UNIVERSITY OF CALGARY. New and Improved Methods for Performing Rate-Transient Analysis of Tight/Shale Gas. Reservoirs. Morteza Nobakht A THESIS

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1 UNIVERSITY OF CALGARY New and Improved Methods for Performing Rate-Transient Analysis of Tight/Shale Gas Reservoirs by Morteza Nobakht A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CHEMICAL AND PETROLEUM ENGINEERING CALGARY, ALBERTA SEPTEMBER 2014 Morteza Nobakht 2014

2 ABSTRACT Analysis of long-term linear flow periods associated with tight/shale gas production has received much attention in recent literature as a means of obtaining information about stimulation efficiency (for example, the product of fracture half-length and square root of permeability referred herein as the linear flow parameter). In this study, first, new methods for analyzing production data from a fractured well in tight/shale gas reservoir producing under a constant flowing pressure, a constant production rate and variable flowing pressure/production rate in the absence or presence of desorption and gasslippage are presented. It is shown that the current formulation of linear flow analysis results in an overestimation of linear flow parameter for constant flowing pressure production. It is also found that the shape of square-root-of-time plot depends on the production rate for constant gas rate production. Secondly, the effects of completion heterogeneity (i.e., all fracture lengths are not the same) of a multi-fractured horizontal well (MFHW) are studied and a method is developed for extending hybrid forecasting methods developed for homogeneous completions to heterogeneous completions. The methodology developed is also applied for multi-well analysis of MFHWs. It is found that ignoring the heterogeneity of the completion can have a large impact on the long-term forecast of these wells. Thirdly, a new set of dimensionless type curves is developed for one of the most commonly used conceptual models for MFHWs. With these dimensionless type curves, the early linear flow (early-time half slope) and boundary-dominated flow (late-time unit ii

3 slope) coincide for different geometric ratios and the transition between these two regimes depends on the geometry of the reservoir and completion. Finally, the applicability of flowing material balance (FMB) analysis for calculating contacted gas-in-place (CGIP) in linear-flow dominated systems is investigated. It is found that even when dealing with a gas reservoir that is depleted significantly, FMB underestimates the original gas-in-place (OGIP). We also investigated whether applying a corrected material balance pseudo-time, calculated using the average pressure in the region of investigation as opposed to average reservoir pressure, will correct for underestimation of OGIP for the case that the reservoir is significantly depleted. iii

4 ACKNOWLEDGMENTS Many people have helped me through different chapters of my life and it would not have been possible for me to be at this stage to write this doctoral dissertation without their help and support. I would like to thank my supervisor, Professor Chris Clarkson, for his help, guidance, encouragement and trust in me. I really admire his understanding of my situation that allowed me to balance my family life, my career and my PhD. I have learnt much from him and he made this a very fun and rewarding experience. I need to thank my heroic wife, Samane, who carried the significant load of our family in the past four years which allowed me to pursue my passion and dream without sacrificing our family life. We are blessed with two children, Diba and Naveed, who were born during my study and without her support, it would have been very hard if not impossible for me to achieve this academic milestone and amazing personal life. I would like to thank my parents who helped me to be the person I am today. They encouraged me to dream big, taught me that hard work always pays off and how to overcome setbacks by working harder towards the next big goal. They sacrificed their happiness and lives for mine and that of my two brothers and sister, without which I would have not been where I am today. I would like to extend my gratitude to the Mattar family, especially Louis Mattar and Sheila Mattar who have been so kind and supportive to our family. They have treated Samane and myself like their children and Diba and Naveed as their grandchildren. Their emotional support means a lot to us and has been priceless. iv

5 I would like to thank Dr. Danial Kaviani for his help with the simulation work. More specifically, he helped with the simulation conducted for Chapter 3 and also verifying that the results of the analytical method used for generating type curves in Chapter 5 are in good agreement with those obtained from numerical simulation. I would like to thank Dr. Ray Ambrose, Jerry Youngblood and Rod Adams who helped with studying the impact of completion heterogeneity on long-term forecast (presented in Chapter 4). In addition to brainstorming and discussion, they provided the data for field case study and evidence for bi-wing fracture for the original paper. The latter is excluded from this thesis, since it was not part of my contribution, and the former is presented in this thesis. I also would like to thank all my friends who created a warm and friendly environment during my study at the University of Calgary. Especially, I would like to thank Dr. Hamed Reza Motahari, Hamid Behmanesh, Farhad Qanbari, Jesse Williams- Kovacs and Dr. Hashem Salari. I would like to thank my former colleagues at Fekete Associates Inc. particularly Louis Mattar, Mehran Pooladi-Darvish and Dave Anderson for shaping my career and the fruitful discussions on the subject of rate-transient analysis. Last but not least, I would like to thank ConocoPhillips for their support of this research. I also would like to thank George Petrosky, Kevin Raterman and Sheila Reader for their feedback on different aspects of this research which resulted in the improvement of the overall research. v

6 DEDICATION I dedicate this dissertation to my family, especially My wife, Samane, for her love, support and being my hero; My parents, Mostafa and Manzar, for all they have done for me, my brothers and sister; My daughter, Diba, and son, Naveed, for being amazing and joyful; My brothers and sister for helping me to be a better person. vi

7 TABLE OF CONTENTS ABSTRACT... ii ACKNOWLEDGMENTS... iv DEDICATION... vi TABLE OF CONTENTS... vii LIST OF TABLES... x LIST OF FIGURES... xii LIST OF SYMBOLS, ABBREVIATIONS, NOMENCLATURE... xxiv CHAPTER 1 INTRODUCTION Possible Conceptual Models Linear Flow Analysis Motivation Outline of the Thesis... 9 CHAPTER 2 IMPROVED LINEAR FLOW ANALYSIS: EFFECTS OF PRESSURE-DEPENDENT PROPERTIES OF GAS Abstract Introduction Derivation/Analysis Methods Constant Rate Boundary Condition Constant Flowing Pressure Variable Rate/Variable Flowing Pressure Data Validation of Analytical Methods Constant Rate Production Constant Flowing Pressure Variable Rate/Variable Flowing Pressure Data Discussion Impact of Distance of Investigation on Constant Flowing Pressure Analysis Application to Ultra-Low Permeability Reservoirs Approximate Solution for Constant Flowing Pressure Analysis Differences Between Constant Flowing Pressure and Constant Rate Production Linear Flow Importance of Pseudo-Time Summary vii

8 CHAPTER 3 IMPROVED LINEAR FLOW ANALYSIS: EFFECTS OF GAS SLIPPAGE AND DESORPTION Abstract Introduction Derivation Validation Discussion Effect of Slippage on Linear Flow Analysis Effect of Desorption on Linear Flow Analysis Effect of Slippage on Gas Production Impact of Distance of Investigation Assumptions/Limitations Summary CHAPTER 4 EFFECT OF COMPLETION HETEROGENEITY Abstract Introduction Development of the Model for Heterogeneous Completion Single-Well Analysis Validation Effect of Completion Heterogeneity on b-value Case Study Multi-Well Analysis Sensitivity of Forecast to b-value EUR Based upon Time EUR Based upon Economic Limit Rate Evolution of b-value During Boundary-Dominated Flow Summary CHAPTER 5 NEW TYPE CURVES FOR ANALYZING HORIZONTAL WELL WITH MULTIPLE FRACTURES IN SHALE GAS RESERVOIRS Abstract Introduction Type Curve Generation for Scenario Considerations for Gas Application of Material Balance Time for Linear Flow Analysis Analysis Method Discussion Flow Regimes Contribution from Outer Reservoir Long-Term Production Forecast Case Study Summary CHAPTER 6 ESTIMATION OF CONTACTED AND ORIGINAL GAS-IN-PLACE FOR RESERVOIRS EXHIBITING LINEAR FLOW Abstract viii

9 6.2 Introduction Basic Model Review of Current Methods for Calculating OGIP/CGIP Comparison of Different Methods for Calculating OGIP/CGIP Correction of Different Methods for Calculating OGIP/CGIP Square-Root-of-Time Plot/Distance of Investigation Correction of FMB Method for Gas Reservoirs Field Case Study Discussion Summary CHAPTER 7 CONCLUSIONS Contributions and Conclusions Recommendations REFERENCES APPENDIX COPYRIGHT PERMISSIONS ix

10 LIST OF TABLES Table 2.1. Input parameters for numerical simulation for different cases used to validate the methodology developed for constant gas rate production. The blank cells indicate that the value for that parameter is the same as that of Case Table 2.2. Input parameters for numerical simulation for different cases used to validate analytically-derived correction factor for constant flowing pressure production. The blank cells indicate that the value for that parameter is the same as that of Case Table 2.3. Input parameters used for numerical simulation for different cases used to evaluate Equation (2.15) Table 3.1. Input parameters to numerical simulation for different cases used to validate methodology presented in this study Table 3.2. Comparison among fracture half-lengths calculated using five different methods for Cases Table 3.3. Comparison among fracture half-lengths calculated using three different methods for Cases Table 4.1. Input parameters used for numerical simulation to validate the methodology for heterogeneous completion presented in this chapter Table 5.1. Flow regimes observed in the reservoir geometry shown in Figure 5.3(b). To apply this table to the geometry shown in Figure 5.3(a), y e is the spacing x

11 between horizontal wells and x e is the spacing between fractures along the horizontal well Table 6.1. Input parameters used for numerical simulation of different cases used in this study. The blank cells indicate that the value for that parameter is the same as that of Case xi

12 LIST OF FIGURES Figure 1.1 Possible combinations of reservoir/hydraulic fracture encountered for tight oil/shale gas reservoirs (Modified from Clarkson and Pederson, 2010) Figure 2.1. Base geometry used for derivation of analytical methods - a hydraulicallyfractured well in the center of a rectangular reservoir Figure 2.2. Square-root-of-time plot for a constant rate case generated using numerical simulation Figure 2.3. Square-root-of-time plot for Cases Figure 2.4. Square-root-of-time plot for Cases Figure 2.5. Square-root-of-time plot for Cases Figure 2.6. Square-root-of-time plot for Cases Figure 2.7. Corrected square-root-of-time plot for Cases Figure 2.8. Corrected square-root-of-time plot for Cases Figure 2.9. Corrected square-root-of-time plot for Cases Figure Corrected square-root-of-time plot for Cases Figure Fracture half-lengths calculated using the method proposed in this study for Cases 1 14 (simulation input provided in Table 2.1). The expected value of x f =250 ft is shown on the plot by the dashed horizontal line Figure Comparison between calculated correction factors obtained using the new method from this study (using Equation (2.22a)) and Ibrahim and Wattenbarger (2005; 2006) with expected correction factor for c f = xii

13 Figure Comparison between calculated correction factors obtained using the new method from this study (using Equation (2.22a)) and Ibrahim and Wattenbarger (2005; 2006) with expected correction factor for c f = /psi Figure Comparison between calculated correction factors obtained using the new method from this study (using Equation (2.22a)) and Ibrahim and Wattenbarger (2005; 2006) with expected correction factor for c f = /psi Figure Gas rate and flowing pressure profile for Example Figure Square-root-of-time plot for Example 1. The line is passed through earlytime data points Figure Normalized pressure versus linear superposition time plot for Example 1. The line is passed through early-time data points Figure Normalized pressure versus linear superposition pseudo-time plot for Example Figure Normalized pressure versus linear superposition pseudo-time plot for Example 1. Equation (2.16) and Equation (2.7) are used to calculate the gasin-place in the region of investigation for Constant Pressure and Constant Rate datasets, respectively Figure Gas rate and flowing pressure profile for Example Figure Normalized pressure versus linear superposition pseudo-time plot for Example Figure Gas rate and flowing pressure profile for Example xiii

14 Figure Normalized pressure versus linear superposition pseudo-time plot for Example Figure Pressure distribution in the reservoir for Case 5 in Table 2.2 after 313 days (p i =10,000 psi, p wf =3,000 psi, k=0.1 md and c f =0). Distance of 2,500m from the fracture represents the reservoir boundaries parallel to the fracture Figure Semi-log derivative plot for Case 5 in Table 2.2 (p i =10,000 psi, p wf =3,000 psi, k=0.1 md and c f =0) Figure The end of linear flow obtained using the pressure profile in the reservoir, (t elf ) O, versus the end of linear flow calculated using Equation (2.27), (t elf ) C Figure Comparison between correction factors calculated using the methods of this study (using Equation (2.30)) with expected correction factor for c f = Figure Comparison between correction factors calculated using the methods of this study (using Equation (2.30)) with expected correction factor for c f = /psi Figure Comparison between correction factors calculated using the methods of this study (using Equation (2.30)) with expected correction factor for c f = /psi Figure Comparison between linear flow correction factors for constant flowing pressure obtained using Equation (1.3), dashed line, and those obtained using Equation (2.34), solid line Figure A plot of g c g versus pressure obtained from the gas properties used for the cases presented in this study xiv

15 Figure 3.1. A hydraulically-fractured vertical well in the center of a rectangular reservoir Figure 3.2. Permeability ratio calculated using Equation (3.6) for three different permeabilities Figure 3.3. Fracture half-lengths calculated using the methodology presented in this study (with both Equation (3.24) and Equation (3.27)) for different numerically-simulated cases. The expected value of x f =250 ft is shown on the plot by the dotted horizontal line Figure 3.4. Comparison of fracture half-lengths calculated using different methods for Cases 1 3. The expected value of x f =250 ft is shown on the plot by the dotted horizontal line Figure 3.5. Comparison of fracture half-lengths calculated using different methods for Cases 4 6. The expected value of x f =250 ft is shown on the plot by the dotted horizontal line Figure 3.6. Comparison among fracture half-lengths calculated using different methods for Cases The expected value of x f =250 ft is shown on the plot by the dotted horizontal line Figure 3.7. Comparison among fracture half-lengths calculated using different methods for Cases The expected value of x f =250 ft is shown on the plot by the dotted horizontal line Figure 3.8. Comparison between gas rates in the presence and in the absence of slippage for p i =2,000 psi, p wf =200 psi and k=0.01 md (Cases 1 and 7) xv

16 Figure 3.9. Comparison between gas rates in the presence and in the absence of slippage for p i =2,000 psi, p wf =200 psi and k=0.001 md (Cases 2 and 8) Figure Comparison between gas rates in the presence and in the absence of slippage for p i =2,000 psi, p wf =200 psi and k= md (Cases 3 and 9).. 94 Figure Comparison between gas rates obtained from numerical simulation with slippage effect and those obtained from calibrated numerical model without slippage effect. Only the first year data is used for history matching Figure Comparison between gas rates obtained from numerical simulation with slippage effect and those obtained from calibrated numerical model without slippage effect Figure Comparison between cumulative production obtained from numerical simulation with slippage effect and that obtained from calibrated numerical model without slippage effect Figure A plot of Q fyd, defined in Equation (3.30), versus y D, defined in Equation (A.3) Figure 4.1. (a) A hydraulically-fractured vertical well in the center of a rectangular reservoir. (b) Example of a homogeneous multi-fractured horizontal well Figure 4.2. (a) Example of a heterogeneous multi-fractured horizontal well used to explain the concept of dividing a heterogeneous completion into different divisions based the duration of linear flow. (b) The schematic shown in Figure 4.2(a) divided into three divisions with different duration of linear flow xvi

17 Figure 4.3. (a) Example of a heterogeneous multi-fractured horizontal well used for validating the procedure presented in this study. (b) The schematic shown in Figure 4.3(a) divided into five divisions with different duration of linear flow (Modified from Ambrose et al. (2011)) Figure 4.4. Comparison among the rates obtained from heterogeneous model, numerical simulation and homogeneous model with b= Figure 4.5. Comparison between the hyperbolic decline exponent, b, values obtained from best fit and those calculated from Equation (4.19) Figure 4.6. A schematic of multi-fractured horizontal well for the field case study. The heterogeneity of the completion was identified from production log Figure 4.7. The schematic shown in Figure 4.6 divided into seven divisions with different duration of linear flow (Modified from Ambrose et al. (2011)). 127 Figure 4.8. Comparison among forecasts obtained using heterogeneous model, homogeneous model with b=0.8 and homogeneous model with b=0.5. Note that homogeneous forecast with b=0.8 is an approximation for heterogeneous forecast Figure 4.9. Reservoir/Completion geometry used in this study for analysis Figure Gas production rate versus time plot for Well 1 and Well 2 in Figure Figure Square-root-of-time plot for Well 1 and Well 2 in Figure Figure Reservoir/Completion geometry shown in Figure 4.9 is divided into different divisions xvii

18 Figure Reservoir/Completion geometry obtained from the analysis of production data ignoring the communication between Well 1 and Well Figure Plot of EUR( b 1.3) EUR( b 0.5) versus t t elf. b = 0.5 is for homogeneous completion and b = 1.3 is for very heterogeneous completion Figure Plot of EUR( b 0.8) EUR( b 0.5) versus t t elf. b = 0.5 is for homogeneous completion and b = 0.8 is for slightly heterogeneous completion Figure Comparison among simulated rates for Case 1 and rates obtained using the simplified method for b = 0, 0.1, 0.2, 0.3, 0.4 and 0.5 (modified from Nobakht et al. (2012b)) Figure 5.1. Possible combinations of reservoir/hydraulic fracture encountered for tight oil/shale gas reservoirs (Modified from Clarkson and Pederson, 2010) Figure 5.2. Schematic diagram (log-log plot) illustrating flow regime sequence for a horizontal well completed with multiple transverse hydraulic fractures in a homogeneous single-porosity reservoir (Clarkson and Beierle, 2011). t* is time for liquids or pseudo-time for gas Figure 5.3. (a). Conceptual model for multi-fractured horizontal well (Scenario 5 in Figure 5.1) and (b) Geometry used to generate type curves in Figures Figure 5.4. q D versus t Dxf type curve for the reservoir geometry shown in Figure 5.3(b) when y e = x e Figure 5.5. q D versus t Dxf type curve for the reservoir geometry shown in Figure 5.3(b) when y e = 2x e xviii

19 Figure 5.6. q D versus t Dxf type curve for the reservoir geometry shown in Figure 5.3(b) when y e = 6x e Figure 5.7. q D versus t Dxf type curve for the reservoir geometry shown in Figure 5.3(b) when y e = 10x e Figure 5.8. q D versus t Dxf type curve for the reservoir geometry shown in Figure 5.3(b) when y e = 20x e Figure 5.9. q DM versus t DM type curve for the reservoir geometry shown in Figure 5.3(b) when y e = x e Figure q DM versus t DM type curve for the reservoir geometry shown in Figure 5.3(b) when y e = 2x e Figure q DM versus t DM type curve for the reservoir geometry shown in Figure 5.3(b) when y e = 6x e Figure q DM versus t DM type curve for the reservoir geometry shown in Figure 5.3(b) when y e = 10x e Figure q DM versus t DM type curve for the reservoir geometry shown in Figure 5.3(b) when y e = 20x e Figure Semi-log pressure derivative plot for the reservoir geometry shown in Figure 5.3(b) for two different values of x e. In both cases, x f =200 ft and y e is infinite Figure Normalized rate ratio versus t Dxf for the reservoir geometry shown in Figure 5.3(b) when y e = x e Figure Normalized rate ratio versus t Dxf for the reservoir geometry shown in Figure 5.3(b) when y e = 2x e xix

20 Figure Normalized rate ratio versus t Dxf for the reservoir geometry shown in Figure 5.3(b) when y e = 6x e Figure Normalized rate ratio versus t Dxf for the reservoir geometry shown in Figure 5.3(b) when y e = 10x e Figure Normalized rate ratio versus t Dxf for the reservoir geometry shown in Figure 5.3(b) when y e = 20x e Figure The production data for field case study plotted on y e =2x e type curve Figure Comparison between forecast rates obtained from the type curves presented in this study and the numerical simulation. The expected ultimate recovery (EUR) obtained using type curve and numerical modeling are 3.6 Bscf and 3.5 Bscf, respectively Figure 6.1. (a) A hydraulically fractured vertical well in the center of a rectangular reservoir. (b) Schematic of a multi-fractured horizontal well in a rectangular reservoir. (c) Stimulated Reservoir Volume for the multi-fractured horizontal well shown in Figure 6.1(b). (d) The region that each fracture drains Figure 6.2. Flowing material balance analysis using the 10-year production data for Case Figure 6.3. Pressure distribution in the reservoir after 350 days for Case 1 (p i =10,000 psi, p wf =3,000 psi, k=0.1 md and c f =0) Figure 6.4. Flowing material balance analysis using the first 350 days of production data for Case xx

