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University of Southern California Dissertations and Theses
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Aspects of fracture heterogeneity and fault transmissivity in pressure transient modeling of naturally fractured reservoirs
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Aspects of fracture heterogeneity and fault transmissivity in pressure transient modeling of naturally fractured reservoirs
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INFORMATION TO USERS This manuscript has been reproduced from the microfihn master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter free, while others may be from any type o f computer printer. The quality o f this reproduction is dependent upon the quality of the copy suhmittcd. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back o f the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Infonnaticn Company 300 Noith Zeeb Road, Ann Aibor MI 48106-1346 USA 313^61-4700 800/S21-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ASPECTS OF FRACTURE HETEROGENEITY AND FAULT TRANSMISSIVITY IN PRESSURE TRANSIENT MODELING OF NATURALLY FRACTURED RESERVOIRS by Abdullah Hamed Al-Ghamdi A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Petroleum Engineering) May 1998 Copyright 1998 Abdullah Al-Ghamdi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9902764 Copyright 1998 by Al-Gluundi, Abdullah Hamed All rights reserved. UMI Microform 9902764 Copyright 1998, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES. CALIFORNIA 90007 Tnis dissertation, written by ......................................... under the direction of h.±s....... Dissertation Committee, and approved by all its members, has been presented to and accepted by Tne Graduate School, in partial fulfillment of re- quirements for the degree of DOCTOR OF PHILOSOPHY Graduate Studies Date ...Aprü J.7;_19% DISSERTATION QQMMÎTTEE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dedication This dissertation is dedicated to my parents, Hamed and Sharee&h, for their boundless love and ceaseless support, my sister Azzah for her love and encouragement, my sons, Magid and Hamed, and ray daughter Ghada for their spirit and for bringing the greatest love and joy to my life. u Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments I would like to express my deepest indebtedness to professor Iraj Ershaghi, my academic advisor and Ph.D. committee chairman, for his continual support, guidance and encouragement throughout the course of my graduate study at USC. I would also like to convey my profound appreciation to my dissertation committee members, professors Yanis Yortsos and Charles Sammis, for their time serving as committee members, reviewing this manuscript and making constructive remarks and comments. Personal thanks to professors Elmer Dougherty and Donn Gorsline and Dr. Bruce Davis, for their time and effort serving on my guidance committee. I would like to extend my loyal thanks to Saudi Aramco and Aramco Services Company for providii% financial, and administrative support. This study was also supported by the Center for Study of Fractured Reservoirs at USC. ui Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Dedication ii Acknowledgments iii List of Tables viii List of Figures ix Nomenclature xv Abstract xvii Introduction and S co p e.......................................................................................... 1 1.1 Introduction.................................................................................................... 1 1.2 Objective and sc o p e .......................................................................................2 Physical and Geological Aspects of N F R s............................................................ 5 2.1 Characteristics of fractured ro ck s..................................................................5 2.2 Fracture classifications...................................................................................7 2.3 Brittle rock failure and mechanics of fracturing and faulting .......................8 2.3.1 State of stress and rock deformation................................................8 2.3.2 Mohr stress circle and mechanical conditions of fracturing ........... 9 2.3.3 Coulomb criteria of brittle rock fracture ........................................ 10 2.4 Geological and physical characteristics o f fractures................................... 11 2.5 Multiple fracture sets and fracture network m odels................................... 12 2.6 Fracture detection methods.......................................................................... 13 2.6.1 Core inspection................................................................................ 13 2.6.2 Geophysical methods ......................................................................14 2.6.2.1 Seismic surveys .....................................................14 2.Ô.2.2 Electrical and electromagnetic methods ........................15 2.6.3 Conventional well lo g s ....................................................................15 2.6.4 Borehole imaging logs......................................................................16 IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.7 Geological and petro-physical aspects of faults..........................................17 2.7.1 Physical characteristics of fault zones ............................................18 2.7.2 Textural classifications of &ult zones ............................................18 2.7.3 Structural fault classifications.........................................................19 2.7.4 Hydrological characteristics of feult zones.....................................20 2.7.5 Sealii% and conductive faults ........................................................ 22 Well Test Analysis of Naturally Fractured Reservoirs.................................... 24 3.1 An overview of conceptual modeling ........................................................ 24 3.2 Uncertainties associated with model selection process...............................25 3.2.1 Limitations of analytical models ....................................................25 3.2.2 Insensitivity and non-uniqueness of model response...................... 26 3.2.3 Limitation to one type of heterogeneity......................................... 27 3.3 An overview of modem test interpretation................................................28 3.3.1 Graphical test analysis using diagnostic plots.................................29 3.3.2 Type curve matching....................................................................... 32 3.3.3 Nonlinear parameter estimation......................................................33 3.4 Complexities of pressure transient analysis of N F R s.................................33 3.5 Double porosity m odel................................................................................34 3.6 Modification to the double porosity m o d el................................................37 3.6.1 Double permeability m odel.............................................................37 3.7 Pressure analysis of naturally fi-actured reservoirs (NFR) ........................ 38 Representation of NFRs by Dual Fracture M o d e l...........................................43 4.1 A bstract.......................................................................................................43 4.2 Introduction.................................................................................................43 4.3 Triple porosity m odel..................................................................................45 4.4 Proposed new models..................................................................................46 4.4.1 Dual fi-acture model ....................................................................... 46 4.4.2 Modified triple porosity model ...................................................... 49 4.4.3 New dual fi-acture m odel.................................................................50 4.4.4 Pressure derivative diagnostic plots................................................52 4.5 Discussion ...................................................................................................57 4.5 Conclusion...................................................................................................59 Well Test Analysis of Faulted R eservoirs..........................................................60 5.1 An overview of sealing faults in oil reservoirs........................................... 60 5.2 Partially communicating faults ...................................................................61 5.3 Pressure transient analysis of faulted reservoirs.........................................62 Analytical Models for Faulted Homogeneous and N F R s................................ 65 6.1 The principle of superposition and method of images .............................. 65 6.2 Analytical solutions of homogeneous and NFRs with sealing faults .... 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.2.1 Analytical solutions for homogeneous model with sealing faults 68 6.2.2 Analytical solution for double porosity model with sealing faults 69 6.3 New models for NFRs with totally sealing 6 u lts........................................71 6.3 .1 Double permeability model with sealing faults..............................71 6.3.2 Dual fracture and triple porosity models with sealing faults .... 72 6.3.3 Pressure transient analysis of the new proposed m odels...............74 7 Analytical Model for NFRs with Partially Communicating F a u lts .................77 7.1 Model description and mathematical derivation..........................................77 7.2 Mathematical representation of the new m odel..........................................78 7.2.1 Region-1 (active-well region, x>0) ...............................................80 7.2.2 Region-2 (non-active region, x < 0 ):...............................................81 7.2.3 Initial and boundary conditions..................................................... 81 7.3 Dimensionless parameters.............................................................................82 7.4 Dimensionless flow equations...................................................................... 84 7.4.1 Region-1 (active region, x > 0 )........................................................84 7.4.2 Region-2 (non-active region, x>0).................................................84 7.4.3 Dimensionless initial and boundary conditions.............................. 85 7.5 Method of solution....................................................................................... 85 7.5.1 Flow equations in the Laplace domain...........................................86 7.5.2 Flow equations in the Fourier dom ain...........................................87 7.6 Model solution in the Fourier dom ain.........................................................88 7.7 Model solutions in the Laplace domain .....................................................89 7.7.1 Dimensionless pressure distribution in active well region.............89 7.7.2 Dimensionless pressure distribution in observation well region . 90 8 Pressure Drawdown Analysis ................................................................................92 8.1 Dimensionless drawdown pressure distribution in the active well ............92 8.2 Incorporation of skin damage and wellbore storage effects.......................93 8.3 Model validation and asymptotic models.................................................... 93 8.4 Pressure drawdown at the active well (region-1, x>0): ............................. 94 8.4.1 Effects of fault transmissivity ratio ...............................................95 8.4.2 Characteristic patterns o f pressure drawdown response 103 8.4.3 Effects of fault distance ( d p ) ........................................................107 8.4.4 Effects of fracture storativity (< d ) ...............................................112 8.4.5 Effects of inter-porosity flow parameter (X) ...............................119 8.4.6 Effects of skin damage (S) and wellbore storage (Cg) ............. 123 9 Pressure Interference Analysis ........................................................................... 132 9.1 Dimensionless pressure distribution in region-1 ........................................ 132 9.2 Dimensionless pressure distribution in region-2........................................ 133 9.3 Pressure interference tests in the active well region-1 ............................. 133 VI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9.3.1 Effects of controlling parameters on interference tests in region-1 .........................................................................................133 9.3.2 Effects of fault-well configuration and geom etry........................136 9.4 Type curve M atching................................................................................ 136 9.4.1 Type curves for interference tests in region-1 ............................ 137 9.5 Pressure interference tests in the observation well region-2...................144 9.5.1 Effects of controlling parameters on interference tests in region-2 .........................................................................................144 9.5.2 Type curves for interference tests in region-2 ............................ 147 9.5.3 Effects of fault-well configuration and geom etry........................147 10 Conclusions and Recommendations for Future W ork .................................156 10.1 Conclusions...............................................................................................156 10.2 Recommendation for future w o rk ............................................................161 Appendix-A Mathematical D erivation.......................................................................163 Appendix-B Computer Program ................................................................................170 Appendix-C Asymptotic M odels..................................................................................175 Appendix-D Fit Information for a^ and e Correlation............................................. 177 Appendix-E Relationship between X and X " ..............................................................180 Appendix-F Fit Information for d^ and m C orrelation...........................................181 Appendix-G Relationship between Cp and Cp ........................................................183 R eferences......................................................................................................................... 184 vu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables Table -1.1: Descr^tion of proposed conceptual models for the representation of NFRs and faulted N F R s............................................................................ 3 Table-3.1 : Application of different diagnostic plots to various reservoir models and flow regimes......................................................................................... 30 Table-4.1: Number o f time cycles to the end of the transition period since the first data point..............................................................................................58 Table-8.1: Drawdown tests to investigate the effect of fault communication parameter (e ) on transient pressure response............................................ 96 Table-8.2: Drawdown tests to investigate the effect of fault distance to the active well (dp) on transient pressure response ................................................. 107 Table-8.3: Drawdown tests to investigate the effect of fi'acture storativity ratio (œ) on transient pressure response............................................................113 Table-8.4: Drawdown tests to investigate the effect of inter-porosity flow parameter (X) on transientpressure response...........................................119 Table-8.5: Drawdown tests to investigate skin damage (S) and wellbore storage (Cn) effects on transient pressure response............................................. 124 Table-9.1: Interference tests in region-1 to investigate the effect of fault communication parameter (e ) and well locations (0<x<b) ..................... 134 Table-9.2: Interference tests in region-1 to investigate the effect of fault communication parameter (e) and well locations (0<b<x) ..................... 135 Table-9.3: Interference tests in region-2 to investigate the effect of fault communication parameter (e ) and well locations (b=500) ................... 145 Table-9.4: Interference tests in region-2 to investigate the effect of fault communication parameter (e ) and well locations (x=-500)................... 146 Table-9.5: Interference tests in region-2 to investigate the effect o f fault and well locations (b=250) .................................................................................... 152 Table-9.6: Interference tests in region-2 to investigate the effect o f fault and well locations (x=-250).................................................................................... 153 vm Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures Figure-1.1: Reservoir model matching and process {after Home [1]).............................2 Figure-2.1 : Modes of fracturing, tensile fractures or joints (mode-I), and shear fractures or 6ults (mode-II and mode-III), {after C. H. Scholtz /V40................................................................................................7 Figure-2.2: Normal and shear stress components of a force acting on a plane.................8 Figure-2.3: Graphical representation of two-dimensional state of stress by Mohr circle diagram [19]........................................................................................10 Figure-2.4: General fault classification based on the displacement in the fault blocks (a)direction of the strike of the fault plane, (b) direction of the dip of the fault plane, (c) actual or net displacement {after Uemura and Mizutani [27]).......................................................................................21 Figure-2.5: Slip dip faults (i): right (A-B) and left-handed slip fault (A-B), (ii) strike slip faults: gravity or normal feult (A-B) and reverse or thrust fault (A-C), {after Uemura and Mizutani [27])..........................................21 Figure-2.6: Rotational feult classification (i) hinge feult (ii) cylindrical fault {after Uemura and Mizutani [27])........................................................................21 Figure-2.7: Sealing faults in homogeneous reservoirs.................................................... 22 Figure-2.8: Partially communicating faults in homogeneous reservoirs......................... 22 Figure-3.1: Development o f straight line representing infinite-acting radial response on semi-log plot for drawdown test............................................. 31 Figure-3.2: Development o f straight line representing infinite-acting radial response on derivative plot for drawdown test........................................... 31 Figure-3.3: NFR models that can behave as a homogeneous reservoir (a)&(b), composite reservoir (c), isotropic reservoir (d), vertically fractured well (e), or as an NFR exhibiting the double porosity behavior (f), {after Cinco-ley [39]).................................................................................. 35 Figure-3.4: Idealization of the double porosity model fi)r the representation of NFRs {After Warren and Root [5]).............................................................36 Figure-3.5: Production through fracture network at wellbore for the double porosity model.............................................................................................. 38 Figure-3.6: Production through both fracture network and matrix system for the double permeability model........................................................................... 38 Figure-3.7: Pressure drawdown response in NFR with double porosity behavior on semi-log plot............................................................................................40 IX Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-3.8: Pressure drawdown response in NFR with double porosity behavior on log-log and derivative plot.......................................................................40 Figure-3.9: Effects of (1) and (m) on semi-log plot........................................................41 Figure-3.10: Effects o f (k) and (eo) on derivative plot......................................................41 Figure-3.11 : Double permeability behavior on semi-log plot............................................42 Figure-3.12: Double permeability behavior on derivative plot..........................................42 Figure-4.1 : Idealization of dual fracture model for the representation of NFRs............ 44 Figure-4.2: Triple porosity model.................................................................................... 45 Figure-4.3: General response of the triple porosity model on the pressure derivative plot............................................................................................... 46 Figure-4.4: Model-1, dual fracture representation by the modified triple porosity model............................................................................................................. 48 Figure-4.5: Model-2a, dual fracture model with production through the macro fiacture system....................................................................................48 Figure-4.6: Model-2b, dual fracture model with production through both macro and microfracture systems............................................................................ 48 Figure-4.7: General response of model-1......................................................................... 53 Figure-4.8: General response of model-2a........................................................................53 Figure-4.9: General response of model-2b....................................................................... 54 Figure-4.10: Typical response of dual fracture system (model-2b)...................................54 Figure-4.11 : The combined response of the two fracture systems....................................56 Figure-4.12: Early trough representing micro fracture response........................................56 Figure-4.13 : Delay o f the micro fracture response..............................................................57 Figure-5.1: Different foult configurations discussed in the literature (a) single fault, (b) parallel faults, (c) perpendicular faults, (d) angular faults............61 Figure-5.2: Doubling of the infinite-acting slope due to the presence of a sealing fault ............................................................................................................. 63 Figure-5.3: Doubling of the derivative value due to the presence sealing faults.............63 Figure-5.4: Pressure deflection on semi-log plot due to the presence of a partially communicating fault..................................................................................... 64 Figure-5.5: Existence of a hump on the derivative plot due to the presence of a partially communicating fault........................................................................64 Figure-6.1: Constant pressure boundary as represented by pressure field (after Home [58])................................................................................................... 67 Figure-6.2: No-flow boundary as represented by pressure field {after Horne [58J). ......................................................................................................................67 Figure-6.3: Pressure drawdown test of the double porosity model with totally sealing fault on semi-log plot........................................................................70 Figure-6.4: Pressure drawdown test of the double porosity model with totally sealing fault on derivative plot......................................................................70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-6.5: Pressure drawdown response of the double permeability model in the presence of a totally sealing fault on semi-log plot......................................75 Figure-6.6: Pressure drawdown response of the double permeability model in the presence of a totally sealing fault on derivative p lo t..................................75 Figure-6.7: Pressure drawdown response of the dual fracture model in the presence of a totally sealing fault on semi-log plot......................................76 Figure-6.8: Pressure drawdown response of the dual fracture model in the presence of a totally sealing fault on pressure derivative plot.....................76 Figure-6.9: Pressure drawdown response of the triple porosity model in the presence of a totally sealing fault on semi-log plot......................................77 Figine-6.10: Pressure drawdown response of the triple porosity model in the presence of a totally sealing fault on derivative plot....................................77 Figure-7.1 : Schematic representation of the proposed model......................................... 79 Figure-7.2: Representation of the semipermeable fault zone...........................................79 Figure-7.3: Different configurations and well locations for pressure drawdown and interference tests of the new model....................................................... 91 Figure-8.1 : Active well location such that the inter-well distance is equal to the well radius for pressure drawdown tests......................................................94 Figure-8.2: Pressure drawdown test of the double porosity model with partially communicating fault on semi-log plot.......................................................... 97 Figure-8.3: Pressure drawdown test of the double porosity model with partially communicating fault on derivative plot........................................................ 97 Figure-8.4: Magnification of the hump region representing partially communicating faults on derivative plot.......................................................98 Figure-8.5: Maximum point of the hump and relationship to the value of fault communication parameter (e)........................................................................99 Figure-8.6: Relationship between the value of maximum point of the hump and the fault communication parameter (e) is shown in..........................................100 Figure-8.7: Relationship between the maximum point of the hump and (e) based on the dimensionless pressure derivative plot............................................ 101 Figure-8.8: Relationship between the maximum point of the hump and (e) based on the amplitude ratio (a^).......................................................................... 101 Figure-8.9: Fit of regression model of Equation-8.6 to the original data based on relationship between (e) and (a^)................................................................ 102 Figure-8.10: Pressure drawdown test of the new model resulting in pattera-I on semi-log plot................................................................................................ 104 Figure-8.11 : Pressure drawdown test of the new model resulting in pattem-I on derivative plot.............................................................................................. 104 Figure-8.12: Pressure drawdown test of the new model resulting in pattem-II on semi-log plot................................................................................................ 105 XI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-8.13: Pressure drawdown test o f the new model resulting in pattem-II on derivative plot..............................................................................................105 Figure-8.14: Pressure drawdown test o f the new model resulting in pattem-III on semi-log plot................................................................................................106 Figure-8.15: Pressure drawdown test o f the new model resulting in pattem-III on derivative plot..............................................................................................106 Figure-8.16: The effect of fault distance (dp) of a sealing fault model on pressure drawdown behavior on semi-log plot.........................................................108 Figure-8.17: The effect of fault distance (dp) of a sealing fault model on pressure drawdown behavior on derivative plot.......................................................108 Figure-8.18: The effect of fault distance (dp) of a non-sealing fault model on pressure drawdown behavior on semi-log plot.......................................... 109 Figure-8.19: The effect of fault distance (dp) of a non-sealing fault model on pressure drawdown behavior on derivative plot........................................ 109 Figure-8.20: The effect of fault distance (dp) of no fault model on pressure drawdown behavior on semi-log plot.........................................................110 Figure-8.21: The effect of fault distance (dp) of no fault model on pressure drawdown behavior on derivative plot.......................................................110 Figure-8.22: Relationship between X and X ,* and fault distance (dp).............................112 Figure-8.23 : The effect of fracture storativity (o) of sealing fault model on pressure drawdown behavior on semi-log plot.........................................................114 Figure-8.24: The effect of fracture storativity (©) of sealing fault model on pressure drawdown behavior on derivative plot.......................................................114 Figure-8.25: The effect of fracture storativity (< o) of non-sealing fault model on pressure drawdown behavior on semi-log plot.......................................... 115 Figure-8.26: The effect of fracture storativity (©) of non-sealing fault model on pressure drawdown behavior on derivative plot........................................ 115 Figure-8.27: The effect of fracture storativity (®) of no fault model on pressure drawdown behavior on semi-log plot.........................................................116 Figure-8.28: The effect of fracture storativity (ca) of no fault model on pressure drawdown behavior on derivative plot.......................................................116 Figure-8.29: Relation between the depression ratio (d^) of the trough and the fracture storativity (<o).................................................................................117 Figure-8.30: Best fit of regression model of Equation-8.9 to the original data based on relationship between (©) and (d^).........................................................118 Figure-8.31 : The effect of inter-porosity flow parameter (X) of sealing fault model on pressure drawdown behavior on semi-log plot..................................... 120 Figure-8.32: The effect of inter-porosity flow parameter (X .) of sealing fault model on pressure drawdown behavior on derivative plot................................... 120 Figure-8.33: The effect of inter-porosity flow parameter (X .) of non-sealing fault model on pressure drawdown behavior on semi-log plot.......................... 121 Figure-8.34: The effect of inter-porosity flow parameter (X .) of non-sealing fault model on pressure drawdown behavior on derivative plot........................ 121 xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-8.35: The effect of inter-porosity flow parameter (A .) o f no feult model on pressure drawdown behavior on semi-log plot..........................................122 Figure-8.36: The effect of inter-porosity flow parameter (A .) o f no fault model on pressure drawdown behavior on derivative plot........................................122 Figure-8.37: Relationship between Cp and Cp* and 6ult distance............................ 124 Figure-8.38: The effect of both skin damage (S) and wellbore storage (Cp) for pattem-I on semi-log plot...........................................................................125 Figure-8.39: The effect of both skin damage (S) and wellbore storage (Cg) for pattem-I on derivative plot.........................................................................125 Figure-8.40: The effect of both skin damage (S) and wellbore storage (Cg) for pattem-n on semi-log plot......................................................................... 126 Figure-8.41: The effect of both skin damage (S) and wellbore storage (Cg) for pattem-n on derivative plot....................................................................... 126 Figure-8.42: The effect of both skin damage (S) and wellbore storage (Cg)for pattem-in on semi-log plot........................................................................127 Figure-8.43: The effect of both skin damage (S) and wellbore storage (Cg) for pattem-III on derivative plot......................................................................127 Figure-8.44: The increasing effect of both skin damage (S) and wellbore storage (Cg) for pattem-I on semi-log plot............................................................129 Figure-8.45: The increasing effect of both skin damage (S) and wellbore storage (Cg) for pattem-I on derivative plot..........................................................129 Figure-8.46: The increasing effect of both skin damage (S) and wellbore storage (Cg) for pattem-II on semi-log plot...........................................................130 Figure-8.47: The increasing effect of both skin damage (S) and wellbore storage (Cg) for pattem-II on derivative plot.........................................................130 Figure-8.48: The increasing effect of both skin damage (S) and wellbore storage (Cg) for pattem-m on semi-log plot......................................................... 131 Figure-8.49: The increasing effect of both skin damage (S) and wellbore storage (Cg) for pattem-in on derivative plot....................................................... 131 Figure-9.1 : Active and observation well locations and fault distance for pressure interference tests in region-1...................................................................... 136 Figure-9.2: Type curves generated by Table-9.1 (0<x<b), for interference tests in region-1 (sealing fault, e=0.0)....................................................................138 Figure-9.3 : Type curves generated by Table-9.1 (0<x<b), for interference tests in region-1 (non-sealing fault, e=0.20).......................................................... 139 Figure-9.4: Type curves generated by Table-9.1 (0<x<b), for interference tests in region-1 (no fault, e=20.0)......................................................................... 140 Figure-9.5: Type curves generated ly Table-9.2 (0<b<x), for interference tests in region-1 (sealing fault,e=0.0)..................................................................... 