The Pennsylvania State University. The Graduate School. Department of Energy and Mineral Engineering

Transcription

1 The Pennsylvania State University The Graduate School Department of Energy and Mineral Engineering ANALYTICAL INVESTIGATION OF UNCONVENTIONAL RESERVOIR PERFORMANCE DURING EARLY-TRANSIENT MULTI-PHASE FLOW CONDITIONS A Dissertation in Energy and Mineral Engineering by Miao Zhang 2016 Miao Zhang Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2016 i

2 The dissertation of Miao Zhang was reviewed and approved* by the following: Luis F. Ayala H. William A. Fustos Family Professor in Energy and Mineral Engineering Professor of Petroleum and Natural Gas Engineering Associate Department Head for Graduate Education Dissertation Advisor Chair of Committee Zuleima T. Karpyn Professor of Petroleum and Natural Gas Engineering Quentin E. and Louise L. Wood Faculty Fellow in Petroleum and Natural Gas Engineering Shimin Liu Assistant Professor of Energy and Mineral Engineering Mathieu Stienon Professor of Mathematics *Signatures are on file in the Graduate School ii

3 ABSTRACT Unconventional gas resources accounted for more than 50% of total U.S. gas production in 2012 and its contribution is expected to increase to 75% by 2040 (EIA, 2015). In these unconventional gas reservoirs, reservoir and fluid characteristics can be significantly different from those in conventional resources, rendering traditional production data analysis methods inadequate. Those effects include long-time transient periods due to ultra-low permeability, pressure-dependent permeability and exemplified large capillary pressure. Development of reliable analysis methods to successfully capture these complex effects demands the formulation of new solutions to the governing flow equations which consider these complex nonlinearities. It is the interest of this study to develop more rigorous performance models for these types of systems derived from fundamental governing flow equations. This study presents a series of novel and rigorous semi-analytical solutions to the governing partial differential equations applicable to single-phase gas and multiphase flow in unconventional reservoirs. Focusing on early-transient periods, the proposed semi-analytical method utilizes similarity theory to transform the system of nonlinear PDEs to ordinary differential form, which is later solved via shooting method coupled Runge-Kutta numerical integration. The work starts with early-transient single-phase gas flow in linear and radial flow regimes under constant pressure and rate production conditions, followed by its direct extension to multiphase flow system using the black-oil fluid formulation. The application of the proposed multiphase flow solution to actual production highlighting producing gas-oil-ratio prediction is also discussed. Additionally, the proposed semi-analytical solution is proven capable of solving the multiphase flow equations under fully compositional fluid formulation. In the last chapter, capillary pressure effects a multiphase flow effect widely recognized to be significant in unconventional system due to nano-scale pore size is studied using the proposed semi-analytical method. Besides studying capillary pressure as an additional pressure drop on fluid flow, the effect of capillary pressure on phase behavior and properties is also analyzed. All the results in this work are validated by matching with finelygridded commercial numerical simulator. iii

4 Table of Contents LIST OF FIGURES... vi LIST OF TABLES... ix NOMENCLATURE... x ACKNOWLEDGEMENTS... xiii INTRODUCTION Similarity Solutions for Gas Flow during Early-Transient Periods Chapter Summary Background Similarity Solutions for Gas Early Transient Flow in a Linear System Linear Flow: Constant-pressure Solution Linear Flow: Constant-rate Solution Similarity Solutions for Gas Early Transient Flow in a Radial System Radial Flow: Constant-pressure Solution Radial Flow: Constant-rate Solution Concluding Remarks Similarity Solution for Multiphase Flow during Early-Transient Periods Chapter Summary Background Governing Flow Equations Similarity solutions for multiphase early transient flow in a linear system Linear flow: Constant pressure solution Linear flow: Constant flow rate solution Similarity solutions for multiphase early transient flow in a radial system Radial flow: Constant bottom hole pressure solution Radial flow: Constant flow rate solution On the So-p path and its interplay with thermodynamics and phase mobility Pressure-saturation relationships under various conditions Compositional paths in PX diagram for the black-oil pseudocomponent formulation Concluding remarks Constant GOR as an Infinite-Acting Effect in Multiphase Systems Chapter Summary Background GOR calculation for saturated multiphase reservoirs Synthetic case studies Model Validation Constant infinite-acting GOR vs. Effect of no-flow outer-boundary Non-linear flow effects (radial flow) Effect of bottom-hole specification Concluding Remarks Similarity-based Study of Flowing and In-Situ Compositions in Multiphase Reservoirs during Early Transient Periods Chapter Summary Background Similarity solution to compositional multiphase flow equations Solving system of ODEs for pressure (p) and moles per unit pore volume (N i ) iv

5 4.3.2 Solving system of ODEs for pressure (p) and overall molar composition (z i ) Synthetic case studies Model Validation Flowing vs. In situ composition Effect of initial and bottomhole condition on composition variation Compositional vs. black-oil fluid formulation Concluding Remarks Similarity-based Analytical Analysis of Capillary Pressure Effects on Recovery from Unconventional Reservoirs Chapter Summary Background IFT-independent capillary pressure effects Synthetic case study: gas condensate Reference pressure selection Capillary pressure gradient vs. fluid property effects Compositional paths in PX diagram for the black-oil pseudocomponent formulation under pc- gradient and property effects IFT dependent capillary pressure effects Model Validation: Bakken oil Synthetic case study: gas condensate Recommended semi-analytical solution for capturing amplified capillary pressure effects in unconventional multiphase systems Concluding Remarks CONCLUSIONS AND RECOMMENDATIONS References Appendix A: Linear 1-D gas flow equation in similarity variables Appendix B: Radial 1-D gas flow equation in similarity variables Appendix C: Linear 1-D multiphase flow equation in similarity variables Appendix D: Radial 1-D multiphase flow equation in similarity variables Appendix E: Similarity Transformation to 1-D Linear Compositional Equations Appendix F: Similarity Solution to 1-D Radial Compositional Equations Appendix G: Approximate solution by neglecting secondary derivative of saturation Appendix H: Solving multiphase flow equations under IFT-independent capillary pressure effect Appendix I: Solving gas condensate flow under capillary gradient effect using oil phase as reference pressure Appendix J: Solving multiphase flow equations capturing capillary property effect and neglecting capillary gradient effect Appendix K: Solving multiphase flow equations under IFT-dependent capillary pressure effect Appendix L: Extrapolation of pseudocomponent properties above dewpoint Appendix M: Procedure for Generating Black-Oil Properties from solved composition profiles Appendix N: Supplementary plots v

6 LIST OF FIGURES Figure 1-1 Pressure vs. similarity variable for constant-pressure p wf,sp=500 psia linear system... 9 Figure 1-2 Flow rate response for linear constant-pressure condition Figure 1-3 Bottomhole pressure responses for linear constant-rate cases Figure 1-4 Pressure profiles for linear constant-rate case: q gsc,sp= 5 MSCF/D Figure 1-5 Flow rate response for radial constant-pressure cases Figure 1-6 Pressure distribution for radial constant-pressure case: p wf,sp= 500 psia Figure 1-7 BHP response for radial constant-rate cases Figure 1-8 Pressure distribution for radial constant-rate case: q gsc,sp=350 MSCF/D Figure 2-1 PVT properties of gas condensate fluid Figure 2-2 Relative permeability curves used for all cases Figure 2-3 Pressure profiles for constant BHP specification in terms of the similarity variable Figure 2-4 Saturation profiles for constant BHP specification in terms of the similarity variable Figure 2-5 Pressure profiles in space for four time points under of linear constant BHP flow with BHP = 500 psia Figure 2-6 Saturation profiles in space for four time points under of linear constant BHP flow with BHP = 500 psia Figure 2-7 Simulated flow rates in time for linear, constant BHP reservoir condition Figure 2-8. Simulated saturation-pressure relationships for various constant BHP specifications Figure 2-9 Pressure profiles in space at day 98 for four flow rates in the case of linear constant gas flow Figure 2-10 Saturation profiles in space at day 98 for four flow rates in the case of linear constant gas flow Figure 2-11 Simulated bottom hole pressure in time with various flow rate specifications Figure 2-12 Pressure profiles at t=100 days in radial constant BHP case Figure 2-13 Saturation profiles at t=100 days in radial constant BHP case Figure 2-14 Simulated flow rate in time for radial constant BHP reservoir condition Figure 2-15 Pressure response at various time pointes in radial constant BHP case in term of similarity variable p wf = 500 psia Figure 2-16 Pressure profiles at t=60 days in radial constant rate case Figure 2-17 Saturation profiles at t=60 days in radial constant rate case Figure 2-18 Simulated bottom hole pressure for four gas flow rates in radial constant gas flow rate scenario Figure 2-19 Effect of bottomhole pressure specification on p-so relationship Figure 2-20 Effect of Initial pressure on p-so relationship Figure 2-21 Different S oc curves investigated Figure 2-22 Effect of critical saturation on p-so relationship Figure 2-23 P-X diagram of black-oil gas psuedocomponent and iso-saturation contour Figure 2-24 Effect of initial pressure on pressure-composition paths Figure 2-25 Effect of bottomhole condition on pressure-composition paths Figure 2-26 Effect of critical oil saturation on pressure-composition paths Figure 3-1 Pressure profile for gas condensate linear flow under p wf = 1000 psia specification Figure 3-2 Saturation profile for gas condensate linear flow under p wf = 1000 psia specification Figure 3-3 Pressure profile for volatile oil linear flow under p wf = 1000 psia specification Figure 3-4 Saturation profile for volatile oil linear flow under p wf = 1000 psia specification Figure 3-5 Producing GOR comparison of gas condensate example for p wf = 1000 psia scenario Figure 3-6 Producing GOR comparison of volatile oil example for p wf = 1000 psia scenario vi

7 Figure 3-7 Linear vs. radial infinite-acting GORs for gas condensate example Figure 3-8 Linear vs. radial infinite-acting GORs for volatile oil example Figure 3-9 GOR comparison under different p wf specifications under radial regime Figure 3-10 Predicted vs. simulated GORs for q gsc = 80 MSCF/D under radial regime Figure 3-11 Pressure-saturation relationship of gas condensate example under different pwf specifications Figure 3-12 Producing GOR comparison of gas condensate example for different p wf scenarios Figure 3-13 Producing GOR changes with p wf for gas condensate example Figure 4-1 Phase envelope of 5-component gas condensate fluid Figure 4-2 Relative permeability curves Figure 4-3 Pressure profiles of validation example Figure 4-4 In-situ composition profiles of validation example Figure 4-5 In situ composition profiles of validation example Figure 4-6 Pressure-saturation profile of validation example Figure 4-7 Producing wellstream composition history Figure 4-8 Producing GOR comparisons Figure 4-9 Molar rate of each component in surface-gas Figure 4-10 Molar rate of each component in surface-oil Figure 4-11 Flowing and in-situ composition comparison Figure 4-12 Flowing and in-situ composition comparison Figure 4-13 Effect of initial pressure on the flowing and in-situ composition of C Figure 4-14 Effect of initial pressure on the flowing and in-situ composition of C Figure 4-15 Effect of initial pressure on saturation-pressure relationship Figure 4-16 Effect of drawdown on the flowing and in-situ composition of C Figure 4-17 Effect of drawdown on the flowing and in-situ composition of C Figure 4-18 Effect of drawdown on saturation-pressure relationship Figure 4-19 Pressure profiles solved by proposed fully compositional and black oil solutions Figure 4-20 Saturation-pressure relationship predicted by fully compositional and black oil solutions Figure 4-21 Surface-gas and oil flow rates predictions by fully compositional and black oil solutions Figure 4-22 Converted compositional solution to black oil properties compared with black oil solution 92 Figure 4-23 Converted compositional solution to black oil properties compared with black oil solution 93 Figure 5-1 Relative permeability curves Figure 5-2 Saturation-dependent capillary pressure curve Figure 5-3 Pressure profiles under Pc effect (po reference) Figure 5-4 Saturation profiles under Pc gradient effect (po reference) Figure 5-5 Saturation-pressure relationship under Pc effect (po reference Figure 5-6 Gas flow rate under Pc gradient effect Figure 5-7 Oil flow rate under Pc gradient effect Figure 5-8 Producing GOR under Pc gradient effect Figure 5-9 Pressure comparisons for Pc gradient effect using oil phase pressure as reference in gas condensate system Figure 5-10 Pressure profiles for Pc gradient effect using difference reference pressures Figure 5-11 Saturation-pressure relationships for Pc gradient effect using difference reference pressures Figure 5-12 Saturation-dependent p cgo curves input Figure 5-13 Saturation-pressure relationships under difference capillary pressure curves Figure 5-14 Pressure comparisons under difference capillary pressure curves Figure 5-15 Mobility comparisons under different Pc curves vii

8 Figure 5-16 Mobility comparisons under different Pc curves Figure 5-17 Pressure profiles under Pc gradient and property effects (Set A) Figure 5-18 Saturation-pressure relationships under IFT-independent Pc gradient and property effects (Set A) Figure 5-19 Composition paths under IFT-independent Pc gradient effect (Set A) Figure 5-20 Composition paths under IFT-independent Pc property and gradient effect (Set A) Figure 5-21 Evaluating phase densities under IFT-independent Pc property effect (Set A) Figure 5-22 Flow chart for capillary pressure and fluid property calculation in proposed solution Figure 5-23 Relative permeability curves of the Bakken oil example Figure 5-24 Normalized p cgo/ift curve for Bakken oil example Figure 5-25 Pressure profile comparisons for under difference Pc effects Figure 5-26 Solved pressure-saturation relationship under difference Pc effects Figure 5-27 Gas/oil capillary pressure profile obtained using proposed semi-analytical method Figure 5-28 Effective interface radii profile obtained using proposed semi-analytical method Figure 5-29 Oil production rate comparisons for under different Pc effects Figure 5-30 Gas production rate comparisons for under different Pc effects Figure 5-31 GOR comparisons for under different Pc effects Figure 5-32 Normalized p cgo/ift curve for synthetic gas condensate example Figure 5-33 Two sets of relative permeability curves used in synthetic case study Figure 5-34 Pressure profiles for gas condensate case A (p i = 4000 psia, S oc = 0.1) Figure 5-35 Saturation-pressure relationships for gas condensate case A (p i = 4000 psia, S oc = 0.1) Figure 5-36 Gas rate comparison for gas condensate case A (p i = 4000 psia, S oc = 0.1) Figure 5-37 Oil rate comparison for gas condensate case A (p i = 4000 psia, S oc = 0.1) Figure 5-38 Gas/oil capillary pressure profile for gas condensate case A (p i = 4000 psia, S oc = 0.1) Figure 5-39 Effective radii profiles for gas condensate case A (p i = 4000 psia, S oc = 0.1) Figure 5-40 Pressure-saturation relationship comparison for gas condensate case B (p i = 4000 psia, S oc = 0.3) Figure 5-41 Gas rate comparison for gas condensate case B (p i = 4000 psia, S oc = 0.3) Figure 5-42 Oil rate comparison for gas condensate case B (p i = 4000 psia, S oc = 0.3) Figure 5-43 Pressure-saturation relationship comparison for gas condensate case C (p i = 3000 psia, S oc = 0.3) Figure 5-44 Gas rate comparison for gas condensate case C (p i = 3000 psia, S oc = 0.3) Figure 5-45 Oil rate comparison for gas condensate case C (p i = 3000 psia, S oc = 0.3) Figure 5-46 Pressure profiles under Pc effects for Bakken oil Figure 5-47 Saturation-pressure relationships under Pc effects for Bakken oil Figure 5-48 Pressure profiles under Pc effects for synthetic gas condensate case C Figure 5-49 Saturation-pressure relationships under Pc effects for synthetic gas condensate case C viii

