Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
University of Calgary
PRISM: University of Calgary's Digital Repository
Graduate Studies The Vault: Electronic Theses and Dissertations
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Data (Unpublished doctoral thesis).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
University of Calgary graduate students retain copyright ownership and moral rights for their
thesis. You may use this material in any way that is permitted by the Copyright Act or through
licensing that has been assigned to the document. For uses that are not allowable under
copyright legislation or licensing, you are required to seek permission.
Downloaded from PRISM: https://prism.ucalgary.ca
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.
III
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
IX
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
XII
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
XIII
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
XIV
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
XV
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
XVI
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
XVII
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
XVIII
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
XIX
(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
CHAPTER 1 INTRODUCTION 2
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
CHAPTER 1 INTRODUCTION 3
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.
CHAPTER 1 INTRODUCTION 4
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
CHAPTER 2 LITERATURE REVIEW 6
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,
CHAPTER 2 LITERATURE REVIEW 7
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).
CHAPTER 2 LITERATURE REVIEW 8
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
CHAPTER 2 LITERATURE REVIEW 9
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
CHAPTER 2 LITERATURE REVIEW 10
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
CHAPTER 2 LITERATURE REVIEW 11
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).
CHAPTER 2 LITERATURE REVIEW 12
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.
CHAPTER 2 LITERATURE REVIEW 13
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.
CHAPTER 2 LITERATURE REVIEW 14
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).
CHAPTER 2 LITERATURE REVIEW 15
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.
CHAPTER 2 LITERATURE REVIEW 16
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.
CHAPTER 2 LITERATURE REVIEW 17
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).
CHAPTER 2 LITERATURE REVIEW 18
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
CHAPTER 2 LITERATURE REVIEW 19
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).
CHAPTER 2 LITERATURE REVIEW 20
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.
CHAPTER 2 LITERATURE REVIEW 21
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.
CHAPTER 2 LITERATURE REVIEW 22
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
CHAPTER 2 LITERATURE REVIEW 23
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)
CHAPTER 2 LITERATURE REVIEW 24
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,
CHAPTER 2 LITERATURE REVIEW 25
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
CHAPTER 2 LITERATURE REVIEW 26
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
Producer BHP changes, skin
changes, and well shut-ins
impact the results
Fluid and medium properties
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
Producer BHP changes, skin
changes, and well shut- ins
impact the results
Fluid and medium properties
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)
Producer BHP changes, skin
CHAPTER 2 LITERATURE REVIEW 27
changes, and well shut-ins
impact the results
Fluid and medium properties
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
CHAPTER 2 LITERATURE REVIEW 28
Well productivity, GOR and
effective permeabilities should
be constant
Diffusivity constant should be
high
MPI based
connectivity
Kaviani and Valkó. (2010)
GOR and effective
permeabilities should be
constant
Diffusivity constant should be
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)
Well productivity, GOR and
effective permeabilities should
be constant
Diffusivity constant should be
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
CHAPTER 3 SENSITIVITY ANALYSIS 30
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
CHAPTER 3 SENSITIVITY ANALYSIS 31
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,
CHAPTER 3 SENSITIVITY ANALYSIS 32
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
CHAPTER 3 SENSITIVITY ANALYSIS 33
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
CHAPTER 3 SENSITIVITY ANALYSIS 34
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
CHAPTER 3 SENSITIVITY ANALYSIS 35
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
CHAPTER 3 SENSITIVITY ANALYSIS 36
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
CHAPTER 3 SENSITIVITY ANALYSIS 37
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
CHAPTER 3 SENSITIVITY ANALYSIS 38
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
CHAPTER 3 SENSITIVITY ANALYSIS 39
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
CHAPTER 3 SENSITIVITY ANALYSIS 40
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
CHAPTER 3 SENSITIVITY ANALYSIS 41
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.
CHAPTER 3 SENSITIVITY ANALYSIS 42
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
CHAPTER 3 SENSITIVITY ANALYSIS 43
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
CHAPTER 3 SENSITIVITY ANALYSIS 44
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.