141 Figure-9.6: Type curves generated hy Table-9.2 (0<b<x), for interference tests in region-1 (non-sealing feult,e=0.20)........................................................... 142 xiu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-9.7: Type curves generated by Table-9.2 (0<b<x), for interforence tests in region-1 (no fault, s f =20.0)..........................................................................143 Figure-9.8: Active and observation well locations and fault distance for pressure interference tests in region-2...................................................................... 144 Figure-9.9: Type curves generated by Table-9.3 (b=500 ft), for interference tests in region-2 (non-sealing fault, 6=0.20)....................................................... 148 Figure-9.10: Type curves generated by Table-9.3 (b=500 ft), for interference tests in region-2 (no fault, e=20.0)......................................................................149 Figure-9.11 : Type curves generated by Table-9.4 (x=-500 ft), for interference tests in region-2 (non-sealing fault, e=0.20)..................................................... 150 Figure-9.12: Type curves generated by Table-9.4 (x=-500 ft), for interference tests in region-2 (no fault, 6=20.0)..................................................................... 151 Figure-9.13 : Type curves generated by Table-9.5 (b=250 ft), for interference tests in region-2 (non-sealing fault, 6=0.20)....................................................... 154 Figure-9.14: Type curves generated by Table-9.6 (x=-250 ft), for interference tests in region-2 (non-sealing fault, 6=0.20)....................................................... 155 XIV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Nomenclature =amplitude ratio, b =x-coordinate o f the active-well location, (ft), c =total compressibility, (psi '). C =wellbore storage constant. Cd =dimensionless wellbore storage constant based on r„. Cd* =dimensionless wellbore storage constant based on L. dp =distance from the producing well to the fault, (ft), dg =depression ratio. Ei =exponential integral function. e =exponential function. f(s) =function defined by equation-7.43. ^ =Fourier transformation operator, h =formation thickness, (ft), kp =effective permeability of fault zone, (md). k^, =effective permeability of matrix system, (md). Ko =modified Bessel fimction of second kind and zero order. If =efiective horizontal thickness of fault, (ft). L =inter-well distance, (ft). LAV =least absolute value. LS =least square. S E =Laplace transformation operator. ml =first semi-log slope change for infinite-acting reservoir. m2 =second semi-log slope of pressure change for image well. P =pressure, (psi). q =production rate, (RB/D). r^ =wellbore radius, (ft). s =Laplace transfi>rm variable. S =skin factor, t =time. Tp =specific transmissibility of fault zone, (kp hp / Ip p). Tf =specific transmissibility of fracture system (kf hf / L p). u =variable o f integration. V =volumetric leakage rate per unit length o f fault, (RB/D-ft). w =pressure in Fourier space. X =x-axis coordinate, y =y-axis coordinate. Z =fiinctions defined by equation-7.46 and 7.47. XV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Greek Letters a =inter-porosity shape factor, (ft '). Ô =symbol for Dirac delta fimction. e =fault communication parameter. T | ^hydraulic diftusivity. K =fracture conductivity thickness. X =inter-porosity flow parameter based on r^. X* =inter-porosity flow parameter based on L. |i =viscosity, (cp). < |) =porosity. o =normal stress. T =shear stress. 0) =ft"acture storativity ratio. ^ =Fourier transform variable. Subscripts c =critical. D =dimensionless. F =fault or macrofracture. f ^fracture or micro fracture. i =initial. m =matrix. n =1 or 2. r =ratio. t =total. w =well. X =x-direction. y =y-direction. z =z-direction. 1 =region-l. 2 =region-2. Subscripts * =dimensionless Parameter based on r„ XVI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract The objective of this study is to develop new and improved analytical conceptual models for the characterization of naturally fractured reservoirs (NFR) in the presence of other related heterogeneities such as microfracture network, totally sealing and partially communicating feults. Fractures and 6ults constitute important categories of reservoirs heterogeneities that may result from sequential geologic and tectonic disturbances and their coexistence in the same reservoir is quite common. First, a dual fracture system is proposed as a more realistic representation of fracture networks in NFRs where two different sets of fractures, macrofractures and microfractures are present in addition to the matrix. This model is a general one and can easily be modified to produce other conceptual models for NFRs including triple porosity, double permeability, and dual porosity models. For the representation of sealing and non-sealing faults in NFRs, three new models are introduced. The first two are generated by ^plying the principle of superposition and the method of images to the double permeability and the dual fracture models, to account for the presence of a totally sealing fault. The last model is a general one that incorporate the effect of partially communicating faults on the pressure behavior of NFRs. For the dual fracture model, delineation of fractures into two broad categories, macrofractures and microfractures, represents a step forward toward a more realistic representation of NFRs. The model provides an explanation for the observation of early pressure support emanating from a network of microfi-actures and often mistakenly attributed to the tight matrix. On derivative plots, the pressure support of micro fractures is similar to that of the matrix, but transition zones form at a substantially earlier time. The model provides explanation for the observation, or the lack of observation of single or double transition periods in test data from NFRs and suggests that extended test duration are required for the detection of the tight matrix rocks. xvu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Solutions to the new models developed for the representation of totally sealing 6ults in NFRs were obtained applying the methods of images and superposition to the Laplace- space solutions of both double permeability and dual fracture models. It was evident that the time to double the slope, vsdiich indicates the presence of sealing 6ults in NFRs, is dependent on fault distance (dp), inter-porosity flow parameter (X) and fracture storativity ratio (©). For the last model of NFRs with partially communicating faults, two solutions are presented for the pressure distribution in the two regions on either side o f fault. Representation of the 6ult was accomplished by considering a semi-permeable linear barrier o f infinite extent and negligible storage capacity, that divide the reservoir into two regions. Region-1, located on the right side of the fault, contains the producing well while region-2 is located on the other side of the fault plane and contains only the observation well. On semi-log plot, the typical response of this model shows three straight lines appearing at early, middle and late times, separated by two transition periods. The first one results in an increased slope characteristic of no-flow boundaries, while the second transition period is that of NFRs. On the derivative plot, the response is also exhibited by three straight lines at early, middle and late times respective^, separated by the same two transition periods. For sealh% 6ults, the 6ult transition period shows a gradual and steady increase of derivative value until it becomes double that of the first straight line. For non-sealing faults however, the first transition period resembles a hump where the derivative value reaches a maximum point then drops back to the infinite-acting straight line. For faults at close distance to the well, slope changes occur before the arrival of the transition period of NFRs which arrives at a later times (pattem-I). For faults at large distance, slope change occurs after the arrival of the transition period of NFRs (pattem-III). For distances in between, the slope behavior in both early time and middle times may be influenced resulting in complicated pattem (pattem-II). For communicating fruits, there is a critical fault communication parameter (8^=10.0) above which the second straight line is an extension of the first one and the fault presence, for all practical purposes, is undetectable. The amplitude o f the hump was found to be inversely related to e. Graphical and empirical correlations were developed, relating the amplitude ratio (a^) of the hump to e. These correlations can be used to calculate e with xvm Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. considerable accuracy from the ratio of amplitude (the value of the maximum point of the hump to the value of the infinite-acting response). It is also found that the minimum point of the depression representing NFRs response is inversely related to m. Graphical and empirical correlations were also developed relating the depression ratio (d,) of the trough to m and can be used to calculate m with accuracy. Solutions for regions-1 and 2 were also analyzed first by conducting pressure interference tests in an observation well located in region-1, followed by interference tests in region-2 across the fault. For the first case, tests were run to examine effects of the controlling parameters as well as various well configurations with respect to the feult location. Pressure response in an observation well across the fault does not depend on the fault distance to either active nor observation well, but only on the inter-well distance. New type curves were generated ly solutions for regions-1 and 2 . These type curves can be used to analyze pressure interference test data in order to estimate the defining parameters including the fault communication parameter (e), fault distance (dp), inter-porosity flow parameter (A .) and fracture storativity ratio (®). Similar families of type curves can be generated to include the effects of wellbore storage coefiBcient (Cp) and skin factor (S). XIX Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction and scope 1.1 Introduction Pressure transient tests are among the most powerful tools used for the evaluation of reservoir properties and near wellbore conditions. Valuable information can be obtained about the well conditions, such as wellbore storage, wellbore damage and improvement, in addition to information about formation rock and fluid properties, reservoir pressure, size, continuity, and heterogeneities. Interpretation methods are based on the assumption that the system being tested behaves according to a given well and reservoir conditions. The selection of an appropriate conceptual model that will regenerate the system’s response is the initial objective for well test analysts. The response o f a particular conceptual model is governed by its controlling parameters such as permeability, storativity, skin factor, wellbore storage coefiBcient, distance to boundaries, reservoir limits etc. The estimation of unknown reservoir parameters from observed well pressure (recorded field data) is an “inverse problem” that requires matching the observed response to that of a specific conceptual model as illustrated in Figure-1.1. Once a reliable match is obtained and a proper model has been identified and validated, the ultimate goal of well test analysis can be accomplished by solving the inverse problem of parameter estimation. Numerous analytical conceptual models, incorporating various types of reservoir heterogeneities, have been introduced in the literature for interpretative purposes. Generally speaking, the majority of existing models are limited to one type of heterogeneity which constitute an oversimplification of reality through idealistic assumptions of homogeneous and 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. field input reservoir parameters k,s,C reservoir response P t Match model response model parameters k.s.C model input P t Figure-1.1: Reservoir model matching and process {after Home [1J). or isotropic storage and flow units. Rock formations in oil reservoirs are highly heterogeneous in nature, and many types of heterogeneity are developed within the same reservoir as a result of past geologic processes. Fractures and foults constitute important categories of reservoir heterogeneities that may result from the same geologic and tectonic disturbances and their coexistence in the same reservoir is a common place. This study focuses on the development of new conceptual models that takes into account the presence of geologically related heterogeneities in naturally fl-actured reservoirs (NFR). 1.2 Objective and scope In this study, improved and more realistic analytical conceptual models, for the characterization of NFRs, in the presence of other related heterogeneities, are developed and their exact analytical solutions are presented in Laplace space. First, the dual fincture system was proposed by Al-Ghamdi and Ershaghi [2] as a more realistic representation of the fl-acture network in NFRs where two dififerent sets of fractures, macroflactures and microfiactures system are present in addition to one matrix system. This model is a general one and can easily be modified to produce other conceptual models for NFRs including triple porosity [3], double permeability [4], and dual porosity [5] models. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For the representation o f geologic &ults (both sealing and non-sealing) in naturally fractured reservoirs (NFR), three new conceptual models are introduced. To account for the presence of totalfy sealing 6ults in the proximity of the tested well, the first two models are generated by applying the principle of superposition and the method of images [6] to the double permeability and the dual fiucture model respectively. The last model is a more general one that incorporates the effect o f a non-sealing fault, also called partially communicating or conductive fault, on the pressure behavior ofNFRs. A list of these new proposed models is shown in Table-1.1 with brief description of the underlying theoretical basis and controlling parameters. Table -1.1: Description of proposed conceptual models for the representation of NFRs and faulted NFRs # Type of model Type of NFR model Type of fault Flow & storage units Controlling parameters 1 NFR Dual fracture None 2 fracture systems & matrix Àf , À„ , (Op, (Of , Kp , S , C q 2 NFR Triple porosity None Fracture & 2 matrix systems Ân/ ' ^ 2 ' ^ml , Kf , S , C q 3 Faulted NFR Double permeability Sealing Fracture & matrix A , , (0 , K f, dp , S , C q 4 Faulted NFR Dual fracture Sealing 2 fracture systems & matrix kf, k„, (Op, (Of , Kp , d p , S , Cp 5 Faulted NFR Triple porosity Sealing Fracture & 2 matrix systems ^ 1 ’ ^ 2 , K f , (Of > ^ m l 'dp, S , Cp 6 Faulted NFR Double porosity Non- Sealing Fracture, feult & matrix systems k , (0 , dp, s , S , Cp Analysis of the generated transient pressure response of these models is conducted on Industry Standard Diagnostic Plots (ISDP). Conventional graphical analysis is conducted on Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. semi-log, log-log plots and by type curve matching. Pressure derivative plot analysis is also employed and emphasized in particular as it is proven to improve the accuracy of test interpretation [7-12] including pattern recognition and parameter estimation processes. In chapter 2, geological aspects ofNFRs are reviewed in relation to the origin of fractures and 6ults and the mechanisms of both fracturing and faulting processes. In chapter 3, fundamental aspects related to conceptual modeling and pressure analysis of naturally fractured reservoirs are highlighted and discussed. In chapter 4, the new conceptual model of dual fracture systems is presented as a step forward toward a more realistic representation of the fracture network in NFRs. chapter 5 is a brief discussion of conceptual modeling and pressure analysis of faulted reservoirs, chapter 6 includes the representation o f completely sealing faults or no flow linear boundaries in naturally fractured reservoirs. New analytical solutions for different NFR models with tota% sealing foults are presented, including the dual fractured model developed in chapter 4. chapter 7 presents the mathematical basis, derivation and the exact analytical solutions for the transient pressure behavior ofNFRs with partially communicating faults. The new solutions are all presented in the Laplace domain and have been numerically inverted using Stehfest algorithm [13] to generate their corresponding pressure response. For the last model ofNFRs with partially communicating faults, two solutions are presented for the pressure distribution in the two regions on either side of the non-sealing fault. Solution for the active region provides us with the ability to conduct drawdown and build-up tests at the active (producii^) well, in addition to pressure interference tests in the producing region on the same side of the feuh. On the other hand, solution for the inactive region allows us to conduct interference tests across the non-sealing fault where pressure is recorded at an observation (inactive) well located on the other side of the fault. In chapter 8, pressine drawdown behavior at the active well is generated and analyzed, while pressure interference tests conducted in both regions of the reservoir are discussed and analyzed in chapter 9. Final comments, concluding remarks and recommendations for future work are highlighted in chapter 10. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Physical and Geological Aspects of NFRs 2.1 Characteristics of fractured rocks Geological and hydrological characterizations of fractures are the subject of extensive inter-disciplinary research due to their vital role in numerous engineering, geotechnical, and hydrological applications. Fractures represent both opportunities and difQculties associated with their existence in all types of geologic rocks and play a deterministic role in the fluid transport phenomenon below the earth’s surface. Characterization of subsurface fluid flow in fractures is crucial in many underground applications such as isolation of toxic and hazardous waste, mining, and rock excavations. Fluid flow in subsurface fractures can also affect numerous surface applications such as the structural stability of tunnels, bridges, and dams. Furthermore, many economically significant underground reservoirs are formed in fi-actured rocks including, petroleum, geothermal, and water supply reservoirs, where fracture properties such as size, density, orientation, conductivity and connectivity, are crucial to reservoir economics. Hydrocarbon resources residing fractured and fissured rocks, known as naturally fi’ actured reservoirs (NFR), constitute a significant portion of the worldwide hydrocarbon reserves and production. Table-2.1 lists location and lithology of some o f the major NFRs throughout the world. Fracture characterization starts with the identification and location of hydraulically significant fractures that might act as flow conduits (conductive fractures) or flow barriers (sealing faults), that will control the overall fluid transport process of the reservoir and influence the flow dynamics of the total system’s at the field scale. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table-2.1: Geographical distribution and lithology of major producing NFRs of the world Geographical location Country or area Rock type Europe France Limestone Norway Chalk United Kingdom Sandstone Far East Philippines Limestone Middle East Iran Limestone Iraq Limestone Kuwait Sandstone Oman Limestone Qatar Limestone Saudi Arabia Limestone North Africa Libya Limestone & Sandstone North America Canada (Alberta) Limestone Mexico Limestone USA (California) Complex lithology (Siliceous rocks) USA (Oklahoma) Dolomite USA (Texas) Limestone South America Brazil Sandstone Venezuela Conglomerate Brazil Sandstone Existence of fractures and friults are common in most geological structures and an understanding of the mechanisms leading to their formation is of overriding significance to any discussion concerned with oil and gas production from naturally fractured reservoirs (NFR). This Chapter is an overview of the in ^rtan t geological, geophysical and hydrological aspects of fi-actures and fractured rocks. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2 Fracture classifications Based on the nature o f the displacement discontinuity, fractures can be divided into three major classifications: ( 1) tensile fractures or joints, (2 ) shearing fractures or faults, and (3) closing firactures or pressure solution surfaces [14]. This study is mainly concerned with the characterization of hydraulically significant fractures and faults created by mechanical breakage of brittle rocks. Therekre, discussion will only focus on the first two classes of fi-actures and the geological processes responsible for their formation. Tensile fiuctures (also called dilating, opening or parting fractures) can be visualized as two fracture surfaces with normal displacement discontinuity (direction of displacement is perpendicular to the surfaces). These types are called mode-I fractures in engineering and fracture mechanics. Shear fractures, also referred to as faults, are shear displacement discontinuities (fracture surfaces move predominantly parallel to each other). Mode-II or mode-in shear fixtures result when relative movement of the two surfaces is perpendicular and parallel to the fracture front respectively (Figure-2.1). Mixed-mode fractures are characterized by a combination of fracturing modes. The most common combinations of fiacturing modes are 6 ulted joints and jointed faults which are kinematically similar to faults and joints. However, because of the compounding effects of the different processes involved, the resulting fiacture systems o f mixed mode origin may posses contrasting geophysical and hydrological properties (fracture geometry, size, shape, orientation, connectedness and internal structures). mode-III mode-I mode-U Figure-2.1 : Modes o f fracturing, tensile fractures or joints (mode-I), and shear fractures or faults (mode-U and mode-IU), {after C. H. Scholtz [14]). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3 Brittle rock failure and mechanics of fracturing and faulting Rocks in the upper part of the lithosphere (schizosphere), where temperature and pressure are relatively low, behave as brittle or semi-brittle solids that deform by elastic fracturing when subjected to stress. The mechanism of brittle rock failure is manifested by the initiation and propagation of fissures and fi-acture cracks which may occur at various scales ranging from microscopic to continental. The initiation and propagation of fissures and fractures, the mechanism of brittle rock is controlled by many factors including rock type, depth, magnitude and state of stress field, and rock elastic properties. 2.3.1 State of stress and rock deformation When an external force is applied to brittle solid rocks, specific state of stress is produced inside it causing it to deform by mechanical fracturing. Stress in the earth crust results from the interaction of the various lithostatic (gravity), thermal, and tectonic forces [15-18]. Stress acting on a plane can be analyzed into two components, normal stress (a) acting perpendicular to the plane and shear stress (x) acting parallel to the plane (Figure 2.2). The relative significance of these two stresses depends on the angular relationship between the direction of the acting force and the surface. The state of stress at any given point can be defined by the three principal stresses acting normally on the three perpendicular planes passing through that point. By convention, the three principal stress components are known as maximum stress (ct,), intermediate stress (O;), and a minimum stress (0 3). Figure-2.2: Normal and shear stress components of a force acting on a plane. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3.2 Mohr stress circle and mechanical conditions of fracturing According to GrifBth theory, fiactures and 6 ults are created by strains that arise from stress concentrations around flaws, heterogeneities, and physical discontinuities [19]. The conditions for the initiation of fracture can be determined by applying one of the criteria of brittle rock failure. Several theories have been proposed for the mechanical criteria for the initiation of fractures in rocks. This study however will only focus on the most fundamental theories underlying both fracturing and Suiting processes. If a three-dimensional stress state is expressed by the three principal stresses (< r,> C T 2> a 3), then a and t for an arbitrary plane making an angle 0 with the a, axis are defined by the following equations: o, + < T . o, -o , o = - L - J . - - l —J-c o s(2 6 ) (1) a, -o, T = —— -sin (2 8 ) (2 ) Combining Equations-1.1 and 1.2 will produce a circular equation defining the well known Mohr stress circle (Figure 2.2): / \ 2 / \ 2 0 , - 0 3 o ---------- I 2 I 2 J (3) The two dimensional state of stress can be graphically represented, by plotting Equation-1.3 on rectangular coordinate system with normal stress (o) and shear stress (t) on the x-axes and y-axes respectively, to generate Mohr stress circle diagram. The center of the circle is located at a distance equal to (a|+0 3)/2 , while the radius has a value of (a,-(T3)/2 . Mohr [19] found that the relationship between a normal stress (c t ) and a shear stress (x) acting on the fracture plane is |x|=!/(cy). For every rock type, there is a characteristic curve defined by the equation |x|=^o), which represents the elastic properties and expresses the failure criterion for that type of rock. The values o f o and x for an arbitrary angle (0) may be given by coordinates on the stress circle and fracturing is expressed as a point of contact between the curve expressed by |xl=y(cr) and Mohr’s stress circle. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Liae «fftaetne Figure-2.3: Graphical representation of two-dimensional state of stress by Mohr circle diagram [19]. 2.3.3 Coulomb criteria of brittle rock fracture Coulomb [19] discovered that the relationship between a and x at the time of fiacturing can be approximated with a straight line known as line of fi-acture (Figure-2.3). The equation describing this fi-acture line is |x|=Xo+crtan(a), where Xgand a are defined as the cohesion coefficient and angle of internal fiiction (coefficient o f internal fiiction) respectively, and are specific constants for each material. According to the Coulomb theory of rock failure, a conjugate set of shear fiactures will develop once the state o f stress is such that the lines of fiacture become tangential to the Mohr circle. This can be accomplished either by increasing the maximum stress (ct,), or decreasing the minimum stress (C T j). In conclusion, one can say that brittle rocks failure and the initiation of fi-acture originate fiom strains that arise fiom stress concentrations and take place according to various combinations of stress state and rock elastic properties. Different episodes of folding, Suiting, uplift, subsidence, and tectonic plate movements can result in multi-fracture systems representing these different sequential geological processes. Moreover, various rock types within the same reservoir structure will deform differently and fracturing may only be regional 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and limited to specific rock types with brittle behavior. Naturally fi-actured reservoirs (NFR) are highly heterogeneous systems with complex stochastic distribution of physical fi-acture properties such as size, shape, density, orientation, and hydraulic conductivity. Description of this complex nature o f heterogeneities ofNFRs is highlighted in the next section. 2.4 Geological and physical characteristics of fractures Fracture geometry depends on how it propagates and terminates and is controlled by factors such as geology o f the fiactured rock mass, stress distribution, loading conditions, and interactions with preexisting fiactures [20]. In layered sedimentary rocks, fi-actures are usually formed perpendicular to the layers. Once the fi-acture propagation fi-ont reaches the top boundary of the layer, it propagates laterally and becomes nearly perpendicular to bedding. Fractures in adjacent layers commonly join to form bath-ways for interlayer fluid flow. The existence of thin shale laminae between layers may cause fiacture termination resulting in hydraulic isolation. This is usually the case for thick shale layers causing fi-actures and joints to be contained in certain stratigraphie units. Fracture sets comprise a number of ^proximately parallel fiactures of the same type, origin and age [20]. Fracture sets can be described by the areal/vertical extent, orientation, spacing or density, physical connections between individual fi-actures and fi-acture aperture (size of opening measured normal to the fi-acture walls). Among the physical characteristics of fi-acture sets, the spacing of individual fi-actures (particularly for joints) has attracted the most attention. It has been suggested that fi-acture spacing depends on the rock’s elastic properties. In layered rocks, it is believed that spacing is also proportional to the thickness of the layer. Recent experimental and numerical studies indicates that spacing is the result of dynamic processes and temporal fluctuation in stress fields, strain magnitude, strain rate, and loading cycle. It has also been suggested that fi-acture spacing may evolve spatially by selective growth of existing fi-actures and cracks. Interaction between fi-actures influences how they are spaced and clustered, hence their connectedness to form fi-acture networks. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 Multiple fracture sets and fracture network models The formation of one fracture set promotes and controls the Initiation of additional sets by providing new stress concentrators and barriers for the deforming rocks. The occurrence of multiple sets representing different episodes of fracturing is the rule rather than the exception in fractured rocks. The simplest multiple fracture pattern is comprised o f two sets of different geometries (size, density, orientation, etc.), where the shorter and discontinuous fractures link longer and more continuous fractures. In isotropic rocks, the intersection angle between the two sets is non-orthogonal when formed under different stress fields. Two orthogonal sets can also be formed simultaneously under a same stress system and orthogonal fracture sets are frequently observed in nature [2 0 ]. Acuna and Yortsos [21] presented a method to generate two and three-dimensional fracture networks using fractal geometry to represent the fracture heterogeneity. Acuna et al. [2 2 ] presented a single-phase, two-dimensional fracture network simulator based on fractal interpretation of the fractured medium to study the pressure transient response of this type of network. Sammis et al. [23,24] presented field evidence that fractures in the geysers geothermal reservoir have finctal geometry and that fracture distribution is scale independent over a wide range o f scales (centimeters to kilometers). Models conceptualization of flow systems in fractured rocks (NFR) should include all of the essential physical and geological characteristics of fi-actures. Natural fractures are highly inhomogeneous and have spacial stochastic distributions with respect to their geometries. Fractures in a network are never identical or equally significant and the overall network maybe dominated by a single fracture or group of regional and localized fi-actures. It is obvious that no single model can represent all the possible variation of natural fractures properties. Spatial and temporal fluctuation of fracture characteristics in terms of their geology, geometry and hydraulics pose many problems that need to be overcome in order to build useful models. The introduction of numerous conceptual models for the representations of naturally fi-actured reservoirs have shed some light on similarities and differences among their characteristic pressure response. General review of the existing conceptual models for the representation of NFRs (including double porosity, double permeability and triple 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. porosity) is discussed in chapter 3. One common over-simplistic assumption shared by these models is homogeneous and isotropic fracture network. In chapter 4, a dual fracture model (two fracture sets) is introduced as a more general model to distinguish between macrofracture and microfracture systems or networks existing in the same reservoir. 2.6 Fracture detection methods Fractures constitute anomalous concentrations of void space in rocks and their geometry affects both fluid flow dynamics and overall physical properties of the bulk rock. Fracture detection methods rely on the fact that fractures are essentially two-dimensional surfaces commonly organized into sets or networks with a preferred spatial orientation (anisotropic) that can be an important characteristic for detection purposes [25]. Numerous detection methods are based on the fact that fractures are commonly near vertical in sedimentary rocks, with a single preferred azimuth, or two orthogonal azimuths. However, as discussed earlier, multiple sets of fractures or fracture networks with variable orientations can be created by successive fracturing episodes in tectonically active areas. Fracture detection during exploration and earfy drilling stages is a very important task that will greatly impact planing of future reservoir development. During the drilling stage, the presence of fractures is reflected by certain indicators such as loss of circulation, mud cut by oil and gas and change in rate of penetration. Comprehensive techniques for locating flow systems in fractured rocks are based on the analysis of several detection methods including core sampling, geochemical analysis of produced water samples and hydraulic and tracer tests, in addition to geophysical well log and seismic surveys. 