21 Figure 6.5. Productivity index versus gas cumulative production calculated using the first 350 days of production data for Case Figure 6.6. Square-root-of-time plot for the first 350 days of production data for Case Figure 6.7. Square-root-of-time plot for Case 1. The end of linear flow is estimated to be 370 days (dashed vertical line) based on the time at which the data diverge from the linear flow line Figure 6.8. Inverse of DER versus time for the first 1,000 days of Case 1. DER is the semi-log derivative of rate-normalized pseudo-pressure. The end of linear flow occurs at 195 days, which is shown by a dashed vertical line Figure 6.9. Contacted gas-in-place calculated from flowing material balance versus time for Case Figure Comparison between contacted gas-in-place calculated from flowing material balance and Equation (6.3) at different times for Case Figure Comparison between contacted water-in-place calculated from flowing material balance and water-equivalent of Equation (6.3) at different times for Case Figure Comparison among contacted gas-in-place calculated from flowing material balance for Case 3 (p wf =3,000 psi), Case 5 (p wf =6,000 psi) and Case 6 (p wf =9,000 psi) and Equation (6.3) Figure Square-root-of-time plot in dimensionless format for bounded reservoir shown in Figure 6.1(a) (solid line on the plot) and infinite acting linear flow (dashed line on the plot) xxi

22 Figure The ratio of observed t elf to t elf calculated using Equation (6.16) plotted against linear flow correction factor for Cases 1, 8 11 (symbols). The solid line is the correlation between observed duration of linear flow, expected duration of linear flow and linear flow correction factor shown in Equation (6.22) Figure Comparison among contacted gas-in-place calculated from Equation (6.5) for Case 3 (p wf =3,000 psi), Case 5 (p wf =6,000 psi) and Case 6 (p wf =9,000 psi) and Equation (6.3) Figure Comparison among contacted gas-in-place calculated from Equation (6.25) for Case 3 (p wf =3,000 psi), Case 5 (p wf =6,000 psi) and Case 6 (p wf =9,000 psi) and Equation (6.16) for calculating distance of investigation Figure Modified flowing material balance plot using the first 350 days of production data in Case 1. The normalized cumulative production is calculated using Equation (6.26) instead of Equation (6.7) Figure Flowing material balance analysis for field case study. The normalized cumulative production is calculated using Equation (6.7) Figure Productivity index versus cumulative production for field case study Figure Square-root-of-time plot for field case study Figure Modified flowing material balance plot for Field Case Study. The normalized cumulative production is calculated using Equation (6.26) instead of Equation (6.7) Figure Effect of flowing pressure on the location of p D =0.2 and p D =0.5 in a water (a slightly compressible fluid) reservoir xxii

23 Figure Effect of flowing pressure on the location of p D =0.2 in a gas reservoir xxiii

24 LIST OF SYMBOLS, ABBREVIATIONS, NOMENCLATURE A Area of the region of investigation or Drainage area, ft 2 A x Area exposed to linear flow, ft 2 A cj Surface area to flow in division j in a heterogeneous completion, ft 2 (A c ) T Total surface area to flow, ft 2 b Hyperbolic decline exponent, dimensionless, or Slippage factor, psi b' Intercept of 1 q versus t plot, 1/Mscf/day b a b pss B B g B gi * B gi Apparent gas slippage factor, psi Inverse of productivity index, psi/stb Liquid formation volume factor, bbl/stb Gas formation volume factor, ft 3 /scf Gas formation volume factor at initial reservoir conditions, ft 3 /scf Gas formation volume factor at initial reservoir conditions adjusted to account for desorption, ft 3 /scf c f Formation compressibility, psi -1 c g Gas compressibility, psi -1 c t Total compressibility, psi -1 CFIP Contacted fluid-in-place, STB CGIP Contacted gas-in-place, scf CWIP Contacted water-in-place, STB d Half of spacing between fractures along the horizontal well, ft d j Half of the distance between fractures in division j in a heterogeneous completion, ft D Diffusion coefficient, ft 2 /D D D D elf D elfj Drawdown parameter defined in Equation (1.4) and Equation (6.19), fraction Decline rate at the end of linear flow, 1/day Decline rate at the end of linear flow for division j in a heterogeneous completion, 1/day xxiv

25 D i EUR f CP F CD G G p h k k a k ai k r k ri k L e m Decline rate at the start of hyperbolic forecast, 1/day Expected ultimate recovery, Mscf Correction factor for constant flowing pressure analysis, dimensionless Dimensionless fracture conductivity, dimensionless Gas-in-place in the region of influence, scf Cumulative gas production, scf Net pay thickness, ft Permeability, md Apparent permeability, md Apparent permeability at initial pressure, md Permeability ratio defined as the ratio of apparent permeability, k a, to the liquidequivalent permeability, k, dimensionless Permeability ratio at initial pressure, dimensionless Liquid-equivalent permeability, md Length of horizontal well, ft Slope of square-root-of-time plot (a plot of normalized pressure versus square root of time), psi 2 day 1/2 /Mscf cp m CP Slope of 1 q versus t plot for early linear flow, day1/2 /Mscf 1 m CPj Slope of versus t plot for division j in a heterogeneous completion, q day 1/2 /Mscf m S Slope of normalized pressure versus linear superposition time plot, psi 2 day 1/2 /Mscf cp m' Slope of normalized pressure versus * t a plot, psi 2 day 1/2 /Mscf cp m' CP Slope of 1 q versus * t a plot, day 1/2 /Mscf M g n N N p Gas molecular weight, lbm/lbm-mol Number of fractures in a multi-fractured horizontal well Original-fluid-in-place in Equation (6.9), STB Liquid cumulative production, STB xxv

26 OGIP Original gas-in-place, scf OWIP Original water in-place, STB p Pressure, psi p p D p i p L p m p p p pi p pwf p wf p sc * p pi Average pressure in the region of influence, psi Dimensionless pressure, dimensionless Initial pressure, psi Langmuir pressure, psi Mean pressure, psi Pseudo-pressure, psi 2 /cp Pseudo-pressure at initial pressure, psi 2 /cp Pseudo-pressure at flowing pressure, psi 2 /cp Flowing pressure, psi Pressure at standard conditions, psi Modified pseudo-pressure to account for slippage effect at initial pressure, psi 2 /cp * p pwf Modified pseudo-pressure to account for slippage effect at flowing pressure, q q D q Dd q DM q j q elf q elfj q f Q Q elf Q fyd psi 2 /cp Liquid rate, STB/day, or Gas rate, Mscf/day Dimensionless rate defined in Equation (5.1), dimensionless Dimensionless rate defined in Equation (5.3), dimensionless Dimensionless rate defined in Equation (5.5), dimensionless Gas rate of division j in a heterogeneous completion, Mscf/day Production rate at the end of linear flow, Mscf/day Production rate at the end of linear flow for division j in a heterogeneous completion, Mscf/day Economic rate limit, Mscf/day Gas cumulative production, Mscf Gas cumulative production at the end of linear flow, Mscf The relative contribution of region between fracture and y D to the total production, fraction xxvi

27 Q r S S g S w t t a Volume of gas produced between the end of linear flow and end of forecast, Mscf Number of divisions in a heterogeneous completion Fluid saturation in Equation (6.9), fraction Gas saturation, fraction Water saturation, fraction Time, days Pseudo-time, days * t a Corrected pseudo-time, days t a, SL Linear superposition pseudo-time, day 1/2 t c t ca Material balance time, days Material balance pseudo-time, days * t ca Modified material balance pseudo-time, days t Dxf t Dxe t DM t elf t elfj Dimensionless time defined in Equation (5.2), dimensionless Dimensionless time defined in Equation (5.4), dimensionless Dimensionless time defined in Equation (5.6), dimensionless Duration of linear flow, days Duration of linear flow for division j in a heterogeneous completion, days t SL Linear superposition time, day 1/2 (t elf ) c End of linear flow calculated using Equation (2.27), days (t elf ) o End of linear flow obtained using the pressure profile in the reservoir, days (t elf ) Expected Expected duration of linear flow calculated from Equation (6.16), days (t elf ) Observed Observed end of linear flow based on 1% deviation from linear flow solution, days T Reservoir temperature, ºR Temperature at standard conditions, R T sc V L x e x f Langmuir volume, scf/ton Reservoir width or the spacing between fractures along the horizontal well, ft Fracture half-length in x-direction, ft (x f ) T Total fracture half-length, ft xxvii

28 x fj Sum of half-length of all fractures in division j in a heterogeneous completion, ft ( x ) Half-length of the longest fracture, ft f max X match y y 10% y D y e Y match Z Z sc Z * Z ** * Ratio of t c (for liquids) or t ca (for gas) to t DM at the selected match point Distance from the fracture or distance of investigation, ft The distance that its pressure drop (observed from simulation) is 10% of the maximum pressure drop, ft Dimensionless distance defined in Equation (A.3), dimensionless Reservoir length or the spacing between horizontal wells, ft Ratio of q p p i wf (for liquids) or p pi q p pwf match point Gas compressibility factor Gas compressibility factor at standard conditions Gas compressibility factor, adjusted to account for desorption (for gas) to q DM at the selected Modified Z-factor that accounts for desorption, water influx, formation compressibility Greek Symbols α j β j g γ j η Ratio of drainage area of division j in a heterogeneous completion to the total drainage area, fraction Parameter defined in Equation (4.12) for division j in a heterogeneous completion Reservoir gas specific gravity (air=1) Product of α j and β j, which is a measure of surface area to flow in division j in a heterogeneous completion to total surface area to flow, fraction Hydraulic diffusivity, ft 2 /day Porosity, fraction g Gas viscosity, cp ρ B Shale bulk density, g/cm 3 xxviii

29 1 CHAPTER 1 INTRODUCTION Due to continuous advances in drilling and completion techniques as well as the overwhelming energy demands worldwide, the exploitation of unconventional reservoirs is increasingly attractive for the oil and gas industry. Unconventional reservoirs are those that cannot be produced at economic flow rates or that do not produce economic volumes of oil and gas without assistance from massive stimulation treatments like hydraulic fracturing or special recovery processes and technologies like steam injection in heavy oil reservoirs. Typical unconventional reservoirs are tight oil and gas sands, coalbed methane, heavy oil, and gas shales (Holditch, 2003). Among the above-mentioned types of unconventional reservoirs, this study focuses on analyzing production data from tight/shale gas reservoirs. Due to the ultra-low permeability of these reservoirs, it is challenging, if not impossible, to produce economically from them using conventional methods. Recent advances in drilling and completion technology have allowed commercial exploitation of ultra-low permeability gas reservoirs. Horizontal wells completed with multiple-fracturing stages are the most popular method for exploiting shale gas reservoirs. Therefore, development of analysis methods for analyzing production data from these wells has gained tremendous attention in the last decade. The success of horizontal wells with multiple fractures in shale reservoirs is due to the creation of a large contacted surface area which, in effect, compensates for the low permeability in these reservoirs.

30 2 1.1 Possible Conceptual Models The use of multi-fractured horizontal wells is expected to create a complex sequence of flow regimes (Chen and Raghavan, 1997; Clarkson and Pederson, 2010). The proper interpretation of these flow regimes is necessary for obtaining information about the hydraulic fracture stimulation and the reservoir. As a result, the understanding of completion/reservoir geometry is critically important. Figure 1.1 shows several possible conceptual models for well/reservoir/hydraulic fracture combinations that can be used to analyze tight/shale gas/oil (Clarkson and Pederson, 2010). Below are descriptions of the different scenarios shown in Figure 1.1. Scenario 1 represents an openhole horizontal well in a single porosity reservoir. This scenario is likely ineffective in shale gas reservoirs with ultra-low permeability due to the relatively small contacted surface area between wellbore and reservoir. Scenario 2 represents an openhole horizontal well in a naturally-fractured reservoir (i.e., dual porosity). This scenario may be applicable to a horizontal well completed in a naturally-fractured reservoir or a multi-fractured horizontal well where a complex hydraulic fracture network has been created (Clarkson and Pederson, 2010). In scenario 3, the stimulated reservoir volume (SRV) (Mayerhofer et al., 2010) is limited to a region near the horizontal well, and the region outside the SRV is single porosity. The stimulated reservoir volume can be considered as enhanced permeability or as a naturally fractured reservoir. The enhanced permeability in the SRV is created from multi-fracturing the horizontal well and represents an induced hydraulic fracture network (including reactivation of existing natural fractures) and natural fractures. In scenario 4, the region outside the SRV is naturally-fractured and everything else is the same as scenario 3. Because of multi-fracturing the horizontal well, the

31 3 SRV and the reservoir outside SRV would have different fracture spacing/permeability and fracture porosity (Clarkson and Pederson, 2010). Scenarios 5 8 are the same as 1 4, but with discrete hydraulic fractures. Note that scenario 7 is similar to the conceptual model that Ozkan et al. (2011) used as the basis of their Trilinear Flow solution for analyzing shale gas wells.

32 Figure 1.1 Possible combinations of reservoir/hydraulic fracture encountered for tight oil/shale gas reservoirs (Modified from Clarkson and Pederson, 2010). 4

33 5 1.2 Linear Flow Analysis Among the flow regimes observed in a multi-fractured horizontal well, early linear flow to fractures is the dominant flow regime observed in most fractured tight/shale gas wells, which may continue for several years. Analysis of long-term linear flow periods associated with shale gas production has received much attention in recent literature as a means of obtaining information about stimulation efficiency. The most popular method for analyzing linear flow is the square-root-of-time plot, a plot of rate-normalized pseudo-pressure (RNP) for gas, p pi p q pwf, versus square root of time, where p pi and p pwf are pseudo-pressures at initial reservoir pressure and flowing pressure, respectively and q is gas rate. It is documented in the literature that linear flow appears as a straight line on the square-root-of-time plot (Wattenbarger et al., 1998; El-Banbi and Wattenbarger, 1998). The slope of this line is used in the literature to calculate x f k using Equations (1.1) and (1.2) for gas production under constant flowing pressure and constant rate production constraints, respectively, in field units (Wattenbarger et al., 1998; El-Banbi and Wattenbarger, 1998): x x f f k k 315.T 4 1, (1.1) h c m g t g t i 200.T 8 1. (1.2) h c m i In these equations, x f is fracture half-length, k is the permeability, T is the reservoir temperature, h is the net pay thickness, ϕ is the reservoir porosity, µ g is gas viscosity, c t is total compressibility (subscript i refers to initial conditions), and m is the slope of the square-root-of-time plot.

34 6 Ibrahim and Wattenbarger (2005; 2006) observed that the use of Equation (1.1) overestimates the value of xf k when analyzing linear flow for constant flowing pressure production. They proposed to multiply x f k obtained using Equation (1.1) by an empirically-obtained correction factor, f CP, for constant flowing pressure production: 2 CP D D f. D. D, (1.3) where D D is the drawdown parameter and is related to pseudo-pressure at initial pressure, p pi, and pseudo-pressure at flowing pressure, p pwf, using Equation (1.4): D D ppi ppwf. (1.4) p pi Equations (1.1) and (1.2) are for analyzing either constant flowing pressure or constant gas rate production. When dealing with real production data, neither the flowing pressure nor the gas rate is usually constant. Therefore, the solutions for constant flowing pressure or constant gas rate are only approximations when analyzing real production data with variable rate and variable flowing pressure. Analyzing linear flow is made more difficult by the fact that the equations for calculating x f k are different for constant flowing pressure (i.e., Equation (1.1)) and constant gas rate production (i.e., Equation (1.2)). In the literature from the last decade, equations based on constant flowing pressure solution are being used for calculating x f k for real production data from multifractured horizontal wells, because most researchers feel that the constant flowing pressure assumption is more realistic.

35 7 1.3 Motivation One of the challenges industry is facing is to determine the amount of hydrocarbon that will be produced in the future as well as how much hydrocarbon will be left behind in the ground after the economic production is complete. The other challenge is optimizing the production and drilling schemes in these reservoirs to achieve the best results economically. Therefore, there is a need to develop new methods for analyzing production data from multi-fractured horizontal wells. In the last decade, there has been a lot of research to develop methods for analyzing the performance of these wells and quantifying reservoir and hydraulic fracture properties as well as forward models. As a result, significant advances have been made in the development of different methods for production data analysis. As discussed by Clarkson and Beierle (2011) and Clarkson (2013), the analysis methods that have been commonly used for unconventional oil and gas reservoirs can be categorized as follows: 1. Straight Line Methods (Wattenbarger et al., 1998; Ibrahim and Wattenbarger, 2005; 2006; Mattar et al., 2006; Cheng et al., 2009; Clarkson and Beierle, 2011; Song et al., 2011; Song and Ehlig-Economides, 2011; Samandarli et al., 2012). 2. Type Curve Methods (Agarwal et al., 1999; Amini et al., 2007). 3. Analytical and Numerical Simulation (ex. Larsen and Hegre (1991), and Raghavan et al. (1997) for conventional reservoirs, and Medeiros et al. (2008), Ozkan et al. (2011), and Bello and Wattenbarger (2008) for unconventional reservoirs). 4. Empirical Methods (ex. Arps, 1945; Ilk et al., 2008; Valkó, 2009; Mattar and Moghadam, 2009).

36 8 5. Hybrid Methods (ex. Fetkovich, 1980; Kupchenko et al., 2008; Nobakht et al., 2012b), which are the combination of empirical and analytical methods. The unique storage and transport properties of unconventional gas reservoirs require that conventional petroleum engineering methods be modified to account for these unique characteristics (Clarkson et al. 2012). For example, as pointed out by Clarkson et al. (2012), a fundamental problem with the application of conventional production data analysis to ultra-low permeability reservoirs is that current methods were derived with the assumption that flow can be described with Darcy's law. This assumption may not be valid for tight/shale gas reservoirs, as they contain a wide distribution of pore sizes, including in some cases nanopores (Loucks et al. 2009) which are pores that are at the nanometer scale. Therefore, the mean-free path of gas molecules may be comparable to or larger than the average effective rock pore throat radius causing the gas molecules to slip along pore surfaces. Gas slippage, along with other flow regimes depending on pore size, results in non-darcy flow behaviour (Javadpour, 2009), which is not accounted for in conventional production data analysis. Therefore, there is a need to evaluate existing methods proposed in the literature for analyzing production data from shale gas reservoirs. The focus of this work is to evaluate these existing methods and modify them where necessary and establish more accurate/practical methods for reservoir characterization and long-term forecasting. In particular, the focus of this thesis is on straight line methods, type curve methods and hybrid methods.

37 9 1.4 Outline of the Thesis This thesis consists of 7 chapters. A brief overview of the structure of the thesis, and a description of the contents of each chapter, is provided below. The thesis is presented in paper format, with the majority of the content of the chapters having been published in peer-reviewed literature. Copyright permissions have been obtained from the respective journals to allow reproduction in the thesis. Chapter 1, as provided above, gives an introduction to the thesis research topic. Chapter 2 provides improvements to linear flow analysis by considering the effects of pressure-dependent gas properties for constant rate, constant flowing pressure and variable rate/flowing pressure production. In this chapter, it is demonstrated that not incorporating pressure-dependent gas properties in linear flow analysis results in errors for all three boundary conditions (i.e., constant rate, constant flowing pressure and variable rate/flowing pressure). Chapter 3 investigates the effects of unique tight/shale gas reservoir properties (i.e., gas slippage and desorption) on linear flow analysis for the constant flowing pressure boundary condition. From this work, it was found that when slippage is ignored in the analysis of extremely low-permeability reservoirs, x f k (and hence, fracture halflength) is significantly over-estimated. It was also shown that as permeability decreases, the slippage contribution to production increases. Chapter 4 discusses the effect of completion heterogeneity (unequal fracture lengths) on the long-term production forecast for a multi-fractured horizontal well (MFHW). The concept of analyzing production data from multiple MFHWs is also presented in this chapter. It is shown that the completion heterogeneity impacts the long-

38 10 term production forecast and expected ultimate recovery (EUR) in MFHWs. It is also demonstrated how multi-well analysis is useful for cases where the adjacent wells are in communication. In Chapter 5, conceptual models for well/reservoir/hydraulic fracture combinations were first presented and the impact of various reservoir types/induced hydraulic-fracture geometries upon the sequence of flow-regimes that could be encountered for shale gas reservoirs were discussed. New sets of dimensionless type-curves for one scenario of fracture geometry and reservoir type, which yield more unique results than those presented previously, were developed and applied. This work also presented a new method for evaluating the contribution from the outer reservoir SRV, where SRV is the drainage volume between hydraulic fractures for the scenario of interest. Chapter 6 compares different methods being used in the industry for calculating contacted gas-in-place (CGIP) and original gas-in-place (OGIP) in low-permeability systems. In particular, flowing material balance (FMB) analysis is compared with other methods based upon the linear flow plot and distance of investigation. It is found that FMB underestimates the total OGIP, even after significant reservoir depletion. FMB analysis is improved by incorporating a material balance pseudo-time calculated using the average pressure in the region of investigation, as opposed to average reservoir pressure. Lastly, OGIP/CGIP estimation using the square-root-of-time plot is enhanced by correcting the distance of investigation equation and accounting for drawdown effects. Without these corrections, this method tends to over-estimate OGIP. Lastly, Chapter 7 provides the overall conclusions from this thesis work and also discusses future work that could be carried out to supplement the findings of this thesis.