9 LIST OF TABLES Table 1-1 Summary of similarity variables and ODEs for gas flow The Linear Case... 6 Table 1-2 Reservoir formation and gas fluid properties... 8 Table 1-3 Similarity variables and principle ODE form for gas flow The Radial Case Table 2-1 Reservoir formation and gas condensate fluid properties for case studies Table 2-2 System of first-order ODEs and boundary conditions for gas condensate linear flow Table 2-3 System of first-order ODEs and boundary conditions for gas condensate radial flow Table 3-1 Reservoir and Fluid Properties Table 4-1 Compositional data for synthetic gas condensate example Table 4-2 Reservoir and Fluid Properties Table 5-1 Reservoir and Fluid Properties for gas condensate example ix

10 NOMENCLATURE A w = cross sectional flow area at the inner boundary location, ft 2 B g = gas formation volume factor, RB/SCF B o = oil formation volume factor, RB/STB c g = gas compressibility, 1/psi f 1 = Runge-Kutta method dependent function f 2 = Runge-Kutta method dependent function f 3 = Runge-Kutta method dependent function f mg = mass fraction of gas phase f mo = mass fraction of oil phase f ng = molar fraction of gas phase f no = molar fraction of oil phase F g = mass flux of surface-gas pseudocomponent, lbm/ft F o = the flux of surface-oil pseudocomponent, lbm/ft h = reservoir thickness, ft k = absolute permeability, md k rg = gas phase relative permeability k ro = oil phase relative permeability m(p) = single-phase gas pseudopressure, psi 2 /cp m g = surface-gas pseudocomponent pseudopressure, psi.scf/(rb.cp) m o = surface-oil pseudocomponent pseudopressure, psi.stb/(rb.cp) m i = i-th component, pseudopressure, psi.lbmole/cp MW gsc = gas molecular weight at standard condition, lbm/lbmole MW osc = oil molecular weight at standard condition, lbm/lbmole N i = moles of i-th component per reservoir pore volume, lbmole/ft 3 N i,o = moles of i-th component per reservoir pore volume at initial condition, lbmole/ft 3 n i = molar rate of i-th component, lbmole/d p = pressure, psia p sc = standard condition pressure, psia p ref = reference pressure, psia p i = initial reservoir pressure, psia p cgo = gas-oil capillary pressure, psi p g = gas phase pressure, psia p o = oil phase pressure, psia p dew = dewpoint pressure, psia p wf = well bottom-hole pressure, psia p wf,sp = specified well bottom-hole pressure, psia q gsc = surface-gas rate, SCF/D q gsc,sp = specified surface-gas rate, SCF/D q osc = surface-oil rate, STB/D r p = effective pore radius, nm r = radial-cylindrical distance, ft r w = wellbore radius, ft R s = solution gas-oil-ratio, SCF/STB R v = volatile oil-gas-ratio, STB/SCF S o = oil phase saturation S g = gas phase saturation S oc = critical oil saturation S o,i = initial oil saturation x

11 T = reservoir temperature, F or R T sc = standard condition temperature, F or R t = time, day t sp = specified time of interest, day v t = unit pore volumes occupied by reservoir fluid, ft 3 v o = pore volumes occupied by reservoir oil phase per pore volume, ft 3 v g = pore volumes occupied by reservoir gas phase per pore volume, ft 3 W gsc = accumulation term of the surface-gas pseudocomponent, SCF/RB W osc = accumulation term of the surface-oil pseudocomponent, STB/RB W i = accumulation term of i-th component, lbmole/ft 3 x = linear distance, ft x f = fracture half length, ft x i = molar fraction of i-th component in oil phase X = Boltzmann independent similarity variable for radial flow geometry, ln (ft/(md.day) 0.5 ) y i = molar fraction of i-th component in gas phase Y = dependent variable for similarity method in single-phase gas flow, psi 2 /(cp.day 0.5 ) Y g = dependent variable for similarity method for surface-gas pseudocomponent equation, SCF/(RB. cp.day 0.5 ) Y o = dependent variable for similarity method for surface-oil pseudocomponent equation, STB/(RB. cp.day 0.5 ) Y i = dependent variable for similarity method for i-th component equation, lbmole/(ft 3. cp.day 0.5 ) z i = overall composition (molar) of i-th component z i,o = overall composition (molar) of i-th component at initial condition z g = overall composition (mass) of surface-gas pseudocomponent z o = overall composition (mass) of surface-oil pseudocomponent Greek α = similarity stretching exponent (independent variable) β = similarity stretching exponent (dependent variable) γ = similarity stretching exponent (dependent variable) η = Boltzmann independent similarity variable for linear flow geometry, ft/(md.day) 0.5 ξ = similarity multiplier λ gsc = mobility of the surface-gas pseudocomponent, SCF/(RB*cp) λ osc = mobility of the surface-oil pseudocomponent, STB/(RB*cp) λ i = mobility of the i-th component, lbmole/(rb*cp) λ go = mobility of the surface-gas pseudocomponent contributed by reservoir oil phase, SCF/(RB*cp) λ oo = mobility of the surface-oil pseudocomponent contributed by reservoir oil phase, STB/(RB*cp) λ og = mobility of the surface-gas pseudocomponent contributed by reservoir gas phase, SCF/(RB*cp) λ gg = mobility of the surface-oil pseudocomponent contributed by reservoir gas phase, STB/(RB*cp) μ g = gas phase viscosity, cp μ o = oil phase viscosity, cp ρ g = gas phase density at reservoir condition, lbm/ft 3 ρ o = oil phase density at reservoir condition, lbm/ft ρ gsc = gas density at standard condition, lbm/scf ρ osc = oil density at standard condition, lbm/scf ρ gsc = gas molar density at standard condition, lbmole/scf φ = porosity σ = interfacial tension, dyne/cm ω gg = mass fraction of surface-gas pseudocomponent in reservoir gas phase xi

12 ω og = mass fraction of surface-oil pseudocomponent in reservoir gas phase ω go = mass fraction of surface-gas pseudocomponent in reservoir oil phase ω oo = mass fraction of surface-oil pseudocomponent in reservoir oil phase χ i = parachor of i-th component χ g = parachor of surface-gas pseudocomponent χ o = parachor of surface-oil pseudocomponent Acronyms BDF Boundary Dominated Flow BHP Bottom Hole Pressure BVP Boundary-Value Problem CCE Constant Composition Expansion CVD Constant Volume Depletion FVF Formation Volume Factor GOR Gas Oil Ratio IVP Initial Value Problem. IFT Interfacial Tension LRG Liquid-rich Gas ODE Ordinary Differential Equation OGIP Original Gas in-place PDA Production Data Analysis PDE Partial Differential Equation xii

13 ACKNOWLEDGEMENTS My deepest gratitude goes to my academic adviser and mentor, Dr. Luis F. Ayala, for his professional guidance and continuous support throughout my graduate study. I would never be able to pursue or complete this doctorate without his supervision and encouragement. My thankfulness extends to Dr. Zuleima Karpyn, Dr. Shimin Liu, and Dr. Mathieu Stienon for their interest and time in serving as members of my doctoral committee. Their suggestions and contributions to this work are highly appreciated. I wish to express my gratitude to the John and Willie Leone Family Department of Energy and Mineral Engineering, members of the Unconventional Natural Resources Consortium (UNRC) at The Pennsylvania State U., George H. Deike, Jr. Research Grant in the College of Earth and Mineral Sciences, and the Foundation CMG for funding our work. I thank Dr. Mathew D. Becker for his help in preparing the manuscript and figures as a co-author of some of the publications that were part of this work. I would like to specially thank Dr. Pichit Vardcharragosad for continuous support and unfailing friendship throughout my graduate study. Finally, I deeply dedicate this dissertation to my parents and my fiancé Qian Sun for their love and support. xiii

14 INTRODUCTION Exploitation of unconventional liquid-rich gas (LRG) reservoirs, including shale gas, tight gas, and coalbed methane, is becoming increasingly important to the energy supply in North America. From 2005 to 2013, in the United States alone, natural gas production increased by 35%, accounting for an increase from 23% to 28% of the natural gas share of total energy consumption in the USA during that time (EIA 2015). Despite the increase in demand, unconventional LRG reservoirs exhibit a number of technical challenges that are fundamentally different from conventional reservoirs. They are directed related to the extensive early-transient infinite acting behavior exhibited by LRG systems, the presence of liquid dropout, and ensuing multi-phase flow of gas and condensate. Additional complexities in these unconventional systems include formation permeability (Javadpour 2009), flow regimes resulting from well completion (e.g. multistage fractured horizontal wells) (Clarkson 2013a), and behavior of hydrocarbons in place as a result of nano-scale pores (Loucks et al. 2009, Jin and Firoozabadi 2015). Because of these fundamental differences, production analysis methods traditionally employed for conventional reservoir analysis fail to successfully estimate and forecast production behavior in LRG systems and new formulations are needed for unconventional reservoir system analysis (Clarkson et al., 2012, Clarkson 2013a, Clarkson 2013b). Production data analysis (PDA) methods aim to quantitatively interpret well production data and generate predictions of fundamental characteristics of the reservoir in question, often by invoking analytical solutions to the governing reservoir flow equations (Ilk, Anderson et al. 2010). Information that can be estimated using PDA techniques includes estimated ultimate recovery (EUR), original gas in place (OGIP), fracture characteristics, reservoir permeability, and/or stimulated reservoir volume. For gas reservoirs analysis, PDA techniques traditionally assume single-phase gas flow and many do not consider liquid-phase dropout that is often observed during the production life of LRG reservoirs. In many studies, analysis of single-phase gas flow routinely involves linearization of the non-linear gas flow equations by expressing the equations in terms of pseudopressure (Al-Hussainy et al.,1966) and pseudotime (Agarwal 1979). This approach subsequently transforms the governing equations to a form that can be solved with existing methods for non-compressible fluid flow, and has been traditionally employed to develop type-curve or straight-line relationships for late-time boundary dominated flow (BDF) in both linear and radial regimes (Fetkovich 1980, Fraim 1987, Wattenbarger, El-Banbi et al. 1998). These traditional approaches are strongly biased toward single-phase and boundary-dominated analysis. For early-transient analysis of single-phase gas flow, an extension of this approach to infinite-acting analysis has been proposed in which pseudovariables are evaluated in a region of influence (Anderson and Mattar, 2007; Nobakht et al., 2011). To account for multiphase flow effects, two-phase pseudopressure and pseudotime have been introduced (Camacho and Raghavan, 1989; Sureshjani and Gerami, 2011) and also applied to early-transient linear 1

15 flow coupled with the region of influence concept (Behmanesh et al., 2015). Despite these efforts, the twophase pseudovariable transformation approach for multi-phase flow presents an important limitation: pressure-saturation data (typically estimated in the laboratory) and/or producing gas-oil-ratio data are required to be obtained a priori in order to evaluate pseudopressure and pseudotime. An additional difficulty is that laboratory-estimated pressure-saturation results could misrepresent the actual behavior of the reservoir, especially in the case of LRG reservoirs (Whitson and Sunjerga, 2012). This study utilizes a similarity-based method to solve the governing partial differential equations (PDE) of early-transient flow behavior in LRG. The similarity solution which is first introduced by Boltzmann in 1894 is a method developed based on the algebraic symmetry of PDEs that allowed them to be transformed and solved in terms of ordinary differential equations (ODE). Without the use of pseudovariables to linearize the PDEs, the solutions of governing nonlinear PDEs are directly achieved by solving the transformed nonlinear ODEs; and thus are found particularly useful in solving highly nonlinear problems to many physical problems in diverse fields, such as heat and mass transfer and fluid dynamics (O Sullivan and Pruess, 1980; O Sullivan, 1981; Pruess et al., 1987; Doughty and Pruess,1990 and 1992). Previous studies using similarity solution to study multiphase flow in LRG are limited to approximate analytical solutions (Bøe et al., 1989; Clarkson and Qanbari, 2015; Behmanesh et al. 2015b) due to the difficulty in obtaining closed-form analytical solutions from the associated multiphase equations with multiple non-linear terms. In the present work, the transformed system of nonlinear ODEs involved in multiphase flow is solved via numerical ODEs solver Runge-Kutta method. In this way, straightforward yet rigorous semi-analytical solutions are obtained without the use of additional approximation or linearization procedure. Chapter 1 employs the similarity theory to solve non-linear PDEs of single-phase gas flow under both linear and radial geometries and constant pressure and constant rate specifications. The proposed similaritybased semi-analytical method is fully able to handle the associated non-linearities in gas flow--pressuredependent gas properties, such as viscosity and compressibility. Proving the proposed similarity-based semi-analytical method to be a robust and rigorous solution, Chapter 2 successfully extends it to solve the multiphase flow diffusivity equations under black-oil formulation for linear and radial flow regimes under constant bottomhole pressure and constant rate conditions. The nonlinear pressure- and saturationdependent terms involved in the system of multiphase PDEs are successfully handled by proposed solution which solves pressure and saturation simultaneously and rigorously. Using the proposed solution, we also explore the sensitivity of pressure-saturation relationship under the effects of bottomhole flowing pressure, initial reservoir pressure, and relative permeability characteristics. Chapter 3 applies the multiphase semianalytical solution presented in Chapter 2 to GOR predictions of unconventional reservoirs under various 2

