injector and producer). Figure 2-9 demonstrates the

map of these connectivity values for both homogeneous and heterogeneous system. A

sealing fault between an injector and a producer decreases the ’s between them, relative

to the homogenous ’s.

Figure ‎2-9 The map on the left shows the ’s‎estimated‎for‎a‎5×4‎homogeneous‎synthetic‎field.‎The‎

length of the arrow is proportional to the value of the coefficient. The map on the right shows the ’s‎

estimated for a 5×4 synthetic field with a sealing fault. From Albertoni and Lake (2003).

Although the ’s are estimated by linear regression, in the diffusivity filter formulation

the pressure formula has been applied in an infinite reservoir. Thus, this model has a

potential nature. The results show that the ’s are independent of the injection rates,

supporting this view. Any change in producer BHPs, new completions, and high GOR

violate the model assumptions. The model is applicable under constant operational

conditions and the diffusivity constant should be high (≈>106 md.psi/cp for a

homogenous 5×4 reservoir). Non-waterflooding production and effective permeability

should be constant.


2.4.3 MPI-based Connectivity

Valkó et al. (2000) suggested the linear productivity model as a matrix form (Appendix

1) for a multiwell homogeneous rectangular reservoir of uniform thickness, h, porosity, ,

permeability k, with constant single phase fluid viscosity, , and total compressibility, ct:

1

1 11 12 1 1

2 21 22 2 2

1

1 2

0 0

0 02

0 0

A Influence matrix

N

N

N N N NN N

q rate vector

J

q a a a s

q a a a skh

q a a a s

1

2

wf

wf

wf N

d drawdown vector

MPI matrix

p p

p p

p p

................................................( 2-4)

where iq is production rate (or injection rate with negative sign) of well i, 1

2

7.08

,

aij’s are influence functions (a function of location and boundary), si is the skin factor of

well i, p is the average reservoir pressure, and iwfp is the BHP of the producer (injector)

i. Multiwell productivity index (MPI) matrix is a generalized form of the productivity

index when more than one well exists in the reservoir.

They did not mention that there is any sign of connectivity in the influence matrix until

Kaviani and Valkó. (2010) used this matrix to infer interwell heterogeneity. They

subtracted the homogeneous reservoir influence matrix (calculated by a formula based on

the well locations and reservoir boundaries, Appendix 1) from the actual reservoir

influence matrix estimated by production-injection history, and mapped the resulting

matrix (heterogeneity matrix) to show the degree of heterogeneity between each pair

(Figure 2-10).


Figure ‎2-10 Normalized connectivity map (map of heterogeneity matrix, Aoptimized-Ahomogeneous); the

gray features are barriers and the white feature is a conduit. For a homogeneous case, the

connectivity values are zero. From Kaviani and Valkó (2010).

In this method, the effects of well locations, reservoir boundary and well conditions such

as skin are decoupled from the interwell heterogeneity assessment. The heterogeneity

matrix is also independent of operational conditions such as well shut-ins, new well

completions, or converting wells from producer to injector. Kaviani and Valkó (2010)

observed that larger changes in producer BHPs result in a better estimation of the

heterogeneity matrix. This model is based on pressure propagation; hence, the

connectivity values are rate (material) independent and are representative of the interwell

heterogeneity. Small permeabilities violate the model assumption in comparison to the

methods in which regression is used. In other words, it is important that the diffusivity

constant should be high enough. GOR and effective permeabilities should be constant.

All producer and injector rates and BHPs are needed to evaluate heterogeneity matrix.

2.4.4 Capacitance Model (CM)

Yousef et al. (2006) coupled a linear productivity model with material balance and

applied superposition in space to obtain a general relation of the flow rates between each

producer with surrounding injectors (see Appendix 2 for the model derivation). As we

explained in Chapter 1, the relationship includes three sets of parameters in the final

model, a weight factor (’s) which quantifies the degree of connectivity between each


injector-producer pair, a time constant (’s) which quantifies the degree of storage

between well pairs, and a weight factor (’s) for the effects of producer BHPs (Equation

2-5).

0 0( ) ( )

0 0

1 1

ˆ pj kj

t t t ti I k K

j pj j ij ij kj wf k wf k wf kj

i k

q t q t e w t v p t e p t p t

....................( 2-5)

where ˆjq t is estimated total production rate of producer j, ijw t and wf kjp t are

(Yousef et al. 2006; Kaviani et al. 2012):

1

1

( )

m m

ij ij

t t t tn

ij i m

m

w t e e w t

.....................................................................................( 2-6)

1

1

m m

kj kj

t t t tn

wf kj wf k m

m

p t e e p t

..............................................................................( 2-7)

The weight factor ij indicates the connectivity for the ij well pair, ij is the time constant

for the medium between injector i and producer j, ijw t is the convolved or filtered

injection rate of injector i on producer j, wf kjp t is the convolved BHP of producer k on

producer j, kjv is a coefficient that determines the effect of the changing BHP of producer

k on producer j, 0jq t is the initial total production rate of producer j, pj is the resultant

time constant of the primary production component and kj is the time constant between

producers k and j.

They mapped both ’s and ’s to demonstrate interwell connectivity. In a homogenous

reservoir, the ’s and ’s are functions of well locations and reservoir boundaries;

assuming production wells have no skin (Figure 2-11).


Figure ‎2-11 Map of the ’s‎(left)‎and‎the‎’s‎(right)‎for‎a‎homogeneous‎5×4‎system;‎the‎length‎of‎the‎

arrow is proportional to the or values. From Yousef et al. (2006).

Yousef et al. (2009) used both the - plot in a log-log format and the flow storage

capacity plot (F-C plot) to interpret geological features such as permeability trends,

barriers, and fractures, and to provide a more integrated connectivity analysis (Figure 2-

12).

Figure ‎2-12 The left plot shows a log-log plot of the ’s‎versus‎’s.‎I04‎and‎I05‎are‎located‎in‎a‎higher‎

permeability layer. The right plot shows a schematic of a different trend of the F-C curve estimated

from the CM parameters, according to the corresponding geological features around a producer.

From Yousef et al. (2009).

Liang et al. (2007) used the CM and, neglecting the BHP variations for all producers,

obtained satisfactory results for the long time behavior between injectors and producers.

They plotted optimal rates under different revenue objectives by coupling this model with

a fractional flow model.


Sayarpour et al. (2009) used the CM in three ways:

1) One for each producer.

2) One for the field (assuming one lumped injector and one lumped producer exist

in the field i.e. treating the field as a tank).

3) One for each injector-producer pair (the same as shown in the work of Yousef et

al.).

They introduced analytical solutions to the fundamental differential equations of the CM

based on superposition in time (case 1 and 2), and superposition in time and then in space

(case 3). They calculated production rates based on two assumptions:

1) Linear variation of BHP during time intervals, but stepwise changes in the

injection rate.

2) Linear variation of both the injection rate and BHP during a consecutive time

interval.

Finally, they coupled the model with a fractional flow model to estimate the oil

production rate, to optimize the value of oil produced by adjusting the water injection

rates.

To estimate representative parameter values within the analysis time window using the

CM, the reservoir conditions must satisfy the following assumptions:

1) Known or constant producer BHP’s; if producers’ BHP’s change and are

unknown, we cannot ignore the BHP term in the CM equation. Ignoring this term

leads to inaccurate results.

2) Constant number of producers; shutting in a producer or adding a new producer is

similar to having a large change in that producer’s BHP. Since we do not have

this change of BHP to input into the CM equation, the obtained results are

incorrect.

3) Constant producer productivity indices; the CM solution is obtained by taking

integration from an ordinary differential equation (Appendix 2) in which producer

productivity indices are constant.


4) Slightly compressible fluid; linear productivity model used in the derivation of the

CM equation is based on slightly compressible fluid assumption.

5) Near-unit mobility ratio; a changing mobility ratio changes the producers’

productivity, which deteriorates the accuracy of the CM results.

6) Constant operational conditions; any change of producers’ skin changes their

productivity during the analysis window and gives misleading results.

7) High diffusivity constant; in chapter 3 we will show that having a diffusivity

constant ≈>106 md.psi/cp guarantees that the homogenous 5×4 reservoir has

passed transient regime and the CM results are stable and repeatable.

If the above reservoir conditions are met, and we have sufficient and accurate data, the

CM parameters will be constant during the analysis window and independent of the

injection rates. In practical field cases, however, we may not satisfy all these

assumptions. Modified versions of the CM are available, which are more robust to

deviations from the above assumptions. Kaviani et al. (2012), for example, showed that

applying the segmented CM can overcome the problem of unmeasured fluctuating BHP

data if the number of major BHP changes is limited. They developed a multi-stage

algorithm to optimize the number of segmentation times at which one or more of the

producers’ BHP changes. In the case of a varying number of producers, Kaviani et al.

(2012) proposed the compensated CM to solve the problem. They applied the concept of

virtual wells when one or more of the producer wells are shut in or a new production well

is added to the reservoir. Jensen et al. (2011) showed the application of the compensated

CM for cases where the productivity index of a producer changes.

is a rate-independent (static) measure of connectivity between the injector-producer

pairs that depends on the location of the wells, reservoir boundary, well conditions, and

interwell heterogeneity. For rectangular homogeneous reservoirs, Kaviani and Jensen

(2010) derived the ’s analytically using the MPI model (Valkó et al., 2000). To do so,

they divided the influence matrix (Equation 2-4) into four parts, considering vector w is

a matrix of injector rates with minus sign and q is a matrix of producer rates, 1

2 kh


and, for simplicity in notation, they considered that the skin matrix is added to the

influence matrix and named the new matrix [A],

1

wfi inj con

T

wfp con prod

p p A A w

p p A A q

........................................................................................( 2-8)

Then, they calculated the CM ’s analytically. The detail of the derivation is represented

in Appendix 3. The final matrix form of the analytical ’s is shown in Equation 2-9.

1

1

1

1 1

1 1[ ]

1 1

p p p i

p i

p p

T

prod conN N N N T

N N prod con

prodN N

A AA A

A

............................................( 2-9)

where pN is the number of producers and iN is the number of injectors. They subtracted

the analytical ’s from optimized ’s in an attempt to decouple the effect of well location

and boundary, and to evaluate the absolute interwell connectivity.

homoptimized ogeneous ....................................................................................................( 2-10)

They applied the model to a real field case and favorably compared their results of

with geological maps.

Soroush (2010) used the concept of the CM and derived another model in which the

injection rate of each injector is a function of the production rate of all producers and

injection well BHPs (Equation 2-11, also see Appendix 4 for the model derivation).

0 0

* ** * *

0 0

1 1

ˆ pi ri

t t t tj K r I

i pi i ji ji ri wf r wf r wf ri

j r

w t w t e q t v p t e p t p t

.........( 2-11)

where ˆiw t is the predicted injection rate of injector i,

jiq and wf rip t are:

1

* *

1

( )

m m

ji ji

t t t tn

ji j m

m

q t e e q t

.................................................................................( 2-12)

1

* *

1

m m

ri ri

t t t tn

wf ri wf r m

m

p t e e p t

..........................................................................( 2-13)


where I is the number of injectors, K is the number of producers, weight factor ji

indicates the connectivity for the ji well pair, ji is the time constant for the medium

between producer j and injector i, jiq t is the convolved or filtered injection rate of

producer j on injector i, wf rip t is the convolved BHP of injector r on injector i, *

riv is a

coefficient that determines the effect of changing the BHP of injector r on injector i,

0iw t is the initial injection rate of injector i, pi is the resultant time constant of the

initial injection solution and ri is the time constant between injector r and i.

Weight factors and time constants in this model are different in value compared to the

CM model, but they are representative of interwell connectivity as well. The existence of

a horizontal producer, the shutting in of a producer, and work over do not affect these

parameters.

2.5 Conclusions

Table 2-1 summarizes the limitations of above mentioned methods.

In statistically-based methods, the reservoir fluid and rock properties are not used in the

model, and flow physics is ignored. Therefore, any change on well condition such as

producers’ BHP changes, skin change, and well shut-ins violate the results. None of these

methods feature any mention of fluid phase and flow regime for the range of applicability

of the models. There are few examples of real field applications for this category. When

we want to investigate the existence of any correlation between injector and producer

rates, we assume that any change in production rate occurs as a result of injection rate

change considering a time lag. Therefore, non-waterflooding changes, which are common

in real field cases, give misleading results. For example, obtaining negative weight

factors confirms the above conclusion.

In material (fluid) propagation-based methods, the amount of fluid and fluid saturation

profile are the basis for analysis. Therefore, they are rate dependent and they are not

representative of medium heterogeneity if the heterogeneity effects are not sufficiently

large. Depending on both type of injection rate variation and magnitude of injection rates,


the time to breakthrough for each producer and well pair connectivity change. The tracer

test is very common in the fields.

In potential (pressure) change propagation-based methods, pressure disturbance is the

basis for analysis. Methods of this group better represent the characteristics of the

medium, such as permeability changes regardless of the amount of fluid injected. Among

these methods, the CM has been tested for a variety of synthetic and real field data.

However, in the CM model non-linear regression is used at the end of calculations; the

CM is derived from flow equations of pressure disturbance and has a potential nature.

Consequently, the ’s are independent of the injection rate. For example, we obtain non-

zero values for non-adjacent wells, even when there is no hydraulic connection. To see

the difference between these categories we assume a reservoir with 2 producers (P01 and

P02) and 2 injectors (I01 and I02) along a straight line where injectors are between the

producers and P01 and P02 are in equal distance from I01 (Figure 2-13). We assume a

constant operational condition for these producers. The CM connectivity values of I01-

P01 and I01-P02 are equal (0.5). However, weight factors from the second category may

result in a high value of I01-P01 relative to I01-P02 depending on well rates. I02

decreases (or may disconnect) the hydraulic connectivity between I01 and P02. We

believe that the results of the first category for this homogenous case are affected by

producer BHPs as well and may not be always the true results.

Figure ‎2-13 Homogeneous 2×2 reservoir; P01 and P02 are in equal distance (x) from I01.

In the following chapters, the range of applicability of the CM and accuracy of the results

will be analyzed. Afterward, it will be shown how the CM can be enhanced or adapted

for non-ideal situations.

P02I02P01

x x

I01


Table ‎2-1 Limitations of different methods in the literature

Method References Limitations

Spearman rank

correlation

Heffer et al. (1997)

Refunjol and Lake (1999)

Soeriawanata and Kelkar (1999)

Fedenczuk et al. (1998)

Provides spurious correlations

for injector-injector and

producer-producer pairs

Is easy to use but it does not

always create correct

correlations

Producer BHP changes, skin

changes, and well shut-ins

impact the results

Fluid and medium properties

are not mentioned

Artificial neural

network

Panda and Chopra (1998)

Demiryurek et al. (2008)

Needs long data history

Few examples are in the

literature and the details of

input data are not mentioned



impact the results


are not mentioned

Extended Kalman

Filter

Liu and Mendel (2009)

Effect of N injectors is obtained

on only one producer

Few examples exist in the

literature


changes, and well shut- ins

impact the results


are not mentioned

Wavelet Approach Jansen and Kelkar (1997)

Lee et al. (2011)

Few examples exist in the

literature (there are few

examples to validate their

model with streamline based

method and the CM)




impact the results


are not mentioned

Streamline based

method

Thiele and Batycky (2006)

Jensen et al. (2004)

Results are rate dependent

Non-adjacent wells have zero

weights

Needs information about fluid

and reservoir properties

Non-reactive tracer

test

Albertoni and Lake (2003) Results are rate dependent

Non-adjacent wells have zero

weights

Results depend on period in

which the tracer is injected and

produced

BHP based

connectivity

Dinh and Tiab (2008) Flow rates are constant at the

observation wells

Total injection rate should be

constant.

Flow rate at the observation

wells is maintained by injection.

Well productivity, GOR and

effective permeabilities should

be constant

New well and well shut-in

impact the results

Is not practical in the field

because injectors and

producers BHP are not

frequently measured

MLR Albertoni and Lake (2003) Applicable under constant

operational conditions

New well and well shut-in

impact the results

Producing BHP should be

constant or non-waterflooding

production should be constant




be constant

Diffusivity constant should be

high

MPI based

connectivity

Kaviani and Valkó. (2010)

GOR and effective

permeabilities should be

constant


high

CM Yousef et al. (2006, 2009)

Liang et al. (2007)

Sayarpour et al. (2009)

Kaviani et al. (2012)

Jensen et al. (2011)

Kaviani and Jensen (2010)

Soroush (2010)

Applicable under constant

operational conditions (the

reverse CM does not have this

limitation)



be constant


high

CHAPTER 3 SENSITIVITY ANALYSIS 29

CHAPTER 3 CM SENSITIVITY ANALYSIS

3.1 Introduction

To calculate the CM parameters, we need to determine the values that minimize the

differences between the predicted and measured production rates over the analysis

window ( 2

1 1

ˆn K

j m j m

m j

q t q t

). This can be done using a non-linear solver. In ideal

conditions, the CM parameters obtained by the solver will be constant during the analysis

window and independent of the rates. Errors, however, will arise in non-ideal situations

and their effects on the and estimates must be evaluated. By non-ideal situations, we

mean the violation of reservoir and statistical assumptions. In Section 2.4.4, we

mentioned seven assumptions for reservoir conditions using the CM. In this chapter, we

only investigate the sensitivity of the CM to the following parameters:

1) Changes in the diffusivity constant

2) Sampling time

3) Reservoir area

4) Number of producers

5) Number of data (injection and production rates) for the analysis window

6) Noise

To do so, we introduce a dimensionless number called the CM number to generalize our

results.

We also show that using the bootstrap, an uncertainty measurement technique, can reveal

the uncertainty of the CM results and provide accurate error assessments.

In this chapter, however, we do not consider the following issues in our sensitivity

analysis:

1) Changing operation conditions; i.e. a changing number of producers, skin, and

producer BHPs


2) Effect of varying mobility ratio; in Chapter Six, we will show that at large

mobility contrasts (≈ 1000), analyzing the data after 0.4-0.5 PV of injection leads

to stable CM results. In other words, by excluding the first 0.4-0.5 PV of

waterflooding data from the analysis, we will get stable and repeatable and

values.

3) Uncertainty of each connectivity coefficient individually; all the results of this

chapter only provide a collective accuracy evaluation of all ’s or ’s in each run.

4) The effects of variance, magnitude, and collinearity of injection rates. Like other

perturbation based methods, the CM performance depends on the variance of

input signals. If the variance of injection rates is small, the signal to noise ratio

may not be sufficient for the CM to accurately estimate connectivity. In addition,

a strong correlation of injection rates leads to poor performance of the CM in

identifying well interactions.

5) One of the non-ideal situations is the existence of deviated or horizontal wells in

the reservoir. While the well geometry should not affect the repeatability and

accuracy of the CM results, we might have a small shift in the error assessment

plots. We did not consider any non-vertical well in the sensitivity analysis of this

chapter. The effect of a horizontal well on the connectivity results will be

discussed in Chapter 4.

Although we expect the bootstrap can provide good uncertainty estimates for the above-

mentioned problems, investigating them in a separate study may provide better

understanding of the effects of these factors.

The results of the analysis provide a useful guide for the reader to estimate the expected

accuracy of the method before applying it to field data. This work also assists the CM

users to assess repeatability and stability of the results and understand the effects of noise

after applying the model.

3.2 Sensitivity to the Diffusivity Constant

In Chapter 2, small dissipation or a large diffusivity constant was listed as one of the

assumptions of the CM. Here, we quantify how much error we may expect in estimating


the CM parameters if the diffusivity constant changes. The diffusivity constant is defined

as

t

k

c

..........................................................................................................................( 3-1)

where k is the permeability, is the porosity, is the viscosity, and ct is the total

compressibility. The effect of different components of the diffusivity constant on CM

performance may not be exactly the same. For example, increasing permeability might

not produce exactly the same change as decreasing porosity. The reason for this is that

compressibility and porosity only affect the dissipation of the system. Permeability and

viscosity, however, also affect the productivity indices of the wells. Thus to better

understand the relation of the diffusivity constant on the CM performance, we analyzed

the effect of these parameters separately.

To investigate the effect of these parameters on and , we applied the CM on several

synthetic cases, changing one parameter while keeping other parameters constant. We

used a commercial numerical simulator (Eclipse 100TM). As we expect and to be

independent of the injection rates, we ran the simulations for each synthetic case with 10

different sets of injection rates. For example, for the cases with 5 injectors, the injection

data were selected from two sets of random injection rates (Figure 3-1). By switching the

injection rates, we generated 5 sets of injection rates (three sets from Figure 3-1, left and

two sets from Figure 3-1, right), and to ensure the independence of the results from the

magnitude of the injection rates, we generated 5 more sets of injection rates where the

rates are larger (5 times larger for the first three sets and 70% larger for the other two

sets). To prevent errors associated with varying fluid properties in the reservoir, we ran

the simulations for a single phase. To ensure the sufficiency of the number of required

data relative to the number of parameters to be evaluated in the CM, we ran the

simulations for 384 months, for a 5×4 case. We named the ratio of the number of flow

measurements to the number of model parameters as L which equals 32 for this case. We

set all producer BHPs constant during the analysis period. We also assumed no well is

stimulated within the analysis period. Since field data are in general available monthly,


we applied the CM to monthly rates (we explain the effect of sampling time in Section

3.3). Both homogeneous and heterogeneous systems are considered.

Figure ‎3-1 The injection rates are selected from the above injection rate profiles. Three sets of rates

were generated based on the left figure, and 2 sets were generated based on the right figure. To

investigate the probable effect of higher rates on the results, we generated 5 more sets of rates by

multiplying the first 3 sets of rates by 5, and the other 2 rates by 1.7.

Case 3.1. This base case is a 5×4 homogeneous reservoir. The locations of the wells are

shown in Figure 3-2 (locations are the same as the base case of other reports in the

literature, as listed in Section 2.4). Reservoir and simulation parameters are presented in

Table 3-1.

Figure ‎3-2 Location of the vertical wells for Case 3.1; 5 injectors and 4 producers are located in a

homogenous reservoir.

0

200

400

600

800

1000

0 100 200 300 400

Inje

ctio

n r

ate

, rb

/day

Months

0

200

400

600

800

1000

1200

0 100 200 300 400

Inje

ctio

n r

ate

, rb

/day

Months

I01 I02

I03

I04 I05

P01

P02 P03

P04


Table ‎3-1 Reservoir and simulator parameters used for the Case 3.1

Parameter Value

, fraction 0.18

Horizontal k, md 10

Vertical k, md 1

ct, psi-1

2×10-6

, cp 0.5

Model dimensions 93×93×1

Grid size, ft 26.667

First we investigated the effect of changing permeability on and . We ran the reservoir

simulator for all 10 sets of injection rates for 0.1 < k < 200 md. To assess the consistency

of the CM results for each permeability value, we calculated the coefficient of variation

(CV) of the ’s and ’s of each well pair for different injection rates. The algorithm of

this calculation is depicted in Figure 3-3. We used the median of these CV’s as a measure

of stability of the CM parameters. Based on our results, the ’s are stable at k > 1 md and

they are unstable at small permeabilities. For the ’s we have instability at both large and

small permeabilities (Figure 3-4). The error level of the estimated rates also increases at

small permeabilities (Figure 3-5).

Figure ‎3-3 The algorithm shows the summary of the procedure to calculate the CV of CM

parameters and for each case.

Apply CM

Reservoir

model

Inj. rates set 1

.

.

.

.

.

.

Calculate the median of CV

Calculate prod. rates using

Eclipse for each

case

Inj. rates set 2

Inj. rates set 3

Inj. rates set 10

Calculate the CV of

each parameter

from different

set of injection

rates


Figure ‎3-4 The median of CV of the estimated ’s‎and‎’s‎is‎shown‎versus‎permeability‎for‎a‎

homogeneous system (Case 3.1).

Figure ‎3-5 By decreasing the permeability, the prediction error of the estimated rates increases to

1% (Case 3.1).

Two factors cause the poor performance of the CM at k < 1 md. First, one of the main

assumptions of the CM is the constant productivity index. This assumption is not valid in

the transient region and, to get a constant productivity index, the data should be taken

0

0.1

0.2

0.3

0.4

0.5

0.6

0.1 1 10 100 1000

Me

dia

n o

f CV

Permeability, mD

0

0.2

0.4

0.6

0.8

1

1.2

0.1 1 10 100 1000

Ab

s e

rro

r, %

Permeability, mD


after this period. At small permeabilities, monthly measurements will be affected by

transient flow. If we increase the sampling time, we obtain better results (Section 3.3).

The second reason for this poorer performance, as discussed by Yousef (2006), is that at

large dissipations the shifted injection signal will deteriorate and lose a part of its high

frequency content. This weak signal will result in poorer estimation of the model

parameters. Since the performance of the CM is more sensitive to the ’s rather than ’s,

inaccurate ’s may deteriorate the accuracy of the ’s at low permeability values.

For k > 15 md, the error is small and the values are stable. The values, however, show

poor stability and by changing the injection rates they have larger changes (Figure 3-4).

There are two causes for this behavior. First, since at large permeabilities the values

decrease to a few days, the monthly sampling time is insufficient to estimate these small

shifts (by decreasing the sampling time we can enhance the stability of the values,

Section 3.3). The second reason for the low accuracy of ’s at large permeabilities is the

CM has a smaller sensitivity to the values at large k. By taking the derivative of the

estimated production rate with respect to we have

1

1

21

ˆ ( )

m n m n

ij ij

t t t t

nj m n m n

ij i m

mij ij

q t t t e t t ew t

...........................................................( 3-2)

At small ’s i.e. large cases, this derivative approaches 0. This contributes to the low

accuracy of values even at small sampling times.

To investigate the effect of changing compressibility on the CM performance, we ran

Case 3.1 for different ct’s from 1 × 10-7

to 2 × 10-3

psi-1

and estimated the CM parameters

for each compressibility value (Figure 3-6). Similar to our analysis of the permeability

effect, by decreasing the compressibility, the ’s become stable and ’s are unstable at

both extremes. To have a better comparison of the effects of permeability and

compressibility, we calculated the diffusivity constant for varying permeability and

compressibility cases and by plotting the results of each case based on their diffusivity

constant value, we observed that their behavior over a wide range of diffusivities are very

similar to each other (Figure 3-7). The main difference in the results is at ’s smaller than


106

md.psi/cp, where the CM results are unstable. This suggests that changing the

diffusivity constant either from permeability or compressibility has the same effect on the

CM performance.