2.6.1 Core inspection Core examination is one of the most obvious and direct analysis method for characterizing fracture properties. However, it has several major drawbacks. First, cores are always of small volume sample that are only representative of rocks in the immediate vicinity of the wellbore. Increasing the number of core samples obtained can be prohibitively expensive. Secondty, cores are most reliable when obtained from hard crystalline rocks where 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the integrity of samples is preserved. However, small core samples are likely to have been damaged or altered during drilling operations. Furthermore, core recovery is usually poor in the intensely fractured regions o f the reservoir. Therefore, core studies may not be reliable for determining characteristics o f dominant fractures or areas of intensely firactured zones. Even with these drawbacks, cores sangles can provide important direct measurement of hydrological and geological properties of firactures and host rocks. Properties like fracture apertures can be estimated by measuring the diameter of isolated crystals deposited on the fracture faces. On the other hand, core analysis is most effective when it is correlated to geophysical well logs from the borehole. Combinations of core and well logs analysis can provide information may not be obtained from either approach alone. Well logs can also be used to determine fracture properties in intervals where core is unreliable or missing. 2.6.2 Geophysical methods Geophysical methods cover a wide range of sophisticated seismic, electrical, and electromagnetic techniques that are routinely used to locate fractures in the subsurface and measure their relevant physical and hydrological properties. 2.6.2.1 Seismic surveys Because elastic properties o f rocks can be determined at greater ranges than electric or electromagnetic properties, seismology is the technique most widely used to explore deep subsurface systems. Seismo logical investigations utilize both compression or P waves, in which the rock deforms in the direction of the wave, and shear or S waves, in which the rock deforms transverse (or perpendicular) to the direction of wave travel [25]. In reflection seismology, a controlled seismic source (or a closely spaced array of sources) imparts energy into the ground. The energy travels through the subsurface rocks and reflects off anomalous features (such as stratigraphie layers, lithologie boundaries, faults and fractures), and returns to the surface where it is received and recorded. In conventional two- dimensional (2-D) surveys, the sources and receivers are aligned along a straight line on the surfece that will result in a 2-D vertical cross section of the subsurface below. In 3-D surveys. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. seismic sources and receivers are arranged in a 2-D areal pattern on the surface. As a result, the investigated region is a 3-D volume of subsur&ce reservoir. The same principles apply even if the survey is not conducted at the earth's surface but from a tunnel or borehole. 2.6.2 2 Electrical and electromagnetic methods Detection of water-filled fractures by electrical and electromagnetic methods is possible because water-filled fi-actures generally have higher electrical conductivities than intact rock. This higher conductivity is attributed to fractures connectedness, higher mobility o f conductive ions in the water saturating the fi-acture, and existence of conductive clay minerals present as fracture linings or as alteration products in the adjacent rocks. These effects are more pronounced if the surrounding rocks have low porosities and permeabilities, hence, fracture detection by electrical and electromagnetic techniques are most effective in tight rocks with low matrix porosity. 2.6.3 Conventional well logs Conventional well logs are used for determining the properties o f the rocks adjacent to the wellbore. As a general class of measurements, well logs have both specific advantages and drawbacks. Their advantages include consistent one-dimensional profile of rock properties correlated with depth, in-situ measurements of rock and fluid properties, and multiple combinations of various independent measurements that can be used to solve for multiple independent variables. On the other hand, their disadvantages are indirect measurements of properties, measure properties of wellbore rocks which are usually altered by drilling, measurements influenced by borehole fluid, and their directional bias because they sample along the direction of the wellbore rather than in 3-D. One of the recognized drawbacks in geophysical logging of fractured rocks is the effect o f sample volumes on resolution. One class of logging tools, known collectively as micro-logging devices, has been designed to improve vertical resolution for the detection of thin beds and fractures. These logging tools are modified versions of conventional neutron, gamma density, and electrical resistivity devices, with closely spaced sources and receivers 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. or closely spaced electrodes embedded in pads that are pressed against the borehole wall. Micro-logging tools are used extensively in the petroleum industry to estimate the properties of mudcake and the invaded zone immediately behind the mudcake. In spite of their success in identifying fractures, micro-resistivity logs have been recently superseded by modem loggii^ tools capable of producing images of borehole wall such as the formation micro scanner and other borehole wall imaging devices which will be discussed later. Other types of logs such as temperature logs can sometimes be used to provide information about fluid flow in fractures. Temperature logs measure temperature in the well as a function of depth. When borehole temperatures are at equilibrium with the local wall rocks, temperature logs accurately reflect local formation temperature. When water is circulating along the wellbore, departures from local geothermal gradient indicate where water is entering/exiting the borehole. Fluid conductivity logs provide similar information because changes in fluid conductivity may indicate where entry or exit of water induces sharp contrasts in dissolved solids contents. Temperature and fluid conductivity logs obtained during production or injection may therefor be effective methods for identifying the depths at which producing fractures intersect boreholes. Conventional impeller flow-meter measurements can also be an efiflcient logging tool for identifying hydraulically conductive fractures intersected by boreholes. However, modem logging techniques also utilize recently-developed high-resolution flow meters (such as heat- pulse, electromagnetic, acoustic-doppler, and laser-doppler tools) that can provide more accurate flow measurements and have the capability to investigate conductivities of individual factures or groups of factures and their relationship to dominant flow paths in the reservoir. 2.6.4 Borehole imaging logs Borehole wall imaging is one of the most direct and effective methods for detecting fractures intersecting boreholes. Imaging tools, based on optical, acoustic, or electrical techniques, are utilized to provide images of the borehole wall. Optical devices include conventional cameras and television cameras. These cameras are used extensively in shallow 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. water wells and other near-sur&ce applications for borehole and casing inspection tasks. Fracture detection applications are often limited because cameras require clear borehole fluids. Furthermore, because the data are not available in digital form, they are not amenable to processing. Nevertheless, under the right conditions, these imaging tools can produce striking images of fiactures. Digital borehole scanner (DBS) is a new tool that provides high resolution, true-color images of a borehole wall that are digitally recorded to yield detailed measurements of fiacture properties identification of open fiactures, mineral-filled fi-actures, zones of discoloration alteration due to chemical or mechanical rock processes. 2.7 Geological and petro-physical aspects of faults Faults and fiacture are oftentimes associated with each other since they are usually created by the same geological processes. Faults are planes of shear failure (fi-actures) which exhibit noticeable signs of differential movement of the rock masses on either side of the fault plane. Points formerly in contact have been dislocated and displaced along the fi-acture plane. Faults occur in many structural situations, ranging from minor faults associated with folding to major fouhs which constitute zones of weakness in the earth’s crust and form the borders of mountain ranges. Conditions for the initiation o f faults can be determined by applying one of the criteria of brittle rock failure. The Navier-Coulomb criterion [19] of brittle failure is based on the concept that brittle shear 6 ilure will occur along a surface when the shear stress acting on that plane is large enough to overcome the cohesive strength of the material plus the fi-ictional resistance to the movement. Geometries of single foults are similar to those of fracture geometries, however, they are much thicker than firactures. Trace geometry of feult is feirly straight when traversed along the direction of slip, but a wavy geometry is common in the perpendicular direction. The location, orientations, and spacings of major slip surfaces in many fault zones are largely dictated by preexisting flaws. In sedimentary rocks, feult zones commonly form along bedding planes. In igneous and metamorphic rocks, fault zones commonly form along pre-existing planes of weakness such as compositional layering, metamorphic foliation, joints, and dikes. 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.7.1 Physical characteristics of fault zones Fault zones are invariably heterogeneous. The overall shape o f a fault zone is somewhat irregular, and its thickness is variable. Fault zones in some sandstones develop by the sequential formation of closely spaced sets o f small foults that collectively form a thick mass of crushed rocks and wide 6 ult zones. Fractures of various modes may also develop at the margin of the foult plane, complicating its physical and hydrological characteristics. Both opening-mode and shearing-mode fractures are common in fault zones. Orientation and distribution of these fractures along foult zones are typically nonuniform and tend to evolve differently in different rock types. Variations in the overall shape of a fault zone result in heterogeneous stress fields along a fault zone, which in turn will lead to a nonuniform fracture distribution along the fault plane. Moreover, stress fields change with time, and fault zones can be reactivated resulting in new fracturing episode and more complex geology. Composition o f deformed materials found in fault zones can also be highly heterogeneous, with blocks o f sparsely fractured rock juxtaposed against lenses of gouge (mechanically and/or chemically altered fine-grained material). Variety of precipitated minerals are also known to deposit in fractures and cavities along fault zones. 2.7.2 Textural classifications of fault zones Sibson [26] discussed the various types of faulted rocks and deformation mechanisms associated with them and presented textural classifications of fault zones (Table-2.2). The main textural divisions are between random fabric and foliated types, and between cohesive and in-cohesive rocks. Subdivisions within the in-cohesive types are based on grain size, ranging from breccia to gouge. The cohesive types are divided based on the tectonic reduction of grain size from that of the host rock and the fraction of fine-grained matrix relative to lithic fi-agments and residual, coarse crystals. Sibson divided the mechanisms of faulting into two main types, "elastico-fiictional," in which the micro-mechanism is primarily brittle fracture, and "quasi-plastic," in which the mechanism involves some degree of crystalline plasticity. The elastico-fiictional processes include brittle fracture o f the host rock, fiictional (abrasive) wear, and cataclastic deformation of the gouge or breccia. 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Random fabric Fault breccia (visible fragments > 30% of rock mass] Fault gouge (visible fragments < 30% of rock mass) IÎ • I I Ji Ü -g II Pscudotachylyie Foliated Foliated gouge Crush breccia Fine crush breccia Crush micTobrecda ( fragments > 0.5 cm) (0.1 < fragments < 0.5 cm) (fragments < 0.1 cm) Protocataclasite Cataclasite Ultracataclasite Protomylonile Mylonite (JItramylonite O-IO 10-50 50-90 90-100 Blastomylonite Table-2.2: Sibson’s textural classification of fault zone rocks, {after C. H. Scholtz [14]). 2.7.3 Structural fault classifications Several classifications of faults are in use. Often times more than one classification is necessary to give a more accurate and con^lete description of the feult zone. Generally, faults can be classified as either translational or rotational based in the type of movement, and may be fiirther subdivided according to the direction of the relative displacement of the rock masses on opposite sides of the fault zone. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For 6 uks created by translational motion, Figure-2.4 summarizes classification based on the direction of the apparent relative movement of the fault blocks [27]. Strike slip faults occur if displacement is in the direction of the strike of fault plane, while dip slip faults have displacement in the direction of the dip. Faults exhibiting movement in both strike and dip slip directions are known as oblique slip faults. Further classifications of both dip slip and strike slip 6 ults are illustrated by Figure-2.5 (i) and (ii) respectively. For dip slip fiiults (Figure-2.5-i), if the hanging-wall (the block of rock which lies above the 6 uk surâce ) moves downward in he dip-slip direction with respect to the fi)Ot-wall (the block below), the 6 ult is classified as “normal fault”. If the hanging-wall moves upward relative to the foot-wall, the structure is classified as a “reverse fault” if the dip angle is greater than 45 degrees, or a “thrust feuk” if the dip angle is less than 45 degrees. Strike slip feults (Figure-2.5-0 are also further subdivided into either right or left-handed slip faults. Finally, faults created by rotational motion are divided into either hinge or cylindrical faults as depicted by Figure-2.6 (i) and (ii) respectively. Hinge 6 ults occur where the axis of rotation is perpendicular to the fault plane while cylindrical faults occur where the axis of rotation is parallel to the fault plane. 2.7.4 Hydrological characteristics of fault zones Hydraulic properties of individual fault zones can vary significantly along the zone depending on many factors including shape of fault sur&ce and magnitude of the local compressive stress acting across it, in addition to the spatial distribution of fi-acture geometries, fracture filling and gouge materials. Generally, fault hydraulic conductivity depends on rock type, its grain-scale fabrics and differences in the inherent macroscopic heterogeneities and flaws. In highly porous sandstones, permeability of feult zones can be several orders of magnitude less than the host rock, and will act as a flow barrier preventing hydraulic communication between the rocks masses across the feult. On the other hand. In low porosity rocks, the permeability of the feult zone tends to exceed that of the host rock. These contrasts in hydraulic conductivities can dominate the subsurface hydrology of fractured rocks. 2 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-2.4: General &ult classification based on the displacement in the fault blocks (a)direction of the strike of the fault plane, (b) direction of the dip of the fault plane, (c) actual or net displacement {after Uemura and Mizutani [27J). Figure-2.5: Slip dip faults (i): right (A-B) and left-handed slip fault (A-B), (ii) strike slip faults: gravity or normal fault (A-B) and reverse or thrust fault (A-C), {after Uemura and Mizutani [27]). Figure-2.6: Rotational fault classification (i) hinge fault (ii) cylindrical fault {after Uemura and Mizutani [27]). 2 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.7.5 Sealing and conductive faults As mentioned earlier, numerous classifications of faults, based on different physical, geological or mechanical properties, are in use by geologist and geophysicists to categorize various structures and textural characteristics o f fault zones. For reservoir engineering purposes and from a fluid flow point of view, geologic faults are divided into sealing (Figure- 2.7) and non-sealing (Figure-2.8) faults according to their hydrological conductivities and their ability to transmit fluids through the fault zone. Figure-2.7: Sealing faults in homogeneous reservoirs. Figure-2.8: Partially communicating feults in homogeneous reservoirs. 2 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Sealing (or non-conductive) faults act as impermeable barriers (no-flow boundaries). On the other hand, non-sealing faults (also known as semi-sealing, semi-conductive, or partially communicating faults) are characterized by their ability to transmit fluids and establishing vertical and/or lateral hydraulic communication. The term conductive faults is mostly used to describe the vertical conductivities of fault zones, where fluid normally flows upward along the fault zone (the direction of fluid flow is parallel to the 6 uk plane), whereas the terms non-sealing and partially communicating faults are commonly used to describe the lateral conductivities of fault zones, where fluid is transmitted across fault zone (direction of fluid flows is perpendiculai' to the fault plane). Characterization of the sealing nature of &ults is a matter often left to the interpretive capability of pressure transient tests. Models available in literature for detection and analysis of pressure response of naturally ft-actured reservoirs with sealing faults are limited to that of the double porosity models. In chapter 6 , effects of sealing faults on other conceptual models for NFRs are investigated. These include the double permeability model (known as double layer model), in addition to dual fracture and triple porosity models derived in chapter 4. Because faulting and fracturing mechanisms, involved in rock deformation, usually lead to the development of extensive fractures along the feult zone, semi-permeability of faults are more common in NFRs, and hydraulic communication across the fault zone is likely to occur through fracture network cutting across the fault. In chapter 7, the well known double porosity model is modified to incorporate the effects o f such partially communicating faults on transient pressure behavior ofNFRs. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Well Test Analysis of Naturally Fractured Reservoirs 3.1 An overview of conceptual modeling Conceptual reservoir models are the basis for any reservoir engineering study and constitute the backbone of pressure test analysis. The selection of the correct reservoir model is of paramount importance and cannot be overstated. Once the proper model has been selected, various interpretation techniques and nonlinear estimation methods can be employed to obtain quantitative descriptions of important reservoir and near wellbore conditions. Conventional interpretation methods utilize graphical analysis by industry standard diagnostic plots (ISDP) including semi-log, log-log and derivative plots, in addition to type curve matching. Considerable improvements in the overall accuracy of modem well test interpretation were achieved by the introduction and application o f computer-aided analysis methods. Modem well test analysis utilizes the recent advancement in computer speed and efficiency to apply numerical analysis methods and estimation techniques to soKe for unknown reservoir parameters. In computer-aided well test analysis, the initial stage of pattem recognition, model identification and validation, can be significantly improved by the application of automated type curve matching in combination with neural network models and artificial intelligence techniques to accelerate and enhanced the overall efficiency and accuracy of model selection process. Automated type curve matching methods can also be combined with nonlinear parameter estimation algorithm such as least squares (LS) and least absolute value (LAV), 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to ensure most accurate match (between field response and selected model response) is obtained and used to solve for unknown reservoir parameters. 3.2 Uncertainties associated with model selection process Regardless of interpretation techniques and estimation methods used, the accuracy of estimated reservoir parameters can be influenced by many other variables, however, the validity of the well test itself is totally dependent on two fimdamental factors, the accuracy of measured field data and the applicability of selected reservoir model. While the solution to unacceptable data is simply to run another test, the solution to inappropriate conceptual model is much more complicated. Three major difiSculties or shortcomings encountered during pattem recognition and model identification process. First, the limited number of available interpretational models that are restricted to pre-specified setting and idealized conditions. Second, the non uniqueness of model response and finally, the limitation of majority of existing heterogeneous reservoir models to one type of heterogeneity. 3.2.1 Limitations of analytical models Actual reservoirs are highly heterogeneous systems, characterized by the existence of immeasurable number o f realizations occurring at all scales. Consequently, an infinite number of different combinations and variations are possible and every reservoir is uniquely constructed. While we are confi'onted with the task of adequately characterizing such infinitely complex systems, the number of existing analytical conceptual models available in literature is relatively very small. Furthermore, analytical modeling of various flow problems in oil reservoirs require overwhelming simplifications made through idealized assumptions with respect to reservoir rock and fluid properties and limited to a pre-specified type of reservoir configuration and conditions that may or may not exist in actual reservoirs. Moreover, the establishment of an unequivocal applicability of a specific conceptual model to the overall field performance is not indicative o f its applicability to local areas throughout the field. Large oil fields in particular, are usually divided into several separate 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. areas based on various geological, geographical, economical or operational considerations. Such isolated and separate sections of the same reservoir exhibit different and unrelated behavior as a result of spacial variations of reservoir properties and multiple models may be needed to characterize these separate regions independently. Solutions to the problem of limited reservoir model is progressively made available by the constant advances in analytical modeling leading to the development of new and improved conceptual models that incorporate various types heterogeneities. Numerous ana^ical models have been introduced for the characterization of different types of reservoirs including NFRs, multi-layered reservoirs, laterally/radially multi-region composite reservoirs in addition to reservoirs with linear/radial barriers and discontinuities such as faults, pinch- outs, fluid contacts and changes in-situ reservoir properties. 3.2.2 Insensitivity and non-uniqueness of model response The second problem, the non-uniqueness of model response, is one of the major challenges facing well tests analysts as pointed out by Ershaghi and Woodbury [28]. In particular, similar or even identical pressure behavior is observed for completely different reservoir models when analyzed on industry standard diagnostic plots (ISDP). The question they addressed is whether the formation of a straight line on a given pressure-time plot is indicative of the corresponding flow regime. Their conclusion was that pseudo-straight lines may be observed and interpreted erroneously especially during the exploratory phase when there is inadequate field data to support a particular reservoir model. The uncertainty associated with the non-uniqueness of model’s response can generally be summarized by our inability to identify appropriate models and/or distinguish between similar patterns of pressure behavior. Numerous reservoirs models developed for conceptually different flow problems may exhibit exactly the same pressure pattem. For example, a naturally fractured reservoir with both matrix and fl-acture systems contributing to wellbore production will exhibit response identical to a two-layer system with inter-layer cross flow and large permeability contrast. 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Another example of ambiguous model response is illustrated by noting that the doubling of slope on semi-log plot for drawdown or Homer plots for buildup tests does not necessarily indicate the presence of a sealing fault or no-flow linear boundary. This is due to the 6 ct that doubling o f slope can be exhibited by several models under particular reservoir flow regimes. Doubling of semi-log slope can also be observed for two-region radially composite reservoir, where the permeability of the outer region is half that of the inner region around the well. Doubling of semi-log slope will also develop for other reservoir models such as NFRs under unsteady state (USS) or gradient condition for matrix fracture interaction, and may even be observed for multi-layer systems. If we consider the additional variable effects of skin damage, wellbore storage, test duration and measurement error, the non-uniqueness problem is further complicated. A combination of proper well test design and integration of various sources of information from geological, and simulation studies, will play a significant role in overcoming these difficulties associated with pattem recognition and model selection. A computer-aided approach to well test interpretation using neural network has been proposed by many authors [29,30] for improved pattem recognition of complex reservoirs. Juniardi and Ershaghi [31] and Ershaghi et al. [32] suggested that the application of neural network methodology, based on automated type-curve matching using derivative plots, can enhance our ability to identify the applicability of a specific reservoir model and analyze pressure tests data obtained from reservoirs with complex heterogeneities such as naturally fractured and/or faulted reservoirs. 3.2.3 Limitation to one type of heterogeneity One major oversimplification of reservoir heterogeneities is caused by the limitation of most of the available reservoir models to one type of heterogeneity. Successful analysis of heterogeneously complex reservoirs is dependent on reliable identification o f the most prominent and hydraulically dominating heterogeneities, whose presence will overshadow all other insignificant features, and invariably influence the overall reservoir behavior as reflected by the generated response. 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reservoir heterogeneities are created by structural disturbances that can be classified as of primary nature, formed during the primary stage of rock formation, or secondary disturbances created by subsequent geologic and tectonic processes. Conceptual reservoir models with heterogeneities of primary origin include layered, composite and anisotropic reservoirs. On the other hand, conceptual models involving heterogeneities created by secondary processes include faulted and naturally fi-actured reservoirs. Because both faults and fi-actures are created by the same geologic processes, their coexistence in the same reservoir is commonplace. The development of new conceptual models that takes into account the presence of geologically related heterogeneities has become an area of emphasis in analytical modeling of oil and gas reservoirs. The main objective of this study is to fiacus on development o f conceptual models for the characterization of NFRs in the presence of other related heterogeneities including microfi-acture network, sealing faults, and partially communicating faults. It is beneficial, however, to first review important aspects of the theoretical foundation underlying the application and limitation of well test analysis techniques. 3.3 An overview of modern test interpretation Most of the standard interpretive techniques discussed in the literature are graphical procedures based on conventional (also called traditional or manual) analysis methods that include semi-log and log-log plots and type curve matching. Modem well test analysis has been greatly enhanced by the use of pressure derivative plots which was introduced by Bourdet et aL [7]. Derivative plot anatysis is emphasized throughout this study as it is proven to magnify small pressure changes, differentiate between responses o f various models, define a clear recognizable pattern for various flow periods and improve the overall accuracy of test interpretation and the estimation o f relevant reservoir parameters [7-12]. Modem pressure tests interpretation has kept pace with the giant leaps taken by the computer industry and the advances in microcomputer technologies and their computational capabilities. During the past two decades, numerous algorithms and con^)uter-aided numerical analysis procedures have revolutionized our abilities to conduct well test analysis with 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. accuracy. In computer-aided anafysis, pattern recognition, model identification and validation processes can be significantly improved by the application of automated type curve matching in combination with neural network models and artificial intelligence techniques to enhance the overall efBciency and accuracy of model selection. Automated type curve matching methods can also be combined with nonlinear parameter estimation algorithms such as least squares (LS) and least absolute value (LAV), to ensure that most accurate match (between the recorded field data and the selected reservoir model) is obtained and used to solve for unknown reservoir parameters. 3.3.1 Graphical test analysis using diagnostic plots In spite of the rapid evolution of modem well test analysis and computer-aided techniques, conventional graphical analysis of semi-log and log-log plots are still the most widely used methods to analyze pressure tests. Table-3.1 lists some of the most important applications of conventional analysis of semi-log, log-log plots to various types of reservoir models and different flow regimes. The semi-log plot analysis of pressure drawdown (MDH plot) and buildup tests (Homer plot) is founded on the existence of a straight line representing infinite-acting radial response (Figure-3.1). Recognition o f this straight line make it possible to estimate average reservoir transmissibility fi-om its slope and average reservoir pressure fi'om y-axis intercept. For homogeneous systems, the beginning of the infinite-acting straight line is usually determined by plotting pressure/time data on log-log plots where the combined effects of wellbore storage and skin are reflected by an early region with a unity slope straight-line. As a rule of thumb, the beginning of the infinite-acting straight line is one and a half cycle beyond the first deviation of the unit-slope line from the 45° line. On pressure derivative plots, the infinite-acting radial response is reflected by a horizontal straight line parallel to the time axis (constant derivative value, equal to 0.5 for dimensionless pressure vs. dimensionless time plots), while the wellbore storage and skin effects are manifested by an early hump masking the early reservoir response (Figure-3.2). 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table-3.1: Application of different diagnostic plots to various reservoir models and flow regimes Reservoir model or flow regime Characteristics Appropriate plot Infinite-acting radial flow (drawdown) Semi-log straight line {p vs. logf/UD, (semi log plot, also called MDH plot) Infinite-acting radial flow (buildup) Homer semi-log straight line (p vs.log[(t+Al)/àtD, (Homer plot) Wellbore storage dominated period Straight line (p vs. t), or Unit slope on (log[Ap] vs. log[At]) (Jogl/SpI vs. loglàlD Finite conductivity fracture (bilinear flow) Straight line with slope (m)= 0.25 on (Iog[Ap] vs. log[At]) (loglApI vs. logfAtf), or (4P vs. Infinite conductivity fracture (linear flow) Straight line with slope (m)=0.5 on (log[Ap] vs. log[At]) (loglApI vs. log|At|), or (Ap vs. At®^ Double porosity (PSS) Two parallel straight lines (m,=m2) separated by an S-shaped transition (p vs. log|At|), (semi-log plot) Double porosity (USS) Two parallel straight lines (m,=m3 ) separated by a straight line transition (2 m2=m,=m3) (p vs. log|At|), (semi-log plot) Single sealing fault (no-flow boundary) Two straight lines (m2=2m,) (p vs. iog|At|), (semi-log plot) Two perpendicular fault Three straight lines (m3 = 2 m2=4 m,) (p vs. log(Atl), (semi-log plot) Closed boundary (PSS) Pressure is linear with time (p vs. At), (Cartesian plot) Constant pressure boundary Flat line on all plots of (pressure vs. time) (p vs. At), (p vs. logjAtl), (ioglApI vs. iog|At|) 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. £ 20 Wellbore storage and skin dominated region 15 10 5 infinite-acting radial response 0 lE+OI IE+02 1E403 lE-HM IE+05 IE+06 lE^O? lE-HW IE+09 lE+IO A  M A â m TD Figure-3.1: Development of straight line representing infinite-acting radial response on semi-log plot for a drawdown test. 0.1 Infinite-acting radial response Wellbore storage and skin —^ dominated region , lE-fOI IE+02 IE+03 lE-HM IB 0. 01 IE+06 IE+07 lE-KM IE+09 lE+IO TD Figure-3.2: Development of straight line representing infinite-acting radial response on derivative plot for a drawdown test. 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3.2 Type curve matching Conventional semi-log analysis only uses the straight line portion of the recorded data to estimate the unknown reservoir parameters such as permeability (k), skin factor (S) and average reservoir pressure. However this straight line, as previously discussed, is usually masked at early times by the combined skin and wellbore storage effects. In cases where no pressure data is obtained beyond the early period dominated by wellbore storage effects, then pressure interpretation can only be made by type curve matching procedure. Pressure build up (BU) and drawdown (DD) tests (where pressure data is recorded at the same active well) are usually analyzed and cross-validated by both semi-log and type curve matching. However, interference tests (where pressure is recorded at an observation well located at some distance) can only be analyzed by type curve matching technique. Type curves are generated from solutions to flow problems under specific flow conditions and numerous standard type curves have been introduced for the analysis of various reservoir model including homogeneous, isotropic, naturally fiactured, faulted, layered, and composite reservoirs. By matching recorded test data to specific type curve, unknown reservoir parameters can be estimated. Log-log type curve is constructed by plotting the logarithm o f some dimensionless pressure group function versus the logarithm of a dimensionless time (usually logCPolvs.logftD/rD*]) on log-log paper. Type curve analysis using pressure derivative plots or simultaneous presentation of both log-log and derivative plots have improved pattern recognition and parameter estimation. Type curve analysis can provide us with the ability to estimate up to 3 unknown parameters, two from time and pressure axis and one from the matched curve. Traditionally, the matching was performed manually by plotting the data on tracing paper, then physically sliding the tracing paper over the t>pe curve so that it will superimpose a particular curve. Nowadays, type curve matching is more eflBciently and accurately done using automated or computer-aided approach. 