16 flow regime and well production constraints. By solving the equations for pressure and saturation, the GOR trend and value can be fully predicted prior to availability of production data. Following the evaluation of GOR under infinite-acting effect using proposed solution, we further discuss the GOR responses under the effect of closed outer boundary by comparing with finely-gridded numerical simulation results. In addition to the multiphase solution presented in Chapter 2 that uses black-oil formulation, Chapter 4 solves governing PDEs written for each component under a fully compositional formulation. Pressure and compositions profiles are accurately solved using the proposed similarity-based semi-analytical approach. Solving governing compositional equations, we provide two options in this study: 1) solving for overall composition and 2) solving for the overall molar density of each component. In Chapter 5, we focus on capturing the effect of capillary pressure not only on fluid flow but also on fluid properties in the multiphase unconventional systems, in the context of a black-oil fluid formulation. Similarity theory coupled Rung-Kutta solver are also implemented. We compare the solved results using the proposed method considering two assumptions of capillary pressure: interfacial-tension-independent and interfacial-tension-dependent Pc effects. The first three chapters of this dissertation correspond with a series of three papers either published or insubmittal. By order of chapter appearance, these papers are: Zhang, M., Vardcharragosad, P., Ayala H., L.F., The similarity theory applied to early-transient gas flow analysis in unconventional reservoirs. Journal of Natural Gas Science and Engineering, v. 21, pp Zhang, M., Becker, M.D., Ayala H., L.F., A Similarity Method Approach for Early-Transient Multiphase Flow Analysis of Liquid-Rich Unconventional Gas Reservoirs. Journal of Natural Gas Science and Engineering, v. 28, pp Zhang, M., Ayala, L.F., Analytical Study of Constant Gas-Oil-Ratio Behavior as an Infinite- Acting Effect in Unconventional Multiphase Reservoir Systems. SPE Journal. In press. My co-authorship in the following paper implies the principal role in the methodology development and model validation: Becker, M.D., Zhang, M., Ayala H. L.F. On The Pressure-Saturation Path in Infinite-Acting Unconventional Liquid- Rich Natural Gas Reservoirs. Manuscript submitted: Journal of Natural Gas Science and Engineering. 3

17 1 Similarity Solutions for Gas Flow during Early-Transient Periods 1.1 Chapter Summary Production data analysis of unconventional reservoirs focuses on early transient behavior analysis within these systems, which are affected by pressure-dependent rock and fluid properties. However, available analytical solutions for early transient behavior analysis are strictly applicable to liquid systems. For gas systems with pressure-dependent fluid properties, the pseudopressure is known not to fully linearize the associated partial differential equation the use of pseudotime as a mean to account for the remaining nonlinearity is strictly applicable to late-time boundary-dominated conditions. Attempts have been made at correcting the pseudotime concept to match early-transient data; however, such early transient gas solutions remain dependent on how the region of influence is defined and how average pressure is calculated. In this work, the similarity theory is applied to solve these non-linear diffusivity equations describing early transient behavior for both linear and radial geometries and constant pressure and constant rate specifications. It is shown that the similarity theory is fully able to handle the associated non-linearities and, for each case, transforms the partial differential equation into a system of ordinary differential equations (ODEs), which are straightforwardly solved using well-known ODE solvers. Study results demonstrate that the early transient behavior of gas systems, which dominate the behavior of unconventional reservoirs, can be successfully captured through similarity via Runge-Kutta. 1.2 Background In 1894, Boltzmann introduced a special solution procedure for the concentration-dependent diffusion problem, which transformed of the associated partial differential equation (PDE) into a simple ordinary differential equation (ODE) amenable to analytical solution. In 1950, Birkoff recognized that Boltzmann s work was based on the algebraic symmetry of PDEs that allowed them to be transformed and solved in terms of ODEs. The technique is known as the similarity method, commonly applied in classical textbooks to analytically solve some important transport phenomena problems (Carslaw and Jaeger, 1959; Crank, 1980; Bird et al., 2012). Dresner (1983) provides a detailed discussion on similarity solutions and their extensive applications to many physical problems in diverse fields, such as heat and mass transfer and fluid dynamics. The similarity method has proven particularly useful in solving highly nonlinear problems without linearization. For example, O Sullivan and Pruess (1980) and O Sullivan (1981) proposed similarity solutions to highly nonlinear multi-phase flow problems in geothermal well test analysis. Pruess et al. (1987) 4

18 worked on the same problems but focused on moving evaporation front. Doughty and Pruess (1990, 1992) further extended the original similarity solutions to problems involving heat transfer and different fluids such as nuclear waste and air. In the petroleum field, similarity method has been applied to analyze early transient behavior of fluid flow under radial flow regime leading to the well-known exponential integral solution used in well test analysis. Similarity method has also been applied in the study of multiphase flow in solution gas-drive and gas condensate reservoirs (Bøe et al., 1989; Vo 1989; Raghavan, 1993). Ayala and Kouassi (2007) presented a Runge-Kutta-based general similarity solution applicable to early-transient multiphase flow conditions in radial gas condensate reservoirs producing at constant rate. More recently, Qanbari and Clarkson (2013) and Chen and Raghavan (2013) independently presented a similarity-based methodology for infinite-acting gas linear flow analysis for constant bottomhole pressure specification. Qanbari and Clarkson (2013) further proposed the implementation of an iterative technique to solve the resulting ODE in terms of an infinite integral. Behmanesh et al. (2013) also derived similarity-based ODE forms applicable to the linear flow geometry for multiphase flow analysis in gas condensate reservoirs. In this study, we showcase the general yet unique nature of the similarity method as applied to the study of a system with strong non-linearities such as infinite-acting gas flow in radial and linear flow geometries produced under constant-rate and constant-pressure production conditions. The nonlinearities considered in this work are pressure-dependent gas properties, such as viscosity and compressibility, but the methodology is shown to be general and readily applicable to other types of non-linearities as required. Infinite-acting conditions are known to be present for extended period of times in unconventional reservoirs; thus the technique is particularly suitable to obtain analytical solutions for the study of such systems. More importantly, the system of ODEs solved in this study yields unique solutions regardless of the strong nonlinearities associated with gas flow. Unique solutions are shown for each of the initial and boundary conditions and flow geometries considered in this study, which are discussed in detail in this chapter. 1.3 Similarity Solutions for Gas Early Transient Flow in a Linear System Appendix A shows that the governing PDE of gas linear flow can be transformed to an ODE for infiniteacting conditions by identifying the appropriate similarity variables η (single independent variable) and Y (single dependent variable). It is also shown that the resulting ODEs and similarity variables take different forms depending on the prevailing inner boundary condition, which in this study include the constantpressure and constant-rate boundary conditions. Table 1-1 presents the similarity variables and resulting ODE forms applicable to gas early-transient linear flow analysis. It is noted that the principal ODEs in Table 1-1 (Equation A-19 and Equation A-23) could be readily and analytical solved by direct integration 5

19 for liquid systems (Raghavan, 1993). However, these ODEs are nonlinear in this study due to the presence of the pressure dependent term μ g (p)c g (p) in their right hand side. The classical fourth-order Runge-Kutta (R-K) method with adaptive step-size (Press et al., 2007) is implemented to generate the semi-analytical solutions for these ODEs. The build-in ODE solver function ode45 provided by MATLAB can also be used for this purpose. By using this R-K technique, pressure-dependent terms in the principal ODEs do not need to be linearized. The application of R-K techniques entails the definition of a set of first-order ODEs. In this study, as shown in Table 1-1, this is accomplished by reducing the original second-order ODE to a set of two first-order ODEs by simply rewriting them in term of two new dependent variables or functions namely f 1=Y and f 2=dY/dη also illustrated in the table Linear Flow: Constant-pressure Solution Equations (A-19) to (A-21) in Appendix A represent the second-order ODE and associated boundary conditions describing constant-pressure, infinite-acting linear flow analysis using similarity variables. As shown in Table 1-1, this system can be reduced to the following set of first-order ODEs by defining f 1 = Y and f 2=df 1/dη to yield: df 1 dη = f 2 df 2 dη = φμ gc g η k 2 f 2 Equation 1-1 with: Equation 1-2 f 1 (η = 0) = m(p wf,sp ) for the η-based inner boundary condition and: Equation 1-3 lim f 1 = 0 η Equation 1-4 For the η-based outer boundary condition. In these equations, η = 0 represents the wellbore condition (x = 0, any time) and η represents both initial conditions (t = 0) and the far from the wellbore condition (x ). Table 1-1 Summary of similarity variables and ODEs for gas flow The Linear Case 6

20 Y, ODE dependent variable η, ODE independent variable Type of inner-boundary production specification Constant-pressure Constant-rate m(p) m(p)/ t x/ t x/ t Principle ODE (second-order) and boundary conditions d 2 Y = φμ gc g dη 2 k η dy 2 dη Y(η = 0) = m(p wf,sp ) lim Y = 0 η (Eq. A-19) (Eq. A-20) (Eq. A-21) d 2 Y = φμ gc g dη 2 k ( 1 2 Y η 2 dy ) dη ( dy ) = q gsc,spρ gsc d η η=0 A w k lim Y = 0 η (Eq. A-23) (Eq. A-24) (Eq. A-25) Equivalent set of first-order ODEs and boundary conditions [ f 1 = y ; f 2 = df 1 dη ] df 1 dη = f 2 (Eq. 1-1) df 2 = φμ gc g dη k η f 2 2 (Eq. 1-2) f 1 (η = 0) = m(p wf,sp ) (Eq. 1-3) lim η f 1 = 0 (Eq. 1-4) df 1 dη = f 2 (Eq. 1-6) df 2 = φμ gc g dη k ( 1 2 f 1 η 2 f 2) (Eq. 1-7) f 2 (η = 0) = q gsc,spρ gsc A w k (Eq. 1-8) lim η f 1 = 0 (Eq. 1-9) This set of ODEs can be straight forwardly integrated via the Runge-Kutta method within the 0 < η < domain (Press et al., 2007). For numerical purposes, a finite boundary value must be selected for the upper limit of integration (η max ) where the condition lim f 1 = 0 (which also implies η η max lim f 2 = 0) is to be η η max satisfied. For the cases investigated, the value η max = 1000 proved fully adequate and capable to capture the required outer boundary condition. Naturally, the final solution must remain unaffected by the choice of η max. An additional consideration is that the Runge-Kutta method typically marches from the beginning (η = 0) to the end (η = η max ) of the domain. This requires that the values of both f 1 and f 2 must be known at a starting point (η = 0). While f 1 is known and specified by the boundary condition in Equation 1-3, an initial guess must be made for f 2 at η = 0 before the solution can be marched forward to η = η max. Once f 1 and f 2 are known at η = 0, values of f 1 and f 2 are sequentially calculated from η = 0 to η = η max via the Runge-Kutta solver. The proper choice of f 2 at η = 0 corresponds to the one that allows f 1 and f 2 values to vanish (f 1 = f 2 = 0, within a tolerance) at η = η max. A single-variable Newton Raphson search is best 7

21 implemented to accelerate the updating of f 2 at η = 0. Following Equation A-22, gas rate at the inner boundary (gas production) can be readily calculated once the value of f 2 at η = 0 is attained using this procedure. It follows that: q gsc (η = 0) = A wk ρ gsc 1 t f 2 η=0 Equation 1-5 where A w is the cross sectional flow area at the inner boundary location. This equation suggests that the slope of 1/q gsc vs. t can be used to evaluate completion and reservoir properties (A w k) which is routinely done for infinite-acting linear flow data. It should be noted, however, that the value of f 2 η=0 plays an important part in this calculation. Table 1-2 Reservoir formation and gas fluid properties Initial pressure, p i 3,000 psia Permeability, k 0.1 md Porosity, φ 0.1 Inner boundary cross sectional flow area, A w 1,000 ft 2 (linear case) Reservoir thickness, h 50 ft Gas specific gravity 0.65 Temperature, T 200 F In order to evaluate the ability of the proposed semi-analytical method to predict infinite-acting, constantpressure, gas linear flow production, similarity solutions are compared against results from finely- (logarithmically) gridded, single-well numerical reservoir simulation (CMG-IMEX, 2012). Input data are presented in Table 1-2, which will remain the same for all other cases presented in this paper to facilitate comparisons. Reservoir outer boundary was placed at 50,000 ft in the simulator to avoid boundary effects in the early data. Gas fluid properties are calculated using Dranchuck and Abou-Kassem (1975) for Z-factor, Lee et al (1966) for gas viscosity, Abou-Kassem et al (1990) for gas isothermal compressibility, and Sutton (1985) for pseudo-critical property calculations. Figure 1-1 demonstrates that a unique pressure profile can be obtained, regardless of time and location, when results are plotted in terms of the similarity variable (η = x/ t) for the case p wf,sp = 500 psia. In Figure 1-1, numerical results generated from the numerical simulator are compared against the similarity-based solution proposed in this study. Total simulated time was 3,000 days, at which point pressure at outer boundary was still equal to initial pressure confirming the infinite-acting nature of the data. In this figure, pressure profiles reported by the simulator correspond to pressure vs. time data collected at fixed locations x=100 ft and x=500 ft and pressure vs. distance data registered at times t=365 days and t=2072 days. The 8

22 excellent agreement demonstrates the validity of the proposed approach. In addition, Figure 1-2 presents the flow rate decline predicted by the semi-analytical Runge-Kutta methodology for a number of bottomhole flowing pressure (BHP) specifications (p wf,sp=500, 1,500, 2,000 and 2,500 psia). Markers in this figure represent the flow rate response generated by the numerical simulator and solid lines represent predictions from the proposed similarity solution. Again, excellent matches are observed, further confirming the validity and applicability of the proposed solution to gas linear flow analysis under constantpressure production specifications Proposed solution 365 days days Grid block #100 Grid block # Figure 1-1 Pressure vs. similarity variable for constant-pressure pwf,sp=500 psia linear system 9

23 q gsc (SCF/D) p wf =500psia p wf =1500psia p wf =2000psia p wf =2500psia Proposed solution t(days) Figure 1-2 Flow rate response for linear constant-pressure condition Linear Flow: Constant-rate Solution Appendix A demonstrates that the similarity-based ODE describing constant-rate, infinite-acting linear flow is given by Equation A-23 and the corresponding boundary conditions are stated in Equations A-24 and A-25. By defining f 1 = Y and f 2=df 1/dη, as shown in Table 1, this system can be reduced to the following set of first-order ODEs: df 1 dη = f 2 df 2 dη = φμ gc g ( 1 k 2 f 1 η 2 f 2) Equation 1-6 with the following inner and outer boundary conditions: Equation 1-7 f 2 (η = 0) = q gsc,spρ gsc A w k Equation