We also investigated the effect of varying viscosity and porosity. As expected, the results

of varying viscosity were identical to varying permeability and varying porosity is almost

identical to the compressibility changes. To test the validity of the preceding results for

heterogeneous cases, we ran Case 3.2.

Figure ‎3-6 At large compressibilities, both ’s‎and‎’s‎have‎large‎CV’s‎for‎different‎injection‎rates.‎At‎

small compressibilities, however, only values are unstable (Case 3.1).

10-3 10-4 10-5 10-6 10-7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Me

dia

n o

f C

V

Total compressibility, psi-1


Figure ‎3-7 Combining the results of Figure 3-4 and 3-6 shows that the effects of compressibility and

permeability are almost the same (Case 3.1).

Case 3.2. This case is a 5×4 heterogeneous reservoir (Figure 3-8). The general reservoir

properties are similar to Case 3.1; however, three barriers and one conduit exist in the

system. To investigate the sensitivity of the model to permeability for this case, we

multiplied the permeability of all the cells by a single factor. For example, at k = 4 md,

we set the channel permeability to 40 md and the barrier permeability to 0.02 md. By

running the simulator for different injection rates and calculating the CM parameters for

each case, we observed that by changing the permeability multiplier, the ’s and ’s

behaved the same way as they did for the homogeneous case (Figure 3-9). Although we

only tested the results for one heterogeneous case, we expect that our findings are valid

for other levels and distributions of heterogeneity.

105 106 107 108 109 1010

0

0.1

0.2

0.3

0.4

0.5

0.6

Me

dia

n o

f CV

Diffusivity constant, md.psi/cP

τ based on ct

λ based on ct

τ based on k

λ based on k


Figure ‎3-8 In Case 3.2, three barriers and one channel exist in the reservoir.

Figure ‎3-9 Similar to the homogeneous case, in Case 3.2 ’s‎are‎unstable at both high and low

permeabilities and the ’s‎are‎unstable at small permeability values.

3.3 Sensitivity to the Sampling Time

To observe the effect of the sampling time we decreased it to 5 days. Figure 3-10 shows

that the range of stable ’s and ’s shifts to higher permeabilities. At large permeabilities,

we have a smaller error for values as in the case of large sampling time since the

k=400 md

k=0.2 mdk=40 md

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.1 1 10 100

Me

dia

n o

f CV

Permeability, mD


sampling time is small enough to estimate small values. However, this range of stability

for both ’s and ’s is not shifted by six-fold (it is less than six-fold).

When we increase the sampling time to 300 days, again this range of stability for both

’s and ’s is not shifted to the lower permeabilities by ten-fold (it is less than ten-fold).

Thus, although we expect the sampling time to have the same effect as permeability, its

effect is slightly different, particularly at the extremes (Figure 3-11).

Figure ‎3-10 At 5-day sampling, the range of stable ’s‎and‎’s‎shifts‎to‎higher‎permeabilities‎(Case‎

3.1).

Figure ‎3-11 The median of CV of the estimated ’s‎and‎’s‎is‎shown‎versus‎sampling‎time for a

homogeneous system (Case 3.1).

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1 10 100

Me

dia

n o

f C

V

Permeability, mD

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 10 100 1000

Me

dia

n o

f C

V

Sampling time, days


3.4 Sensitivity to the Reservoir Area

We also calculated the CV of ’s and ’s versus reservoir area while the other parameters

are kept at the base-case conditions (Figure 3-12). Similar to the analysis of the porosity

or total compressibility effect, by increasing the inverse of area, the ’s become stable

and ’s are unstable at both extremes.

Figure ‎3-12 Trend of CV’s‎for‎both‎’s‎and‎’s‎versus‎the‎reservoir‎area‎is‎similar‎to‎the trend of

CV’s‎versus‎total‎compressibility‎(Case‎3.1).

3.5 Sensitivity to the Number of Producers

One of the other parameters that affects the CM results is the number of producers. It

may appear that the number of producers only affects the number of data (one

measurement at each time step for each well) and the number of parameters; however, it

also affects CM performance. Reducing the number of producers increases the drainage

area of each producer and this leads to larger values. On the other hand, since the

number of injectors (and, in general, injection rates) has no effect on the CM results,

changing the number of injectors will not change the model performance.

0

0.1

0.2

0.3

0.4

0.5

0.6

1.0010.00100.001000.00

Med

ian

of

CV

Reservoir area/106, ft2


Case 3.3. This case is similar to Case 3.1 but with only two producers, P01 and P02.

Comparing the CV’s obtained from the two-producer situation and Case 3.1, we observe

that the window of stable ’s and ’s increases to 4 - 35 md, instead of 2 - 17 md. We

tested this for two more situations, with different well patterns where the well locations

are changed. Although the interwell distances also affect the values, this effect is small

compared to the effect of changing numbers of wells. When we have two producers

instead of four producers, the transient period lasts longer. Therefore, the window of

stable results shifts to the right. To assess the results for a larger number of wells and

different reservoir parameters, we tested the procedure on an 8×8 heterogeneous

reservoir.

Case 3.4. This case is an 8×8 heterogeneous reservoir (Figure 3-13), where four barriers

and one channel exist. Similar to the previous cases, after running the simulator for 10

different injection rates for 400 months and calculating the CM parameters, we generated

the CV plot for permeabilities from 1 to 40 md (Figure 3-14). The general trend of both

’s and ’s is similar to the previous cases. For this case, the pore volume is 2.3 times

larger than Case 3.1. On the other hand, the number of producers is twice what is found

in Case 3.1. Thus, we expect to have a similar stable range of CM parameters for both

cases. However, the range for Case 3.4 is slightly narrower. In this case, since we have a

larger number of wells, the values are, in general, smaller. This leads to larger CV’s

(Figure 3-15), and in total we will have a narrower range for stable CM parameters. If we

exclude the and values corresponding to small ’s, the estimated values will be close

to the Case 3.1 results. For example, by excluding the ’s < 0.1 the median of CV of for

k = 4 md becomes 0.01, which is very close to the value we have for Case 3.1 at k = 3

md.


Figure ‎3-13 Permeability map of Case 3.4; four barriers and one channel exist in the system.

Figure ‎3-14 For Case 3.4; since both the area and number of producers are twice those of Case 3.1,

we expect to have the same range of stable parameters as Case 3.1. However, this range is slightly

narrower than Case 3.1.

k=400 md

k=0.2 md

k=40 md

0 900 ft

0

0.1

0.2

0.3

0.4

0.5

1 10 100

Me

dia

n o

f C

V

Permeability, mD


Figure ‎3-15 For well pairs with smaller values, the uncertainty in the estimated ’s‎increases.‎Here,‎

the CV of the estimated ’s‎for‎Case‎3.4‎at‎k = 1 md is plotted.

One of the constraints of the CM is the summation of ’s for each injector. For two

producers this adds an inequality for two unknowns (’s of each injector). For a larger

number of producers, the effect of this constraint decreases. For example, for eight

producers this constraint adds one inequality for eight unknowns. This results in a smaller

ratio of equations to unknowns for a larger number of wells, and so there is a larger

uncertainty.

3.6 CM Number

In the previous sections, we discussed how changing the reservoir parameters, sampling

time, reservoir area, and numbers of wells affect the CM performance. Here, we propose

a new parameter, called the CM number, to generalize the results of previous sections.

The form of the CM number is based on the following observations.

1. As shown in Figure 3-7, the diffusivity constant summarizes the effect of its

components in one parameter.

2. Increasing the sampling time has a similar effect to increasing the diffusivity

constant, so t should appear as a product.

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.1 0.2 0.3 0.4

CV


3. The area of the reservoir has the same effect as porosity. Thus and A should be

together in the CM number.

4. The number of producers has the same effect as increasing the permeability, so it

is also multiplying t.

Thus, we can define the CM number C as

0.006328t

k tKC

A c

............................................................................................................( 3-3)

where the coefficient is used for field units (md, days, ft2, cp, and psi

-1). Note that, if we

have only one well in the system, this parameter will be equivalent to the dimensionless

time as defined in transient well testing (Lee et al. 2003). Dimensionless time is

proportional to diffusivity constant multiplied by time over reservoir area. In other words,

dimensionless time is related to the ability of fluid flow in a specific volume. A highly

connected medium has large dimensionless time (or CM number) and vice versa. Plotting

the median of CV versus C for four producers (Figure 3-16), we observe that the CM

parameters are stable and repeatable for 0.3 < C < 10. To confirm the behavior of C from

this plot, we also generated some synthetic cases with the same C but changing the non-

permeability parameters. As expected, the results are in good correspondence with each

other. We will generate our sensitivity plots based on C in the following sections. Based

on these results, with the CM number of any field case, we can refer to these plots and

estimate the expected error.


Figure ‎3-16 By calculating the CM number (C) for different reservoir conditions, sampling time,

reservoir area, and well numbers, the CM results are stable and repeatable for 0.3<C<10.

3.7 Sensitivity to the Number of Data

In any parameter estimation problem, a sufficient number of measurement data is a key

concern in the stability of the results. If the number of samples is small compared to the

number of model parameters, the model may not evaluate the connectivities correctly. In

such a case, although the model might fit the flow rate data, the ’s and ’s could be

inaccurate. This issue and a suitable value for L, the ratio of the number of data to the

number of parameters, have been discussed in the literature. For example, Haykin (2009)

suggests that the relative accuracy of the model is equivalent to 1 / L. For the results

described above, we used L = n×K / [2×I×(K + 1)] and a large number of time samples n

to reduce the possible effects of small sample numbers on the results e.g., L = 32 for Case

3.1. In the analysis of field data, however, the number of available data may be much

smaller, e.g., L = 2. Thus we need to understand how the model accuracy changes with L.

For Case 3.1, the number of model parameters is 48. We selected the first 30, 48, and 96

time steps of the data for the analysis, to give L= 2.5, 4, and 8, respectively. After

applying the CM, we calculated the median CV for the ’s and ’s (Figure 3-17). As

0

0.1

0.2

0.3

0.4

0.5

0.6

0.1 1 10 100

Me

dia

n o

f C

V

CM number

tau from different parameters 5x4

lambda from different parameters 5x4

tau from perm 5x4

lambda from perm 5x4

tau from 8x8

lambda from 8x8


expected, compared to the case with 384 time steps (L = 32), the CV’s are much larger

and by decreasing the number of samples, the CV’s increase. Interestingly, even with a

small number of data at CM numbers larger than 3, estimation is stable. However, the

range of small CV for becomes narrower, and at L = 2.5, the values are unstable for all

CM number values.

Figure ‎3-17 By decreasing the length of analysis window, the range of stable ’s‎(left)‎and‎’s‎(right)‎

will be shorter.

We also used the analytical values of the ’s, which are only a function of well locations

and boundaries (see Section 2.4.4), to test the correctness of the estimated values for

different numbers of data. We define the average absolute difference (AAD) as

1 1

ˆ

AAD

I Kt

ij ij

i j

IK

..........................................................................................................( 3-4)

where ˆij is the estimated ij using the CM and

t

ij is the true value of ij calculated

analytically using Equation 2-9. We use the AAD as a criterion for accuracy of the

estimated ’s for each case (Figure 3-18). As expected, even for a large number of data

(L = 32), the AAD becomes large for C < 0.7 and this is in agreement with the median

CV’s results, where we observed unstable estimates of ’s at small CM numbers (small

permeabilities). As a rule-of-thumb, AAD 1/L overestimates the error by a large

margin, except at C = 0.07 where AAD 0.1/L is more appropriate. In general, larger L

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.01 0.1 1 10 100

Me

dia

n o

f C

V

CM number

L=32

L=8

L=4

L=2.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.01 0.1 1 10 100

Med

ian

of

CV

CM number

L=32

L=8

L=4

L=2.5


reduces the errors but, unless C < 0.7, little advantage is gained by having L > 4. We also

calculated AAD versus CM number and L by changing permeability and parameters

except permeability in the CM number and observed similar results (Figure 3-19).

To test the validity of our findings for heterogeneous cases, we repeated the same

procedure for Case 3.2 and the results were very similar to the results of the

homogeneous case.

Figure ‎3-18 Including more data leads to a more accurate estimation of ’s‎(Case‎3.1).‎At‎small‎CM‎

numbers, errors increase as L decreases.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.01 0.10 1.00 10.00 100.00 1000.00

AA

D o

f

CM number

L=2.5

L=4

L=8

L=32


Figure ‎3-19 By calculating the CM number by changing parameters except permeability (Mix), we

observed that the trend of increasing the AAD of by decreasing the CM number is similar to what

we had by changing permeability.

3.8 Sensitivity to Noise

Up to this point, we have examined the effects of the CM number and the number of data

for the case where the measurements were all noise-free. Practically, however, some

noise may exist in the data because of human or measurement errors. This noise may lead

to poor CM performance in estimating the production rates and unrepresentative sets of

connectivity parameters. To study the effect of noisy data, we added uniformly

distributed white noise for Case 3.1, ranging in amplitude from 5 to 40 percent. That is, a

20% error on an interval of 50 samples from 4 wells means the noise is a set of 200 (=

4×50) samples from a uniform distribution within 0.9 and 1.1, and the production rate of

each producer at each time step is multiplied by a value from this set. These error levels

are similar to those observed by Dong et al. (2009) in their assessment of flow rate

measurement errors of two-phase (oil and water) flows.

Figure 3-20 shows the median CV of ’s and ’s for different injection rates versus

numbers of samples, amounts of noise, and CM number. As expected, increasing

amounts of noise reduce parameter stability. At low noise (5%) and a medium number of

0

0.01

0.02

0.03

0.04

0.05

0.06

0.1 1 10 100

AA

D o

f

CM number

L=2.5, 5x4

L=4, 5x4

L=8, 5x4

L=32, 5x4

Mix, L=32, 5x4

L=4, 8x8

L=8, 8x8


samples (L = 8), the range of stable values is very narrow (Figure 3-20 c); for the ’s

we have a broader stable range. At large noise levels (40%), however, the estimated ’s

for all CM numbers at L = 4 and 8 are unstable, and only at L = 32 they are acceptable

(below 20% for C > 0.7). Putting all this together (Figure 3-21), the AAD is below 0.1 for

every situation except C = 0.7, L = 2.5, and noise more than 15%. While we saw little

benefit for the noise-free case of having L > 4 (Figure 3-18), there is a systematic

reduction in AAD as L increases when noise is present. For example, at C = 0.7, with L =

4, and 10% noise, we expect to have 0.03 deviation in the estimated ’s. However, if L =

8, this deviation decreases to 0.02.


0

0.5

1

1.5

2

2.5

0.01 0.1 1 10 100

Me

dia

n o

f C

V

CM number

noise=5%

noise=10%

noise=20%

noise=40%

Ideal

0

0.2

0.4

0.6

0.8

1

1.2

0.01 0.1 1 10 100

Me

dia

n o

f C

V

CM number

noise=5%

noise=10%

noise=20%

noise=40%

Ideal

0

0.5

1

1.5

2

2.5

0.01 0.1 1 10 100

Me

dia

n o

f C

V

CM number

noise=5%

noise=10%

noise=20%

noise=40%

Ideal

0

0.2

0.4

0.6

0.8

1

1.2

0.01 0.1 1 10 100

Me

dia

n o

f C

V

CM number

noise=5%

noise=10%

noise=20%

noise=40%

Ideal

0

0.5

1

1.5

2

2.5

0.01 0.1 1 10 100

Me

dia

n o

f C

V

CM number

noise=5%

noise=10%

noise=20%

noise=40%

Ideal

0

0.2

0.4

0.6

0.8

1

1.2

0.01 0.1 1 10 100

Me

dia

n o

f C

V

CM number

noise=5%

noise=10%

noise=20%

noise=40%

Ideal

(a) (b)

(c) (d)

(f) (e)

L= 4 L= 4

L= 8 L= 8

L= 32 L= 32


Figure ‎3-20 By adding noise, CM parameter errors increase. In (a), (c), and (e) the median of CV of

’s is shown at L = 4, 8, and 32 respectively. (b), (d), and (f) show the median of CV of ’s‎at‎L = 4, 8,

and‎32‎respectively.‎By‎“ideal”‎we‎mean‎the‎noise-free case with a large number of samples. By

introducing noise to the data, estimates become more variable for small and medium numbers of

samples. However, values are stable at small number of samples with moderate noise (10%).

Figure ‎3-21 This figure shows AAD of ’s‎versus‎different‎amounts‎of‎noise,‎number‎of‎data,‎and‎CM‎

number.

3.9 Error Assessment Using the Bootstrap

From the above results, if we know the general reservoir properties and the uncertainty in

the measurements, we can estimate the uncertainty in the CM parameters. In practice,

however, we may not have some of this information. A practical way to assess the

uncertainty in the estimated connectivity parameters is to apply the bootstrap, which is a

sampling with replacement technique (Efron and Tibshirani, 1994). In this method, we

evaluate the estimator error based on the performance of the model on several subsets of

observations derived from the original dataset. These subsets of observations have the

same number of flow rate measurements as the original dataset; however, some of the

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 10 20 30 40

Ave

rage

ab

solu

te e

rro

r in

Noise, %

L=32, C= 27.84

L=32, C=2.78

L=32, C=0.70

L=8, C=27.84

L=8, C=2.78

L=8, C= 0.70

L=4, C=27.84

L=4, C=2.78

L=4, C=0.70

L=2.5, C=27.84

L=2.5, C=2.78

L=2.5, C=0.7


original samples are excluded and replaced with other measurements. For example, a

subset could have the measurements at the 9th time step excluded, while the

measurements at the 28th time step are retained and get double-weight. This technique

has been applied previously in petroleum engineering for reserve estimation (Jochen and

Spivey 1996; Cheng et al. 2010).

To begin, we amend the objective function by adding weights gi,

2

1 1

ˆn K

m j m j m

m j

B g q t q t

............................................................................................( 3-5)

This function is the same as described in Section 3.1 when g = 1.

For the bootstrap, we will change the values of g. We generate a set of random numbers

having integer values between 1 and the number of time steps n in the dataset (Figure 3-

22). This set has n samples and any number may be repeated. Based on these numbers,

we assign values to gm according to the following pattern:

• If a time step m is not selected, gm = 0.

• If a time step m is selected once, gm = 1.

• If a time step m has been selected more than one time, the value of gm will be the

total number of times it has been selected.

Now we apply the CM on this new set of data and determine the CM parameters by

minimizing B. (Note that there is no constraint on gi and this does not affect the

minimization process.) Then, we repeat the resampling and apply the CM for 50 times.

We calculate the sum of the standard deviations of the ’s between well pairs for all these

runs. Figure 3-22 summarizes this procedure for a set of n = 10 time steps.


Figure ‎3-22 In the bootstrap technique, based on selected subsamples of data, we assign the

appropriate weight to each time step. Then we apply the CM for each scenario and finally we

calculate the standard deviation of the estimated ’s.

We applied this procedure for Case 3.1, and for various combinations of permeabilities,

injection rates, number of samples, number of wells, and amount of noise. We also

calculated the AAD of without applying the bootstrap for each case. Comparing the

estimated standard deviation from the bootstrap and the AAD of we observe that they

are strongly correlated (Figures 3-23, 3-24). In other words, at large standard deviations

obtained by the bootstrap, we have large AADs. The scatter of the points diminishes as

L increases, particularly at large standard deviations (Figure 3-23). Figures 3-23 and 3-24

suggest that, independent of the number of wells or heterogeneity in the reservoir,

evaluating the standard deviation from the bootstrap provides an AAD estimate for the

’s. For example, if the standard deviation is 0.1, the expected AAD in the estimated ’s

will be about 0.06.

1

2

3

4

5

6

7

8

9

10

1, 2, 2, 4, 5, 7, 8, 9, 10, 10

2, 2, 4, 6, 6, 7, 7, 8, 9, 9

1, 4, 4, 4, 6, 7, 10, 10, 10, 10

3, 5, 5, 7, 8, 9, 9, 9, 9, 10

.

.

.

.

.

.

Calculate

standard

deviation of

the estimated

’s

Assign the weights

Apply the CM

Apply the CM

Apply the CM

Assign the weights

Assign the weights

Assign the weights Apply the CM

All the

time

steps

Subsampled time

steps


Figure ‎3-23 Standard deviation of the estimated ’s‎using‎the‎bootstrap‎correlates‎well‎with‎the‎

AADs.

Figure ‎3-24 Independent of the number of wells, by applying bootstrap and estimating the standard

deviation, we can estimate the error in estimated ’s.‎

We also calculated the average of the estimated ’s for each injector-producer pair from

each bootstrap iteration i.e. ij = ij)p for p = 1, 2, …, 50. Comparing the difference of

0

0.05

0.1

0.15

0.2

0 0.05 0.1 0.15

Stan

dar

d d

evi

atio

n o

f 5

0 b

oo

tstr

aps

average absolute error from all data

L=2.5

L=4

L=8

0

0.05

0.1

0.15

0.2

0 0.05 0.1 0.15

Stan

dar

d d

evi

atio

n o

f 5

0 b

oo

tstr

ap


5x4 homogeneous

5x4 heterogeneous

8x8 heterogeneous


this average from the true values, ij - t

ij with the one derived using all the data, ˆij

- t

ij , we observe that ij is less biased than the estimated from all the data (Figure 3-

25). This bias is larger at larger AAD’s and could be up to 50%. Thus, using the averaged

’s of several bootstrap iterations could provide more representative values of the

interwell connectivity. This result accords with the known result that the bootstrap can

provide estimates with smaller bias (Efron and Tibshirani, 1993).

Figure ‎3-25 The error in the estimated ’s‎from‎averaging‎results‎of‎several‎bootstraps‎(vertical‎axis)

is, in general, more accurate than ones obtained from a single run of CM (horizontal axis).

One of the main questions in applying the bootstrap technique is the necessary number of

resamplings. In general, a larger number of resamplings provides a more representative

estimate of the bootstrap standard deviation. Considering the required CPU times,

however, large resampling numbers are not feasible. One way to estimate this number is

to track the standard deviation values for the bootstraps and determine where this number

becomes constant. For the studied cases, we plotted the maximum difference in the

estimated standard deviation of bootstrap at different numbers of resamplings with the

one from 50 resamplings (Figure 3-26). We observed that by increasing L, this difference

becomes constant for smaller numbers of resamplings. For cases with a larger number of

wells (8×8), the standard deviation decreases faster, showing we need a smaller number

of resamplings for larger fields. This suggests that, with 10 to 20 resamplings, we can

0

0.05

0.1

0.15

0 0.05 0.1 0.15

av

erag

e ab

osl

ute

err

or

fro

m

aver

agin

g 5

0 b

oo

stra

p s

amp

les


5x4 homogeneous

5x4 heterogeneous

8x8 heterogeneous


obtain a good estimate of the bootstrap standard deviation. We suggest that for each case

users track these changes to obtain reliable estimates of the standard deviation.

Figure ‎3-26 Applying 10 to 20 resamplings, we can get a good estimation of the standard deviation of

the bootstrap. For larger number of samples, we need a smaller number of resamplings.

In the case of changing the number of producers during the analysis period (e.g., new

wells or conversions), we need to use the compensated CM. If we do not apply the proper

method, or the data is not of good quality (e.g. unreported short term shut-ins, skin

changes, or BHP changes), the bootstrap technique can reveal the problem. For example,

for Case 3.1, at k = 1 md, with 48 months of data and one producer shut-in at 24 months,

the standard deviation of the bootstrap is 0.102 and the AAD is 0.08. However, for the

case without shut-in, this standard deviation is only 0.002.

3.10 Field Examples

In the last seven years, the CM has been used to evaluate connectivity and/or manage

waterfloods for several published field studies (Table 3-2).

0

0.01

0.02

0.03

0.04

0.05

0.06

0 10 20 30 40 50

Max

imu

m d

iffe

ren

ce f

rom

std

of

50

re

sam

plin

gs

Number of resamplings

L=2.5, 5x4

L=8, 5x4

L=2.5, 8x8

L=8, 8x8


Table ‎3-2 Selected field cases analyzed with the CM; we devoted a number to each field to show in the contour plot.

Number in plot Reference Field

1 Weber (2009) East Wilmington (California)

2A Yousef (2006) CDSN (Argentina)

2B Yousef (2006) Magnus (North Sea)

2C Yousef (2006) SWCF (West Texas)

2D Yousef (2006) North Buck Draw

3 Kaviani (2009) Williston Basin

4A Sayarpour (2008) Reinecke Field

4B Sayarpour (2008) East Wilmington (California)

4C Sayarpour (2008) McElroy

4D Sayarpour (2008) MESL Field

5 Izgec and Kabir (2010) USG Synthetic Field

By calculating C and L from the available information, we can estimate the uncertainty of

evaluation for these eleven cases (Figure 3-27, 3-28), assuming the flow rate data are

noise-free.

Nine of the eleven fields have C > 0.9, so that the estimates would be expected to be

accurate with CV < 0.05 (Figure 3-27). Four of those nine, however, have data

limitations (L < 5.6) which increase the errors. Time and more measurements, however,

will bring these evaluations into lower error regions of the plot. Fields 2D and 3, with C

< 0.9, approach the unconventional classification, so that further data will only

marginally improve the errors. Only three of the eleven fields have C and L values

which are conducive to small-error estimates (Figure 3-28). More time and data will

bring another four or five fields into the low-error region with CV < 0.05.

This analysis suggests that about half of the fields analyzed with the CM and reported in

the literature have small and errors and therefore probably contributed to the

successful results. The reports show that Case 3 has low R2 values and Case 5 has high

R2 values which are both in agreement with our maps. Cases 1, 2A and 2C performed

better than expected. Although Case 2B has small error according to our maps, Yousef

(2006) reports the wells suffered frequent shut-in periods and this, therefore, could

explain the low R2 values. More generally, the reports show good performance of the CM

and we may have set the region of acceptable CM performance (CV < 0.05) too tightly.


As more reports of field analysis appear, we expect to better identify the regions of

acceptable C and L values.

Figure ‎3-27 Contour plot of median CV’s‎of‎’s‎at‎different‎CM‎numbers‎and‎L; for the source of

each point, see Table 3-2. Most of the cases have stable ’s.