39 11 CHAPTER 2 IMPROVED LINEAR FLOW ANALYSIS: EFFECTS OF PRESSURE-DEPENDENT PROPERTIES OF GAS Abstract Many tight/shale gas wells exhibit linear flow, which can last for several years. The most commonly used method to analyze linear flow is the square-root-of-time plot, a plot of normalized pressure versus square root of time. Linear flow is expected to appear as a straight line on this plot and the slope of this line can be used to calculate the product of fracture half-length and square root of permeability. In this chapter, linear flow from a fractured well in tight/shale gas reservoir under constant rate, constant flowing pressure and variable rate/variable flowing pressure constraints are studied. For the case of constant rate constraint, it is shown analytically that the shape of the square-root-of-time plot depends on the production rate. It is also shown that depending on production rate, the square-root-of-time plot may not be a straight line during linear flow; the higher the production rate the earlier in time the plot deviates from the expected straight line. This deviation creates error in the analysis. To address this issue, a new analytical method is developed for analyzing linear flow data for the constant gas rate production constraint. The method is then validated using a number of numerically- 1 This chapter is the combination of modified version of followings: Nobakht, M. and Clarkson, C.R A New Analytical Method for Analyzing Production Data from Shale Gas Reservoirs Exhibiting Linear Flow: Constant Rate Boundary Condition. SPE Reservoir Evaluation & Engineering, 15 (1): Nobakht, M. and Clarkson, C.R A New Analytical Method for Analyzing Production Data from Shale Gas Reservoirs Exhibiting Linear Flow: Constant-Flowing-Pressure Boundary Condition. SPE Reservoir Evaluation & Engineering, 15 (3): Nobakht, M. and Clarkson, C.R Analysis of Linear Flow in Shale Gas reservoirs: Rigorous Corrections for Fluid and Flow Properties. Journal of Natural Gas Science and Engineering, 8:

40 12 simulated cases. Excellent agreement is found between the fracture half-lengths obtained from the new analytical method for the constant rate case (provided permeability is known) and those input into numerical simulation. For the case of constant flowing pressure constraint, it is shown that using the slope of square-root-of-time plot results in an overestimation of fracture half-length. The degree of this overestimation is influenced by initial pressure, flowing pressure and formation compressibility. An analytical method is presented to correct the slope of the square-root-of-time plot to improve estimation of fracture half-length. The method is validated using a number of numerically-simulated cases. The newly-developed fullyanalytical method for the constant flowing pressure case results in a reliable estimate of fracture half-length, if permeability is known. Finally, we present a method for analyzing linear flow for real production data, where neither flowing pressure nor gas rate is constant. The method is validated using three numerically-simulated cases. It is found that this method works well for the three cases provided. 2.2 Introduction The dominant flow regime observed in most fractured tight/shale gas wells is linear flow, which may continue for several years. The square-root-of-time plot, a plot of normalized pressure versus square root time, is probably the most important plot for analyzing linear flow (Anderson et al., 2010). It is reported in the literature that linear flow appears as a straight line on the square-root-of-time plot. The slope of this line can

41 13 be used to calculate the product of fracture half-length and square root of permeability. Therefore, permeability must be known to estimate fracture half-length, and vice versa. In this chapter, linear flow for constant gas rate production, constant flowing pressure production and variable rate/variable flowing pressure production are studied. For constant gas rate production, it is shown that gas production rate affects the shape of the square-root-of-time plot. It is also shown that depending on the gas rate, the squareroot-of-time plot may not be a straight line during linear flow. As the square-root-of-time plot is not a straight line, a methodology is presented to calculate the product of fracture half-length and square root of permeability. For constant flowing pressure production, the correction of the slope of square-root-of-time plot for constant flowing pressure is analytically derived. The method is then validated by comparing its results against test cases which are built using numerical simulation. The effects of initial pressure, flowing pressure, permeability and formation compressibility are investigated. Finally, we present a method for analyzing linear flow for variable rate/variable pressure data. 2.3 Derivation/Analysis Methods The base reservoir geometry that is used to develop the new analytical methods for analyzing transient linear flow for wells subject to constant gas rate and constant flowing pressure constraints is shown in Figure 2.1; this corresponds to a single hydraulicallyfractured vertical well in the center of a rectangular reservoir, or possibly to a single fracture stage in a multi-fractured horizontal well. It is assumed that the fracture has infinite conductivity and there is no skin.

42 14 2y Fracture y e x e Figure 2.1. Base geometry used for derivation of analytical methods - a hydraulicallyfractured well in the center of a rectangular reservoir. For constant flowing pressure and constant gas rate production, xf k is related to the slope of square-root-of-time plot (a plot of normalized pressure vs. square root of time) using Equations (1.1) and (1.2) respectively. These equations are based on liquid flow theory; past researchers have used pseudo-pressure to account for pressuredependent properties of gas. However, this substitution alone is not sufficient. Pseudotime should also be incorporated to account for changing gas compressibility (Fraim and Wattenbarger, 1987; Agarwal et al., 1999; Anderson and Mattar, 2005), provided that the correct reference pressure is used. In other words, to obtain the correct value of x f k, the slope of the normalized pressure plotted against square root of pseudo-time, defined in Equation (2.1), should be used in Equations (1.1) and (1.2): t a t dt gcti. (2.1) 0 c g t Here, g and c t are gas viscosity and total compressibility evaluated at the average reservoir pressure. Traditionally, the average pressure in the whole reservoir is used to

43 15 calculate the pseudo-time (Fraim and Wattenbarger, 1987; Agarwal et al., 1999). However, as indicated by Anderson and Mattar (2005), this introduces significant errors especially in low-permeability reservoirs. Anderson and Mattar (2005) suggested using the average pressure in the region of influence to calculate pseudo-time. The pseudo-time calculated using average pressure in the region of influence is called corrected pseudotime (Anderson and Mattar, 2005). In this study, the corrected pseudo-time is used for the model derivation. To calculate corrected pseudo-time using Equation (2.1), the average pressure in the region of influence, p, needs to be calculated. For this purpose, The following material balance equation is used (Moghadam et al., 2011): Z p p Gp (1 ) G. (2.2) i ** ** Zi Here, p i is initial pressure, ** ** Z and Z i are modified Z-factors introduced by Moghadam et al. (2011) at average pressure in the region of influence and initial pressure, respectively, G p is cumulative gas production and G is contacted gas-in-place (i.e., gasin-place in the region of influence). Using the volumetric equation for gas, G is calculated as follows: Ah Sgi G, (2.3) B gi where A is the area of the region of influence, h is the net pay thickness, ϕ is the reservoir porosity, S gi is initial gas saturation and B gi is initial gas formation volume factor. For the reservoir geometry shown in Figure 2.1, area of the region of influence, A, is related to distance of investigation, y, and fracture half-length, x f, as:

44 16 A2x y 4x y (2.4) e f Equations (2.2) (2.4) will be used for deriving analytical methods for analyzing linear flow in the following sections Constant Rate Boundary Condition For the case of constant rate production, the cumulative production, G p, at time t is: 3 t 3 Gp 10 qdt 10 qt. (2.5) 0 The unit of q in Equation (2.5) is Mscfd and the conversion factor of 10 3 is used in this equation to convert G p to scf. When a well is producing with constant production rate, the distance of investigation, y, can be obtained from the following equation during the linear flow period (Wattenbarger et al., 1998): kt y (2.6) c g t i Combining Equations (2.3), (2.4) and (2.6), the contacted gas-in-place for constant rate production for the reservoir geometry shown in Figure 2.1 becomes: h Sgixf k G t. (2.7) B c gi g t i The unit of G in this equation is scf. Substituting G p and G from Equation (2.5) and Equation (2.7), respectively, into Equation (2.2) leads to: gi g t i ** i gi f p p 1000qB c i (1 t ). (2.8) ** Z Z. h S x k

45 17 Equation (2.8) demonstrates that the average pressure in the region of influence is changing with time when the well with reservoir geometry shown in Figure 2.1 is producing with constant rate. Equation (2.8) can be used to calculate the average pressure in the region of influence at different times, which in turn can be used to calculate corrected pseudo-time, t * a, using Equation (2.1). The normalized pressure can then be plotted against * t a and the slope of this plot, m', can be used to obtain x f k from the following equation: x f k 200.T 8 1. (2.9) h c m g t i However, in order to calculate the average pressure in the region of influence from Equation (2.8), x f k should be known. Further, the average pressure in the region of influence is required to calculate the corrected pseudo-time, and the slope of normalized pressure versus * t a plot is needed to calculate x f k from Equation (2.9). This makes analysis of linear flow for constant gas rate production an iterative process. Approximation Solution To approximate the relationship between corrected pseudo-time and time for constant gas rate production during transient linear flow, the following assumptions are made: 1. The gas is ideal (Z=1). Using the definition of gas compressibility, this assumption leads to: 1 1 dz 1 cg. (2.10) p Zdp p

46 18 2. Gas viscosity is not changing with pressure. 3. Total compressibility is dominated by gas compressibility, i.e., c t S c. (2.11) g g Combining Equations (2.1), (2.8), (2.10) and (2.11) and using constant gas viscosity assumption, the corrected pseudo-time, t * a, becomes: gi f 2000qBgi * gct i ta t t t. (2.12) h S x k From this equation we see that at early time, t * a t and at late time, t * a t. This means that at early time, the square-root-of-time plot is a straight line for constant gas rate production linear flow and eventually it deviates from a straight line. The deviation from the early time straight line on square-root-of-time plot (using real time) is shown in Figure 2.2. Equation (2.12) also demonstrates that the shape of square-root-of-time plot depends on the flow rate with this newly-defined pseudo-time, a plot of normalized pressure versus * t a plot forms a straight line during linear flow.

47 Normalized Pressure, (10 6 psi 2 /cp) / (MMscf/day) Square Root of Time, days 0.5 Figure 2.2. Square-root-of-time plot for a constant rate case generated using numerical simulation. Analysis Method Based on the derivation above, an analysis method is presented here to analyze linear flow for the geometry shown in Figure 2.1 when the well is producing at constant gas rate. As shown in the previous section, t * a t at early times under some assumptions. This means that the slope of square-root-of-time plot for early times can be used to provide an initial guess for xf k in the following iterative procedure: 1. Plot normalized pressure versus t on Cartesian coordinates and place a line through early-time data points. Determine the slope of the line, m. 2. Calculate xf k using Equation (1.2).

48 20 3. Calculate the average pressure in the region of influence at different times using Equation (2.8). 4. Calculate the corrected pseudo-time, t * a using Equation (2.1) which incorporates the average pressure in the region of influence at different times. 5. Plot normalized pressure versus * t a on Cartesian coordinates and place a line through the linear flow data points. Determine the slope of the line, m'. 6. Calculate xf k using Equation (2.9). 7. Continue steps 3 6 until xf k converges Constant Flowing Pressure Because it is assumed that the fracture has infinite conductivity and there is no skin, the linear flow for constant flowing pressure production can be represented by: p pi p q pwf m t. (2.13) Here, p pi and p pwf are pseudo-pressures at initial reservoir conditions and flowing pressure, respectively. Because we are dealing with constant flowing pressure production, p pi p pwf is constant and therefore, using Equation (2.13), the cumulative production at time t is: 3 3 t 210 ( ppi ppwf ). (2.14) 0 Gp 10 qdt t m The unit of q in Equation (2.13) is Mscfd and the conversion factor of 10 3 is used in Equation (2.14) to convert G p to scf. When a well is producing under constant flowing

49 21 pressure, the distance of investigation, y, can be obtained from the following equation during the linear flow period (Wattenbarger et al., 1998): kt y (2.15) c g t i Combining Equations (2.3), (2.4) and (2.15), the contacted gas-in-place for constant flowing pressure production, using the reservoir geometry shown in Figure 2.1, becomes: h Sgixf k G t. (2.16) B c gi g t i The unit of G in this equation is scf. Substituting G p and G from Equation (2.14) and Equation (2.16), respectively, into Equation (2.2) leads to: gi pi pwf g t i ** i gi f p p 2000 B ( p p ) c i (1 ). (2.17) ** Z Z. mh S x k From this equation, we see that the average pressure in the region of influence is not time-dependent for constant flowing pressure production. As the average pressure in the region of influence is constant, using Equation (2.1), the corrected pseudo-time, t * a, becomes: * g t i a gct t c t. (2.18) This means that the corrected pseudo-time has a linear relationship with time. Equation (2.18) also shows that the slope of normalized pressure versus * t a plot, m', and the slope of normalized pressure versus t plot (i.e., the square-root-of-time plot), m, have the following relationship for constant flowing pressure production:

50 m c g t i m. (2.19) gct 22 In order to get the correct value for xf k when gas is being analyzed, the slope of normalized pressure versus * t a plot should be used in Equation (1.1). However, if the traditional form of the square-root-of-time plot is to be used, which uses real-time and not pseudo-time, the following equation can be used to calculate xf k from the slope of the square-root-of-time plot, m: x f k 315.T 4 1 c g t i m h gc t gc. (2.20) t i A correction factor, f CP, to improve the values of x f k calculated from the slope of the traditional square-root-of-time plot, can be defined as: f CP c g t i. (2.21) gct This equation indicates that the correction factor is related to the average pressure in the region of influence and initial pressure. Substituting x f k from Equation (2.20) and B gi ZT p i into Equation (2.17) results in: i p p ( Zgc i t) i( ppi ppwf) gct ** ** Z Zi Sgi pi gct i. (2.22a) Equation (2.22a) demonstrates that the average pressure in the region of influence depends on initial pressure, flowing pressure and gas properties. This equation can be

51 23 solved to obtain average pressure in the region of influence and then the correction factor, f CP, can be calculated using Equation (2.21). To improve xf k obtained from linear flow analysis, xf k calculated from Equation (1.1) can be multiplied by f CP. Note that Equation (2.22a) contains the average pressure in the region of influence and gas properties (modified Z-factor, viscosity and total compressibility) at this average pressure as well. To solve this equation, g(p) defined in Equation (2.22b) is plotted versus pressure to find the pressure at which g(p) becomes zero: p p ( Zgc i t) i( ppi ppwf) gct gp ( ) ** ** Z Zi Sgi pi gct i. (2.22b) Variable Rate/Variable Flowing Pressure Data When analyzing real production data, neither the flowing pressure nor the gas rate may be constant. Therefore, the solutions for constant flowing pressure or constant gas rate are only approximations when analyzing real production data with variable rate and variable flowing pressure. Analyzing linear flow is made more difficult by the fact that the equations for calculating x f k are different for constant flowing pressure (i.e., Equation (1.1)) and constant gas rate production (i.e., Equation (1.2)). It should be mentioned that Equation (1.1) is most often used in the literature to calculate xf k when analyzing real production data from multi-fractured horizontal wells, because most researchers feel that the constant flowing pressure assumption is more representative in this case. This assertion comes from the observation that while flowing pressure usually

52 24 changes rapidly during early production, it often stabilizes to near-constant values at later times. Historically, superposition time for a specific flow-regime (e.g. bilinear, linear, radial, boundary-dominated flow) is used to account for changing rate with time. For the case of linear flow, the linear superposition time, t SL, at any time t n is defined as (Clarkson and Beierle, 2011): ( q q ) t ( t t ). (2.23) n j j1 SL n j-1 j1 qn Equation (2.23) is applied to allow the use of constant rate solutions to the flow equations for liquid flow, as is often done in well-test analysis the form varies by flow- regime. A plot of rate-normalized pressure versus linear superposition time could then be used for analyzing linear flow for constant flowing pressure, constant rate and variable rate/flowing pressure production. The linear flow portion of the data forms a straight line on this plot and the slope of the line, m S, can be used to calculate xf k : x f k 200.T 8 1. (2.24) h c ms g t i Because the superposition function converts the variable rate data to the corresponding constant rate equivalent, Equation (2.24) has the same form as Equation (1.2) (i.e., for constant rate) except that m is replaced by m S. Thus with this formulation, there is no confusion whether to use constant rate or constant pressure equation for calculating xf k, which is valuable when analyzing real production data.

53 25 However, because of pressure-dependent properties of gas, the calculation of superposition time should really be done in terms of (corrected) pseudo-time, t a, rather than time: ( q q ) t ( t t ). (2.25) n j j1 a, SL a, n a, j-1 i1 qn Here, t a, SL is linear superposition pseudo-time. The slope of normalized pressure versus linear superposition pseudo-time can be used in Equation (2.24) to calculate x f k. Calculation of corrected pseudo-time to be used in Equation (2.25) depends on the average pressure in the region of influence, p, at different times. p can be calculated using Equation (2.2), where the gas-in-place in the region of influence, G, is calculated using Equation (2.16) for constant flowing pressure and Equation (2.7) for constant rate production. The problem with calculating linear superposition pseudo-time (i.e., t a, SL ) for the variable rate/variable flowing pressure case is estimation of gas-in-place in the region of influence at different times; the gas-in-place in the region of influence is different for constant flowing pressure (Equation (2.16)) and constant gas rate production (Equation (2.7)). To address this issue, we propose the following procedure to establish the correct equation to use for G: (i) Plot p pi p q pwf versus square root of time on Cartesian coordinates. Place a line through the early-time data points and determine the slope of the line, m. (ii) Using the slope determined in Step (i), calculate xf k using Equation (1.1) (i.e., constant flowing pressure equation).

54 26 (iii) Using the slope determined in Step (i), calculate xf k using Equation (1.2) (i.e., constant rate equation). (iv) Plot p pi p q pwf versus linear superposition time on Cartesian coordinates. Place a line through the early-time data points and determine the slope of the line, m S. (v) Using the slope determined in Step (iv), calculate xf k using Equation (2.24). (vi) If xf k from Step (v) is closer to that from Step (ii) compared to that from Step (iii), calculate G using Equation (2.16). Otherwise, calculate G using Equation (2.7). Following the determination of the correct form of G (for constant rate or constant flowing pressure production), we propose the following procedure to analyze linear flow for variable rate/variable flowing pressure data: 1. Plot p pi p q pwf versus linear superposition time on Cartesian coordinates. Place a line through the early-time data points and determine the slope of the line, m S. 2. Using the slope determined in Step 1, calculate xf k using Equation (2.24). 3. Calculate the average pressure in the region of influence at different times using gas material balance in Equation (2.2) and G obtained from steps (i) to (vi) above. 4. Calculate the corrected pseudo-time from Equation (2.1) and by using the average pressure in the region of influence calculated in Step 3.

55 27 5. Plot p pi p q pwf versus linear superposition pseudo-time (calculated using Equation (2.25)) on Cartesian coordinates. Place a line through the early-time data points and determine the slope of the line, m' S. 6. Using the slope determined in Step 5, calculate xf k using Equation (2.24). 7. Continue Steps 3 6 until xf k converges. 2.4 Validation of Analytical Methods Constant Rate Production To validate the analysis method presented above for constant gas rate production, synthetic production profiles (1 year production period) were generated using a black-oil simulator, assuming a single porosity reservoir. The common parameters among all the test cases are as follows: T=120 ºF, h=100 ft, ϕ=10%, S g =100%, γ g =0.65, x f =250 ft, x e =500 ft, y e =5,000 ft and c f =0. The input data for initial pressure, gas flow rate and permeability for the numerical simulation cases are given in Table 2.1. The blank cells in this table indicate that the value for that parameter is the same as that of Case 1. To model the hydraulic fracture in the numerical simulation, we added a high permeability grid in the x-direction at the center of the reservoir. The permeability of this grid is chosen large enough to have a fracture conductivity of F CD =400. In other words, the hydraulic fracture is assumed to have infinite conductivity in these simulated cases (i.e., negligible pressure drop along the fracture). Logarithmic gridding (maximum geometric ratio of 2) was used to model pressure transients accurately. The pressure difference between adjacent grids is kept below 15% to ensure the reliability of the reservoir

56 28 simulation results. In addition, the signatures of expected flow regimes (transient linear followed by boundary-dominated flow) for the hydraulically-fractured well geometry shown in Figure 2.1 were observed on the semi-log derivative plot (not shown here), suggesting the gridding used is adequate. Table 2.1. Input parameters for numerical simulation for different cases used to validate the methodology developed for constant gas rate production. The blank cells indicate that the value for that parameter is the same as that of Case 1. Case p i (psi) q (Mscfd) k (md) 1 10, , , , , , , , ,000

57 29 Figures show the normalized pressure for gas versus square root of time for Cases 1 14, with different combinations of initial pressure and permeability. These figures demonstrate that the conventional square-root-of-time plot (using real time) for constant gas rate production is a function of production rate. The higher the rate, the sooner the data diverge from early-time straight line behavior. Ibrahim and Wattenbarger (2005; 2006) similarly observed a rate-dependence for the conventional square-root-oftime plot. We can deduce from Equation (2.8) that increasing production rate reduces the average pressure in the region of influence, and consequently reduces the corrected pseudo-time. Each case is then analyzed using the methodology proposed in this study to calculate xf k. After xf k converges, the average pressure in the region of influence is calculated using Equation (2.8) and corrected pseudo-time values are calculated using Equation (2.1). The normalized pressure versus square root of corrected pseudo-time for different cases is shown in Figures These figures show that using square root of corrected pseudo-time reduces the dependency of square-root-of-time plot on gas production rate. For each case, the fracture half-length is obtained from the calculated value of x f k and the input permeability for the simulation. The fracture half-lengths calculated from this method for Cases 1 14 (using simulation input given in Table 2.1) are shown in Figure This figure clearly shows that the fracture half-lengths calculated using the new analytical method agree very well with the expected value of x f ( 250 ft).