24 lim f 1 = 0 η Equation 1-9 Again, this set of ODEs can be straight forwardly integrated via the Runge-Kutta method within the 0 < η < η max domain (with η max = 1000, as stated earlier). Also, Runge-Kutta requires starting values of f 1 and f 2 at η = 0 before it can march from η = 0 to η = η max. In this constant-rate scenario, f 2 is known and specified by Equation 1-8 but the starting f 1 must come from an initial guess bottomhole pressure guess. The proper choice of f 1 at η = 0 is the one that make f 1 and f 2 go to zero (within a tolerance) at η = η max. A single-variable Newton Raphson search is typically implemented to speed up f 1-updating at η = 0. Once f 1 η=0 is known, the associated BHP at the inner boundary condition is calculated. It should be recalled that the definition of f 1 is m/ t for constant rate systems (see Table 1-1), as opposed to m in constant pressure systems. Therefore, a unique, universal pressure profile such as the one presented in Figure 1-1 is no longer possible for the constant-rate gas solution. It would be possible, however, for a liquid system for which the viscosity-compressibility product in Equation 1-7 would be constant and a unique profile for f 1 would follow. For the gas case, Equation 1-7 is non-linear because of the presence of the μ g c g term. In these cases, values of time are required to map any value of f 1 to the pressure needed to evaluate the pressure-dependent μ g c g coefficient. As a result, gas linear flow solution under a constant-rate constraint will yield time-dependent profiles for both f 1 and pressure. In order to verify applicability of the proposed similarity solution for infinite-acting gas linear flow a under constant-rate constraint, BHPs at different constant-rate specifications (q gsc,sp = 1, 2, 5 and 7 MSCF/D) are generated and investigated using numerical simulation. Figure 1-3 compares BHP predictions from the proposed similarity method (solid lines) against results reported from numerical simulation (as markers). Excellent matches are observed for all conditions. In Figure 1-4, pressure versus distance profiles are examined for the case q gsc,sp = 5 MSCF/D for times t=30, 150, 365 and 730 days. Data points shown as markers correspond to numerically simulated results. They are compared against the semi-analytical solution using our proposed method shown in solid lines. Agreement is excellent, which further corroborates the strength of the proposed approach. 11

25 p(psia) BHP(psia) q gsc =1MSCF/D q gsc =2MSCF/D 1600 q gsc =5MSCF/D q gsc =7MSCF/D Proposed solution t(days) Figure 1-3 Bottomhole pressure responses for linear constant-rate cases days 150 days days 730 days Proposed solution x(ft) r(ft) Figure 1-4 Pressure profiles for linear constant-rate case: qgsc,sp= 5 MSCF/D 12

26 1.4 Similarity Solutions for Gas Early Transient Flow in a Radial System Appendix B derives the similarity-based principle ODEs, along with boundary conditions, applied to gas radial flow for both constant-pressure and constant-rate constraints in terms of the similarity variables η and y. These equations are summarized in Table 1-3. Unlike the linear case (Table 1-1), similarity variables and principle ODE are identical for both production constraints in radial systems. For the radial case, the f 1 and f 2 functions used to reduce the second-order principle ODE to a set of first-order ODEs are defined as f 1=Y and f 2=η df 1/dη. Y, ODE dependent variable η, ODE independent variable Principle ODE (second-order) Table 1-3 Similarity variables and principle ODE form for gas flow The Radial Case Type of inner-boundary production specification Constant-pressure Constant-rate (η w = 0 if line-source wellbore representation is used) d dη dy (η ) = φμ gc g dη k m(p) r/ t η 2 dy 2 dη (Eq. B-10) Principle ODE boundary conditions Y(η = η w ) = m(p wf,sp ) (Eq. B-12) lim Y = 0 η (Eq. B-11) (η dy d η ) η=η w = q gsc,spr w lim Y = 0 η A w k ρ gsc (Eq. B-13) (Eq. B-11) Equivalent set of first-order ODEs [ f 1 = Y ; f 2 = η df 1 dη ] df 1 = f 2 dη η df 2 = φμ gc g dη k η 2 (Eq. 10) 2 f 2 (Eq. 11) First-order ODEs boundary conditions f 1 (η = η w ) = m(p wf,sp ) (Eq. 1-12) f 2 (η = η w ) = q gsc,spr w A w k ρ gsc (Eq. 1-15) lim η f 1 = 0 (Eq. 1-13) lim η f 1 = 0 (Eq. 1-16) 13

27 1.4.1 Radial Flow: Constant-pressure Solution Appendix B shows that the similarity-based ODE describing constant-pressure, infinite-acting radial flow is Equation B-10 with boundary conditions in Equations B-11 and B-12. As stated in the Appendix B, the inner boundary condition is imposed at η = η w in this study, rather than at η = 0, in order to make the constant-pressure solution possible. As a result, as shown below, the resulting solution becomes available for a specified time of interest (t sp). In order to make this second order ODE system amenable to a Runge- Kutta integration, the functions f 1 = m(p)=y and f 2= η df 1/dη are defined yielding the following set of first-order ODEs and boundary conditions (see Table 1-3): df 1 dη = f 2 η df 2 dη = φμ gc g η 2 k 2 f 2 with the associated initial and boundary conditions can also be written in term of f 1 given by: Equation 1-10 Equation 1-11 f 1 (η = η w ) = m(p wf,sp ) lim f 1 = 0 η Equation 1-12 Equation 1-13 As before, the set of ODEs is integrated via Runge-Kutta by marching the solution from η = η w (a value that is fixed once the time of interest t sp is specified, with η w = r w / t sp ) to η = η max (where η max = 1000, as discussed earlier). Starting values of f 1 and f 2 at η = η w are required, with f 1 ηw known and specified by Equation 1-12 but with f 2 ηw needed to be discerned by Newton-Raphson updating as shown during gas linear flow analysis for the constant-pressure condition. Once f 2 ηw is identified, gas flow rate predictions at the wellbore location directly follow from Equation B-14: q gsc (η = η w ) = A wk r w ρ gsc f 2 ηw Equation 1-14 It is noted again that this Runge-Kutta implementation yields time-specific profiles (i.e., applicable to the selected t= t sp) which makes it possible to solve the constant-pressure, similarity-based ODE. 14

28 q gsc (SCF/D) The applicability of the proposed constant-pressure, radial similarity solution for infinite-acting gas flow is now verified using numerical simulation on the basis of the same reservoir and fluid properties given in Table 1-1. All input parameters remain the same with the exception of the presence of a much reduced inner boundary cross sectional flow area (A w) corresponding to a r w = 0.25 ft wellbore radius (A w = 2πr w h for radial flow). Wellbore flow rate and reservoir pressures are generated by numerical simulation for infiniteacting transient radial flow under the following drawdown specifications: p wf,sp = 500; 1,500; 2,000; and 2,500 psia. In Figure 1-5, numerically simulated wellbore gas rates (shown in markers) are compared with predicted rates from the proposed solution (solid lines). In Figure 1-6, reservoir pressure responses at different times (t=1, 30, 150 and 730 days) for the p wf,sp = 500 psia specification are presented: numerical responses are highlighted with markers and the similarity-based solutions are shown with solid lines. All scenarios show excellent agreement among them p wf =500psia p wf =1500psia p wf =2000psia p wf =2500psia Proposed solution t(days) Figure 1-5 Flow rate response for radial constant-pressure cases 15

29 q (SCF/D) gsc p(psia) day days 150 days 730 days Proposed solution r(ft) Figure 1-6 Pressure distribution for radial constant-pressure case: pwf,sp= 500 psia Radial Flow: Constant-rate Solution For the constant-rate radial solution, the applicable similarity-based second-order ODE remains the same as the preceding constant-pressure radial case (Equation B-10, see Appendix B and Table 1-3). Thus, the equivalent set of first-order ODEs with f 1 = m(p)=y and f 2= η df 1/dη remain the same (Equations 1-10 and 1-11, see Table 1-3). The appropriate initial and boundary conditions written in term of f 1 and f 2 are given by: f 2 (η = η w ) = q gsc,spr w A w k ρ gsc lim f 1 = 0 η Equation 1-15 Equation 1-16 The proposed development uses the finite-wellbore approximation where the inner boundary condition is specified at η = η w, successfully employed in the preceding section and discussed in Appendix B. The analogous similarity solution for the more classical constant-rate, line-source wellbore approximation (where the inner boundary condition is specified at η w = 0, r w = 0) was presented in Ayala and Kouassi 16

30 q gsc BHP(psia) (SCF/D) (2007) and is not further elaborated upon here. It should be noted, however, that the advantage of the linesource representation is that the ensuing Runge-Kutta procedure directly yields a unique, time-independent f 1 and f 2 versus η profile applicable to all times and radial distances. When the inner boundary condition is specified at η = η w 0, the resulting Runge-Kutta f 1 and f 2 versus η profile is only applicable to the time of interest (t sp ) used to calculate η w (η w = r w / t sp ). Both methodologies can be shown to yield virtually identical pressure versus distance responses. When the finite-wellbore approximation is used, the Runge- Kutta procedure is matched forward from η = η w to η = η max (η max = 1,000). As always, starting points for f 1 and f 2 are required at η = η w. The latter is known from Equation 1-15, and f 1 ηw is subject to Newton- Raphson updating as explained in previous sections. Figure 1-7 shows BHP responses calculated from finely-gridded numerical simulation for the constant-rate specifications q gsc,sp = 100, 200, 300 and 350 MSCF/D. Figure 1-8 displays reservoir pressure profiles for the q gsc,sp = 350 MSCF/D scenarios at times t =1, 30, 365 and 730 days. Numerical predictions are shown in markers and the proposed similarity-based solution is shown with solid lines. Excellent agreement is observed for both BHP (Figure 1-7) and reservoir pressure response profiles (Figure 1-8). This shows that the proposed similarity-based solution is able to fully capture early-transient behavior of gas flow under a constant-rate constraint in a 1-D radial-cylindrical system using finite-wellbore approximations q gsc =100 MSCF/D 1000 q gsc =200 MSCF/D q gsc =300 MSCF/D q gsc =350 MSCF/D Proposed solution t(days) Figure 1-7 BHP response for radial constant-rate cases 17

31 p(psia) day days 150 days 730 days Proposed solution r(ft) Figure 1-8 Pressure distribution for radial constant-rate case: qgsc,sp=350 MSCF/D 1.5 Concluding Remarks This study demonstrates that the similarity theory is a powerful tool readily available to the analyst to investigate early-time performance in unconventional gas reservoirs. The methodology can be successfully applied to solve early-transient, infinite-acting problems of interest for gas reservoirs exhibiting radial and linear flow geometries, both for constant-rate and constant-pressure production constraints. Nonlinearities in the principal second-order ODEs such as pressure-dependent compressibility and viscosity are straightforwardly handled by the Runge-Kutta method without the introduction of the concept of pseudotime. Solutions for all scenarios are rigorously derived and verified through number of case studies. Average relative errors, relative to full-scale reservoir simulation, are calculated to be less than 0.5% for all cases and scenarios presented in this study. Results indicate that the proposed method, based on the use of similarity theory and Runge-Kutta method, can fully capture early transient behavior of reservoirs with pressure-dependent gas properties. Successful application of this method in the gas system suggests that it can be extended to handle more complex problems of unconventional reservoirs such as multi-phase or pressure-dependent permeability, as already shown by Ayala and Kouassi (2007) for the case of constantrate production under multiphase flow conditions in gas condensate reservoirs. 18

32 2 Similarity Solution for Multiphase Flow during Early-Transient Periods 2.1 Chapter Summary In this chapter, a novel extension of the similarity method is developed to forecast multiphase flow behavior in unconventional liquid-rich gas (LRG) systems. The methodology uses the black-oil fluid formulation and considers linear and radial flow regimes under constant bottomhole pressure (BHP) and constant flow rate well conditions. Using this method, the system of governing partial differential equations is reduced to a system of ordinary differential equations solved by well-known Runge-Kutta techniques without the need for linearization. It is demonstrated that reservoir pressure and saturation behavior can be forecast simultaneously, thereby eliminating the need for pressure-saturation or producing gas-oil-ratio data as inputs to the model. In all cases explored, the similarity results compared well to numerically generated reservoir data for a variety of well BHP and gas flow rate specifications. Calculated well production metrics also successfully matched data sets, indicating that this approach can be straightforwardly extended to estimate production metrics of interest, such as liquid and gas production rates, gas-oil-ratio, or stimulated reservoir volume. Results strongly suggest that the method developed here provides a rapid and robust alternative to numerical simulation for forecasting of LRG reservoir systems. Additionally, we explore the sensitivity of pressure-saturation relationship under the effects of bottomhole flowing pressure, initial reservoir pressure, and relative permeability characteristics. Results are compared to constant composition expansion (CCE) and constant volume depletion (CVD) tests, and it is demonstrated that these tests provide very poor estimations of saturation-pressure path for all cases considered. Following the results of the sensitivity study, we use the concept of the compositionallyextended black-oil formulation to further study the physical mechanisms that control saturation-pressure path behavior, which are elucidated through an investigation of the in-situ and flowing compositions of surface-gas and oil pseudocomponents. The results of this section indicate that traditional methods for estimation of the saturation-pressure path in liquid-rich unconventional systems cannot account for the influence of system parameters on liquid-phase dropout and condensate accumulation. 2.2 Background Despite their increased role in the global energy supply, LRG systems present a number of technical challenges. Among them, traditional production data analysis methods fail to successfully estimate and forecast production behavior in these systems. This failure is directly related to the extensive early-transient 19

33 infinite acting behavior exhibited by these systems, and the added complexities involved with liquid dropout and ensuing multi-phase flow of gas and condensate. Traditional approaches are strongly biased toward single-phase and boundary-dominated analysis; and when multiphase flow is considered, pressuresaturation data (typically estimated in the laboratory) and producing gas-oil-ratio data are required input data. The similarity method transforms the governing partial differential equations (PDEs) for fluid flow into ordinary differential equations (ODEs) that are written in terms of a single independent variable that combines time and space (O'Sullivan 1981, Doughty and Pruess 1990). This method provides a distinct advantage in that well-known ODE solvers can be employed to solve the resulting system of equations and linearization of the equations is not required. This approach has been specifically applied to unconventional reservoir analysis by a number of studies where the resulting semi-analytical solutions were demonstrated to match synthetically generated cases in a number of flow regimes and well conditions (Chapter 1 of this dissertation, Behmanesh et al. 2015). Similarity transformations have also been undertaken to solve multiphase flow equations in gas condensate reservoirs; however, in those studies, pressure was the only dependent variable considered and pressure-saturation data was also required (Boe et al. 1989, Behmanesh et al. 2015), resulting in the same limitations described above. To our knowledge, no analytical production analysis method developed to date has considered pressure and saturation behavior as separate dependent variables that can be resolved simultaneously and without a priori availability of gas-oil ratio data. The present work aims to fill this knowledge gap by extending a modification of the similarity method, previously developed in the context of single-phase unconventional gas reservoirs in Chapter 1, to multiphase LRG reservoirs. Here, similarity solutions to the governing equations for multi-phase flow were derived for two inner boundary conditions (constant bottom hole pressure (BHP) and constant gas flow rate) in each of the two traditional flow regimes (linear and radial). In all cases, pressure and saturation were simultaneously considered as separate dependent variables, thereby eliminating the potential issues associated with inputting pressure-saturation data. Following case-by-case analytical development, similarity method results were compared to synthetic data generated by a finely-gridded numerical simulation. In each flow regime and boundary condition, the model s ability to predict PDA metrics of interest (e.g. gas flow rate and well bottomhole pressure) was also explored to demonstrate the capabilities of the proposed method. 20