CHAPTER 3 SENSITIVITY ANALYSIS 45
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
CHAPTER 3 SENSITIVITY ANALYSIS 46
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
CHAPTER 3 SENSITIVITY ANALYSIS 47
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
CHAPTER 3 SENSITIVITY ANALYSIS 48
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
CHAPTER 3 SENSITIVITY ANALYSIS 49
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.
CHAPTER 3 SENSITIVITY ANALYSIS 50
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
CHAPTER 3 SENSITIVITY ANALYSIS 51
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
CHAPTER 3 SENSITIVITY ANALYSIS 52
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.
CHAPTER 3 SENSITIVITY ANALYSIS 53
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
CHAPTER 3 SENSITIVITY ANALYSIS 54
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
average absolute error from all data
5x4 homogeneous
5x4 heterogeneous
8x8 heterogeneous
CHAPTER 3 SENSITIVITY ANALYSIS 55
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
average absolute error from all data
5x4 homogeneous
5x4 heterogeneous
8x8 heterogeneous
CHAPTER 3 SENSITIVITY ANALYSIS 56
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
CHAPTER 3 SENSITIVITY ANALYSIS 57
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.
CHAPTER 3 SENSITIVITY ANALYSIS 58
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
CHAPTER 3 SENSITIVITY ANALYSIS 59
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
CHAPTER 3 SENSITIVITY ANALYSIS 60
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
CHAPTER 4 THE CM AND HORIZONTAL WELLS 62
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
CHAPTER 4 THE CM AND HORIZONTAL WELLS 63
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
Model dimensions 93×93×5
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
CHAPTER 4 THE CM AND HORIZONTAL WELLS 64
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
CHAPTER 4 THE CM AND HORIZONTAL WELLS 65
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.
CHAPTER 4 THE CM AND HORIZONTAL WELLS 66
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).
CHAPTER 4 THE CM AND HORIZONTAL WELLS 67
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
Horizontal well Horizontal well
CHAPTER 4 THE CM AND HORIZONTAL WELLS 68
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.
CHAPTER 4 THE CM AND HORIZONTAL WELLS 69
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
CHAPTER 4 THE CM AND HORIZONTAL WELLS 70
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).
CHAPTER 4 THE CM AND HORIZONTAL WELLS 71
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).
CHAPTER 4 THE CM AND HORIZONTAL WELLS 72
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
CHAPTER 4 THE CM AND HORIZONTAL WELLS 73
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
CHAPTER 4 THE CM AND HORIZONTAL WELLS 74
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
CHAPTER 4 THE CM AND HORIZONTAL WELLS 75
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* *
CHAPTER 4 THE CM AND HORIZONTAL WELLS 76
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
Horizontal well Horizontal well
* *
CHAPTER 4 THE CM AND HORIZONTAL WELLS 77
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
CHAPTER 4 THE CM AND HORIZONTAL WELLS 78
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.
CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY 80
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
CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY 81
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.
CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY 82
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
CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY 83
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
CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY 84
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
CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY 85
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.
CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY 86
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
CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY 87
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
CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY 88
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
CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY 89
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
CHAPTER 5 NEAR WELL AND INTERWELL HETEROGENEITY 90
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).
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 92
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
Model dimensions 93×93×1
Grid size, ft 26.667×26.667×30
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 93
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
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 94
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
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 95
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
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 96
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
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 97
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
Start time of window, months
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
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 98
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
Start time of window, months
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 99
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
Start time of window, months
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 100
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.
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 101
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
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 102
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
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 103
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.
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 104
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
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 105
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.
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 106
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
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 107
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.
CHAPTER 6 THE CM IN HEAVY OIL RES. - WORMHOLE ASSESSMENT 108
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:
CHAPTER 7 THE CM IN TIGHT FORMATIONS 110
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.