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3.3 Nonlinear parameter estimation Automated type-curve matching and non-linear parameter estimation methods have proven to o&r a significant inçrovement over conventional semi-log analysis and type curve matching techniques. During the last two decades, nonlinear parameter estimation techniques along with constrained optimization methods have extensively been used in well test analysis. The estimation of unknown reservoir parameters fi’ om an observed well response is an “inverse problem" that requires matching the observed pressure response to a model which is a function of the unknown parameters. Because all of the reservoir models in use today are nonlinear functions of these parameters, nonlinear parameter estimation procedure is required. In the past, one of the major difficulties with automated well test interpretation was the evaluation of the reservoir response function in closed form. Rosa & Home [33] showed that by numerical inversion fi"om Laplace space of both fimction values and their gradients, it is possible to fit well test date to the most complicated mathematical models by the least square methods, resulting in significantly improved efficiency and accuracy of the parameter estimation process than is possible with conventional semi-log and log-log methods [34-38]. Advantages of automated well test analysis techniques over conventional graphical estimation methods include the following: achieving higher resolution than both semi-log and type-curve matching analysis, elimination of the error associated with choosing incorrect straight lines or match, minimization of numerical or procedural error, the ease in handling of multiple-flow-rate-history, the ability to match complex reservoir models, the ability o f handling several unknown parameters and assigning confidence interval to the estimated parameters. 3.4 Complexities of pressure transient analysis of NFRs Naturally fi-actured reservoirs (NFR), as discussed as earlier in chapter 2, are heterogeneous systems with stochastic distribution with respect to its physical, geometrical, and hydrological properties. Due to their complex nature, NFRs may exhibit pressure responses that may resemble other conceptual models. Cinco-Ley [39] presented a discussion 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the applications and limitations of pressure transient tests in the evaluation of NFRs and the different responses exhibited that resemble a variety of conceptually different models. A naturally fractured reservoir may display a response similar to that o f a homogeneous reservoir when the reservoir is intensely fractured with small matrix blocks (Figure-3.3.a) or when the fracture network contains the bulk of oil volume (Figure-3.3.b). When the reservoir is only fractured locally in a region around the producing well (Figure- 3.3.C), NFRs exhibit a behavior similar to that of a radially composite reservoir. NFRs with parallel fracture planes (Figure-3.3.d) will result in elliptical flow system with pressure response identical to anisotropic reservoirs. When the well is located near a major linear fracture plane such as a leaky feult (Figure-3.3 .e), wellbore pressure response will resemble that of a hydraulically fractured well. Finally, the classic response of the double porosity model is displayed when the fractured network is isotropic throughout the reservoir and constitutes the onfy channel ft>r flow into the producing well while the matrix system acts as a supporting reservoir containing the bulk of oil in place (Figure-3.3.f). Natural^ fractured reservoirs may also exhibit responses similar to other conceptual models such as the double permeability or layered reservoir model introduced by Bourdet et al. [4] where the matrix-to-matrix conductivity is not negligible and the matrix system contributes to the production at the wellbore. NFR may also exhibit a behavior similar to that of triple porosity [3] or dual fr-acture models [2] where there exist two distinct matrix systems or two different fracture systems respectively. 3.5 Double porosity model Over the past forty years, numerous analytical models have been introduced for the characterization of naturally fractured reservoirs from pressure transient tests. The most widefy used model is the one proposed by Barenblatt and Zeltov [40] and introduced to the petroleum literature by Warren and Root [5]. The idealized model of Warren and Root (Figure-3.4) consists of two distinct porosities of different flow and storage capacities, therefore carrying the designation of “double porosity model”. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-3.3: NFR model that can behave as a homogeneous reservoir (a)&(b), composite reservoir (c), isotropic reservoir (d), vertically fractured well (e), or as an NFR exhibiting the double porosity behavior (f), {after Cinco-ley [39]). 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Idealized Actual Figure-3.4: Idealization of the double porosity model for the representation of NFRs {After Warren and Root [5]). The primary porosity consists of the matrix system while the secondary porosity represents the fracture network which is made up of equidistant orthogonal fracture planes that divides the matrix system into cubic blocks of equal volume. The fracture system contains a small fraction of the indigenous oil (usually less than 2%), yet with hydraulic conductivities superior to that of the matrix system it constitute the primary channel for production at wellbore. On the other hand, the matrix system while containing the bulk of oil in place provides pressure support to the fracture system. Warren and Root indicated that two additional parameters are needed to characterize naturally fractured reservoirs. The first parameter, fracture storativity ratio (c d ), is defined as the ratio of storage capacity of the fi’ acture to that of the total system. The second parameter is the inter-porosity flow parameter (1), which is proportional to the ratio of the matrix permeability to that of the fi-acture multiplied by a shape frictor (a). 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.6 Modification to the double porosity model The double porosity model as proposed by Warren and Root is based on the pseudo steady state condition for the interaction between the matrix and the fracture system. This assumption has an advantage of the mathematical formulation and analysis of the flow problem and a disadvantage in that the pressure within the matrix blocks is assumed to be uniformly distributed in space. Other modifications of the matrix-fr-acture interaction were proposed hy many authors. Streltsova [41] proposed the average pressure gradient condition throughout the matrix blocks where the diflusivity equation is responsible for the pressure to be spatially distributed within the matrix blocks. DeSwaan [42] and Najurieta [43] proposed the unsteady state condition for the matrix-fracture interaction. Under this condition, the pressure in the matrix system is assumed to be an unsteady state fimction of the pressure in the fracture system. Kazemi [44] formulated a finite radial numerical model of naturally fractured reservoir with uniform fiacture distribution. The model consists of a set of horizontally spaced matrix layers separated ly horizontal fracture planes. Kazemi’s model also assumes an unsteady state condition for the matrix-fracture interaction. 3.6.1 Double permeability model Bourdet [4] introduced the concept of double permeability for the representation of layered reservoirs with inter-layer cross-flow. Similar to the double porosity, the double permeability model is often applied for the characterization of naturally fractured reservoirs. However, while only the fracture network is producing at the wellbore in case of the double porosity (Figure-3.5), in the double permeability model, the primary porosity plays another important role by establishing matrix to matrix conductivity throughout the reservoir and contributing to production at the wellbore (Figure-3.6). In addition to (X .) and (<a), a new parameter (k) is introduced, which represents the ratio of the fi-acture conductivity thickness to that of the total system. When (ic=l), no production is coming from the matrix system and the fixture network is the only communicating channel to the wellbore and the model break down to the well known double porosity model. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Layer-1 (M a trix ) Layer-2 (Fracture) i - Figure-3.5: Production through fracture network at wellbore for the double porosity model. Layer 1 Layer 2 4 - Figure-3.6: Production through both fracture network and matrix system for the double permeability model. 3.7 Pressure analysis of naturally fractured reservoirs (NFR) Diagnostic plots have been extensively applied and still are widely used to analyze the pressure response of NFRs. On the semi-log plots of both pressure drawdown and buildup tests, the double porosity behavior is reflected by two parallel straight lines separated by a transition period indicating matrix re-pressurization of fracture network (Figure-3.7). The first 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. line indicates radial infinite-acting response in the fracture system, while the second straight line represents the response of the total system. On derivative plot (Figure-3.8), the response is manifested by two infinite-acting straight lines separated by a trough (depression) indicating the transition period. Inter porosity flow parameter (K) determinesthe time corresponding to onset o f transition period while its depth and duration are controlled by fracture storativity ratio (m) as shown on both semi-log (Figures-3.9) and derivative plots (Figures-3.10). Under pseudo steady state conditions, the pressure during the transition period on semi-log plots shows a curvature with an inflection point. On a derivative plot, this is reflected by deeper depression (trough) where the minimum value corresponds to the inflection point on semi-log plot. On the other hand, gradient or unsteady state assumptions will result in a linear segment during the transition period with no inflection point. The slope of this transition line was found by Streltsova [41] to be equal to one-half the slope of the early and late-time straight lines. On derivative plot, this straight line segment is reflected by a horizontal straight line where the pressure derivative value is also equal to one-half the value of early and late-time straight lines representing infinite acting response. Depending on the value of fracture conductivity-thickness ratio ( k), the pressure response of the double permeability model is identical to that of double porosity model at one asynptotic limit, and to a homogeneous reservoir on the other. Pressure drawdown behavior is illustrated on semi-log (Figure-3.11) and derivative plots (Figure-3.12). On semi-log plot, the response is characterized by two parallel straight lines separated by transition period similar to that of Warren and Root’s model at one limit (k=1 .0), to one straight line representing an infinite acting homogeneous response at the other ( k= 0 ). On the derivative plot, the response is similarly manifested by the double porosity behavior with a frU ty developed transition period at one limit to one infinite acting straight line representing the homogeneous radial infinite-acting model (k=1.0). 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 14 12 m 10 8 e m 4 m, = m 2 1E +04 1E +10 TD Figure-3.7: Pressure drawdown response in NFR with double porosity behavior on semi-log plot. 100 £ I g g " O 0.1 0.01 1E «00 1E +04 1E+10 1E*08 TD Figure-3.8; Pressure drawdown response in NFR with double porosity behavior on log-log and derivative plot. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 12 10 (û = 0.001 < D = 0.01 1 E -H I8 1E*02 TD Figure-3.9; Effects o f (X) and (m) on semi-log plot. 0.1 C D = 0.01 0.01 C D = 0.001 0. 001 1E*10 TD Figure-3.10: Effects o f (A .) and (to) on derivative plot. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 10 \ i i d . 1 E -H I2 1E403 1E4M 1E«05 1 E « 4 W 1E««7 lE + O B 1E +09 1E +10 TD Figure-3.11 : Double permeability behavior on semi-log plot. K=0. 0.1 ÏÊ+02 1E*03 1E*04 1E«0S I 1 1 1 * lE-me 1E*07 1E«08 1E 409 1E +10 TD 0.01 Figure-3.12: Double permeability behavior on derivative plot. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Representation of NFRs by Dual Fracture Model 4.1 Abstract Dual fiacture models, developed by Al-Ghamdi and Ershaghi [2], are examined as a more realistic alternative to dual porosity models for the representation of naturally fi*actured reservoirs. A major component of the fi-acture system is the network of microfi*acture which by virtue of their lower permeability respond somewhat later than the macrofi"actures. A delineation of microfiracture response versus matrix response is made using the proposed conceptual models. It is demonstrated that the response of microfi*actures may at times be mistakenly attributed to matrix. 4.2 Introduction Studies published on diagnostic plots of pressure transient test data indicate strong similarities among certain cases of conceptual reservoir models. In particular, diagnostic plots expected for naturally &actured reservoirs are oftentimes not developed because of either inadequate test duration or wellbore controlled conditions. A major question in the testing of naturally fi’ actured reservoirs is explanation for causes of non-development of transition period [45]. This transition was predicted by Warren and Root [5] in their dual porosity conceptualization of naturally firactured reservoirs. Other researchers have also predicted the transition periods for layered type response as well as systems of triple porosity. In the dual porosity conceptualization, an assumption is made as to the nature of flow units with inter porosity properties. Specifically, two types of flow units are considered. First is a system of 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tight matrix with substantial storativity for fluid. The second unit is the network of fractures with high fluid conductivity. In this study, the above model is extended to a more realistic one where the effects of microfractures are also included. The objective is to predict response duration for these subsets and develop guidelines for interpretations of pressure transient test data misinterpreted because of the selection of an inappropriate model. In this chapter, new conceptual models are proposed to differentiate between the micro and the macrofractures. Dual fracture systems (Figure-4.1) consisting of macro fractures and micro fractures are introduced as the basis of the reservoir architecture. The theoretical basis of the proposed model is developed and the anticipated pressure transient response on the pressure derivative plot is then compared to those of the existing models. Both the double porosity and the triple porosity models predict transition periods reflecting matrix support to the fracture system. In actual field tests, for certain tests these transition periods may be observed. However, there are cases where the response of naturally fractured reservoirs have lacked a clear definition for matrix support. One purpose of this study is to ascertain the similarities and differences between the support from tight matrix and that of the more permeable micro fractures. Figure-4.1 ; Idealization of the dual fracture model for the representation of NFRs. 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 Triple porosity model The triple porosity model consists of two different matrix systems with different properties. One basic assunqition of the triple porosity is that the two matrix systems are not in communication with each other (Figure-4.2). The model uses two (ks) and two (cos) relating each matrix system to the fracture system. The matrix with larger permeability will respond first, followed by the response o f the tighter matrix at a later stage. The general response of this model on the pressure derivative plot (Figure-4.3) shows three horizontal line segments separated by two troughs representing transition periods when each matrix type provides pressure support to the system. The three line segments correspond to the fracture response, the fracture and matrix-1 response, and the response of the total system, respective^. This model also assumes an unsteady state (gradient flow) between the fiacture and each matrix system. This last assumption will only influence the transition period by limiting the depth o f depression to a value o f 0.25 (Figure-4.3) which is half the value corresponding to the infinite acting response. The concept of triple porosity model can be further extended to the dual fracture systems. The concept of triple porosity model has been tested with X values in the range of (10'^-10"’) représentât matrix inter-porosity flow. However, the new proposed model of dual fiactures, the inter-porosity flow between the two systems of fiactures has values hi the range of (10'-10^), uidicating higher permeability for the micro fracture system. Matrix 1 Macrofracture Matrix 2 -1 - - f Figure-4.2: Triple porosity model. 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0^ lim e 2 I 0 JS lim e USS (Gradient) m atrix-fracture I E + 1 2 td Figure-4.3: General response o f the triple porosity model on the pressure derivative plot. 4.4 Proposed new models Two of the conceptual models that can be employed to represent dual fracture system are discussed here. The first model is similar to that of the triple porosity but with the microfiacture system replacing one of the matrix systems. This model assumes no inter porosity flow between the microfracture and the matrix systems, yet both support the macro fracture system. The second model assumes pressure support from the matrix to the micro fi-actures which in turn support the macro fractures. The macro fractures and the microfiactures both contribute to the overall hydraulic conductivity and to the production at the test well. 4.4.1 Dual fracture model Fractures and fissures occur in many sedimentary rocks at all scales. If the fiacture size distribution can be delineated into two broad categories representing macrofractures and micro fractures, then the macro fracture system will dictate the very early time response of pressure transient tests. The response of the micro fracture system will only be distinguished 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. if the ratio of the microfracture permeability to that of the macrofracture is small (A^<0.001). Otherwise the two fiacture system will respond practically at the same time causing the pressure transient response to be similar to that of the double porosity model with longer and a more steady extension of the first straight line representing the combined response of the two fi-acture systems. The first model to be considered (model-1) is similar to that of the triple porosity model with the microfi-acture system interacting with the macrofiracture system but not with the matrix (Figure-4.4). This model also assumes that production at the wellbore is primarily fi-om the macrofi^acture system. In the second model, the microfi-actures play an additional role by receiving support fi-om the matrix and transmitting support to the macro fi-acture system. The matrix system will only provide pressure support to the microfiractures and cannot flow directly the macrofiacture. This model assumes pseudo steady-state flow between the two fi-acture systems and between the matrix and the microfiracture systems. Unlike model-1, the second model (model-2a) allows interaction of the micro fi-acture system with both the macrofiactuie and the matrix system. The second model can be divided into two sub-models. Model-2a (Figure-4.5) assumes that only the macrofiracture system produces at the wellbore, while in model-2b, the assumption is made that production is contributed fi-om both macro and microfiacture systems (Figure-4.6). The contribution of each fiacture system wül be proportional to its permeability ratio. While model-2b is the most general, the three models are equivalent to double porosity, triple porosity, or double permeability models under certain limiting conditions. For example the assumption that the storativity of the microfi-acture system is zero will change the above models to the double porosity model. Assuming that the storativity of the matrix system is zero and changing the range of the X and < a for the microfiacture to that of another matrix system will change the above models to the double porosity model for both model-1 and model-2.a. Model-2.b, on the other hand, will be equivalent to the double permeability. Changing the range o f the values of A , and m for the microfiacture to that of another matrix system will produce the triple porosity model for all of the three new model. 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Matrix I Macrofracture t Microfracture Figure-4.4: Model-1, dual fiacture representation by the modified triple porosity model. Matrix Microfracture Macrofracture Figure-4.5: Model-2a, dual fi-acture model with production through the macro fiacture system. Matrix Microfracture Macrofracture Figure-4.6: Model-2b, dual fi-acture model with production through both macro and microfi-acture systems. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4.2 Modified triple porosity model The first model as mentioned is similar to that o f the triple porosity model. Solution of this model was derived by Abdassah and Ershaghi [3]. This solution was not intended fisr X values outside the range (10‘^-10 ’) which is representative of the two matrix systems. In the dual fiacture system, the microfi'acture system replaces one of the matrix systems with X values in the range of (lO '-lO'^). The dimensionless pressure solution (including the effects o f wellbore storage and skin) in the Laplace space is: ^o(x) ^o(X) s(l+5.C„*:„(X )) (1) 1 T / \ / \ 1 - 1 s tanh I 1 - 1 1 “ / J V l “V j / . \ - L - i .5 tanh 1 — -1 l “>" J N I ^ ] 0.5 (2) Where the dimensionless parameters are defined as: P fd = Inkph _ {Pi -Pp) kpt = (kh)p i9c,h)p + i<^c,h\ (3 ) (4 ) (5 ) (6) 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (kh)j- 2 Or = ----------------------- (tpC,/»V + ((pC,/7)^ This Laplace space solution of model one is the same as that of the triple porosity model which was derived using both Laplace and Hankie transformations, for detailed derivation of the above solution the reader is referred to reference [3]. 4.4.3 New dual fracture model The dimensionless equations of this model (in Laplace space) describing the flow in the macrofracture, microfracture and the matrix systems respectively are: P fd ~ P fd ~ ^ /^ P jd ~P fd^ (9 ) ( 1 = + ^f(P/D - P fd) ~K (Pn,D ~ P p ) (10) (1 -(Ù^-(ÛJ.)S = -KiPmD-Pjü) (11) The dimensionless solution of this model in the Laplace domain including the effects of skin and wellbore storage is: C o C S .C o ) '- ; ' s\ f.C g + f V(^) 1 ] (12) Y(^) I ) 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y(s) = Kflj + 1 -K K4, +1 -K (42 - 1 ) (K ^ ia ^ is) ) ^ S ) (a, -1) ( < ( o , (s) ) ^5) % ) = 02(^l(*2)) / X (ÛS +Xj- +A 2 \ , 1-K K , (13) (14) (15) (16) (17) \ / X +Xj. (ÛS +Xj. \ -A 2 I [ 1-K K J / A= X tDJ + A y 1-K #, -1 + — , 1 a, = 1 + — ^ X X=[oy5+X„ - 4X : T k (1 - k ) X - ( 1 - k ) ct, X-(1 - k)c t 2 (1 -(û^-(ùp)s (18) (19) (20) (21) (22) The dimensionless parameters of this model are defined differently fi'om those of model-1 and are shown in Appendix-A. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4.4 Pressure derivative diagnostic plots The Laplace solution o f both model-1 and model-2 were inverted numerically using the Stehfest's algorithms [13] to generate the corresponding pressure derivative diagnostic plots. The analytical representation of these new models is numerically calibrated against asymptotical models and parameter sensitivity studies are conducted to examine the expected pressure response on corresponding diagnostic plots. For model-1, wherein macrofractures provide the only hydraulic communications to the wellbore, the expected general response is shown on the derivative plot (Figure-4.7). This response predicts an early microfracture pressurization period followed by a matrix- controlled domain. Superposition of the microfracture response with micro fractures can distort the early time infinite acting behavior. Similar behaviors are obtained for models-2a and 2b as shown in Figure-4.8 and Figure-4.9 respectively. Note in particular that the similarity of macrofracture permeability (kp) and macrofracture permeability (kj can predict an almost horizontal trend at early times masking the contribution of microfractures. A delayed matrix support period can be erroneously interpreted as an overall response of a homogeneous reservoir. Also note the differences between the two transition zones of the three models (models-1, 2a and 2b) according to the underlying matrix-micro fracture-macro fracture interaction and their permeability contrast. Since model-1 is developed under the assumption of USS media interaction, the depth of both transition zones are limited to the asymptotic minimum of 0.5. While models-2a and 2b are developed under PSS interaction, they differ in that model only the macrofracture is contributing to production at the wellbore. This results indifferent first transition periods resembling the double permeability behavior for model-2a (Figure-4.8) and the double porosity behavior for model-2b (Figure-4.9). The general response o f the microfracture system, consisting of three line segments separated by two transition zones, are fiuther seen in Figure-4.10. The three lines represent the responses of the macrofracture, macro and microfiactures, and that of the total system respectively. Thick solid lines correspond to the case of no skin damage (8=0) and downhole pressure measurement (Cg=0). 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. r I 9 \ n = - =1.6x10 03r = n . O X 1 U 09fn“ 5 .1 X iO 0.1 L — 1E+00 1E10 1E8 1E+02 1E4 1E6 td Figure-4.7; General response o f model-1 . 10.00 2 Xf= 1C F 3. A.f= 10' 4. Af= 10. 0.01 100 1E8 1E4 1E6 1E10 1 td Figure-4.8: General response o f model-2a. 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10.00 2. X r= 10^ 3. Xf= 10 4 . Xf- 1 0® 0.10 0.01 1E6 1 100 1E4 1E8 1E10 t d Figure-4.9: General response of model-2b. 10.000 1.000 F 0.010 f 0.001 s =10 Cd= 100 / / V K = 0 .9 9 \ / Xf = 10" V 1 0 ® • tOp* 1 0 ® --------------- 1 ----- 00f= ------- H - 1 0 ® ------------ 1 -------------- 1 -------------- 1 100 1E4 1E6 t d 1E8 1E10 Figure-4.10: Typical response of dual fracture system (model-2b). 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As indicated from the previous Figures, the macro fracture network influence the very early time pressure response. The presence of the microfracture support can only be observed if XfO.OOl, otherwise both macrofracture and microfracture systems will practically respond at the same time, with longer and better defined extension of the first straight line representing the total fracture response (Figure-4.11). In such cases, the shape of this response is similar to that of the double porosity model where only two straight line segments are observed, representing the response of the combined total fracture systems and the response of the total system respectively. For small values of microfi-acture to macrofi-acture inter-porosity flow parameter (Xf<0.001), the micro fracture response will be distinguished from that of the macro fracture by observing relatively an early trough on the derivative plot (Figure-4.12) indicating the transition period between the two fiacture systems and could be mistakenly considered to be representing a matrix-fracture interaction. This transition trough will be further delayed on the time axis (Figure-4.13) for smaller X f approaching values representative of matrix-fi-acture inter-porosity flow. Considering the effects of skin and wellbore storage, it is clear that this early trough is very likely to be masked unless the combined effects are too small. The matrix response on the other hand will come at later times. Initially for instances where X f is very small (representing large contrast between micro fracture and macrofracture systems), the trough representing the matrix response may never be observed within normal test duration. Extended pressure tests and downhole pressure recording may be necessary and have the potential to further explore the complex nature of naturally fi-actured reservoirs as represented by the dual fracture model. 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10.000 1.000 F S 0.010 ' 0.001 s = 1 0 C d= 100 K = A .f = 10 V / K r 10® ■ 03 (T = 10® - - 1 ■ 09 f= 10® 1 1 --------------1 -------------- 100 1E4 1E6 td 1E8 1E10 Figure-4.11 : The combined response of the two fracture systems. 10.000 1.000 K = 0 .9 9 S 0.100 Xf= 10 10 0.010 cop= 10 0.001 1E8 1E10 Figure-4.12: Early trough representing micro fracture response. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10.000 1.000 ? s 0.010 • 0.001 s =10 / Cd= 100 / 1C= 0 .90 \ / 10= \ / \ n = 1»! tOf= 10 ■ H ------ fijf= 10^ ------ 1------- — H ---------------1 --------------- 100 1E4 1E6 td 1E8 1E10 Figure-4.13: Delay of the microfracture response. 4.5 Discussion Conceptual models representing naturally fractured reservoirs by the formulations discussed here, predict several cycles of time data, before the tight matrix can be recognized. Examination of a number of actual pressure buildup and drawdown tests (Table-4.1) that exhibit a double porosity behavior for naturally fractured reservoirs, indicates pressure support of the matrix develops in relatively short time (1-3 cycles) since the first point recording. Considéra^ the large expected contrast between the matrix permeability and that o f the fiacture resulting in an inter-porosity flow parameter A , in the range of ( 10'^-1 O’), one can predict that the matrix support will actually require more time to develop. The exhibition of an early transition period can be attributed to the presence of the micro fracture system with considerably larger permeability than that of the matrix. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.1 - Number of time cycles to the end of the transition period since the first data point Source Test # of cycles Comments Warren and Root |5| BU 2 < n < 3 FIgnre-Dl Warren and Root |S| BU 1 < n < 2 Fignre-D2 Crawford (46| BU 1 <n<2 Test A Crawford (46| BU I < n < 2 Test B Crawford [46| BU 1 < n < 2 Test-C Crawford [46| BU 1 < n < 2 Test D Crawford [46| BU K n < 2 Test-E Strobel |47| DD 2<n < 3 Well-2 Strobel |47| BU 3< n <S Well-5 Streltsova (41 j DD 3< n < 4 — Bonrdet|4j BU 2<n < 3 Double Perm. In the dual fracture model, the response of the so called micro fractures is characterized by a transitional period similar to that of the matrix. On the pressure derivative plot, a trough is developed at a much earUer time (tp=10‘-10'‘). On the other hand the response of the matrix is manifested by a second trough that comes at a later stage. The dual fracture model provides an explanation for many field tests where reservoirs known to be naturally firactured are responding in a way similar to that of homogeneous formatioiL In wells with high skin and wellbore storage, the first trough is very likely to be masked. If the permeability of the reservoir rock is very low (tight matrix) with representing the matrix in the range of (10'^-10 ’), the second trough may require days or weeks and may never be detected within realistic test durations. 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 Conclusion The results of the pressure analysis of the dual fracture model can be summarized in the following: - The delineation of fractures into two broad categories, macrofractures and microfractures, is a step forward towards a more realistic representation of naturally fractured reservoirs. - On the pressure derivative plot, the pressure support of the microfractures is similar to that of the matrix and the presence of micro fractures can lead to the formation of transition zones at substantially earlier time. These zones may be mistakenly interpreted as matrix support. - The proposed models provide an explanation for the observation of early pressure support emanating from a network of microfractures and often attributed to the tight matrix rocks. - The models also provide a general explanation for the observation or lack of observation of single or double transition periods on the test data, from naturally fractured reservoirs. - The concept of dually fracture reservoir can lead to a better estimation of reservoir parameters including the partition coeffrcients corresponding to the volumetric contribution of macro and microfractures in addition to the matrix. - The proposed models suggest that the pressure response of the tight matrix rocks require extensive test duration to be observed. A more realistic design of pressure test duration can be implemented for improved characterization of naturally fractured reservoirs. - The observation of single or double troughs in NFR should be examined against the dual fracture model. - Finally, the influence of the microfractures support can be best exhibited with down-hole recording and by minimizing the effect of wellbore storage. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Well Test Analysis of Faulted Reservoirs 5.1 An overview of sealing faults in oil reservoirs The idea of detectii^ the presence of a sealing fault through pressure transient testing was first introduced by Homer [48] who reported that the presence of a linear no-flow barriers such as sealing faults will result in the presence of two straight lines on the semi-log plot of pressure buildup and drawdown tests, where the slope of the second line is double that of the first. Homer also presented a method of calculating the well distance to the fault fi"om the time corresponding to the intersection of the two straight lines. Davis and Hawkins [49] developed a graphical method for calculating the fault distance fi"om the time corresponding to the intersection of the two straight lines on the semi log Homer’s plot. Gray [50] and Earlougher and Kazemi [51] investigated difficulties associated with 6ult detection fix>m buildup tests and outlined criteria for minimum producing time prior to buildup tests necessary for the development o f this doubling of slope on Homer’s plot. Kabir [52] investigated the effects of wellbore storage and skin damage on pressure behavior in the presence of faults and linear no-flow barriers. Other authors [53-57] investigated the pressure behavior o f homogeneous reservoirs in the presence of multi-feult systems with various fault geometries as illustrated in Figure-5.1. The application of pressure derivative plots to drawdown and buildup tests in faulted reservoirs have also proven to increase the accuracy of the model recognition and overall test interpretation processes, hence, improving the validity of calculated test results 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o o E 0 L O ° 1 0 0 ^ Figure-5.1: Different fault configurations discussed in the literature (a) sii^le fault, (b) parallel faults, (c) perpendicular faults, (d) angular faults. 5.2 Partially communicating faults The effects of lateral partial communication between strata across conductive faults on the pressure behavior in an otherwise infinite-acting homogeneous and isotropic reservoir have been investigated both numerically by Stewart et al. [58] and analytically by Yaxley [59] and Ambastha et al. [60] Stewart et al. [58] investigated the effects of the existence of a partially communicating fault on the pressure response of drawdown and interference tests using a two-dimensional single-phase numerical simulator. Modeling of the partially communicating fault was accomplished by considering a semi-permeable linear vertical barrier of reduced permeability and thickness and negligible capacity. Yaxley [59] and Ambastha et al. [60] took different approaches for the analytical representation of partially communicating faults in their homogeneous and two-layer composite reservoirs respectively. In a way similar to one proposed by Stewart et al., Yaxley 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. represented the non-sealing 6uk zone as an infinite semi-permeable linear plane of negligible capacity and reduced thickness. Lateral communication across the fault was also accomplished by imposing a linear Darcy flow between the two sides of the fault. Ambastha, on the other hand, represented the partially communicating fault as an infinitesimal-thickness skin boundary located between the two regions of his double-region linear composite reservoir model. 53 Pressure transient analysis of faulted reservoirs Semi-log analysis of pressure drawdown and buildup response of wells in the proximity of a sealing fault or no-flow boundary is reflected by the existence of a second straight line with slope twice that o f the first line (Figure-5.2) representing the infinite-acting behavior (m,=2m|). On dimensionless semi-log plots, the two straight lines have slopes m,= 1.151 and m,= 2.303 respectively. On the derivative plot, the presence of a totally sealing fault is also reflected by the doubling of the pressure derivative value. On dimensionless derivative plots, the two horizontal straight lines have pressure derivative values o f (0.5) and (1.0) respectively (Figure-5.3). Conclusions resulting fi"om the studies of partially communicating faults suggest that the pressure behavior will resemble inverted forms of the characteristic response of naturally fi-actured reservoirs. On the semi-log plots, two parallel straight lines are separated by a transition zone curve opposite to that of an NFR (Figure-5.4). On dimensionless semi-log plots, the two straight lines have slopes m,=m2= l. 151. On the derivative plot, partially communicating 6ults are manifested by the existence of a “hump” and two straight lines representing the infinite acting response (Figure-5.5). On dimensionless derivative plots, the two horizontal straight lines have pressure derivative values o f (0.5) while the transition zone hump has an asymptotic maximum value of (1.0). 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 m2 m2 IS 10 m l m l 5 0 I H I M IE401 IE+02 IE403 lE-HM lE+OS IE406 IE407 lE-HW IE-H)9 lE+IO TD Figure-5.2: Doubling of the infinite-acting slope due to the presence of a sealing fault. 0.1 0.01 IE40I IE+02 IE+03 IE+04 lE+OS IE406 lE+07 IE400 IE-H)9 lE+IO TD Figure-5.3: Doubling of the derivative value due to the presence sealing faults. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 25 m2 20 m l lE+IO IE402 IE406 lE-HW lE-HM TD c=0.0 Figure-5.4: Pressure deflection on semi-log plot due to the presence of a partially communicating fault. khl, 0.1 lE+IO lE-02 IE+02 lE-HM TD e=0.0 0.5 Figure-5.5: Existence of a hump on the derivative plot due to the presence o f a partially communicating fault. 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 Analytical Models for Faulted Homogeneous and NFRs 6.1 The principle of superposition and method of images This chapter deals with the development of new analytical solutions for different models of naturally fractured reservoirs with totally sealing fault. Solutions are obtained by applying the superposition principle and method of images to both double permeability and dual fracture models to include the effect of no flow boundary on their pressure response. The superposition principle is a powerful technique applied in reservoir engineering to generate complex reservoir response using simple solutions of basic model. It states that the response of a linear homogeneous system to a number of perturbation is exactly equal to the sum of each perturbation as if they were present by themselves. Applying the superposition principle, pressure drop in a multi-well system can be represented by adding up the pressure drops caused by each individual well. The superposition principle can also be applied to construct the pressure response of variable or multiple rate tests in a single well from the simple constant rate solution. Superposition principle in conjunction with the method of images are applied to basic reservoir models to simulate the existence of specific flow boundaries. Raghavan [61] described the method of images as a procedure for distributing sources and sinks (production and injection wells) in an infinite porous media so that bounding planes that are either no-flow or constant pressure are formed in the porous medium. A constant pressure boundary can be represented by placing an identical and equidistant image well, with a flow rate of opposite sign, on the other side o f the boundary (Figure-6.1 ). For the case of no-flow boundaries such sealing faults, the flow rate of image well has the same sign (Figure-6.2). 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure-6.1 : Constant pressure boundary as represented by pressure field {after Horne [62]). Figure-6.2: No-flow boundary as represented by pressure field {after Horne [62]). 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.2 Analytical solutions of homogeneous and NFRs with sealing faults Application of the method of images and superposition principle to simulate the presence of a sealing fault or other linear no-flow barriers was first investigated by Homer [48] for homogeneous reservoir and by Khatchatoorian and Ershaghi for NFRs [63]. 6.2.1 Analytical solutions for homogeneous model with sealing faults Homer constmcted his real space solution for pressure buildup behavior in well with a totally sealing 6uk in an otherwise infinite acting homogeneous reservoir utilizing the basic constant rate line-source solution. Kabir [52] incorporated effects of both skin factor (S) and wellbore storage (C^) on pressure behavior by deriving the dimensionless line-source pressure drawdown solution and applying convolution scheme in the Laplace space. Similarly, the pressure drawdown solution for cylindrical-source well, producing at constant rate in an infinite homogeneous reservoir with linear no-flow barrier, can be expressed in Laplace domain to include the effects of skin and wellbore storage: J f.C g + 1 (1) Where s is the Laplace variable and F(s) is defined by: F(s)= K ^(/7)^S (2) N Ws)= —-------—------- (3) Kg and K, are the modified Bessel functions of second kind and of zero and first order respectively and dp is the perpendicular well distance to the fault. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.2.2 Analytical solution for double porosity model with sealing faults Khatchatoorian and Ershaghi [63] investigated the effect of the presence of sealing faults in naturally fractured reservoirs (NFR) by applying the method of images and superposition to the double porosity model of Warren and Root. The dimensionless pressure drawdown solution, for cylindrical-source well in an infinite NFR with totally sealing linear fault, can be similarly expressed in Laplace domain, including skin and wellbore storage effects, by the following set o f equations: 1 5 f.Cg + I F(s) (4) (5) (6) where f(s) is the matrix-fracture transform fiinction defined by: A s ) = (ÙS - (1 -m )a +X (7) Typical pressure drawdown test of the above model is illustrated on both semi-log (Figure-6.3) and derivative plots (Figure-6.4). Investigating the pressure response of the above model, Khatchatoorian and Ershaghi showed that depending on the fault distance to the well (dp), the presence of a totally sealing fault can affect the nature and duration of the transition zone of the double porosity model which in turn will influence the estimation of the inter-porosity parameters (A .) and fracture storativity (œ). 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 S 20 m2 10 m2 m l 1 E -K M TD Figure-6.3: Pressure drawdown test of the double porosity model with totally sealing fault on semi-log plot. 100 10 0.1 0.01 1E + 04 TD Figure-6.4: Pressure drawdown test of the double porosity model with totally sealing fault on derivative plot. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Juniardi and Ershaghi [31] emphasized the complicated nature of the pressure behavior in naturally fiactured and Suited reservoirs and outlined difficulties in the application of neural network and pressure derivative pattern recognition techniques to such complex reservoirs. They also presented a table of dimensionless pressure solutions in Laplace space for ten different variation o f Suited and NFR models obtained by the application of the method images and superposition principle. The pressure response for homogeneous and NFR models with various combinations of linear, angular, parallel and perpendicular 6ult systems were analyzed on diagnostic plots. Ershaghi et al. [32] proposed the use of multiple sub-neural networks derived from a single conceptual model to improve pattern recognition process of such complex reservoir models. They also presented a similar table o f analytical pressure solution for 12 different faulted reservoir models, adding solutions for bounded homogeneous and bounded naturally fractured reservoirs with totally sealing faults. 6.3 New models for NFRs with totally sealing faults The methods of images and superposition are similarly applied to the Laplace-space analytical solutions of both double permeability and dual fracture models to investigate the effect of totally sealing fault on their transient pressure behavior. Analysis of the generated pressure drawdown response of these new models is also conducted on both semi-log and derivative plots. 6.3.1 Double permeability model with sealing faults The dimensionless pressure drawdown solution, for cylindrical-source well in an infinite NFR of double permeability behavior and with totally linear sealing fault, is expressed in Laplace domain by the following set of equations: ............................... (g) 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y{s) = (<ij - 1 ) ( C { O j W ) * S ) ( a , - 1 ) ( C ( o , W ) * S ) (9) V(-s) = KOy +1 - K («%2 - 1 ) ) +5) (a, -1 ) (C (< J, W ) +S) (10) grOO, V _ ^0(^|) ^o( ^ 1*,) — .2 J_ 2 a , = - I (l-(o)f +X ^ (Û S +1 I -K K .2 2 2 Oj- — ' ( (1 -C0)5 + X (ÙS +X 1-K ' = 1,2 + A -A (11) (12) (13) A= 1-K K k ( 1 - k) (14) a , = 1 + y [ ( 1 - o) ) 5 - ( 1 - K ) o f (15) #2=1+ —[(1 -co>5 - ( 1 - K ) a (16) 6.3.2 Dual fracture and triple porosity models with sealing faults Finally, the application of the method o f images and superposition principle to the Laplace-space pressure drawdown solution o f the dual fracture model to incorporate the presence of a totally sealing linear fruit will yield the following set of equations: 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. v(f) = Y (j ) = KOj + 1-K J f.C p + vW 7(f) K a , +1 - K (a^-\) iK^ia^is) )+S) {a,-I) (s) ) +3) (17) (18) (19) ^00, K^(c,)^K^(2d^c) ' = 1,2 (20) I / X +)y (ÙS + X j- +A 2 \ , 1-K K , / (21) 1 X ms + A y \ -A 2 \ I 1-K K J / (22) A = % + A y C Û 5 + A y 1 - K 4% y k ( 1- k) (23) a, = 1 + — a , = 1 + — X= [to-y - X -(1 - k) o, X -(1 -K)02 CD 5 + X _ ■K] (24) (25) (26) The above equations for the dual fracture model can be modified as described in chapter 4 to yield equivalent expressions for the triple porosity model. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.3.3 Pressure transient analysis of the new proposed models Pressure drawdown response of the double permeability model in the presence of a totally sealing fault is shown on semi-log (Figure-6.5) where two parallel straight lines are separated by the transition period of matrix re-pressurization of the fracture system. Also indicated is an early slope increase that is not clearly conclusive as to the presence of a nearby sealing fault. While semi-log plot showed no definite indication of the fault response, the pressure derivative plot (Figure-6.6), proven to be have superior diagnostic capabilities, shows an early doubling of derivative value representing the presence of a totally sealing fault. The fault’s response arrive at the very early time due to the its close proximity to the producing well, followed by the characteristic response of NFR (two infinite-acting horizontal straight lines separated by a transition period). Pressure drawdown response of the dual fracture model, in the presence of a totally sealing fault is also shown on semi-log (Figure-6.7). The early part of the plot shows curves that are ambiguous and inconclusive whereas the late time response shows two parallel straight lines, separated by the transition period of NFRs. Once again, semi-log plot show no definite indication of the fault response due to the its close proximity to the producing well. However, the pressure derivative plots (Figure-6.8) shows an early doubling of derivative value representing the presence of a totally sealing feult in addition to another early transition period of the macrofracture-microfracture interaction. Pressure drawdown response of triple porosity model with a totally sealing fault is identical to that of the dual fracture model with the exception of a totally developed first transition period as shown on semi-log (Figure-6.9) and derivative plots (Figure-6.10). Similar to the conclusions arrived at by Khatchatoorian and Ershaghi [63], the time for doubling of slope corresponding to the presence o f the sealing response is found to be dependent on the 6uk distance (d^) as well as on the other controlling parameters such as (X) and (o) of NFRs used. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 Figure-6.5; Pressure drawdown response of the double permeability model in the presence of a totally sealing feult on semi-log plot. 100 0.1 0.01 Figure-6.6: Pressure drawdown response of the double permeability model in the presence of a totally sealing fault on derivative plot. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2S 2 0 IS 10 1E +10 TO Figure-6.7: Pressure drawdown response of the dual fracture model in the presence of a totally sealing fault on semi-log plot. 100 10 -o 0.1 — 1E*00 1E*04 TO Figure-6.8: Pressure drawdown response of the dual fructure model in the presence of a totally sealing fault on pressure derivative plot. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2S 2 0 IS 10 1E+10 1E*00 1 E -H M TD Figure-6.9: Pressure drawdown response of the triple porosity model in the presence of a totally sealing fault on semi-log plot. 100 10 1E«M TO Figure-6.10: Pressure drawdown response of the triple porosity model in the presence of a totally sealing fault on derivative plot. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 7 Analytical Model for NFRs with Partially Communicating Faults 7.1 Model description and mathematical derivation As illustrated in Chapter 2, fault zones are usually a few feet to hundreds of feet wide and are characterized by the existence of sub-parallel 6uks and intensive fracturing. The presence of this highfy fractured rocks in the vicinity of the fault zone in a naturally fractured reservoir can lead to pressme communication and lateral fluid movement across the fault. The effect of non-sealing or partially communicating fault on pressure transient tests of homogeneous reservoirs was first investigated analytically by Yaxley [59] who presented the ana^cal solutions for the two regions on either side of the 6ult in the real time domain. Yaxley represented the non-sealing fault zone as a semi-permeable linear plane of infinite extent and negligible storage capacity. Communication across the fault was accomplished by assuming linear flow between the two regions of the model. This means a continuous pressure drop, across the fault, proportional to the instantaneous leakage rate through the fault plane. An approach similar to that of Yaxley is utilized for representation of the partially communicating fault to study the effect of the presence of such fault on transient pressure behavior of naturally fractured reservoirs. The solution of the new model is ultimately presented in the Laplace domain where the effects of skin and wellbore storage can be easily incorporated. The same double porosity model of Warren Root [5] is also used here for the new model under pseudo-steady state inter-porosity flow condition. Representation of the partially communicating fault was accomplished by considering a semi-permeable linear barrier of infinite extent and negligible storage capacity as first suggested by Steward et al. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [58], who also suggested that modeling of partially communicating faults in such a manner give it the essential property o f imposing linear flow pattern at the fault. 7.2 Mathematical representation of the new model Most of the analytical models available in literature utilize the difflisivity equation in its radial form. However, the flow problem described by the new model is more easily represented and solved in Cartesian coordinate system. Figure-7.1 is a schematic representation of the proposed model The 6uk plane is represented by linear semi-permeable barrier coinciding with the y-axis (x=0). Region-1, located on the right side of the fault plane (x>0), contains the producing or active well and will be referred to as the active well region. On the other hand, the observation well region (region-2) is located on the left side of the fault plane (x<0) and contain only one observation well. Each of these two regions require separate formulations of the diftusivity equation that will coupled by the boundary conditions at the fault plane(x=0). The mathematical derivation of the new model is based on the following assumptions: 1. Infinite-acting NFR in both x and y directions. 2. Single infinite line-source production well located at (b,0). 3. Single-phase reservoir fluid of low and constant compressibility. 4. Homogeneous and isotropic matrix system with high storativity. 5. Continuous and uniform fi-acture system of superior hydraulic conductivities. 6. Fracture network is the only channel of communication across the fault. 7. Fracture network is the sole contributor to production at the active well. 8. Pseudo steady state inter-porosity interaction between the matrix and fracture. 9. The fault plane is of infinite extent with negligible storage capacity. 10. Linear lateral fluid flow in the positive x-direction through the fault plane. 11. Fluid leakage rate though the fault is proportional to the instantaneous pressure gradient across the fault (Figure-7.2). 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Y-Axis X - A x i s Figure-7.1 : Schematic representation of the proposed model. T ,= Unit length If 1^ Figure-7.2: Representation of the semipermeabie fault zone. 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.2.1 Region-1 (active-well region, x>0) The diffusivity equation describing the pressure behavior in a naturally fractured reservoir with a semi-permeable linear barrier is further divided into fluid flow in the fracture and flow in the matrix system: a) Flow in fracture: (1) Here the constant flow-rate inner boundary condition is incorporated by introducing an infinite-line-source production well o f strength (q/<j>c^). This is accomplished by adding the term (q/h) 8(x-b) ô(y) to the left side of Equation-7.1, where the Dirac delta function ô(x- b) 0(y), representing the active well at (x=b) and (y=0), is defined by the following properties: 8(x) = Dirac delta function. 5 (x -6 ) = 0 , fo r x*b. j'ô(x-b)dx = 1 (2) b) Flow in matrix: d t = - a, (khi ml i^Pn-Ap„.) (3) Here the fluid transfer from matrix to fracture systems of both regions is under pseudo-steady-state (PSS) conditions. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.2.2 Region-2 (non-active region, x<0): Although this side of the reservoir contains no producing wells, it responds to the pressure depletion of region-1 by transmitting fluids through the fracture system across the fault plane t (x=0). a) Flow in fracture: 7 2 dx~ dy^ dt b) Flow in matrix: dt ikh). «2------- i ^ p . - ^ n , 2 ) (5) 7.2.3 Initial and boundary conditions Initially the reservoir pressure is assumed to be equal to initial pressure, in both fracture system as well as matrix system, of both regions on each side of the fault. The reservoir is also assumed to be infinite-acting in both x and y-directions in both fracture and matrix systems: At /= 0 , = , « = 1,2 As |y|-oo, = , « = 1,2 (6) (7) (8) 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The last two boundary conditions are defined at the fault plane (x=0). The first condition ensures mass balance through the 6uk plane while the second condition defines the flow resistance o f the semi-permeable barrier and states that the fluid leakage rate through the feuk is proportional to the instantaneous pressure difference between the two sides of the fault plane and the specific fault transmissivity: (9) 7.3 Dimensionless parameters The following dimensionless parameters are defined and used throughout: , * A ' 2nkf.hf„ Pün--— ^ i P r P n ) < " = '.2 (12) RV- (13) To - Y (14) ^0 ■ Y (15) 82 ReprocJucecJ with permission of the copyright owner. Further reproctuction prohibitect without permission. (16) kp hp L " 1 ^ < * * > Here, the characteristic length (L) is defined as the distance between the producing well and the observation well, or: L^=y^+(|b-x|)^ as shown in Figure-7.1. For interference tests, tliis definition of L is applicable to both region-1 as well as region-2. However, because of convenience, for the case of pressure build-up and drawdown tests, L is defined to be equal to the perpendicular distance (b) between the fault plane and the active well. In Equation-7.18, the feuk communication parameter (e) is defined as the ratio of the specific transmissivity of the feult to that of the fincture system. This new parameter is similar to the fault specific transmissivity ratio introduced by Yaxley [59] to describe the communicating nature o f the fault zone. This parameter is used because it allows the model to be extended to the case of unequal formation thickness on opposite sides of the fault. For the case of constant reservoir thickness the new parameter is reduced to one similar to the fault conductivity used by Stewart et al. [58] in modeling partially communicating faults in reservoir simulation. The fault communication parameter (e) takes on values between zero and infinity representing the two limiting cases of a totally sealing fault and no fault respectively. Values between these two limit represent the cases o f non-sealing or partially communicating feult. More insight into the role o f e is presented on the following chapters. 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.4 Dimensionless flow equations Using the above dimensionless parameters, the difhisivity equations and its initial and boundary conditions are next expressed in the dimensionless fom tL 7.4.1 Region-1 (active region, x>0) a) Flow in fracture: ^PjDl + + 2 X0(Xq-6o) ÔOo) = Q > ^ - K i (J>„DI -P/DI ) dxjj dyà (19) b) Flow in matrix: © ml ( dp„a,] dtr. ^ml (PmDI P/DI ) (20) 7.4.2 Region-2 (non-active region, x>0) a) Flow in fracture: dxl dyl (21) b) Flow in matrix: dt D ^m2^PmD2 P/D2^ (22) 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.4.3 Dimensionless initial and boundary conditions '0=0, PjDn=PmDn=^ , « = 1,2 (23) As |Xoh=o, Pjtj„=P„Dn=^ ’ " = 1’2 (24) bol Pjjy„ = p„u„ =0 , n = 1,2 (25) ^PjDl I _ ^PjD2 I (26) -P ^ .) (27) 7.5 Method of solution The technique used to solve the system of partial differential equations defining the new model above is similar to that used by many authors [59,60,64] who have dealt with the analytical treatment of reservoirs with linear discontinuities. The solutions of the new model are obtained by performing dual transformation on the dimensionless equations, initial and boundary conditions. The Laplace transform is first applied with respect to the dimensionless time followed by Fourier transfisrm with respect to the y-variable. ^Pon^^D’yo^h)] = jPon^ = P o n ^ ^ D ^ y (28) 0 = I Pon^ (29) This double transfi)rmation yield a system of ordinary differential equations in Fourier space where they can be readily solved to give the dimensionless pressure solutions in the 85 ReprocJucecJ with permission of the copyright owner. Further reproctuction prohibitect without permission. Laplace-Fourier domain. The final expressions for the distribution is obtained by applying the integral inversion formula [65] for the complex Fourier transform to invert the solution back to the Laplace domain. 7.5.1 Flow equations in the Laplace domain Applying the Laplace transform to Equations-7.19 to 7.22 yields: ^ ^ P /D l ~ P/DI ~ ^m l ^ m O l ~PjD l^ ~ PmDI ~ ~^m l ^ m D l ~P jD I^ ^ ^P /D 2 ~ P/D2 ~ ^m2 ^PmD2 ~P/D2 ) ^ m 2 ^ PmD2 ~ ~^m2 C Pm D ’ ~PjD2 ) (30) (31) (32) (33) the boundary conditions become: As Pfr>n^PmDn=^ ’ " = ^’2 A s , PjD„ = PmOn = ^ , « = 1,2 ^P fP t I _ ^PjV2 I dXr ^P/D2 I _ . 1,0=0 ~ ^ ^P/Dl P/D2 ) (34) (35) (36) (37) substituting Equations-7.31 and 7.33 in Equations-7.30 and 7.32 respectively yields: (OnS - (l-© „ )5 +X.„,) m/ lit (38) 8 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V P/D2 (39) 7.5.2 Flow equations in the Fourier domain Applying the Fourier transform as defined by Equation-7.29, to the difilisivity equations in Laplace space (Equations-7.38 and 7.39} and boundary conditions (Equations- 7.34 to 7.37) yields: -*o) /(^)n = I ■ > + X , (40) (41) (42) The boundary conditions become: As |xol-oo, = , « = 1,2 dw, •JD I dXr I x p = 0 ■ ® ( ^JDl ^/D2 ) (43) (44) (45) The resulting Fourier space Equations-7.40 and 7.41 are ordinary differential equations (ODE) that are functions of the x-variable only, and therefore can easily be solved as shown in the next section. 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.6 Model solution In the Fourier domain The solutions o f the previous ODE’s are W q ,, and W fj,, for region-1 and 2 are: (46) (47) where: exp (48) ^ +2e exp t/»(4(6o + l^ol) (49) Now that solutions are available in Fourier space, the next step is to invert these solution back Laplace space. The Fourier inversion o f w^, and w^, can be performed by inverting Z, and Z, independently: (SO) (51) where: f ( Z , ) e ‘^ - d £ , Z 7 C J (52) 2tc J (53) 8 8 ReprocJucecJ with permission of the copyright owner. Further reproctuction prohibitect without permission. therefore: = / 7 exp = -\K . '^D cos(^j/o)d^ (54) - / f , V ? t W ^ 2 e -exp exp c o si^^)d ^ -\l^tfn(s) (6o+|Xo|) = -!- («0 [ i / / ,w [yl *(»D * l%l )")]) - — ^ ' • / ' {bp < ■ |jc y | ) - 2 eu V ^ C s ) ( ÿ i ^ J < /u (55) 7.7 Model solutions In the Laplace domain The inversion of the above solutions back to the Laplace domain will allow us to use the well known convolution scheme to incorporate he effects of skin damage and wellbore storage as suggested by Agarwal et al. [66]. The Laplace solutions can then be inverted numerically using Stehfest algorithm to generate the corresponding diagnostic plots. 7.7.1 Dimensionless pressure distribution in active well region The pressure drawdown distribution in the fracture network in the active-well region can be obtained using Equations-7.50 to 7.55: 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 8 ^ 2 G (56) - 2eu du \^diere Kq is the modified Bessel o f the second kind of zero order. The dimensionless pressure solution for region-1, as shown above, contains three different terms. The first term represents the pressure drop caused by the producing well while the second term represents the effect of the image well on the other side of the fault. The last term of the equation containing the integral represents the effect o f the partially communicating fault. The integral expression defined in the new solution can be evaluated numerically using Romberg integration [67]. This integral expression is improper with an upper limit equal to infinity and an integrand that is a product of exponential and modified Bessel fimctions. This integral is very similar to the integral form of the El function of the constant rate line-source solution in that both are improper integrals with upper limits equal to infinity and integrands that fall off exponentially [68]. Special subroutines fi’ om Numerical recipes [69] were employed to take care of these properties of the integral and to perform the integration with adequate accuracy. Appendix-B contains a Pascal program used to generate the dimesionless pressure response in the two regions on either side o f the fault. 7.7.2 Dimensionless pressure distribution in observation well region The dimensionless pressure drawdown distribution in the fi-acture network of the non active region is similarly obtained using Equations-7.50 to 7.55: ■ f ‘ ibp* \X [} \) 2c u du (57) 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The above solution for region-2 contains onfy one integral term similar to the last term of the solution of region-1 indicating that the pressure distribution in the non-producing region is highly dependent on the foult transmissivity ratio ( e ) which controls the fluid leakage rate through the fault. It can be seen from the above equation that when the fault is completely sealing (e=0), the pressure drop in the non-active region will be equal to zero. Solutions for both regions of the new model will be analyzed in the following chapters, first by conducting pressure drawdown tests at the active well in the next chapter. Finally interference tests in an observation well located on the same side of the fault (region- 1) followed by interference tests in an observation well located across the fault (region-1) are conducted and analyzed in Chapter 9. The different configurations and well locations of these tests can be visualized as portrayed in the four cases in Figure-7.3. R e g io n - 2 R e g io n - 1 case.1 -------------L -------^ ^ActiKwell Ohenatiamwell eate-2 -------- L -------► case-3 case-4 © Fa«It Figure-7.3: Different configurations and well locations for pressure drawdown and interference tests of the new model. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 8 Pressure Drawdown Analysis 8.1 Dimensionless drawdown pressure distribution in the active well The new model provides us with two different solutions for region-1 and region-2 on either side of the fault plane. In this chapter, the first solution for the active well region (Equation-7.56), originally derived for interference test in region-1, is modified to produce the pressure solution at the active well which can be used to conduct pressure buildup and drawdown tests. Pressure drawdown response at the active well can be obtained by setting L=b, x=b-r„ and y=0, hence: {b^-x^)=r^o=rJb (1) (6o+ x^)= 2-r^o (2) Therefore, the dimensionless pressure solution in Laplace space becomes: ^ D / = Y ^0 [ ] ^ [l/ÂW (2 ^ e . j* u^du . (3) 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Simulated pressure drawdown tests at the active well using the above solution were run to investigate the e&cts of the controlling parameters and test the new model against its asymptotic models as discussed in the following sections. 8.2 Incorporation of skin damage and wellbore storage effects Incorporating the effects of skin and wellbore storage at the active well can be accomplished using the well known convolution scheme in Laplace space as suggested by Agarwal et al. [66] and the final pressure response can be expressed as shown below: where S is the skin factor and Cq is the dimensionless wellbore storage: C ^ ___________C__________ 27c[((pc,/j)^+((pc,/i)„„]L2 The above solutions of the new model in Laplace space are inverted numerically using the Stehfest algorithm to generate the corresponding dimensionless pressure response. Both semi-log and derivative diagnostic plots will be employed to analyze the pressure behavior of the new model with more emphasis on the pressure derivative plot throughout this study as it is proven to magnify the small pressure changes that cannot be otherwise seen on conventional semi-log and log-log plots. The incorporation of the wellbore storage and skin effects at the active well by the application of the convolution scheme in Laplace space as described by Equations-8.4 and 8.5 can also be extended to an observation well for the case of interference tests (in both regions-1 and 2). 8.3 Model validation and asymptotic models The drawdown solution of the new model can be tested against its limiting cases which include the following five asymptotic limits: 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. •NFR with sealing fault. .NFR (no fault). .Homogeneous reservoir with non-sealing feult. .Homogeneous reservoir with sealing feult. Homogeneous reservoir (no fault). Theses limiting cases of the new model can be derived mathematically from the drawdown pressure solution at the active well as shown in Appendix-C. 8.4 Pressure drawdown at the active well (region-1, x>0): Both the active well and the observation well are represented by points on the x-y coordinate system. The drawdown solution (Equation-8.3) is obtained from the pressure solution in region-1 by bringing the observation well closer to the point representing active well location such that the inter-well distance is equal to the well radius (Figure-8.1). Drawdown tests at the active well will be conducted using Equation-8.3 to investigate the various effects of the parameters involved. Region-2 Region-1 ease-1 ♦ - b -U A c tb e e e ll case-2 case-3 " ------------ " --------------- 1 case-4 0 n m h Figure-8.1 : Active well location such that the inter-well distance is equal to the well radius for pressure drawdown tests. 