34 2.3 Governing Flow Equations In this section, semi-analytical similarity solutions to the modified black oil model for liquid-rich gas reservoirs were developed for four scenarios: constant BHP and constant gas flow rate in each of the linear and radial regimes. In all cases, separate analytical developments were undertaken, as the similarity solutions are dependent on initial and boundary conditions. Following solution derivations for each case, presented in the following sections, simulations were conducted under two specified well conditions (constant BHP or constant rate) for each of two flow regimes (linear and radial). All simulations were conducted using a single set of PVT properties which were generated by applying a Peng-Robinson equation of state to a ten-component gas condensate fluid through a standard constant-volume-depletion (CVD) calculation (Figure 2-1) (Peng and Robinson 1976). A single set of relative permeability curves was implemented for all case studies undertaken (Figure 2-2). Relevant reservoir parameters used in this investigation are presented in Table 1, which consider a liquid-rich gas fluid found initially saturated. To generate test case data and validate similarity solutions, corresponding numerical simulations were conducted using a commercial black-oil simulator (CMG, 2012). Table 2-1 Reservoir formation and gas condensate fluid properties for case studies Absolute Permeability, k(md) 0.01 Porosity, φ (fraction) 0.1 Rock Compressibility, c R(1/psi) 0 Fracture Half Length, x f (ft) 250 Reservoir thickness, h(ft) 50 Wellbore radius, r w (ft) 0.25 Dew Point Pressure, p dew(psia) Initial pressure, p i (psia) Temperature, T(F) 250 Maximum CVD Liquid Dropout, fraction

35 B o (RB/STB), R s (MSCF/STB) and B g (RB/MSCF) R v (STB/MSCF) R v B g B o R s p (psia) Figure 2-1 PVT properties of gas condensate fluid Figure 2-2 Relative permeability curves used for all cases. 22

36 For multiphase flow analysis using a black oil fluid formulation, the governing partial differential equations for surface-gas and surface-oil are written as, respectively: [( k rg k ro + R μ g B s ) p] = φ g μ o B o k t (S g S o + R B s ) g B o k rg [( k ro + R μ o B v ) p] = φ o μ g B g k t (S o S g + R B v ) o B g Equation 2-1 Equation 2-2 For this preliminary study, capillary pressure and rock compressibility have been neglected to showcase the fundamentals of the proposed methodology, a routine assumption in production data analysis methods for unconventional reservoirs (Behmanesh et al. 2015b; Hamdi et al. 2015). These additional non-linearities, however, may be also incorporated without changes to the underlying similarity transformation technique. Here, the pressure, p, and oil saturation, S o, functions are treated and solved independently using the solution methodology described below. In this study, since we are primarily concerned with liquid-rich gas reservoirs during the early-transient flow stage, an infinite acting system is assumed where outer boundary conditions can be written as: lim p = p i and lim S o = S o,i x x Equation 2-3 To investigate cases in which gas condensate begins to form and to verify the similarity solutions under multi-phase flow conditions, all case studies are considered to begin at the reservoir dew-point, p dew. Therefore, reservoir initial conditions can be written as: p(x, t = 0) = p dew and S o (x, t = 0) = S o,i Equation 2-4 In the following sections, inner boundary conditions are presented on a case-by-case basis based on the well condition simulated. In all cases, a one-dimensional (linear or radial) reservoir model is assumed, inner boundary conditions are formulated based on the fracture/well condition of interest and case-specific derivations of the similarity solutions are presented in which pressure and saturation profiles are solved independently. Appendices C and D provide the detailed implementation of the similarity theory to the 1D linear and radial cases, respectively. 23

37 2.4 Similarity solutions for multiphase early transient flow in a linear system Linear flow: Constant pressure solution For the linear constant pressure solution, the inner boundary condition is specified by: p(x = 0, t) = p wf,sp Equation 2-5 Considering the mathematical model given by Equations 2-1 to 2-5, Appendix C presents the mathematical transformation of the model in terms of the similarity variables (η = x φ m ) and Y (Y = kt γ). Under a constant bottomhole pressure (BHP) specification, Appendix C demonstrates that the similarity transformation results in the following mathematical model, expressed in terms of the pressure derivative explicitly: d dη (λ dp gsc dη ) = η dw gsc 2 dη t d dη (λ dp osc dη ) = η dw osc 2 dη Equation 2-6 Equation 2-7 λ gsc, λ osc, W gsc and W osc are pressure- and saturation-dependent mobility and accumulation terms of surface-gas and surface-oil pseudocomponents: k ro λ gsc = k rg + R μ g B s g μ o B o λ osc = k ro μ o B o + R v k rg μ g B g W gsc = S g B g + R s S o B o W osc = S o B o + R v S g B g Equation 2-8 Since the resulting mathematical model in similarity space is a system of second-order ordinary differential equations (ODEs) in pressure, Equations 2-6 and 2-7 can be rewritten as an equivalent system of three ODEs by introducing the first derivative of pressure as a third dependent variable, λ dp gsc. The resulting dη 24

38 system of ODEs for three dependent variables, f 1, f 2, and f 3 is presented in the Constant BHP column of Table 2-2. Notably, applying the Boltzmann transformation requires that the initial and outer boundary conditions for distance and time, Equations 2-3 and 2-4, collapse to a single boundary condition for the Boltzmann variable η (i.e. where η ; see B.C.s in Table 2-2). Appendix C also provides an alternative reduced system of 1 st -order ODEs that can be solved in terms of a two-by-two matrix for both of the two production constraints. Table 2-2 System of first-order ODEs and boundary conditions for gas condensate linear flow Constant BHP Constant qgsc f1 p f2 So f3 λ gsc dp/dη df 1 dη df 1 dη = 1 f λ 3 gsc df 2 dη df 3 dη B.C.s df 2 dη = df 1 dη η 2 (λ W gsc osc p λ W osc gsc p ) df 1 dη (λ λ gsc osc p df 1 dη (λ λ gsc λ osc λ osc S gsc ) + η o S o 2 (λ osc W gsc S o df 3 dη = η 2 ( W gsc df 1 p dη + W gsc df 2 S o dη ) f 1 (η = 0) = p wf,sp lim f 1 = p i and limf 2 = S oi η η λ λ osc gsc p ) λ gsc W osc S o ) df 2 dη = R 2 L 1 df 1 L 2 R 1 dη (See Eqs. C-24 to C-27 for additional function definitions) df 3 dη = 1 ( W gsc λ gsc p + W gsc df 2 S o dη / df 1 dη ) ( 1 2 m g + η 2 λ df 1 gsc dη ) f 3 (η = 0) = q gsc,sp t 2x f h φk lim f 1 = p i and limf 2 = S oi η η Following the solution approach of Chapter 1 for a single-phase gas reservoir system, an adaptive-size fourth-order Runge-Kutta method was implemented to solve this system of nonlinear first-order ODEs subject to the resulting boundary conditions ( Constant BHP column of Table 2-2). This study has implemented both the Runge-Kutta algorithm provided by Press et al. (2007) and the built-in ode45 function in MATLAB R2015a (The MathWorks Inc., Natick MA) and achieved identical results. By using 25

39 the Runge-Kutta solution method, the highly-nonlinear pressure- and saturation-dependent terms in the principal ODEs are resolved numerically, and thus, do not need to be linearized. In the resulting boundary conditions (Table 2-2), η = 0 represents an infinite-conductivity fracture at x = 0 and the infinite-time condition (t ), and η represents both the initial condition (i.e. t = 0) and the infinite acting boundary condition (i.e. x ). For the purposes of model implementation, the infinite boundary condition in Table 2-2 is replaced by a large but finite outer boundary value η max where the condition of infinite-acting (pressure transient is not felt) is satisfied. To find the best suitable value of η max, in general, it is suggested to use an iterative scheme to search the η max that allows pressure gradient (or flux, f 3 in Table 2-2) vanishes at η = η max (i.e. f 3 η=η max f 3 η=0 = 0, within a tolerance). In this study, η max = 250 was chosen for linear systems. This choice of η max proved to be appropriate maintaining the infinite-acting boundary condition in all cases investigated. Notably, this value of η max differs from that in Chapter 1 (η max = 1000 in Chapter 1) due to the different definition of η here, which is normalized by porosity and permeability in this chapter (η = x φ kt ). It is also noted that Constant BHP boundary conditions summarized in Table 2-2 result in a two-point boundary value problem with constraints on both ends of the domain. This requires the iterative implementation of the Runge-Kutta routine via the shooting method (Press et al. 2007). In this case, the values of two dependent functions (f 1 and f 2) are prescribed on the outer boundary (η = η max ) and a third boundary condition must be also satisfied at the well condition, i.e. f 1 (η = 0) (see Table 2-2). As a result, f 3 (η = η max ) at the outer boundary is to be refined iteratively until the resulting value for f 1 (η = 0) has converged to the specified BHP upon completion of the Runge-Kutta integration. Secant method is implemented to carry out the iterative scheme in this chapter. Symbols represent numerically generated results for the case specified in the legend and lines represent the corresponding similarity solutions in all the figures in this section. The proposed multiphase similarity model was employed to simulate four different BHP parameterizations and simulation results were compared against numerically generated synthetic reservoir responses. Specified well BHPs in the four cases were 500, 1000, 1500, and 2000 psia and the initial reservoir pressure was defined to be the fluid dew point pressure at 250 F, psia (Table 2-1) and results are presented in Figures 2-3 to 2-6. These figures demonstrate that, in all cases, the proposed similarity solution provided nearly identical results to the numerically generated cases for pressure and saturation profiles in terms of the similarity variable, (Figure 2-3 and 2-4). The availability of these unique profiles in similarity space allows them to be translated to a myriad of time and distance profiles by a simple transformation of the similarity variable to the independent variable of interest (in this case, t or x). This capability is 26

40 demonstrated here, where the similarity solution for BHP of 500 psia in Figure 2 is translated to pressure and saturation versus distance profiles for four different time points of interest: 90, 181, 273, 350 days as presented in Figures 2-5 and 2-6. In all cases, results match the numerically generated reservoir pressure and saturation data very closely, indicating the ability of the similarity method to capture both properties accurately in space and time by solving for unique similarity profiles. It must be highlighted that pressure and saturation are treated as separate dependent variables and thus independently resolved by proposed method. Previous work in the area of multi-phase flow linear analysis had utilized Boltzmann transformations to solve the multi-phase flow equations, but the pressure-saturation relationship was used as an input parameter required to be known before model implementation (Behmanesh et al. 2015). In this study, it is demonstrated that the saturation response during infinite-acting multiphase conditions may and should be predicted independently and should not be required as model input. To better showcase the potential and validity of this similarity approach in the forecasting of infinite-acting multiphase production conditions, flow rate predictions for constant-bhp linear flow were also compared to numerically generated data. For a given set of reservoir properties, including fracture half-length (x f), reservoir permeability (k), porosity ( ) and thickness (h), well gas flow rates are straightforwardly calculated using (f 3 ) η=0 values resulting from the similarity solution, given that according to Darcy s law, for the case of surface-gas rate: q gsc = 2x f kh (λ gsc (p, S o ) p x ) x=0 = 2x fh kφ (f 3 ) η=0 t Equation 2-9 A similar expression can be readily derived for surface-oil rates. Predicted gas and oil flow rates calculated using similarity compare very well to numerically generated cases for all BHP specifications (refer to Figure 2-7 for gas rate predictions). Proposed method is not only able to provide reliable predictions of reservoir pressure and saturation profiles (Figures 2-3 to 2-6) but also reliable flow rate predictions for production data analysis (Figure 2-7). These calculations can be readily extended to predict other parameters of interest such as expected gas-oil-ratios and/or stimulated reservoir volumes. Figure 2-8 further demonstrates that, because of the method s ability to simultaneously and independently solve for pressure and saturation profiles in space and time, reliable estimation of reservoir-specific pressure-saturation relationship can also be derived. Note that actual pressure-saturation relationships are not unique to the given fluid sample but can also be BHP-dependent unlike what would be implied by laboratory-generated saturation-pressure data. 27

41 Oil Saturation Pressure (psia) η [ft*(md*day) -0.5 ] Figure 2-3 Pressure profiles for constant BHP specification in terms of the similarity variable η [ft*(md*day) -0.5 ] 500 psi 1000 psi 1500 psi 2000 psi Similarity Solution Figure 2-4 Saturation profiles for constant BHP specification in terms of the similarity variable 28

42 Oil Saturation Pressure (psia) Distance from fracture (ft) Figure 2-5 Pressure profiles in space for four time points under of linear constant BHP flow with BHP = 500 psia Distance from fracture (ft) 90 day 181 Day 273 Day 350 Day Similarity Solution Figure 2-6 Saturation profiles in space for four time points under of linear constant BHP flow with BHP = 500 psia 29

43 Oil Saturation Gas Flow Rate (SCF/D) 1.00E E E Time (d) Similarity Solution 500 psi 1000 psi 1500 psi 2000 psi Figure 2-7 Simulated flow rates in time for linear, constant BHP reservoir condition Pressure (psia) 500 psi 1000 psi 1500 psi 2000 psi Similarity Solution Figure 2-8. Simulated saturation-pressure relationships for various constant BHP specifications 30

44 2.4.2 Linear flow: Constant flow rate solution For the case of the linear flow regime with a constant flow rate specification, the solution methodology develops differently compared to the previous case. Details are provided in Appendix C. While the model governing equations defined by Equations 2-1 to 2-4 remain similar, the inner boundary condition is prescribed by Darcy s law applied at the wellbore in terms of the near-well pressure gradient, which in terms of the similarity variable and for a constant gas flow rate specification is written as: q gsc,sp (x = 0, t) = 2x f hk (λ gsc p x ) x=0 = 2x f h φk t γ 1 ( dy g dη ) = const Equation 2-10 Appendix C shows that for this boundary condition, =1 and the following system of second order ODEs results: d dη (λ dp gsc dη ) = ( 1 ( W gsc λ gsc p + W gsc S o ds o dη dη dp )) (1 2 m g + η 2 λ dp gsc dη ) d dη (λ dp osc dη ) = ( 1 ( W osc λ osc p + W osc S o ds o dη dη dp )) (1 2 m o + η 2 λ dp osc dη ) Equation 2-11 Equation 2-12 Rewriting the equations as a system of first order ODEs results in the equation set in Table 2-2, column Constant q gsc. By the same token, the inner boundary condition in Equation 2-10 can be rewritten in terms dp of λ gsc (i.e. f3 in the system of first order ODEs) to result in: dη dp (λ gsc dη ) = q gsc,sp t sp η=0 2x f h kφ Equation 2-13 where the outer boundary conditions are equivalent to those in the linear constant BHP case (Table 2-2). Two major differences arise in the similarity solution for constant flow rate in the linear domain, each with separate implications for the solution of the resulting equations. It is first noted that Equations 2-11 and 2-12 (and those in Table 2-2) are inherently different from 2-6 and 2-7 presented in the previous case as they are now also functions of the pseudopressure of the gas and oil components (m g and m o, defined in equations C-10 and C-11, which would be unknown a priori). Secondly, the inner boundary constraint in this case became time-dependent: see Equation 2-13 which now depends on a user-specified time, t sp. This type of time-dependent inner boundary condition was also described by Chapter 1 in the case of single-phase 31