0.050.05

0.05

0.10.1

0.1

0.20.2

0.2

0.30.3

0.3

0.40.4

0.4

0.5

L

CM

Nu

mb

er

0 5 10 15 20 2510

-1

100

101

102

1

2A

2B2C

2D

3

4A

4B

4C

4D

5


Figure ‎3-28 Contour plot of median CV’s‎of‎’s‎at‎different‎CM‎numbers‎and‎L; for description of

each point see Table 3-2. Compared to the ’s‎(Figure‎3-28), a smaller number of cases has stable ’s.

3.11 Conclusions

The results of this chapter confirm that both CM parameters and are affected by a

number of factors including fluid and reservoir properties (diffusivity constant), sampling

time, reservoir area, number of measurements, and measurement noise. Several of these

factors can be aggregated into two dimensionless numbers, the CM number, C, and the

ratio of number of measurements to the number of model unknowns, L.

If we have enough samples:

a. When 0.3 < C, estimates of are stable and have AAD < 0.1/L.

b. When 0.3 < C < 10, estimates of are stable and have a small variability.

The effect of flow measurement noise decreases with more measurements, but L = 4

appears sufficient to give stable values.

0.050.050.05

0.05

0.050.05

0.10.1

0.1

0.1

0.10.1

0.2

0.20.2

0.2

0.2

0.30.3

0.3

0.3

0.40.4

0.4

0.5

0.5

0.5

0.50.5

0.5

0.6

0.6

L

CM

Nu

mb

er

0 5 10 15 20 2510

-1

100

101

102

1

2A

2B2C

2D

3

4A

4B

4C

4D

5


The bootstrap is a useful tool for analyzing CM performance, especially when there is a

lack of information about reservoir properties and uncertainty in the measurements:

a. About 20 resamplings are adequate to derive estimates and errors.

b. Bootstrap-derived variabilities correlate well with the AAD.

c. Averaged ’s from the bootstrap are less biased than ’s obtained using all the

data.

Maps of C and L values from eleven literature reports where the CM was used suggest

that about half of the cases gave conditions where estimates have small variabilities.

Several cases were limited by too few data. The estimates are likely to be more

variable. Unconventional reservoirs will be challenging for CM analysis.

The work in this chapter represents a collective combined effort with Danial Kaviani. He

obtained the median CV’s, AADs, and bootstrap results for the mentioned synthetic field

cases. We interpreted the results together and worked on mentioned published field cases.

CHAPTER 4 THE CM AND HORIZONTAL WELLS 61

CHAPTER 4 THE CM AND HORIZONTAL WELLS

4.1 Introduction

When a horizontal producer exists in a reservoir, the values of the CM parameters are

changed and they represent a different response to the reservoir heterogeneity. Therefore,

the interpretation of reservoir heterogeneity using the CM parameters becomes a

challenge.

In this chapter, we investigate the effect of horizontal producers via different synthetic

field examples. We analyze the effect of horizontal well length and trajectory on the CM

parameters. Then we suggest two methods to decouple the horizontal well geometry

during heterogeneity evaluation. In the first method with an analytical approach, we

calculate ’s of a homogenous reservoir with a horizontal well and subtract them from

optimized ’s (Equation 2-10) to produce a revised parameter '. In the second method,

we apply the reverse CM which is explained in Section 2.4.4 to exclude the effect of

horizontal producers and show that the reverse CM parameters do not change when a

horizontal producer exists in the system. Finally, we examine our approaches in a

heterogeneous synthetic field with a horizontal well and compare the results with the case

where all the producers are vertical. We do not, however, cover the following conditions

in this chapter:

1) More than one horizontal producer in the reservoir; our previous results show that

the conclusions are very similar to the ones obtained here.

2) Positioning a horizontal well near the reservoir top; previous results of perforating

horizontal wells in other layers lead to the same conclusions as presented here.

4.2 Horizontal Well Effect on the CM Parameters

If we have one horizontal producer in a reservoir - for example, a homogenous 5×4

synthetic field in which all other producers are fully penetrating vertical wells - we expect

much higher amounts of production from the horizontal well, assuming all the producers’

BHPs are kept at an identical value. The reason is that the drainage area of the horizontal


well is larger than the vertical wells’ drainage area. In other words, the shape of the high

pressure drop area in the vicinity of the horizontal wells is a large ellipse, while for

vertical wells, it is a circle with a radius of a few feet. Consequently, the productivity of

the horizontal well would be larger, and larger productivity results in larger ’s. If we

focus on the CM equation (Figure 4-1), the high amount of production of a horizontal

well increases its ’s to enforce material balance. Consequently, reduced production rates

of vertical wells decrease their ’s. The ’s are relative values and = 1 for each

injector in a closed system. Therefore, we expect this reduction of vertical wells ’s since

the ’s of horizontal well increase. To examine this effect quantitatively, we compare

Cases 4.1 and 4.2.

Figure ‎4-1 Schematic of increasing production rate of horizontal well and its effect on the CM

parameters; subscript H and V stand for a horizontal and vertical well respectively. C(t) stands for

non-waterflood terms. High amounts of production from horizontal well increase the ’s‎of‎that‎

horizontal well and decreases the ’s‎of‎vertical‎wells.

Case 4.1. To observe the effect of a horizontal well on the CM parameters

quantitatively, we first build a synthetic field case where all the producers are vertical.

This case is a 5×4 homogeneous reservoir with reservoir and simulation parameters

presented in Table 4-1. Locations of the wells are the same as in Case 3.1 (Figure 3-2).

We used the first set of injection rates (Figure 3-1) in Case 3.1 in our simulation model.

To ensure the results are independent of injection rates, we used 400 months of

waterflood (L=32) for the simulation. We used a commercial numerical simulator (CMG

IMEX). Figure 4-2 depicts the maps of ’s and ’s and Table 4-2 (left) shows the ’s

evaluated for this case. All the ’s and ’s are symmetric in respect to the well locations.

' ( )iV iV Vq t w t C t

' ( )iH iH Hq t w t C t


Table ‎4-1 Reservoir and simulator parameters used for Case 4.1

Parameter Value

, fraction 0.18

Horizontal k, md 40

Vertical k, md 4

ct, psi-1

2×10-6

, cp 0.5


Grid size, ft 50×50×12

Figure ‎4-2 Map of ’s (left) and ’s (right) for Case 4.1; all the wells are fully penetrating vertical

wells.

Case 4.2. This case is similar to Case 4.1, except that P01 is a W-E two-branch

horizontal well with a length of 550ft toward injector I01 and I02 which is drilled in the

bottom layer (Figure 4-3). The P01-I01 well spacing is 1500 ft. In the simulation

model the horizontal well is located in 11 grid blocks, with one perforation in each

grid block. We chose the W-E two-branch horizontal well as a base of our analysis,

since we can see the symmetry of the CM parameters with respect to the well’s

location, so it is easier to interpret the trend of CM parameter changes. We will analyze

the other well trajectories in Section 4.3.

I01 I02

I03

I04 I05

P01

P02 P03

P04

=0.5

I01 I02

I03

I04 I05

P01

P02 P03

P04

=10


Figure ‎4-3 W-E cross section of the simulation model for the Case 4.2; P01 is a two-branch horizontal

well with a length of 550ft which is drilled in the bottom layer.

When we plot the ’s of Case 4.2, the horizontal well ’s are enhanced by about 80%. ’s

toward vertical wells decrease relatively (Figure 4-4 left, Table 4-2 right). We also

observed that, depending on well locations, the amount of reduction of ’s for all vertical

wells is not the same. For example, for I01 and P03 the reduction is about 50% while I04

and P04 have a 12% reduction. The reason for this is that a horizontal well affects the

pressures between well pairs which are closer to it, such as I01 and P03, rather than

between well pairs which are far from it, such as I04 and P04.

Figure ‎4-4 Map of ’s (left) and ’s (right) for the Case 4.2; producer P01 is a W-E two-branch

horizontal well with a length of 550 ft.

I01 I02

I03

I04 I05

P01

P02 P03

P04

=0.5

I01 I02

I03

I04 I05

P01

P02 P03

P04

=10

Horizontal well Horizontal well


Table ‎4-2 ’s evaluated for the Case 4.1 (left) and the Case 4.2 (right); right table shows ’s are enhance for P01 and decreased for P02, P03, and P04.

P01 P02 P03 P04

0.322 0.322 0.177 0.178

0.322 0.178 0.322 0.178

0.250 0.250 0.250 0.250

0.177 0.322 0.178 0.32

0.178 0.178 0.322 0.321

P01 P02 P03 P04

0.591 0.223 0.081 0.104

0.594 0.081 0.223 0.102

0.457 0.175 0.175 0.193

0.327 0.268 0.124 0.281

0.330 0.123 0.267 0.280

On the other hand, the results show that the horizontal well and vertical wells ’s

decrease by approximately 30% and 20%, respectively (Figure 4-4, right). The reason

for this is that (according to the definition, Equation 4-1) ’s are generally proportional to

the injector-producer well pair pore volume and inversely proportional to the producer

productivity. Assuming we have an injector-producer well pair, if the producer is a

horizontal well, productivity is much larger relative to the case where the producer is a

vertical well. Although the pore volume is also increased, the ratio of pore volume over

productivity results in a smaller . Consequently, the horizontal well decreases the pore

volume of other well pairs. A similar argument applies to vertical producers. However, this

reduction is smaller (20%) relative to a horizontal well ’s reduction (30%).

t PcV

J ...........................................................................................................................( 4-1)

A cross plot of ’s and ’s for the Case 4.2 versus ’s and ’s for the Case 4.1, confirms

these conclusions (Figure 4 - 5). Thus by mapping the ’s, the effect of horizontal well

P01 is less noticeable. We focus on ’s rather than ’s in the following sections.


Figure ‎4-5 Cross plot of ’s‎(left) and ’s (right) for Case 4.2 versus Case 4.1; horizontal well ’s‎are‎

enhanced and vertical well ’s‎are‎decreased.‎On‎the‎other‎hand,‎all‎’s‎decrease.

4.3 Well Trajectory Effect

So far, we have assumed the base case of a two-branch W-E horizontal well. To

investigate the effect of well trajectory, we examine the effect of a one-branch horizontal

well, a two-branch horizontal well in four different directions, and a deviated well.

4.3.1 One-branch Horizontal Well

Case 4.3. In this case, all the properties and locations are the same as Case 4.2 except

the trajectory of horizontal well P01, which is a one-branch horizontal well in W-E

direction toward injector I01 with a length of 550 ft (Figure 4-6). The results of ’s are

similar to Case 4.2, except between I01 and P01, which increases 95% instead of 80%,

and between I02 and P01 which increases only 70% percent instead of 80% (Figure 4-

7, left). There is not a large difference between ’s of this case relative to Case 4.2 (Figure

4-7, right).


Figure ‎4-6 W-E cross section of the simulation model for Case 4.3; P01 is a one-branch horizontal

well with a length of 550 ft which is drilled in the bottom layer.

Figure ‎4-7 Map of ’s (left) and ’s (right) for Case 4.3; producer P01 is a one-branch horizontal well

with a length of 550 ft.

4.3.2 Horizontal Well Direction

Case 4.4. In this case, four different directions of P01 are considered: P01 as a two-

branch horizontal well in a W-E direction, a SW-NE direction, a S-N direction and a SE-

NW direction. All the other conditions are the same as in Case 4.2. For all of these

directions, ’s of horizontal well versus ’s of that well if it were vertical are mapped in

Figure 4-8. The results show that the direction of the horizontal well does not have a

I01 I02

I03

I04 I05

P01

P02 P03

P04

=0.5

I01 I02

I03

I04 I05

P01

P02 P03

P04

=10



major effect on the ’s. Only the ’s from one injector are shown in Figure 4-8. A

similar trend could be observed for the other injectors.

Figure ‎4-8 Effect of well direction; P01 is a two-branch horizontal well: four different orientations

were considered, including W-E, SW-NE direction, S-N direction and SE-NW directions (Case 4.4).

4.3.3 Deviated Wells

Case 4.5. In this case we assume the producer deviation is less than 90. To do so, in our

simulation model we incline well P01 toward I01 with the angle of 76 (in each layer

from the top to the bottom every perforation is made with one grid shift toward I01 to

reach this deviation, Figure 4-9). We observe the ’s are smaller than horizontal well

ones in the Case 4.2 (Figure 4-10). The reason is that the length of this deviated well is

about 257 ft which is not drilled horizontally, so the well drainage area is not as large as

seen in Case 4.2. Meanwhile there are only 5 perforations per well (for Case 4.2, there

were 11 perforations per well). Therefore, the productivity of that well is smaller.


Figure ‎4-9 W-E cross section of the simulation model for Case 4.5; P01 is a 76 deviated well.

Figure ‎4-10 Map of ’s (left) and ’s (right) for Case 4.5; producer P01 is a 76 deviated well.

4.4 Well Length Effect

To investigate the effect of horizontal well length on the CM parameters, we ran Case

4.6.

Case 4.6. In this case, 7 different situations are considered: P01 as a vertical well, a two-

branch W-E horizontal well with lengths of 550ft, 1100ft, and 1650ft; and a one-branch

horizontal well with lengths of 550ft, 1100ft, and 1650ft. If the and versus the

length of the horizontal well P01 (two-branch W-E horizontal well in Figure 4-11, one-

branch horizontal well in Figure 4 - 12) are plotted, the following observations are

I01 I02

I03

I04 I05

P01

P02 P03

P04

=0.5

I01 I02

I03

I04 I05

P01

P02 P03

P04

=10

Deviated well Deviated well


made. As expected, the ’s of the horizontal wells increase as the length increases. The

increase, however, is not linear. The ’s for other pairs decrease. On the other hand, all

the ’s decrease as the length of the horizontal well increases. The length of the

horizontal well affects the ’s and ’s, such that for the first 500 ft increase in the length,

horizontal well ’s increase about 46% and vertical producers ’s decrease about 30%,

while all the ’s decrease about 10%. However, for the one- branch case, as expected,

the rate of increase of between I01 and P01 is much larger than other horizontal well

’s. Moreover, the rate of decrease of between I01 and P01 is much larger than other

horizontal well ’s. We saw a similar trend for other trajectories which are mentioned in

Section 4.3. Those results are not presented here. Generally, the nonlinear trend of ’s

and ’s with respect to the horizontal well indicates that converting a vertical well to a

horizontal well has large effect on producers’ productivity (because the drainage area of the

horizontal well is larger than the vertical well drainage area). When the length of the

horizontal well is large enough (in Figures 4-11 and 4-12 about 500 ft), a further increase

of the horizontal well’s length does not result in a large change in producers’ drainage

areas.

Figure ‎4-11 (left) and (right) versus length of horizontal well (two-branch W-E horizontal well);

As‎P01’s‎length‎increases,‎its‎’s‎are‎increasing‎and‎the‎vertical‎well‎’s‎are‎decreasing.‎However,‎by‎

increasing the horizontal well length, all the ’s‎are decreasing (Case 4.6).


Figure ‎4-12 (left) and (right) versus length of horizontal well (one-branch horizontal well toward

I01); the trend is similar to the Figure 4-11, except the rate of change of ’s‎and‎’s‎between‎P01‎and‎

I01 (closest injector) which is higher (Case 4.6).

4.5 Analytical Method

In Section 2.4.4, we explained that Kaviani and Jensen (2010) derived an analytical formula

based on the MPI model to calculate ’s (Equation 2-9). Afterwards, they decoupled the

effects of well location and boundary by calculating ′’s (Equation 2-10). Although this

formula is for vertical wells, we want to extend it for horizontal wells. To do so, we

discretize each horizontal well into a series of vertical wells along the horizontal well

trajectory. In Figure 4-13, P01 is divided into 11 vertical wells (each vertical well

in one grid block in the simulation model). Consequently, the ’s can be calculated

from Equation 2-9 between 14 producers and 5 injectors (Table 4-3).


Figure ‎4-13 Horizontal well P01 is divided into 11 vertical producers (each vertical well in one grid

block).

To calculate the ’s between the horizontal well P01 and each injector, we sum the ’s

of all the discrete vertical wells (the ’s of all 11 vertical wells) to obtain the horizontal

well ’s (Table 4-4). If we plot the cross plot of CM versus analytical ’s we see that this

method is reasonably accurate with less than 1% error on average (Figure 4-14). There is

a small difference between the observed ’s in Table 4-2 (right) and the analytical results

in Table 4-4. To get an accurate result we worked on both the simulation model and

analytical approach. In the simulation model, we increased the number of grids by dividing

every grid to three grid blocks to diminish the effect of the grid size. On the other hand, in

the analytical model, the number of vertical wells per length of horizontal well should

approach infinity. Thus, if we compare the new simulation model results with a large

number of vertical wells per length of horizontal well in the analytical model, the

analytical ’s will approach the simulation model results (Figure 4-15). We believe that if

the simulation gridding becomes smaller, the sum of absolute error (the vertical axis in

Figure 4-15) approaches zero.

I01 I02

I03

I04 I05

P01

P02 P03

P04


Table ‎4-3 Horizontal well P01 is divided into 11 vertical producers and ’s are calculated between 14 producers and 5 injectors.

P01 P02 P03 P04

0.041 0.221 0.045 0.051 0.061 0.077 0.110 0.039 0.038 0.039 0.042 0.221 0.080 0.102

0.041 0.080 0.039 0.038 0.039 0.042 0.054 0.045 0.051 0.061 0.077 0.080 0.221 0.102

0.033 0.174 0.034 0.035 0.039 0.045 0.061 0.034 0.035 0.039 0.045 0.174 0.174 0.192

0.023 0.267 0.024 0.026 0.029 0.035 0.049 0.023 0.024 0.026 0.030 0.267 0.124 0.280

0.023 0.124 0.023 0.024 0.026 0.030 0.040 0.024 0.026 0.029 0.035 0.124 0.267 0.280

Table ‎4-4 ’s of all 11 vertical wells are summed up to obtain ’s between horizontal well and each injector.

Figure ‎4-14 Cross plot of the optimized ’s‎using‎the‎CM‎(Case 4.2) versus the analytical ’s‎using

the MPI; P01 has a length of 550 ft.

P01 P02 P03 P04

0.597 0.221 0.080 0.102

0.597 0.080 0.221 0.102

0.461 0.174 0.174 0.192

0.329 0.267 0.124 0.280

0.329 0.124 0.267 0.280


Figure ‎4-15 By allowing the number of producer elements to approach infinity and increasing the

number of grid blocks in the simulation model, the difference between analytical and optimized ’s‎is‎

minimized.

This analytical approach is applicable for any other trajectories mentioned in Section 4.3.

For example, if we have a well trajectory in any direction, we apply the discretization

along the well on that direction. If we have a deviated well, we consider the horizontal

length of the well and apply the discretization along the horizontal length of the well.

4.6 Applying the Reverse CM

As we mentioned in Section 2.4.4, Soroush (2010) used the concept of the CM and

derived another model in which the injection rate of each injector is a function of the

production rate of all producers and injection well BHPs (Equation 2-11). Weight factors

and time constants in this model are different in value compared to the CM model, but

they are representative of interwell connectivity as well. In this section we will show that

the existence of a horizontal producer does not affect the reverse CM parameters. To do

so we applied the reverse CM for Case 4.1 and plotted the ’s and ’s in Figure 4-16. In

this case as well, the ’s and ’s are a function of well locations and a reservoir

boundary in a homogenous reservoir. To map these values, each line which is

5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

sum

of

absolu

te e

rror

betw

een lam

bdas o

f C

M a

nd M

PI

number of producers along horizontal well

sum of absolute error vs number of producers


proportional to the ’s or ’s is drawn from each producer toward the injectors (instead

of drawing the line from each injector toward the producers as used in the CM). Referring

to the Appendices 3 and 4, the CM ’s are obtained from prodA and T

conA but the reverse

CM ’s are obtained from injA and conA . Although T

conA is transpose of conA , there is no

relationship between prodA and injA . Therefore, there is no mathematical relationship

between the CM ’s and the reverse CM ’s.

Figure ‎4-16 Map of the ’s‎(left)‎and‎the‎’s‎(right)‎for‎Case 4.1 using the reverse CM; the length of

the arrow is proportional to the or values.

If we plot the map of both ’s and ’s using the reverse CM for Case 4.2 (Figure 4-17),

we see that the horizontal well does not impact these values. The reverse CM can be

applied for other well trajectories with a similar conclusion.

P01

P02 P03

P04

I01 I02

I03

I04 I05

=0.5

P01

P02 P03

P04

I01 I02

I03

I04 I05

=10* *


Figure ‎4-17 Map of the ’s‎(left)‎and‎the‎’s‎(right)‎for‎the‎Case 4.2 using the reverse CM; the

horizontal well does not impact these values.

4.7 Heterogeneous Reservoir

Case 4.7. In this case, an example of a reservoir with a barrier and fracture is considered

as a simple heterogeneous reservoir. P01 is a two-branch horizontal well with a length

of 550 ft and there is an impermeable barrier between I01 and P01 with around zero

permeability and a fracture between P04 and I05 with very large permeability in

the order of a thousand (100 times the system permeability, Figure 4-18). When the

results are compared with the homogeneous case, the ′ between I01 and P01 has a large

negative value, showing a low connectivity area affected by the barrier. In contrast, the

effect of the fracture is recognized by a positive ′ between I05 and P04. Figure 4-18

shows the map of ′ if P01 is a vertical well (left) and the map of ′ if P01 is a horizontal

well (right). Comparing the ′’s for the system of vertical wells and the ′’s for the

system with horizontal well(s) shows that the effect of the horizontal well(s) on the ′’s

has been diminished. To calculate the ′’s we used our analytical method of discretization

in Section 4.5. We can also use the second approach which is mentioned in Section 4.6

and obtain the reverse CM ′’s. Calculating ′’s for the reverse CM is similar to the

ones in the CM by using Equation 2-10 with analytical ’s calculated for the reverse CM

from Appendix 4. Figure 4-19 shows very similar results for the existence of a barrier and

fracture. In other words, ′ between I01 and P01 is negative and ′ between I05 and

P01

P02 P03

P04

I01 I02

I03

I04 I05

=0.5

P01

P02 P03

P04

I01 I02

I03

I04 I05

=10


* *


P04 is positive. Moreover, by comparing the left and right plots in Figure 4-19, we

conclude that the horizontal producer does not impact the reverse CM ′’s (or ’s).

Note that in the calculation of the reverse CM ′’s, there is no need to discretize the

horizontal producer. If we have a horizontal injector, however, for calculation of the

reverse CM ′’s (not the CM ′’s), discretization is necessary.

Figure ‎4-18 The ′’s‎for‎the‎system‎of‎vertical‎wells‎(left)‎and‎the‎′’s‎for‎the‎system‎with‎horizontal‎

well(s) (right); ′’s‎in‎black‎color‎have‎a positive value and ′’s‎in‎red‎color‎have‎a negative value.

The blue rectangle shows a barrier with permeability close to zero and the green rectangle shows a

fracture with permeability of about 100 times the system permeability (Case 4.7).

I01 I02

I03

I04 I05

P01

P02 P03

P04

´=0.1´=-0.1

I01 I02

I03

I04 I05

P01

P02 P03

P04

´=0.1´=-0.1

Horizontal well Vertical well


Figure ‎4-19 The ′’s‎for‎the‎system‎of‎vertical‎wells‎(left)‎and‎the‎′’s‎for‎the‎system‎with‎horizontal‎

well(s) (right) using the reverse CM; the blue rectangle shows a barrier with permeability close to

zero and the green rectangle shows a fracture with permeability of about 100 times the system

permeability (Case 4.7).

4.8 Conclusions

As expected, a horizontal well increases the values associated with it and decreases the

’s of vertical wells in a reservoir. The ’s of the horizontal wells increase as the length

of that horizontal well increases and the ’s for other pairs decrease. The trend,

however, is not linear. All values decrease as the length of the horizontal well increases.

The trajectory of the horizontal well does not have a major effect on the ’s. Our results

are valid for deviated wells with any angle; however, the effect of a deviated well on the

CM parameters is smaller than for a horizontal well.

Two methods are suggested for heterogeneity investigation when there is a horizontal

producer in the system. The use of an analytical approach is suggested here by

discretizing the horizontal well to calculate ′ results to decouple the horizontal well

effect, well location and boundary. The second method is using the reverse CM to

evaluate connectivity parameters since a horizontal producer does not impact the reverse

CM parameters and there is no need to discretize the horizontal well.

P01

P02 P03

P04

I01 I02

I03

I04 I05

´=0.1´=-0.1

P01

P02 P03

P04

I01 I02

I03

I04 I05

´=0.1´=-0.1

Horizontal well Vertical well

*

*

*

*

CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY 79

CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY

5.1 Introduction

In general, injection and production data contain connectivity information and the CM

can decouple the rate-dependent components of these data and provide connectivity

information. Because the CM equation is basically derived from coupling material

balance with a linear productivity model for the drainage area of each producer,

considering radial flow in this drainage area, any heterogeneity close to the producer

wellbore has more effect on the production rate and, consequently, the connectivity

parameters, than a similar heterogeneity far from the wellbore. This near producer

heterogeneity could be a geological feature or permeability change due to wormhole

development or due to the well stimulation. In this chapter, we describe in detail the

effect of interwell connectivity and near well connectivity in the CM results, and

conclude that the near well connectivity impact is dominant. Then, we introduce one

method to assess interwell heterogeneity and two methods to analyze near well

heterogeneity. At the end we demonstrate the relationship between the values and

wellbore skin.

5.2 Near Wellbore Effect

Assuming there is one producer in a radial flow system, most pressure drops occur near

the producer wellbore which we call the “rapidly drained area”. All of the streamlines

transect this rapidly drained area (Figure 5-1). Thus, any barrier or channel which is

located in this small area has a greater effect on production than if the event were located

close to the boundary of total drainage area. In a system of one injector and one producer,

we use the term near well connectivity area instead of rapidly drained area. Also, the area

between two wells is called the interwell connectivity area (Figure 5-2). Similarly, the

effect of any channel or barrier near the producer is dominant on production and

consequently, connectivity parameters in comparison to those far from that producer.


Figure ‎5-1 Schematic subdivision of drainage area into a rapidly drained area and total drained

area; a large number of streamlines traverse a segment which is located in the rapidly drained area

(right). The largest pressure drop occurs in the rapidly drained area (left).