CHAPTER 7 THE CM IN TIGHT FORMATIONS 111
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
Model dimensions 93×93×1
Grid size, ft 26.667
CHAPTER 7 THE CM IN TIGHT FORMATIONS 112
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
CHAPTER 7 THE CM IN TIGHT FORMATIONS 113
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
CHAPTER 7 THE CM IN TIGHT FORMATIONS 114
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)
CHAPTER 7 THE CM IN TIGHT FORMATIONS 115
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)
CHAPTER 7 THE CM IN TIGHT FORMATIONS 116
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
CHAPTER 7 THE CM IN TIGHT FORMATIONS 117
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
CHAPTER 8 MULTIWELL COMPENSATED CM 119
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.
CHAPTER 8 MULTIWELL COMPENSATED CM 120
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):
CHAPTER 8 MULTIWELL COMPENSATED CM 121
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 [].
CHAPTER 8 MULTIWELL COMPENSATED CM 122
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)
CHAPTER 8 MULTIWELL COMPENSATED CM 123
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
CHAPTER 8 MULTIWELL COMPENSATED CM 124
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
CHAPTER 8 MULTIWELL COMPENSATED CM 125
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
Model dimensions 93×93×1
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
CHAPTER 8 MULTIWELL COMPENSATED CM 126
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
CHAPTER 8 MULTIWELL COMPENSATED CM 127
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
CHAPTER 8 MULTIWELL COMPENSATED CM 128
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
CHAPTER 8 MULTIWELL COMPENSATED CM 129
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
CHAPTER 8 MULTIWELL COMPENSATED CM 130
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
CHAPTER 8 MULTIWELL COMPENSATED CM 131
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
CHAPTER 8 MULTIWELL COMPENSATED CM 132
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
CHAPTER 8 MULTIWELL COMPENSATED CM 133
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).
CHAPTER 9 FIELD APPLICATION 135
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
Averageoil in 68 F, cp 746
Averageoil in 104 F, cp 283
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.
CHAPTER 9 FIELD APPLICATION 136
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.
CHAPTER 9 FIELD APPLICATION 137
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
CHAPTER 9 FIELD APPLICATION 138
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
CHAPTER 9 FIELD APPLICATION 139
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)
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)
CHAPTER 9 FIELD APPLICATION 140
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.
CHAPTER 9 FIELD APPLICATION 141
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
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
CHAPTER 9 FIELD APPLICATION 142
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
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
CHAPTER 9 FIELD APPLICATION 143
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
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
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
CHAPTER 9 FIELD APPLICATION 144
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=-
CHAPTER 9 FIELD APPLICATION 145
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.
CHAPTER 9 FIELD APPLICATION 146
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.
CHAPTER 9 FIELD APPLICATION 147
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
CHAPTER 9 FIELD APPLICATION 148
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
CHAPTER 9 FIELD APPLICATION 149
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
CHAPTER 9 FIELD APPLICATION 150
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
CHAPTER 9 FIELD APPLICATION 151
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
CHAPTER 9 FIELD APPLICATION 152
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)
Predicted Production
True Production
0 100 200 300 400 500 6000
100
200
300
400
500
600
Predicted Production (rb/day)
Tru
e P
rodu
ctio
n (r
b/da
y)
Well 8-17
CHAPTER 9 FIELD APPLICATION 153
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
CHAPTER 10 CONCLUSIONS AND RECOMMENDATIONS 155
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.
CHAPTER 10 CONCLUSIONS AND RECOMMENDATIONS 156
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
CHAPTER 10 CONCLUSIONS AND RECOMMENDATIONS 157
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
REFERENCES
Albertoni, A. and Lake, L. W. 2003. Inferring Interwell Connectivity Only from Well-
Rate Fluctuations in Waterfloods. SPEREE 6(1): 6-16.
Cheng, Y., Wang, Y., McVay D.A., and Lee, W.J. 2010. Practical Application of a
Probabilistic Approach to Estimate Reserves Using Production Decline Data. SPE Econ
& Manag.2 (1): 19-31.
Demiryurek, U. Banaei-Kashani, F. and Shahabi, C. 2008. Neural-Network Based
Sensitivity Analysis for Injector-Producer Relationship Identification. SPE 112124.
Dinh, A. and Tiab, D. 2008. Interpretation of Interwell Connectivity Tests in a
waterflooding system. SPE 116144.