45 constant-rate gas flow. As a result, the proposed similarity solution for this case could represent an approximate but potentially useful solution. Implementation of the ODE solver is similar to the previous section, with the exception of the equation differences noted above. Since the boundary condition cannot be expressed exclusively in terms of the similarity variable, similarity solutions were dependent on time and were calculated individually at the time points of interest by specifying the corresponding inner boundary condition in Equation For each specified time, the solver progression was initiated at the outer boundary, η max = 250, and calculated inwards until reaching the inner boundary ( η = 0 ). During the Runge-Kutta numerical integration, pseudopressures m g and m o are calculated as the two-phase pseudopressure difference between current pressure level and that at outer boundary (p = p i). Because pressure and saturation profiles within such pressure interval are solved with Runge-Kutta progression starting from outer boundary, m g and m o --which are functions of both p and S o--are calculated rigorously in this semi-analytical method. The shooting method described in the previous section was employed to solve the BVP defined by the equations in Table 2-2. To confirm the performance of the proposed approximate similarity solution for the linear constant gas flow rate case, pressure and saturation profiles for various flow rates (q gsc,sp of 10 MSCF/D, 25 MSCF/D, 50 MSCF/D, and 75 MSCF/D) were compared to numerically generated data. Results showed that the proposed method provided very close matches to reservoir pressure profiles throughout the domain (Figure 2-9) and oil saturation values near the inner boundary (i.e. as x 0; Figure 2-10) for each flow rate investigated. It was also observed that the liquid saturation predictions were slightly underestimated in reservoir regions that experienced the greatest rate of condensate accumulation (Figure 2-10, with maximum deviations of 9.1 %). Notably, the method does accurately forecast bottomhole pressures at the fracture for all flow rates (Figure 2-11). These results suggest that despite the approximate nature of this similarity formulation that relies on a time-dependent inner-boundary, the resulting solution was fully able to provide meaningful forecasts and reveal significant insights about the behavior of the physical system. This finding is consistent with those from previous works, where it was shown that similarity theory can prove useful even for certain problems that cannot be expressed exclusively in terms of similarity variables, but where the transformed mathematical model may still allow the analyst to gain useful physical insight into system behavior (Dresner 1983 and Chapter 1 of this dissertation). 32

46 Oil Saturation Pressure (psia) Distance from fracture (ft) 10 MSCF/D 25 MSCF/D 50 MSCF/D 75 MSCF/D Similarity Solution Figure 2-9 Pressure profiles in space at day 98 for four flow rates in the case of linear constant gas flow Distance from fracture (ft) 10 MSCF/D 25 MSCF/D 50 MSCF/D 75 MSCF/D Similarity Solution Figure 2-10 Saturation profiles in space at day 98 for four flow rates in the case of linear constant gas flow 33

47 Well Bootomhole Pressure (psia) Time (d) 10 MSCF/D 25MSCF/D 50MSCF/D 75MSCF/D Similarity Solution Figure 2-11 * Simulated bottom hole pressure in time with various flow rate specifications 2.5 Similarity solutions for multiphase early transient flow in a radial system The transformation of the multiphase black oil equations in 1-D radial-cylindrical coordinates to similarity space is shown in Appendix D. Briefly, starting from governing multiphase equations applicable to the radial flow regime, the similarity-based ODEs for infinite-acting conditions were derived in terms of the similarity variable X = ln (r φ ). In the radial system, the logarithm of the similarity variable η is kt implemented to address the near-wellbore steep pressure gradients experienced in radial regimes. Following similar development from the previous sections, both constant BHP and constant gas flow rate conditions resulted in the identical system of first-order ODEs, written in terms of the dependent functions f 1, f 2 and f 3 and provided in Table 2-3. Details on equation definitions and model development are provided in Appendix D. For the purposes of the present work, a well in the center of the radial domain of finite radius, r w, was considered for both radial flow cases investigated. This implies that the transformed inner boundary conditions in both cases are time dependent. A vanishing well radius (line source) could be assumed instead, which is particularly useful for the analysis of constant rate specifications (see, for example, Boe et al. 1989, Ayala and Kouassi 2007). In this work we demonstrate that the similarity method can be applied to the more physically relevant finite well radius case for both constant BHP and rate constraints. *Figure 2-11 replaces Figure 7 in Zhang et al. (2016). Figure 7 in Zhang et al. (2016) was erroneously generated for the condition of x f = 25 ft, k = 0.1 md and φ =

48 Table 2-3 System of first-order ODEs and boundary conditions for gas condensate radial flow Constant BHP Constant qgsc f1 p f2 So f3 λ gsc dp/dx df 1 dη df 2 dη df 3 dη df 2 dx = df 1 exp (2X) W gsc (λ dx 2 osc p df 1 dx (λ osc df 3 dx df 1 dx = 1 f λ 3 gsc λ gsc λ λ osc S gsc ) + o S o = exp (2X) 2 λ W osc gsc p ) df 1 dx (λ λ gsc osc p exp (2X) 2 ( W gsc df 1 p dx + W gsc df 2 S o dx ) λ λ osc gsc p ) W gsc W (λ osc λ osc S gsc ) o S o B.C.s f 1 (X = X w ) = p wf,sp lim f 1 = p i and lim f 2 = S o,i η η f 3 (X = X w ) = q gsc,sp 2πkh lim f 1 = p i and limf 2 = S o,i η η Radial flow: Constant bottom hole pressure solution Following the similarity transformation described in Appendix D, the inner boundary condition for the constant bottom hole pressure radial flow case can be expressed as: p(x = X w ) = p wf,sp X w (t sp ) = ln (r w φ kt sp ) Equation 2-14 Equation 2-15 Because of its definition, X w is strictly a function of the user-specified time, t sp, for a given finite well radius, r w. Because of this, solutions to the ODEs become only valid for the time of interest specified. The derived similarity solution in the radial constant BHP case is presented in Table 2-3. For numerical simulation of the resulting ODEs in all radial cases, the outer boundary was chosen to be X max = 5, which demonstrated to preserve the infinite acting condition at the outer boundary in all cases investigated. X w was specified according to Equation 2-15 as a function of the time of interest t sp. The ODEs were then solved numerically 35

49 Pressure (psia) via Runge-Kutta, by initiating the solution from the outer boundary X max, and iterating via the shooting method until p(x w ) was sufficiently close to the specified BHP, p wf,sp. The similarity solution was utilized to resolve pressure and saturation profiles for specified BHPs of 500, 1000, 1500, and 2000 psia in a well with radius of 0.25 ft. In each case, pressure and saturation profiles were resolved at approximately quarterly (i.e. ~90 day) intervals and compared to numerically generated reservoir response curves, but for brevity, only the results at t=100 days, are shown (Figures 2-12 and 2-13). These results confirm that the similarity solution was fully able to capture, and solve independently for, the expected pressure and liquid saturation response. Simulation results also demonstrated that the similarity solution was able to successfully forecast gas flow rates (Figure 2-14) Radius (ft) 500psia 1000psia 1500psia 2000psia Similarity solution Figure 2-12 * Pressure profiles at t=100 days in radial constant BHP case *Figure 2-12 replaces Figures 8(A) in Zhang et al. (2016). Figure 8(A) in Zhang et al. (2016) was erroneously generated for the condition x f = 25ft, k = 0.1md and φ =

50 Gas Flow Rate 9SCF/D) Pressure (psia) Radius (ft) 500psia 1000psia 1500psia 2000psia Similarity Solution Figure 2-13 * Saturation profiles at t=100 days in radial constant BHP case Time (d) 500 psia 1000 psia 1500 psia 2000 psia Similarity Solution Figure 2-14 Simulated flow rate in time for radial constant BHP reservoir condition *Figure 2-13 replaces Figures 8(B) in Zhang et al. (2016). Figure 8(B) in Zhang et al. (2016) was erroneously generated for the condition x f = 25ft, k = 0.1md and φ =

51 Pressure (psia) Similarity Variable X [ln(ft/sqrt(md.day))] 24d 65d 100d 200d Similarity Solution Figure 2-15 * Pressure response at various time pointes in radial constant BHP case in term of similarity variable pwf = 500 psia To further elucidate the behavior of the time-dependent similarity boundary condition, pressure profiles in similarity space from five time points from the 500 psia case were compared to numerical results (Figure 2-15). While the similarity solution effectively captures the pressure profile in each case, the effect of the time-dependent boundary condition X w is observed in the leftmost (i.e. most negative) portion of the similarity solution domain. As time increases, X w shifts to the left. Because of this domain shift, resulting similarity profiles become dependent on the time point considered. However, despite the time-dependency built into the X w variable, comparisons between the similarity results and numerically-generated cases indicate that the solution framework still provides reliable liquid saturation and pressure forecast for each time of interest. As demonstrated in all previous cases, the ability for the proposed similarity method to reliably and independently predict pressure and liquid saturation during infinite-acting conditions is a significant step forward for production data analysis methods applied to LRG systems, which to date had relied on user-provided saturation vs. pressure data for predictions Radial flow: Constant flow rate solution The radial constant rate solution derivation follows very similarly to the radial constant pressure solution, and is detailed in Appendix D. The set of first order ODEs remains identical to the constant BHP case, as *Figure 2-15 replaces Figure 9 in Zhang et al. (2016). Figure 9 in Zhang et al. (2016) was erroneously generated for the condition x f = 25ft, k = 0.1md and φ =

52 Pressure (psia) well as the outer boundary conditions (Table 2-3). The main difference arises in the treatment of the constant rate well condition at the inner boundary which becomes: q gsc,sp (r = r w, t) = 2πkh (λ gsc r p r ) r=r w = const Equation 2-16 In this case, the well boundary condition is implemented by defining the f 3 function at the inner boundary as f 3 (X w ) = q gsc,sp 2πkh, and the Runge-Kutta shooting method iterates until f 3(X w ) converges to the target flow-rate dependent boundary condition. After employing the time-prescribed inner boundary X w (Equation 2-14), the problem is solved using a similar approach to that described in the previous section. This formulation could also be developed under the assumption of zero well radius, in which case the time dependence of the inner boundary condition would be removed (i.e. Ayala and Kouassi, 2007). To confirm the performance of the similarity method in the radial regime with a constant flow rate well condition, four cases were investigated, where q gsc,sp was set at 2.5 MSCF/D, 5 MSCF/D, 7.5 MSCF/D, and 10 MSCF/D and well BHP response was compared to numerically generated results (Figure 2-16). In all cases, the BHP well response was accurately captured by the similarity solution. In addition, pressure and saturation profiles for each time point considered matched numerical results (Figure 2-17 and 2-18), indicating that the similarity solution also performed well in the radial constant gas flow rate case Radius (ft) Figure 2-16 Pressure profiles at t=60 days in radial constant rate case 39

53 Well BHP (psi) Oil Saturation Radius (ft) 2.5 MSCF/D 5 MSCF/D 7.5 MSCF/D 10 MSCF/D Similarity Solution Figure 2-17 Saturation profiles at t=60 days in radial constant rate case Time (d) 2.5 MSCF/D 5 MSCF/D 7.5 MSCF/D 10 MSCF/D Similarity Solution Figure 2-18 Simulated bottom hole pressure for four gas flow rates in radial constant gas flow rate scenario 40

54 2.6 On the So-p path and its interplay with thermodynamics and phase mobility As demonstrated in previous section, pressure -saturation relationship is highly-dependent on bottomhole conditions in both linear and radial system under constant-pressure and -rate specifications. In this section, taking the example of linear flow under constant-pressure production, the effects of constant bottomhole flowing pressure, initial reservoir pressure, and relative permeability characteristics on pressure-saturation path are explored. Results are compared to constant composition expansion (CCE) and constant volume depletion (CVD) tests, which are routinely invoked to estimate saturation-pressure path. It should be noted that, the CCE liquid saturation shown in this section is directly calculated as the ratio between volume of condensed liquid divided by the cell volume which expands with time. Some authors normalize CCE liquid dropout volume by the volume of the cell at the fluid dew point (i.e. Walsh and Lake 2003, Pedersen and Christensen 2007), and the resulting profile is referred by CCE w/ normalization in following figures Pressure-saturation relationships under various conditions We first examine the effect of bottomhole and initial conditions on pressure-saturation relationship. Relative permeability curves implemented in Figures 2-19 and 2-20 use van Genuchten tuning parameter 0.8 for modified Corey correlation and S oc = 0.1 (van Genuchten, 1980). Figure 2-21 indicates that, during infinite acting conditions, for reservoirs under identical initial pressure (p i = 4000 psia), saturation-pressure curves strongly depend on well drawdown conditions. CCE and CVD results shown in Figure 2-21 provided poor predictions of saturation-profile paths under various bottomhole flowing pressures and cannot account for the effect of drawdown pressure on liquid dropout. The saturations in the near-fracture region (i.e. at reservoir pressures near p wf) decreased with decreasing specified bottomhole flowing pressures. This behavior occurs as a result of increased condensate flux and condensate revaporization that occurs at lower BHP. Under decreased BHP (i.e. increased drawdown level), the reservoir experiences minimum pressures that are increasingly lower than the fluid dewpoint, leading to greater pressure gradients in the two-phase region. These greater fluid gradients create increased condensate flux within the reservoir, which mitigates condensate buildup in the near-fracture region, resulting in a lower oil saturation-pressure profile. Furthermore, increased condensate revaporization at lower BHP also contributes to lower reservoir saturation levels, as exemplified in the case of p wf = 150 psia. In that particular case, the reservoir fluid is completely revaporized at low pressures, as indicated by the saturation-pressure path returning to S o = 0. These results suggest that decreased bottomhole flowing pressure specification (i.e. greater drawdown and pressure gradients in the reservoir) has the capability to 41

55 mitigate the constraining effects of condensate buildup on gas flow as a result of increased fluid flux and condensate revaporization. In the results shown in Figure 2-22, it is evident that, during infinite-acting conditions, reservoir undersaturation level increases the height of the saturation-pressure curve for identical drawdown levels. This behavior occurs regardless of oil and gas relative permeability data. This is because increasing undersaturation with identical bottomhole specification leads to increased gas flux within the reservoir, caused by the higher pressure gradient between the undersaturated (p > p dew) and saturated (p <= p dew) reservoir regions. More liquid is condensed from the increase gas flux, resulting in higher oil saturation. As expected, CVD and CCE experimental data yield poor predictors of reservoir behavior for these scenarios, as it does not account for the increasing liquid dropout with increased reservoir undersaturation. These features of the saturation-pressure path strongly depend on fluid mobility within the reservoir, effects that cannot be estimated with CVD and CCE tests. For the case of the initial reservoir pressure closest to the dew point of the fluid-in-place (i.e. pi = 3000 psia), the CCE (with normalization) results provide a not-sobad prediction of the saturation profile in regions of the reservoir with pressure from ~2000 psia to the fluid dew point (i.e. p dew = psia), similar to the results of Behmanesh et al. (2015a). CCE w/ norm. CVD CCE Figure 2-19 Effect of bottomhole pressure specification on p-so relationship 42