Figure ‎5-2 Schematic subdivision of the area between one injector-producer pair; the near producer

area has more effect on production and connectivity parameters.

5.3 Interwell Connectivity Assessment

The ′ estimated using the CM reflects the heterogeneity in the interwell scale. Thus, the

effect of all the features that exist in the interwell region will be lumped in the ′. In this

manner, depending on the size, permeability, and distance of the features we may obtain

positive or negative ′’s in a specific direction. In the lumping process, the weight of the


near producer heterogeneity will be larger than the other parts. We called the features

close to the producer as near well, or more precisely, near producer heterogeneity that has

a large effect on the ′’s. To describe this concept quantitatively, we have defined the

synthetic cases below.

Case 5.1. This is a 5×4 homogeneous case similar to Case 4.1, but with a base

permeability of 100 md. The permeability at the location of producers P01 and P02 is

1000 md and at the location of producers P03 and P04 is 10 md. In the simulation model

we change the permeability of only nine grid blocks around each producer. However,

when there is no feature between well pairs, the ′’s toward P01 and P02 are positive and

the ones toward P03 and P04 are negative (Figure 5-3, left). The relatively large values of

′’s shows a strong impact of near producer heterogeneity on the CM results. Knowing

that for a homogeneous case 0.18 < homogeneous < 0.32 and for heterogeneous case we can

have 0 < heterogeneous < 1, then -0.18 < ′ (heterogeneous - homogeneous) < 0.68. Comparing the

results in Figure 5-3 (left), with these extremes shows that negative connectivity values

are close to the extreme minimum value of -0.18 and positive connectivity values are

about 30% of the extreme maximum value of 0.68. In other words, the effect of near

producer heterogeneity is very large in the CM results.

To see the effect of interwell connectivity, we consider a skin value in the MPI equation

to calculate ′ from Equation 2-10. If we have prior knowledge about the near producer

permeability zones, by applying the skin factor formula (Equation 5-1) and considering

the value of this apparent skin in calculation of the homogeneous (Equation 2-9), the

adjusted ′’s are very close to zero.

1 ln s

s w

rks

k r

..........................................................................................................( 5-1)

k is reservoir permeability, ks is near well permeability, rs is radius of this near well

permeability, and rw is well radius. To drive Equation 5-1, area around the wellbore is

divided into 2 regions with permeability of ks (inner region) and k (outer region). We

assume the flow in each region is governed by radial flow equation.


In Figure 5-3 (right), the apparent skin is included in the calculation and the effect of near

producer permeability is diminished. The results illustrate that there is no special feature

between well pairs.

Figure ‎5-3 Base permeability is 100 md and near wellbore altered permeabilities are 1000 md for P01

and P02 and 10 md for P03 and P04 (left). Applying the apparent skin diminishes near producer

heterogeneity effect (Case 5.1).

We changed the location of these permeability spots in Figure 5-3 from a producer to

interwell location and then around the injectors. We ran the simulation and calculated the

′’s. Our results were very close to zero. This shows that the interwell and near injector

heterogeneity have a very small effect on ′’s. The larger the distance of these spots from

the producers, the smaller their effect will be on the connectivity parameters.

In field cases, in practice we might not have an accurate estimation of the size of the near

producer permeability, and obtaining these adjusted ′’s is not as straightforward.

However, considering such effects in our analysis will give a more reliable interpretation

of connectivity parameters. In the following example, we use geostatistical software

(SGeMS) to build a more complicated heterogeneous reservoir, which is closer to the real

field cases.

k = 1,000 mD

k = 10 mD

k = 100 mD

I01I02

I03

I04I05

P01

P02 P03

P04

= 0.5'= -0.5

I01I02

I03

I04I05

P01

P02 P03

P04

´= 0.5'= -0.5


Case 5.2. This case is also a 5×4 homogeneous case, similar to Case 4.1, but with the

permeability distribution generated using geostatistical simulation (SGeMS). After

simulation, we calculated ′’s. As we observed, the ′ map is not representative of

connectivity (Figure 5-4, left). For example, ′’s of I01-P01 and I05-P03 are

approximately in the same order of magnitude, but the high permeability zone around

producer P03 is much larger in size and in magnitude. These near producer effects are the

dominant factors in the values of ′’s. If we apply the method of apparent skin, we can

diminish this effect and the results are better representations of interwell heterogeneity

(Figure 4-5 right). However, the low permeability area close to the I05 still cannot be

well determined.

Figure ‎5-4 The figure on the left shows ′’s‎are‎less‎affected‎by‎the‎interwell‎features.‎If‎we‎apply‎

apparent skin the ′’s‎could‎be‎better‎representative of interwell heterogeneity (right). Permeability

distribution is generated by SGeMS (Case 5.2).

In complicated heterogeneous fields such as Case 5.2, interpretation is even more

difficult because not only are the ′’s more influenced by the near well permeability than

by the interwell permeability, but also they are still relative values. In comparison to the

connectivity values with the permeability trend, we calculated the normalized (weighted)

average of interwell permeability for each well pair and compared it with the ′’s. We

devoted more weight to the permeabilities near the producers, than to those far from the

producers. Moreover, we normalized the values to one around each injector to have

I01 I02

I03

I04 I05

P01

P02 P03

P04

´=0.1

´=-0.1

k, mD

100

90

80

70

60

50

40

30

20

10

1


relative values comparable with the ′’s. Figure 5-5 shows the cross plot of the ′’s

obtained by the CM using an appropriate pseudo skin factor for homogenous ’s versus

weighted interwell permeability. The left plot shows approximately zero values for the

′’s and the right plot shows that these values are nearly correlated. The errors in the right

plot may be due to the method of averaging or amount of weight associated in the

method.

Figure ‎5-5 Figures show the cross plot of ′’s‎using‎pseudo skin versus normalized interwell

permeability for Cases 5.1 (left) and 5.2 (right).

5.4 Near well Connectivity Assessment

As we concluded in the previous section, near producer heterogeneity masks the effect of

interwell heterogeneity. We tried to diminish the effect of near producer heterogeneity by

the means of using apparent skin. However, this is a rough estimation and the effect is not

totally removed since the CM parameters are less sensitive to the interwell heterogeneity

than near well heterogeneity. Therefore, if there is some noise in the data or the

heterogeneity size is not large enough relative to well distances, the results would be

more complicated. These results suggest, however, that we could use the CM for

assessing near producer connectivity, which is more dominant in the results. In the

following sections, we introduce two methods to evaluate near producer heterogeneities.

-0.008

-0.004

0

0.004

0.008

-0.008 -0.004 0 0.004 0.008

''s

ob

tain

ed

usi

ng

psa

ud

o s

kin

normalized interwell permeability

P01

P02

P03

P04

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-0.2 -0.1 0 0.1 0.2

''s

ob

tain

ed

usi

ng

pse

ud

o s

kin

normalized interwell permeability

P01

P02

P03

P04


5.4.1 Median and Interquartile Range

We can calculate the median of ′’s for each producer to predict whether or not they are

located in a high-connectivity region. Also, we calculate the interquartile range (IQR) as

a measure of the effects of directionality and relative distance to reservoir boundaries.

The median and IQR of ′’s for each producer are calculated for Case 5.1 (Figure 5-6).

Note that the circle size around each well is considered schematically to compare

negative or positive values to each other. The positive median values (0.208) for P01 and

P02 show that they are located in a high permeability area. On the other hand, negative

median values (-0.208) for P03 and P04 reveal that they are located in a low permeability

area. All producers have the same IQR value, showing that there is no directional effect

on median values, and relative distance to the boundary is the same for all producers

since the well configuration is symmetric. For Case 5.2 (Figure 5-7), we evaluated the

median and IQR of ′’s of each producer using both CM and reverse CM. Positive

values of the median for I01, P01, P03, I04, and P04 illustrate these wells are located in

high permeability areas. On the other hand, the negative values of the median for I02,

P02, I03, and I05 show that they are located in low permeability areas. The larger

magnitude of these values matches the size and magnitude of permeable areas generated

in the simulation model. Similar IQR’s of all the ′’s is showing that the trend of

permeability is smooth.


Figure ‎5-6 The median and interquartile range (IQR) of the′ values for each producer (right); the

red circle signifies a positive median and the blue circle indicates a negative median (Case 5.1).

Figure ‎5-7 The median and interquartile range (IQR) of the′ values for each producer (right); the

red circle signifies a positive median and the blue circle indicates a negative median (Case 5.2).

5.4.2 Equivalent Skin Factor

One of the problems of the method of calculating medians is the complexity of

directionality which is inside these values. In other words, the median of ′ for P01 in

Figure 5-7 is a result of only five ′ toward five injectors (five directions) and is not the

median of ′ over whole the angles around P01. Although we decouple the effect of well

k = 1,000 mD

k = 10 mD

k = 100 mD

I01I02

I03

I04I05

P01

P02 P03

P04

M=0.208

IQR=0.107

M=0.208

IQR=0.107

M=-0.208

IQR=0.107

M=-0.208

IQR=0.107

I01 I02

I03

I04 I05

P01

P02 P03

P04

k, mD

100

90

80

70

60

50

40

30

20

10

1

M=0.053

IQR=0.031

M=-0.009

IQR=0.005

M=-0.018

IQR=0.002

M=0.003

IQR=0.012

M=-0.087

IQR=0.035

M=0.003

IQR=0.005

M=-0.032

IQR=0.008

M=0.021

IQR=0.027

M=0.006

IQR=0.026


location and reservoir boundaries by using ′ instead of , they are relative values in

which any change of one value affects another one around each injector. For example, a

high permeability area between I01 and P02 in Figure 5-7 (left) affect ′ between I01 and

P01. We propose another method for which we can omit these problems. If we coupled

the CM with the MPI, in such a way that the ’s in the CM are substituted with a

homogenous analytical equation ’s derived from the MPI, we can directly evaluate the

equivalent skin around each well. To do so, we add in our code new parameters called

“equivalent skin” in each diagonal index of the influence matrix and count them as the

parameters to be optimized along with the values. Figure 5-8 summarizes the flowchart

of this procedure. The idea is to incorporate all the heterogeneities in the near well area in

these equivalent skins. The number of ’s is equal to the (number of injectors)×(number

of producers), but the number of equivalent skins is equal to the number of producers.

Thus, the number of parameters to be optimized decreases by (number of

injectors)×(number of producers-1).

Figure ‎5-8 Flowchart used in the CM-MPI code; equivalent skin will be optimized via Matlab.

We refer to this new method as the CM-MPI method. Therefore, the advantage of the

CM-MPI method relative to the CM method is that the effect of directionality and relative

distance from the boundaries will be decoupled and we have fewer parameters to be

Initial guess:SP01, SP02, SP03 …

Using MPI Calculate :P01-I01, P01-I02, P02-I01 …

Predict:qP01, qP02, qP03 …

New Skin values using Matlab optimizer by minimizing

objective function

CM+MPI


optimized. The equivalent skin values are representative of the near well connectivity.

We calculated the equivalent skin for each producer for the Case 5.1 (Figure 5-9). The

results show that both P01 and P02, which have high near-well permeability, have

negative equivalent skins of -4.35. On the other hand, P03 and P04, which are located in

low permeability regions, have positive and large equivalent skins of 43. We can validate

these values by comparing them to the values obtained from Equation 5-1 considering an

approximate radius for those near well features. The equivalent skins for P01 and P02 are

-4.37 and for P03 and P04 are 43.73 using this equation. ks values are assumed to be 1000

md and 10 md respectively. rs is assumed to be a circle with a radius of 2 grid blocks.

These values agree very well with the computed skins shown in Figure 5-9. We also

evaluated the equivalent skin for the Case 5.2 (Figure 5-10). To obtain the equivalent skin

around injectors, we used the reverse CM as well. Negative values of equivalent skin for

I01, P01, P03, I04, and P04 indicate that these wells are located in high permeability

areas. On the other hand, positive values of equivalent skin for I02, P02, I03, and I05

show that they are located in low permeability areas. The larger magnitude of these

values matches the size and magnitude of permeable areas generated in the simulation

model.

Figure ‎5-9 Equivalent skin obtained using the CM-MPI algorithm; the red circle signifies negative

skin and the blue circle indicates positive skin (Case 5.1)

k = 1,000 mD

k = 10 mD

k = 100 mD

I01I02

I03

I04I05

P01

P02 P03

P04

S=-4.35

S=-4.35 S=43.11

S=43.26


Figure ‎5-10 Equivalent skin obtained using the CM-MPI algorithm; the red circle signifies negative

skin and the blue circle indicates positive skin (Case 5.2).

5.5 -skin Relationship

To explain the relation between the and producer skins, we first introduce an example

in Case 5.3.

Case 5.1. This is a 5×4 homogeneous case similar to the Case 4.1, except the skin of P01

is changing three times. We assume that other producers’ skins are constant. We plot the

skin of P01 versus reciprocal of the between P01 and all the injectors. Figure 5-11

shows that the relationship is linear with R2

= 1.

Figure ‎5-11 Plot of skin versus 1/for P01; calculated R2’s

equal one confirm that the relationship is

linear.

I01 I02

I03

I04 I05

P01

P02 P03

P04

k, mD

100

90

80

70

60

50

40

30

20

10

1

S=-3.03 S=2.31

S=2.58

S=-1.44 S=7.5

S=-1.85

S=3.20 S=-2.48

S=-1.47

-4

-2

0

2

4

6

8

10

12

0 5 10 15

skin

1/lambda

i1

i2

i3

i4

i5


We can derive this linear relationship analytically by simplifying the analytical equation

of values for a simple system of 1 injector and 2 producers. Then we can generalize it

for more complicated well systems, knowing that ’s are not a function of the number of

injectors. The derivation details and final equations are presented in Appendix 5. This

linear relationship between the reciprocal of ’s of a producer and its skin confirms the

strong relation between the and near producer connectivity. For example, changing P01

skin from -2 to 10 can change between P01 and I05 about 65%.

5.6 Conclusions

Near producer heterogeneity has a large effect on CM connectivity parameters. By

applying the skin factor formula and calculating the adjusted ′, we can have a better

estimate of interwell connectivity. However, the CM results are less sensitive to interwell

connectivity than near well connectivity. A better solution is to apply the CM to assess

near well heterogeneity. The method of calculating the median and IQR is suggested to

evaluate near well heterogeneity. However, the effect of directionality and relative values

of ′ makes the results complicated. To remove this problem, the CM and MPI are

coupled to evaluate the equivalent skin instead of ′. A linear relationship between the

reciprocal of ’s of a producer and its wellbore skin confirms the strong relation between

the and near producer connectivity.

CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 91

CHAPTER 6 THE CM IN HEAVY OIL RESERVOIRS -

WORMHOLE ASSESSMENT

6.1 Introduction

So far we have assumed that the mobility ratio is approximately 1 in our calculations.

Therefore, saturation of the phases will have no effect on producer productivity or

connectivity parameters, and they will stay constant. If mobilities of the injected and

produced phases are different, depending on the saturations, we will have a different

average mobility at each point of the reservoir that changes over time. In this manner, the

producer productivity or connectivity parameters change at each time step. On the other

hand, decreasing the productivity of the producers closer to the injectors decreases their

’s. Consequently, the ’s of distant wells increase. This is equivalent to having a

positive skin around all producers. In this chapter, we first show how much the mobility

changes affect the connectivity parameters. Then, we discuss how much error we may

expect when applying the CM in heavy oil waterfloods and how we can reduce this error.

As the oil is produced in heavy oil reservoirs, sand near the producer wellbore may be

produced as well. This creates extended open areas called wormholes. We apply the CM

to assess wormhole development at the end of this chapter.

6.2 Connectivity Evaluation in Heavy Oil Reservoirs

We know that in heavy oil reservoirs, depending on the injection or production rates, the

connectivity parameters change over time. In the following synthetic cases, we can track

these changes. At first, it may seem that using all the data is the best choice to get the

most representative set of connectivity parameters. This is because in general, having a

longer period of data (large L in Chapter 3) leads to a more accurate estimation of

parameters. However, in heavy oil reservoirs, this might not be the best choice, since the

parameters change over the time. To determine the most representative time window, for

some synthetic cases, we apply the CM to different windows and compare the accuracy

of the estimated connectivity parameters. We have applied the CM in several synthetic

reservoirs with a non-unit mobility ratio and with both vertical and horizontal well(s).


Case 6.1. This is a 5×4 homogeneous case where the general reservoir properties are

presented in Table 6-1 and well locations are similar to Case 4.1. The oil viscosity is 20

cp (mobility ratio = 10). We ran the case for 290 months (L=23) and calculated the

connectivity parameters at each time step (Figure 6-1). Based on these results, the major

changes (which is less than 10%) in ’s are at the first 0.4 PV of the waterflood, and after

that, the changes in the connectivity parameters become smaller (less than 5%). We also

observed that even after injecting 2 PV, the ’s are still slightly (about 3%) different from

the analytical ones. In this case, average viscosity of total produced fluid is higher than

the unit-mobility case. Considering the saturation changes with respect to time around the

producers, average permeability of total produced fluid decreases. If we assume that

productivity is proportional to the permeability and inversely proportional to the

viscosity, we have an overall decrease in productivity. Reduction in producers’

productivity acts as an apparent positive skin. We can test this observation with a case of

single phase waterflood for which all the producers have an appropriate positive skin

(Figure 6-2). At first 0.4 PV we can consider this decrease in producers’ productivity

with an equivalent skin of +2 (Figure 6-2, left). After that, the decrease in producers’

productivity is in order of an equivalent skin of +1 (Figure 6-2, right) and even less than

+1 after injecting 2PV. Note that at first 0.4 PV the waterflood impact on the production

rate is less than after 0.4 PV injection so the equivalent skin is even larger at the

beginning. After this period, the saturation profile is stabilized.

Table ‎6-1 Reservoir and simulator parameters for Case 6.1

Parameter Value

, fraction 0.18

Absolute k, md 40

Oil end point k, md 36

Water end point k, md 9

Coil, psi-1

5×10-6

Cwater, psi-1

1×10-6

Crock, psi-1

1×10-6

Irreducible oil saturation, faction 0.35

Irreducible water saturation, faction 0.2

oil, cp 20

water, cp 2


Grid size, ft 26.667×26.667×30


Figure ‎6-1 ’s‎variations‎over‎time‎for‎Case‎6.1‎(mobility‎ratio=10).‎As‎time‎increases,‎the‎’s‎

converge to the ideal ones. Each solid line shows the between a well pair. The dashed lines show the

ideal ’s‎(from‎unit-mobility ratio).

Figure ‎6-2 ’s‎map‎when‎all‎the‎producers‎have‎a‎skin‎of‎+2‎(left)‎and‎+1‎(right).

Case 6.2. This is a 5×4 homogeneous case where the general reservoir properties and

well locations are similar to Case 6.1. However, the oil viscosity is 2000 cp (mobility

ratio = 1000). We simulate the case for 290 months (L=23). Similar to the previous case,

we calculated the ’s for each time step (Figure 6-3). Compared to the lighter oil case, the

’s became stable faster. This could be because the saturation profile will be stabilized

faster. Therefore, the producers’ productivity does not change in this period of injection.

0.15

0.17

0.19

0.21

0.23

0.25

0.27

0.29

0.31

0.33

0.35

0 0.5 1 1.5 2

Injected PV

I01 I02

I03

I04 I05

P01

P02 P03

P04

´=0.02´=-0.02

I01 I02

I03

I04 I05

P01

P02 P03

P04

´=0.02´=-0.02


The stabilized ’s in this case have an error of 10%. Compared to Case 6.1 (with the

stabilized ’s error of 3%) the producers’ productivity decreases further because the

average viscosity of produced fluid is higher. If we assume an equivalent skin for each

producer in Case 6.2, an skin of +2 is appropriate after ’s are stabilized (Figure 6-2,

left).

Figure ‎6-3 ’s‎variations‎over‎time‎for‎the Case 6.2 (mobility ratio=1000); as time increases, the ’s‎

becomes stable. Unlike Case 6.1., the ’s‎at‎the‎last‎time‎step‎do‎not‎converge‎to‎the‎ideal‎ones.‎Each‎

solid line shows the between a well pair. The dashed lines show the ideal ’s‎(from‎the unit-mobility

ratio).

To compare the estimated ’s at each time step with the ideal ones (from mobility ratio =

1), we used AAD (Equation 3-4) as well. Based on the results obtained from the cases

above, we observed that the AAD will not exceed 0.02 (Figure 6-4). This may imply that

selecting the window from any stage of the floods will lead to an error smaller than 0.02

in the estimated ’s. Considering an AAD of 0.01 for mr=10 and 0.015 for mr=1000,

after 1 PV of the waterflood the results are repeatable. We also expect that for mobility

ratios larger than 1000, AAD should be larger than 0.05.

0.15

0.17

0.19

0.21

0.23

0.25

0.27

0.29

0.31

0.33

0.35

0 0.5 1 1.5 2

Injected PV


Figure ‎6-4 The AAD variations over time for the Cases 6.1 (mobility ratio=10) and 6.2 (mobility

ratio=1000) based on the data for each time step; the AAD for these cases never exceeds 0.02.

Since the CM parameters are changing within the window specifically at early times, the

obtained error from different windows with the same size might be different. For this

purpose, we looked at the estimated ’s from different windows of the data set. Figure 6-

5 shows the estimated ’s for Case 6.1 with a 50 months length window (L=4) starting

every 10 months (i.e. 1-50, 11-60, etc). As we observed, before 20 months the ’s are not

stable. Error in the ’s after 40 months and before 60 months is less than 21%, and after

that, it is less than 6%. In the case with heavier oil (Case 6.2) the ’s become stable

faster, and after 40 months the error is less than 10% (Figure 6-6). However, after 125

months, the ’s of the lighter oil case are closer to the unit-mobility ones (Figure 6-7).

Using water end point permeability of 9 md, the CM number is 3.1 (Equation 3-3). Figure

3-17 shows that with L = 4 and C = 3.1 median of CV’s is 0. This means that after

producing all the mobile oil, the results are stable (after 60 months). Before this period,

however, the ’s are changing because the producers’ productivity is changing (average

viscosity and average permeability are changing). Note that with L = 4 and C = 3.1,

Figure 3-18 suggests that AAD should be 0. The AAD value in Figure 6-7, however, does

not approach zero because the stabilized ’s are different than analytical ’s (due to the

reduced producers’ productivity in non-unit-mobility case). We also observe that

selecting the very early window of the waterflood may provide misleading results

0

0.005

0.01

0.015

0.02

0 0.5 1 1.5 2

AA

D

Injected PV

mr=10

mr=1000


specifically for Case 6.1 (mr = 10). Although, based on Figure 6-4, the estimated

connectivity parameters from each time step are smaller than 0.02, since the ’s are

fluctuating within the window, the estimation of connectivity parameters from these data

may lead to a larger error. For Case 6.2 (mr = 1000), the ’s are closer to the unit-

mobility one at the very stages of the flood and that could be an appropriate interval to

understand the heterogeneity. Therefore, we conclude that for the prediction of future

rates, the late ’s could be more representative.

Figure ‎6-5 ’s‎variations‎by‎moving‎the‎analysis‎window‎for Case 6.1 (mobility ratio=10); applying

the CM to early data will lead to unstable results. Each solid line shows the between a well pair.

The dashed lines show the ideal ’s‎(from‎the unit-mobility ratio).

0.15

0.17

0.19

0.21

0.23

0.25

0.27

0.29

0.31

0.33

0.35

0 50 100 150 200 250

Start time of window, months


Figure ‎6-6 ’s‎variations‎by‎moving‎the‎analysis‎window‎for‎the Case 6.2 (mobility ratio=1000);

applying the CM only on very early data will lead to unstable results. Each solid line shows the

between a well pair. The dashed lines show the ideal ’s‎(from the unit-mobility ratio).

Figure ‎6-7 The AAD variations by moving the analysis window for Case 6.1 (mobility ratio=10) and

6.2 (mobility ratio = 1000); at a very late time the lower mobility ratio provides less variable ’s.‎

0.15

0.17

0.19

0.21

0.23

0.25

0.27

0.29

0.31

0.33

0.35

0 50 100 150 200 250


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 50 100 150 200 250

AA

D

Start time of window

mr=10

mr=1000


For both of the previous cases, the injectors started to inject all at the same time and their

average rates were not significantly different from each other. To investigate the

sensitivity of the results to the injection rates, we looked at two other scenarios for the

Case 6.2.

Case 6.2.1. We assumed that injector I03 (the center injector) was shut-in for the first 100

months. Based on the results, the estimated ’s from windows (Figure 6-8) are not

significantly different from Case 6.2.

Figure ‎6-8 ’s‎variations‎by‎moving‎the‎analysis‎window‎for‎the‎case‎where‎I03‎was‎shut-in for the

first 100 months (Case 6.2.1); the results are not significantly different from Case 6.2. Each solid line

shows the between a well pair. The dashed lines show the ideal ’s‎(from‎unit-mobility ratio).

Case 6.2.2 We assume that injectors I02 and I04 (two corner wells) are shut-in for the

first 100 months. The results of this case (Figure 6-9) are close to those of Case 6.2;

however, we observe that the differences between the ’s of some of the well pairs are

greater. For example in Case 6.2.2, the largest ’s (which are related to the closer well

pairs) vary from 0.300 to 0.335, a difference of 0.035 or about 10%, and the smallest ’s

(which are related to the distant well pairs) vary from 0.165 to 0.200, which is a

difference of 0.035 or about 18%. For the case of closer well pairs, the upper limit of

these ’s (0.335) is related to the injectors that were opened from the first month. Since

0.15

0.17

0.19

0.21

0.23

0.25

0.27

0.29

0.31

0.33

0.35

0 50 100 150 200 250



these injectors have started their injection at the beginning, they are the main injectors

which stabilized the saturation profile. Therefore, they play a main role in the

productivity of the closer producers. Consequently, the effect of shut-in injectors on the

productivity of closer producers would be less (0.3) than Case 6.2 (which is 0.32).

Because the ’s are relative values, the ’s between shut-in injectors and distant

producers would be higher (0.2) than Case 6.2 (which is 0.18). For Case 6.2.1 (where

injector I03 was shut-in), since the distances of the injector to all the producers are equal,

we did not observe this feature.