Dinh, A. and Tiab, D. 2013. Analytical Determination of Interwell Connectivity Based on
Flow Rate Fluctuations in Waterflood Reservoirs. SPE 164481.
Dong, F., Tan, C., and Zhang, F. S. 2009 Flow rate measurement of oil-water two-phase
flow based on V-cone flow meter, J. Physics Conference Series, 147, paper 012059, doi
10.1088/1742-6596/147/1/012059.
Efron, B., and Tibshirani, 1994, An introduction to the bootstrap, Chapman and Hall,
New York.
Fedenczuk, L. and Hoffmann, K. 1998. Surveying and Analyzing Injection Responses for
Patterns with Horizontal Wells. SPE 50430.
Gherabati, S.A. Hughes, R.G., Zhang, H., and White, C.D. 2012. A Large Scale Network
Model To Obtain Interwell Formation Characteristics. SPE 153386.
Haykin, S. 2009. Neural Networks and learning machines. Third ed., Prentice-Hall Inc.,
Upper Saddle River, New Jersey.
Heffer, K. J., Fox, R. J., McGill, C. A. and Koutsabeloulis, N. C. 1997. Novel Techniques
Show Links Between Reservoir Flow Directionality, Earth Stress, Fault Structure and
Geomechanical Changes in Mature Waterfloods. SPEJ 2(2): 91-98.
163
Izgec, O. and Kabir, C.S. 2010. Understanding Reservoir Connectivity in Waterfloods
Before Breakthrough. . Journal of Petroleum Science and Engineering75 (1-2): 1-12.
Jansen, F. E. and Kelkar, M. G. 1997. Application of Wavelets to Production Data in
Describing Interwell Relationships. SPE 38876.
Jensen, J. L., Lake, L. W., Bui, T. D., Al-Yousef, A., Gentil, P. 2004. Interwell
Connectivity and Diagnosis Using Correlation of Production and Injection Rate Data in
Hydrocarbon Production. Annual Report, NETL 4010.
Jochen, V.A. and Spivey, J.P. 1996. Probabilistic Reserves Estimation Using Decline
Curve Analysis with the Bootstrap Method. SPE 36633.
Kaviani, D., 2009. Interwell Connectivity Evaluation from Well Rate Fluctuations: A
Water Flooding Management Tool. PhD Dissertation, Texas A&M University.
Kaviani, D. and Jensen, J.L. 2010. Reliable Connectivity Evaluation in Conventional and
Heavy Oil Reservoirs: A Case Study From Senlac Heavy Oil Pool, Western
Saskatchewan. SPE 137504.
Kaviani, D., Jensen, J.L., and Lake, L.W. 2012. Estimation of Interwell Connectivity in
the Case of Unmeasured Fluctuating Bottomhole Pressures. Journal of Petroleum Science
and Engineering, 90-91 (1): 79-95.
Kaviani, D. and Valkó, P.P. 2010. Inferring Interwell Connectivity Using Multiwell
Productivity Index (MPI). Journal of Petroleum Science and Engineering 73 (1-2): 48-58.
Lee, K., Ortega, A., Mohammad Nejad, A. and Ershaghi, I. 2011, An Active Method for
Characterization of Flow Units Between Injection-Production Wells by Injection-Rate
Design. SPEREE 14(4): 453-465.
Lee, W.J., Rollins, J.B., and Spivey, P. 2003. Transient Pressure Testing. SPE Textbook
Sereis, Richardson, TX.
Liang, X., Weber, D.B., Edgar, T.F., Lake, L.W., Sayarpour, M., and Al-Yousef, A.
2007. Optimization of Oil Production Based on a Capacitance Model of Production and
Injection Rates. SPE 107713.
164
Lines, L., Chen, S., Daley, P.F., Embleton, J. and Mayo, L. 2003. Seismic Pursuit of
Wormholes. The leading Edge, May issue, 459-461.
Lines, L., Agharbarati, H., Daley, P.F., Embleton, J., Fay, M., Settari, T., Vasheghani, F.,
Wang, T. and Zhang, A. 2008. Collaborative Methods in Enhanced Cold Heavy Oil
Production. The leading Edge, 1152-1156.