58 30 Normalized Pressure, (10 6 psi 2 /cp) / (MMscf/day) q=0.25 MMscfd q=0.5 MMscfd q=1 MMscfd q=2.5 MMscfd q=4 MMscfd Square Root of Time, days 0.5 Figure 2.3. Square-root-of-time plot for Cases q=0.25 MMscfd q=0.5 MMscfd q=1 MMscfd Normalized Pressure, (10 6 psi 2 /cp) / (MMscf/day) Square Root of Time, days 0.5 Figure 2.4. Square-root-of-time plot for Cases 6 8.

59 q=0.25 MMscfd q=0.5 MMscfd q=1 MMscfd Normalized Pressure, (10 6 psi 2 /cp) / (MMscf/day) Square Root of Time, days 0.5 Figure 2.5. Square-root-of-time plot for Cases Normalized Pressure, (10 6 psi 2 /cp) / (MMscf/day) q=0.25 MMscfd q=0.5 MMscfd q=1 MMscfd Square Root of Time, days 0.5 Figure 2.6. Square-root-of-time plot for Cases

60 32 Normalized Pressure, (10 6 psi 2 /cp) / (MMscf/day) q=0.25 MMscfd q=0.5 MMscfd q=1 MMscfd q=2.5 MMscfd q=4 MMscfd Square Root of Corrected Pseudo-Time, days 0.5 Figure 2.7. Corrected square-root-of-time plot for Cases q=0.25 MMscfd q=0.5 MMscfd q=1 MMscfd Normalized Pressure, (10 6 psi 2 /cp) / (MMscf/day) Square Root of Corrected Pseudo-Time, days 0.5 Figure 2.8. Corrected square-root-of-time plot for Cases 6 8.

61 q=0.25 MMscfd q=0.5 MMscfd q=1 MMscfd Normalized Pressure, (10 6 psi 2 /cp) / (MMscf/day) Square Root of Corrected Pseudo-Time, days 0.5 Figure 2.9. Corrected square-root-of-time plot for Cases Normalized Pressure, (10 6 psi 2 /cp) / (MMscf/day) q=0.25 MMscfd q=0.5 MMscfd q=1 MMscfd Square Root of Corrected Pseudo-Time,, days 0.5 Figure Corrected square-root-of-time plot for Cases

62 Calculated Fracture Half-Length, ft Case Number Figure Fracture half-lengths calculated using the method proposed in this study for Cases 1 14 (simulation input provided in Table 2.1). The expected value of x f =250 ft is shown on the plot by the dashed horizontal line Constant Flowing Pressure The test cases used to validate the correction factor derived analytically in this chapter are built with the same methodology as that presented above for constant gas rate production test cases. The common parameters among all the test cases for constant flowing pressure are as follows: T=120 ºF, h=100 ft, ϕ=10%, S g =100%, γ g =0.65, x f =250 ft, x e =500 ft and y e =5,000 ft. The input data for initial pressure, flowing pressure, permeability and formation compressibility for the numerical simulation cases are given in Table 2.2. The blank cells in this table indicate that the value for that parameter is the same as that of Case 1.

63 35 For each case, first, the average pressure in the region of influence is calculated from Equation (2.22a) and the correction factor is analytically calculated using Equation (2.21). Then, x f k is calculated using the slope of the square-root-of-time plot in Equation (1.1) and it is compared with the expected value for x f k (i.e., input to numerical simulation). Finally, the expected correction factor is calculated as the ratio of expected xf k to calculated xf k.

64 36 Table 2.2. Input parameters for numerical simulation for different cases used to validate analytically-derived correction factor for constant flowing pressure production. The blank cells indicate that the value for that parameter is the same as that of Case 1. Case p i (psi) p wf (psi) k (md) c f (1/psi) 1 10,000 5, , , , , , , , , , , ,000 1, ,000 1, ,000 1, , , , , , , , , , , , , ,

65 37 Comparisons between analytically-calculated and expected correction factors for different values of formation compressibilities are shown in Figures For c f =0 and c f = /psi (Figure 2.12 and Figure 2.13, respectively), the analytical method presented in this study underestimates the correction factor whereas Equation (1.3) (i.e., empirically-derived correction factor from Ibrahim and Wattenberger (2005; 2006)) overestimates the correction factor. However, the average of analytically-obtained correction factors and those obtained using Equation (1.3) agrees well with the expected correction factors. Figure 2.14 shows that for c f = /psi, in general, both methods underestimate the correction factor. However, analytically-obtained correction factors are in better agreement with expected correction factors compared to those obtained from Equation (1.3). This is because the correlation developed by Ibrahim and Wattenbarger (2005; 2006) does not include the effect of formation compressibility, whereas the analytical method presented in this study considers the effect of formation compressibility through the total compressibility.

66 Ibrahim and Wattenbarger (2005; 2006) This Study y=x Calculated f CP Expected f CP Figure Comparison between calculated correction factors obtained using the new method from this study (using Equation (2.22a)) and Ibrahim and Wattenbarger (2005; 2006) with expected correction factor for c f = 0.

67 Ibrahim and Wattenbarger (2005; 2006) This Study y=x Calculated f CP Expected f CP Figure Comparison between calculated correction factors obtained using the new method from this study (using Equation (2.22a)) and Ibrahim and Wattenbarger (2005; 2006) with expected correction factor for c f = /psi.

68 Ibrahim and Wattenbarger (2005; 2006) This Study y=x 1 Calculated f CP Expected f CP Figure Comparison between calculated correction factors obtained using the new method from this study (using Equation (2.22a)) and Ibrahim and Wattenbarger (2005; 2006) with expected correction factor for c f = /psi Variable Rate/Variable Flowing Pressure Data Example 1. The flowing pressure and gas rate data for this case are shown in Figure The flowing pressure is input into the numerical simulation and the gas rate is calculated assuming the reservoir geometry shown in Figure 2.1 with p i =5,000 psi, T=120 ºF, h=100 ft, ϕ=6%, k=0.2 md, S g =100%, γ g =0.65, x f =1,000 ft, x e =2,000 ft, y e =10,000 ft and c f =0. The square-root-of-time plot for this case is shown in Figure Using the slope of the line passed through early-time data points and k=0.2 md, the fracture half-lengths calculated using Equation (1.1) and Equation (1.2) are x f =1,566 ft and x f =997 ft, respectively. Figure 2.17 shows the plot of rate-normalized pressure

69 41 versus linear superposition time for this case. Using the slope of the line in this plot and k=0.2 md, the fracture half-length from Equation (2.24) becomes x f =1,030 ft. We see that Equation (1.2) provided a fracture half-length that is closer to that calculated using Equation (2.24), which suggests that this case can be handled using the constant rate method for calculating G. From Figure 2.17 we see that a line can be drawn through early-time data points, but this could also easily be done for late-time data. However, the early-time straight line is the appropriate one in this example. The reason that the data does not form one straight line in Figure 2.16 and Figure 2.17 is because of the use of time instead of pseudo-time. Because the change in gas compressibility with pressure is ignored in these figures a nonlinear plot results. 12 Gas Rate Flowing Pressure Gas Rate, MMscf / D Flowing Pressure, psi Time, Days Figure Gas rate and flowing pressure profile for Example 1.

70 Normalized Pressure, (10 6 psi 2 /cp)/(mmscf/d) Square Root of Time, Days 0.5 Figure Square-root-of-time plot for Example 1. The line is passed through earlytime data points.

71 Normalized Pressure, (10 6 psi 2 /cp)/(mmscf/d) Linear Superposition Time, Days 0.5 Figure Normalized pressure versus linear superposition time plot for Example 1. The line is passed through early-time data points. Using the constant-rate method for calculating G (Equation (2.7)), combined with the subsequent 7-step procedure provided for analyzing variable rate/variable flowing pressure data, the calculated fracture half-length is x f =1,024 ft. Figure 2.18 shows a plot of normalized pressure versus linear superposition pseudo-time after x f k converges. The advantage of using Figure 2.18 for analysis is that the data form a straight line and there is no question about where to draw the line. It should be mentioned that even if we start with xf k calculated from the slope of the line passed through late-time data points in Step 1 of the 7-step procedure, x f k converges to the same value as that shown in Figure 2.18.

72 Normalized Pressure, (10 6 psi 2 / cp)/(mmscf / D) Linear Superposition Pseudo-Time, Days 0.5 Figure Normalized pressure versus linear superposition pseudo-time plot for Example 1. Finally, to demonstrate the importance of using the correct equation for calculating the gas-in-place in the region of influence at different times, Figure 2.19 shows a plot of normalized pressure versus linear superposition pseudo-time obtained using the parameters entered into the numerical simulation for this example. The difference between two datasets shown on this plot is the equation that is used to calculate the gasin-place in the region of influence. For constant flowing pressure, Equation (2.16) is used, whereas for constant rate, Equation (2.7) is used. This plot clearly shows that the data do not form a straight line throughout the linear flow when Equation (2.16) (for constant flowing pressure) is used for calculating the gas-in-place in the region of

73 45 influence. On the other hand, the data form a straight line when Equation (2.7) is used for calculating the gas-in-place in the region of influence. 200 Constant Rate Distance of Investigation Constant Pressure Distance of Investigation Normalized Pressure, (10 6 psi 2 /cp)/(mmscf/d) Linear Superposition Pseudo-Time, Days 0.5 Figure Normalized pressure versus linear superposition pseudo-time plot for Example 1. Equation (2.16) and Equation (2.7) are used to calculate the gas-in-place in the region of investigation for Constant Pressure and Constant Rate datasets, respectively. Example 2. The flowing pressure and gas rate data for this case are shown in Figure The rate is calculated from the input flowing pressure using simulation of the reservoir geometry shown in Figure 2.1 with p i =2,000 psi, T=120 ºF, h=300 ft, ϕ=6%, k=0.015 md, S g =100%, γ g =0.65, x f =320 ft, x e =640 ft, y e =3,500 ft and c f =0. The fracture half-lengths calculated (with permeability known) using Equation (1.1) and Equation (1.2) are x f =411 ft and x f =261 ft, respectively and the fracture half-length

74 46 calculated using Equation (2.24) is x f =368 ft. Therefore, Equation (2.16) (for constant flowing pressure) is used in Step 3 to calculate gas-in-place in the region of influence at different times. Figure 2.21 shows a plot of normalized pressure versus linear superposition pseudo-time after x f k converges. Using k=0.015 md, x f =319 ft is obtained from the plot, which is in a very good agreement with the expected value of x f =320 ft. 6 Gas Rate Flowing Pressure Gas Rate, MMscf/D Flowing Pressure, psi Time, Days Figure Gas rate and flowing pressure profile for Example 2.

75 Normalized Pressure, (10 6 psi 2 /cp)/(mmscf/d) Linear Superposition Pseudo-Time, Days 0.5 Figure Normalized pressure versus linear superposition pseudo-time plot for Example 2. Example 3. The flowing pressure and gas rate data for this case are shown in Figure The flowing pressures are obtained from a well completed in the Barnett shale, and the rates are calculated from the flowing pressures using simulation of the reservoir geometry shown in Figure 2.1 with p i =4,680 psi, T=120 ºF, h=300 ft, ϕ=8%, k=0.005 md, S g =100%, γ g =0.65, x f =900 ft, x e =1,800 ft, y e =4,000 ft and c f =0. The fracture half-lengths calculated (with permeability known) using Equation (1.1) and Equation (1.2) are x f =1,178 ft and x f =750 ft, respectively and the fracture half-length calculated using Equation (2.24) is x f =1,131 ft. Therefore, Equation (2.16) (for constant flowing pressure) is used in Step 3 to calculate gas-in-place in the region of influence at different times. Using the procedure proposed in this study, the fracture half-length is calculated to

76 48 be x f =900 ft, which is exactly the same as the value input to numerical simulation. Figure 2.23 shows the plot of normalized pressure versus linear superposition pseudo-time for this case after xf k converges. 40 Gas Rate Flowing Pressure Gas Rate, MMscf/D Flowing Pressure, psi Time, Days Figure Gas rate and flowing pressure profile for Example 3.

77 Normalized Pressure, (10 6 psi 2 /cp)/(mmscf/d) Linear Superposition Pseudo-Time, Days 0.5 Figure Normalized pressure versus linear superposition pseudo-time plot for Example Discussion Impact of Distance of Investigation on Constant Flowing Pressure Analysis Although, we have provided an analytical method for correcting the overestimation of xf k from Equation (1.1), the new method leads to an underestimation of x f k. In this section, the cause of this underestimation is investigated. Using Equation (2.15), the end of linear flow can be calculated from the following equation: ye 2 elf , (2.26) kt c g t i where y e is reservoir length and t elf is the duration of linear flow. Equation (2.26) can be rewritten as follows:

78 50 t elf ye gct i k 2. (2.27) Case 5 in Table 2.2 (p i =10,000 psi, p wf =3,000 psi, k=0.1 md and c f =0) is chosen to evaluate Equation (2.27). The end of linear flow is estimated to be 313 days for this case using Equation (2.27). Figure 2.24 shows pressure profile in the reservoir after 313 days, calculated from numerical simulation. Although Equation (2.27) predicts that the end of linear flow (or start of boundary-dominated flow) is after 313 days, this figure indicates that, the pressure propagation reaches the boundaries before 313 days (pressure at the boundary, distance of 2,500 ft from the fracture, is dropped from initial pressure of 10,000 psi to ~8,000 psi). Figure 2.25 shows the semi-log derivative versus time plotted in log-log coordinates. This figure also shows that the linear flow ended before 313 days for Case 5 in Table 2.2. Therefore, there is a possibility that Equation (2.15) underestimates the distance of investigation, which causes underestimation of the average pressure in the region of influence. This can explain underestimation of the correction factor using the method introduced in Equation (2.21).

79 51 10,000 9,000 8,000 Pressure, psi 7,000 6,000 5,000 4,000 3, ,000 1,500 2,000 2,500 Distance from Fracture, ft Figure Pressure distribution in the reservoir for Case 5 in Table 2.2 after 313 days (p i =10,000 psi, p wf =3,000 psi, k=0.1 md and c f =0). Distance of 2,500m from the fracture represents the reservoir boundaries parallel to the fracture.

80 t =180 days t =313 days Semilog Derivative, (10 6 psi 2 /cp)/(mmscf/d) Time, Days Figure Semi-log derivative plot for Case 5 in Table 2.2 (p i =10,000 psi, p wf =3,000 psi, k=0.1 md and c f =0). To evaluate Equation (2.15) for calculating the distance of investigation for constant flowing pressure boundary condition in the reservoir geometry shown in Figure 2.1, a number of test cases were built using numerical simulation. The input data for initial pressure, flowing pressure and permeability for these cases are given in Table 2.3. The formation compressibility is zero and the other parameters not listed in Table 2.3 are as follows: T=120 ºF, h=100 ft, ϕ=10%, S g =100%, γ g =0.65, x f =250 ft, x e =500 ft, y e =5,000 ft and c f =0. For each case, first, the end of linear flow is calculated using Equation (2.27). Then, the end of linear flow is obtained using the pressure distribution profile in the reservoir. The end of linear flow in this method is defined as the time at which the pressure drop at the upper and lower reservoir boundaries reach 10% of the maximum

81 53 pressure drop (i.e., p 01. p p ). For example, for p i =10,000 psi and p wf =3,000 i wf psi, the end of linear flow is assumed to be reached when the pressure at the boundaries reach 9,300 psi. Figure 2.26 shows a plot of the end of linear flow obtained using the pressure profile in the reservoir, t Equation (2.27), linear regression: elf o, versus the end of linear flow calculated using t. Based on this data, t is correlated to elf t t elf o c elf c elf o t by applying the. (2.28) elf c Table 2.3. Input parameters used for numerical simulation for different cases used to evaluate Equation (2.15). p i (psi) p wf (psi) k (md) 10,000 1, , , , , , ,000 1, , , , ,

82 (t elf ) O, days (t elf ) C, days Figure The end of linear flow obtained using the pressure profile in the reservoir, (t elf ) O, versus the end of linear flow calculated using Equation (2.27), (t elf ) C. Using Equation (2.28), the following equation can be obtained to calculate the distance of investigation: kt y 10% , (2.29) c g t i where y 10% is the distance that its pressure drop is 10% of the maximum pressure drop. Using Equation (2.29) instead of Equation (2.15) for derivation presented in the previous sections, Equation (2.22a) will be changed to the following equation:

83 55 p p ( Zgc i t) i( ppi ppwf) gct ** ** Z Zi Sgi pi gct i. (2.30) The comparison between analytically-calculated correction factors using Equation (2.30), instead of Equation (2.22a), and expected correction factors for different values of formation compressibilities are shown in Figures It can be seen that using Equation (2.30) significantly improves the analytically-calculated correction factors Calculated f CP Expected f CP Figure Comparison between correction factors calculated using the methods of this study (using Equation (2.30)) with expected correction factor for c f = 0.

84 Calculated f CP Expected f CP Figure Comparison between correction factors calculated using the methods of this study (using Equation (2.30)) with expected correction factor for c f = /psi.

85 Calculated f CP Expected f CP Figure Comparison between correction factors calculated using the methods of this study (using Equation (2.30)) with expected correction factor for c f = /psi Application to Ultra-Low Permeability Reservoirs Multi-fractured horizontal wells are the most commonly used method for exploiting shale gas reservoirs. Because massive hydraulic fractures are typically created, the dominant flow regime observed in these wells is linear flow to fractures that may last for a long time because of extremely low permeability of shale gas reservoirs. Therefore, it is of practical interest to validate the methodology presented in this study for application in multi-fractured horizontal wells. Linear flow analysis results in the product of total fracture length (or surface area to flow) and square root of permeability. If the permeability is known, the total fracture length can be obtained. This means that when analyzing linear flow for a hydraulically-

86 58 fractured vertical well, the length of the fracture can be obtained, whereas linear flow analysis for a multi-fractured horizontal well results in total fracture length (i.e., the summation of lengths of all the fractures). This is essentially the difference between analyzing linear flow for hydraulically-fractured vertical wells and horizontal wells with multiple fractures. Based on this discussion, Equations (1.1) and (1.2) can be used to calculate the product of total fracture length and square root of permeability using the slope of the data on square-root-of-time plot. For constant flowing pressure production, this value can be multiplied by the correction factor calculated using the method presented in this study or that calculated from Equation (1.3). The permeability for the test cases presented in Tables 2.2 and 2.3 are more appropriate for tight-gas reservoirs. To validate the applicability of the method presented in this study for shale gas reservoirs, a new test case was built which is the same as Case 24 in Table 2.2 except c f =0 and the permeability is reduced to k=100 nd. Using the slope of the square-root-of-time plot, which is not shown here, in Equation (1.1) and k=100 nd, the fracture half-length is calculated to be 325 ft. The correction factor obtained using the method presented in this study (i.e., Equation (2.21) using average pressure in the region of influence calculated from Equation (2.30)) for this case is Therefore, the improved fracture half-length is =250 ft, which is the same as the expected value of 250 ft (i.e., the input to numerical simulation). This demonstrates the validity of the method presented in this study for the permeability range expected for shale gas reservoirs. For comparison, the correction factor calculated using the Ibrahim and Wattenbarger method (Equation (1.3)), is 0.83, which corresponds to a fracture halflength of 270 ft.