As we observed, even in the case where two corner injectors started their injection later,

the AAD was less than 0.015. Considering possible noise in the data, this deviation is

negligible. The results above imply that, by selecting a proper window, varying injection

rates do not have a major effect on the ’s for homogeneous cases.

Figure ‎6-9 ’s‎variations‎by‎moving‎the‎analysis‎window‎for‎the‎case‎where‎I02‎and‎I04‎were‎shut-in

for the first 100 months (Case 6.2.2); in comparison to previous cases, we observed a slightly different

trend in the ’s.‎Each solid line shows the between a well pair. The dashed lines show the ideal ’s‎

(from the unit-mobility ratio).

0.15

0.17

0.19

0.21

0.23

0.25

0.27

0.29

0.31

0.33

0.35

0 50 100 150 200 250



Up to this point, we investigated the effect of varying mobility on the CM performance

only for homogeneous cases with non-stimulated vertical wells. In the following case

example, we will look at a heavy oil case with a horizontal well.

Case 6.3. This is a 5×4 homogeneous case where the general reservoir properties and

well locations are similar to Case 6.2. The oil viscosity is 2000 cp (mobility ratio = 1000)

and L=23. However, P01 is a W-E two-branch horizontal well with a length of 550 ft. For

heavy oil horizontal wells, we expect that the ’s are different from the ’s of the unit

mobility ratio case. As is the case in a system with vertical wells, producers’ productivity

decreases in non-unit-mobility cases because produced fluid viscosity increases and

produced fluid permeability decreases. Figure 6-10 shows the trend of AAD of ’s from a

unit mobility ratio for a 50 months window (L=4). Comparing the late time AAD from

the horizontal well example and the vertical wells (Case 6.2), we observed that the effect

of non-unit mobility ratio on ’s of the horizontal well system is much more pronounced.

If we consider a horizontal well as a multiple infinite vertical well along its trajectory

(Section 4.5), the non-unit-mobility ratio decreases all these imaginary producers’

productivity. Therefore, the summation of this reduction is more than a system with

vertical wells. Meanwhile, Figure 6-10 shows that the ’s in the vertical well system

approach stable values after 50 months, whereas the ’s in horizontal well system

approach stable values after 150 months. This is because the horizontal well acts as a line

sink rather than a pint sink. Therefore, the saturation profile will be stabilized within a

longer time period, in comparison to the vertical well system in which the wells are like a

point sink.


Figure ‎6-10 Plot of average absolute error in for M=1000 relative to M=1 for both a system with

and without horizontal well; x axis is the starting time of every 50 month time interval in which the

CM is applied, the ’s‎in‎a vertical well system approach stable values after 50 months, whereas the

’s‎in‎horizontal‎well‎system‎approach‎stable‎values‎after‎150‎months.‎The average absolute error in

for horizontal well system is larger than that of vertical well system.

6.3 Wormhole assessment

One of the potential applications of connectivity evaluation is identifying the effect of

production in the permeability of the reservoir. Since the CM could easily identify near

producer heterogeneity, it could be a useful tool for determining the permeability

enhancement or loss for the producers over time. For example, in a cold heavy oil

production with sand, wormholes develop around producers. To evaluate the application

of the CM for this case, we analyze the data for the period after the wormhole was

developed. Depending on the time interval of investigation, the CM gives us different

assessments about the wormhole development. For instance, if we apply the CM to the

whole time interval, it only detects the presence of a wormhole around the specific

producer. On the other hand, if we apply the CM on several subintervals of time, it

estimates the equivalent skin associated to that wormhole and the rate of wormhole

growth. This section shows how this information can be derived from the CM.

6.3.1 Wormhole detection

The following case is a simple example of the application of the CM on wormhole

identification.

0

10

20

30

40

50

0 50 100 150 200 250A

vera

ge a

bso

lute

err

or i

n

, %Start time of window, months

Horizontal well

Vertical well


Case 6.4. This is a 5×4 case with the properties the same as Case 6.2, but wormholes are

developed around P01 (Figure 6-11 left). To model the wormhole, we assumed that the

cells with wormhole have a permeability two times larger than the average reservoir

permeability. Considering that the width of the cells is 26.667 ft, this increase in

permeability is equivalent to a 600 times permeability enhancement in features with the

size of one inch. After simulation we calculated ′’s. As expected, we observed increased

connectivity towards P01. Since the wormhole has been developed in different directions,

we do not observe any difference in the ′’s of P01 and different injectors; e.g. ′I01-P01 =

′I02-P01. We also ran another case where the wormhole is developed only towards one of

the injectors (Figure 6-11 right). By calculating the ′’s, we observed that the ′’s are not

considerably different from the previous case. For example, for this case the difference

between ′I01-P01 and ′I02-P01 is only 0.01, which is negligible. This is largely because of

the near producer component of ′’s, which plays the main role in its magnitude.

Therefore, we conclude that the CM cannot predict wormholes’ direction. Lines et al.

(2003, 2008) used seismic data to detect zones of foamy oil associated with wormholes

and estimate the preferred direction of the wormholes.

Figure ‎6-11 CM is a robust tool to detect the presence of a wormhole (Case 6.4); however, the

wormhole geometry has a subtler effect (left vs. right).

I01I02

I03

I04I05

P01

P02 P03

P04

´=0.1´=-0.1

k=80 md

k=40 md

k=80 md

k=40 md

I01I02

I03

I04

I05

P01

P02 P03

P04

´=0.1´=-0.1


6.3.2 Equivalent skin associated with the wormhole

We know that there is a linear relationship between the reciprocal of ’s of a producer

and its wellbore skin (Section 5.5). To evaluate the effective (or equivalent) skin change

caused by the wormhole, we explain the following case.

Case 6.5. This is a 5×4 case with properties identical to what was seen in Case 6.4, where

wormholes are developed around P01. The only difference is that we simulated a

wormhole around P01 by using a 4- branch horizontal well where the length of branches

is growing evenly in the reservoir (Figure 6-12). However, the permeability and

complexity of a real wormhole is not considered here; this is a very simple model to

assess the wormhole growth. To plot the type curve of 1/ versus skin (Figure 6-13) we

applied the CM-MPI for this case at the beginning of waterflooding over a short time

subinterval (considering enough samples to get a stable result for example L > 4) and

continue running that stepwise toward the end of the time interval. Using this type curve

has the following aspects.

1- Using CM, we can estimate the equivalent skin of the wormhole at any time for the

specific well configuration (Figure 6-13). In other words, by knowing the y-axis values

from the CM results, the skin associated with the wormhole can be predicted from

reading the x-axis value.

2- If any producer’s changes over time, it means that the changing value of the skin

may be associated with wormhole development, and is not due to geological

heterogeneity. On the other hand, if a producer’s is constant over time, no wormhole is

identified.

3- We can use the type curve method for a heterogeneous system, too. The curves will be

shifted in the plot and we need to first apply CM-MPI one time over the whole interval to

generate the type curve. We assume that a wormhole is developing only in one well,

otherwise the type curves would not be straight lines. Therefore, using the type curve

method for a reservoir with wormhole developments in several wells is an approximation.


Figure ‎6-12 Simple wormhole model; P01 is a 4 - branch horizontal well where the length of branches

is growing evenly in the reservoir (Case 6.5).

Figure ‎6-13 Type curves to evaluate equivalent skin associated with the wormhole for any specific

time; negative skin values in x-axis is associated to the wormhole (Case 6.5).

-5 0 5 10 150

2

4

6

8

10

12

14

skin

1/lam

bda

01-P01

02-P01

03-P01

04-P01

05-P01


6.3.3 Rate of wormhole growth

We used Case 6.5 and applied the CM-MPI to calculate the ’s as a function of

horizontal well length (Figure 6-14). In other words, for each length of horizontal well at

any time we calculated the ’s. Results show that rate of change of ’s versus wormhole

length becomes nearly constant after some distance away from borehole. Thus we have:

dcte

d length

.............................................................................................................( 6-1)

or

d d lengthcte

d t d t

.................................................................................................( 6-2)

The procedure is similar to that described in the previous section. We apply the CM at the

beginning of waterflooding over a short interval (considering enough number of samples

to get a stable result L > 4) and continue running the CM stepwise toward the end of the

time interval. Then, we use already generated type curves for the specific well

configuration and estimate the wormhole length at any time (Figure 6-15). Note that the

wormhole length equals the sum of all branch lengths. Similar to what was discussed in

the previous section, this type curve method can be extended to heterogeneous systems.

Although this method is for a system of wormholes in more than one well, it does give an

approximation.


Figure ‎6-14 Rate change of ’s‎with respect to length of wormhole stabilizes after the wormhole

grows some distance away from the borehole. In this plot, only the ’s‎of‎I01‎are‎shown.‎Other‎

injector ’s‎have‎the‎same‎trend‎(Case‎6.5).

Figure ‎6-15 Type curve generated to evaluate equivalent wormhole growth for any specific time for a

homogenous 5-injector 4-producer system; wormhole length (summation of all branches length) is

divided by the length of the reservoir (Case 6.5).

0 100 200 300 400 500 600 7000

0.5

1

1.5

2

2.5

3x 10

-3

d(lam

bda)/

d(length

)

wormhole (hw) length, ft

wormhole growth

P01

P02

P03P04

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

lam

bda

dimensionless wormhole length

P

P

P

P


There are a few methods in the literature to evaluate wormhole length. Liu and Zhao

(2005) calculated the maximum wormhole length versus time during the fast growth

period (Figure 6-16). Both plots in Figure 6-16 show that the wormhole had grown at

several different speeds during the fast growth period. We tried to simulate a wormhole

using a 4-branch horizontal well and compare our results with the CM results (Figure 6-

17). Our simulated wormhole growth is approximately similar to the model of Liu and

Zhao. We increase the total length of the horizontal well branches by 25 ft every 4

months. We used a 50 months length window (L=4) at 4 specific times to evaluate the

’s. Then we used Figure 6-15 to estimate the wormhole length at each time. Figure 6-17

shows that there is a difference between our simulation results and the CM results. The

reason is that the ’s are changing during the analysis periods.

Figure ‎6-16 Liu and Zhao (2005) calculated maximum wormhole length versus time during the fast

growth period for 2 wells.


Figure ‎6-17 Comparison of the CM results and the simulation model (Case 6.5).

6.4 Conclusions

We have applied the CM in several synthetic reservoirs with a non-unit mobility ratio and

with both vertical and horizontal well(s). In non-unit mobility ratio systems, producers’

productivity is reduced mostly due to the high average produced fluid viscosity.

Therefore we should expect an AAD (≈ 5%) for the ’s. This effect is greater when we

have horizontal well(s) in the system (up to 30% error in average). At large mobility

contrasts (≈ 1000), analyzing the data after 0.4-0.5 PV of injection leads to stable CM

results. In other words, by excluding the first 0.4-0.5 PV of waterflooding data from the

analysis, we will get stable values. Higher mobility ratio results in more stable ’s at

early time but a larger error at a late time of waterflood.

Using the CM we can detect the existence of a wormhole, although it is difficult to

recognize its direction. Using generated type curves, we can evaluate the wormhole

equivalent skin and the rate of wormhole growth. Since the ’s are changing during

wormhole development, the CM results approximate the wormhole length.

0

50

100

150

200

250

0 20 40 60 80

Wo

rmh

ole

s to

tal l

en

gth

, ft

Time, months

simulation

CM

CHAPTER 7 THE CM IN TIGHT FORMATIONS 109

CHAPTER 7 THE CM IN TIGHT FORMATIONS

7.1 Introduction

In Chapter Three, we explained that the CM parameters have very small variability when

the CM number is between 0.3 and 10. In low permeability reservoirs (tight formations),

however, this number is smaller than 0.3. The CM has an error since the flow regime is

transient and not in a pseudo steady state. In this chapter, we extend the MPI for the

transient regime and calculate the connectivity parameters analytically. We will show that

these parameters are time dependent and a function of the diffusivity constant. At the end,

we will derive a new model which is equivalent to the CM but, works in a transient

regime.

7.2 Transient MPI

As we explained in Section 2.4.3, Valkó et al. (2000) suggested that the linear

productivity model can take on a matrix form. They assumed a rectangular reservoir and

suggested that pressure distribution in the reservoir during the pseudo steady state can be

evaluated for one well using the influence function (Equation 7-1).

1, , , , ,2

D D wD wD eD

Bp p x y a x y x y y q

kh

.....................................................................( 7-1)

where the influence function, a, for a homogeneous reservoir is given by

2 2

21

, , , ,

12 2 cos cos

3 2

D D wD wD eD

D wD mDeD D wD

meD eD

a x y x y y

y y tyy m x m x

y y m

..........................................( 7-2)

cosh cosh

sinh

eD D wD eD D wD

m

eD

m y y y m y y yt

m y

.......................................( 7-3)

Then they presented Equation 7-1 as a matrix form:

1

2sd A D q

kh

....................................................................................................( 7-4)

where the matrices are as follows:


1 1 11 12 1 1

2 2 21 22 2 2

1 2

0 0

0 0, , ,

0 0

wf N

wf N

s

wf N N N N NN N

p p q a a a s

p p q a a a sd q A D

p p q a a a s

If we define J as the productivity index, we can simplify Equation 7-1 as shown below:

( , )q J p p x y ...........................................................................................................( 7-5)

where

1

2

, , , ,D D wD wD eD

khJ

Ba x y x y y

......................................................................................( 7-6)

and we can define [J] for Equation 7-4 as a matrix form:

[ ]q J d ..........................................................................................................................( 7-7)

where

1

1

2[ ] s

khJ A D

B

..................................................................................................( 7-8)

In Equation 7-6, the influence function (a) is a function of location, boundaries, and

wellbore radius. We have a similar equation to Equation 7-5 for transient flow if we

define J as shown below:

2

1

2

t w

khJ

c rBEi

kt

..................................................................................................( 7-9)

where Ei function is a function of wellbore radius, reservoir and fluid properties.

Assumptions of the Ei function include a homogenous and isotropic reservoir with

uniform thickness. The fluid should be slightly compressible and the wells must fully

penetrate the entire reservoir’s thickness. The wells have a small radius and drain an

infinite area. All the Ei function assumptions are similar to the MPI model assumptions

(Section 2.4.3) except the latter one (infinite acting assumption). As long as the pressure

disturbances of all wells do not reach the boundary we will use Equation 7-9. However,

the transition from this period to the pseudo steady state period is hard to model, and we

do not consider it in this chapter.


In a multiwell system (Equation 7-8), the influence matrix ([A]) is a function of location,

boundary and wellbore radius. We can define a similar Equation as 7-8 for a multiwell

system during the transient regime as:

1

1

2[ ] s

khJ E D

B

...............................................................................................( 7-10)

where

11 12 1

21 22 2

1 2

N

N

N N NN

Ei Ei Ei

Ei Ei EiE

Ei Ei Ei

where the Ei indices for a homogeneous reservoir are given by

2 2 2

( )

[( ) ( ) ],

t j i j i t wij i j ii

c x x y y c rEi Ei Ei Ei

kt kt

......................................( 7-11)

Note that like influence matrix [E] matrix is symmetric.

Case 7.1. This is a one injector, one producer homogenous synthetic field with the

properties mentioned in Table 7-1. We ran the simulation model with a reservoir

permeability of 100 md, then applied both MPI and transient MPI to predict average

reservoir pressure (Figure 7-1). The results show that both models can accurately predict

average reservoir pressure. We decreased the permeability of the reservoir to 0.1 md and

ran the simulation again. This time the MPI could not predict the average reservoir

pressure, while the transient MPI prediction is satisfactory (Figure 7-2).

Table ‎7-1 Reservoir and simulator parameters for the Case 7.1

Parameter Value

, fraction 0.18

Horizontal k, md 100, 0.1

ct, psi-1

2×10-6

, cp 0.5


Grid size, ft 26.667


Figure ‎7-1 Average reservoir pressure using MPI (left) and transient MPI (right) with permeability

of 100 md for the Case 7.1; the blue dots indicate simulated average pressure, while the red shows

predicted average pressure.

Figure ‎7-2 Average reservoir pressure using MPI (left) and transient MPI (right) with permeability

of 0.1 md for the Case 7.1; the blue line shows simulated average pressure and the red line shows

predicted average pressure.

7.3 Connectivity Parameters in Transient Regime

As we explained in Section 2.4.4, Kaviani and Jensen (2010) calculated the ’s

analytically using the MPI model (Equation 2-9). We use the same method for deriving

transient MPI ’s. We divide the Ei matrix into four components (Equation 7-12),

considering w is a vector of injector rates with a minus sign and q is a vector of

producer rates, 1

2 kh

and, for simplicity in notation, we consider the skin matrix is

added to the Ei matrix and name the new matrix as [E],

1

wfi inj con

T

wfp con prod

p p E E w

p p E E q

......................................................................................( 7-12)

0 2000 4000 6000 8000 10000500

600

700

800

900

1000

Time, day

Avera

ge p

ressure

, psia

0 2000 4000 6000 8000 10000500

600

700

800

900

1000

Time, day

Avera

ge p

ressure

, psia

0 2000 4000 6000 8000 100000

1

2

3

4

5

6

7

8x 10

4

Time, day

Avera

ge p

ressure

, psia

0 2000 4000 6000 8000 100000

1

2

3

4

5

6

7

8x 10

4

Time, day

Avera

ge p

ressure

, psia


Then we can calculate the CM ’s analytically for the transient period. The detail of this

derivation is presented in Appendix 6. The final matrix form of analytical ’s is shown in

Equation 7-13.

1

1

1

1 1

1 1[ ]

1 1

p p p i

p i

p p

T

prod conN N N N T

N N prod con

prodN N

E EE E

E

..........................................( 7-13)

where pN is the number of producers and iN is the number of injectors.

Comparing Equations 2-9 (pseudo steady state ’s) and 7-13 (transient ’s) and

considering heterogeneity in the reservoir, we can conclude that, at early time of

production, we have ’s which are variable with time; they then reach a constant value

when the boundary of the reservoir is reached (Figure 7-3). In moderate to high

permeability reservoirs, however, we directly obtain a pseudo steady state

Transienttrend is more obvious in tight formations.

Figure ‎7-3 At early time is a function of time then it approaches to a constant value when reservoir

boundary is reached.

Case 7.2. This is a 5×4 homogenous synthetic field with the properties mentioned in

Table 7-1 and a permeability of 100 md. We also multiplied the grid size by 10 to see the

effect of transient flow more precisely. We did not run the simulation model this time.

We just calculated transient ’s and pseudo steady state ’s (Figure 7-4). At time zero, all

transient ’s are 0.25. In other words, at the beginning of production, the effect of all

injectors are negligible or the same on each producer. When the time elapses, transient

approaches pseudo steady state .

Pseudo steady state = F (location, boundary, producers’ skin, rw, heterogeneity)

Transient = F (relative well distance, producers’ skin, rw, diffusivity constant, time, heterogeneity)

Early time

Reservoir boundary is touched


Figure ‎7-4 Transient and pseudo steady state is calculated for the Case 7.2. The left figure is

between I01 and P01; the middle figure is between I01 and P03; and the right figure is between I03

and P01.

Figure 7-3 shows that the transient ’s are also a function of the diffusivity constant. We

calculated transient ’s with respect to permeability at 300 days (Figure 7-5). In very low

permeabilities, all transient ’s are 0.25. The effect of all injectors on each producer is

negligible. By increasing the permeability, all the transient ’s approach a constant value

(pseudo steady state ’s).

Figure ‎7-5 Transient is calculated versus permeability after 300 days. The left figure is between

I01 and P01; the middle figure is between I01 and P03; and the right figure is between I03 and P01.

Both transient and pseudo steady state ’s change with respect to the reservoir area. To

see this trend, we plotted transient ’s versus area in Figure 7-6. We discovered that the

smaller the reservoir area, the more pronounced the effect of injectors. By increasing the

reservoir area, all the transient ’s approach 0.25 again indicating a negligible short-term

effect caused by injection.

0.24

0.25

0.26

0.27

0.28

0.29

0.3

0.31

0 2000 4000 6000 8000 10000

lam

bd

a (

1,1

)

time, day

transient lambda

pss lambda

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

0.26

0 2000 4000 6000 8000 10000

lam

bd

a (

1,3

)

time, day

transient lambda

pss lambda

0.24

0.25

0.26

0 2000 4000 6000 8000 10000

lam

bd

a (

3,1

)

time, day

transient lambda

pss lambda

0 2 4 6 80.25

0.26

0.27

0.28

0.29

0.3

k, md

lam

bda(1

,1)

0 2 4 6 80.2

0.21

0.22

0.23

0.24

0.25

k, md

lam

bda(1

,3)

0 2 4 6 80.24

0.245

0.25

0.255

0.26

k, md

lam

bda(3

,1)


Figure ‎7-6 Transient is calculated versus reservoir area after 300 days and a permeability of 100

md. The left figure is between I01 and P01; the middle figure is between I01 and P03; and the right

figure is between I03 and P01.

In Chapter 3 we defined a dimensionless number (CM number, Equation 3-3) to

generalize the behavior of the CM parameters. At small CM numbers due to the transient

flow effect, median CV’s and AAD are not negligible. If we calculate the analytical ’s

in transient regime and plot it versus CM number (Figure 7-7), at C=1 transient ’s

approach pseudo steady state ’s. This is in agreement with the results of our sensitivity

analysis in Chapter 3 (Figures, 3-17, 3-18), where L approaches infinity, and the median

CV’s and AAD approach 0 at C>1.

Figure ‎7-7 Transient is calculated versus CM Number. The left figure is between I01 and P01;

the middle figure is between I01 and P03; and the right figure is between I03 and P01.

Assuming a reservoir with one producer at the center, the time of investigation can be

calculated from Equation 7-14.

0 0.5 1 1.5 2 2.5 3 3.50.25

0.26

0.27

0.28

0.29

0.3

Area/109, ft2

lam

bda(1

,1)

0 0.5 1 1.5 2 2.5 3 3.50.2

0.21

0.22

0.23

0.24

0.25

Area/109, ft2

lam

bda(1

,3)

0 0.5 1 1.5 2 2.5 3 3.50.24

0.245

0.25

0.255

0.26

Area/109, ft2

lam

bda(3

,1)

10-4

10-3

10-2

10-1

100

101

0.25

0.26

0.27

0.28

0.29

CM Number

lam

bda(1

,1)

10-4

10-3

10-2

10-1

100

101

0.21

0.22

0.23

0.24

0.25

CM Number

lam

bda(1

,3)

10-4

10-3

10-2

10-1

100

101

0.24

0.245

0.25

0.255

0.26

CM Number

lam

bda(3

,1)


2

4

inv

inve

esti

stigatio

gati

n

ont

r

........................................................................................................( 7-14)

where is the diffusivity constant (Equation 3-1), and r is the radius of investigation. If

we assume the reservoir area equals 3.14r2.

12.56investigation

At

..........................................................................................................( 7-15)

where A is the reservoir area. Comparing Equation 7-15 with Equation 3-3 in SI unit,

0.08C K .......................................................................................................................( 7-16)

where K=1 for this case. It means that at C>0.08, the pseudo steady state flow is reached.

For the case of 4 producers, we may pass a transient regime at C>0. 32. This value is

smaller than what we obtained in this chapter. One reason might be using the Equation 7-

15, which approximates the time of investigation for the rectangular reservoirs.

7.4 Transient CM

We can derive an equation of production rate of each producer as a function of injection

rate of all injectors and BHP of producers similar to the CM equation using MPI. Kaviani

(2009) derived this equation for the pseudo steady state regime. We derive it for the

transient regime since the CM equation has some errors in this regime and we may use

this developed equation (transient CM) instead of the ordinary CM (pseudo steady state

CM). The detail of derivation is in Appendix 7. Equation 7-17 gives the final equation of

transient CM.

1

1

1

1

1

1 1

1

1

1

1 1

1

( 1 1)

1 1

1)

1 1(

p

p p p i

p p

p p

p

p p

i prod N

T

prod conN N N N T

prod con

prodN N

prodN N

prod piN

prodN N

q p L E

E ELE E w

E

EL E I p

E

....................................( 7-17)

where


1

2

1 1exp( 1 1 )

p pprodN N

t p

L E tcV

..........................................................................( 7-18)

Note that this L is different from L in Chapter 3. We can also convert initial pressure term

to initial production rate term (Appendix 7). We can write equation 7-17 in terms of ’s.

1

1

1

1

1

1 1

1

1

1

1 1

1

1 1

1 1

1)

1 1(

p

p p p i

p i

p p

p p

p

p p

i prod N

T

prod conN N N N

prodN N

prodN N

prodN N

prod piN

prodN N

q p L E

E Ew E w

E

EL E I p

E

L

...........................................( 7-19)

If we have a pseudo steady state regime, the E matrices approach constant values and the

only parameter which is a function of time is L. Equation 7-19 is not the same as the CM

and it does not have values. However, we can use the optimization toolbox to evaluate

the parameters of this equation for the pseudo steady state regime in homogenous and

heterogeneous reservoirs. In the transient period, the E matrices are a function of time.

Therefore, we should use the above equation to predict rate. Initial reservoir pressure and

the ratio of diffusivity constant over total compressibility times total pore volume can,

however, also be estimated. For heterogeneous systems during a transient period, we

cannot apply this equation.

7.5 Conclusions

The transient MPI is developed for tight formations with small permeability.

Connectivity parameters are a function of time in a transient regime. After some days, the

connectivity parameters approach constant values. The higher the permeability or the

smaller the reservoir size, the shorter the transient times, and this results in connectivities

reaching the constant value sooner. For CM numbers larger than one, the model says that

the transient connectivities approach constant values (pseudo steady state connectivities).

We developed an equation similar to the CM which is applicable for a transient period in

homogenous reservoirs.

CHAPTER 8 MULTIWELL COMPENSATED CM 118

CHAPTER 8 MULTIWELL COMPENSATED CM

8.1 Introduction

Field maintenance procedures, such as shut-ins and work-overs, cause production rate

changes which are not caused by injection rate fluctuations but which mislead

connectivity estimators such as CM. We have developed a method in this chapter which

is tolerant to changes caused by external factors. This method, called the Multiwell

Compensated Capacitance Model (MCCM), is based on the superposition principle. It

can analyze injection and production data when producers’ skin factors change, new

producers are added, or active producers are shut-in. The MCCM also deals with another

common problem in field data, which is when there are frequent producer shut-ins within

sampling intervals (mini-shut-ins). For example, a producer is shut-in for a few days

when flow rates are measured every month. By deriving the MCCM equations using

average rates, we have developed an efficient approach to overcome this problem.