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8.4.1 Effects of fault transmissivity ratio Typical pressure drawdown response of the new model (with reservoir parameters as summarized in Table-8.1) is shown on both semi-log plots and derivative plots (Figures-8.2 and 8.3 respectively). On each plot, twelve curves are generated corresponding to a wide range of values for fault communication parameter (e=0.0, 0.01,0.02, 0.05,0.1, 0.2, 0.5, 1.0,2.0,5.0,10,20, 50,100,200, 500), representing the case of totally sealing fault (s^O.O) at one limit, to the case of totalty communicating fault or no fault (£->«>) at the other. Values between these two limits represent the response of partially communicating faults. On the dimensionless semi-log plot (Figure-8.2), three straight lines appear at early, middle and late time respectively, separated by two transition periods. The first transition period results in an increased slope characteristic o f no flow boundaries, while the second transition period is that of NFRs. All twelve curves have the same first straight line with a slope of m,=1.151 representing the infinite-acting behavior in the firacture system. The slope ofthe second straight line is equal to the first slope (m,=m,=1.151) for (e>0.0) representing the case of partially or totally communicating fault, and m2=2m,=2.303 for the case of a totally sealing fault (e=0.0). However, the slope of the third line is always equals to the second for all cases (m3=ni2), as expected for NFRs. The ratio of the slope of the second straight line (or third) to that of the first line is 2:1 for the case of totally sealing fault (e=0.0), while the ratio of the two slopes is 1:1 for the case of partially or totally communicating fault. On the dimensionless derivative plot (Figure-8.3), three straight lines are reflected at early, middle and late times respectively, and separated by the two transition periods. The three straight lines represents periods of fi-acture system response, fault influenced region and total system, respectively. The first transition period is that indicating the presence of a fault while the second transition period is that of NFRs. For sealing fiiults, the first transition period shows a gradual and steady increase of the derivative value until it becomes double that of the first line (fi"om 0.50 to 1.0). For partially communicating faults however, the first transition period resemble a hump or a trough as expected where derivative value reach a maximum point then drop back to same value of the early straight line (0.5). 95 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. Table-8.1: Drawdown tests to investigate the effect of fault communication parameter (e) on transient pressure response Drawdown Tests 8 dr (0 s Cd * T est# I 0 50 1 X 10-* 0.01 0 0 Test # 2 0.1 50 ! X 10-* 0.01 0 0 Test # 3 0.2 50 I X 1 0 - * 0.01 0 0 Test # 4 0.5 50 1 X 1 0 - * 0 .0 1 0 0 T est# 5 1.0 50 1 X 1 0 - * 0 .0 1 0 0 Test # 6 2.0 50 1 X 10-* 0.01 0 0 T est#? 5.0 50 1 X 1 0 - * 0.01 0 0 Test # 8 10 50 1 X 1 0 - * 0.01 0 0 Test # 9 20 50 1 X 1 0 - * 0.01 0 0 Test # 10 50 50 1 X 1 0 - * 0.01 0 0 Test# 11 100 50 1 X 1 0 ^ 0.01 0 0 Test# 12 500 50 1 X 1 0 - * 0.01 0 0 96 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. s 18 16 14 1 2 1 0 8 6 4 IE-04 IE-03 IE-02 IE-01 1E+00 1E +01 1E+02 1E+03 1E+04 1E+05 »E=0 to ' 0.1 2 0 O ’ S O T D O J too 2 0 0 — 500 Figure-8.2: Pressure drawdown test of the double porosity model with partially communicating fault on semi-log plot. I S 1 0 b=son C D = 0.0 S D =0.0 (W=0.5 Sealing Fault 1 N o Fault 0.1 0.01 IE-04 IE-03 IE-02 IE-01 1E+00 1E + 01 1E+02 1E+03 1E+04 1E+05 >E=0 1 0 • 0.1 2 0 0 2 50 T D O J too I 200 • 5 0 0 Figure-8.3: Pressure drawdown test of the double porosity model with partially communicating fault on derivative plot. 97 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. All twelve curves similarly share the same first straight line representing the early infinite-acting response in the fiacture system. Once the fault presence starts appearing, three characteristic shapes are observed representing the three cases of no fault, sealing fault and partially communicating or non-sealing fault. The derivative value of the second straight line is equal to the first for (s>0.0) representing the case of partially or totally communicating fault, while for the case of a totally sealing feult (8f = 0.0) the derivative value is doubled. As can be seen fi-om Figures-8.3, for the case o f partially communicating faults, there is a critical communication foult parameter (8^=10.0) above which the second straight line is an extension of the first line and the fault presence for all practical purposes is undetectable. This is more clearly shown by the derivative plot (Figure-8.3) where the hump almost disqjpear for fouhs with communication fault parameter 8,.>10.0 (Magnification of the hump is shown in Figure-8.4). The last curve (e=500.0) shows only two different lines with equal derivative values at early and late times and separated by one transitional period of naturally fi’ actured reservoirs (no hump representing partially communicating fault). Sealing Fault 0.9 0.8 0.7 0.6 0 5 " " 1E.03 1 E + 0 0 1E +01 1 E 4 1 T D 1 E 4 2 200 — S O O 100 Figure-8.4: Magnification of the hump region representing partially communicating faults on derivative plot. 98 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. Close examination of the hump region confirms that for e>=10.0, the amplitude of the hump (the rise of the highest or maximum point on the hump from the infinite-acting straight line) is so small that it is for all practical purposes undetectable (Figure-8.5). It is also evident that the amplitude of the hump is inversely related to the foult communication parameter (e). While the amplitude is largest for small fault communication parameter (e-^O), it is lowest approaching zero for large fault communication parameter (e— > < » ) as shown in Figure-8.6. The lower limit of the foult communication parameter is (e=0) for the case of sealing fault. The maximum derivative value on the dimensionless plot is 1.0 which is double that of the early infinite-acting response where derivative is equal to 0.50. As indicated earlier for fault communication parameter larger than the critical value (e>=10.0), the maximum derivative value is equal to 0.50 indicating no foult response. For values in between these two limits (0<e<ej, the maximum derivative value lies between the two limits (0.50 and 1.0) as can be seen from Figure-8.5 and 8.6. b=50ft C D =0.0 SD=0.0 tw =0.5 0.9 0.8 M axim um 0.7 0.6 T D M ax IE-01 T D 1E+01 1 E + 0 0 IE-02 E=0 — 0.1 0.5 1 0 0 200 — S O O Figure-8.5: Maximum point of the hump and relationship to the value of fault communication parameter (e). 99 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 0.75 0.7 0.65 a. 0.6 0.55 0.5 Figure-8.6: Relationship between the value of maximum point of the hump and the fault communication parameter (e) is shown in. When using the dimensionless pressure derivative plot, the relationship between the maximum point o f the hump and fault communication parameter (e) is shown in Figure-8.6. Knowing the dimensionless pressure derivative value of the maximum point of the hump, one can use Figure-8.7 to calculate the value of the fault communication parameter (e). Because well test data are collected in field unites and analysis are made with dimensional pressure and time values, the ratio of amplitude (the value of the maximum point of the hump) to the value of the infinite-acting response (equals 0.5 for dimensionless plot) is better suited. Therefore, allowing the application of the relationship to the analysis of both dimensional and dimensionless pressure derivative plots. A graphical correlation was developed (Figure-8.8), relating the amplitude ratio (a^) of the hump as defined earlier to the fault communication parameter (e). This correlation can be used to calculate e with maximum accuracy fi-om the ratio of amplitude (the value of the maximum point of the hump) to the value of the infinite-acting response. 100 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 0.9 a 0.8 0.7 0.6 # N |— 1E2 IE-4 IE-3 IE-2 1E1 1E3 IE-1 1E0 Figure-8.7: Relationshÿ between the maximum point of the hump and (e) based on the dimensionless pressure derivative plot. 1E 3 1E2- 1 E 1 1E 0 ( O IE -1 IE-2- - IE-3- Ç 1E -4 + 1 1 .S 2 1.4 A m pi M u d # R atio 1 .6 Figure-8.8: Relationship between the maximum point of the hump and (e) based on the amplitude ratio (a^). 101 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. Using nonlinear regression and curve fitting programs, an empirical correlation was also developed (for e ranging between K T * and 100) relating the amplitude ratio (a^) to the fault communication parameter (e) as follows: 8 =0.386 ( 2 - a ,) '" * 0.988 (6) Although the above equation is simply an approximation of the curve of Figure-8.8 that is less accurate than the graphical correlation, it still is applicable with excellent accuracy and very convenient for programming purposes. Figure-8.9 shows the accuracy of the fit of regression model of Equation-8.6 to the original input data (Figure-8.8). Detailed fit information are presented in Appendix-D. » 1 Oe-01 1 6 17 T .9 2 0 1 .3 16 1 0 1 1 1.4 1 5 Amplitude Ratio Figure-8.9: Fit of regression model of Equation-8.6 to the original data based on relationship between (e) and (a^). 1 0 2 Reproduced with permission ofthe copyright owner. Further reproduction prohibited without permission. 8.4.2 Characteristic patterns of pressure drawdown response In addition to drawdown tests run to investigate the effects of e on pressure response, a number o f case studies were conducted with the main variables being dp, X , m , Cg and S. These terms are defined in the nomenclature. In general, for linear boundaries at close distance to the well (dp<100 ft), slope changes reflecting the linear boundary occur before the arrival of the transition period of NFR and the total system response which arrive at a later times. For faults at large distance to the well (dp>500 ft), slope changes reflecting the linear boundary occur after the arrival of the transition period of NFR. For faults at distance to the well (100<dp<500 ft), the slope behavior in both the early time and the middle time may be influenced and result in a complicated pattern. As shown in Figures-8.2 and 8.3, fi)r a dp=50 ft, the linear boundary effect is seen first in the early time data without distorting the late time response for all the cases of Partially communicating faults (e>0). For totally sealing fault however, early fault response will continue distorting the transition period and late time response. This type of pattern, where the feuk reflection is seen before the transition period of NFR and, is referred to as Pattem-I. Under certain X and ©, there is an early time fault response. For reservoirs with very tight matrix (X*=10'’ to 10*’), or with the linear boundary very close to the well (dp<100 ft), the early feult reflection resuhii^ in pattem-I may be observed (Figures-8.10 on semi-log plot and Figure-8.11 on derivative plot). For reservoirs with less tight matrix (X*=10‘® to 10'^), or with the linear boundary further away fi’ om the well (100<dp<500 ft), the fault reflection (feuh transition period) will coincide with the transition period of NFR, resulting in distortion to both transition periods as shown in Figures-8.12 on semi-log plot and Figure-8.13 on derivative plot. This pattern where the two transition zones interfere with each other is called pattem-n. Finally, for reservoirs with permeable matrix system (X*=10^ to 10^), or with the linear boundary far away firom the well (dp>500 ft), the feult reflection (foult transition period) will only arrive at late time after the transition period of NFR and the total system response has been observed. This resulting pattern, where the transition period of NFR is observed at early time, is shown in Figures-8.14 on semi-log plot and Figure-8.15 on derivative plot and will be referred to as pattem-III. 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 14 1 2 IE-04 IE-03 IE-02 IE-01 1E+00 1E +01 1E+02 1E+03 1E+04 1E+05 T D e = 0 2 0 0 S C O too Figure-8.10: Pressure drawdown test of the new model resulting in pattem-I on semi-log plot. 1 0 b=son C D =0.0 SD=0.0 tw =0.5 Sealing F auK N o Fault 0 .1 m a ii 0.01 IE-04 IE-03 IE-02 IE-01 1E+00 1E +01 1E+02 1E+03 1E+04 1E+05 T D 0.2 e = 0 2 0 0 5 0 0 too Figure-8.11 : Pressure drawdown test of the new model resulting in pattem-I on derivative plot. 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. b=2oo n C D =0.0 SD =0.0 fw =0.5 Sealing Fault 1 2 ■ N o Fault IE-04 IE-03 IE-02 IE-01 1E+00 1E +01 1E+02 1E+03 1E+04 1E+05 T D E=0 — 0 2 2 0 0 Figure-8.12: Pressure drawdown test of the new model resulting in pattem-II on semi-log plot. I 1 Sealing Fault N o Fault 0 .1 b=200 It C D =0.0 SD =0.0 f W = O .S 0.01 ' ' " " " t m u I I H— .Iiiiiia I i i i i ^ IIIIH* n if n il— IE-04 IE-03 IE-02 IE-01 1E+00 1E + 01 1E+02 1E+03 1E+04 1E+05 T D • e=0 - 0 2 - 200 Figure-8.13: Pressure drawdown test of the new model resulting in pattem-n on derivative plot. 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 b=sooo ft C0=0.0 S0=0.0 nw = 0.5 Sealing Fault s N o Fault IE-05 IE-04 IE-03 IE-02 IE-01 1E+00 1E+01 1E+02 1E+03 1E+04 T D I e = 0 — 0 2 2 0 0 Figure-8.14; Pressure drawdown test of the new model resulting in pattem-in on semi-log plot. I S 1 Sealing Fault No Fault 0 .1 b = S O O O It C D =0.0 SD =0.0 tw =0.5 0.01 IE-05 IE-04 IE-03 IE-02 IE-01 1E+00 1E+01 1E+02 1E+03 1E+04 T D * c = 0 — 0 2 2 0 0 Figure-8.15: Pressure drawdown test o f the new model resulting in pattem-ni on derivative plot. 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8.4.3 Effects of fault distance (dp) Drawdown tests were also run to investigate the effect of feult distance (dp) on pressure behavior of the new model as summarized in Table-8.2. Different distances to the fault were used (ranging from 10 to 5000 ft) to generate and examine their corresponding pressure response on diagnostic plots. Table-8.2: Drawdown tests to investigate the effect of fault distance to the active well (dp) on transient pressure response Drawdown tests E dp X * (D s Cd* Test# 1 0.2 10 I X 10^ 0.01 0 0 Test # 2 0.2 20 I X 10-* 0.01 0 0 Test # 3 0.2 50 1 X 10-* 0.01 0 0 Test # 4 0.2 100 1 X 10^ 0.01 0 0 Test # 5 0.2 200 1 X 10-* 0.01 0 0 Test # 6 0.2 500 1 X 10-* 0.01 0 0 T est#? 0.2 1000 1 X 10-* 0.01 0 0 Test # 8 0.2 2000 I X 10-* 0.01 0 0 Test # 9 0.2 5000 I X 10-* 0.01 0 0 Although Table-8.2 only shows test run with partially communicating fault (e=0.2), the other two asymptotic models of sealing and no fault are also generated by assigning different values to the feult communication parameter (e=0 and 200 respectively).Pressure response of these three models are shown on both semi-log and derivative plots in Figures- 8.16 and 8.17 for sealing feuk (efO), Figures-8.18 and 8.19 for non-sealing fault (e=0.2) and Figures-8.20 and 8.21 for the case of no fault (e=200). 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 C D =0.0 SD =0.0 n m = 0 .5 2 0 SeaRng Fault 15 1 0 IE-04 1 E + 0 0 1E+06 IE-02 1E + 02 1E+04 T D 2 0 50 1 0 0 ---- 2 0 0 5 0 0 ------- 1000 ------ 2000 5 0 0 0 Figure-8.16: The effect o f fault distance (dp) of a sealing fault model on pressure drawdown behavior on semi-log plot. C D =0.0 S D =0.0 «1=0.5 Sealing Fault 0 .1 0.01 IE-04 1E+06 IE-02 1 E + 0 0 1E + 02 1E+04 T D b=10 f t — 20 ---- 2 0 0 50 100 5 0 0 ------ 2000 1000 5 0 0 0 Figure-8.17: The effect of fault distance (dp) of a sealing fault model on pressure drawdown behavior on derivative plot. 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s 1 8 C D =0.0 SD =0.0 n*=0.5 16 Non-Sealing 14 1 2 1 0 8 6 4 2 0 • — lE-04 1E-02 1 E + 0 0 1E+02 1E+04 1E+06 ib = 1 0 ft 2 0 5 00 1 0 0 0 T D 5 0 ICO 2 00 — 2 0 0 0 5 0 0 0 Figure-8.18: The efifect of fault distance (dp) of a non-sealing fault model on pressure drawdown behavior on semi-log plot. C D =0.0 SD =0.0 rw = 0.5 0.1 Non-Sealing 0.01 IE-04 IE-02 1 E + 0 0 1E+02 1E+04 1E+06 T D b=lO f t 20 50 100 — 200 500 -------- 1000 200 0 5 0 0 0 Figure-8.19: The effect o f fault distance (dp) of a non-sealing fault model on pressure drawdown behavior on derivative plot. 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 C D =0.0 SD=0.0 n« (= 0 .5 16 14 N o Fault 1 2 1E +06 1 E + 0 2 1E+04 IE-04 IE-02 1 E + 0 0 T D -------- 200 2 0 100 b=10 f t 50 5 0 0 0 500 - 1000 2000 Figure-8.20: The effect of fault distance (dp) of no fault model on pressiu’ e drawdown behavior on semi-log plot. N o Fault C D =0.0 SD=0.0 iw = O .S 0.1 0.01 1E +06 IE-04 1E-02 1E+02 1E+04 1 E + 0 0 T D -------- 200 100 50 -------- 1000 5 0 0 0 500 ----- 2000 Figure-8.21: The effect of fault distance (dp) of no fault model on pressure drawdown behavior on derivative plot. 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It can be seen from Figures-8.16 to Figure-8.21, the effect of the fault distance (dp) on the pressure behavior is reflected by its relationship to the inter-porosity flow parameter (k) as defined by Equation-7.16 in the dimensionless parameter section. As indicated previously at the beginning o f this chapter, for the case o f drawdown tests (L=b) and the inter-porosity flow parameter (A .) becomes: where b is defined as the x-coordinate of the producing well location, which is equal to the feult distance (dp). Therefore since the model uses k that is defined based on the fault distance (dp), instead of k* which is based on the well radius ( r j , the value of k will change depending on the fault distance to the well (dp). Hence, the relationship between k and k* can be defined as: l = k ’ (8) This dependence is most exhibited in Figures-8.20 and 8.21 for the case of no fault (e=200), where the feult high conductivity make it virtually undetectable (no fault response) except through its ef&ct on k and the resuhii^ effect on the time needed for the onset of the transition period of the NFR. For small fault distance (dp<100 fl;), the value of A , will be close to that of A .* , and the effect is minimal where the onset time for the transition period will not change dramatically. On the other hand, if fouh distance is large (dp>500 ft), A . may be several order of magnitude larger than A,, and the effect on onset time for the transition period will be significant. The relationship between A , and A .* and fault distance (dp) is also shown graphically in Figure-8.22 and in tabulated form in Appendix-E, for (rw=0.5 ft). I ll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1E4 1E2 1E 0 IE-2 1E-4 IE-6 IE-8 1E 3 1E 1 1E4 1E 2 IE-67 IE-49 Figure-8.22: Relationship between X and X* and fault distance (dp). 8.4.4 Effects of fracture storativity ( o d) Drawdown tests were also run to investigate the effect of fracture storativity (a) on pressure behavior of the new model as summarized in Table-8.3. Values of o)=0.001, 0.002,0.005,0.01,0.02 and 0.05 were used to examine the fracture storativity effects on both semi-log plots and on derivative plots for all the three cases of sealing fault (e=0), partially communicating fault (e=0.2) and no fault model (e=200). Although Table-8.3 only shows test nm with partially communicating fault (e=0.2), the other two asymptotic models of sealing and no fault are also generated by assigning d if ^ n t values to the feuk communication parameter (e=0 and 200 respectively).The effects o f fracture storativity (co) on transient pressure drawdown behavior for the three cases are shown by semi-log plot (Figures-8.23 ,8.25 and 8.27) and derivative (Figures-8.24, 8.26 and 8.28) for sealing feult (e=0), partially communicating feult (e f = 0.2) and no fault model (e=200) respectively. 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table-8.3: Drawdown tests to investigate the effect of fracture storativity ratio (©) on transient pressure response Drawdown tests G dp X* C D S Co* T e st# ! 0.2 50 I X 1 0 -* 0.001 0 0 T est#2 0.2 50 1 X 1 0 -* 0.002 0 0 Test # 3 0.2 50 1 X 1 0 -* 0.005 0 0 Test# 4 0.2 50 1 X 10-* 0.01 0 0 T est#5 0.2 50 I X 1 0 -* 0.02 0 0 Test # 6 0.2 50 1 X 1 0 -* 0.05 0 0 The efifect of fracture storativity (m) on transient pressure drawdown behavior is best exhibited by the derivative plot (Figure-8.24 and 8.26) where it is evident that © will not only influence the onset time and depth of depression of the transition period, but also the onset time for the fault response (the hump in the case of non-sealing fault). It is found that for smaller values of fr-acture storativity (©), the depth of transition zone (trough for NFRs) is larger while the onset time is shorter for both transition periods (hump for non-sealing fault and trough for NFRs). Figures-8.23 to 8.28 are generated for small well distance to the fault (dp=50 ft). It is worth noting that for the case of sealing faults (e=0), the depth of depression of the transition period (for NFRs) is less than that of the other two cases o f non-sealing (e=0.2) and no fault model (e=200). 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 0 C D =0.0 SD =0.0 tw =0.5 b=50ft Sealing Fault IE-02 IE-04 1E + 00 1E+02 1E+04 1E+06 T D — 0 = 0 .0 0 1 0.002 0.005 0 .0 1 0 .0 2 0.05 Figure-8.23: The efiFect o f fracture storativity (ca) of sealing fault model on pressure drawdown behavior on semi-log plot. CD=0.0 SD=0.0 nw =0.5 b=SOtt 0 .1 0 .0 1 Sealing F ault 0.001 IE-04 IE-02 1E + 00 1E+04 1E+02 T D (0= 0.001 0.002 0.005 — 0 .0 1 0 .0 2 0.05 Figure-8.24: The effect of fracture storativity (ca) of sealing fault model on pressure drawdown behavior on derivative plot. 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 C D =0.0 SD =0.0 c w = O .S b=SO (t Non-Sealing Fault 1E+02 1E +04 1E+06 IE-04 IE-02 1E+00 T D - 0.005 < D = O .O O I 0.002 0.05 0 .0 1 0 .0 2 Figure-8.25: The effect of fracture storativity (to) of non-sealing fault model on pressure drawdown behavior on semi-log plot. 0 .1 0 .0 1 N on-Sealing 0.001 IE-02 1E+02 1E + 04 1E+06 IE-04 1E+00 T D 0.005 0.002 ■ ' - 1 0 = 0 .0 0 1 0.0 5 0 .0 1 0 .0 2 Figure-8.26: The effect of fracture storativity (œ) of non-sealing fault model on pressure drawdown behavior on derivative plot. 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 1 0 N o Fault 1E +04 1E+06 IE-02 1E+02 IE-04 1E+00 T D 0.005 0.002 ( 0 = O .O O I 0.05 0 .0 1 0 .0 2 Figure-8.27: The efifect of fiiacture storativity (a>) of no fault model on pressure drawdown behavior on semi-log plot. 0 .1 C D =0.0 SD =0.0 rw = O .S b=50ft 0 .0 1 N o Fault 0.001 1E+06 IE-02 1 E + 0 2 1E +04 IE-04 1E+00 T D 0.002 0.005 (□=0.001 0.05 0 .0 1 0 .0 2 Figure-8.28: The efifect of fracture storativity (co) of no fault model on pressure drawdown behavior on derivative plot. 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is also evident that the relationship between the minimum point o f the depression is inversely related to of the fracture storativity (©). While the depth o f the depression is largest for small fracture storativity (®), it is lowest approaching zero for large. Once again, the ratio of depression (the value of the minimum point of the transition zone) to the value o f the infinite-acting response (equals 0.5 for dimensionless plot) is similarly better suited for practicality purposes, allowing the application o f the relationship to the analysis of both dimensional and dimensionless derivative plots. A graphical correlation was developed (Figure-8.29), relating the depression ratio (d^) of the trough to the fracture storativity (ca). Knowing the dimensionless pressure derivative value of the minimum point o f the transition zone, one can use Figure-8.29 to calculate the value of < D with significant accuracy from the depression ratio (the value of the minimum point of the trough to the value of the infinite-acting response). nil 0 .1 0 .0 1 0.001 0.0001 0.001 IE-05 0.0001 0 .0 1 0 .1 Figure-8.29: Relation between the depression ratio (d^) o f the trough and the fracture storativity (®). 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Using the data of Figure-8.29 and nonlinear regression and curve fitting programs, an empirical correlation was also developed relating the fiacture storativity (o) to the depression ratio (dg) of the trough as follow: to=0.270(3.537r«(i/„)‘ ‘^ ^ (9) The above equation is an approximation of the curve o f Figure-8.29 that is still applicable with almost the same accuracy as the original graphical correlation. Figure-8.30 shows the accuracy of the fit of the regression model as defined by Equation-8.9 to the original data (Figure-8.29). Detailed fit information of the above correlation is also tabulated and presented in Appendix-F. 8 D m p f B A R i o n R a t i o Figure-8.30: Best fit o f regression model of Equation-8.9 to the original data based on relationship between (m) and (d^). 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8.4.5 Effects of inter-porosity flow parameter (X .) Drawdown tests were also run to investigate the efifect of matrix-fracture inter- porosity flow parameter (X .) on pressure behavior of the new model as summarized in Table- 8.4. Values of X.=l,10‘,10’ ^,10'^,10"‘ and 10'^ were used to examine the efifect of changing X , on both semi-log plot (Figures-8.37, 8.39 and 8.41) and on derivative plot (Figures-8.38, 8.40 and 8.42) for all the three cases of sealing fault (e f=0), partially communicating fault (s=0.2) and no fault model (^200) respectively . With small fault distance (dp=50 ft) and well radius (r, =0.5 ft), these X , values corresponds to X,*=10‘ ^,10'^,10'^,10'\10 * and 10 ’ respectively. The effect of varying X , is reflected by a similar pressure response exhibited by the tests run to examine the effect of fault distance (dp) in Figures-8.16 to 8.21. These effects ar manifested by the different time of arrivals of the transition period onset (reflecting NFRs) or equally the time for the arrival of the minimum depression point. Table-8.4: Drawdown tests to investigate the effect of inter-porosity flow parameter (X .) on transient pressure response Drawdown tests G dp X * ( D s Cd* T e st# l 0.2 50 1 xlQ-* 0.01 0 0 Test #2 0.2 50 1x10^ 0.01 0 0 Test # 3 0.2 50 1 xIO-* 0.01 0 0 Test # 4 0.2 50 1 X to " 0.01 0 0 Test # S 0.2 50 I xIO^ 0.01 0 0 Test # 6 0.2 50 1 xtO^ 0.01 0 0 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s 20 CD=0.0 SD=0.0 rw = O .S b=SOft 18 Sealing Fault 16 14 1 2 1 0 8 6 4 1E+02 1E + 04 IE-02 1 E + 0 0 T D • >.•=IE-09 IE-06 IE-08 IE-05 IE-07 0.0001 Figure-8.31 : The effect of inter-porosity flow parameter (A .) of sealing fault model on pressure drawdown behavior on semi-log plot. I S 1 0 CD=0.0 SD=0.0 nw =0.5 b=50R Sealing Fault 1 0 .1 0 .0 1 — 1E-04 IE-02 1E + 04 1E+00 1E+02 1E+06 T D A .» = IE -0 9 IE-06 IE-08 IE-05 IE-07 0.0001 Figure-8.32: The effect of inter-porosity flow parameter (A .) of sealing fault model on pressure drawdown behavior on derivative plot. 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 C D =0.0 SD =0.0 n»=0.5 b=SO ft 1E+04 1E+06 IE-04 1E+00 1 E + 0 2 IE-02 T D IE-07 il*=IE-09 0 .0 0 0 1 — IE-06 Figure-8.33: The effect of inter-porosity flow parameter (X) o f non- sealing 6ult model on pressure drawdown behavior on semi-log plot. C D =0.0 SD =0.0 tw =0.5 b=50ft 0 .1 Non-Sealing 0 .0 1 1E+06 IE-04 1E+00 1E+02 1E+04 IE-02 T D IE-07 A .«= IE -09 IE-08 0.0001 — IE-06 IE-05 Figure-8.34: The effect of inter-porosity flow parameter (X) of non sealing 6ult model on pressure drawdown behavior on derivative plot. 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s 1 2 C D =0.0 SD=0.0 n*=0.5 b=SOIt N o Fault 1 0 8 6 4 2 * — 1E-04 1E+04 1E+06 1E-02 1E+00 1E+02 T D X ,*= IE -09 lE-06 lE-08 lE-05 lE-07 0.0001 Figure-8.35: The effect of inter-porosity flow parameter (X) of no fault model on pressure drawdown behavior on semi-log plot. 0 .1 N o Fault 0 .0 1 1E+04 1E+06 IE -0 2 1E+00 1E+02 IE-04 T D X .*=lE -09 IE -08 IE-07 IE-06 0.0001 IE-05 Figure-8.36: The effect of inter-porosity flow parameter (X) of no feult model on pressure drawdown behavior on derivative plot. 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8.4.6 Effects of skin damage (S) and weilbore storage (Cg) Equation-8.4 and 8.5 were used to examine the effects of both skin damage (S) and weilbore storage (Cg) at the active well on the pressure drawdown behavior of the new model. As indicated previously at the beginning of this chapter, for the case of drawdown tests (L=b), therefore the dimensionless weilbore storage (Cp) is: C = __________C__________ ^ 27t[((pc,/»)^+((pc,/j)„]62 Therefore since the model uses Cp that is defined based on dp instead of Cp* which is based r^, the value of Cp will change depending on the fault distance to the well (dp). Hence, the relationship between Cp and Cp* can be defined as: (11) For small feult distance (dp<100 ft), the value o f Cp will be close to that of Cp*. On the other hand, if feult distance is large (dp>500 ft), Cp may be several order of magnitude larger than Cp*. The relationship between Cp and Cp* and fault distance (dp) is also represented by graphical correlation shown in Figure-8.37. Tabulated relationship is also included in Appendix-G. Various combination of skin damage (S) and weilbore storage (Cp) were used (as summarized in Table-8.5) to test the extent of their masking effect and the magnitude of the resulting distortion of the pressure response on both semi-log and derivative plots. For tests- 1, 4 and 7 on Table-8.5, the combined effects of skin and weilbore storage on the transient pressure behavior at the active well are shown on both semi-log plot (Figures-8.38, 8.40 and 8.42) and on derivative plot (Figures-8.39, 8.41 and 8.43) for all the three cases of sealing fault (8=0), partially communicating fault (s=0.2) and no fault model (8=200) respectively. Depending on their magnitude, weilbore storage and skin effects will result in the masking of the early time portion of the recorded pressure data leading to erroneous parameters estimation or even incorrect pattern recognition of the actual reservoir model. 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1E4 1E2 1E0 IE-2 1E-4 1E -S IE-8 1E 1 1E 2 1E 3 IE* C D * = IO O 1 0 0 0 10000 100000 1000000 Figure-8.37: Relationship between Cp and Cg* and fault distance. Table-8.5: Drawdown tests to investigate skin damage (S) and weilbore storage (Cp) effects on transient pressure response Drawdown tests e dp X * ( D s Cd * Test# 1 0.2 50 1 X 10^ 0.01 I KP Test # 2 0.2 50 1 X 1Q -* 0.0! 5 1 0 ^ Test # 3 0.2 50 1 X 1Q -* 0.01 1 0 1 0 * Test # 4 0.2 200 I X 10^ 0.01 1 1 0 * Test # 5 0.2 200 I X 10"^ 0.01 5 1 0 ^ Test # 6 0.2 200 1 X iO^ 0.01 10 1 0 * T est#? 0.2 5000 I X 10^ 0.01 1 1 0 * Test # 8 0.2 5000 1 X IQ -® 0.01 5 1 0 ^ Test #9 0.2 5000 1 X 10^ 0.01 1 0 1 0 * 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 b=SOft C D =0.1 SD=1.0 tw =0.5 Sealing Fault 15 1 0 N o Fault 5 0 IE-04 IE-03 IE-02 IE-01 1E+00 1E+01 1E+02 1E+03 1E+04 1E+05 T D >E=0 0 2 2 0 0 Figure-8.38: The e&ct of both skin damage (S) and weilbore storage (Cd ) for pattem-I on semi-log plot. Sealing Fault No Fault 0 .1 b = S O R C D =0.1 SD=1.0 iw =0.5 0 .0 1 0.001 IE-04 IE-03 IE-02 IE-01 1E+00 1E+01 1E+02 1E+03 1E+04 1E+05 T D 2 0 0 e=0 ' 0 2 Figure-8.39: The effect of both skin damage (S) and weilbore storage (Cd ) for pattem-I on derivative plot. 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 0 b =200 n CD =6.3E-3 SD =1.0 n u F O .S Sealing Fault N o Fault IE-04 IE-03 IE-02 IE-01 1E+00 1E + 01 1E+02 1E+03 1E+04 1E+05 T D E=0 — OJJ 2 0 0 Figure-8.40: The efifect of both skin damage (S) and weilbore storage (Cp) for pattem-n on semi-log plot. Sealing Fault r N o Fault b=200 ft CD=6.3E-3 SD=1.0 r w = O .S 0 .0 1 IE-04 IE-03 IE-02 IE-01 1E+00 1E + 01 1E+02 1E+03 1E+04 1E+05 T D 2 0 0 Figure-8.41: The efifect of both skin damage (S) and weilbore storage (Cp) for pattem-n on derivative plot. 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 0 b = S O O O ft CD =1E-5 S0=1.0 rw =0.5 Sealing Fault N o Fault 0 I iiiaW I iiiW iiiiwl I I lim a i iiiia* i iiiia< i iiiW m i l .......id IE-05 1E-04 IE-03 IE-02 IE-01 1E+00 1E+01 1E+02 1E+03 1E+04 T D E=0 — OJ 2 0 0 Figure-8.42: The efifect of both skin damage (S) and weilbore storage (Cp)for pattem-III on semi-log plot. 2 1 0 Sealing Fault 1 N o Fault 0 .1 b = S O O O ft C D =1E -5 SD =1.0 n w = O .S 0.01 m iiea I I" "" H iiiaa iiiiiaa ....... I i niaa « m iW iiiii^ ' 1E-05 IE-04 IE-03 IE-02 IE-01 1E+00 1E+01 1E+02 1E+03 1E+04 T D • e=0 — 02 2 0 0 Figure-8.43: The efifect of both skin damage (S) and weilbore storage (Cp) for pattem-III on derivative plot. 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figures-8.38 and 8.39 were generated with parameters as shown in test # I resulting in pattem-I. Even with small values of skin and wellbore storage (S and Cp), the early time straight line, the 6ult transition period and the middle times straight line are all obscured. Because of their arrival at late times for pattem-I response, only the transition depression of the naturally fractured reservoir (NFR) and the late time straight line remain unaffected. Figures-8.40 and 8.41 were generated by parameters as shown in test # 4 resulting in pattem-n response. In this case, the hump representing partially communicating fault is split in two halves by the depression zone of the naturally fractured reservoir, and the masking of the wellbore storage and skin effects only covers the early times straight line with part of the early parts of the two transition periods (for the fault response and the naturally fractured reservoir response respectively). Figures-8.42 and 8.43 were generated by parameters as shown in test # 7 which result in pattem-III response with the transition period of the naturally fractured reservoir (depression) arriving before that of partially communicating faults (hump). The masking effects of skin and wellbore storage covers the early times straight line with part of the transition period of naturally fractured reservoir. The fault response is not masked since it arrives at late times for pattem-III. The nine draw down tests on Table-8.5 are shown on both semi-log plot and on derivative plot as shown in Figures-8.44 and 8.45, Figures-8.46 and 8.47 and Figures-8.48 and 8.49 for pattem-I, II and III respectively. All nine tests are run for partially communicating faults (6=0.2), as shown in Table-8.5, with increasing skin and wellbore storage effects. Tests #1,2 and 3 are run for pattem-I, test #4,5 and 6 are run for pattem-H and test #7,8 and 9 are run for pattem-III. As can be seen from these nine graphs (Figures-8.44 and 8.49), increasing skin and wellbore storage effects will extend their duration to middle or even late times, leading to the masking of the characteristic features of the model response. It is therefore highly recommended that pressure measurement are recorded downhole in order to minimize wellbore storage effect and capture the early time portion of the model response. 