87 Approximate Solution for Constant Flowing Pressure Analysis In this section, a simplified form of Equation (2.30) is obtained using the following assumptions, which are the same assumptions used to derive the simplified relationship between corrected pseudo-time and time for constant flowing pressure production: 1. The gas is ideal (Z=1). Using the definition of gas compressibility, this assumption leads to: c g 1 1 dz 1. (2.10) p Zdp p 2. Gas viscosity is not changing with pressure. Using the definition of pseudopressure, p 2 p ppi 2 dp pdp Z 2 i. (2.31) g gi gi 3. Total compressibility is dominated by gas compressibility, i.e., c t S c. (2.11) g g Equation (2.32) is the simplified form of Equation (2.30), which is obtained by combining Equations (2.10), (2.11), (2.30) and (2.31): ppi ppwf pi p p i( ). (2.32) p p pi As gas viscosity is assumed to be constant and gas compressibility is assumed to be inversely proportional to pressure, Equation (2.21) simplifies to: f CP c c p g t i ti. (2.33) gct ct pi Combining Equations (2.32) and (2.33), we will end up with the following equation:

88 60 3 CP CP D f f D 0, (2.34) where D D is the drawdown parameter defined in Equation (1.4). Equation (2.34) demonstrates that under the assumptions presented above, the correction factor, f CP, only depends on the drawdown parameter defined by Ibrahim and Wattenbarger (2005; 2006). In Figure 2.30, a comparison is made between the correction factor obtained from Equation (1.3) and Equation (2.34). This figure shows that the correction factors obtained using Equation (2.34) and the empirical values calculated from the correlation developed by Ibrahim and Wattenbarger (2005; 2006) are very close for an ideal gas with constant viscosity when gas compressibility dominates the total compressibility. 1 Equation (1.3) Equation (2.34) 0.95 Correction Factor Drawdown Parameter Figure Comparison between linear flow correction factors for constant flowing pressure obtained using Equation (1.3), dashed line, and those obtained using Equation (2.34), solid line.

89 Differences Between Constant Flowing Pressure and Constant Rate Production Linear Flow Comparing the findings between linear flow for constant rate and constant flowing pressure boundary conditions, the following differences are observed in gas reservoirs: 1. The average pressure in the region of influence is constant (Equation (2.17)) for constant flowing pressure, whereas the average pressure in the region of influence is time dependent and decreases with time for constant rate production. 2. The corrected pseudo-time has a linear relationship with time (Equation (2.18)) for constant flowing pressure production. However, for constant rate production, corrected pseudo-time is almost equal to time at early time and becomes smaller than time as time increases. In other words, the relationship between corrected pseudo-time and time is not linear for constant rate production. 3. For constant flowing pressure, the square-root-of-time plot is a straight line, whereas for constant rate production, the square-root-of-time plot may not be a straight line, if corrected pseudo-time is not used, and its shape depends on gas production rate. 4. For constant rate production, the proposed analysis method is iterative, whereas for constant flowing pressure, the procedure presented is not iterative and it only involves finding a correction factor Importance of Pseudo-Time In our analytical approach, pseudo-time is used to account for changing gas viscosity and gas compressibility ( g c g ) with pressure. The product of gas compressibility

90 62 and gas viscosity is the term that causes the diffusivity equation for gas to be non-linear, as it changes drastically with pressure. Figure 2.31 shows a plot of g c g versus pressure obtained using the gas properties used for the cases presented in this study. It is obvious from this figure why using pseudo-time is important in gas reservoirs. It should be mentioned that for a well operated such that changes in g c g throughout the reservoir are not significant, then using time as opposed to (corrected) pseudo-time may not introduce noticeable errors. This will happen when wells are producing under a low drawdown condition. On the other hand, for wells that are producing under high drawdown condition, it is expected that g c g changes significantly throughout the reservoir. In this case, if the change in g c g with pressure is not incorporated into the analysis (i.e., using time as oppose to (corrected) pseudo-time), then a larger fracture half-length and/or permeability is required to match the production data to account for the extra energy caused by increasing gas compressibility with (partial) reservoir deletion. We therefore recommend using pseudo-time for analyzing multi-fractured horizontal wells in tight/shale gas reservoirs because most of these wells are producing under high drawdown to maximize the production. For rate-restricted wells, one can look at the plot of g c g with pressure and (based on the initial pressure and flowing pressure data) decide whether it is necessary to use pseudo-time or not.

91 63 1.E-03 1.E-04 g c g, cp psi -1 1.E-05 1.E Pressure, psi Figure A plot of g c g versus pressure obtained from the gas properties used for the cases presented in this study. 2.6 Summary In this chapter, transient linear flow of gas from a well producing at a constant rate, constant flowing pressure or variable rate/pressure was studied. First, the constant rate case was studied. It was shown analytically that gas rate can drastically affect the shape of the square-root-of-time plot and the conventional square-root-of-time plot (using real time) may not be straight line during linear flow. The rate dependence of the square-rootof-time plot was ascertained using a number of numerically-simulated cases. Corrected pseudo-time was then used to correct the square-root-of-time plot to a straight line, simply by plotting normalized pressure against square root of corrected pseudo-time. Using corrected pseudo-time drastically reduced the dependence of square-root-of-time

92 64 plot on gas rate. A method for calculating x f k from the square-root-of-time plot for constant gas rate was presented. The values for xf k obtained from this method were in good agreement with the entries to numerical simulation. An analytical method to calculate a correction factor for calculating xf k from the slope of square-root-of-time plot for constant flowing pressure was then presented. It was demonstrated that for the reservoir geometry shown in Figure 2.1, the average reservoir pressure in the region of influence is constant during linear flow. The method was then validated with numerical simulation. It was found that, in general, the correction factors obtained using the method proposed (using Equation (2.15) for calculating distance of investigation) are lower than those expected. It was also found that Equation (2.15) underestimates the distance of investigation for linear flow. This equation was then modified to match the distance of investigation observed in simulation. Using the modified equation for distance of investigation, the analytically-calculated correction factors were in good agreement with those expected. Then, it was shown that for a case of ideal gas with constant viscosity, if total compressibility is dominated by gas compressibility, the correction factor only depends on the drawdown parameter defined in Equation (1.4). Finally, a method was presented to analyze linear flow for real production data, where the flowing pressure and gas rate are changing with time. The method was validated using three numerically-simulated cases. It was found that this method works well for these three cases.

93 65 CHAPTER 3 IMPROVED LINEAR FLOW ANALYSIS: EFFECTS OF GAS SLIPPAGE AND DESORPTION Abstract Multi-fractured horizontal wells are currently the most popular method for exploiting low-permeability tight and shale gas reservoirs. Production data analysis is the most widely used tool for analyzing these reservoirs for the purpose of reserves estimation, hydraulic fracture stimulation optimization, and development planning (Ambrose et al., 2011). However, as pointed out by Clarkson et al. (2012), a fundamental problem with the application of conventional production data analysis to ultra-low permeability reservoirs is that current methods were derived with the assumption that flow can be described with Darcy's law. This assumption may not be valid for tight/shale gas reservoirs, as they contain a wide distribution of pore sizes, including in some cases nanopores (Loucks et al., 2009). Therefore, the mean-free path of gas molecules may be comparable to or larger than the average effective rock pore throat radius causing the gas molecules to slip along pore surfaces. This results in gas slippage (non-darcy flow), which is not accounted for in conventional production data analysis. Clarkson et al. (2012) modified the pseudo-variables used for analyzing gas reservoirs in production data analysis to account for slippage. They demonstrated that if the effect of slippage is not considered, it leads to noticeable errors in reservoir 1 This chapter is the modified version of Nobakht, M., Clarkson, C.R. and Kaviani, D New and Improved Methods for Performing Rate-Transient Analysis of Shale Gas Reservoirs. SPE Reservoir Evaluation & Engineering, 15 (3):

94 66 characterization. Clarkson et al. (2012) also mentioned that even after using the modified pseudo-variables, the values for permeability and fracture half-length do not exactly match the input data to simulation. In this paper, a methodology to properly analyze the production data from a fractured well in tight/shale gas reservoir producing under a constant flowing pressure in the presence of desorption and slippage is presented. This method uses new pseudo-time definition instead of conventional pseudo-time currently being used in production data analysis. The method is validated using a number of numerically-simulated cases. It is found that the newly-developed analytical method results in a more reliable estimate of fracture half-length or contacted matrix surface area, if permeability is known. 3.2 Introduction The analysis of linear flow has gained significant attention in the last decade because this is often the dominant transient flow regime observed for horizontal wells (cased or open hole) with multiple fractures. Nobakht and Clarkson (2012b) studied the transient linear flow for constant flowing pressure production in detail and explained that overestimation of x f k using the slope of square-root-of-time plot in Equation (1.1) is due to the fact that the square-root-of-time plot does not account for changing gas viscosity and gas compressibility, which are incorporated into pseudo-time. They therefore developed an analytical method to correct for overestimation of x f k calculated from conventional linear flow analysis. Clarkson et al. (2012) noted that the common assumptions used for the development of conventional production data analysis are often not true for

95 67 unconventional reservoirs with extremely low permeability. One of the limitations pointed out by Clarkson and co-workers is the existence of gas-slippage (non-darcy flow) in low-permeability reservoirs. In low-permeability reservoirs, the gas molecules may slip along the pore surfaces (i.e., the gas velocity at pore surfaces is not zero) and cause a flux in addition to that due to the viscous flow expressed by Darcy s law. Because of this additional flux, apparent gas permeability, k a, becomes higher than the liquid-equivalent permeability, k, of the same porous medium. Clarkson et al. (2012) used the following pseudo-pressure, Equation (3.1), and pseudo-time, Equation (3.2), to include slippage effect in production data analysis: k, (3.1) i r ppi ppwf 2 p pdp pwf gz t kdt r t a gct i 0 c. (3.2) g t In these equations, * indicates modified variables to account for slippage and k r is permeability ratio defined as the ratio of apparent permeability, k a, to the liquidequivalent permeability, k. The gas-slippage effect is commonly corrected for using the Klinkenberg slippage factor (Klinkenberg, 1941), b: k a k 1 b p m. (3.3) Here, p m is the mean pressure. Klinkenberg (1941) showed that k obtained from this relationship is in fact the true permeability of the reservoir (or sample). Using Equation (3.3), k r becomes: k k b a r 1 k p. (3.4) m

96 68 Traditionally, the slippage factor is considered to be constant (Klinkenberg, 1941; Jones and Owens, 1979; Heid et al., 1950; Sampath and Keighin, 1982). Assuming the gas transport is controlled by concentration (Fick s law) and pressure (Darcy s law), Ertekin et al. (1986) developed an apparent or dynamic gas slippage factor, b a. For single phase flow, the following equation was developed to calculate apparent slippage factor in field units (Ertekin et al., 1986): b a pcggd. (3.5) k In this equation, c g is gas compressibility and D is the diffusion coefficient. From Equation (3.5), it is expected that the apparent gas slippage factor depends on pressure, temperature and gas properties. Javadpour (2009) showed that sensitivity of the apparent gas permeability to temperature is negligible. For the purposes of this study, the following equation will be used to calculate k r and include the dynamic-slippage factor: k r 1 b a. (3.6) p Ertekin et al. (1986) related the gas diffusion coefficient to gas molecular weight, M g, and liquid-equivalent permeability, k, using the following equation in field units: D k M. (3.7) g Recently, Javadpour (2009) and Civan (2010) introduced rigorous methods for calculating apparent permeability change as a function of Knudsen number. In these methods, the dominant flow regime (continuum flow, slip flow, transitional flow or free molecular flow) is predicted as a function of pore size, pressure, temperature and gas properties. Note that in these methods, a dynamic slippage factor can be calculated using

97 69 k r, i.e., the ratio of apparent permeability to liquid-equivalent permeability, in Equation (3.6) and solving for b a (Civan, 2010; Clarkson et al., 2012). Clarkson et al. (2012) mentioned that even after using the modified pseudovariables defined in Equations (3.1) and (3.2), the values for permeability and fracture half-length do not exactly match the input data to simulation. In this chapter, a methodology to improve analysis of production data from a fractured well in a tight/shale gas reservoir, producing under a constant flowing pressure constraint in the presence of desorption and slippage, is presented. The method is validated using a number of numerically-simulated cases. It is found that the newly-developed analytical method results in a more reliable estimate of fracture half-length (or contacted matrix surface area in the case of complex fracturing), if permeability is known. 3.3 Derivation In this section, a methodology to analyze transient linear flow from a fractured well in a tight/shale gas reservoir producing with a constant flowing pressure in the presence of desorption and slippage is presented. This method represents an extension of the Nobakht and Clarkson (2012b) method to account for not only changing gas properties with pressure, but also desorption and slippage, by incorporating pseudo-time into analysis. As with the previous work, the pressure-dependent properties incorporated into pseudo-time are evaluated at the average pressure in the region of influence. The base reservoir geometry used is shown in Figure 3.1, which is the same as that used in Chapter 2 (i.e., a hydraulically-fractured vertical well with infinite fracture conductivity and no skin in the middle of a rectangular reservoir).

98 70 2y Fracture y e x e Figure 3.1. A hydraulically-fractured vertical well in the center of a rectangular reservoir. Linear flow theory (Wattenbarger et al., 1998; El-Banbi and Wattenbarger, 1998) indicates that at a constant flowing pressure, a plot of inverse gas rate versus t in Cartesian coordinates is a straight line. The slope of this line can be used to calculate xf k, where x f is fracture half-length and k is the permeability, using the following equation for gas (Wattenbarger et al., 1998; El-Banbi and Wattenbarger, 1998): x f k 315.T h c p p m g t i pi pwf CP. (3.8) In this equation, T is the reservoir temperature, h is the net pay thickness, ϕ is the reservoir porosity, µ g is gas viscosity, c t is total compressibility (subscript i refers to initial conditions) and p pi and p pwf are pseudo-pressures at initial reservoir pressure and flowing pressure, respectively and m CP is the slope of 1 q versus t plot. In the presence of slippage, the following equation should be used instead of Equation (3.8):

99 71 x f k 315.T m h gc p t pi p pwf i CP, (3.9) where k is the liquid-equivalent permeability and * p pi and * p pwf are the modified pseudopressures to account for slippage at initial reservoir pressure and flowing pressure, respectively, calculated using Equation (3.1). As it is assumed that the fracture has infinite conductivity and there is no skin effect - linear flow for constant flowing pressure can therefore be represented by: 1 mcp t q. (3.10) Using Equation (3.10), the cumulative production at time t is: CP 3 3 t 210 Gp 10 qdt t. (3.11) 0 m The unit of q in Equation (3.10) is Mscf/D and the conversion factor of 10 3 is used in Equation (3.11) to convert G p to scf. When a well is producing under constant flowing pressure, the distance of investigation, y, can be obtained from the following equation during the linear flow period (Wattenbarger et al., 1998): y. ai 0 159, (3.12) k t c g t i Where k ai is the apparent permeability at initial pressure. Apparent permeability at initial pressure, k ai, is used in this equation because the pressure propagation is occurring against initial pressure and therefore, permeability at initial pressure is used in Equation (3.12) to calculate the distance of investigation. Using the definition of k ai, kai k k ri, Equation (3.12) becomes:

100 72 y. ri (3.13) k k t c g t i Here, k ri is the permeability ratio at initial pressure and k is the liquid-equivalent reservoir permeability. The contacted gas-in-place (i.e., gas-in-place in the region of influence) including desorption is: Sgi VLp i G Ah B. (3.14) B gi p L p i Here, A is the area of the region of influence, S gi is initial gas saturation, B gi is initial gas formation volume factor, ρ B is bulk density, V L is Langmuir volume, p L is Langmuir pressure and p i is initial reservoir pressure. It is assumed that the gas content follows the Langmuir isotherm (Yang 1987). Equation (3.14) can be represented as: Ah S G, (3.15) B gi * gi where * B gi is gas formation volume factor, adjusted to account for desorption effect, and is defined as (King, 1993; Clarkson et al., 2007): 1 1 B VLpi B B S p p * gi gi gi L i. (3.16) Using the definition of area of the region of influence, A 2xey 4xfy, and replacing y from Equation (3.13) results in: h Sgixf kkri G t. (3.17) B c * gi g t i The unit of G in this equation is scf. The average pressure in the region of influence, p, can be calculated using the following equation (Moghadam et al., 2011):

101 73 Z p p Gp (1 ) G. (3.18) i ** ** Zi Here, ** Z and i ** Z are modified Z-factors introduced by Moghadam et al. (2011) at average pressure in the region of influence and initial reservoir pressure, respectively. Substituting G p and G from Equation (3.11) and Equation (3.17), respectively, into Equation (3.18) leads to: * 2000Bgi gct i i ** i CP gi f ri p p (1 ). (3.19) ** Z Z. m h S x k k This equation shows that the average pressure in the region of influence is not timedependent for constant flowing pressure production. Nobakht and Clarkson (2012b) reported this finding in the absence of slippage and desorption. As the average pressure in the region of influence is constant, using Equation (3.2), the corrected pseudo-time, t * a, becomes: k c t. (3.20) * r g t i a gct t We see from Equation (3.20) that corrected pseudo-time has a linear relationship with time. Equation (3.20) also leads to the conclusion that the slope of 1 q versus * t a plot, m CP, and the slope of 1 q versus t plot, m CP, have the following relationship: m k c r g t i CP mcp. (3.21) gct

102 74 In order to obtain an accurate value for xf k when gas is being analyzed, the slope of 1 q versus * t a plot, m CP, should be used in Equation (3.9) (Nobakht and Clarkson, 2012b). Therefore, through application of Equation (3.21), the following equation can be used to calculate xf k from the slope of 1 q versus t plot, m CP: x f 315.T kr gct i h gct p pi p pwf mcp gct i k. (3.22) Comparing Equation (3.9) and Equation (3.22), the correction factor, f CP, that should be applied to improve the estimate of x f k calculated from the slope 1 q versus t plot is: f CP k c r g t i. (3.23) gct Note that Equation (3.23) accounts for changing gas properties with pressure, slippage and desorption. Substituting x f k from Equation (3.22) and * Z * gi i T B into p Equation (3.19) results in: p ** Z Z kk c * p ( Z gc i t) i( ppi ppwf) gc t ** i Sgi pi ri r g t i. (3.24) i Here, * Z is gas compressibility factor, adjusted to account for desorption effect (King, 1993; Clarkson et al., 2007). Assuming oil, water and formation compressibilities are

103 75 negligible, water influx is negligible and gas saturation=100%, Clarkson et al., 2007): * Z becomes (King, 1993; Z Z* ZTpscVL B 1 Z T p + p sc sc L. (3.25) In this equation, p sc, Z sc and T sc are pressure, gas compressibility factor and temperature at standard conditions, respectively. Equation (3.24) shows that the average pressure in the region of influence depends on initial pressure, flowing pressure, reservoir temperature, gas properties, gas saturation and k r defined in Equation (3.6). Equation (3.24) can be solved to obtain average pressure in the region of influence and then the correction factor, f CP, can be calculated using Equation (3.23). To improve linear flow analysis, xf k calculated from Equation (3.9) can be multiplied by f CP. This is similar to Nobakht and Clarkson (2012b) procedure. Note that in the absence of slippage (i.e., k r =1) and desorption (i.e., * Z Z ), Equation (3.24) is identical to Equation (2.22a). If k r =1, Equation (3.24) becomes independent of reservoir permeability and therefore, correction factor calculated from Equation (3.23) is independent of permeability. In the presence of slippage, the permeability ratio calculated from Equation (3.6) depends on reservoir permeability (liquid-equivalent) and as a result, average pressure in the region of influence calculated from Equation (3.24) depends on the reservoir liquid-equivalent permeability. 3.4 Validation To validate the methodology proposed in this study for analyzing linear flow in tight/shale gas reservoirs, a number of test cases were built using numerical simulation.

104 76 The common parameters among all the test cases are as follows: p i =2,000 psi, T=120 ºF, h=100 ft, ϕ=10%, S g =100%, γ g =0.65, x f =250 ft, x e =500 ft, y e =5,000 ft and c f =0. For cases in which desorption is included, V L =89 scf/ton, p L =540 psi and ρ B =2.47 g/cm3 are used as Langmuir volume, Langmuir pressure and shale bulk density, respectively. The input data for flowing pressure and permeability for the numerical simulation cases are given in Table 3.1. Gas slippage is incorporated in the simulation using permeability ratio, k r, calculated at different pressures from Equations (3.5) (3.7), as pressure-dependent transmissibility multipliers (as per Clarkson et al., 2012). For cases in which slippage is not considered, k r =1 is entered as transmissibility multiplier. A plot of k r versus pressure for k=0.01 md, k=0.001 md and k= md is shown in Figure 3.2. As expected, at a constant permeability, the permeability ratio increases by reducing the pressure. This figure also shows that for a given pressure, the permeability ratio increases by reducing the permeability.

105 77 Table 3.1. Input parameters to numerical simulation for different cases used to validate methodology presented in this study. Case p wf (psi) k (md) Slippage Desorption Yes No Yes No Yes No Yes No Yes No Yes No No No No No No No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes No Yes No Yes No Yes No Yes No Yes

106 k= md k=0.001 md k=0.01 md 2.2 Permeability Ratio Pressure, psi Figure 3.2. Permeability ratio calculated using Equation (3.6) for three different permeabilities. For each case, first, the average pressure in the region of influence is calculated from Equation (3.24) and the correction factor is calculated using Equation (3.23). Then, xf k calculated using Equation (3.9) (using the slope of 1 q versus t plot) is multiplied by the calculated correction factor. Finally, x f is calculated for each case using the permeability entered in the simulation (i.e., liquid-equivalent permeability). The calculated x f using the method developed in this study for Cases 1 21 are shown in Figure 3.3. The calculated values for fracture half-length vary between 240 ft and 247 ft, which are very close to the value entered into numerical simulation (i.e., x f 250 ft ). However, all the calculated fracture half-lengths are less than the expected value.