We will show that in several synthetic cases with varying skin, long term shut-in, and

frequent mini-shut-ins, the MCCM successfully determined the true connectivity

parameters and predicted the production rates accurately.

8.2 CM and Compensated CM (CCM)

To calculate connectivity parameters, as we mentioned in Section 2.4.4, Yousef et al

(2006) coupled linear pseudo steady state productivity with material balance and applied

superposition to a system of injectors and producers. He solved the resulting differential

equation to predict total fluid production (Equation 2-5). When the number of producers

changes (a producer is shut-in or a new producer is added), the connectivity values

change and the time interval should be divided into two parts: the before- and after-event

intervals. In practice, however, dividing the data set in this manner may lead to very short

intervals that are not useful for CM analysis (small L). To avoid this problem, the

compensated CM (CCM) has been developed (Kaviani et al. 2012). In the CCM, a shut-

in producer is treated as an open producer in which all the produced fluid is re-injected

from a “virtual injector” at the same location. The new parameter added to the model is


the connectivity of the virtual injector with the other producers. After shutting-in the

producer x, we have:

0

( ) ( ) ( )

0

1

( ) ( )

n

pj

t tn

x x x

j n j ij ij n

i

q t q t e w t

..............................................................................( 8-1)

Where ( ) ( )x

j nq t is the predicted production rate of producer j when producer x is shut-in,

( )x

ij is the new interwell connectivity coefficient after shutting-in producer x and

( ) ( )x

ij nw t is the shifted injection rate of injector i with respect to producer j when producer

x is shut-in. We can calculate ( )x

ij as:

( )x

ij ij xj ix ...............................................................................................................( 8-2)

where xj is defined as the interwell connectivity coefficient between the virtual injector

at the location of producer x and producer j (Kaviani et al. 2012). If the skin factor of a

producer changes, the ’s between the well pairs will change, too. The CCM has only

been derived and tested for cases with a small number of producer shut-ins. Since we

may deal with cases where a large number of wells receive treatments at the same time,

such as during a workover campaign, we need to generalize the CCM.

8.3 Skin and the CCM

To account for skin changes, we extend a concept used in the CCM. If, instead of

shutting-in a producer, the producer’s skin changes, then we can rewrite Equation 8-2 as:

xs xs

ij ij xj ix ix ..................................................................................................( 8-3)

where xs

ij is the ij after the skin factor of producer x changes, and xs

ix is the

between injector i and producer x after the skin factor of producer x changes. Based on

this formula, the changes in the ’s after the shut-in depend on the producer x’s

connectivity and the change in the ’s of the stimulated wells. We can see that Equation

8-2 is a special form of Equation 8-3 when

0xs

ix ; i.e. producer x is shut-in.


If the skin factor of a producer changes, its production rate and all its ’s will be

multiplied by a constant number. For example, suppose that at the start of the analysis the

skin factors of all the producers are zero. This is an ideal situation where the connectivity

of well pairs is independent of the skin factor. We define the skin segment as the time

interval for each producer where the skin factor stays constant. Thus, for skin segment s

(the time interval where the skin factor of well x is constant but not zero; note that s is not

the value of skin factor), we have:

xs

ixx

ix

s

....................................................................................................................( 8-4)

where x s is the skin coefficient which is the coefficient of all ’s of producer x at skin

segments, ix is the connectivity coefficient between injector i and producer x when all

skin factors are zero, and xs

ix is the ix at this skin segment. Therefore, we can update

the ’s of the other producers when there is a skin change at producer x:

1xs

ij ij xj ix x s ...............................................................................................( 8-5)

xj controls the effect of skin change in producer x on the production rate of producer j.

If producer x is damaged (positive skin), the 1x s and it leads to an increase in the

’s of the other producers. If it has negative skin, the 1x s and it leads to a decrease

in the ’s of the other producers. When producer x is shut-in, 0x s , and Equation 8-

5 will be equivalent to the original definition of the CCM. Thus, the CCM is a specific

case of Equation 8-5, when the skin of producer x is infinite.

8.4 Multiwell CCM (MCCM)

In general, we may have a skin factor for each producer that changes with time. For this

problem, we need to express Equation 8-5 in matrix form. To begin, we define the

following matrices:

[] is the matrix of ’s between injectors and producers (this is the transpose matrix of

[ ]p iN N we used in Equation 2-9):


11 1

1

[ ]

K

I IK

Λ

[] is the matrix of ’s between the producers:

12 1

21 2

1 2

1

1[ ]

1

K

K

K K

Β

Note that [] is, in general, not symmetric. If we include all the producers of the system

when we estimate [], it stays constant over the analysis period so that, by changing the

producer conditions, adding producers, or shutting-in, we do not need to update it.

However, to calculate ’s in case some producers (x1, … xn) are shut-in ( 1 , , nx x

Λ ) we

may need to use a transformed form of [:

1 1 1

1 1

, , , , , ,n n n

n n

T Tx x x x x x

x x x x Λ Λ Λ Β Λ Β ..............................................( 8-6)

where

1

1

, , nx x

xΒ is the x1-st column of the transformed matrix of ’s when producers (x1,

… xn) are shut-in. T denotes the matrix transpose. The elements of this matrix are

calculated by:

1

1

1

, ,det

det

n

shut in shut in

i x jx x

x j shut in

shut in

Β Β

Β

..............................................................................( 8-7)

Where

shut in

shut in

Β is the matrix of ’s of the shut-in producers.

1

shut in shut in

i x j

Β Β is

made by appending two matrices:

1

shut in

i x

Β the matrix of ’s of the shut-in producers

excluding its x-th column and

shut in

j

Β the j-th column of the matrix containing only

rows of the shut-in producers of [].


If instead of changing the number of producers, the skin factors of the producers change,

for each skin segment of the system we need to define two new matrices. If some

producers (x1, … xn) have non-zero skin factors and the others (y1, … ym) do not:

1 2 1 1 1 1 1 1 1 1

2 1 2 2 2 2 1 2 2 2

1 2 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

0 0 0 1 0

0 0 0 0 1

n m

n m

n n n n n n n n

s

x x x x x x x y x x y x

x x x x x x x y x x y x

x x x x x x x y x x ym x

s s s s

s s s s

s s s s

Β

and

1 1 2 1 1 1 1 1 1 1 1

2 1 2 2 2 2 2 1 2 2 2

1 2 1

*

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

0 0 0 1 0

n m

n m

n n n n n n n n n

s

x x x x x x x x y x x y x

x x x x x x x x y x x y x

x x x x x x x x y x x ym x

s s s s s

s s s s s

s s s s s

Β

0 0 0 0 1

Here, for simplicity we showed the definition of these matrices only for the case that all

the stimulated producers are at their skin segment s1; in general, they could be at any of

the skin segments. The matrix of ’s when producers (x1, … xn) are stimulated (

1 , , n sx x

Λ ) will be:

1 1 1

1 1

, , , , , ,n n ns s s

n n

T Tx x x x x x

x x x x Λ Λ Λ Β Λ Β ............................................( 8-8)

where

1

1

, , n sx x

xΒ is the x1-th column of the transformed matrix of ’s when producers

(x1, … xn) are stimulated. The elements of this matrix are calculated by (Appendix 8):

1

* 1

*, ,

( 1) det

det

i

n s

i j

stim stimx

s si x jx x

x y stim

s stim

Β Β

Β ....................................................................( 8-9)


where stim denotes the stimulated wells and xi* is the index of well xi. For example, if it

is the first producer, the index will be 1 and if it is the second one, it will be 2. The

following example clarifies the use of these formulas. Note that, this formula is also valid

for determining 1 , , n s

i j

x x

x x , where both xi and xj are stimulated. If instead of stimulation

the wells are shut-in (x = 0), Equation 8-9 will be equivalent to Equation 8-7.

Example. Assume that we have a 5×4 system. The [] and [] will be:

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

51 52 53 54

[ ]

Λ

and

12 13 14

21 23 24

31 32 34

41 42 43

1

1[ ]

1

1

Β

If producers x1 and x2 are stimulated, for the ’s of the first injector (1j) based on

Equation 8-8, we have:

1,2 1,2 1,2

11 11 2111

1,2 1,2 1,2

12 12 12 22

11 121,2 1,2 1,213

13 13 23

1,2 1,2 1,21414 14 24

s s s

s s s

s s s

s s s

T T TT

To calculate the elements of the matrix of transformed ’s, at first we need to define [s]

and [s*]:

12 1 1 13 1 1 14 1 1

21 2 1 23 2 1 24 2 1

1 1 1 1

1 1 1 1

0 0 1 0

0 0 0 1

s

s s s

s s s

Β

and


1 1 12 1 1 13 1 1 14 1 1

21 2 1 2 1 23 2 1 24 2 1*

1 1 1 1

1 1 1 1

0 0 1 0

0 0 0 1

s

s s s s

s s s s

Β

Then using Equation 8-9 we can calculate the elements of matrix of transformed ’s. For

example,

12 1 1 1 1

1,2 1,21 1

*1 1 21 2 11,2

11 1,2

12 1 11,2

21 2 1

1 1det

( 1) det 1 1

1 1detdet

1 1

s

s si

s

s s

s

s

s

Β Β

Β

and

13 1 1

1,2 1,22 1

*2 3 21 2 1 23 2 11,2

23 1,2

12 1 11,2

21 2 1

1 1det

( 1) det 1 1

1 1detdet

1 1

s

s si

s

s

s s

s

s

Β Β

Β

To show the application of the developed method, we use a synthetic case.

Case 8.1. This is 5×4 case in a heterogeneous reservoir (Figure 8-1) with the general

reservoir properties listed in Table 8-1. All the wells have been shut-in at times and have

also been stimulated (Figure 8-2). The analysis period is 10 years (L ≈ 10). At first, to

calculate the correct ’s, we simulated the case where no well has been shut-in or

stimulated and then used the CM to calculate the ’s. To calculate the ’s, we also ran 4

separate cases where, for each of them, one of the producers has been shut-in for half of

the analysis period. Then we analyzed the data for the current case (with shut-ins and skin

changes) three times: first using the simple CM, second using the

segmented/compensated CM (as described in Kaviani et al. 2012), and then using the

MCCM as described above. Comparing the estimated ’s, the MCCM provides the most

accurate estimates of the ’s (Figure 8-3). The segmented/compensated CM also gives

good estimates. However, the simple CM results are poor. Looking at the ’s, we


observe that the MCCM estimates ’s accurately (Figure 8-4). The

segmented/compensated CM, however, gives poor estimates.

Table ‎8-1 Reservoir and simulator parameters for Case 8.1

Parameter Value

, fraction 0.18

Absolute k, md 40

Oil end point relative k, md 0.9

Water end point relative k, md 0.225

Coil, psi-1

5×10-6

Cwater, psi-1

1×10-6

Crock, psi-1

1×10-6

Irreducible oil saturation, faction 0.35

Irreducible water saturation, faction 0.2

oil, cp 0.5

water, cp 2


Grid size, ft 26.667×26.667×30

Figure ‎8-1 In Case 8.1, three barriers and one channel exist in the reservoir.

I01

I02

I03

I04 I05

P01

P02 P03

P04

k=400 md

k=0.2 md

k=0.2 md

k=0.2 md

k=40 md


Figure ‎8-2 Producers’‎conditions‎change for Case 8.1.

Figure ‎8-3 Applying the MCCM provides the most accurate ’s‎for‎Case‎8.1. Estimated ’s‎using‎the‎

segmented/compensated CM is also good. The simple CM, however, gives poor estimates.

Years 0

P01

P02

P03

P04

Production period

Stimulation

s = -2

s=-1.5

s = -2

s = -2

10

s = -2

s = -1.5

0.0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

Esti

mat

ed

Correct

Simple CM

Seg/Comp CM

MCCM


Figure ‎8-4 Applying MCCM for the Case 8.1, provides accurate estimation of ’s.‎The‎estimated‎’s‎

using the segmented/compensated CM are far from the correct ones.

8.5 Application of MCCM for Mini Shut-ins

The main assumption in the previous section is that the conditions are constant during the

sampling interval, tu-1 to tu for u = 2, 3, …, n. For example, if the data are monthly (tu - tu-

1 = 30 days), we assume that there is no shut-in within a month or, if the producer is shut-

in at a time step, it stays closed throughout that month. In practice, the wells may produce

for a fraction of a sampling period. We call these “mini-shut-ins”. If these incidents are

very few or for only a small fraction of a time step, it will have little effect on the model

performance. However, in many fields, it is common to have mini-shut-ins.

If a producer is open only a fraction, f, of a time step, its average production within that

time step will be approximately f times its production rate in the case where the well has

not been shut-in. Thus, if we have one producer in the system, to handle the mini-shut-in

problem, we can multiply the non-shut-in production rate at this time step by f. If we have

more than one producer in the system, the mini-shut-in of a producer will affect the rates

of the other producers. In this case, depending on the producers’ connectivity, the

0.2

0.3

0.4

0.5

0.6

0.7

0.2 0.3 0.4 0.5 0.6 0.7

Esti

mat

ed

Correct

Seg/Comp CM

MCCM


production rates of the other producers within that time step will increase (Figure 8-5).

Applying the MCCM, we can overcome this problem. In this case, the skin effect is

known and is equal to f. On the other hand, as discussed by Kaviani and Jensen (2010),

since the sampling interval in mini-shut-in case is smaller, the difference between the

average and instantaneous rate might be quite large. So we need to use the average rate

formula for such a case. Here, we derive the average rate formula for the CM and show

the application of MCCM for mini-shut-ins.

Figure ‎8-5 If a producer is shut-in temporarily within a sampling interval, it will lead to an increase

in the production rates of its connected producers.

The CM is originally derived to calculate the instantaneous production rate at the end of

the sampling interval. In practice, however, we have the average production within a

sampling interval. For example, the monthly data are comprised of the average

production rate within a month, and not the rate on the last day of the month. If the

diffusivity constant is large (large mobility and small compressibility), the difference

between the instantaneous and monthly data is negligible. However, at smaller diffusivity

constants or small sampling intervals (as we have in mini-shut-in cases), this difference

becomes important (Figure 8-6). For example, for the well P01 from Case 8.1, at 30 days,

the difference between the instantaneous and average rate is only 8%. However, at 10

0

200

400

600

800

1000

1 5 9 13

Liq

uid

pro

du

ctio

n r

ate

, rb

/day

Months

No mini-shut-in

Mini-shut-in

0

100

200

300

400

500

600

700

800

1 5 9 13

Liq

uid

pro

du

ctio

n r

ate

, rb

/day

Months

0

100

200

300

400

500

600

700

800

1 5 9 13

Liq

uid

pro

du

ctio

n r

ate

, rb

/day

Months

0

200

400

600

800

1000

1200

1 5 9 13

Liq

uid

pro

du

ctio

n r

ate

, rb

/day

Months

P01P03

P04

P02


days this difference is 12%. If the average reservoir permeability increases to 500 md,

these differences will be 2% and 4% for 30 and 10 days, respectively.

Figure ‎8-6 In general, the average rate is different from the instantaneous rate at the end of the time

step and, at smaller diffusivity constants, this difference is larger. The left figure is for k=40 md and

the right figure is for k=500 md.

To calculate the average rate within a sampling interval, we just need to integrate the rate

over the interested period. Assuming a constant injection rate within the sampling

interval and constant producer’s BHP, the average production rate will be (Appendix 9):

1 0 0

0 1

11 1

ˆ ( )

n n

pj pj

t t t ti I

pj j ij n ij n

j n ij ij i n

in n n n

q t w t w tq t e e w t

t t t t

....................( 8-10)

where ˆ ( )j nq t is the estimated production rate using the model at time tn. w′ij is the shifted

injection rate defined as:

1

1

m n m n

ij ij

t t t tn

ij n i m

m

w t e e w t

...............................................................................( 8-11)

To test the applicability and advantages of the suggested method, we used two simulation

cases. The first one considers a case where the producers have mini-shut-ins. The second

one is a case with both skin change and mini-shut-in.

Case 8.2. This is a homogeneous 5×4 case. All the reservoir and fluid properties are

similar to Case 8.1; however, no barrier or channel exists. The analysis interval is 81

months (L ≈ 6), where the producers have several mini shut-ins during this period (Figure

300

400

500

600

700

0 10 20 30

Liqu

id ra

te, r

b/da

y

Time, Days

300

400

500

600

700

0 10 20 30

Liqu

id ra

te, r

b/da

y

Time, Days


8-7). The mini-shut-in periods are based on the production data from a real heavy oil

waterflood case. To analyze the data, at first we applied the simple CM where no mini-

shut-ins are considered in the model. As expected, the estimated connectivity parameters

were not accurate (Figure 8-8) and the average prediction R2 is 0.75 (Figure 8-9). Another

approach to analyze this data using the CM is to exclude the time intervals with mini-

shut-ins. In other words, at the time steps we have a mini-shut-in, and we exclude the

production rate of the mini-shut-in well from the estimated error. In this manner, the

effect of production rate reductions for those producers will be removed; however, it is

not able to modify the production rate of the other producers at this time step. Applying

this approach, the average prediction R2 increases (Figure 8-9) and the estimated ’s will

be slightly more accurate than the previous approach (Figure 8-8). Applying the MCCM

as described above improves the results considerably and we get an accurate estimation

of both ’s and production rate (Figures 8-8 and 8-9).

Figure ‎8-7 Number of shut-in days within sampling intervals for Case 8.2.

0

5

10

15

20

25

30

0 20 40 60 80

Shu

t-in

day

s

Months

P01

P02

P03

P04


Figure ‎8-8 Applying the MCCM provides accurate estimates of ’s‎for Case 8.2.

Figure ‎8-9 Applying the MCCM, the estimated production rate is much more accurate than the other

estimators for Case 8.2.

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

fo

r m

ini-

shu

t-in

cas

e

True

Simple CM

Excluding mini-shut-ins

MCCM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P01 P02 P03 P04

R2

Producer

Simple CM

Excluding mini-shut-ins

MCCM


Note that until now, in all published applications of the CM to field data, either the

simple or compensated/segmented CM has been used. The MCCM can also estimate the

’s. Comparing the estimated ’s for Case 8.2 with the true ones, we observe that the

estimation is relatively poor (Figure 8-10). Note that true ’s can be obtained from

Equation 2-9, assuming the shut-in well or stimulated well is a new injector. Therefore

calculated ’s of that well relative to the other producers would be ’s. Since the number

of data used to estimated ’s are limited to only 34 points (number of mini-shut-ins), and

we have 12 parameters, the number of data is too small to give a robust estimate of the

parameters (L ≈ 3).

Figure ‎8-10 The estimated for the Case 8.2 are relatively inaccurate.

Case 8.3. This case is similar to Case 8.2. However, each of the producers has been

stimulated one time during the analysis period. By applying the MCCM, we could

estimate ’s accurately (Figure 8-11) and the average R2 of the predicted production rate

was 0.998.

0.2

0.3

0.4

0.2 0.3 0.4

fo

r m

ini-

shu

t-in

cas

e

True


Figure ‎8-11 In Case 8.3, the estimated ’s‎using‎the‎MCCM are very close to the correct values.

8.6 Conclusions

By applying the MCCM, we can remove the effect of changes in producers’ conditions

on the results. By adapting the CM for changing skin, we have generalized previous

work to include both well shut-ins (infinite skin) and well treatments for any number of

wells. The MCCM is also able to assess connectivities for both injector-producer and

producer-producer interactions successfully. In the case of mini-shut-ins, by combining

the MCCM and average rate, we can estimate the connectivity parameters accurately.

The work in this chapter represents a collective combined effort with Danial Kaviani. We

derived the equations together. He obtained the MCCM results for the mentioned

synthetic field cases and compared them with the simple CM and CCM results.

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

fo

r C

ase

3

True

CHAPTER 9 FIELD APPLICATION 134

CHAPTER 9 FIELD APPLICATION

9.1 Introduction

In this chapter, we apply the MCCM to a heavy oil (median to heavy oil) field (Marsden)

and a conventional field (Storthoaks).

The Marsden field is located in Saskatchewan and produces from the Sparky Formation.

We considered a time period of 386 months (from June 1979 to July 2011). In this period,

40 injectors and 178 producers are drilled. Some of the producers are actively producing

while some of them are suspended or abandoned. We compare our results with a sand

body map, net pay map, and dye test results of some wells in the southeast part of this

field.

Storthoaks is located in the south-east of Saskatchewan and produces from the north-east

side of the Williston basin. Eight injectors and 15 producers are drilled in this pool. We

use flow rates for 165 months (from January 1998 to September 2011), and compare our

results with a seismic impedance-amplitude map, kh (permeability-thickness) map and

measured production rates.

9.2 Marsden South Field

9.2.1 Field Description

Husky Energy provided data for the Marsden South field to test our methods in this

waterflood case. The data consisted of rates, geological maps (net pay, top pay, and sand

body maps), and dye test results for wells in the southeast part of the field. The Marsden

field is located in Saskatchewan and produces from the Sparky Formation. We infer from

the maps and literature (Morshedian et al., 2012) that the sand bodies are fluvial (point

bars) and the spaces between the sand bodies may be abandoned channels or other, lower

quality deposits. The field properties are summarized in Table 9-1. The average water cut

is 79% so we consider it as a mature waterflood. Well deviations are small. We also used

Accumap and prepared a combined map of sand body and well locations (Figure 9-1).


Table ‎9-1 Marsden south field properties

Parameter Value

Average , fraction 0.28

Median k, md 770

Average water cut, % 79

Average oil rate, bbl/day 9.87

Average GOR, scf/bbl 281

Average produced liquid, bbl/day 156

Average water injection, bbl/day 149

Averageoil in 50 F, cp 1946



Average formation thickness, ft 12

If we assume a total compressibility of 5×10-6

psi-1

and 8 producers per township, we

would have a CM number C ~ 30. If L > 4, the median CV’s of and are 0 and about

0.5 respectively (Figure 3-17). The AAD for is zero (Figure 3-18). Therefore, we

expect a small uncertainty in the connectivity values especially ’s.


Figure ‎9-1 Overlain maps of sand bodies and well locations; red triangles indicate injectors and black

circles represent producers. The names are not actual names of the wells; I = injector, P = producer,

S = suspended and A = abandoned at the time.

9.2.2 Applications of analytical connectivity values

We used analytical connectivity values, assuming a homogeneous reservoir, to determine

constraints for the field-derived connectivity values (Figure 9-2). The results show that

the values are in the range of 0.2 and less. A further application of the analytical values is

to determine a cut off for choosing a window size, which is explained further in the

following section. We also used the analytical values to calculate the ′ values.

Analytical between 0 to 0.2 suggests that ′ should be between -0.2 to 1.


Figure ‎9-2 values calculated from analytical model for equivalent homogenous system; values are

between 0 and 0.2.

9.2.3 Window selection to apply the model

There are 40 injectors and 178 producers over the entire 386 month-history, from June

1979 to July 2011. Applying the MCCM over the entire field is impractical due to the

CPU time burden. The idea of the window is to choose spatial domains - centered on

each production well - beyond which we can assume negligible interaction with other

wells in the field. To do so, we calculated the and (producer-producer interaction)

values from the analytical model to define a domain size for each parameter. We plotted

and with respect to well distance and chose a window radius of 3000 ft for

(producer-injector interactions) and a radius of 2000 ft for (producer-producer

interactions) (Figure 9-3).

=0.3


There is an element of compromise during the selection of the window shape and radius.

The larger the radius, the more likely the CM analysis can detect distant well-well

interactions caused by geobodies, fractures, faults, and other features that have a

preferential orientation. When this information is available, the window shape can be

changed from circular to elliptical to reduce the number of wells in each window. The

maps (e.g., Figure 9-1) do not suggest preferential sand body orientations in this case, so

that a circular shape is acceptable. Given the well density (Figure 9-1), the window size

of approximately 1 mile in diameter should adequately capture the influence of

heterogeneities within and across the sand bodies on the well-well interactions.

Figure ‎9-3 Analytical and versus well distance to determine window size; we selected a cut off of

0.05 for at a 3000 ft distance and a cut off of 0.15 for at a 2000 ft distance.

9.2.4 Selecting the number of producers in each window

When we chose a window location, centered on a producer well, the window will include

both producers and injectors. However, adding any producer could result in the need to

include more surrounding injectors, thus defeating the aim of the window. To avoid this,

we chose only one producer (target producer) for minimizing the objective function and

retained only a few other nearby producers to allow for producer-producer interactions.

9.2.5 Including production hours

Each recorded rate is an average over 30 days. In practice, however, on some days a well

may be shut-in during the month. This can cause inaccurate results when we apply the

simple CM because these months of reduced production are not caused by injection rate


changes. The MCCM has the option to input the production hours and overcome this

problem. Thus, the MCCM was needed to analyze the Marsden South field data as there

were numerous “mini-shut-ins” occurring (Figure 9-4). In applying MCCM, we observed

the R2 of the predicted rates was improved up to 35%, compared to simple CM for

different wells.

Figure ‎9-4 Comparing actual production rate with the model predicted rate ignoring production

hours (left) and including production hours (right).

9.2.6 Using bootstrap technique

We applied the bootstrap technique with about 20 resampling for this field (see Section

3.9).

9.2.7 Comparing ′‎values‎to‎the‎sand‎body‎map

Three locations were selected for detailed connectivity analysis. When we apply the

model to this reservoir, high oil viscosity and non-unit mobility ratio affect the

connectivity values. Once the front arrives, however, the connectivity values become

stable. We applied the model at the end of the waterflooding period where the

connectivity values are stable with small variability. As we mentioned in Section 8.5, is

obtained by Equation 9-1, where the homogeneous calculation is similar to the

calculation, except the target producer is assumed as an injector and its weight factors are

calculated with respect to the other producers. Analytical between 0 to 0.35 suggests

that ′ should be between -0.35 to 1.