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m = 0 .0 1 c = 0 ^ 2 0 b=50ft ni=O .S s 5 • IE-02 1E+00 1E+02 1 E + 0 G IE-04 1E+04 T O 10.10 1 ,0 .1 Figure-8.44: The increasing effect of both skin damage (S) and wellbore storage (Cp) for pattem-I on semi-log plot. 0 .1 0 .0 1 0.001 0.0001 ■ IE-05 IE-02 1E+00 1E+04 1E+06 IE-04 1E+02 T O — 1 . 0 .1 1 0 . 1 0 Figure-8.45: The increasing effect of both skin damage (S) and wellbore storage (Cg) for pattem-I on derivative plot. 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 1= 0.16 co=0.0! e = 0 2 IE-04 IE-03 1E-02 IE-01 1E+00 1E+01 1E+02 1E + 03 1E+04 1E+05 T D 5.0.0625 — 1.0.00625 10.0.625 Figure-8.46: The increasing effect of both skin damage (S) and wellbore storage (Cp) for pattem-II on semi-log plot. 0 .1 0 .0 1 b=200 It tw =0.5 0.001 0.0001 IE-04 IE-03 IE-02 IE-01 1E+00 1E+01 1E+02 1E +03 1E+04 1E+05 T D — — 1.0.00625 5.0.0625 10.0.625 Figure-8.47: The increasing effect of both skin damage (S) and wellbore storage (Cg) for pattem-II on derivative plot. 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 25 2 0 IE-05 IE-04 1E -03 IE-02 IE-01 1E+00 1E +01 1E+02 1E+03 1E+04 T D SD=0.CD=0 — 1.1E -5 5.1 E -4 10.1E -3 Figure-8.48: The increasing efiFect o f both skin damage (S) and wellbore storage (Cg) for pattem-III on semi-log plot. 0 .1 b=5000 n tw =0.5 0 .0 1 IE-05 IE-04 IE-03 IE-02 IE-01 1E+00 1E+01 1E+02 1E+03 1E+04 T D SD=0,CD=0 1.1E -5 5,1 E-4 10.1E-3 Figure-8.49: The increasing effect of both skin damage (S) and wellbore storage (Cg) for pattem-III on derivative plot. 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 9 Pressure Interference Analysis 9.1 Dimensionless pressure distribution in region-1 Interference tests can be performed by measuring the pressure response at an observation well located at some distance from the producing well. The new model allows for the placement of two observation wells on either side of the fault plane in region-1 (active region) and region-2 (inactive region). Pressure interference response at an observation well (both in the active region-1 and the inactive region-2) can be obtained by setting L=|b-x|, and y=0, therefore : (|6o-z^l)=^A[=l (1) (^D +*o) = 1+2*0 (2) Hence, the dimensionless pressure solution in Laplace space becomes: ^/DI = ] + 1 [/? ;ü T( 2xo +1 ^ ’ . I e [^ /,(j ) u] du (2x„^l) The above equation is used for all pressure interference tests in region-1. (3) 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9.2 Dimensionless pressure distribution in region-2 The dimensionless pressure distribution in the fracture network in the non-active region is similarly obtained using equations A-50 to A-55: P = 2 e 2e(2|xo|M) V ’ V , ____ , (4) . f . K , [ , j ^ u \ d u (2|xo| H ) The above solution for region-2 is highly dependent on the fault transmissivity ratio (e) which controls the fluid leakage rate through the fault. When the fault is completely sealing (efO), both the leakage rate through the feult and the pressure drop in the non-active region will be equal to zero. 9.3 Pressure interference tests in the active well region i Solutions for both regions of the new model were analyzed first by conducting simulated pressure interference tests in an observation well located on the same side of the fault (region-1) followed by interference test simulations in an observation well located on the other side of the fault (region-2). 9.3.1 Effects of controlling parameters on interference tests in region-i For an observation well located on the same side of the fault as the producing well (region-1), tests were run to examine the effects of the controlling parameters on the pressure behavior of the new model, as well as the effects of various well configurations with respect to the fault location. Interference tests were simulated to examine the effects of the controlling parameters including fault communication index (e) and fault distance to both active and observation wells. The different values used in these tests in region-1 are summarized in Tables-9.1 and 9.2. The effects of various configurations of active and observation well locations (points b and X respectively) with respect to the fault plane can portrayed as shown in Figure-9.1. 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table-9.1: Interfereace tests in region-1 to investigate the effect of fault communication parameter (e) and well locations (0<x<b) T e s ts # E b X X * Q ) s C o* T est# 1 0 ,0 J , 20 2500 100 1X i(r* 0.01 0 0 T est# 2 0,0.2, 20 2500 200 1 X 10^ 0.0! 0 0 Test # 3 0, OJ, 20 2500 300 1 X 10^ 0.01 0 0 Test # 4 0,0.2, 20 2500 400 1 X 1 Q -* 0.01 0 0 Test # 5 0,0.2, 20 2500 500 1 X IQ -* 0.01 0 0 Test # 6 0,0.2,20 2500 600 1 X 1 0 -* 0.01 0 0 T e st# ? 0,0.2,20 2500 700 1 X IQ -* 0.01 0 0 Test # 8 0,0.2,20 2500 800 1 X 1 Q -* 0.01 0 0 Test # 9 0,0.2,20 2500 900 I X 10' 0.01 0 0 Test # 10 0,0.2,20 2500 1000 1 X IQ -* 0.01 0 0 T est# 11 0,0.2, 20 2500 1100 1 X i(r* 0.01 0 0 Test # 12 0,0.2,20 2500 1200 1 X IC 0.01 0 0 T est# 13 0,0.2,20 2500 1300 1 X IQ -* 0.01 0 0 Test # 14 0,0.2, 20 2500 1400 1 X 1 Q -* 0.01 0 0 Test # 15 0,0.2,20 2500 1500 1 X I C 0.01 0 0 Test #16 0,0.2,20 2500 1600 1 X lO -* 0.01 0 0 Test # 17 0,0.2,20 2500 1700 1 X IQ-' 0.01 0 0 T est# 18 0,0.2,20 2500 1800 1 X IQ -* 0.01 0 0 Test # 19 0,0.2,20 2500 1900 1 X 1 Q -* 0.01 0 0 Test #20 0,0.2,20 2500 2000 1 X lO -* 0.01 0 0 Test #21 0,0.2,20 2500 2100 1 X 1 0 -* 0.01 0 0 Test # 22 0,0.2, 20 2500 2200 1 X 1 Q -* 0.01 0 0 Test # 23 0,0.2,20 2500 2300 I X 1 0 -* 0.01 0 0 Test # 24 0,0.2,20 2500 2400 1 X 1 0 -* 0.01 0 0 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table-9.2: Interference tests in region-1 to investigate the effect of fault communication parameter (e) and well locations (0<b<x) T ests # 8 X b X * 0) s Cd * T est# 1 0,0.2,20 2500 100 1 X 1 0 -* 0.01 0 0 Test # 2 0,0.2,20 2500 200 I X 10^ 0.01 0 0 T e st# 3 0,0.2,20 2500 300 1 X 10^ 0.01 0 0 Test # 4 0,0.2,20 2500 400 1 X 10" 0.01 0 0 Test # 5 0,0.2,20 2500 500 1 X I0-* 0.01 0 0 Test # 6 0,0.2,20 2500 600 I X 10^ 0.01 0 0 T e s t# ? 0,0.2,20 2500 700 1 X IQ -* 0.0! 0 0 Test # 8 0,0.2,20 2500 800 1 X I0-* 0.01 0 0 Test # 9 0, 0.2,20 2500 900 1 X 10^ 0.01 0 0 Test # 10 0,0.2, 20 2500 1000 1 X 1 0 -* 0.01 0 0 T est# I t 0,0.2,20 2500 1100 1 X 10-» 0.01 0 0 Test # 12 0,0.2, 20 2500 1200 1 X 1 0 -* 0.01 0 0 Test # 13 0,0.2,20 2500 1300 1 X 10-” 0.01 0 0 Test # 14 0,0.2, 20 2500 1400 1 X i(y 0.01 0 0 T est# 15 0,0.2,20 2500 1500 1 X I0-» 0.01 0 0 Test # 16 0,0.2, 20 2500 1600 I X 1 0 -* 0.01 0 0 T est# 17 0,0.2, 20 2500 1700 1 X I0-* 0.01 0 0 Test # 18 0,0.2,20 2500 1800 1 X 10-* 0.01 0 0 T est# 19 0,0.2,20 2500 1900 1 X I0-* 0.01 0 0 Test # 20 0,0.2,20 2500 2000 1 X 1 Q -* 0.01 0 0 Test #21 0,0.2, 20 2500 2100 1 X IQ -* 0.01 0 0 Test # 22 0,0.2, 20 2500 2200 I X 10^ 0.01 0 0 Test # 23 0,0.2,20 2500 2300 1 X I C 0.01 0 0 Test #24 0,0.2,20 2500 2400 1 X I0-* 0.01 0 0 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y=0 O kesratw M vcIl y=0 OkeratlaKweil y=0 [Region-2] Fudt (x=0) Figure-9.1: Active and observation well locations and fault distance for pressure interference tests in region-1. 9.3.2 Effects o f fault-well configuration and geometry The location of the observation well can be placed between the fault plane and the active well as shown in cases-1 and 2 (Figure-9.1). In case-1, the observation well is placed between the fault plane and the active well (0<x<b). In case-2 the observation well is gradually moved to different locations by increasing x such that 0<x<b. In case-3, the active well is placed between the fault plane and the observation well (0<b<x), which in turn is also placed at different locations by choosing larger x values such that 0<b<x. 9.4 Type curve Matching Pressure buildup (BU) and drawdown (DD) tests are usually analyzed and cross validated by both semi-log and type curve matching. However, interference tests can only be analyzed by type curve matching technique. Type curves are generated from solutions to flow problems under specific flow conditions. Numerous standard type curves have been introduced for analysis of various reservoir models including homogeneous [70], naturally fractured [71-79], Suited [56-59,80-82], layered [4], and composite reservoirs [61,64]. 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Log-log type curve is constructed by plotting the logarithm of some dimensionless pressure group function versus logarithm of dimensionless time group function (usually log[P[)]vs.log[t[/rD^]) on log-log paper. By matching the recorded test data to a specific type curve, unknown reservoir parameters can be estimated. 9.4.1 Type curves for interference tests in region ! New type curves can be generated by the dimensionless pressure solutions of both region-1 and region-2 (Equations-9.3 and 9.4 respectively). These type curves can be used to analyze pressure interference test data in order to estimate the defining parameters including feult communication index (e), feult distance (dp), inter-porosity flow parameter (X), fracture storativity ratio (<»), wellbore storage coefiBcient (Cg) and skin factor (S). Type curve analysis can provide us with the ability to estimate up to 3 unknown parameters, two from time and pressure axis and one from the matched curve. Because the model has several parameters (six unknowns), families of type curve are needed so that each family of curves is controlled by three parameters only. This can be done by allowing three of the parameters to be constants. Assuming that the fracture storativity ratio, wellbore storage coefficient and skin fector to be constants (oy=0.01, Cp=0 and S=0), the dimensionless pressure solutions of region-1 (Equations-9.3) was used to generate two families of type curves with the main variables being the fault communication index (e), fault distance (dp), inter-porosity flow parameter (A .). Each femily of type curves consists of three log-log plots (log[Pg]vs.log[to]) for the three cases of for sealing fault (e=0.0), non-sealing fault (6=0.2) and for the case o f no fault (e=20) respectively. The first family of type curves (Figure-9.2, 9.3 and 9.4) were generated by simulated tests # 1 to 24 o f Table-9.1, where the observation well is placed between the feuk plane and the active well (0<x<b) as shown in case-1 of Figure-9.1. Similarly the second family of type curves (Figure-9.5,9.6 and 9.7) were generated by tests # 1 to 24 of Table-9.2, where the active well is placed between the fault plane and the observation well (0<b<x) as shown in case-2 of Figure-9.1. 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 CD=0.0 SD=0.0 rw=0.5 b=2500n Tesc»l U : l Sealing Fault Q Q . IE-01 1E+00 1E+03 1E+01 1E+02 13 :3 14 :4 TD 10 20 Figure-9.2: Type curves generated by Table-9.1 (0<x<b), for interference tests in region-1 (sealing fault, e=0.0). 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 T e st# I 1 1 :1 CD=0.0 SD=0.0 rw=0.5 b=2500m Non Sealing Fault Q Q . X,*=I.OE-8 (0= 0.01 e=0.20 ikoi 1E+00 1E+01 1E+02 1E+03 3 13 23 U 24 TD 10 20 Figure-9.3: Type curves generated by Table-9.1 (0<x<b), for interference tests in region-1 (non-sealing fault, e=0.20). 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 T e s t # l I I 21 No Fault o 0. 1 13 23 X * = I . 0 E - 8 (0= 0 .0 1 8= 20.0 CD=0.0 SD=0.0 rw=0.5 b=2500ft 0.1 - IE-01 1E+00 1E+01 TD 1E+02 1E+03 10 20 Figixre-9.4: Type curves generated by Table-9.1 (0<x<b), for interference tests in region-1 (no feult, 8=20.0). 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 CD=0.0 SD=0.0 rw=0.5 x=2500ft Sealing Fault 12 22 Q Q _ 13 23 H 24 e=0.0 IE-01 1E+00 1E+01 TD 1E+02 1E+03 10 20 Figure-9.5: Type curves generated by Table-9.2 (0<b<x), for interference tests in region-1 (sealing fault, e=0.0). 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 CD=0.0 SD=0.0 rw=0.5 x=2500n Test#l II :1 Non Sealing Fault Q Q . 1 13 33 u :4 e=0.20 1 — IE-01 1E+00 1E+01 TD 1E+02 1E+03 10 :o Figure-9.6: Type curves generated by Table-9.2 (0<b<x), for interference tests in region-1 (non-sealing fault, e=0.20). 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. No Fault CD=0.0 SD=0.0 rw=0.5 x=2500tt 0.1 - IE-01 1E+00 1E+01 TD 1E+02 1E+03 Figure-9.7: Type curves generated by Table-9.2 (0<b<x), for interference tests in region-1 (no fault, e=20.0). 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9.5 Pressure interference tests in the observation well region-2 Similar to the ones run for region-1, pressure solution for region-2 was also analyzed by conducting pressure interference tests in an observation well located on the other side of the feuk (region-2). Figure-9.8 shows the location of both active and observation wells with respect to the fault plane. The different values used for the model’s controlling parameters are summarized in Tables-9.3 and 9.4. C4se-1 y=0 case-2 y=0 O kem tim itell case-3 y=0 (Region-2] Figure-9.8: Active and observation well locations and fault distance for pressure interference tests in region-2. 9.5.1 Effects of controlling parameters on interference tests in region-2 Interference tests were run in region-2 to examine the effects o f the controlling parameters including fault communication index (e) and fault distance to both active and observation wells. As shown in Figure-9.8, the effects of various configurations o f active and observation well locations (points b and x respectively) with respect to the fault plane were also investigated. 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table-9.3: Interference tests In region-2 to Investigate the effect of fault communication parameter (e) and well locations (b = = S O O ) Tests# 8 b X X * C O s C o* T est# 1 0.2,20 500 -100 1 X i(y * O .O l 0 0 Test # 2 0.2,20 500 -200 I X 1 0^ 0 .01 0 0 Test # 3 0.2,20 500 -300 1 X 1 0 - » 0.01 0 0 Test # 4 0.2,20 500 -400 1 X I Q - * 0.01 0 0 Test # 5 0.2,20 500 -500 1 X I 0 - * 0.01 0 0 Test # 6 0.2,20 500 -600 I X 1 0 - » 0.01 0 0 Test # 7 0.2,20 500 -700 I X I 0 - * 0.01 0 0 Test # 8 0.2,20 500 -800 1 X 1 0 - » 0.01 0 0 Test # 9 0.2,20 500 -900 1 X 1 0 - » 0.01 0 0 Test » 10 0.2,20 500 -1000 1 X 1 0 - » 0.01 0 0 T est# II 0.2,20 500 -1100 1 X 1 0 - » 0.01 0 0 T est# 12 0.2,20 500 -1200 1 X 1 0 » 0.01 0 0 T est# 13 0.2,20 500 -1300 1 X 1 0 - » 0.01 0 0 Test #14 0.2, 20 500 -1400 1 X 1 0 - » 0.01 0 0 Test # 15 0.2,20 500 -1500 1x10-» 0.01 0 0 T est# 16 0.2,20 500 -1600 1 X 1 0 - * 0.01 0 0 Test # 17 0.2,20 500 -1700 1 X 1 0 - » 0.01 0 0 Test # 18 0.2,20 500 -1800 1 X 1 0 - » 0.01 0 0 Test # 19 0.2,20 500 -1900 1 X 1 0 - » 0.01 0 0 Test # 20 0.2,20 500 -2000 1 X 1 0 - » 0.01 0 0 Test #21 0.2,20 500 -2100 1 X 1 0 - » 0.01 0 0 Test #22 0.2,20 500 -2200 1 X 1 0 - » 0.01 0 0 Test #23 0.2,20 500 -2300 1 X 1 0 - » 0.01 0 0 Test # 24 0.2,20 500 -2400 1 X 1 0 - » 0.01 0 0 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table-9.4: Interference tests in region-2 to investigate the effect of fault communication parameter ( e) and well locations (x=-500) Tests# 8 X b X * C D s C d* T est# 1 0.2,20 -500 100 1 X 1 0 -* 0.01 0 0 Test #2 0.2, 20 -500 200 1 X 10^ 0.01 0 0 T est# 3 0.2,20 -500 300 1 X 10* 0.01 0 0 Test # 4 0.2,20 -500 400 I X 10-* 0.01 0 0 Test # 5 0.2,20 -500 500 1 X IQ -* 0.01 0 0 Test # 6 0.2,20 -500 600 I X I0-* 0.01 0 0 Test # 7 0.2,20 -500 700 ! X 1 0 -* 0.01 0 0 Test # 8 0.2,20 -500 800 1 X 1 0 -* 0.01 0 0 Test # 9 0.2,20 -500 900 1 X 1 0 -* 0.01 0 0 Test # 10 0.2,20 -500 1000 1 X 1 0 -* 0.01 0 0 T est# 11 0.2,20 -500 1100 1 X 1 0 -* 0.01 0 0 Test # 12 0.2,20 -500 1200 1 X 10 -* 0.01 0 0 Test # 13 0.2,20 -500 1300 1 X 1 0 -* 0.01 0 0 Test # 14 0.2,20 -500 1400 1 X 1 0 -* 0.01 0 0 T est# IS 0.2,20 -500 1500 1 X 1 0 -* 0.01 0 0 Test # 16 0.2,20 -500 1600 1 X 1 0 -* 0.01 0 0 Test # 17 0.2, 20 -500 1700 1 X 10 -* 0.01 0 0 Test # 18 0.2,20 -500 1800 1 X 10 -* 0.01 0 0 Test # 19 0.2,20 -500 1900 1 X 1 0 -* 0.01 0 0 Test # 20 0.2,20 -500 2000 1 X 1 0 -* 0.01 0 0 Test #21 0.2,20 -500 2100 1 X 1 0 -* 0.01 0 0 Test # 22 0.2,20 -500 2200 1 X 1 0 -* 0.01 0 0 Test #23 0.2,20 -500 2300 1 X 10 -* 0.01 0 0 Test # 24 0.2,20 -500 2400 1 X 10 -* 0.01 0 0 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9.5.2 Type curves for interference tests in region-2 Similar to interference tests in region-1, assumptions are made that the fracture storativity ratio, wellbore storage coefBcient and skin factor to be constants (ca=0.01, Cg=0 and S=0). Using the dimensionless pressure solution for region-2 (Equations-9.4), two families of type curves were generated with the main variables being the fault communication parameter ( e ) , fault distance (dp), inter-porosity flow parameter (X). Each femily of type curves also consists of only two log-log plots (log[P[,]vs.log[tD]) for the cases of non-sealing fault (e=0.2) and no fault (e=20) respectively, since for the case of sealing feuh (^=0.0), both fluid leakage through the fault and pressure drop in region-2 are equal to zero. The first family o f type curves (Figure-9.9 and 9.10) were generated by tests # 1 to 24 of Table-9.3, where the location of the active well (b) is kept stationary while the observation well distance to the fault (x) is gradually increased by placing the observation well at different locations along the x-axis as shown in case-2 and 3 of Figure-9.8. Similarly the second famUy of type curves (Figure-9.11 and 9.12) were generated by tests #1 to 24 o f Table-9.4, where the location of the observation well is kept stationary while the active well distance to the fault is increasing by placing the active well at different locations along the x-axis as shown in case-2 and 3 of Figure-9.8. 9.5.3 Effects o f fault-well configuration and geometry Similar to conclusions arrived at by Yaxley [59], it was also found that the pressure response in an observation well across the feuk (in region-2) does not depend on the distance of either active nor observation well to the fault (location of points b and x respectively), but only on the inter-well distance (L). Tables-9.5 and 9.6 shows two sets of interference tests with active and observation wells configurations similar to those in Tables-9.3 and 9.4. Although the locations of both active and observation well are different for all the tests in Tables-9.5 and 9.6, the inter-well distance (L) is equal for corresponding tests. Hence, the log-log type curve plots (Figure-9.13 and 9.14) generated by Tables-9.5 and 9.6, are identical with equal pressure drops at any particular time. 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 CD=0.0 SD=0.0 rw=0.5 b=500ft Testl»l II Zl Non Sealing Fault 3 '.3 :3 o CL 1 A.*=I.0E-8 < 0 = 0 .0 1 6= 0.20 0.1 IE-02 IE-01 1E+00 1E+02 1E+01 1E+03 a IB TD 10 20 Figure-9.9: Type curves generated by Table-9.3 (b=500 ft), for interference tests in region-2 (non-sealing fault, e=0.20). 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 Test»! Il :1 No Fault Q C L 1 14 :4 A.*=!.0E-8 o j= O .O I e = 2 0 .0 CD=0.0 SD=0.0 rw=0.5 b=500ft 0.1 ----------------------------------— - 1E-02 1E-01 1E+00 1E+01 TD 1E+02 1E+03 1 0 :0 Figure-9.10: Type curves generated by Table-9.3 (b^SOO ft), for interference tests in region-2 (no fault, e=20.0). 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 CD=0.0 SD=00 rw=0.5 x=-500fl Test#l II 21 Non Sealing Fault o 0. 1 0.1 IE-02 IE-01 1E+00 1E+01 1E+02 1E+03 3 13 23 U 24 TD 10 20 Figure-9.1 1 : Type curves generated by Table-9.4 (x=-500 ft), for interference tests in region- 2 (non-sealing fault, e=0.20). 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 T e a t # l I I : l No Fault Q C L 1 13 23 14 24 A.*=l.0E-8 c i) = O .O I e= 2 0 .0 CD=0.0 SD=0.0 rw=0.5 x=-500ft 8 L B 0.1 ---- IE-02 IE-01 1E+00 1E+01 TD 1E+02 1E+03 10 20 Figure-9.12: Type curves generated by Table-9.4 (x=-500 ft), for interference tests in region- 2 (no fault, e=20.0). 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table-9.5: Interference tests in region-2 to investigate the effect of fault and well locations (b=250) Tests# £ b X X * m s C d* T est# 1 0.2 250 10 1 X 10^ 0 .0 1 0 0 Test # 2 0.2 250 20 1 X I Q - * 0 .0 1 0 0 Test # 3 0.2 250 -30 1 X I 0 - * 0 .0 1 0 0 T e s t# 4 0.2 250 -40 1 X 1 0 * 0 .0 1 0 0 T est# 5 0.2 250 -50 I X 1 0 ^ 0 .0 1 0 0 Test # 6 0.2 250 -60 1 X 1 0 ^ 0 .0 1 0 0 Test # 7 0.2 250 -70 I X 1 Q - * 0 .0 1 0 0 Test # 8 0.2 250 -80 1 X 1 Q - * 0 .0 1 0 0 Test # 9 0.2 250 -90 1 X I Q - * 0 .0 1 0 0 Test # 10 0.2 250 -100 1 X 1 0 - * 0 .0 1 0 0 T e st# II 0.2 250 -110 1 X 1 0 - * 0 .0 1 0 0 Test # 12 0.2 250 -120 1 X t o - * 0 .0 1 0 0 T est# 13 0.2 250 -130 I X I 0 - * 0 .0 1 0 0 Test # 14 0.2 250 -140 1 X 1 0 * 0 .0 1 0 0 Test # 15 0.2 250 -150 1 X 1 0 - * 0 .0 1 0 0 Test # 16 0.2 250 -160 1 X 1 0 - * 0 .0 1 0 0 T est# 17 0.2 250 -170 1 X 1 0 - * 0 .0 1 0 0 Test # 18 0.2 250 -180 1 X 1 0 - * 0 .0 1 0 0 Test # 19 0.2 250 -190 1 X 1 0 - * 0 .0 1 0 0 Test # 2 0 O J 250 -200 1 X 1 0 - * 0 .0 1 0 0 Test #21 0.2 250 -210 1 X 1 0 - * 0 .0 1 0 0 Test #22 0.2 250 -220 1 X 1 0 - * 0 .0 1 0 0 Test # 23 0.2 250 -230 1 X 1 0 - * 0 .0 1 0 0 Test # 24 0.2 250 -240 1 X 1 0 - * 0 .0 1 0 0 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table-9.6: Interference tests in region-2 to investigate the effect of fault and well locations (x=-250) Tests# E X b X * (O s C„* T est# 1 0.2 -250 1 0 1 X 10^ 0 .0 1 0 0 Test #2 0.2 -250 20 1 X I0-* 0 .0 1 0 0 Test #3 0.2 -250 30 1 X 10^ 0.0! 0 0 T est# 4 0.2 -250 40 1 X 1 0 -* 0.01 0 0 T est# 5 0.2 -250 SO I X I0-* 0 .0 1 0 0 Test #6 0.2 -250 60 1 X 1 0 -* 0.01 0 0 T est# 7 0.2 -250 70 1 X i(r* 0.01 0 0 Test #8 0.2 -250 80 I X 1 0 -* 0 .0 1 0 0 Test #9 0.2 -250 90 I X icr» 0.01 0 0 Test # 10 0.2 -250 1 0 0 1 X 1 0 -* 0.01 0 0 T est# 11 0.2 -250 1 1 0 I X 0.01 0 0 T est# 12 0.2 -250 1 2 0 1 X 10^ 0.01 0 0 Test # 13 0.2 -250 130 I X I0-* 0.01 0 0 Test # 14 0.2 -250 140 1 X 1 0 -* 0.01 0 0 Test # 15 0.2 -250 150 I X I0-* 0.01 0 0 Test # 16 0.2 -250 160 I X 10^ 0 .0 1 0 0 Test # 17 0.2 -250 170 1 X 1 0 ' 0.01 0 0 Test # 18 0.2 -250 180 1 X I0-« 0 .0 1 0 0 Test # 19 0.2 -250 190 1 X IQ - * 0.01 0 0 Test # 20 0.2 -250 20 0 1 X I Q - * 0.01 0 0 Test #21 0.2 -250 210 1 X 10^ 0.01 0 0 Test #22 0.2 -250 220 1 X 1 Q - * 0.01 0 0 Test # 23 0.2 -250 230 1 X 1 0 -* 0.01 0 0 Test # 24 0.2 -250 240 1 X 1 Q - * 0.01 0 0 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CD=0.0 -------------------- Tesc»l 11 21 SD=0.0 n*=Q g Non Sealing Fault 12 22 b=250 ft 3 13 23 A 14 24 o Q . b 15 A.*=1.0E-8 6 7 16 17 m =O .O I e=0.20 ' - 1E+00 1E+01 1E+02 18 1E+03 ----- - - TD 10 19 20 Figure-9.13: Type curves generated by Table-9.5 (b^250 ft), for interference tests in region-2 (non-sealing fault, 8=0.20). 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 CD=0.0 SD=0.0 rw=0.5 x=-250ft Test#l 11 21 Non Sealing Fault 3 13 :3 4 14 : 4 Q 0 _ 5 15 X . * = 1 . 0 E - 8 o )= 0 .0 1 e=0.20 1 - — 1E+00 1E+01 1E+02 1E+03 TD 10 : o Figure-9.14: Type curves generated by Table-9.6 (x=-250 ft), for interference tests in region- 2 (non-sealing fault, e=0.20). 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 10 Conclusions and Recommendations for Future Work 10.1 Conclusions Rock formations in oil reservoirs are highly heterogeneous in nature, and many types of heterogeneity are usually developed within the same reservoir as a result of past geologic processes. Because both fouks and fractures are often created ly the same geologic processes, their coexistence in the same reservoir is commonplace. The main objective of this study is to focus on the development of new and improved analytical conceptual models for the characterization of NFRs in the presence of other related heterogeneities such as microfracture networks, sealing faults and partially communicating foults. First, the dual fracture system is proposed as a more realistic representation of the fracture network in NFRs, where two different sets of fi^ctures, macro fracture and micro fracture systems are present in addition to the matrix system. This model is a general one that can easily be modified to produce other conceptual models for NFRs including triple porosity [3], double permeability [4], and dual porosity [5] models. For the representation of sealing and non-sealing geologic faults in naturally fractured reservoirs (NFRs), three new conceptual models are introduced. The first two models are generated by applying the principle of superposition and the method of images [6] to the double permeability and the dual fracture model respectively, to account for presence of totally sealing foults in NFRs. The last model is a more general one that incorporate the effect of a partially communicating fault on the pressure behavior of NFRs. 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ana^rgis of the generated transient pressure response of these models is conducted on semi-log, log-log and pressure derivative plots and by type curve matching. Results of pressure analysis o f the dual fracture model can be summarized in the following remarks: 1. The delineation of fractures into two broad categories, macro fractures and microfractures, is a step forward towards a more realistic representation of naturally fractured reservoirs. 2. On the pressure derivative plot, the pressure support o f the microfractures is similar to that of the matrix and their presence can lead to the formation of transition zones at substantially earlier time. These zones may be mistakenly interpreted as matrix support. 3. The proposed models provides an explanation for the observation of early pressure support emanating from a network of microfractures and often attributed to the tight matrix rocks. 4. The model also provides a general explanation for the observation (or lack of observation) of single or double transition periods on the test data from naturally fractured reservoirs. 5. The concept of dually fracture reservoir can lead to better estimation of reservoir parameters including the partition coefBcients corresponding to the volumetric contribution of macro and micro fractures in addition to the matrix. 6. The proposed model suggests that the pressure response of the tight matrix rocks requires extended test duration to be observed. More realistic design of pressure test can be implemented for improved characterization of naturally fractured reservoirs. 7. Finally, the influence of the microfractures support at early times can be best exhibited with down-hole recording to minimize the effect of wellbore storage. For the next two models, the methods of images and superposition are applied to the Laplace-space analytical solutions of both double permeability and dual fiacture models to investigate the effect of totally sealing fault on their transient pressure behavior. Analysis of 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the generated pressure drawdown response of these new models is conducted on both semi log and derivative plots. Similar to the conclusions arrived at by Khatchatoorian and Ershaghi [63], the time for doubling of slope corresponding to the presence of a sealing fault is found to be dependent on the foult distance (dp) as well as on the other controlling parameters such as (X) and (©) of NFRs. For the last model o f NFRs with partially communicating faults, two solutions are presented for pressure distribution in the two regions on either side of the non-sealing fault. Solution for the active region (region-1) provides us with the ability to conduct simulated drawdown and buildup tests at the active (producing) well, in addition to pressure interference tests in the producing region on the same side o f the fault. On the other hand, solution for the inactive region (region-2) allows us to simulate interference tests across the non-sealing foult, where pressure is recorded at an observation (inactive) well located on the other side of the fault. Similar to the approach utilized by Yaxley [59], the representation of the partially communicating foult in naturally fiactured reservoirs is accomplished by considering a semi- permeable linear barrier of infinite extent and negligible storage capacity located at x=0 and divide the reservoir into two regions. Region-1, located on the right side of the fault plane (x>0), contains the producing or active well. On the other hand, the observation well region (region-2) is located on the other side of the fault plane (x<0) and contains only one observation well. The first solution for the active well region (Equation-7.56) was modified to produce the pressure drawdown solution at the active well. This drawdown solution was tested against its limiting cases which include the following five asymptotic limits: 1. NFR with sealing fault. 2. NFR (no fault). 3. Homogeneous reservoir with non-sealing fault. 4. Homogeneous reservoir with sealing fault. 5. Homogeneous reservoir (no foult). On the dimensionless semi-log plot (Figure-8.2), typical response of the new model shows three straight lines appearing at early, middle and late time respectively, separated by 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. two transition periods. The first transition period results in an increased slope characteristic of no flow boundaries, while the second transition period is that of NFRs. On the dimensionless derivative plot (Figure-8.3), the response is exhibited by three straight lines that are reflected at early, middle and late times respectively, and separated by the same two transition periods. For sealing faults (e=0.0), the first transition period shows a gradual and steady increase of the derivative value until it becomes double that of the first line (firom 0.50 to 1.0). For partially communicating faults (e>0.0), however, the first transition period resemble a hump where derivative value reaches a maximum point then drops back to same value of the early infinite-acting straight line (0.5). It was found that for the case of partially communicating faults, there is a critical communication fault parameter (^=10.0) above which the second straight line is an extension of the first line, and the fault presence for all practical purposes is undetectable. It is also found that the amplitude of the hump is inversely related to the fault communication parameter (e). While the amplitude is largest for small fault communication parameter (e— >0), it is lowest approaching zero for large fault communication parameter (£— ► < » ) . Graphical and empirical correlations were developed, relating the amplitude ratio (a^) of the hump to the fault communication parameter (e). These correlations can be used to calculate e with significant accuracy fi’ om the ratio of the amplitude (the value of the maximum point o f the hump) to the value o f the infinite-acting response. In addition to simulated drawdown tests run to investigate the effects of e on pressure response, a number of case studies were also conducted with the main variables being dp, k, (Ù , Cn and S. In general, for linear boundaries at close distance to the well (dp<lOO ft), the slope change reflecting the linear boimdary occurs before the arrival of the transition period of NFR and the total system response, which arrive at later times. For faults at large distance to the well (dp>500 ft), slope change reflecting the linear boundary occurs after the arrival of the transition period of NFR. For faults at distance to the well where 100<dp<500 ft, the slope behavior at both the early times and the middle times may be influenced resulting in a complicated and inconclusive patterns. 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For reservoirs with very tight matrix (A ,*=10*^ to lO '® ), or with the linear boundary very close to the weU ((*^<100 ft), the early feult reflection resulting in pattem-I may be observed. For reservoirs with less tight matrix (A,*=10'^ to 10'^), or with the linear boundary ftirther away from the well (100<dF<500 ft), the fault reflection will coincide with the transition period of NFR, resulting in distortion to both transition periods. This pattern where the two transition zones interfere with each other is called pattem-II. Finally, for reservoirs with permeable matrix system (k*=10"^ to 10'*), or with the linear boundary fer away from the well (df>500 ft), the fruit reflection will only arrive at late times after the transition period of the NFR and the total system response have been observed. This resulting pattern, where the transition period of NFR is observed at early times is referred to as pattem-IIl. It is also found that the minimum point of the depression representing NFRs is inversely related to the fiacture storativity ratio (to). Both graphical and empirical correlations were also developed relating the depression ratio (d^) of the trough to the fracture storativity ratio (to). Knowing the pressure derivative value of the minimum point of the transition zone, these correlations can be used to calculate the value of to with significant accuracy from the value of the depression ratio (the value o f the minimum point of the trough to the value of the infinite-acting response). Even with small values of skin damage (S) and wellbore storage (Cp), the early and midtile times straight lines are likely to obscured. Increasing skin and wellbore storage effects will extend their duration to middle or even late times, leading to the masking of the characteristic features of the model behavior. It is therefore highty recommended that pressure measurement are recorded downhole in order to minimize wellbore storage effect and capture the early time portion of the model response. Solutions for both regions of the new model were also analyzed first by conducting simulated pressure interference tests in an observation well located on the same side of the fault, followed by simulated interference tests in an observation well located on the other side. For an observation well located on either side of the fault, tests were run to examine the effects of the controlling parameters on the pressure behavior of the new model, as well as the effects o f various well configurations with respect to the fault location. 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. New type curves were generated by the dimensionless pressure solutions of both region-1 and region-2 (Equations-9.3 and 9.4 respectively). Assuming that the fracture storativity ratio (m), wellbore storage coefBcient (C^) and skin factor (S) are constants, the dimensionless pressure solutions of region-1 and region-2 were used to generate several families of type curves with the main variables being the feult communication index (e), fault distance (dp) and inter-porosity flow parameter (X). These Emilies of type curves and others can be used to analyze pressure interference test data in order to estimate the defining parameters of the new model including fault communication index (e), fault distance (dp), inter-porosity flow parameter (X), firacture storativity ratio (©), wellbore storage coefficient (Cg) and skin fector (S). Similar to conclusions arrived at by Yaxley [59], it was also found that the pressure response in an observation well across the feuk (in region-2) does not depend on the distance of either active nor observation well to the fault (location of points b and x respectively), but only on the inter-well distance (L). 