56 CCE w/ norm. CVD Figure 2-20 Effect of Initial pressure on p-so relationship The influence of relative oil phase mobility on the saturation-pressure profile was explored by simulating reservoir evolution under the base case undersaturation and drawdown conditions with three sets of oil relative permeability curves. Relative permeability characteristics of the reservoir were manipulated by modifying the critical oil saturation, S oc, in the van Genuchten (1980) relative permeability correlations with tuning parameter = 0.8 (Figure 2-21). Using this method, the simulated profiles could be compared to a single tunable parameter, S oc. Similarity-based semi-analytical and numerical simulation results (Figure 2-22) suggest that the relative oil mobility has a significant effect on the saturation-pressure path at low pressures, especially near-fracture region. In regions of low reservoir pressure, the oil saturation decreased with decreasing critical oil saturation, suggesting that reduced oil phase mobility promotes greater condensate buildup in the near-fracture region. In regions with greater pressures, however, critical oil saturation did not strongly influence the saturation-pressure profile height. It should be highlighted, again, that CCE and CVD are unable to capture the saturation-pressure path in these scenarios. Furthermore, in the case of variable S oc, these estimations are not able to provide reliable predictions of condensate buildup in the low-pressure regions of the reservoir, behavior that strongly influences wellstream composition and producing GOR. 43

57 Figure 2-21 Different Soc curves investigated CCE w/ norm. CVD CCE Figure 2-22 Effect of critical saturation on p-so relationship 44

58 2.6.2 Compositional paths in PX diagram for the black-oil pseudocomponent formulation In order to examine pressure-saturation relationships by introducing the concept of component relative mobility (flowing composition), it is necessary to relate compositions of the two traditional black-oil pseudocomponents (namely, surface-oil and surface-gas) to black-oil PVT parameters. This subsection presents these relationships in the context of the present work as a function of well-known definitions and assumptions that describe modified black-oil fluid models (Walsh and Lake 2003): ρ o = 5.615ρ osc + ρ gsc 5.615B o ρ g = 5.615ρ oscr v + ρ gsc 5.615B g R s ρ gsc ω go = 5.615ρ osc + R s ρ gsc ρ gsc ω gg = R v 5.615ρ osc + ρ gsc Equation 2-17 Equation 2-18 Equation 2-19 Equation 2-20 where ω go is mass fraction of surface-gas pseudocomponent in reservoir oil phase, ω gg is mass fraction of surface-gas pseudocomponent in reservoir gas phase; ρ o and ρ g are reservoir oil and gas phase densities, respectively. Using Equations 2-17 to 2-20, phase compositions at reservoir conditions can be calculated exclusively in terms of pressure which a consequence of Gibbs Phase Rule applied to a two-phase binary mixture under isothermal conditions. Also, the composition of surface-gas at the fluid dew point is equal to the composition of surface-gas in reservoir gas, ω gg, and the composition of surface-gas at the fluid bubble point is equal to the composition of surface-gas in reservoir oil, ω go. Based on these relationships, the black-oil extended pressure-composition phase envelope at a given temperature (in the case examined here, 250 ºF) can be calculated at pressures below the fluid dew point pressure (in the case examined here, psia). Moreover, in situ oil saturations can be transformed into pressure-surface-gas composition space using an extension of the lever rule: f mg = z g ω go ω gg ω go Equation

59 S o (p, z g ) = 1 f mg ρ o 1 f mg ρ o f mg ρ g Equation 2-22 where f mg and f mo represent the mass fractions of gas and oil phases, respectively and f mo = 1- f mg. Using Equations 2-21 and 2-22, saturation can be mapped into p-z g space, to develop a contour map for analysis of flowing and in-situ compositions of reservoir fluid (Figure 2-23). This plot is usually called PX diagram under compositionally-extended black-oil concept (Nojabaei, 2015). Dew point and bubble point lines were extrapolated up to the critical point (above p dew = psia) in Figure 2-23 using the procedure proposed by Nojabaei and Johns (2015). Detailed procedure for the extrapolation is provided in Appendix L. To relate this PX diagram of black-oil pseudo-component to actual pressure-saturation relationship, the first and simplest in-situ composition profiles we can generate is CVD and CCE paths. The CCE path, as expected, is represented by a vertical line at initial reservoir composition. CVD gas composition continues to decrease due to the continuous removal of saturated gas. Both of their So-p profiles shown in subsection (Figures 2-19 and 2-20) are perfectly tracked in the PX diagram. Figure 2-23 P-X diagram of black-oil gas psuedocomponent and iso-saturation contour 46

60 Another important path we can identify in the PX diagram is the flowing composition. Flowing composition can be computed via the flux of gas and oil pseudocomponents, which can be computed using the relative mobility of the oil and gas phase: F g = (ω gg k rgρ g μ g F o = (ω go k rgρ g μ g + ω go k roρ o ) p μ o x + ω oo k roρ o ) p μ o x Equation 2-23 where F g is the flux of surface-gas pseudocomponent and F o is the flux of surface-oil pseudocomponent. Flowing gas pseudocomponent composition can then be calculated by the ratio between gas flux to the total flux: F g z g = flow F o + F g Equation 2-24 Following the calculation shown above, Figures 2-24 to 2-26 investigate the effects of initial condition, drawdown specification and critical oil saturation on in-situ and flowing composition paths in PX diagram. In Figure 2-24, undersaturation level is indicated by different initial pressures. It demonstrates that under different undersaturation levels, the in-situ composition paths follow very different paths from the dew point where they enter two-phase region. This is the result of different pressure gradient when entering twophase region caused by the difference between p dew and p i. To be more specific, higher pressure difference between initial and dewpoint (p i - p dew) result in increased gas flux when entering two-phase region, leading to increased liquid dropout seen with increasing initial pressure (see Figure 2-19). Figure 2-24 also indicates that the increased liquid dropout in scenarios with greater reservoir initial pressure leads to decreased z g, behavior that occurs because of the greater fraction of reservoir oil in the flowing fluid. Because flow the mass fraction of surface-gas in the oil phase is less than the mass fraction of surface-gas in the gas phase, increased flux of the oil phase contributes to decreased mass fraction of surface-gas in the flowing reservoir fluid. Figure 2-25 illustrates the effect of drawdown (p wf) on in-situ and flowing gas composition paths. It can be seen that the drawdown condition has the greatest impact on the in situ composition near the fracture (p ~ p wf), which results in significant difference in So at fracture as shown by Figure Notably, revaporization occurs in the low pressure region, thereby sharply increasing in-situ z g in Figure This sharp change of in situ composition corresponded with a sharp decrease in saturation-pressure path in that 47

61 particular case. Following complete revaporization as fluid flowing towards fracture, in-situ and flowing composition will merge at dewpoint, the reservoir fluid exists as gas exclusively, and in situ and flowing compositions remain constant and equal. Finally, in the case of changing critical oil saturation, the composition of the in situ and flowing fluids are similar throughout the domain, and only diverge at low pressure levels (i.e. near the fracture, Figure 2-26). This corresponds to the similar saturation-pressure paths observed in Figure At low reservoir pressures, decreased S oc levels corresponded to increased oil mobility, as a result of the increasing relative permeability to oil. At higher S oc levels, greater differences were observed between the in-situ and flowing compositions, indicating that increasing S oc leads to richer in situ fluid and leaner flowing fluid. Notably, in the case of S oc = 0.4, during the majority of the pressure depletion process, S o is very close to S oc, resulting in negligible oil phase mobility. This is also reflected on flowing composition path, which almost overlaps with dewpoint curve as flowing fluid is very close to saturated condition. Taken in concern with the saturation-pressure paths for these cases (Figure 2-26), this result suggests that increased buildup of lowmobility oil in the near-fracture (i.e. low-pressure) region, which was seen with increased S oc levels, was likely exacerbated by the lack of mobility of the reservoir oil phase. This, in turn, resulted in the steeper oil saturation spikes near the sandface for cases with decreased S oc. These results suggest that saturation near the sandface and thus, well-stream gas-oil-ratio (GOR) could be employed to quantify critical oil saturation. In-situ composition Flowing composition Figure 2-24 Effect of initial pressure on pressure-composition paths 48

62 In-situ composition Flowing composition Pwf Pwf Pwf Condensate is fully re-vaporized Figure 2-25 Effect of bottomhole condition on pressure-composition paths S oc = 0.4 Figure 2-26 Effect of critical oil saturation on pressure-composition paths 49

63 2.7 Concluding remarks In this chapter, a novel extension of the similarity method was employed to independently forecast pressure and saturation profiles in space in time for two-phase (gas and condensate) flow in LRG systems. In all cases explored, the reservoir pressure and saturation profiles for the similarity solutions closely matched numerically generated synthetic reservoir data. Specifically, in the linear constant BHP case, equation development resulted in a model that was expressed exclusively in terms of the similarity variable, thereby resulting in a single profile that could be transformed to represent all profiles in time and space. In the cases of linear constant gas flow rate, radial constant BHP, and radial constant flow rate, although the resulting systems of ODEs was time-dependent, and the resulting similarity solution was still able to resolve reservoir behavior at user-specified times. In all cases, simulation via the semi-analytical similarity solution provided a fast and reliable forecasting tool for unconventional LRG reservoir response, and demonstrated to be an apt alternative to full numerical simulation of these systems. It was demonstrated that pressure and liquid saturation behavior can be forecast independently and simultaneously in these systems, thereby eliminating the need for used-provided pressure-saturation or producing gas-oil-ratio data which constituted a significant limitation of currently available production data analysis tools applied to these systems. The influence of reservoir and drawdown conditions on saturationpressure path was examined in linear infinite-acting reservoir systems under constant bottomhole flowing pressure. It was found that reservoir fluid-rock interactions, undersaturation level, and drawdown pressure all have a significant influence on the shapes and heights of the reservoir saturation-pressure path. By doing so, we showcased the semi-analytical methodology that effectively forecasts and predicts all these effects, prior to availability of any production data and without relying on CVD/CCE saturation-pressure data. In all cases, CVD and CCE tests, which are traditionally employed to estimate a priori saturation-pressure relationships for simulation efforts, provided poor predictions of saturation-pressure path. Moreover, the pressure-composition paths for each of the cases under investigation indicate the dependence of saturationpressure path on reservoir fluid and relative pseudocomponent mobility. In these systems, the in situ and flowing psuedocomponent mass fractions and the relative pseudocomponent mobilities are tightly coupled, and each contributes significantly to the level of liquid dropout that occurs in the reservoir. Finally, the sensitivity of reservoir phase behavior to reservoir and drawdown characteristics will also result in high sensitivity of producing GOR to these characteristics. For this reason, future efforts should be directed towards employing knowledge of these relationships coupling these effects into to develop novel multiphase analysis and forecasting techniques for liquid-rich gas reservoir systems. 50

64 3 Constant GOR as an Infinite-Acting Effect in Multiphase Systems 3.1 Chapter Summary This chapter focuses on the discussion of constant producing gas-oil-ratio (GOR) trends observed in many unconventional oil and gas reservoirs. It is a direct application of the semi-analytical multiphase solution proposed in previous chapter to study actual production performance of unconventional wells exhibiting multiphase flow. In this chapter, we analytically corroborate that constant GOR could indeed be expected in depleting reservoir systems exhibiting early-transient infinite-acting flow under constant bottomhole pressure (BHP) production constraint for both linear and radial flow geometries. This effect had been shown to be present in infinite-acting linear systems by previous authors; but this chapter shows how this GOR trend and value can be fully predicted apriori. By solving the equations for pressure and saturation, the GOR trend and value can be fully predicted prior to availability of production data. The results show that a constant GOR effect could be maintained as long as the flow regime remains infinite-acting, and its value varies with BHP specifications for a given reservoir and fluid system. 3.2 Background The producing gas-oil ratio (GOR), as the ratio of gas production to that of oil at standard and surface conditions, has been long recognized as one of the key indicators of reservoir fluid type (McCain, 1994a; 1994b; McCain et al., 2011). GOR trends of production data have also been traditionally correlated to reservoir conditions. During an isothermal depletion process, a common assumption in the production of hydrocarbon reservoirs, a second fluid phase forms (i.e. lighter components vaporize in oil system, and heavier components condense in gas system) when pressure drops and the bubble/dew point is crossed. The difference between vapor and liquid phase mobility results in the composition variation of producing fluid. Therefore, constant GOR behavior is always realized in reservoir systems that remain single-phase at reservoir conditions and away from their saturation pressures. Once reservoir pressure falls below saturation conditions (i.e., below bubble point for oils and dew point for gas condensates), producing GORs are expected to increase owing to the higher mobility of the reservoir gas phase and resulting compositional changes of the in-situ fluid (McCain, 1999a; 1994b). In recent years, constant GOR trends have been repeatedly observed in many unconventional systems. Some authors have suggested that the combination of higher critical gas saturation and suppressed bubblepoint pressure, caused by significant capillary pressure effects in nanopores, can explain long periods of 51

65 relatively flat GOR in unconventional oil systems. Typical examples include the Bakken (Nojabaei et al., 2013) and Eagle Ford fields (Khoshghadam et al., 2015), in which producing GOR was equal to initial GOR of the undersaturated reservoir fluid. Some other authors observed that, even when bottomhole pressure was below saturation (bubble/dew point) conditions, producing GOR remained approximately constant and not equal to undersaturated GOR (Beliveau, 2014; Clarkson and Qanbari, 2015). Further numerical simulation studies using finely-gridded black-oil and compositional numerical simulation models showed that constant BHP multiphase production in a hydraulically fractured configuration, characterized as a one-dimensional (1-D) planar system, indeed leads to a constant GOR signature during infinite-acting conditions (Whitson and Sunjerga, 2012; Tabatabaie, 2014; Behmanesh et al., 2014, 2015a, 2015b). Bøe et al. (1989) had previously presented analytical arguments that indicated that GOR can stabilize at a constant value at late production times during infinite-acting conditions in constant-rate multiphase well tests in a radial-cylindrical system. Behmanesh et al. (2015b) demonstrated that constant-pressure production of multiphase well under linear flow geometry would result in constant GOR behavior. In this study, we first show that flat GOR responses are possible in multiphase unconventional systems even prior to accounting for high capillary pressure and phase behavior effects caused by nanopores in unconventional systems. The constant GOR trend is possible even when two fully mobile phases co-exist at sandface conditions, and lasts as long as infinite-acting conditions prevail which is a typical occurrence in unconventional formations. While the existence of a constant GOR trend during infinite acting conditions have been analyzed by these previous authors, the available studies are limited to numerical simulation analysis and approximate analytical solutions (Bøe et al., 1989; Clarkson and Qanbari, 2015; Behmanesh et al. 2015b). This study is the first one to show how this GOR trend and value can be straightforwardly and rigorously predicted prior to availability of production data by solving the governing flow equations for pressure and saturation simultaneously. In both Bøe et al. (1989) and Behmanesh et al. (2015a) studies, the similarity theory was employed to propose approximate solutions due to the difficulty in obtaining closed-form analytical solutions from the associated multiphase equations with multiple non-linear terms. In this study, the objective is to further extend the similarity method to fully solve the applicable multiphase governing flow equations, from which GOR can be independently calculated. The approach is an extension of a previous study (Zhang et al, 2014), where it was shown that similarity method can successfully capture the early-transient behavior of single-phase gas wells producing under constant-pressure and constant-rate conditions exhibiting linear and radial flow. Following the evaluation of GOR under infinite-acting effect using proposed solution, we further discuss the GOR responses under the effect of closed outer boundary by comparing with finely-gridded numerical simulation results. 52