0 20 40 60 80 100 1200

500

1000

1500

2000

2500

3000

3500

4000

time (month)

Pro

du

ctio

n (

rb/d

ay)

Predicted Production

True Production

500 1000 1500 2000 2500 3000 3500

500

1000

1500

2000

2500

3000

3500

Predicted Production (rb/day)

Tru

e P

rod

ucti

on

(rb

/day)

0 20 40 60 80 100 1200

500

1000

1500

2000

2500

3000

3500

4000

time (month)

Pro

du

ctio

n (

rb/d

ay)


True Production

500 1000 1500 2000 2500 3000 3500

500

1000

1500

2000

2500

3000

3500


Tru

e P

rod

ucti

on

(rb

/day)


homoptimized ogeneous .....................................................................................................( 9-1)

In the first location (Figure 9-5), the target producer (P35) is located on the boundary of a

sand body. The MCCM connectivity values, |′| < 0.1, suggest only a moderate amount

of heterogeneity introduced by the region between the sand bodies. The field net pay map

(Figure 9-8) denotes this inter-sand region as an area of partial erosion. In the second

location (Figure 9-6), we chose a producer (P50) within a sand body and near to a shale

channel. Within the sand body, ′ values are in the range of 0.4, showing the producer is

located in a well-connected area. Connectivity from I35 and I37, across the shale channel,

is smaller than connectivities with injectors I32, I33, and I34, and producers P49 and S44,

which are within the sandbody. Connectivity values for P52 are small in comparison to

P50 (Figure 9-7). This may suggest that P52 might not belong to the indicated sandbody

or that a wormhole is developing around P50.

We also compared our connectivity results with the net pay map (Figure 9-8). The field

net pay map indicates that the region around well P35 has a good reservoir in the

surrounding area. This corresponds with the CM results, which suggest good

connectivity and only a weak anisotropy NS versus EW. Well P50 has better pay to the

south-east and poorer pay to the north and west, which is also in correspondence well

with the CM connectivity results. Well P52 is in an area of decreasing net pay, which also

agrees with the CM results (Figure 9-8).

In general, we see similarities between the sand body locations and connectivities. A

detailed assessment, however, is hindered by our lack of understanding about how the

sand body boundaries were defined and the nature of the regions between the sand

bodies.


Figure ‎9-5 P35 (blue arrow) is located between two sand bodies. The connectivity values are only

slightly different from what would be obtained for a homogeneous reservoir. The distance between

grid lines is one mile (5280 ft). P signifies a producer, I represents an injector, and S signifies a well

currently shut-in.

I01I02

I03 I04

I05

I06I07

I08

I09I10

I11 I12

I13I14

I15I16

I17

I18

I19

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I25I26

I27

I28

I29I30

I31

I32

I33I34

I35 I36

I37I38

I39 I40

I41I42

P01P02

P03

P04P05

P06

P07

P08 P09

P10

P11

P12P13

P14

P15

P16

P17P18

P19

P20

P21

P22 P23

P24

P25

P26

P27

P28

P29

P30

P31P32

P33P34

P35

P36

P37

P38P39

P40

P41

P42

P43

P44P45P46

P47

P48

P49

P50P51

P52 P53

P54

P55

P56

P57

P58

P59

P60

P61

P62

P63

P64

P65

P66 P67

P68P69P70P71

P72 P73 P74 P75

P76

P77

P78

P79

A01A02

A03 A04

A05

A06

A07A08

A09 A10

A11

A12

A13

A14

A15

A16

A17

A18

A19

A20A21

A22

A23

A24

A25

A26

A27A28

A29

A30A31

A32A33

A34

A35

A36

A37

A38

A39

A40

A41

S01

S02

S03

S04

S05 S06

S07

S08 S09 S10

S11

S12

S13

S14

S15

S16

S17

S18

S19 S20

S21

S22

S23

S24S25

S26

S27

S28

S29

S30

S31

S32

S33S34

S35

S36S37

S38

S39

S40

S41

S42

S43S44 S45 S46

S47

S48 S49 S50

S51

S52 S53

S54S55

S56

S57

S58

S59

S60

S61S62

S63

Sand bodies


Figure ‎9-6 P50 (blue arrow) is located within a sand body. The connectivity values are large in

absolute value and it could be a sign of wormhole development.

Figure ‎9-7 P52 (blue arrow) is mapped as being within a sand body. The connectivity values are

small in absolute value.

I01I02

I03 I04

I05

I06I07

I08

I09I10

I11 I12

I13I14

I15I16

I17

I18

I19

I20I21

I22 I23 I24

I25I26

I27

I28

I29I30

I31

I32

I33I34

I35 I36

I37I38

I39 I40

I41I42

P01P02

P03

P04P05

P06

P07

P08 P09

P10

P11

P12P13

P14

P15

P16

P17P18

P19

P20

P21

P22 P23

P24

P25

P26

P27

P28

P29

P30

P31P32

P33P34

P35

P36

P37

P38P39

P40

P41

P42

P43

P44P45P46

P47

P48

P49

P50P51

P52 P53

P54

P55

P56

P57

P58

P59

P60

P61

P62

P63

P64

P65

P66 P67

P68P69P70P71

P72 P73 P74 P75

P76

P77

P78

P79

A01A02

A03 A04

A05

A06

A07A08

A09 A10

A11

A12

A13

A14

A15

A16

A17

A18

A19

A20A21

A22

A23

A24

A25

A26

A27A28

A29

A30A31

A32A33

A34

A35

A36

A37

A38

A39

A40

A41

S01

S02

S03

S04

S05 S06

S07

S08 S09 S10

S11

S12

S13

S14

S15

S16

S17

S18

S19 S20

S21

S22

S23

S24S25

S26

S27

S28

S29

S30

S31

S32

S33S34

S35

S36S37

S38

S39

S40

S41

S42

S43S44 S45 S46

S47

S48 S49 S50

S51

S52 S53

S54S55

S56

S57

S58

S59

S60

S61S62

S63

Sand bodies

I01I02

I03 I04

I05

I06I07

I08

I09I10

I11 I12

I13I14

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I17

I18

I19

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I25I26

I27

I28

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I31

I32

I33I34

I35 I36

I37I38

I39 I40

I41I42

P01P02

P03

P04P05

P06

P07

P08 P09

P10

P11

P12P13

P14

P15

P16

P17P18

P19

P20

P21

P22 P23

P24

P25

P26

P27

P28

P29

P30

P31P32

P33P34

P35

P36

P37

P38P39

P40

P41

P42

P43

P44P45P46

P47

P48

P49

P50P51

P52 P53

P54

P55

P56

P57

P58

P59

P60

P61

P62

P63

P64

P65

P66 P67

P68P69P70P71

P72 P73 P74 P75

P76

P77

P78

P79

A01A02

A03 A04

A05

A06

A07A08

A09 A10

A11

A12

A13

A14

A15

A16

A17

A18

A19

A20A21

A22

A23

A24

A25

A26

A27A28

A29

A30A31

A32A33

A34

A35

A36

A37

A38

A39

A40

A41

S01

S02

S03

S04

S05 S06

S07

S08 S09 S10

S11

S12

S13

S14

S15

S16

S17

S18

S19 S20

S21

S22

S23

S24S25

S26

S27

S28

S29

S30

S31

S32

S33S34

S35

S36S37

S38

S39

S40

S41

S42

S43S44 S45 S46

S47

S48 S49 S50

S51

S52 S53

S54S55

S56

S57

S58

S59

S60

S61S62

S63

Sand bodies


Figure ‎9-8 Comparison of connectivity results and net pay map, Note the change of scales for ′ for

the P35 map (left) and P50/P52 map (right).

9.2.8 Comparing median of ′‎to‎the‎sand‎body‎map‎

Due to the dominant near well connectivity effect on the CM ’s, discussed in Chapter 5,

we evaluated the median of ′ for each producer to obtain a broad view of connectivity

behavior in the Marsden South field. Because the directionality and boundary effect still

exist within these median values, the interquartile range (IQR) of the ′’s for each

producer is also calculated to indicate how variable the ′ values are for each producer.

For example, a location with positive ′ and a small IQR would indicate a producer with

good, omnidirectional connectivity. A producer with a negative ′ and large IQR suggests

a well with generally weak connectivity, but some directions have much better

connectivities than others.

Our results show that negative values are around the boundary between northern sand

bodies and center sand bodies (Figure 9-9), as one would expect if those bodies were

separated by significant shales. Moreover, the northwest sand body has quite low

connectivity, while the east sand body has good connectivity, but the southeast boundary

of the reservoir has low connectivity values. P50 has a very high median value (0.120) in

comparison to the other adjacent wells such as P52 suggesting a wormhole development

around this well. Husky Energy recorded a small amount of sand production (in the

I01I02

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I05

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P03

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P07

P08 P09

P10

P11

P12P13

P14

P15

P16

P17P18

P19

P20

P21

P22 P23

P24

P25

P26

P27

P28

P29

P30

P31P32

P33P34

P35

P36

P37

P38P39

P40

P41

P42

P43

P44P45P46

P47

P48

P49

P50P51

P52 P53

P54

P55

P56

P57

P58

P59

P60

P61

P62

P63

P64

P65

P66 P67

P68P69P70P71

P72 P73 P74 P75

P76

P77

P78

P79

A01A02

A03 A04

A05

A06

A07A08

A09 A10

A11

A12

A13

A14

A15

A16

A17

A18

A19

A20A21

A22

A23

A24

A25

A26

A27A28

A29

A30A31

A32A33

A34

A35

A36

A37

A38

A39

A40

A41

S01

S02

S03

S04

S05 S06

S07

S08 S09 S10

S11

S12

S13

S14

S15

S16

S17

S18

S19 S20

S21

S22

S23

S24S25

S26

S27

S28

S29

S30

S31

S32

S33S34

S35

S36S37

S38

S39

S40

S41

S42

S43S44 S45 S46

S47

S48 S49 S50

S51

S52 S53

S54S55

S56

S57

S58

S59

S60

S61S62

S63

Sand bodies

I01I02

I03 I04

I05

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I08

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I13I14

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I17

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I19

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I22 I23 I24

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I28

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I32

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I37I38

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I41I42

P01P02

P03

P04P05

P06

P07

P08 P09

P10

P11

P12P13

P14

P15

P16

P17P18

P19

P20

P21

P22 P23

P24

P25

P26

P27

P28

P29

P30

P31P32

P33P34

P35

P36

P37

P38P39

P40

P41

P42

P43

P44P45P46

P47

P48

P49

P50P51

P52 P53

P54

P55

P56

P57

P58

P59

P60

P61

P62

P63

P64

P65

P66 P67

P68P69P70P71

P72 P73 P74 P75

P76

P77

P78

P79

A01A02

A03 A04

A05

A06

A07A08

A09 A10

A11

A12

A13

A14

A15

A16

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A18

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A20A21

A22

A23

A24

A25

A26

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A29

A30A31

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A34

A35

A36

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A41

S01

S02

S03

S04

S05 S06

S07

S08 S09 S10

S11

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S13

S14

S15

S16

S17

S18

S19 S20

S21

S22

S23

S24S25

S26

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S30

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S32

S33S34

S35

S36S37

S38

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S43S44 S45 S46

S47

S48 S49 S50

S51

S52 S53

S54S55

S56

S57

S58

S59

S60

S61S62

S63

Sand bodies


bottom of the graduated cylinder) which can be due to wormhole development. The IQR

values are high around the borders due to the sand body boundary effect.

Figure ‎9-9 Median and interquartile range (IQR) of ′‎values‎for several producers.

9.2.9 Analysis of dye test results

We were provided with dye test results for several wells in the southeast sand body,

where connectivity generally appears to be good. They injected dye in some injectors and

recorded the arrival times in a few producers. They mentioned that the injection rates

were fairly steady during the dye test and they sampled the production wells every 15

minutes for the first hour, every half an hour until 4 hours had elapsed, and then every

hour after. Therefore, they recorded more than one arrival time in Figure 9-10. These

arrival times are of the order of a few hours. In some wells, however, they did not detect

the arrival of any dye.

In an injector-producer well system the analytical travel time for the dye is calculated

from Equation 9-2.

p

inj

VTravel time

W .............................................................................................................( 9-2)

-0.0190.088

0.0350.230

-0.0310.014

0.0510.099

0.0660.095

0.0710.053

-0.0050.111

0.1200.194

-0.0200.072

0.0190.084

-0.0090.148

0.0290.095

0.0490.089

-0.0020.069

0.0560.099

-0.0140.185

-0.0190.034

MedianIQR

Blue=+Red=-


where Winj is the rate of water injected and Vp is the volume between injector and

producer swept by the injected water. In principle, the travel time is calculated as Vp

approaches zero as a streamline in a homogeneous system. We calculated travel time

between well pairs using Equation 9-2 and compared them with actual travel time

measured in the field (Figure 9-11). Our results show that the actual travel time values are

smaller than analytical values. Therefore, assuming the reported rates values are correct,

the connectivities between well pairs are larger than the homogenous case. Moreover, I10

relative connectivity to P03 and P07 is larger than I10 to P29.

Finally we compared both and values with dye-test travel times. Our results show that

is weakly correlated with travel time and is strongly correlated to travel time (Figure

9-12). The negative correlation of with travel time is as expected, since better

connectivity (larger ) is associated with larger fluid velocities, although we would

expect to see a stronger correlation. The positive correlation of with travel time is

consistent with our expectations, but the timescales of the two measurements are quite

different. These differences may be caused by wormholes and we would like to obtain

sand production amounts from the operator to test this concept.


Figure ‎9-10 Dye test arrival time for some wells in the southeast sand body; injection started at 9 am.

In some wells they did not detect any dye.


Figure ‎9-11 Comparison of first arrival time calculated from the analytical model and actual first

arrival time of the dye; ellipses identify times from a common injector.

Figure ‎9-12 Correlations of and with dye travel time.

i06

i07

i06

i10

i07

i10

i06

i10

0

2

4

6

8

10

12

0 500 1000 1500

dye

tra

vel t

ime

, hrs

, day

p04

p07

p03

p29

i06

i07

i06

i10

i07

i10i06

i10

0.001

0.01

0.1

1

1 10 100

dye travel time, hrs

p04

p07

p03

p29


9.3 Storthoaks Field

9.3.1 Field Description

Storthoaks is located in the south east of Saskatchewan and produces from the north east

side of the Williston basin. It was discovered in 1962; a waterflooding project began in

1996. The field properties are summarized in Table 9-2. The reservoir was undersaturated

initially. A 3D seismic survey shows no evidence of faults or other tectonic features that

might affect connectivity (Kaviani et al. 2012).

Table ‎9-2 Storthoaks field properties

Parameter Value

Average , fraction 0.16

Median k, md 7

Initial reservoir temperature, °C 40

Average oil rate, bbl/day 309

Average water cut, % 60

Average GOR, scf/bbl 772

Average produced liquid, bbl/day 708

Average water injection, bbl/day 628

Averageoil at 1209 psi (bubble point pressure), cp 1.5

Oil gravity, °API 37

Oil formation volume factor, rbbl/stb 1.19

Solution gas oil ratio, mcf/bbl 380

If we assume a total compressibility of 5×10-6

psi-1

and 10 producers, we would have a

CM number C ~ 0.15. Choosing 165 months (L = 11), the median CV’s of and would

be 0.3 and 0.2 respectively (Figure 3-17) and the AAD of would be 0.06 (Figure 3-18).

Therefore, we expect a small uncertainty in the connectivity values.

9.3.2 MCCM Results

We applied the MCCM to the Storthoaks data and compared the connectivity results with

a seismic impedance-amplitude and kh (permeability-thickness) maps to determine the

degree of correlation between the types of information. We used flow rates for 165

months (from January 1998 to September 2011). Eight injectors and 15 producers are

drilled in the pool but we used 7 injectors and 10 producers. In other words, we ignore

very low rate producers and those wells which are not active during the selected time.

The number of wells is small, so there is no need to select a window. We used the


MCCM with 20 bootstrap resampling. Analytical values between 0 to 0.31 suggest that

′ should be between -0.31 to 1. Analytical between 0 to 0.37 also suggests that ′

should be between -0.37 to 1.

The low impedance is equivalent to high porosity and high impedance corresponds to low

porosity values (Figure 9-13). Figure 9-13 shows that 5 wells located in the SW of the

considered area have high positive ′ values. These wells are within or close to the high

porosity zone. Wells in the center and NE part of the area are mostly located in a low

porosity zone and have low connectivity values. Figure 9-14 depicts the positive

correlation of kh values and connectivity values. In other words, wells with high positive

′ values are in kh values > 300. Comparing the impedance-amplitude map with the kh

map, they are globally correlated. It means that the SW part has a high value of kh and a

low value of impedance. The center of the field has a low value of kh and a high value of

impedance. On the other hand, they are not locally correlated. For example, the

impedance values in the SW part are correlated in the SW-NE direction, but the kh values

in the SW part are correlated in the W-E direction. The connectivity values direction in

the SW part are a better match with the impedance map rather than the kh map.

Figure ‎9-13 Seismic impedance-amplitude map and connectivity results for the Storthoaks field; the

yellow color signifies low impedance and the pink color represents high impedance.

'5_16'

'10_16'

'16_16'

'2_17'

'3_21''4_22'

'6_22'

'4_16'

'11_16'

'12_16'

'14_16'

'191_14_16'

'1_17''8_17'

'9_17'

'2_21'

'3_22'

´(´)=1

´(´)=-1

800

1800

2800


Figure ‎9-14 kh map and connectivity results for the Storthoaks field; the red color signifies high kh

and the blue color represents low kh.

We also calculated the median of ′’s using both the CM and the reverse CM and

compared them with the impedance and kh map (Figures 9-15 and 9-16). The median

values larger than -0.08 in the SW part are in agreement with both maps. The median

values less than -0.08 that are located in the center part are in agreement with the

impedance map and median = -0.023 located in the NW part is consistent with the kh

map. The IQR values equal 0.622 and 0.203 at the corner of the SW part which is very

high relative to the other IQR values, which can illustrate that there should be a large

change in impedance or the kh value.

'5_16'

'10_16'

'16_16'

'2_17'

'3_21''4_22'

'6_22'

'4_16'

'11_16'

'12_16'

'14_16'

'191_14_16'

'1_17''8_17'

'9_17'

'2_21'

'3_22'

´(´)=1

´(´)=-1

100

300

500


Figure ‎9-15 Median of ′’s‎(′’s)‎and impedance map for the Storthoaks field.

Figure ‎9-16 Median of ′’s (′’s) and kh map for the Storthoaks field.

MedianIQR

Blue=+Red=-

-0.0620.121

-0.0750.015

-0.0780.101

-0.0500.150

-0.0870.117

-0.0570.081

-0.0650.051

-0.1040.140

-0.0230.058

0.0000.203

-0.0680.073

0.2080.622

-0.0790.509

-0.1370.054

-0.1110.110

-0.1700.069

-0.0540.360

MedianIQR

Blue=+Red=-

-0.0620.121

-0.0750.015

-0.0780.101

-0.0500.150

-0.0870.117

-0.0570.081

-0.0650.051

-0.1040.140

-0.0230.058

0.0000.203

-0.0680.073

0.2080.622

-0.0790.509

-0.1370.054

-0.1110.110

-0.1700.069

-0.0540.360

800

1800

2800

100

300

500


We also applied the MCCM to predict liquid production rate (Figure 9-17). MCCM

predicts total production rate. Assuming there is no free gas in the reservoir, the liquid

rate equals the total production rate. The model can predict both high production rates

(for well 8-17 in Figure 9-17 with an R2 of 0.84) and low production rates (for well 9-17

in Figure 9-18 left with an R2 of 0.96). The model is also capable of predicting small

fluctuations and shut-in periods (for well 11-16 in Figure 9-18 right with an R2 of 0.98).

Figure ‎9-17 MCCM predicts the total rate and catches the small fluctuations; 8-17 has a high average

rate).

Figure ‎9-18 MCCM can predict the total rate of producers with a low rate (left) and those which are

shut in during the analysis period (right).

0 20 40 60 80 100 120 140 160 1800

100

200

300

400

500

600

time (month)

Pro

duct

ion

(rb/

day)


True Production

0 100 200 300 400 500 6000

100

200

300

400

500

600


Tru

e P

rodu

ctio

n (r

b/da

y)

Well 8-17


9.4 Conclusions

The MCCM was applied to the Marsden South field and the results were compared

favorably with geological maps and dye tests.

a. A windowing technique worked successfully to reduce the model computational

burden where there is a large number of wells.

b. MCCM connectivities, both injector-producer and producer-producer, were consistent

with the net pay and sandbody maps.

c. Since near well heterogeneity has a large effect on connectivity, calculating the median

and interquartile range of ′ is a robust method to evaluate a broad view of connectivity

and to assess wormhole potential.

d. Dye test travel times correlated well with values and weakly with values.

The MCCM was also applied to the Storthoaks field and the results compared favorably

with geological maps and measured rates.

a. Medians of ′’s are larger in the SW part, which are in agreement with both the

impedance and kh maps. The medians of ′’s, however, are smaller at the center of this

field which are in agreement with the impedance map. The median of ′ at the NW part is

consistent with the kh map.

b. The directions of the connectivity values are correlated with the impedance map rather

than the kh map.

b. MCCM can predicts rate fluctuations and shut-ins with an R2 higher than 0.84.

CHAPTER 10 CONCLUSIONS AND RECOMMENDATIONS 154

CHAPTER 10 CONCLUSIONS AND RECOMMENDATIONS

10.1 Conclusions

This work has provided a number of useful results towards the evaluation of light and

heavy oil reservoirs and tight formations. The results are summarized below:

1. Both CM parameters and are affected by a number of factors, including fluid

and reservoir properties (diffusivity constant), sampling time, reservoir area,

number of measurements, and measurement noise. Several of these factors can be

aggregated into two dimensionless numbers, the CM number, C, and the ratio of

the number of measurements to the number of model unknowns, L. Within a

specific range of C and L, the CM results are accurate and repeatable.

2. The bootstrap is a useful tool for analyzing CM performance, especially when

there is a lack of information about reservoir properties and uncertainty in the

measurements.

3. Maps of C and L values from eleven literature reports where the CM was used

suggest that about half of the cases gave conditions where estimates have small

variabilities. Several cases were limited by too few data. The estimates are

likely to be more variable. Unconventional reservoirs will be challenging for CM

analysis.

4. As expected, a horizontal well increases the values of ’s associated with it and

decreases the ’s of vertical wells in a reservoir. The ’s of the horizontal wells

increase as the length of that horizontal well increases and the ’s for other pairs

decrease. The trajectory of the horizontal well does not have a major effect on

the ’s. Two methods are applied for heterogeneity investigation when there is a

horizontal producer in the system.

5. Near producer heterogeneity has a large effect on CM connectivity parameters.

By applying the skin factor formula and calculating the adjusted ′, we can have

better estimates of interwell connectivity. However, the CM results are less


sensitive to the interwell connectivity than near well connectivity. A better

solution is to apply the CM to assess near well heterogeneity. Two methods have

been developed to evaluate near well heterogeneity. There is a linear relationship

between the reciprocal of ’s of a producer and its wellbore skin.

6. We have applied the CM in several synthetic reservoirs with non-unit mobility

ratio and with both vertical and horizontal well(s). The non-unit mobility ratio

affects the ’s especially when the system includes a horizontal well. As a

conclusion at large mobility contrasts, analyzing the data after some PV of

injection leads to stable CM results.

7. Using the CM we can detect the existence of a wormhole, although its direction is

hard to be recognized. Using generated type curves we can evaluate the

wormhole’s equivalent skin and the rate of wormhole growth.

8. Transient MPI is developed for tight formations. Connectivity parameters are a

function of time in the transient regime. After some time, the connectivity

parameters approach constant values. The higher the permeability or the smaller

the reservoir size, the shorter the transient time, resulting in connectivities

reaching the constant value sooner. We developed an equation similar to the CM

which is applicable for the transient period in homogenous reservoirs.

9. Applying the MCCM, we can remove the effect of changes in producers’

conditions on the results. By adapting the CM for changing skin, we have

generalized previous work to include both well shut-ins (infinite skin) and well

treatments for any number of wells. The MCCM is also able to assess

connectivities for both injector-producer and producer-producer interactions

successfully. In the case of mini-shut-ins, by combining the MCCM and average

rate, we can estimate the connectivity parameters accurately.

10. The MCCM was applied to the Marsden South and Storthoaks field and the

results compared favorably with the available data.


10.2 Recommendations

Further work would build on the results obtained here and aid the development of an

important evaluation tool. Topics which would benefit from further work include:

1. Sensitivity analysis on injection rate fluctuation; very small fluctuations in

injection rate can not affect the productivity of the producers and connectivity

values especially when the data is noisy. On the other hands, very large

fluctuations may also cause the poor performance of the CM in identifying well to

well interactions. Yousef (2006) briefly mentioned the collinearity between the

injection rates as one source of error in the results. A comprehensive error

assessment should be carried out on values and trend of injection rates.

2. Sensitivity analysis on the CM results when we have horizontal well(s); a

horizontal well may increase the range of stability of the connectivity results since

it decreases the time of transient regime. Therefore, depending on the horizontal

well length, we expect that the range of stable ’s and ’s shifts to lower CM

numbers.

3. Finding a relationship between permeability and the ’s; permeability is a tensor

but the ’s are relative values in direction toward well pairs. Since the ’s are a

function of producers’ productivity and knowing that there is a linear relationship

between producer’s skin and the reciprocal of the ’s (Section 5.5), we may find a

relationship or correlation between permeability and the ’s.

4. Finding a correlation between the error of the CM results with respect to the

viscosity; in Section 6.2 we discussed 2 mobility ratios (10 and 1000) and

concluded that at the mobility ratio of 1000 the ’s are stabilized faster but the

stabilized ’s of mobility ratio of 10 are more closer to the ideal ’s (from the

unit-mobility ratio). Therefore, by increasing the mobility ratio, we expect similar

results. Knowing the trend of error with respect to viscosity, for any heavy oil

waterflood we can have some idea about the stability of the CM results.

5. Continuing wormhole assessment using CM-MPI; the ’s are changing during

wormholes’ growth and the CM gives us an average value. In Section 6.3.3 we


chose L=4 as an optimum interval to evaluate wormhole length. Decreasing the L

value (especially less than 4) increases the variability of the results (Figures 3-17

& 3-18), but the average of each parameter over that time interval is closer to the

exact value when L is very small. Therefore, for determining an optimum time-

slice length, a sensitivity analysis should be carried out.