107 79 Nobakht and Clarkson (2012b) observed the same result and concluded that this is due to underestimation of distance of investigation calculated using Equation (3.12). They showed that a more accurate estimate of fracture half-length can be obtained using the following equation for distance of investigation: y. ai (3.26) k t c g t i Using Equation (3.26) instead of Equation (3.12) for the derivation of the method presented in this study, Equation (3.24) changes to: p ** Z Z kk c * p ( Z gc i t) i( ppi ppwf) gc t ** i Sgi pi ri r g t i. (3.27) The fracture half-lengths calculated using Equation (3.27) for Cases 1 21 are shown in Figure 3.3 and vary between 250 ft and 259 ft. Although the calculated fracture half-lengths using both Equation (3.24) and Equation (3.27) are very close to the expected value of x f 250 ft, it seems that overall the fracture half-lengths obtained using Equation (3.27) are in better agreement with the expected value. Therefore, for the rest of this paper, we will use Equation (3.27) instead of Equation (3.24) for calculating the average pressure in the region of influence.

108 Equation (3.24) Equation (3.27) 250 Calculated Fracture Half-Length, ft Case Number Figure 3.3. Fracture half-lengths calculated using the methodology presented in this study (with both Equation (3.24) and Equation (3.27)) for different numerically-simulated cases. The expected value of x f =250 ft is shown on the plot by the dotted horizontal line. 3.5 Discussion Effect of Slippage on Linear Flow Analysis The simulated Cases 1 6, in which there is no desorption effect, were analyzed by the following methods: Method 1: Time is used for linear flow analysis, not corrected pseudo-time, and slippage is not incorporated into pseudo-pressure. In this method, Equation (3.8) is used. Method 2: Corrected pseudo-time, with k r =1, is used for linear flow analysis. Slippage is not incorporated into pseudo-pressure and pseudo-time. This is the Nobakht and Clarkson (2012b) method.

109 81 Method 3: Time is used for linear flow analysis, not corrected pseudo-time, and slippage is incorporated into pseudo-pressure. In this method, Equation (3.9) is used. Method 4: Ibrahim and Wattenbarger (2005; 2006) approach is used. Time is used for linear flow analysis, not corrected pseudo-time, and slippage is not incorporated into pseudo-pressure. Fracture half-lengths calculated from Equation (3.8) are multiplied by correction factor calculated from Equation (1.3). Method 5: Corrected pseudo-time is used for linear flow analysis. Slippage is incorporated into both pseudo-pressure and pseudo-time. This method is the proposed method in this study. As there is no desorption for Cases 1 6, Z * =Z is used in Equation (3.27). The fracture half-lengths calculated using these methods are listed in Table 3.2 and also shown in Figure 3.4 (for Cases 1 3) and Figure 3.5 (for Cases 4 6). These figures show that the fracture half-lengths calculated from Method 1 are much higher than the expected value, which is 250 ft for all cases. Also, as the permeability decreases, the degree of overestimation of fracture half-length calculated from this method is increasing. This is because in Method 1, the slippage is not considered in the analysis and as the permeability decreases, the slippage has a greater effect on the production rate. In Method 2, the Nobakht and Clarkson (2012b) method, in which k r =1, is used to calculate fracture half-length. Figure 3.4 and Figure 3.5 show that as permeability increases, fracture half-length calculated from this method becomes closer to the expected value. This is because the slippage effect becomes smaller when permeability

110 82 increases. As this method does not incorporate slippage effects and uses corrected pseudo-time with k r =1, it can be concluded that at higher permeabilities, using corrected pseudo-time, which accounts for changing gas viscosity and compressibility with pressure, is more important than incorporating slippage into analysis. In Method 3, slippage is incorporated into pseudo-pressure and time (not corrected pseudo-time) is used for analysis. Figure 3.4 and Figure 3.5 show that as permeability decreases, fracture half-length calculated from this method becomes closer to the expected value. This is opposite to our observation for Method 2. From this, it can be concluded that at lower permeabilities, pseudo-pressure correction to account for slippage is more important than using corrected pseudo-time, which does not account for slippage. This was expected as at lower permeabilities, the slippage effect becomes stronger and ignoring it significantly affects the analysis. In Ibrahim and Wattenbarger (2005; 2006) approach (i.e., Method 4), gas slippage is not considered in linear flow analysis and the slope of inverse gas rate versus squareroot-of-time plot is used in Equation (3.8). Calculated fracture half-length is then multiplied by correction factor calculated from Equation (1.3). It can be seen from Figure 3.4 and Figure 3.5 that the fracture half-lengths calculated from this method are in good agreement with the input value to numerical simulation (i.e., x f 250 ft ) at k 001. md. However, like Method 1, as the permeability decreases, this method overestimates the fracture half-length. This is expected, as the slippage has more contribution to production at a lower permeability and slippage is not incorporated into this method.

111 83 Finally, Figures 3.4 and 3.5 illustrate that among the five methods presented, Method 5 produces fracture-half lengths that more closely match simulation input ( x f 250 ft ) for all permeability levels. It should be mentioned that although we have used the dynamic-slippage approach suggested by Ertekin et al. (1986) as the method for calculating apparent gas permeability, our approach for correcting slippage effect is general enough that it only needs the apparent gas permeability as a function of pressure. Therefore, any other proposed method that predicts apparent gas permeability as a function of pressure (for example, Javadpour, 2009; Civan, 2010) can be used. The Ertekin et al. (1986) method was used in this study due to its simplicity.

112 84 Table 3.2. Comparison among fracture half-lengths calculated using five different methods for Cases 1 6. Case p wf (psi) k (md) x f1 (ft) x f2 (ft) x f3 (ft) x f4 (ft) x f5 (ft) x f1 : Fracture half-length calculated from Method 1. No correction for pseudo-pressure to account for slippage, and time is used for linear flow analysis, not corrected pseudotime. x f2 : Fracture half-length calculated from Method 2. No correction for pseudo-pressure and pseudo-time to account for slippage. Corrected pseudo-time is used for linear flow analysis (i.e., Nobakht and Clarkson 2012b method). x f3 : Fracture half-length calculated from Method 3. Correction for pseudo-pressure to account for slippage and time is used for linear flow analysis, not corrected pseudotime. x f4 : Fracture half-length calculated from Method 4. No correction for pseudo-pressure to account for slippage and time is used for linear flow analysis, not corrected pseudotime. Ibrahim and Wattenbarger (2005; 2006) correction factor defined in Equation (1.3) is applied. x f5 : Fracture half-length calculated from Method 5. Correction for both pseudo-pressure and pseudo-time to account for slippage and corrected pseudo-time is used for linear flow analysis.

113 Method 1 Method 2 Method 3 Method 4 Method 5 Calculated Fracture Half-Length, ft Case Number Figure 3.4. Comparison of fracture half-lengths calculated using different methods for Cases 1 3. The expected value of x f =250 ft is shown on the plot by the dotted horizontal line.

114 Method 1 Method 2 Method 3 Method 4 Method 5 Calculated Fracture Half-Length, ft Case Number Figure 3.5. Comparison of fracture half-lengths calculated using different methods for Cases 4 6. The expected value of x f =250 ft is shown on the plot by the dotted horizontal line Effect of Desorption on Linear Flow Analysis To properly analyze linear flow in the presence of desorption, the effect of desorption needs to be incorporated into total compressibility which in turn is used in calculating corrected pseudo-time; no correction is required for pseudo-pressure. Assuming oil, water and formation compressibilities are negligible, water influx is negligible and gas saturation is 100%, the total compressibility at any pressure, p, in the presence of desorption becomes (Bumb and McKee, 1988; Clarkson et al., 2007; Moghadam et al., 2011): BV p c. (3.28) B g L L t cg pl + p

115 87 This equation is used in this study for the cases that desorption is included. To study the effect of desorption on linear flow analysis, the simulated Cases were analyzed by the following methods: Method 1: Time is used for linear flow analysis, not corrected pseudo-time. Method 2: Corrected pseudo-time is used for analysis. However, the effect of desorption is ignored. This is the Nobakht and Clarkson (2012b) method. Method 3: Corrected pseudo-time with desorption effect, which is the proposed method in this study, is used for analysis. As there is no slippage effect for Cases 16 21, k r =1 is used in Equation (3.27). The fracture half-lengths calculated using these methods are listed in Table 3.3 and also shown in Figure 3.6 (Cases 16 18) and Figure 3.7 (Cases 19 21). These figures clearly demonstrate that the fracture half-lengths calculated from Method 1 are higher than the expected value of 250 ft. This is because in this method, changing gas compressibility with pressure and the effect of desorption are ignored. These figures also show that for each flowing pressure, the degree of overestimation is not changing with permeability.

116 88 Table 3.3. Comparison among fracture half-lengths calculated using three different methods for Cases Case p wf (psi) k (md) x f1 (ft) x f2 (ft) x f3 (ft) x f1 : Fracture half-length calculated from Method 1. Time is used for linear flow analysis, not corrected pseudo-time. x f2 : Fracture half-length calculated from Method 2. Corrected pseudo-time is used for analysis. However, the effect of desorption is ignored. This is the Nobakht and Clarkson (2012b) method. x f3 : Fracture half-length calculated from Method 3. Corrected pseudo-time with desorption effect, which is the proposed method in this study with k r =1, is used for analysis.

117 Method 1 Method 2 Method 3 Calculated Fracture Half-Length, ft Case Number Figure 3.6. Comparison among fracture half-lengths calculated using different methods for Cases The expected value of x f =250 ft is shown on the plot by the dotted horizontal line.

118 Method 1 Method 2 Method 3 Calculated Fracture Half-Length, ft Case Number Figure 3.7. Comparison among fracture half-lengths calculated using different methods for Cases The expected value of x f =250 ft is shown on the plot by the dotted horizontal line. Comparing the fracture half-lengths calculated using Method 2 and Method 3 in Figure 3.6 and Figure 3, we see that, although both methods slightly overestimate the fracture half-lengths, they result in fracture half-lengths that are in good agreement with the input value to the numerical model. Method 2 overestimates the fracture half-length by approximately 9% and 5% for Cases and Cases 19 21, respectively. Method 3 results in fracture half-lengths close to 250 ft (overestimation by nearly 3% and 0.5% for Cases and Cases 19 21, respectively). The reason that fracture half-lengths from Method 2 are only 9% higher than the expected value of 250 ft, although this method does not incorporate desorption, is the small contribution of desorption to the total

119 91 compressibility for Cases at the average pressure in the region of influence. The small contribution of desorption to total compressibility is due to low initial reservoir pressure (large gas compressibility), relatively low Langmuir volume and the fact that the average pressure in the region of influence is higher than Langmuir pressure. To study the effect of Langmuir volume on the fracture half-lengths obtained from the above-mentioned three methods, a new simulation case is generated which is identical to Case 16, except V L =267 scf/ton (which is three times of that in Case 16). The fracture half-lengths calculated for the new case using Method 1, Method 2 and Method 3 are 340 ft, 302 ft and 261 ft, respectively. This shows that Method 1, Method 2 and Method 3 overestimate the fracture half-length by 36%, 21% and 4%, respectively, for the new case. Clearly, the fracture half-length calculated using Method 3 (i.e., the method proposed in this study) is in a very good agreement with the expected value Effect of Slippage on Gas Production As mentioned previously, in low-permeability reservoirs, the gas molecules may slip along the pore surfaces and causes additional flux on top of the viscous flow expressed by Darcy s law. Because of this additional flux, for the same porous medium, the gas production rate in the presence of slippage is higher than that when slippage is not present. To study the effect of slippage on gas production, Cases 7 9 in which the slippage effect was ignored for simulation (i.e., k r =1) are compared with Cases 1 3. A comparison between the gas rates with slippage effect and without slippage effect for different values of permeability is shown in Figures It can be seen from each of these figures that gas rate is higher in the presence of slippage. Figures also

120 92 show that as permeability decreases, the slippage contribution to production increases. This is because the slippage effect becomes stronger as the permeability decreases. For example, the slippage contributes 5% to gas rate at k=0.01 md, 10% at k=0.001 md and 20% at k= md. 10 With Slippage No Slippage Gas Rate, MMscf/D Time, Days Figure 3.8. Comparison between gas rates in the presence and in the absence of slippage for p i =2,000 psi, p wf =200 psi and k=0.01 md (Cases 1 and 7).

121 93 10 With Slippage No Slippage 1 Gas Rate, MMscf/D Time, Days Figure 3.9. Comparison between gas rates in the presence and in the absence of slippage for p i =2,000 psi, p wf =200 psi and k=0.001 md (Cases 2 and 8).

122 94 1 With Slippage No Slippage Gas Rate, MMscf/D Time, Days Figure Comparison between gas rates in the presence and in the absence of slippage for p i =2,000 psi, p wf =200 psi and k= md (Cases 3 and 9). In practice, a model (numerical or analytical) is calibrated to match the production data (i.e., history matching). The calibrated model will then be used to forecast the production rate. To study the effect of ignoring slippage on long-term production forecast when history matching is conducted, one test case was built using numerical simulation of a 10,000-day production profile (in the presence of slippage). The parameters used for this case are as follow: p i =2,000 psi, T=120 ºF, h=100 ft, ϕ=10%, S g =100%, γ g =0.65, k= md (or k=100 nd), c f =0, x f =250 ft, x e =500 ft, y e =100 ft and p wf =200 psi. Then the first year of the synthetic production was used to calibrate a numerical model that does not account for slippage. All the parameters entered in the new numerical model were the same as those entered into the numerical model used to generate the 10,000-day

123 95 production profile, except for permeability. The permeability of k=140 nd was obtained by finding the best match of the numerical model results to production rates of the original numerical model. As expected, the permeability obtained from the history matching when slippage was ignored in the numerical model is higher than the input value to the original numerical model. A comparison between the rates obtained from original numerical model (with k=100 nd and slippage effect) with those obtained from the calibrated model (with k=140 nd and no slippage effect) for the first year of production is shown in Figure The calibrated numerical model was then used to generate the 10,000-day production profile, which is shown in Figure This figure shows that practically, there is only a small difference in the rates obtained from original numerical simulation with slippage effect and those obtained from calibrated model (without slippage). Cumulative production for these two cases is also shown in Figure 3.13; again there is very little difference between the two cases.

124 96 1 k=100 nd-with Slippage k=140 nd-no Slippage Gas Rate, MMscf/D Time, Days Figure Comparison between gas rates obtained from numerical simulation with slippage effect and those obtained from calibrated numerical model without slippage effect. Only the first year data is used for history matching.

125 97 1 k=100 nd-with Slippage k=140 nd-no Slippage 0.1 Gas Rate, MMscf/D Time, Days Figure Comparison between gas rates obtained from numerical simulation with slippage effect and those obtained from calibrated numerical model without slippage effect.

126 k=100 nd-with Slippage k=140 nd-no Slippage Cumulative Production, MMscf Time, Days Figure Comparison between cumulative production obtained from numerical simulation with slippage effect and that obtained from calibrated numerical model without slippage effect. 3.6 Impact of Distance of Investigation In this chapter, we used the distance of investigation equation (Equation (3.12) and Equation (3.26)) for calculating the gas-in-place in the region of influence. The derived gas-in-place is then used in material balance calculation (Equation (3.18)) to estimate the average pressure in the region of influence. Therefore, the distance of investigation for the purpose of this study should represent the portion of the reservoir that contributes relatively significant to the total production. It should be mentioned that this is different from the range that pressure influence reaches, as this range is infinite. To understand why Equation (3.26) provides a more representative distance of investigation compared to Equation (3.12) (for this study), the pressure profile for a well

127 99 producing at a constant flowing pressure for the reservoir geometry in Figure 3.1 is presented in Appendix. It was found that for constant flowing pressure production of a slightly compressible fluid in the reservoir geometry shown in Figure 3.1, Equation (3.12) represents the position of points at which the pressure has dropped 16% of the maximum pressure drop (i.e., p i p wf ). In other words, the gas-in-place in the region of influence calculated based on Equation (3.12) represents the portion of the reservoir in which its pressure has dropped more than 16% of the maximum pressure drop. This can explain why Nobakht and Clarkson (2012b) used a larger distance of investigation compared to Equation (3.12). On the other hand, the gas-in-place in the region of influence calculated based on Equation (3.26) represents the portion of the reservoir for which its pressure has dropped more than 7% of the maximum pressure drop. This indicates that based on the pressure profile in the reservoir, Equation (3.26) results in more representative gas-in-place for using in material balance calculations compared to Equation (3.12). Note that using Equation (3.26) by Nobakht and Clarkson (2012b), fracture half-lengths calculated from linear flow analysis matched well to those input to numerical simulation. As mentioned above, for the purpose of material balance, the distance of investigation should provide a region that has a relatively significant contribution to the total production. To explain this mathematically, we define a dimensionless parameter Q fyd using Equation (3.29), which is a measure of the relative contribution of region between the fracture and y D to the total production: Q fyd yd 0 0 p dz D p dz D D D. (3.29)

128 100 Using the relation of p D and y D from Equation (A.1) in Appendix, Q fyd becomes: yd erfc( zd) dzd 2 0 y D fyd D D Q y erfc( y ) e 1. (3.30) erfc( z ) dz 0 D D A plot of Q fyd versus y D is plotted in Figure Based on this figure, Q fyd is 0.91 and 0.97 for y D =1 (Equation (3.12) for distance of investigation) and y D =1.28 (Equation (3.26) for distance of investigation), respectively. This indicates that Equation (3.12) represents a region that contributes 91% to the production. On the other hand, the region that contributes 97% to the total production is represented by Equation (3.26). This also shows why Equation (3.12) underestimates the gas-in-place in the region of influence for the purpose of material balance calculation Q fyd y D Figure A plot of Q fyd, defined in Equation (3.30), versus y D, defined in Equation (A.3).

129 Assumptions/Limitations The methodology presented in this chapter is based on some assumptions which result in the limitations listed below: 1. Linear flow. In the present form, the method is strictly applicable to linear flow and cannot be used for analyzing other flow regimes, like radial flow and bi-linear flow. However, the formulation and methodology presented in this study can be extended for analyzing those flow regimes. The method is developed for linear flow as it often is the only flow regime available for analysis in multi-fractured horizontal wells. 2. Constant flowing pressure. This is an acceptable assumption for multi-fractured horizontal wells in tight/shale gas reservoirs as they are often produced under high drawdown condition to maximize production. When there is a significant change in drawdown during life of the well, this method is an approximation only. In this case, superposition time functions (bilinear, linear, radial, boundary-dominated flow) have been used historically to account for changing rate with time. For the case of tight/shale gas reservoirs, the dominant flow regime is linear flow and therefore, linear superposition time (superposition using corrected pseudo-time) is recommended to avoid any misinterpretation 3. Single-phase gas flow. This is also an acceptable assumption in multi-fractured horizontal wells after the fracture clean up period is over, for dry gas reservoirs. 4. Adsorption can be described by the Langmuir isotherm. Although this is the most popular method for modeling adsorption, other isotherm models can be used in Equations (3.14), (3.16), (3.25) and (3.27) if necessary.

130 Summary This chapter presented an analytical method to accurately analyze linear flow for constant flowing pressure in the presence of slippage and desorption. To model gas slippage, we have used the dynamic-slippage approach suggested by Ertekin et al. (1986) to quantify apparent gas permeability changes with pressure. It was shown that for the reservoir geometry shown in Figure 3.1, the average pressure in the region of influence is constant during linear flow and depends on initial pressure, flowing pressure, reservoir temperature, gas properties, gas saturation and the degree of slippage (i.e., permeability). The method was then validated using numerical simulation. Pressure-dependent transmissibility multipliers were used to incorporate apparent gas permeability changes with pressure into simulation. It was found that, in general, the fracture half-lengths obtained using the method proposed in this study are in good agreement with those expected. It was shown that when slippage is ignored, x f k (and hence, fracture halflength) is significantly over estimated. When comparing the impact of slippage corrections in pseudo-pressure and pseudo-time, it was observed that correcting pseudopressure is more important at lower permeabilities, whereas correction to pseudo-time is more important at higher permeabilities. That being said, to properly analyze the production data in tight/shale gas reservoirs, corrections to both pseudo-pressure and pseudo-time is necessary. The analysis method developed in this study is general enough that it only needs the apparent gas permeability (or permeability ratio) as a function of pressure and as a result: (a) any correlation or model that predicts apparent gas permeability as a function of pressure can be used and (b) the method can be applied for analyzing production data

131 103 with geomechanical (compaction) effects, as long as permeability ratio versus pressure is available. Finally, it is shown that as permeability decreases, the slippage contribution to production increases. It was also found that when a model that does not incorporate slippage is calibrated to production data that includes the gas slippage effect (i.e., history matching), the long-term forecast from this model is practically the same as the model that incorporates the slippage effect. This finding demonstrates that including gas slippage is more important for reservoir and fracture characterization than for long-term production forecasting.