10.2 Recommendation for future work Based on the progress made by this study, the following areas are recommended for future work: 1. Improvement of the newly proposed models: i. The dual fracture model developed in Chapter 4 can be further improved by a number of ways including addition of matrix contribution to production at the wellbore and allowing for matrix interaction with both microfi-acture and macro fiacture networks. ii. The dual fincture model, developed to account for two distinct fracture sets, can be further extended to NFRs with multi-fracture sets. This would be equivalent to multi-layer reservoirs with inter-layer cross flow. 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iii. The double permeability and the dual fracture models with sealing 6ult, developed in Chapter 6, can be further improved to include the effect of partially communicating 6ults. iv. The double porosity model with partially communicating fault, developed in Chapter 7, can be further improved by allowing for matrix-matrix communication across the &ult and contribution to production at the wellbore. This would be equivalent to the suggested double permeability model with partially communicating faults. V. Representation of the partial^ communicating 6ult as a semi-permeable linear vertical barrier of reduced permeability and thickness and negligible capacity as suggested by Stewart et aL [58] and Yaxley [59] can also be accomplished in a way similar to the one proposed by Ambastha et al. [60] who represented the partially communicating fault as an infinitesimal-thickness skin boundary located between the two regions of his double-region linear composite reservoir model. 2. Because of the significant advantages of Automated Well Test Analysis techniques over conventional graphical estimation methods, the application of computer-aided interpretation techniques including neural networks for pattern recognition and various nonlinear regression methods will significantly increase the accuracy and efficiency of the parameters estimation process of the new model. 3. Results arrived at during the course of this study should be verified by numerical simulators. Numerical modeling of the partially communicating fault can be accomplished ly considering the non-sealing fault as a semi-permeable linear vertical barrier of reduced permeability and thickness and negligible capacity as suggested by Stewart et al. [58] 4. Real test data obtained in NFRs are to be collected and analyzed according to the techniques suggested for the application of the newly proposed models. 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix-A Mathematical Derivation The dimensionless equations describing the flow in the macrofi’ acture, microfractiire, and the matrix system respectively are: ^ P fd 1 d r FD . ^yFD o '■ o ^Pfd ^ \ - ^ f^ P /D P fD ) (1) (1 -K ) ^PjD ^ 1 dPjD drl '-D ^ '’ d '*'^f^P/D PfD^ ^m^PmD PjD^ (2) œ. àp„o dtr. = -KiPmD-Pjo) (3) The initial and boundary conditions used for the system of equations-A. I to A.3 are: PF=Pf = Pm=Pi (4) At r=nv, ........ Pf =P/=P» CPp) = lim^_. (/y) = lim^_. (j>„) =p, (5) (6) 163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The downhole production rate is: dr a ? (7) Transforming the above equations to the Laplace space, then substituting equation- A.3 in equation-A.2 yields: kV (ùf^S Pfd^ (8) (1 - K )V ^/V = m m (9) where the dimensionless parameters are: ^ 0.000264( j ; k h )t (10) i ' Z k h ) , (11) ''d = E =kph^ + k^hj. (12) (13) E = ((p C ;A )p + ((p c,A ), + ((p c,A ) (14) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 E (15) Q > , E (16) “/ = _ ((PCf^)y E (17) “ m = ( * - ® F - ®/) (18) v - Æ ' - (19) m m E k h " (20) Let: X = Û ) 5 — ^ " to_î+X, m m (21) The solutions to equations-A.8 and A.9 are: ^FD ^F ^ ( ^ ) (22) (23) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 Substituting equations-A.22 and A.23 into equations-A.8 and A.9 yields: [ica^-a)^5 - =0 (24) (25) Solutions are possible when: (26) or: a* - (X - y ) ^ {fOpS+kj.) 1 - K K ^ +Ay)(%-Ay) -A^ k ( 1 - k ) (27) Solutions to equation-A.27 are: + A (28) (x-y ^ ((O f-y-^y 1 - K K -A (29) A = (x~y , (cof-y+y 1 - K K K (1 -K) (30) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 Substituting a / and into equations-A.24 and A-25, obtain: ^F D ~ ^ ( ^ 1 ^ 2 ^ 2 ^ 0 ( ^ 2 ) (31) (32) where: a, = 1 + ^ [ % - ( 1 - K)0(] Kf (33) (34) Applying the boundary condition from equation-A.6, we obtain: B. ( I ^2 ) -^0 ( ^2 ) ( 1 (35) ( 1 - «2 ) ^0 ( ® 2 ) (36) ^2 = - ( I - a, ) ^ ( o , ) (37) 6 = ^ ( 1 -fl, )(Kfl2 + 1 - k ) o 2 ATo(a, )A'j ( 0 2 ) - f ( I - 0 2 ) ( K 0 ; + 1 - K)o, ^ ^ ( 0 2 ) ^ :, ( o ,) . (38) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 167 Therefor, the dimensionless pressure solution in the Laplace domain is: ^ ( a , ) AgCO;) (39) where: o ( « 2 - 1 ) ( K c t , + 1 - k ) P ---------------------------------- "2-^1 (40) X = (a, - 1 ) ( K^2 + 1 - k) a, -^2 (41) Incorporating the effects of skin and wellbore storage, the inner boundary conditions become: r=rw, ........ I dr )r (42) q B = -2 nr dr dr dp^ dt (43) where: = 0.8936 C (44) 168 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The dimensionless solution of this model in the Laplace domain including the effects of skin and wellbore storage is: 1 s\ f.C g + Y(^) )) (45) V(-s) = Ka^ + \ -K Ka, +1 -K ( ^ 2 - 1 ) (K°(o^is) ) *S) (a, -1 ) ( < ( o , is) ) +5) (46) y(j) = < (02 (5 )) < ( a ,( 5 ) ) ( « 2 - I )(< (0 2 (5 ) ) +5) (Û, - 1 ) ( < ( 0 , is) ) +5) (47) Where: < ( G ,) = - o, (A',(o,)) (48) (49) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 Appendix-B C o m p u t e r P r o g r a m The following is a program written in Pascal language to generate the dimensionless pressure distribution in the two regions (active well region-I and observation well region-II) by evaluating Equations-7.56 and 7.57. (* *) ($ N + ) (* THIS PROGRAM CALCULATES THE DIMENSIONLESS PRESSURE DISTRIBUTION, IN THE FRACTURE NETWORK OF A NATURALLY FRACTURED RESERVOIR, FOR REGION-I AND REGION-II ON EITHER SIDE OF A PARTIALLY COMMUNICATING FAULT ♦) program intgrt(ouput); uses windos, wincrt, upawer,ubessel; const days : array [0..6] of String[9] = {'Sunday','Monday','Tuesday', 'Wednesday','Thursday','Friday', 'Saturday'); np=25; type dpmaxtarray = array[1..1,0..25,1..3] of EXTENDED; extarray = array[0..np] of EXTENDED; tparray = array[1..135] of EXTENDED; dparray = array[2..134] of EXTENDED; var hr, mt, sn, hund,yy, mm, dd, dow epsln g,hi,V tdd,pdd, dt,td,tm Dpd, Pmax,tmax,Pmin,tmin 1ml, om, epsl, gma, fs, rslt, cdl, sd, Im, cd, depsl, eta, XX, xd, 1, yd, y, bd, bf, alpha, t, tdl, arg, dpmax, dtmax, pda, result, s, k, rw, trml, trm2, dPmin, dtmin, bbb, xxx, xf h, i,j,m,n, fd, ii, jj , fm,nn,chk,cy,cyy,iy, ik, ix,iz outfile,out outputfile label 100; (*------------------- function to be itegrated— midexp var midexpit :longint; function func(u:EXTENDED) : EXTENDED; var arg3, argl,arg2,top,bot : EXTENDED; label 22; begin argl:=EXP(-(2*epsl*u) ) ; 22 : func:=argl*besskO(sqrt(fs* (sqr (u)+sqr(yd))) ) ; end; :Word; :dpmaxtarray; : extarray; : tparray; :dparray; : extended; :longint; : text ; : string[100]; 170 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( * —----------———— — — ———————— ———————— p i ^ o c G d m r © n i x d © x p "•— ———— ————------------ procedure midexp(aa,bb:extended; var ss: extended; n:longint); var j :longint; X, tnm,sum,del,ddel,b, a : extended; function funk(x:extended) : extended; begin funk:=fune(-In(x))/x; end; begin b:=exp(-aa); a:=0.0; if n=l then begin ss: = (b-a)* funk(0.5*(a+b)); midexpit :=1 end else begin tnm:=midexpit; del:=(b-a)/(3.0*tnm); ddel:=del+del; x:=a+0.5*del; sum:=0.0 ; for j:=l to midexpit do begin sum:=sum+funk(x); x:=x+ddel; sum:=sum+funk(x) ; x:=x+del; end; ss:=(ss+(b-a)*sum/tnm)/3.0; midexpit :=3*midexpit end end; (*——---------- ———————————— procedure polint — ——————-— procedure polint(var xa,ya :extarray; n :longint; X : extended; var y,dy : extended); var ns,m,i :longint; w, hp, ho,dift,dif,den : extended; c,d :''extarray; begin new(c); new(d); ns:=l; di f:=abs(x-xa[1]); for i:= 1 to n do begin dift:=abs(x-xa[i]); if dift<dif then begin ns:=i; dif:=dift;end; c'' [i] :=ya [i] ; d^[i]:=ya[i]; end; y:=ya[ns]; ns:=ns-l; for m:=l to n-1 do begin for i:=l to n-m do begin ho:=xa[i]-x; hp:=xa[i+m]-x; w:=c'' [i+1] -d'' [i] ; den:=ho-hp; if den=0.0 then begin writeln('pause in routine polint '); readln;end; den:=w/den; d'' [i] :=hp*den; c''[i] :=ho*den;end; 171 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. if 2*ns < n-m then dy:=c^[ns+1] else begin dy:=d'' [ns] ; ns:=ns-l;end; y:=y+dy end; DISPOSE(D); DISPOSE(c); end; (*------------------------------- procedure qromo------------------------------*) procedure qromo(a, b:extended; var ss: extended); label 99; const eps=1.0e-6; jmax=21; jmaxp=22; kk=12; km=ll; (* *) type extarrayjmaxp=array[l..jmaxp] of EXTENDED; var i,j ;longint; dss,epss : extended; h, s :''extarrayjmaxp; c,d :^extarray; begin epss:=eps; new(h); new(s); new(c); new(d); h"[l]:=1.0; for j;= 1 to jmax do begin midexp(a,b,s^[j],j) ; if j >= kk then begin for i:=l to kk do begin c'' [i] :=h^ [ j-kk+i] ; d'' [i] :=s^ [j-kk+i] ;end; polint (c'', d' ', kk, 0. 0, ss, dss) ; if abs(dss) < epss*abs(ss) then goto 99; end; s^[j + 1]:=s^ [ j ]; h"[j+l]:=h"[j]/9.0; end; writeln('pause in QROMO - too many steps '); readln; 99: DISPOSE(D); DISPOSE(c); DISPOSE(s); DISPOSE(h); end; (* function for region-1---------------------------- *) function fun(s: extended): extended; var trmll,trm22,trm33,KO,argl,arg2,ss,ds : extended; begin fs:=om*s-(sqr(1ml) /(s-om*s+lml))+lml; trml:=sqrt(fs*(sqr(yd)+sqr(abs(bd-xd)))); trm2:=sqrt(fs*(sqr(yd)+sqr(bd+abs(xd)))); kO :=(besskO(trml)+besskO(trm2)); qromo((bd+abs(xd)), 1.Ge30,rslt); argl:=rslt; trmll :=k0-2*epsl*exp(2*epsl*(bd+abs(xd)))*argl; trm22 := trml 1/s; trm33:= (s*trm22+sd)/(s*(l+s*cdl*(s*trm22+sd))); fun := trm33; end; (*----------------------------function for region-2--------------------------* ) function funl(s: extended): extended; var trmll,trm22,KO, argl, arg2,ss,ds : extended; begin 172 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fs:=om*s-(sqr (1ml) /(s-om*s+lml) ) +lml; qromo ( (bd+abs (xd) ) , 1. 0e30, rslt) ; argl:=rslt; trmll :=2*epsl*exp (2*epsl* (bd+abs (xd) ) ) *argl; trm22 := trmll/s; funl := (s*trm22+sd)/(s*(l+s*cdl* (s*trm22+sd) ) ) ; end; (*----------------------------procedure stehfest----------------------------- *) procedure stef(n:longint;x:EXTENDED; var result : EXTENDED); var i, ih, ic, j, sn, nhi : longint; a, q, fa, fo, e,nh : EXTENDED; label 22; begin if fmoO then GOTO 22; fm:=l; g[0]:=1; nh:=(n/2); nhi:=trunc(nh); for i:=l to n+2 do begin g[i]:=g[i-l]*i; end; hi[1]:=2/g[nhi-l]; for i:=2 to nhi do begin e:=g[2*i]/(g[nhi-i]*g[i]*g [i-1]) ; hi[i]:=sisxy(i, nh)*e; end; for i:=l to n do begin v[i]:=0.0; q:=(i+l)/2; i :=i; if i > nh then begin j:= nhi; end; for k:=trunc(q) to j do begin v[i]:=v[i]+hl[k]/(g[i-k]*g[2*k-i]); end; v[i]:=sisxy(-l,nh+i)*v[i] ; end; 22: fa:=0; a:=0.6931471805599453/x; for i:= 1 to n do begin fa:=fa+v[i] *fun(i*a) ; end; fa:=a*fa; result :=fa; end; (*----------------------------Function Date & Time---------------------------*) function LeadingZero(w : Word) : String; var s : String; begin Str(w:0,s) ; if Length(s) = 1 then s := 'O' + s; LeadingZero := s; end; (*--------------------------------Main Program------------------------------- *) BEGIN bbb:=10; for ix:=l to 1 do begin bbb:=10*bbb; for iz:=l to 1 do begin bf:=iz*bbb; for ik:=l to 3 do begin if iz=l then begin epsl:=0.0; epsln[ix, iz,ik]:=epsl; outputfile: = 'd: \bp\p-well~l\newmdl\a211. out ' ; end; epsl:=0.2; epsln[ix, iz,ik]:=epsl; outputfile : = ' d: \bp\p-well~l\newmdl\a212. out ' ; end; epsl:=200.0; epsln[ix,iz,ik]:=epsl; 173 if ik=l then begin outputfile if ik=2 then begin outputfile if ik=3 then begin Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. outputfile:='d:\bp\p-well~l\newmdl\a213.out'; end; end; assign(outfile,outputfile); rewrite(outfile); (. *) GetDate(yy,mm,dd,dow); Writeln(outfile,'Date=', days[dow],', ', mm:0, '/', dd:0, */', yy:0); GetTime(hr,mt,sn,hund); Writeln (outfile, 'Time=', LeadingZero (hr) LeadingZero (mt)LeadingZero (sn ),'.',LeadingZero(hund)) ; (*-----------------------------------------------------------------------------------------------------------*) xf:=2500; y:=0.0; xx:=xf; rw:=0.5; L:=sqrt(sqr(bf-xx)+sqr(y)); xd:=xf/L; yd:=y/L; bd:=bf/L; om:=0.05; lm:=1.0e-8; 1ml:=lm*sqr(L/rw) ; sd:=0; cd:=00000; cdl:=cd*sqr(rw/L) ; ( * -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------*) writeln(outfile,'epsl=',epsl: 2); writeln(outfile,'lm=',lm:2, ' ',’lml=',1ml: 2, ' ','om=',om: 2); writeln(outfile,'rw=',rw: 2,' ','bd=',bd: 2,' ',’b=',bf:2); writeln(outfile,'sd=', sd:2, ' ','xd=',xd:2,' ', ' yd= ', yd : 2 ) ; writeln(outfile,'cd=',cd:2, ' ','cdl=',cdl:2) ; writeln(outfile,'L=',1: 2, ' ','x=',xx: 2,' ' > ' y= ' / y : 2 ) ; (*----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------*) n:=10; fd:=0 ; k:=l.Oe-4; for h:=l to 15 do begin Ic: =10. 0*k; for j:=l to 9 do begin t:=j*K; tdl:=t; Stef(n,tdl,result); Pda :=result; fd:=fd+l; tdd[fd]:=tdl; pdd[fd]:=Pda; end; end; writeln(outfile,' • , ' = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = • ) ; writeln(outfile,' td pd Deriv ' ) ; writeln(outfile,' dpmin:=1.0; dtmin:=0.0; dpmax:=0.0; dtmax:=0.0; (*-----------------------------------------------------------------------------------------------------------*) for i:=2 to 134 do begin dpd[i]: = ((ln(tdd[i]/tdd[i-1]))*pdd[i+l] ) / ((ln(tdd[i+l]/tdd[i] ) ) *(In (tdd[i+l]/t dd[i-l] ) )) + ((In((tdd[i+1]*tdd[i-l] ) /sqr(tdd[i])))*pdd[i])/((In(tdd[i+l]/tdd[i] ))*(ln(tdd[i]/tdd[i-1])))-((ln(tdd[i+l]/tdd[i]))*pdd[i-l] ) /((In(tdd[i]/tdd[i-1 ]))*(ln(tdd[i+l]/tdd[i-l]))); writeln(outfile,' ',tdd[i]: 4, ' •,pdd[i]: 8 :5,' ’,dpd[i]: 12 : 9) ; end; 100 : close(outfile); (*--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- *) end end end; end. ($n-) 174 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix-C A s y m p t o t i c M o d e l s The drawdown solution o f the new model can be tested against its limiting cases which include the following five asymptotic limits: .NFR with sealing fault. .NFR (no fault). .Homogeneous reservoir with non-sealing fault. .Homogeneous reservoir with sealing fault. Homogeneous reservoir (no fault). Theses limiting cases of the new model can be derived mathematically fi-om the drawdown pressure solution at the active well as shown below. The pressure drawdown solution at the active well for the new model is: = '•.d ] + - 'f'o (2V '- , d)1- — « ^ s . / .K ,[ ,jfjr ) u ] d u . {2b^-r^p) When c O f n =1, ^(s) simplifies to s, and Equation-C.l is reduced to one for partially communicating fault in a homogeneous reservoir: 175 (1) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = ' . d ] . j* e'^^“ . KQ\yf7u\ du (2bp-r^p) When E =0, Equation-C. 1 is reduced to one for NFR with a sealing fault: ^/wDI ~ "r ^0 [ ’'w D ] r, 1 ( 3) When (O fi, =1, ^(s) simplifies to s, and Equation-C.3 is reduced to one for homogeneous reservoir with a sealing fault: ^ (4 ) When bu , Equation-C.3 is reduced to one for a single line-source well in an infinite-acting NFR: (5 ) Where the first term is exactly identical to the dimensionless pressure drawdown solution of the Warren and Root’s model. Once more, when co ^ , = 1, ^(s) simplifies to s, and Equation-C.5 is reduced to one for homogeneous reservoir: (6) 176 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix-D F i t I n f o r m a t i o n f o r a » a n d e C o r r e l a t i o n The following table contains the fit information about the coefBcients and the goodness of fit for the empirical correlation (Equation-8.6) and fit plot (Figure-8.9), developed to estimate the value of the feult communication parameter (e) fi’ om the derivative plot ratio of the hump amplitude value to that of the infinite-acting straight line (3r). The following fit information and results are shown: 1. Values for the fitted variables in the model. 2. The 68%, 90%, 95% and 99% confidence intervals for the fitted variables. 3. A table o f entered and estimated values for all entered data points. 4. Residual table which shows the vertical difiference between the actual and estimated values at all data points. 5. Percent Error table, which sImws the percent error of the estimated dependent variable value compared to the actual values. 6. Absolute Residual table which shows the absolute values of the vertical differences between the actual and estimated values for all data points. 7. Minimum Residual (Min. Residual). 8. Maximum Residual (Max. Residual) 9. Residual Sum of Squares (RSS) which is the sum of the squares of the vertical differences between the actual and estimated values for all data points. 10. Residual Standard Deviation (RSD) which is the Standard Deviation of the vertical differences between the actual and estimated values. 11 Correlation CoefBcient, or R- 12. Standard Error of the Estimate (Standard Error). Table-D.l: Values for the fitted variables and fit information. Variable V alue 68% (+/-) 90% (+/-) 95% (+/-) 99% (+/-) a 0.3862769272 0.0390432162 0.06427324561 0.0780864324 0.1005315434 b 1.475976945 2.066077986 3.401193621 4.132155973 5.319900072 c 0.9879826341 0.01819751463 0.02995688986 0.03639502926 0.04685639169 177 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3r E C alculated e R esidual % Error A bs. R esidual 1.001798 100 100.0385387 -0.03853872387 0.03853872387 0.03853872387 1.002 90 90.01392353 -0.01392352553 0.01547058392 0.01392352553 1.002252 80 80.01561966 -0.01561966464 0.01952458079 0.01561966464 1.002578 70 69.96591375 0.03408624937 0.04869464195 0.03408624937 1.003012 60 59.94521705 0.05478294937 0.09130491562 0.05478294937 1.003618 50 49.95490642 0.04509357625 0.0901871525 0.04509357625 1.004524 40 39.98649573 0.01350427387 0.03376068468 0.01350427387 1.006032 30 30.00416894 -0.004168940705 0.01389646902 0.004168940705 1.009024 20 20.03430138 -0.03430138109 -0.1715069055 0.03430138109 1.017834 10 10.04334064 -0.04334064397 -0.4334064397 0.04334064397 1.019764 9 9.038936725 -0.03893672529 -0.432630281 0.03893672529 1.022164 8 8.032695068 -0.03269506771 -0.4086883464 0.03269506771 1.025224 7 7.026071081 -0.02607108104 -0.3724440148 0.02607108104 1.02926 6 6.018842979 -0.01884297892 -0.3140496487 0.01884297892 1.03482 5 5.011993771 -0.01199377137 -0.2398754274 0.01199377137 1.042964 4 4.005769494 -0.005769494435 -0.1442373609 0.005769494435 1.056132 3 2.994592376 0.005407624336 0.1802541445 0.005407624336 1.08079 2 1.986129171 0.0138708294 0.6935414701 0.0138708294 1.144068 1 0.9800391139 0.01996088608 1.996088608 0.01996088608 1.156322 0.9 0.8800922607 0.0199077393 2.211971033 0.0199077393 1.17085 0.8 0.780515919 0.019484081 2.435510125 0.019484081 1.18837 0.7 0.6813273403 0.01867265972 2.667522817 0.01867265972 1.210036 0.6 0.5821983171 0.01780168292 2.966947153 0.01780168292 1.237398 0.5 0.4837832318 0.01621676817 3.243353635 0.01621676817 1.27336 0.4 0.385765446 0.01423455396 3.55863849 0.01423455396 1.32323 0.3 0.2880944926 0.01190550738 3.968502461 0.01190550738 1.39827 0.2 0.190978995 0.009021005015 4.510502508 0.009021005015 1.528504 0.1 0.09562271025 0.004377289752 4.377289752 0.004377289752 1.547814 0.09 0.08611802284 0.003881977161 4.313307957 0.003881977161 1.568278 0.08 0.07695537516 0.003044624844 3.805781055 0.003044624844 1.590928 0.07 0.06778870438 0.002211295623 3.158993747 0.002211295623 1.617206 0.06 0.05829075622 0.001709243777 2.848739628 0.001709243777 1.646848 0.05 0.04886251354 0.001137486461 2.274972921 0.001137486461 1.681564 0.04 0.03932089642 0.0006791035803 1.697758951 0.0006791035803 1.723252 0.03 0.02968851718 0.0003114828204 1.038276068 0.0003114828204 1.776172 0.02 0.01986047959 0.0001395204079 0.6976020397 0.0001395204079 1.852918 0.01 0.009477205941 0.0005227940587 5.227940587 0.0005227940587 1.863194 0.009 0.008386350809 0.000613649191 6.818324345 0.000613649191 1.874278 0.008 0.007282514379 0.0007174856212 8.968570265 0.0007174856212 1.88631 0.007 0.006167940961 0.0008320590395 11.88655771 0.0008320590395 1.899402 0.006 0.005052237198 0.0009477628021 15.7960467 0.0009477628021 1.913324 0.005 0.003975171423 0.001024828577 20.49657153 0.001024828577 1.927982 0.004 0.002962217809 0.001037782191 25.94455478 0.001037782191 178 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ® R e C alculated e R esidual % Error At)s R esidual 1.94316 0.003 0.002045154541 0.0009548454594 31.82818198 0.0009548454594 1.958772 0.002 0.001246147427 0.0007538525733 37.69262866 0.0007538525733 1.974766 0.001 0.0005908403926 0.0004091596074 40.91596074 0.0004091596074 1.976384 0.0009 0.0005346221869 0.0003653778131 40.59753479 0.0003653778131 1.978008 0.0008 0.0004802094901 0.0003197905099 39.97381373 0.0003197905099 1.979636 0.0007 0.0004277426359 0.0002722573641 38.89390916 0.0002722573641 1.981266 0.0006 0.000377360207 0.000222639793 37.10663216 0.000222639793 1.9829 0.0005 0.0003290846499 0.0001709153501 34.18307001 0.0001709153501 1.984534 0.0004 0.0002831265621 0.0001168734379 29.21835947 0.0001168734379 1.986176 0.0003 0.0002393779637 6.062203627E-05 20.20734542 6.062203627E-05 1.987822 0.0002 0.0001980925695 1.907430494E-06 0.953715247 1.907430494E-06 1.98947 0.0001 0.0001594821007 -5.948210075E-05 -59.48210075 5.948210075E-05 Min. Residual Max. R esidual R SS RSD R2 S td. Error 0.04334064397 0.05478294937 0.01798166387 0.01821861609 0.9999994423 0.0185957314 179 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix-E R e l a t i o n s h i p b e t w e e n A, a n d X * The following table contains the relationship between k, defined based on the fault distance (dp), and k' which is based on the well radius (r,^): Table-E.l; Relationship between Xand X .* . X * X X * 1 X X " X X * X X * X 10 IE-05 0.004 IE-06 0.0004 IE-07 4E-05 IE-08 4E-06 IE-09 4E-07 20 IE-05 0.016 IE-06 0.0016 IE-07 0.00016 IE-08 1.6E-05 IE-09 1.6E-06 30 IE-05 0.036 IE-06 0.0036 IE-07 0.00036 IE-08 3.6E-05 IE-09 3.6E-06 40 IE-05 0.064 IE-06 0.0064 IE-07 0.00064 IE-08 6.4E-05 IE-09 6.4E-06 50 IE-05 0.1 IE-06 0.01 IE-07 0.001 IE-08 0.0001 IE-09 IE-05 60 IE-05 0.144 IE-06 0.0144 IE-07 0.00144 IE-08 0.000144 IE-09 1.44E-05 70 IE-05 0.196 IE-06 0.0196 IE-07 0.00196 IE-08 0.000196 IE-09 1.96E-05 80 IE-05 0.256 IE-06 0.0256 IE-07 0.00256 IE-08 0.000256 IE-09 2.56E-05 90 IE-05 0.324 IE-06 0.0324 IE-07 0.00324 IE-08 0.000324 IE-09 3.24E-05 100 IE-05 0.4 IE-06 0.04 IE-07 0.004 IE-08 0.0004 IE-09 4E-05 200 IE-05 1.6 IE-06 0.16 IE-07 0.016 IE-08 0.0016 IE-09 0.00016 300 IE-05 3.6 IE-06 0.36 IE-07 0.036 IE-08 0.0036 IE-09 0.00036 400 IE-05 6.4 IE-06 0.64 IE-07 0.064 IE-08 0.0064 IE-09 0.00064 500 IE-05 10 IE-06 1 IE-07 0.1 IE-08 0.01 IE-09 0.001 600 IE-05 14.4 IE-06 1.44 IE-07 0.144 IE-08 0.0144 IE-09 0.00144 700 IE-05 19.6 IE-06 1.96 IE-07 0.196 IE-08 0.0196 IE-09 0.00196 800 IE-05 25.6 IE-06 2.56 IE-07 0.256 IE-08 0.0256 IE-09 0.00256 900 IE-05 32.4 IE-06 3.24 IE-07 0.324 IE-08 0.0324 IE-09 0.00324 1000 IE-05 40 IE-06 4 IE-07 0.4 IE-08 0.04 IE-09 0.004 1000 IE-05 40 IE-06 4 IE-07 0.4 IE-08 0.04 IE-09 0.004 2000 IE-05 160 IE-06 16 IE-07 1.6 IE-08 0.16 IE-09 0.016 3000 IE-05 360 IE-06 36 IE-07 3.6 IE-08 0.36 IE-09 0.036 4000 IE-05 640 IE-06 64 IE-07 6.4 IE-08 0.64 IE-09 0.064 5000 IE-05 1000 IE-06 100 IE-07 10 IE-08 1 IE-09 0.1 6000 IE-05 1440 IE-06 144 IE-07 14.4 IE-08 1.44 IE-09 0.144 7000 IE-05 1960 IE-06 196 IE-07 19.6 IE-08 1.96 IE-09 0.196 8000 IE-05 2560 IE-06 256 IE-07 25.6 IE-08 2.56 IE-09 0.256 9000 IE-05 3240 IE-06 324 IE-07 32.4 IE-08 3.24 IE-09 0.324 10000 IE-05 4000 IE-06 400 IE-07 40 IE-08 4 IE-09 0.4 1 8 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix-F F i t I n f o r m a t i o n f o r a n d m C o r r e l a t i o n The following table contains the fit information about the coefiBcients and the goodness of fit for the empirical correlation (Equation-8.9) and fit plot (Figure-8.30), developed to estimate the value of the fi-acture storativity ratio (m ) fi-om the derivative plot ratio of the hump amplitude value to that of the infinite-acting straight line (dp). All the fit information and results are the same as those mentioned in appendix-D. Tabie-F.l: Values for the fitted variables and fit information. V ariable V alue 68% (+/-) 90% (+/-) 95% (+/-) 99% (+/-) a 0.2703847271 0.4856386667 0.7994621433 0.9712773334 1.250460629 b 3.537413238 0.8107143759 1.334604299 1.621428752 2.087491128 c 1.156427697 0.08838946682 0.1455074265 0.1767789336 0.2275921499 Min. R esidual Max. Residual R SS RSD R2 Std. Error -0.0052655597 0.0289264588 0.001297003049 0.0053884728 0.9994417274 0.0055570698 181 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f i > Calculated < d Residual % Error Alas. Residual 0.000164 IE-05 1.134521495E-05 -1.345214952E-06 -13.45214952 1.345214952E-06 0.000287 2E-05 2.167385422E-05 -1.673854221E-06 -8.369271107 1.673854221E-06 0.000392 3E-05 3.10870057E-05 -1.0870057E-06 -3.623352333 1.0870057E-06 0.000494 4E-05 4.062444834E-05 -6.244483408E-07 -1.561120852 6.244483408E-07 0.000592 5E-05 5.008762076E-05 -8.762075958E-08 -0.1752415192 8.762075958E-08 0.000688 6E-05 5.960179404E-05 3.982059622E-07 0.6636766037 3.982059622E-07 0.000783 7E-05 6.922641555E-05 7.735844464E-07 1.105120638 7.735844464E-07 0.000876 8E-05 7.882969924E-05 1.170300765E-06 1.462875956 1.170300765E-06 0.000968 9E-05 8.849038095E-05 1.509619049E-06 1.677354499 1.509619049E-06 0.001018 0.0001 9.380313909E-05 6.196860912E-06 6.196860912 6.196860912E-06 0.001878 0.0002 0.0001906508102 9.349189753E-06 4.674594877 9.349189753E-06 0.00267 0.0003 0.000286677423 1.332257704E-05 4.440859013 1.332257704E-05 0.003443 0.0004 0.0003850499176 1.495008238E-05 3.737520594 1.495008238E-05 0.004194 0.0005 0.0004842002509 1.579974906E-05 3.159949812 1.579974906E-05 0.004928 0.0006 0.0005840184537 1.598154633E-05 2.663591055 1.598154633E-05 0.005649 0.0007 0.0006845408149 1.54591851 IE-05 2.208455016 1.54591851 IE-05 0.006357 0.0008 0.0007853986775 1.460132249E-05 1.825165311 1.460132249E-05 0.007053 0.0009 0.0008864454245 1.35545755E-05 1.506063945 1.35545755E-05 0.007358 0.001 0.0009312822285 6.871777153E-05 6.871777153 6.871777153E-05 0.013964 0.002 0.00197007158 2.992841955E-05 1.496420977 2.992841955E-05 0.019871 0.003 0.002984690103 1.530989697E-05 0.510329899 1.530989697E-05 0.025458 0.004 0.004003147986 -3.147986424E-06 -0.07869966061 3.147986424E-06 0.03079 0.005 0.005021474824 -2.147482381 E-05 -0.4294964762 2.147482381E-05 0.035913 0.006 0.006038653956 -3.865395627E-05 -0.6442326045 3.865395627E-05 0.040862 0.007 0.007054952611 -5.495261134E-05 -0.7850373048 5.495261134E-05 0.045655 0.008 0.008069146543 -6.91465433E-05 -0.8643317912 6.91465433E-05 0.050314 0.009 0.009082089031 -8.208903111 E-05 -0.9121003456 8.208903111 E-05 0.055401 0.01 0.010217599 -0.000217599002 -2.17599002 0.000217599002 0.096411 0.02 0.02042195674 -0.000421956735 -2.109783675 0.000421956735 0.131947 0.03 0.0307033133 -0.0007033132995 -2.344377665 0.0007033132995 0.162717 0.04 0.04067633925 -0.0006763392523 -1.690848131 0.0006763392523 0.190849 0.05 0.0506836819 -0.0006836818988 -1.367363798 0.0006836818988 0.216794 0.06 0.06069048459 -0.0006904845915 -1.150807652 0.0006904845915 0.241033 0.07 0.07073749841 -0.0007374984089 -1.053569156 0.0007374984089 0.263622 0.08 0.080729999 -0.0007299989983 -0.9124987478 0.0007299989983 0.285047 0.09 0.09078909996 -0.000789099961 -0.8767777344 0.000789099961 0.309772 0.1 0.1031280272 -0.003128027196 -3.128027196 0.003128027196 0.469833 0.2 0.2043615781 -0.004361578094 -2.180789047 0.004361578094 0.585698 0.3 0.3052655597 -0.005265559702 -1.755186567 0.005265559702 0.676924 0.4 0.4049789047 -0.004978904749 -1.244726187 0.004978904749 0.751989 0.5 0.5028464356 -0.002846435645 -0.569287129 0.002846435645 0.815561 0.6 0.5985145045 0.001485495532 0.2475825887 0.001485495532 0.87054 0.7 0.6918391127 0.008160887301 1.165841043 0.008160887301 0.918828 0.8 0.7827319562 0.01726804383 2.158505479 0.01726804383 0.961714 0.9 0.8710735412 0.02892645883 3.214050981 0.02892645883 182 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix-G R e l a t i o n s h i p b e t w e e n a n d C q* The following table contains the relationship between , defined based on the fault distance (dp), and which is based on the well radius (r,): Table-G.l; Relationship between C„ and €„*• 4 C|t* c„ Cp* C„ Cp* Co Co* Co Co* Co 10 100 2.5E-01 1000 2.5E+00 10000 2.5E+01 1.0E+05 2.5E+02 1.0E+06 2.5E+03 20 100 6.3E-02 1000 6.3E-01 10000 6.3E+00 1.0E+05 6.3E+01 1.0E+06 6.3E+02 30 100 2.8E-02 1000 2.8E-01 10000 2.8E+00 1.0E+05 2.8E+01 1.0E+06 2.8E+02 40 100 1.6E-02 1000 1.6E-01 10000 1.6E+00 1.0E+05 1.6E+01 1.0E+G6 1.6E+02 S O 100 1.0E-02 1000 1.0E-01 10000 1.0E+00 1.0E+05 1.0E+01 1.0E+06 1.0E+02 60 100 6.9E-03 1000 6.9E-02 10000 6.9E-01 1.0E+05 6.9E+00 1.0E+06 6.9E+01 70 100 5.1E-03 1000 5.1E-02 10000 5.1E-01 1.0E+05 5.1E+00 1.0E+06 5.1E+01 80 100 3.9E-03 1000 3.9E-02 10000 3.9E-01 1.0E+05 3.9E+00 1.0E+06 3.9E+01 90 100 3.1E-03 1000 3.1E-02 10000 3.1E-01 1.0E+05 3.1E+00 1.0E+06 3.1E+01 100 100 2.5E-03 1000 2.5E-02 10000 2.5E-01 1.0E+05 2.5E+00 1.0E+06 2.5E+01 200 100 6.3E-04 1000 6.3E-03 10000 6.3E-02 1.0E+05 6.3E-01 1.0E+06 6.3E+00 300 100 2.8E-04 1000 2.8E-03 10000 2.8E-02 1.0E+05 2.8E-01 1.0E+06 2.8E+00 400 100 1.6E-04 1000 1.6E-03 10000 1.6E-02 1.0E+05 1.6E-01 1.0E+06 1.6E+00 500 100 1.0E-04 1000 1.0E-03 10000 1.0E-02 1.0E+05 1.0E-01 1.0E+06 1.0E+00 600 100 6.9E-05 1000 6.9E-04 10000 6.9E-03 1.0E+05 6.9E-02 1.0E+06 6.9E-01 700 100 5. IE-05 1000 5.1E-04 10000 5.1E-03 1.0E+05 5. IE-02 1.QE+06 5. IE-01 800 100 3.9E-05 1000 3.9E-04 10000 3.9E-03 1.0E+05 3.9E-02 1.0E+06 3.9E-01 900 100 3.1 E-05 1000 3. IE-04 10000 3. IE-03 1.0E+05 3.1E-02 1.0E+06 3. IE-01 1000 100 2.5E-05 1000 2.5E-04 10000 2.5E-03 1.0E+05 2.5E-02 1.0E+06 2.5E-01 2000 100 6.3E-06 1000 6.3E-05 10000 6.3E-04 1.0E+05 6.3E-03 1.0E+06 6.3E-02 3000 100 2.8E-06 1000 2.8E-05 10000 2.8E-04 1.0E+05 2.8E-03 1.0E+06 2.8E-02 4000 100 1.6E-06 1000 1.6E-05 10000 1.6E-04 1.0E+05 1.6E-03 1.0E+06 1.6E-02 5000 100 1.0E-06 1000 1.0E-05 10000 1.0E-04 1.0E+05 1.0E-03 1.0E+06 1.0E-02 6000 100 6.9E-07 1000 6.9E-06 10000 6.9E-05 1.0E+05 6.9E-04 1.0E+06 6.9E-03 7000 100 5.1E-07 1000 5.1E-06 10000 5.1E-05 1.0E+05 5.1E-04 1.0E+06 5. IE-03 8000 100 3.9E-07 1000 3.9E-06 10000 3.9E-05 1.0E+05 3.9E-04 1.0E+06 3.9E-03 9000 100 3. IE-07 1000 3.1E-06 10000 3.1 E-05 1.0E+05 3.1E-04 1.0E+06 3. IE-03 10000 100 2.5E-07 1000 2.SE-06 1 10000 2.5E-05 1.0E+05 2.5E-04 1.0E+06 2.5E-03 183 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. References [ 1 ] Horne, R.N. : “Modem Well Test Anafysis A Computer Aided Approach,” 2 ™ * edition, Petroway, Inc. 1995, 9. [2] Al-Ghamdi, A. and Ershaghi, I.: “Pressure Transient Analysis of Dually Fractured Reservoirs,”Soc. Pet. Eng. J . , (March 1996) Vol 1, No. 1, 93-100. [3] Abdassah, D. and Ershaghi, I.: “Triple Porosity Model for Representing Naturally Fractured Reservoir,” PH.D. Dissertation, USC, (July 1984). [4] Bourdet, D. And Johnston, F: “Pressure Behavior of Layered Reservoir with Crossflow,” Paper SPE 13628, presented at the SPE California Regional Meeting, Bakersfield, CA, March 27-29, 1985. [5] Warren, J. E. and Root, P. J.: “The Behavior of Naturally Fractured Reservoirs” Soc. Pet. Eng. J. (Sep. 1963) 245-255, Trans. AIME, Volume 228. [6] Earlougher, R. 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M.: “Table of Integrals, Series, and Products,” Corrected and Enlarged Edition, Academic Press, Inc., New York City (1980) 532,692, 740, 1177-1187. [66] Agarwaf R. G., Al-Hussainy, R., and Ramey, H.J., Jr.: “An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: I- Analytical Treatment,” Soc. Pet. Eng. J., (Sept. 1970), 279. [67] Press W. H., Flannery B.P., Teukolsky S. A., and Vetter ling W. T.: “Numerical Recipes in Pascal,” Cambridge University Press, NY, (1992)129-138. [68] Abramowitz, M. and Stegun, LA.: “Handbook of Mathematical Functions,” Nat. Bureau o f Standards, 1968, 227-233, 355-435, 479-495. [69] Press W. H., Flannery B. P., Teukolsky S.A., and Vetterling W. T.: “Numerical Recipes in Pascal,” Cambridge University Press, NY, (1992), 90-93. 129-138, 197- 202. [70] Earlougher, R. C.: “Advances in Well Test Analysis,” Volume 5, SPE Monograph Series, (1977), 193-194. [71] Crawford, G. E., Hagedom, A.R., and Pierce, A.E.: “Analysis of Pressure Buildup Tests in a Naturally Fractured Reservoir,” JPT, (November 1976), 1295-1300. [72] Uldrich, David O., and Ershaghi, L: “A Method for Estimating the Inter-Porosity Flow Parameter in Naturalfy Fractured Reservoirs,” SPE Paper 7142, presented at the 1978 California Regional Meeting of the SPE-AIME, San Francisco, CA, April 12- 14, 1978. [73] Mavor, N.J. and Cinco, H.: “Transient Pressure Behavior of Naturally Fractured Reservoirs,” SPE paper 7977, presented at the 1979 California Regional Meeting of the Society of Petroleum Engineers of AIME, Ventura, CA, April 18-20, 1979. 189 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [74] Jalali-Yazdi, Y. and Ershaghi, L: “A Unified Type Curve Approach for Pressure Transient Analysis of Naturally Fractured Reservoirs,” SPE paper 1678, presented at the SPE 62^ Annual Fall Technical Conference and Exhibition, Dallas, Texas, September 27-30, 1987. [75] Cinco-Ley, H. and Samaniego-V., P.: “Pressure Transient Analysis for Naturally Fractured Reservoirs,” SPE paper 11026 presented at the SPE 57^ Annual Fall Technical Conference and Exhibition, New Orleans, LA., September 26-29, 1982. [76] Petak, K. R., Soliman, M.Y., and Anderson, M.F.: “Type Curves for Analyzing Naturally Fractured Reservoirs,” SPE paper 15638, presented at the 6T‘ Annual Technical Conference and Exhibition o f the Society of Petroleum Engineers held in New Orleans, LA., October 5-8, 1986. [77] Serra, J. 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M.: “A Field Example o f Interference Testing Across a Partially Communicating Fault,” SPE Paper 19306, 1989. 190 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. IMAGE EVALUATION TEST TARGET (Q A -3 ) / 'A 1 . 0 l.l 1 .2 5 B Kà |3 j6 1 4 .0 123 1 . 4 1 . 8 1 . 6 150mm y y ^ P P L IE D ^ IIVMGE . Inc ■ ■ 1653 East M ain Street Roctiester. N Y 14609 U SA Ptione: 716/482-0300 . ^ ^ ^ 5 Fax: 716/288-5989 0 1993, Applied Image. Inc.. A il Rights Reserved 4 ^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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Al-Ghamdi, Abdullah Hamed
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Core Title
Aspects of fracture heterogeneity and fault transmissivity in pressure transient modeling of naturally fractured reservoirs
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Doctor of Philosophy
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Petroleum Engineering
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engineering, petroleum,Geology,OAI-PMH Harvest
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Ershaghi, Iraj (
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