6. Identifying wormhole configuration (single event vs. multi-fingered geometry);

using the values as well as the can help since the ’s involve volume of the

flow path. Also, time-lapse analysis can help to identify the wormhole

configuration since there is an interesting correlation between time-lapse seismic

changes and heavy oil production (Lines et al. 2008).

7. Developing type curves for a reservoir with a wormhole growing in more than

one well using the CM-MPI; in this case the trends are not a straight line

(Sections 6.3.2 & 6.3.3).

8. Applying the transient CM to more synthetic fields and real fields; in

heterogeneous reservoirs using the transient CM is an issue since the parameters

are a function of time. Exponential trend of the ’s versus time can be a

reasonable approximation which should be tested.

9. Obtaining more information about Marsden South to strengthen the comparisons

with CM evaluations; for example, obtaining sand production indications for

specific wells and evaluating equivalent skins to compare with CM-predicted

wormholes.

158

NOMENCLATURE

The Variables

AAD = average absolute difference

a[] = influence function, dimensionless

B = modified objective function for bootstrap

C = the CM number, dimensionless

ct = total compressibility, Lt2/m

gm = weight for time step m

h = formation thickness, L

I = total number of injection wells

J = single well productivity index, L4t/m

k = permeability, m2

K = total number of production wells

L = number of samples per model parameter

m = time step number in the CM

n = number of samples (time steps)

pwf = BHP of the producer, m/(Lt2)

p'wf = shifted BHP of the producer, m/(Lt2)

p = average reservoir pressure, m/(Lt2)

ip = initial average reservoir pressure, m/(Lt2)

q = total fluid production rate, L3/t

q = total estimated fluid production rate, L3/t

q' = shifted production rate using the reverse CM, L3/t

159

q(t0) = effect of production prior to the analysis period, L3/t

R2 = coefficient of determination

s = skin factor (skin segment)

t = time, t

Vp = pore volume, L3

w = injection rate, L3/t

w = total estimated injection rate using the reverse CM, L3/t

w' = shifted injection rate, L3/t

xe = reservoir x coordinate, L

ye = reservoir y coordinate, L

xw = well x coordinate, L

yw = well y coordinate, L

xD = x / xe dimensionless x coordinate

yD = y / xe dimensionless y coordinate

Greek Symbols

= interwell connectivity constant between producer/producer well

pair, dimensionless

′ = corrected interwell connectivity with respect to the homogeneous

case, dimensionless

= porosity, dimensionless

= diffusivity constant, L2/t

= interwell connectivity constant between injector/producer well

pair, dimensionless

= interwell connectivity constant between injector/producer well

160

Pair using the reverse CM, dimensionless

′ = corrected interwell connectivity with respect to the homogeneous

case, dimensionless

= fluid viscosity, m/(Lt)

= the coefficient of producers’ BHP term, L4t/m

= the coefficient of producers’ BHP term using the reverse CM,

L4t/m

= skin effect, dimensionless

= time-constant between injector/producer well pair, t

= time-constant between injector/producer well pair using the reverse

CM, t

p = time-constant of the effect of production prior to the analysis

period, t

p = time-constant of the effect of production prior to the analysis

period using the reverse CM, t

Matrices and Vectors

[A] = influence matrix, dimensionless

= matrix of ’s, dimensionless

[E] = Ei matrix, dimensionless

[J] = productivity index, L4t/m

= matrix of ’s, dimensionless

= matrix of ’s using the reverse CM, dimensionless

= transpose of matrix of ’s, dimensionless

w = vector of injection rates, L3/t

161

q = vector of production rates, L3/t

wfip = vector of injectors’ BHP, m/(Lt2)

wfpp = vector of producers’ BHP, m/(Lt2)

Subscripts and Superscripts

con = index of the interaction of injector/producer well pairs in influence

matrix

i = injector index

inj = injector index in the influence matrix

j = producer index

k = producer-BHP index

m = number of the time step

n = number of the time step of interest

prod = producer index in the influence matrix

s = segmentation time index

T = matrix transpose

x = shut-in well index

162

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166

APPENDIX 1

Derivation of MPI Formulas

Derivation of MPI matrix is pointed out here which is developed by Valkó et al. (2000).

Pressure distribution in the reservoir with one well during the pseudo steady state is:

1, , , , ,2

D D wD wD eD

Bp p x y a x y x y y q

kh

................................................................ (A-1-1)

where the influence function a for a homogeneous reservoir is given by

2 2

21

, , , ,

12 2 cos cos

3 2

D D wD wD eD

D wD mDeD D wD

meD eD

a x y x y y

y y tyy m x m x

y y m

..................................... (A-1-2)

cosh cosh

sinh

eD D wD eD D wD

m

eD

m y y y m y y yt

m y

.................................. (A-1-3)

If we have N-well system of production and injection wells, with flow rates at reservoir

condition using superposition theory we have

1

1

, ( , , , , )2

N

D D wDn wDn eD n

n

p p x y a x y x y y qkh

........................................................... (A-1-4)

To apply above formula around well j and regarding the effect of skin around this well

we get

1

1

( , , , , )2

N

wf j wDj wDj wDj wDn wDn eD n j j

n

p p a x y r x y y q s qkh

..................................... (A-1-5)

and in the matrix form,

1

2sd A D q

kh

............................................................................................... (A-1-6)

or

1

1

2s

khq A D d

............................................................................................ (A-1-7)

where matrices are as follows:

167

1 1 11 12 1 1

2 2 21 22 2 2

1 2

0 0

0 0, , ,

0 0

wf N

wf N

s

wf N N N N NN N

p p q a a a s

p p q a a a sd q A D

p p q a a a s

[A] is called influence matrix and MPI matrix is:

1

1

2s

khJ A D

............................................................................................. (A-1-8)

Finally it gives a generalized formula for linear productivity model as a matrix form:

q J d .................................................................................................................... (A-1-9)

168

APPENDIX 2

Derivation of the CM

To calculate connectivity parameters Yousef et al. (2006) coupled linear pseudo steady

state productivity model with material balance. All of the following derivations have

been developed by Yousef et al (2006). Basic formulas are as follows, if we assume there

is one injector-producer pair in the medium,

wfq t J p t p t .............................................................................................. (A-2-1)

t P

dpcV w t q t

dt ................................................................................................... (A-2-2)

where tc is the total compressibility,

PV is the drainage pore volume, p is the average

pressure in PV , w t is the injection rate, q t is the total production rate (water and oil),

wfp t and J are the flowing BHP and productivity index of the producer.

By differentiating equation A-2-1 and substituting in Equation A-2-2 it gives,

wfdpdq dpJ

dt dt dt

.................................................................................................... (A-2-3)

wfdpdqq t w t J

dt dt ........................................................................................ (A-2-4)

The term t PcV

J is called time constant for the drainage volume.

Equation A-2-4 is the first order differential equation and can be solved as follows,

1

t twfdpd

e q t e w t Jdt dt

............................................................................. (A-2-5)

0

0

0

1ttt

wf

t

dpe q t e q t e w J d

d

............................................................ (A-2-6)

0

0

0

tt t t

wf

t

dpeq t q t e e w J d

d

...................................................... (A-2-7)

169

0

0 0

0

tt t t tt

wf

t t

dpeq t q t e e w d Je e d

d

............................................ (A-2-8)

The analytical solution can be simplified using integration by parts as:

0

0

0

0

0

0

tt t t

t

tt t t

wf wf wf

t

eq t q t e e w d

eJ p t e p t e p d

........................................................ (A-2-9)

By generalizing the model for multiple injectors and producers using superposition in

space and discretizing the integrals, the final equation will be:

0 0

0 0

1 1

pj kj

t t t ti I k K

j pj j ij ij kj wf k wf k wf kj

i k

q t q t e w t v p t e p t p t

....... (A-2-10)

where ijw t and wf kjp t are:

1

1

m m

ij ij

t t t tn

ij i m

m

w t e e w t

............................................................................ (A-2-11)

1

1

m m

kj kj

t t t tn

wf kj wf k m

m

p t e e p t

..................................................................... (A-2-12)

where I is the number of injectors, K is the number of producers, weight factor ij

indicates the connectivity for the ij well pair, ij is the time constant for the medium

between injector i and producer j, ijw t is the convolved or filtered injection rate of

injector i on producer j, wf kjp t is the convolved BHP of producer k on producer j, kjv is

a coefficient that determines the effect of changing the BHP of producer k on producer j,

0jq t is the initial total production rate of producer j, pj is the resultant time constant of

the primary production solution and kj is the time constant between producer k and j.

170

APPENDIX 3

Derivation of the Analytical Formula for ’s‎Using‎MPI

Kaviani (2009) split the influence matrix to four components to derive an analytical for

’s (all derivations below have been completed by Kaviani, 2009):

1

wfi inj con

T

wfp con prod

p p A A w

p p A A q

................................................................................... (A-3-1)

1 1T

prod wfp prod conq A p p A A w

............................................................... (A-3-2)

where 1

2 kh

. By coupling Equation A-3-2 with material balance equation we get,

1 12

1 11 1

p i

T

prod wfp prod conN Nt p

dpA p p A A w w

dt cV

....................... (A-3-3)

where pN and

iN are number of producers and injectors respectively.

1

1 12

1 1

1 1 1

1 1

1 1 1

p p

p i p

prodN N

Tt p

prod wfp prod conN N N

A pdp

dt cV A p A A w

...................... (A-3-4)

Ignoring the exponential term:

1 1

1 1 1

1

1 1

11 1 1

1 1

p p i

p p

T


prodN N

A p A A w

pA

.................................. (A-3-5)

Coupling Equation A-3-5 with A-3-2 for production rate,

1

1 1 1

1

1 1

1

1 1 1 1

1

1 1

1 1

1 1

11 1

1

[

1

]1

p p

p

p p

p p i

p p

prodN N

prod wfpN

prodN N

T

prod conN N NT

prod con

prodN N

Aq A I p

A

A A wA A w

A

................................... (A-3-6)

171

Finally, production rates in terms of injection rates and production BHPs are obtained,

1

1

1

1 1

1

1

1

1 1

1 1

1 1

1

1 1

p p p i

p p

p p

p

p p

T

prod conN N N N T

prod con

prodN N

prodN N

prod wfpN

prodN N

A Aq A A w

A

AA I p

A

....................................... (A-3-7)

We can compare these weight factors with connectivity coefficients in the CM:

1

1

1

1 1

1 1

1 1

p p p i

p i

p p

T

prod conN N N N T

prod conN N

prodN N

A AA A

A

................................... (A-3-8)

172

APPENDIX 4

Derivation of the Reverse CM

If we want to formulate iw t in terms of jq t , another set of parameters can be derived

which shows the connectivity between well pairs as well. Injection rate can be calculated

by Equation A-4-1:

i wf iw t J p t p t ............................................................................................ (A-4-1)

where wf ip t and iJ are the flowing bottom hole pressure and injectivity index of the

injector w. By coupling Equation A-4-1 with material balance equation we get,

wf i

i

dpdw dpJ

dt dt dt

.................................................................................................. (A-4-2)

* * wf i

i i i

dpdww t J q t

dt dt .................................................................................. (A-4-3)

where * t Pi

i

cV

J ,

* *

*

*

1i i

t t

wf i

i i

i

dpde w t e q t J

dt dt

.................................................................... (A-4-4)

0

* * *

0

*

0 *

1i i i

tt t

wf i

i i

it

dpe w t e w t e q J d

d

.................................................. (A-4-5)

*0

* *

0

*

0 *

i

i i

tt t t

wf i

i i

i t

dpew t w t e e q J d

d

............................................... (A-4-6)

*0

* * * *

0 0

0 *

i

i i i i

tt t tt t

wf i

i

i t t

dpew t w t e e q d J e e d

d

................................... (A-4-7)

The final analytical solution can be simplified using integration by parts as:

173

*0

* *

0

*0

* *

0

0 *

0 *

i

i i

i

i i

tt t t

i t

tt t t

i wf i wf i wf i

i t

ew t w t e e q d

eJ p t e p t e p d

.................................................... (A-4-8)

By generalizing the model for multiple injectors and producers using superposition in

space and discretizing the integrals we obtain:

0 0

* ** * *

0 0

1 1

pi ri

t t t tj K r I

i pi i ji ji ri wf r wf r wf ri

j r

w t w t e q t v p t e p t p t

...... (A-4-9)

where ( )ijq t and wf rip t are:

1

* *

1

( )

m m

ji ji

t t t tn

ji j m

m

q t e e q t

............................................................................ (A-4-10)

1

* *

1

m m

ri ri

t t t tn

wf ri wf r m

m

p t e e p t

..................................................................... (A-4-11)

where I is the number of injectors, K is the number of producers, weight factor ji

indicates the connectivity for the ji well pair, ji is the time constant for the medium

between producer j and injector i, jiq t is the convolved or filtered injection rate of

producer j on injector i, wf rip t is the convolved BHP of injector r on injector i, *

riv is a

coefficient that determines the effect of changing the BHP of injector r on injector i,

0iw t is the initial injection rate of injector i, pi is the resultant time constant of the

initial injection solution and ri is the time constant between injector r and i. The values

of these parameters are different from the parameters in the CM equation; however, they

also indicate the connectivity of the producer and injector well pairs.

With a similar derivation as in Appendix 3 we can use MPI and obtain injection rates in

terms of production rates and injection BHPs, then reverse CM weight factors:

174

1

1

1

1 1

1

1

1

1 1

1 1

1 1

1

1 1

i i i p

i i

i i

i

i i

inj conN N N N

inj con

injN N

wfi

injN N

inj N

injN N

A Aw A A q

A

AI p

AiA

............................................... (A-4-12)

1

1*

1

1 1

1 1

1 1

i i i p

i p

i i

i conN N N N

c

nj

inj

in

onN N

N Nj

A AA A

A

......................................... (A-4-13)

175

APPENDIX 5

Derivation of the Relationship between and Skin

We simplify the analytical equation of values for simple systems. Then we generalize it

for more complicated well systems. For a system of 1 injector and 2 producers, we

obtained the following equations:

1

2

2

1

2 2

1

2

1 1

1 1

1 1

p

p

p p p

p

p p p

s Cs

λ s A s A

s Cs

λ s B s B

................................................................................. (A-5-1)

For more than 2 producers we have:

1

2

i p1 p1 i p1

p

i p2 p2 i p2

p

1s s s

λ

1s s s

λ

A B

C D

.................................................................................. (A-5-2)

where A, B, C, and D are constants. Thus, we conclude there is a linear relationship

between the skin of a producer and reciprocal of that producer’s . The ’s are not a

function of the number of injectors. Therefore, if we have more than one injector, the

results are the same.

176

APPENDIX 6

Derivation of the Analytical Formula for ’s‎in‎Transient‎Regime

Similar to the method Kaviani (2009) used for deriving analytical ’s we split the E

matrix to four components:

1

wfi inj con

T

wfp con prod

p p E E w

p p E E q

.................................................................................. (A-6-1)

1 1T

prod wfp prod conq E p p E E w

.............................................................. (A-6-2)

where 1

2 kh


1 12

1 11 1

p i

T


dpE p p E E w w

dt cV

...................... (A-6-3)

where pN and


1

1 12

1 1

1 1 1

1 1

1 1 1

p p

p i p

prodN N

Tt p


E pdp

dt cV E p E E w

...................... (A-6-4)

Ignoring the exponential term:

1 1

1 1 1

1

1 1

11 1 1

1 1

p p i

p p

T


prodN N

E p E E w

pE

.................................. (A-6-5)


1

1 1 1

1

1 1

1

1 1 1 1

1

1 1

1 1

1 1

11 1

1

[

1

]1

p p

p

p p

p p i

p p

prodN N

prod wfpN

prodN N

T

prod conN N NT

prod con

prodN N

Eq E I p

E

E E wE E w

E

................................... (A-6-6)

177


1

1

1

1 1

1

1

1

1 1

1 1

1 1

1

1 1

p p p i

p p

p p

p

p p

T

prod conN N N N T

prod con

prodN N

prodN N

prod wfpN

prodN N

E Eq E E w

E

EE I p

E

...................................... (A-6-7)

We can compare these weight factors with connectivity coefficients in the CM:

1

1

1

1 1

1 1

1 1

p p p i

p i

p p

T

prod conN N N N T

prod conN N

prodN N

E EE E

E

.................................. (A-6-8)

178

APPENDIX 7

Derivation of Transient CM

Similar to the Appendix 6 we split the E matrix to four components:

1

wfi inj con

T

wfp con prod

p p E E w

p p E E q

.................................................................................. (A-7-1)

1 1T

prod wfp prod conq E p p E E w

.............................................................. (A-7-2)

where 1

2 kh


1 12

1 11 1

p i

T


dpE p p E E w w

dt cV

...................... (A-7-3)

where pN and


1

1 12

1 1

1 1 1

1 1

1 1 1

p p

p i p

prodN N

Tt p


E pdp

dt cV E p E E w

...................... (A-7-4)

assuming:

2 1

dpc c p

dt ............................................................................................................... (A-7-5)

then,

2 21

1 1

( )exp( )i

c cp p c t

c c ......................................................................................... (A-7-6)

Now if we assume:

12

1 1 11 1

p pprodN N

t p

c EcV

................................................................................ (A-7-7)

and

1 12

2 1 1 11 1 1

p p i

T

prod wfp prod conN N Nt p

c E p E E wcV

............................. (A-7-8)

179

then:

1 1

1 1 1

1

1 1

12

1 1

1 1

1 1 1

1

1 1

1 1 1

1 1

exp( 1 1 )

1 1 1exp

1 1

p p i

p p

p p

p p i

p p

T


prodN N

i prodN Nt p

T


prodN N

E p E E wp

E

p E tcV

E p E E w

E

1

2

1 1( 1 1 )

p pprodN N

t p

E tcV

......... (A-7-9)


1 1

11 1 12

1 1 1

1 11

1 1

1 1 1

1

1 1 1exp( 1 1 )

1 1

1 1 1

1

p p i

p p

p p

p p i

p

T


i prodN Nt pprodN N

prodT


prodN

E p E E wp E t

cVEq E

E p E E w

E

12

1 1 1

1

1

exp( 1 1 )1

p p

p

prod wfpN Nt p

N

T

prod con

E t pcV

E E w

...... (A-7-10)

simplifying it:

1

1 1 1

1

1 1

1

1 1 1

1

1 1

1 12

1 1 1

1 1

1 1

11 1 1

]1 1

1 exp( 1

[

1 )

p p

p

p p

p p i

p p

p p p

prodN N

prod wfpN

prodN N

T

prod conN N N

prodN N

i prod prodN N Nt p

pr

Eq E I p

E

E E w

E

p E E tc V

E

1

1 1 1

1

1 1

1

1 1 1 12

1 1 1

1 1

1

1 1

1 1

11 1 1

]exp( 1 11 1

[

)

p p

p

p p

p p i

p p

p p

prodN N

od wfpN

prodN N

T

prod conN N N

prodN Nt pprodN N

T

prod con

EI p

E

E E w

E tc VE

E E w

.......... (A-7-11)

If we call the exponential term as lag term and call it L:

1

2

1 1exp( 1 1 )

p pprodN N

t p

L E tcV

..................................................................... (A-7-12)

180

1

1 1 1 1

11

1 1

1

1 1 1 1

1

1 1

1 11 )

1 1

11 1 1

]1 1

( [p p

p p

p p

p p i

p p

prodN N

i prod prod wfpN N

prodN N

T

prod conN N NT

prod con

prodN N

Eq p L E L E I p

E

E E wE E w

E

....... (A-7-13)


1

1

1

1

1

1 1

1

1

1

1 1

1

( 1 1)

1 1

1)

1 1(

p

p p p i

p p

p p

p

p p

i prod N

T

prod conN N N N T

prod con

prodN N

prodN N

prod piN

prodN N

q p L E

E ELE E w

E

EL E I p

E

............................... (A-7-14)

We can convert initial pressure to initial production rate:

1

1 11

p

T

i prod i con i piNp E q E w p

.................................................................. (A-7-15)

Then we rewrite the equation A-7-14:

1 1

1

1

1

1 1

1

1

1

1

1 1)

1 1

1)

1

( )

1

(

(

p p p i

p p

p p

p p

T

i prod con i prod pi

T

prod conN N N N T

prod con

prodN N

prodN N

prod

prodN N

q Lq E E w E p

E ELE E w

E

EL E

L

E

1

ppN

I p

............................... (A-7-16)

We can rewrite the equation A-7-14 in terms of ’s as well:

1

1

1

1

1

1 1

1

1

1

1 1

1

1 1

1 1

1)

1 1(

p

p p p i

p i

p p

p p

p

p p

i prod N

T

prod conN N N N

prodN N

prodN N

prodN N

prod piN

prodN N

q p L E

E Ew E w

E

EL E I p

E

L

...................................... (A-7-17)

181

APPENDIX 8

Derivation of the MCCM for Skin Changes

If two producers (x and y) have non-zero skin factor and are at segment s1, the ij at this

segment (ij(xs,ys)) will be:

, , ,xs ys xs xs ys xs ys xs ys ys

ij ij yj iy ij xj ix ............................................................... (A-8-1)

where ,xs ys

yj is the yj where producers x and y are stimulated and is at their segment s.

Replacing Equation 8-5 in Equation A-8-1 we obtain:

, ,

1 11 1xs ys xs ys

ij ij xj ix x yj iy xy ix xs s ................................ (A-8-2)

or

, , ,

11xs ys xs ys xs ys

ij ij ix x xj xy yj iy yjs ............................................ (A-8-3)

and

, ,

1 11 1xs ys xs ys

ij ij yj iy y xj ix yx iy ys s ................................... (A-8-4)

or

, , ,

11xs ys xs ys xs ys

ij ij ix xj iy y yj yx xjs ............................................... (A-8-5)

Combining Equations A-8-3 and A-8-5 we obtain:

, ,

11xs ys xs ys

xj x xj xy yjs ............................................................................ (A-8-6)

and

, ,

11xs ys xs ys

yj y yj yx xjs ............................................................................ (A-8-7)

Combining Equations A-8-6 and A-8-7 we obtain:

, ,

1 1 11 1 1xs ys xs ys

xj xj x xy y yj yx xj xs s s ............................. (A-8-8)

or:

182

1 1,

1 1

1 1

1 1 1

x xj xy yj yxs ys

xj

xy yx x y

s s

s s

............................................................. (A-8-9)

We also have:

, , ,xs ys xs ys xs ys

ix ix xx ix yx iy ............................................................................... (A-8-10)

In a similar procedure as shown in Equations A-8-2 to A-8-9 we can show:

, 1

1 1

11 1 1

xs ys x

xx

xy x yx y

s

s s

.......................................................... (A-8-11)

and

1 1,

1 1

1

1 1 1

x yx yxs ys

yx

xy x yx y

s s

s s

................................................................ (A-8-12)

which are also equivalent to Equation A-8-9.

For the case where three producers are stimulated, in a similar way we can calculate

ij(xs,ys,zs)

which will be in the form of Equation 8-9. If a larger number of wells is

stimulated, it is not easy to reproduce this equation. However, we have tested the

correctness of this equation with the Matlab symbolic tool for cases with four and five

producers stimulated. For a larger number of producers, we have investigated this

numerically, and the equation was exactly correct.

183

APPENDIX 9

Calculating Average Production Rate Using the CM

If the producers’ BHPs are constant, based on the CM we have:

0 1

0

1 1

ˆ ( )

n m n m n

pj ij ij

t t t t t ti I n

j n j ij i m

i m

q t q t e e e w t

............................................... (A-9-1)

This provides the instantaneous production rate at time tn. The average rate at time step n

could be calculated by integrating Equation A-9-1 over tn-1 to tn:

0 1

1

0

1 11

1ˆ ( )

m mn

pj ij ij

n

t t t t t tt i I n

j n j ij i m

i mn n t

q t q t e e e w t dtt t

........................... (A-9-2)

or

1 0 0

1

1

0

1

1 1 1 11

ˆ ( )

1

n n

pj pj

m mn

ij ij

n

t t t t

pj j

j n

n n

t t t tt i I n i I n

ij i m ij i m

i m i mn n t

q tq t e e

t t

e w t e w t dtt t

............................... (A-9-3)

we also have:

1 1

1 1 1 1

1 1

1

...

m nn n n n

ij ij ij ij

n n n n

t t t t t t t tt t t tt

i m i i n i

mt t t t

e w t dt e w t dt e w t dt e w t dt

......... (A-9-4)

or

1

1 1 1 1 1 1

1

1

1 1 1 1...

mn

ij

n

n n n n n n

ij ij ij ij

t tt t

i m

mt

t t t t t t t t

i n n

ij i i i n i n

ij

e w t dt

w t t te w t e w t e w t e w t

.... (A-9-5)

or

1

1

11

1 1

m m n m nn

ij ij ij

n

t t t t t tt t ni n n

i m ij i m

m m ijt

w t t te w t dt e e w t

..................... (A-9-6)

184

In a similar way we have:

1 1 1 1

11 1 1

m m n m nn

ij ij ij

n

t t t t t tt t n n

i m ij i m i m

m m mt

e w t dt e w t e w t

............................... (A-9-7)

Replacing Equations A-9-6 and A-9-7 in Equation A-9-3 we have:

1 0 0

1

1 1 1

0

1

11

1 11

1

ˆ ( )

1

n n

pj pj

m n m n

ij ij

m n m n

ij ij

t t t t

pj j

j n

n n

t t t tI n

i n n

ij ij i m

i mn n ij

t t t tn

i m

m

q tq t e e

t t

w t t te e w t

t t

e e w t

........................... (A-9-8)

or

1 0 0

1 1 1 1

0

1

1

1 1 11

1

ˆ ( )

1

n n

pj pj

m n m n m n m n

ij ij ij ij

t t t t

pj j

j n

n n

t t t t t t t tI n n

ij ij i m i m

i m mn n

i n n n

ij

q tq t e e

t t

e e w t e e w tt t

w t t t

...... (A-9-9)

Replacing the definition of w´ij(tn) from Equation 8-11, we obtain:

1 0 0

0 1

11 1

ˆ ( )

n n

pj pj

t t t tI

pj j ij n ij n

j n ij ij i n

in n n n

q t w t w tq t e e w t

t t t t

.............. (A-9-10)

interwell connectivity evaluation using injection and

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