132 104 CHAPTER 4 EFFECT OF COMPLETION HETEROGENEITY Abstract Shale gas reservoirs have become a significant source of gas supply in North America owing to the advancement in drilling and stimulation techniques to enable commercial development. The most popular method for exploiting shale gas reservoirs today is the use of long horizontal wells completed with multiple-fracturing stages (MFHW). The stimulation process may result in bi-wing fractures or a complex hydraulic fracture network. However, there is no way to differentiate between these two scenarios using production data analysis alone, making accurate forecasting difficult. For simplicity, hydraulic fractures are often considered bi-wing when analyzing production data. A conceptual model that is often used for analyzing MFHWs is that of a homogeneous completion; in which all fractures have the same length. However, fractures of equal length are rarely if ever seen (Ambrose et al., 2011). The production data analysis methods developed so far are aimed to obtain understanding of fracture length, fracture conductivity, stimulated reservoir volume (SRV), contacted gas-in-place and other information for a MFHW being analyzed. 1 This chapter is the combination of the modified version of the followings: Nobakht, M., Clarkson, C.R, Ambrose, R.J., Youngblood, J.E. and Adams, R Effect of Completion Heterogeneity in a Horizontal Well with Multiple Fractures on the Long-Term Forecast in Shale Gas Reservoirs. Journal of Canadian Petroleum Technology, 52 (6): Nobakht, M. and Clarkson, C.R, Multiwell Analysis of Multifractured Horizontal Wells in Tight/Shale Gas Reservoirs. Paper SPE presented at the Canadian Unconventional Resource Conference, Calgary, Alberta, 30 October 1 November. Nobakht, M. and Clarkson, C.R Hybrid Forecasting Methods for Multi-Fractured Horizontal Wells: EUR Sensitivities. Paper SPE presented at the SPE Middle East Unconventional Gas Conference and Exhibition, Abu Dhabi, UAE, January.

133 105 Although single well analysis methods are of tremendous value, the industry also needs analysis methods for analyzing a group of MFHWs. In this chapter, first, production data from a heterogeneous MFHW (i.e., where all fracture lengths are not the same) is studied for reservoirs with extremely low permeability. For this purpose, the simplified forecasting method of Nobakht et al. (2012b), developed for homogeneous completions, is extended to heterogeneous completions. For one specific case, the Arps decline exponent is correlated to the heterogeneity of the completion. It is found that, as expected, Arps decline exponent (used after the end of linear flow) increases with the heterogeneity of the completion. It is also shown that ignoring the heterogeneity of the completion can have a material effect on the long-term forecast. Secondly, analysis methods are developed for a group of multiple MFHWs. For this purpose, the methods developed for single-wells with a heterogeneous completion are found to be very useful for cases where adjacent wells are in communication (ex. fracturing one well affected the production of the adjacent wells). It is demonstrated how these methods help engineers diagnose and characterize the communication between MFHWs; further the same methods can be used to optimize fracture stimulation job sizes and spacing between horizontal wells in tight/shale gas plays. Finally, the sensitivity of expected ultimate recovery (EUR) for horizontal wells with multiple fractures to decline exponent is studied. This is very important from the reserves evaluation perspective due to uncertainty in the decline exponent, b. One of the causes of this uncertainty is reservoir/completion heterogeneity. It is found that for a time-based forecast (duration of forecast is specified), the ratio of EURs for two different

134 106 specified values of decline exponent depends on the ratio of economic life time of a well to the duration of linear flow. On the other hand, this EUR ratio depends on the ratio of rate at the end of linear flow to economic rate limit for economic limit-based forecast (economic rate limit is specified). 4.2 Introduction Multi-fractured horizontal wells (MFHWs) are now the primary method for exploiting low- and ultra-low permeability gas and oil reservoirs. Analytical, empirical and hybrid (i.e. combined analytical and empirical) approaches for forecasting these wells are in their infancy. Simple, yet rigorous methods are in demand from the industry to assist with forecasting of horizontal wells with multiple fractures, which is a critical step for reserve estimation and development planning. Analytical approaches, such as numerical and analytical simulation have been discussed in the literature (ex. Van Kruysdijk and Dullaert, 1989; Larsen and Hegre, 1991; Raghavan et al., 1997; Medeiros et al., 2008; Ozkan et al., 2011), as have purely empirical approaches (for example. Ilk et al., 2008 and Valko, 2009). Hybrid methods (Kupchenko et al., 2008; Bello and Wattenbarger, 2010; Nobakht et al., 2010; 2012b) have also been investigated, but until recently, have only focused on simple (homogeneous) hydraulic fracture geometries. In a previous study (Ambrose et al. 2011), a method was proposed to forecast MFHWs where hydraulic fractures are unequal in length ( heterogeneous completions ), which is considered the norm for actual field cases. Although planar (low-complexity) hydraulic fracture geometry for each stage was considered, the method may be applicable for more complex geometries as well. Like the Nobakht et al. (2010; 2012b) method, this

135 107 technique combines linear flow theory (linear flow being the dominant transient flowregime for many tight/shale wells), with Arps hyperbolic decline technique for boundary-dominated flow. Because of unequal length of fractures, Ambrose et al. (2011) suggested to use a hyperbolic decline exponent larger than 0.5 for a heterogeneous completion and observed that the final Arps hyperbolic decline exponent was a qualitative function of completion heterogeneity. In this chapter, we continue to investigate the effect of completion heterogeneity on the long-term production forecast of a MFHW. Using the concept presented by Ambrose et al. (2011), we present the equations for extending the Nobakht et al. (2010; 2012b) hybrid forecasting method to heterogeneous completions. The method is general enough to be extended to any other hybrid forecasting method. The method is then validated by comparing its results against a numerically-simulated test case. Very good agreement is found between the forecast rates obtained using the new method and the numerically-simulated rates throughout the well life. In the next step, we develop a relationship between Arps hyperbolic decline exponent and the heterogeneity of a completion for a specific case. A field case study is presented to compare the forecasts obtained when heterogeneity in the completion is considered and ignored. It is shown that ignoring the heterogeneity of the completion can have a great effect on the long-term forecast of these wells. Using the Arps hyperbolic decline method with an exponent adjusted to account for heterogeneity provides a rate profile that is in good agreement with the more rigorous method. Next, the method is extended to analyze multi-fractured horizontal wells that are adjacent to each other.

136 108 Finally, the sensitivity of expected ultimate recovery (EUR) for horizontal wells with multiple fractures to decline exponent is studied. 4.3 Development of the Model for Heterogeneous Completion Single-Well Analysis Nobakht et al. (2010; 2012b) developed a simplified forecasting method for the reservoir geometry shown in Figure 4.1(a) under constant flowing pressure and then extended that to the reservoir geometry shown in Figure 4.1(b). The method was developed for constant flowing pressure production and combined linear flow theory during transient flow and Arps decline during boundary-dominated flow. This procedure involves plotting inverse gas rate versus square root of time on Cartesian coordinates. The linear flow portion of the data should form a straight line with the following equation: 1 q m t b. (4.1) CP In this equation, m CP is the slope of inverse gas rate versus square-root-of-time plot and b' is the intercept that represents the additional pressure drop caused by skin or finite conductivity of fractures. Using the slope of the line in Equation (4.1), the duration of linear flow, t elf, can be calculated using the following equation in field units (Wattenbarger et al., 1998): t elf 2 Ah gct m i CP ( ppi ppwf ). (4.2) 200.T 6

137 109 In this equation, A is the drainage area, h is the net pay thickness, is the reservoir porosity, g is gas viscosity, c t is total compressibility, subscript i refers to initial reservoir conditions, p pi and p pwf are pseudo-pressures at initial pressure and flowing pressure, respectively and T is the reservoir temperature. The duration of linear flow can also be calculated from the following equation using reservoir permeability, k, and reservoir length, y e (Wattenbarger et al., 1998): t elf ye gct i k 2. (4.3) For the purpose of this study, Equation (4.2) is used for calculating t elf when the deviation from linear flow in not observed in the available production history of a well. For the linear flow duration (t t elf ), the rate can be calculated based on the line through the linear flow portion of the data and using the following equation: 1 q. (4.4) m t b CP

138 110 (a) Fracture y e x e (b) Figure 4.1. (a) A hydraulically-fractured vertical well in the center of a rectangular reservoir. (b) Example of a homogeneous multi-fractured horizontal well. To forecast rates during boundary-dominated flow, Nobakht et al. (2010; 2012b) used Equation (4.5) which is Arps hyperbolic decline immediately after t elf : q q elf. (4.5) 1/b 1 bdelf ( t telf ) In this equation, q elf and D elf are production rate and decline rate at the end of the linear flow period, respectively, and b is hyperbolic decline exponent. The production rate at the end of the linear flow can be calculated using t = t elf in Equation (4.4) and the decline rate at the end of linear flow can be calculated as follows (Nobakht et al., 2010; 2012b):

139 111 D elf 1 m m t b t CP. (4.6) CP elf 2 elf Nobakht et al. (2012b) used a decline exponent of 0.5 for a volumetric gas reservoir. It should be mentioned that in Nobakht et al. (2012b) method, there is a step change in the decline exponent at t = t elf. For example if b'=0 in Equation (4.1), the decline exponent changes from b=2 to b=0.5 at the end of linear flow. Although in reality the decline exponent is a continuous function (i.e., no step change occurs at t = t elf ), Kupchenko et al. (2008) showed that the sudden drop in b-value at the end of linear flow does not create major errors in their forecasting method. Nobakht et al. (2012b) observed that the forecast rates match the simulated rates very well by switching to b=0.5 at the end of linear flow. However, there were cases that using b=0.5 overestimate the rates at late times. More discussion on the decline exponent is provided later in this chapter. Ambrose et al. (2011) suggested that using b=0.5 in a volumetric gas reservoir is only applicable when dealing with a homogeneous completion (for example Figure 4.1 (b)) in a MFHW, where all hydraulic fractures are equal in length (and height). They then introduced the concept of heterogeneous completion as a completion where not all the fracture lengths (or fracture surface areas) are the same. This model is a more real representation of actual completions within a MFHW. A schematic of a heterogeneous completion is shown in Figure 4.2(a). Ambrose et al. (2011) proposed to divide a heterogeneous completion into different systems with different durations of linear flow. To explain this concept, let us consider the completion shown in Figure 4.2(a), consisting of three fractures (1, 2 and 3) of different lengths. It is assumed that these fractures are equidistant, which is a practical and reasonable assumption. The distance between the fractures is 2d. Transient linear

140 112 flow to the fracture faces will occur until the pressure pulse has traveled a distance d. At this point, pure transient linear flow will end (t = t elf ) and subsequently the flow will consist of a mixture of transient linear flow and boundary-dominated flow. The data will no longer appear as a straight line on square-root-of-time plot. Some regions will be in boundary-dominated flow, while others will still be in transient. For this conceptual model, the flow sequence can be described as: When t < t elf : All regions are in Transient flow. There are NO regions in boundary-dominated flow. When t elf < t < 4t elf, where 4 comes from (2d/d) 2 : The following Regions have entered boundary-dominated flow: ELMF, KPQN, ACHB (red color in Figure 4.2(b)). The following Regions are still in transient linear Flow: FMNG, DKLE, GQRH, COPD. When 4t elf < t < 25t elf, where 25 comes from (5d/d) 2 : The following Regions have entered boundary-dominated flow: FMNG, DKLE (green color in Figure 4.2(b)). The following Regions are still in Transient Linear Flow: GQRH, COPD. When t > 25t elf : The following Regions have entered boundary-dominated flow: GQRH, COPD (Blue color in Figure 4.2(b)) The following Regions are still in Transient Linear Flow: None

141 113 Based on the time that different regions enter boundary-dominated flow, the heterogeneous completion shown in Figure 4.2(a) can be divided into three divisions shown in Figure 4.2(b). (a) (b) A C D K O P Division 1 Division 2 E I L Division F J M B G H N Q R Figure 4.2. (a) Example of a heterogeneous multi-fractured horizontal well used to explain the concept of dividing a heterogeneous completion into different divisions based the duration of linear flow. (b) The schematic shown in Figure 4.2(a) divided into three divisions with different duration of linear flow.

142 114 The procedure developed by Nobakht et al. (2012b) can be applied to any of the divisions to calculate the production rate of each division at different times. To do this, a number j is associated with each division which indicates the ratio of drainage area of that division to the total drainage area. Using this definition, 2dj xfj j, (4.7) 2 L ( x ) e f max where d j is half of the distance between two adjacent fractures in division j, xfj is the sum of fracture half-lengths in division j, (x f ) max is the half-length of the longest fracture and L e is the length of horizontal well. Note that xfj is equal to half of the surface area to flow in division j divided by net pay and flow to both sides of a fracture needs to be considered here. For example, for the completion geometry shown in Figure 4.2(b), d 1 =d, d 2 =2d, d 3 =5d and 2 xf1 KN+(LM+IJ)+(IJ+EF)+CH (IJ is added twice because of flow to both sides of the fractures). Assuming m CP is the slope of inverse gas rate versus square-root-of-time plot before fractures start to interfere, it is inversely proportional to the total surface area to flow, (A c ) T (Wattenbarger et al., 1998): (A ) =4 hx ( ) c T f T 1. (4.8) m CP Here, (x f ) T is the summation of fracture half-lengths of all fractures (or total fracture halflength). The proportionality constant depends on reservoir properties and flowing pressure. For the purpose of the derivation in this section, it is important to calculate m CPj, which is the slope of inverse gas rate versus square-root-of-time plot for division j. Using the same concept as Equation (4.8), the surface area to flow in division j, and m CPj are related as follows: Acj,

143 115 A =2h cj 1 xfj. (4.9) mcpj As the proportionality constants in Equation (4.8) and Equation (4.9) are the same, combining Equations (4.7) (4.9) results in, m CPj 2 d j( xf) T mcp. (4.10) L ( x ) j e f max Assuming the fractures are spaced evenly along the horizontal well with fracture spacing of 2d, the length of the horizontal well, L e, is related to number of fractures, n, and fracture spacing, 2d, as L e = 2nd. Using this in Equation (4.10), m CPj 1 mcp, (4.11) j j with j defined as: ( x ) d f max j. (4.12) xfd j Here, x f is the average fracture half length, which is calculated by dividing the sum of all fracture half-lengths by the number of fractures. To calculate the duration of the linear flow for any division j, t elfj, the constant flowing pressure linear flow distance of investigation is used. Assuming t elf is the duration of the linear flow for the first division, which corresponds to the pressure pulse traveling the fracture spacing of d, t elfj becomes: t elfj d j telf. (4.13) d Knowing the slope of inverse gas rate versus square-root-of-time plot, m CPj from Equation (4.11), and the duration of linear flow for each division, t elfj from Equation

144 116 (4.13), the simplified forecasting method of Nobakht et al. (2012b) that was developed for homogeneous completion now can be applied to calculate the production rate, q j, for each of the divisions at different times: 1 q q 1 U( t t ) U( t t ). (4.14) elfj j elfj 1/b elfj mcpj t 1 bdelfj( t telfj) Note that for simplicity, Equation (4.14) is derived assuming that the line through inverse gas rate versus square-root-of-time plot passes through the origin. In Equation (4.14), U( t t elfj ) is the unit step function, which is zero for t < t elfj and one for t t elfj. In addition, D elfj and q elfj are respectively the decline rate and production rate at the end of the linear flow period for division j and can be calculated using the following equations: D elfj 1 2t, (4.15) elfj q elfj 1. (4.16) m t CPj elfj Combing Equations (4.11), (4.14) and (4.16) and assuming j = j j, the production rate at any time for any division j becomes: qj j 1 U( t telfj) U( t t 1 elfj) /b mcp t. (4.17) mcp telfj 1 bdelfj( t telfj) Finally, the total rate production can be calculated as follows: r r q qj j 1 U( t telfj) U( t t 1 elfj) /b j 1 j 1 mcp t.(4.18) mcp telfj 1 bdelfj( t telfj)

145 117 Here, r is the number of divisions. It can be shown that r j1 1 j. Using the development presented above, the following procedure can be used for forecasting in a heterogeneous completion with even fracture spacing: 1. Divide the completion into different divisions in which each division has a value for duration of linear flow. 2. Calculate j from Equation (4.7), j from Equation (4.12) and j = j j for each of the divisions defined in Step Plot 1 q versus t on Cartesian coordinates and place a line through the linear flow data points. Determine the slope, m CP, of this line. 4. Determine the duration of linear flow for the first division, t elf. a. If there is deviation from linear flow (which is an indication of some sort of depletion) in the plot of inverse gas rate versus square root of time, note the point of deviation as t elf. b. If data in the plot of inverse gas rate versus square root of time still shows transient linear flow, specify a drainage area, A. Use A 1 = A and m CP1 = m CP / 1 as A and m CP, respectively, in Equation (4.2) to calculate the duration of linear flow for the first division. 5. Calculate the duration of linear flow for different divisions using Equation (4.13). 6. Assume a value for decline exponent, b, and calculate total production rate using Equation (4.18). In practice, many tight gas and shale gas wells are producing at the highest possible drawdown, and b values that approach 0.5 are anticipated for each division (Okuszko et al., 2007).

146 Validation To validate the methodology presented above, a numerically-simulated case was built for the completion geometry shown in Figure 4.3(a) using the parameters shown in Table 4.1 with economic limit of 50 Mscf/D. To model the hydraulic fractures in the numerical simulation, we added high permeability grids in the y-direction. The permeability of these grids are chosen large enough to have F CD >200 for fracture conductivity of individual fractures. In other words, the hydraulic fracture is assumed to have infinite conductivity in these simulated cases (i.e., negligible pressure drop along the fracture). Logarithmic girding was used to model pressure transients accurately. The spacing between fractures is 100 ft and the fracture half-lengths (from left to right) are as follows: 215, 270, 115, 325, 185, 270, 60, 195 and 325 ft. Using the method presented above, 5 divisions with different durations of linear flow can be created for this completion as shown in Figure 4.3(b). The dotted lines in this figure are the mid-point between the two adjacent fractures (or boundary) in each division. The first year of the synthetic production data was then used for analysis. The data was analyzed as discussed in Steps 1 to 6 in the forecasting procedure presented in the previous section. Note that a value of b=0.5 was used to forecast rates during boundary-dominated flow of each of the divisions. Then the calculated rates using this simple procedure were compared to the original numerically-generated synthetic rates.

147 119 Table 4.1. Input parameters used for numerical simulation to validate the methodology for heterogeneous completion presented in this chapter. p i (psi) 3,000 p wf (psi) 300 k (nd) 250 S g (%) 100 S w (%) 0 (%) 10 h (ft) 100 T (ºF) 120 γ g 0.65 x e (ft) 900 y e (ft) 650 c f 0

148 120 (a) (b) Division 1 Division 2 Division 3 Division 4 Division 5 Figure 4.3. (a) Example of a heterogeneous multi-fractured horizontal well used for validating the procedure presented in this study. (b) The schematic shown in Figure 4.3(a) divided into five divisions with different duration of linear flow (Modified from Ambrose et al. (2011)).

149 121 Figure 4.4 shows the comparison among the heterogeneous forecast rates obtained using the above-mentioned six steps, the numerically-simulated data, and the homogeneous forecast rates with b=0.5 in Equation (4.5) for t > t elf. This figure shows that the heterogeneous forecast is in good agreement with the numerically-simulated data during the economic life of the well. Unlike the heterogeneous forecast, the homogeneous forecast with b=0.5 underestimates the rate after the end of linear flow of the first division. As mentioned above and also discussed by Ambrose et al. (2011), this is because using b=0.5 implies that the entire reservoir is in boundary-dominated flow after the end of linear flow, which is not the case in heterogeneous completions. It is worth mentioning that the cumulative production after 4,000 days for numerically-simulated data, heterogeneous forecast and homogeneous forecast with b=0.5 are 974, 965 and 836 MMscf.