Liu, F. and Mendel J. M. 2009. Forecasting Injector-Producer Relationships from
Production and Injection Rates Using an Extended Kalman Filter. SPEJ 14(4) 653-664.
Liu, X. and Zhao G. 2009. A Fractal Wormhole Model for Cold heavy Oil production.
JCPT 44(9): 31-36.
Morshedian, A., MacEachern, J. A., and Dashtgard, S. E., 2012, Integrated Ichnology,
Sedimentology, and Stratigraphy of the Lower Cretaceous Sparky Alloformation
(Mannville Group), Lloydminster Area, west-central Saskatchewan, Canada, Bull. Can.
Pet. Geol. 60 (2): 69-91.
Panda, M. N. and Chopra, A. K., 1998. An Integrated Approach to Estimate Well
Interactions. SPE 39563.
Refunjol, T. and Lake L. W. 1999. Reservoir Characterization Based on Tracer Response
and Rank Analysis of Production and Injection Rates. Reservoir Characterization Recent
Advances, AAPG Memoir 71, p. 209-218.
Sayarpour, M. 2008. Development and Application of Capacitance/Resistive Models to
Water/CO2 Floods. PhD dissertation, The University of Texas at Austin, Texas.
Sayarpour, M., Kabir, C.S., and Lake, L.W. 2009. Field Applications of Capacitance-
Resistive Models in Waterfloods SPEREE 12 (6): 853-864. SPE 114983.
Soeriawinata, T. and Kelkar, M. 1999. Reservoir Management Using Production Data.
SPE 52224.
Soroush, M., 2010. Investigation of Interwell Connectivity Using Injection and
Production Fluctuation Data in Water Flooding Projects. MSc Dissertation, the
University of Calgary - Petroleum University of Iran.
165
Thiele, M. R. and Batycky, R. P. 2006. Using Streamline-Derived Injection Efficiencies
for Improved Water flood Management, SPEREE 9(2): 187-196.
Valkó, P. P., Doublet, L.E., and Blasingame, T.A. 2000. Development and Application of
the Multiwell Productivity Index (MPI). SPEJ 5(1): 21-31.
Weber, D., Edgar, T.F., Lake, L.W., Lasdon, L., Kawas, S., and Sayarpour, M. 2009.
Improvements in Capacitance-Resistive Modeling and Optimization of Large Scale
Reservoirs. SPE 121299.
Yousef, A.A., Gentil, P., Jensen, J.L., and Lake, L.W. 2006. A Capacitance Model to
Infer Interwell Connectivity from Production and Injection Rate Fluctuations. SPEREE
9(5): 630-646.
Yousef, A.A, Jensen, J.L., and Lake, L.W. 2009. Integrated Interpretation of Interwell
Connectivity Using Injection and Production Fluctuations. Mathematical Geosciences,
41(1): 81-102.
Yousef, A.A. 2006. Investigating Statistical Techniques to Infer Interwell Connectivity
From Production and Injection Fluctuations. PhD dissertation, The U. of Texas at Austin,
Texas.
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
prod wfp prod conN N N
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
. By coupling Equation A-6-2 with material balance equation we get,
1 12
1 11 1
p i
T
prod wfp prod conN Nt p
dpE p p E E w w
dt cV
...................... (A-6-3)
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
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
prod wfp prod conN N N
prodN N
E p E E w
pE
.................................. (A-6-5)
Coupling Equation A-6-5 with A-6-2 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
Eq E I p
E
E E wE E w
E
................................... (A-6-6)
177
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
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
. By coupling Equation A-7-2 with material balance equation we get,
1 12
1 11 1
p i
T
prod wfp prod conN Nt p
dpE p p E E w w
dt cV
...................... (A-7-3)
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
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
prod wfp prod conN N N
prodN N
i prodN Nt p
T
prod wfp prod conN N N
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)
Coupling Equation A-7-5 with A-7-2 for production rate,
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
prod wfp prod conN N N
i prodN Nt pprodN N
prodT
prod wfp prod conN N N
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)
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
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)