150 Heterogeneous Forecast Simulation Data Homogeneous Forecast 1000 Gas Rate, Mscf/D ,000 2,000 3,000 4,000 Time, days Figure 4.4. Comparison among the rates obtained from heterogeneous model, numerical simulation and homogeneous model with b= Effect of Completion Heterogeneity on b-value Ambrose et al. (2011) suggested that the Nobakht et al. (2010) method can be applied in heterogeneous completions using a hyperbolic decline exponent larger than 0.5 after the end of linear flow of the first division. As the forecasting procedure presented for heterogeneous completions is validated against numerical simulation, it will be used to determine the b-value that best matches the data after the end of linear flow of the first division in a heterogeneous completion. For the purpose of this study, we focused on heterogeneous completions with seven divisions; the duration of linear flow for the first division is one year and the distance between fractures, 2dj, in divisions 2 7 are respectively 2, 3, 4, 5, 6 and 12 times of the fracture spacing along horizontal well. 100

151 123 different distributions of γ 1, γ 2, γ 3, γ 4, γ 5, γ 6 and γ 7 ( 7 j1 1) were generated along with the 10,000-day gas production rate profile for these 100 cases. The Arps hyperbolic decline curve model was fit to the rate after 1 year (i.e., the duration of linear flow for the first division) and a value for decline exponent obtained for each case. The calculated value of decline exponent was then correlated to values of γ j as follows: b (4.19) Figure 4.5 demonstrates the good agreement obtained between the calculated values of decline exponent and those obtained from Equation (4.19). Equation (4.19) shows that as the heterogeneity increases (i.e., increasing γ 2, γ 3, γ 4, γ 5, γ 6 and/or γ 7 or decreasing γ 1 ), larger decline exponent needs to be used after t elf for forecasting using hyperbolic decline. This is because as heterogeneity increases, a greater portion of the reservoir stays in transient linear flow after the first division goes to boundary-dominated flow with b=0.5. This illustrates the importance of considering heterogeneity for the purpose of long-term production forecast in a MFHW. It is worth mentioning that when there is no heterogeneity (i.e., all γ 2, γ 3, γ 4, γ 5, γ 6 and γ 7 are zero), the decline exponent calculated from Equation (4.19) is 0.49, which agrees well with the value of 0.5. Note that Equation (4.19) is not a universal correlation, as it is developed based on some assumptions for t elf, d j /d and duration of the forecast and it only can be used in situations where these assumptions are valid. j

152 124 2 Estimated b-value using Equation (4.19) b-value Obtained from Best Fit Figure 4.5. Comparison between the hyperbolic decline exponent, b, values obtained from best fit and those calculated from Equation (4.19). 4.6 Case Study For this case study, a multi-fractured horizontal in the Barnett shale well was analyzed. The well is producing under high drawdown condition (between 90% and 95% drawdown) and therefore, we assume constant flowing pressure production. The horizontal well length is 1,400 ft and it has 14 fractures. A schematic of the completion is shown in Figure 4.6. The heterogeneity of the completion was identified from the production log assuming that any variation in production along the well is due to fracture half-length variation (no change in permeability and fracture conductivity); the relative lengths of the fractures are assigned based on their % of total production. Figure 4.7 shows the same completion divided into seven regions. The values of j for division 1 7

153 125 are 71.5%, 15.8%, 6.7%, 1.8%, 2.4%, 1.2% and 0.6%, respectively. The duration of linear flow for the first division is almost one year (analysis is not shown here) and the distance between fractures, 2d j, in divisions 2 7 are respectively 2, 3, 4, 5, 11 and 23 times of the fracture spacing along horizontal well, 2d. Using the values of j from this example in Equation (4.19), it is expected that using b=0.8 accounts for heterogeneity of the completion. Figure 4.8 shows the comparison among forecasts obtained using heterogeneous model, homogeneous model with b=0.8 and homogeneous model with b=0.5. It can be seen from this plot that, as expected, the rates obtained from homogeneous model with b=0.5 are lower than those calculated using heterogeneous model. This figure also shows that there is a very good agreement between the rates obtained from homogeneous model with b=0.8 (which is approximation for heterogeneous forecast) and heterogeneous model. Finally, the expected ultimate recovery (EUR) after 10,000 days calculated using heterogeneous model, homogeneous model with b=0.8 and homogeneous model with b=0.5 are 1030, 1010 and 810 MMscf, respectively. Using the homogeneous model with b=0.5 underestimates the EUR in this case by 20%.

154 Figure 4.6. A schematic of multi-fractured horizontal well for the field case study. The heterogeneity of the completion was identified from production log. 126

155 127 Division 1 Division 2 Division 3 Division 5 Division 6 Division 7 Division 4 Figure 4.7. The schematic shown in Figure 4.6 divided into seven divisions with different duration of linear flow (Modified from Ambrose et al. (2011)).

156 128 10,000 Production Data Heterogeneous Forecast Homogeneous Forecast with b=0.8 Homogeneous Forecast with b=0.5 1,000 Gas Rate, Mscf/D ,000 10,000 Time, days Figure 4.8. Comparison among forecasts obtained using heterogeneous model, homogeneous model with b=0.8 and homogeneous model with b=0.5. Note that homogeneous forecast with b=0.8 is an approximation for heterogeneous forecast. 4.7 Multi-Well Analysis The concept of dividing the systems into different divisions based on different duration of linear flow can be used when analyzing a group of multi-fractured horizontal wells. To demonstrate this, the two horizontal wells with multiple fractures shown in Figure 4.9 are considered. The fractures for both wells are identical and the spacing between fractures along both horizontal wells is 100 ft. Then a numerically-simulated case is created for this reservoir/completion geometry using the following parameters: p i =3,000 psi, T=120 ºF, h=100 ft, ϕ=10%, S g =100%, γ g =0.65, k = 250 nd, c f =0, x f =200 ft, x e =500 ft, y e =700 ft and p wf = 300 psi. The hydraulic fractures are modeled using high permeability grids in the y-direction to create negligible pressure drop along the fractures.

157 129 Figure 4.10 shows a plot of gas rate versus time obtained from simulation for both wells. It should be mentioned that both wells start the production at the same time. The squareroot-of-time plots for the two wells using the first 400 days of production are shown in Figure As expected, it can be seen that the data for both wells form a straight line initially, confirming linear flow to fractures. Using the slope of these lines, x f k is calculated to be and md 1/2 ft for well 1 and well 2, respectively, using Equation (1.1). Note that for simplicity, we did not include the linear flow correction factor (Ibrahim and Wattenbarger, 2006; Nobakht and Clarkson, 2012b) to correct for overestimation of xf k using Equation (1.1). Well 1 y e Well 2 x e Figure 4.9. Reservoir/Completion geometry used in this study for analysis.

158 Well 2 Well 1 1 Gas Rate, MMscf/D Time, Days Figure Gas production rate versus time plot for Well 1 and Well 2 in Figure Well 2 Well Normalized Pressure, (psi 2 /cp)/(mmscf/d) Square Root of Time, Days 0.5 Figure Square-root-of-time plot for Well 1 and Well 2 in Figure 4.9.

159 131 Using the semi-log derivative plot (not shown here), the data diverges from halfslope linear flow after almost 50 days for both wells. To understand the cause of deviation from linear flow, the reservoir/completion geometry shown in Figure 4.9 is divided into three divisions as shown in Figure Based on this figure, the deviation from half-slope linear flow is related to the end of linear flow for the first division when the distance of investigation is 25 ft (half distance between fractures in Division 1 in Figure 4.12). Using y=25 ft and t=50 days in the distance of investigation equation for constant flowing pressure linear flow, the permeability is calculated to be almost 280 nd, which agrees wells with k=250 nd used for simulation. Using x f k values calculated from linear flow analysis for the two wells and permeability of 280 nd, total fracture halflength for well 1 and well 2 become 925 ft and 1156 ft, respectively, which corresponds to almost 232 ft half-length for each of the fractures. Knowing the spacing between horizontal wells and the lengths of individual fractures, one can determine the communication between wells and use the results for optimizing future frac jobs and well spacing.

160 132 Figure Reservoir/Completion geometry shown in Figure 4.9 is divided into different divisions. The analysis explained above was conducted knowing that there is some communication between wells. It is worthwhile analyzing the data with methods currently in the literature for single well analysis. For this purpose, we only consider well 1. It is well known in the literature that the data in the square-root-of-time plot diverge from straight line when fractures within a multi-fractured horizontal well start to interfere. Therefore, if one only considers well 1 by itself, 50 days (the time at which data on the semi-log derivative plot for well 1 diverge from straight line) can be interpreted as the time that the distance of investigation is 50 ft (i.e., half of the fracture spacing for well 1). Therefore, using y=50 ft and t=50 days in the distance of investigation equation for constant flowing pressure linear flow, the permeability is calculated to be 1,120 nd. Using this permeability, the half-length of each fracture is calculated to be 116 ft, which is just

161 133 about half of the expected value of 200 ft. Using this methodology, the reservoir/completion geometry that is obtained from analysis is as shown in Figure 4.13, which shows no communication between wells. Comparing Figure 4.13 and Figure 4.9 shows how ignoring the communication between wells results in misinterpretation. Well 1 y e Well 2 x e Figure Reservoir/Completion geometry obtained from the analysis of production data ignoring the communication between Well 1 and Well Sensitivity of Forecast to b-value We observe from the discussions above for heterogeneous completions, the decline exponent can vary between 0.5 and 2 after the fractures start to interfere in a multifractured horizontal well due to different fracture lengths and/or unequal fracture spacing along the horizontal well. As shown in this chapter, it is possible to use a single b-value after the end of linear flow to match the heterogeneous forecast and therefore it is important to study the relative impact of b-value used after the end of linear flow on long-

162 134 term production forecast for linear-flow dominated wells. The forecasting procedure of Nobakht et al. (2012b) can be used to investigate the sensitivity of EUR to decline exponent, b, which is being used for forecasting during boundary-dominated flow EUR Based upon Time The relationship between cumulative production and time for hyperbolic decline is as follows: Q 1 q 1 bdt i b, (4.20) i bd i where q i is the production rate at the start of the hyperbolic forecast period, b is the hyperbolic decline exponent, D i is decline rate corresponding to q i and t is time since the start of the hyperbolic forecast. Because the hyperbolic forecast starts after the end of linear flow, we will use the hyperbolic decline equation in the form shown in Equation (4.21), which is obtained from Equation (4.20) by re-initializing the time at t = t elf : 1 q elf 1 Q 1 1 bdelf t telf b. (4.21) 1 bdelf In this equation, Q is the volume produced during boundary-dominated flow using hyperbolic decline. Using Equation (4.1) with b'=0, the cumulative production at the end of linear flow (i.e., t = t elf ) is: 1 2 Q qdt dt t elf telf telf 0 0 elf. (4.22) m m CP t CP To obtain the EUR at the end of the forecast (when t t ), the volume produced elf using Equation (4.21) can be added to the cumulative production at the end of linear flow calculated from Equation (4.22). Combining Equations (4.4), (4.6), (4.21) and (4.22):

163 135 2 telf 1 b t EUR m CP 1 b 2 t elf 1 1 b. (4.23) Here, t is the economic life of the well. Equation (4.23) shows that the ratio between EUR values obtained for two different values of decline exponent depends on the value of t t elf, given that the two decline exponents are specified. Note that Equation (4.23) is derived by ignoring the intercept of 1 q versus t plot (i.e., b'=0 in Equation (4.1)). To show the EUR sensitivity plots, b = 0.5, b = 0.8 and b = 1.3 are chosen based on the Ambrose et al. (2011) observations that b = 0.5 is for homogeneous completion, b = 0.8 is for slightly heterogeneous completion and b = 1.3 is for very heterogeneous completion. EUR( b 1.3) EUR( b 0.5) and EUR( b 0.8) EUR( b 0.5) versus t t elf are shown in Figure 4.14 and Figure 4.15, respectively. These plots can be used to study the sensitivity of forecast to the value of decline exponent. For example, it can be seen from Figure 4.15 that for t 30t, EUR for b = 0.8 is almost 20% higher than EUR for b = 0.5. elf

164 EUR (b =1.3) / EUR (b =0.5) t /t elf EUR( b 1.3) t Figure Plot of versus. b = 0.5 is for homogeneous completion EUR( b 0.5) telf and b = 1.3 is for very heterogeneous completion.

165 EUR (b =0.8) / EUR (b =0.5) t /t elf EUR( b 0.8) t Figure Plot of versus. b = 0.5 is for homogeneous completion EUR( b 0.5) telf and b = 0.8 is for slightly heterogeneous completion EUR Based upon Economic Limit Rate When the economic limit rate, q f, is known, the volume produced using hyperbolic decline is calculated using the following equation: b i q Q q q 1 b D i 1b 1b i f. (4.24) Therefore, the volume produced during boundary-dominated flow ΔQ for hyperbolic decline is: b elf q Q q q 1 b D elf 1b 1b elf f. (4.25)

166 138 Using q elf and D elf from Equation (4.4) and Equation (4.6) respectively (with b'=0), the EUR for hyperbolic decline becomes: EUR b1 2 telf 1 qelf 1 1 mcp 1 b qf. (4.26) Equation (4.26) shows that the ratio between EUR values obtained for two different values of decline exponent depends on the ratio of the rate at the end of linear flow to the economic rate limit, given that the two decline exponents are specified. When EUR is being calculated based upon economic limit rate, Equation (4.26) can be used to investigate the sensitivity of EUR to decline exponent Evolution of b-value During Boundary-Dominated Flow Nobakht et al. (2012b) observed that the forecast rates from their method matched the simulated rates very well and that the proposed method overestimated the rates at late times. In an effort to find the cause of the rate overestimation when using their method, rates obtained from numerical simulation for one of the cases (with relatively higher permeability) and rates obtained using the simplified method for b=0, 0.1, 0.2, 0.3, 0.4 and 0.5 were plotted against time on a log-log plot by Nobakht et al. (2012b), as shown in Figure They observed that the b value is actually not constant at 0.5 but gradually decreases towards zero (i.e., exponential decline) but their simplified method uses constant b-value to forecast the rates, which was the cause for the overestimation of the rates at late times. However, as shown by Nobakht and Clarkson (2012c), this error is of little economic consequence particularly for tight gas and shale gas reservoirs where this deviation occurs late in time.

167 b=0.5 b=0.4 b=0.3 b=0.2 b=0.1 b=0 Simulated Data Rate (MMscfd) Time (days) Figure Comparison among simulated rates for Case 1 and rates obtained using the simplified method for b = 0, 0.1, 0.2, 0.3, 0.4 and 0.5 (modified from Nobakht et al. (2012b)). This time-dependent behavior of Arps decline exponent has also been reported by others in the literature (Ayala and Peng, 2013). Ayala and Peng (2013) confirmed the varying behavior of the decline exponent during boundary-dominated flow. They observed that there is a time window early during boundary-dominated flow of a gas well, which they called it hyperbolic window, at which the decline exponent remains constant. Ayala and Peng (2013) also showed that the constant decline exponent character is lost at later times and the decline approaches exponential decline. It should be mentioned that the value of the decline exponent in the hyperbolic window is a function of drawdown and gas properties (Ayala and Peng, 2013). From their work,

168 140 b=0.5 can be justified analytically when the well is wide open (100% drawdown) and the product of gas viscosity and gas compressibility is inversely proportional to gas density. Therefore, since many tight gas and shale gas wells are producing at the highest possible drawdown, b values that approach 0.5 are anticipated, which is consistent with Okuszko et al. (2007) work. 4.9 Summary In this chapter, first, we expand on the work of Ambrose et al. (2011) and develop a rigorous ( hybrid ) method for forecasting multi-fractured horizontal wells with a heterogeneous completion. We first demonstrate how the hybrid method developed by Nobakht et al. (2012b) for homogeneous completions can be modified to account for heterogeneity. We then developed a method for adjusting the Arps decline exponent to account for heterogeneity. The robustness of the new hybrid method is demonstrated using a simulated case and its applicability through comparison to an actual field case. Comparisons between forecasts using the new hybrid method that accounts for heterogeneity and that assuming a homogenous completion demonstrates the magnitude of error that can be expected with the latter. It is shown that the new hybrid model can be simplified to yield a simple two-step Arps hyperbolic curve similar to Ambrose et al. (2011). Secondly, it was shown how the concept of dividing a heterogeneous completion into a number of divisions based on their end of linear flow can be helpful when analyzing production data from two multi-fractured horizontal wells in communication. It was also shown that ignoring the communication between wells when exists results in

169 141 misleading information about permeability, fracture half-length and reservoir/completion geometry. Finally, the simplified forecasting method of Nobakht et al. (2012b) was used to study the sensitivity of expected ultimate recovery (EUR) to decline exponent used after the end of linear flow. This is important for reserve evaluation because of uncertainty in decline exponent due to factors like adsorption and heterogeneity in completion. It was found that for two different specified values of decline exponent, the ratio between their EURs depends on the ratio of economic life of the well to the duration of linear flow for time-based forecast and the ratio of the rate at the end of linear flow to the economic rate limit for economic limit-based forecast.

170 142 CHAPTER 5 NEW TYPE CURVES FOR ANALYZING HORIZONTAL WELL WITH MULTIPLE FRACTURES IN SHALE GAS RESERVOIRS Abstract Economic production from shale gas plays is now possible using horizontal wells (cased or open hole) with multiple fractures. A number of well-performance behaviors can be seen for these wells depending on reservoir behavior and induced hydraulicfracture geometry. In this chapter, conceptual models for well/reservoir/hydraulic fracture combinations are first presented. Next, we discuss the impact of various reservoir types/induced hydraulic-fracture geometries upon the sequence of flow-regimes that could be encountered for shale gas reservoirs. In addition, we develop and present new sets of dimensionless type-curves for one of the conceptual models. The newly-developed type curves in this study yield more unique results than those presented previously. With these dimensionless type curves, the early linear flow (early-time half slope) and boundary-dominated flow (late-time unit slope) fall on top of each other and the transition between these two regimes depends on the geometry of the reservoir and completion. Using the type curves as a guide, we then present the flow regimes that are expected for different values of horizontal well length, number of fractures, length of the fractures and spacing between horizontal wells. We also present a new method for evaluating the contribution from the outer reservoir (beyond the fracture length). Finally, 1 This chapter is the modified version of: Nobakht, M., Clarkson, C.R. and Kaviani, D. (2013) New Type Curves for Analyzing Horizontal Well with Multiple Fractures in Shale Gas Reservoirs, Journal of Natural Gas Science and Engineering, 10:

171 143 using the characteristics of new dimensionless parameters and the new type-curve set, we present a simple and practical procedure for long-term forecasting in multi-fractured horizontal wells. 5.2 Introduction Horizontal wells with multiple fractures are currently the most popular completion technology for exploiting tight/shale gas reservoirs. The use of multi-fractured horizontal wells is expected to create a complex sequence of flow regimes (Chen and Raghavan, 1997; Clarkson and Pederson, 2010). The proper interpretation of these flow regimes is necessary for obtaining information about the hydraulic fracture stimulation and the reservoir. As a result, understanding the completion/reservoir geometry is critically important. Figure 5.1 shows several possible conceptual models for well/reservoir/hydraulic fracture combinations that can be used to analyze tight/shale gas/oil (Clarkson and Pederson, 2010). The descriptions of these different scenarios was explained in Chapter 1. The sequence of flow regimes is complex in multi-fractured horizontal wells and the proper interpretation of these flow regimes is necessary for obtaining information about the hydraulic fracture stimulation and the reservoir. The focus of this chapter is scenario 5 in Figure 5.1; the sequence of flow regimes that may be expected for this scenario have been described by Chen and Raghavan (1997). These flow regimes are summarized in Figure 5.2 using the semi-log derivative signature on log-log plot for the case that the hydraulic fractures are equally-spaced along the horizontal well. The early bilinear flow (quarter slope) may occur in finiteconductivity fracture systems due to linear flow within the fractures and linear flow in the

172 144 formation perpendicular to the fractures. An early linear flow period (half slope) caused by flow perpendicular to the fractures then develops, followed in some cases by early radial flow (zero slope) around the fractures. This latter flow regime may not occur, depending upon the spacing and length of the fractures (Chen and Raghavan, 1997). During the early linear flow period, flow across the fracture tips is negligible whereas it becomes considerable during early radial flow (Chen and Raghavan, 1997). Until the end of early radial flow, the fractures behave independently (i.e., fractures do not communicate and the production from each fracture is independent of the other fractures). The compound linear flow (half slope) period occurs after the early radial flow. This is the period during which the fractures start to interact and interfere. The flow during this period is perpendicular to the vertical plane containing the horizontal well (Chen and Raghavan, 1997). This flow period is followed by pseudo-radial (late radial) flow in which the flow across the tips of the horizontal well dominates. Finally, just as for any closed reservoir, boundary-dominated flow (unit slope) is the last flow regime. In this chapter, first, we develop and present new sets of dimensionless type curves for scenario 5 in Figure 5.1(e). The type curves are then used to investigate the flow regimes that can be expected in scenario 5 for different fracture spacing, horizontal well spacing and length of fractures. Secondly, a method is presented to evaluate the contribution from the outer reservoir (i.e., region outside SRV, where SRV is the drainage volume between hydraulic fractures). The time at which the outer reservoir starts to contribute is translated into a dimensionless time. This time is found to be independent of horizontal well spacing and fracture spacing and it depends on reservoir properties and length of the fractures. Finally, a simple and practical procedure is

173 145 presented for using the new type curves developed in this study for long-term forecasting in multi-fractured horizontal wells. The application of the method is then demonstrated with a field case study. Figure 5.1. Possible combinations of reservoir/hydraulic fracture encountered for tight oil/shale gas reservoirs (Modified from Clarkson and Pederson, 2010).