66 3.3 GOR calculation for saturated multiphase reservoirs In very low permeability reservoirs, wells are usually completed with several stages of hydraulic fracturing and linear flow may be observed to last for a long time after production has started. In this analytical study, the following simplifications are used in order to arrive to the proposed semi-analytical solution: 1) reservoir undergoes isothermal depletion (effect of temperature variation on fluid properties is neglected); 2) Conductivity of fractures is infinite, and fluid flow can be represented as 1-dimensional linear flow; 3) Flow to the hydraulic fracture comes from a matrix with no natural fractures and thus represented as single porosity/permeability; 4) Capillary pressure and pressure-dependent reservoir properties (permeability and porosity) are not considered. Using the black-oil fluid formulation, the governing partial differential equations of surface-gas and oil components in a 1-D linear system are (see Appendix C): d dη (λ dp gsc dη ) = η dw gsc 2 dη d dη (λ dp osc dη ) = η dw osc 2 dη Equation 3-1 Equation 3-2 subject to infinite-acting, constant bottomhole pressure boundary conditions described in Appendix C. Appendix C and Chapter 2 have shown that by applying Boltzmann variable η = x φ, these governing kt partial differential equations (Equation 3-1 and 3-2) can be transformed into the system of simultaneous nonlinear ordinary differential equations (ODEs) summarized in Table 2-2 thus enabling GOR predictions in infinite-acting linear multiphase flow systems. Details regarding the solution technique shooting method coupled Runge-Kutta integration can be found in Chapter 2. By using the Runge-Kutta solver, the highly-nonlinear pressure/saturation-dependent terms (λ gsc, W gsc, λ osc and W osc ) in the resulting ODEs do not need to be linearized. These terms are directly calculated as functions of pressure and saturation during the Runge-Kutta numerical integration. In the infinite-acting boundary conditions (Equation C-6), η = 0 represents the infinite-conductivity fracture (x = 0), and η represents both initial condition (t = 0) and far away from wellbore condition (x ). After solving for p and S o from system of ODEs shown in Table 2-2, pressure and saturation profiles in terms of the similarity variable η are obtained, from which producing gas-oil-ratio (GOR) is straightforwardly calculated to be: 53

67 GOR = q 2x f kh (λ gsc (p, S o ) p gsc x ) x=0 = q osc 2x f kh (λ osc (p, S o ) p = x ) x=0 k rg k ro μ g B + R s g μ o B o k ro k rg ( μ o B + R v o μ g B g) η=0 = f(p, S o ) η=0 Equation 3-3 Given that this system of equations admits a similarity solution, it follows that p = p(η) and S o = S o (η) only. As a result, at any value of η, there can only exist one value of p and S o and a unique S o vs. pressure relationship exists during infinite-acting flow. Unique profiles of pressure and saturation functions of Boltzmann variable η lead to fixed values of λ gsc (p, S o ) and λ osc (p, S o ) at any point η in the domain, including the fracture location (η = 0). Therefore, the GOR predicted at the inner-boundary condition or fracture location (η = 0) is bound to remain constant for infinite-acting constant BHP production in linear systems, regardless of whether co-existing phases are mobile or immobile at those conditions. Such GOR value will be uniquely prescribed by the single values that k rg μ g B g, k ro μ o B o, R s, and R v terms must take at η = 0 (see Equation 3-3). It is noted that when reservoir liquid mobility is zero ( k ro μ o B o = 0), GOR = (1/R v ) η=0. When reservoir gas mobility is zero ( k rg μ g B g = 0), GOR = (R s ) η=0. When both phases are mobile, GOR will be the result of the mobility-weighted ratio of (R s ) η=0 to (R v ) η=0 presented in Equation 3-3. In all these cases, predicted GOR value will remain the same as long as infinite-acting conditions remain. For comparison purposes, we also discuss the producing GOR prediction in 1-D radial flow under constant rate and BHP constraints in this chapter; the similarity-based semi-analytical solution for radial system is provided in Appendix D. This comparison provides useful information about what to expect when flow geometries are not linear during infinite-acting conditions. As shown in Appendix D, the inner boundary in radial geometry is represented by a time-dependent similarity variable X w ( =ln (r w φ/(kt sp )) in order to honor the finite-radius wellbore assumption. GOR = q 2πkh (rλ gsc (p, S o ) p gsc r ) r=r = w q osc 2πkh (rλ osc (p, S o ) p = r ) r=r w ( k rg k ro + R μ g B s g μ o B o k ro μ o B o + R v k rg μ g B g) X=X w = f(p, S o ) X=Xw Equation 3-4 An important observation made from Equation 3-4 is that constant BHP production in radial regime may yield a somewhat time-dependent GOR. We will further compare this infinite-acting radial GOR behavior with that of linear systems in the section

68 3.4 Synthetic case studies To validate the proposed similarity solution, pressure and saturation profiles obtained using the proposed semi-analytical solution are then compared against finely-gridded numerical data from a commercial numerical simulator (CMG-IMEX). For the purposes of this validation, the input reservoir properties displayed in Table 3-1 have been used, and the numerical model was constructed on the basis of a onedimensional logarithmic-grid linear system with one well located at one end of the reservoir domain. Both gas condensate and volatile oil cases are presented in this section. Some supplementary plots are provided in Appendix N. The gas-condensate case is a ten-component fluid with dewpoint equal to psia at reservoir temperature (250 F) and 22% maximum CVD liquid dropout volume. Black oil properties of this gas condensate fluid are generated using the Peng-Robinson Equation of State (Peng and Robinson, 1976) and are the same as the fluid used in Chapter 2. The implemented set of relative permeability curves is shown in Figure H-1. The volatile-oil example is taken from the base case presented by Whitson and Sunjerga (2012) for which bubble point pressure is equal to 4250 psia at 235 F. Black oil properties are provided by Figure H-2. Relative permeability curves used for this volatile oil example uses Corey s correlation with a Corey s exponent of 2.5 and critical oil saturation S oc = 0.20 (Figure H-1). Table 3-1 Reservoir and Fluid Properties Absolute Permeability, k(md) 0.01 Porosity, φ (fraction) 0.03 Rock Compressibility, c R(1/psi) 0 Fracture Half Length, x f (ft) 125 Reservoir thickness, h(ft) 50 Wellbore radius, r w (ft) Model Validation Using the reservoir characterization shown in Table 3-1, performance predictions for both the gas condensate and volatile oil examples are obtained for a bottom-hole pressure condition maintained at 1000 psia, which corresponds to ~3000 psia difference between initial and bottmohole conditions observed in field cases (e.g. Clarkson and Qanbari, 2015). For simulation in the numerical model, reservoir outer boundary is placed at 10,000 ft to minimize boundary effects in the early-time data. Figures 3-1 and 3-2 display pressure and saturation profile predictions, respectively, predicted by proposed semi-analytical methodology (continuous line) and numerical simulation (markers) for the gas condensate example. Figure 3-3 and 3-4 present the same predictions for the volatile oil case. As previously discussed, unique pressure and saturation profiles are indeed obtained as a function of the similarity variable. In all figures, semi-analytical predictions are compared against numerical simulation predictions using p and S o 55

69 vs. time data collected at arbitrary fixed locations (25 and 50 ft away from the wellbore) as well as at arbitrary fixed times (3 rd and 30 th day). Simulation results displayed in markers demonstrate that, pressure/saturation changes at any location and at any time fall on top of unique pressure and saturation profiles when expressed in terms of the similarity variable η. The agreement between the predicted profiles and numerical simulation is excellent. It is important to note that since the critical condensate (S oc ) is equal to 0.10 for the gas condensate fluid example, the condensate phase is fully mobile at sandface conditions (see Figure 3-2, where (S o ) η=0 > 0.10). Gas and liquid phases are also fully mobile at sandface conditions for the volatile oil case (see Figure 3-4). Figures 3-1 to 3-4 highlight the unique S o vs. pressure relationship during that is developed during infinite acting linear flow. As shown, these unique pressure and saturation profiles lead to fixed values of pressure and saturation at any point η in the domain, including the fracture location (η = 0). At the fracture location, η = 0, it is clear that the values of pressure and saturation remain constant as long as infinite acting constant BHP conditions prevail. Therefore, as per Equation 3-3, producing GOR must remain constant in these systems. time = 3 days time = 30 days distance = 25 ft distance = 50 ft Figure 3-1 Pressure profile for gas condensate linear flow under pwf = 1000 psia specification 56

70 time = 3 days time = 30 days distance = 25 ft distance = 50 ft Figure 3-2 Saturation profile for gas condensate linear flow under pwf = 1000 psia specification time = 3 days time = 30 days distance = 25 ft distance = 50 ft Figure 3-3 Pressure profile for volatile oil linear flow under pwf = 1000 psia specification 57

71 time = 3 days time = 30 days distance = 25 ft distance = 50 ft Figure 3-4 Saturation profile for volatile oil linear flow under pwf = 1000 psia specification Constant infinite-acting GOR vs. Effect of no-flow outer-boundary To further study the constant GOR behavior versus the effect of the presence of a no-flow outer boundary, Figures 3-5 and 3-6 reproduce the same scenarios above for a linear system in which the outer boundary is found only 500 ft away from the fracture such that the onset of boundary-dominated flow (BDF) starts sooner. Information about pressure at outer boundary is also plotted as function of production time to detect the onset of BDF. In both figures, a perfect match is observed between calculated GOR and reported values from the numerical simulator during infinite-acting flow. Comparison of these two infinite-acting multiphase GORs (27,551 SCF/STB in Figure 3-5 and 50,550 SCF/STB in Figure 6, which matches Whitson and Sunjerga s reported oil gas ratio) against undersaturated GOR values (1/R vi = 7,661 SCF/STB for the gas condensate fluid in Figure H-2 and R si =4,000 SCF/STB for volatile oil case in Figure H-3), it is clear that infinite-acting multiphase GORs can be several times higher than expected initial GORs for the undersaturated fluid. In Figures 3-5 and 3-6, numerical and analytical GOR predictions result in horizontal lines at early times, reinforcing the observation that the GOR is invariant to production time under infinite-acting flow evidenced by the reported constant outer boundary pressure at early times. As the pressure transient reaches the outer boundary, a monotonic increase of GOR starts to observed, confirming that the constant-gor 58

72 effect would only last as long as infinite-acting conditions continue. As discussed above, this constant GOR effect is the direct consequence of the unique pressure-saturation relationship that results during infiniteacting conditions for linear systems as long as the BHP remains constant. Notably, a nearly-constant GOR even during BDF has also been reported for linear flow by Clarkson et al. (2014). This constant BDF GOR is likely due to extensive depletion that the reservoir had experienced before BDF begins, yielding very low flow rates where GOR changes may be difficult to detect. Calculated infinite-acting GOR Figure 3-5 Producing GOR comparison of gas condensate example for pwf = 1000 psia scenario Calculated infinite-acting GOR Figure 3-6 Producing GOR comparison of volatile oil example for pwf = 1000 psia scenario 59

73 3.4.3 Non-linear flow effects (radial flow) To further investigate GOR behavior during infinite-acting conditions for non-linear flow scenarios, Figures 3-7 and 3-8 display the infinite-acting GOR evolution for both linear and radial geometries. While the linear solution describes flow in a 1-D cartesian stencil, radial flow is a representation of a type of 2-D cartesian flow. Broken lines in both figures represent calculated analytical GOR in radial constant-bhp flow (see semi-analytical method for radial flow details in Appendix D), while markers report results from logarithmical-gridded radial-cylindrical numerical model. Solid lines in both figures represent GOR behavior during linear flow for the same fluid and reservoir characterization. Initial GOR for the undersaturated fluid is also displayed for comparison purposes. 2-D cartesian (radial), infinite-acting GOR behavior, in both examples, is clearly time-dependent. However, it is important to note that the GOR evolution happens over an extremely short early-time window (0.001 to 1 day) before stabilizing around constant, undersaturated GOR values. The figures also highlight that infinite-acting GOR behavior in radial regimes is bounded by the values of linear and undersaturated GORs. Notably, very-early infinite-acting radial GORs start at linear GOR values indicating that very-early radial flow is equivalent to linear flow during extremely short times when pressure transients are limited to extremely small regions around the wellbore. Such small time windows are of no practical interest and thus radial GOR values are expected to quickly stabilize around constant, near undersaturated values during latetime infinite-acting flow conditions. The linear and radial GOR comparison presented in Figure 3-7 and 3-8 provides an important insight into GOR behavior for non-linear geometries. Whitson and Sunjerga (2012), for example, numerically demonstrated through a numerical example that GOR of a 2-D non-fully penetrating planar fracture system would initially overlap with pure-linear, constant GOR predictions at early times; but GOR would then decrease once non-linear flow starts i.e, once the area beyond the tip of the fracture started to contribute to flow. This is consistent with the observations in Figures 3-7 and 3-8 where it is shown that decreased GORs are observed once non-linear flow starts. To provide a more comprehensive picture regarding GOR predictions for radial regime, Figure 3-9 and 3-10 further present the simulated results of the gas condensate example generated for a small 1-D radial cylindrical reservoir (r e = 250 ft) with permeability equal to 0.1 md, for both constant BHP and rate constraints, while other properties remain identical to that used in Figures 3-7 and 3-8. Agreement between the proposed method and numerical simulation during early-transient (infinite-acting) is excellent. Discrepancies between predicted and numerically simulated GOR are only present when boundary effects are evident as should be expected. In Figure 3-10, before BHP drops below dewpoint, producing GOR remains constant and equal to undersaturated GOR. After BHP drops below the dewpoint, GOR stabilizes 60

74 GOR (SCF/STB) at the late infinite-acting period and the stabilized value is also close to initial GOR where the bottomhole is above the dewpoint. This is consistent with the findings by Bøe et al. (1989), who had previously argued that GOR would indeed stabilize at a constant value during late infinite-acting production times for constant-rate multiphase production in a radial-cylindrical system. Radial GORs shown in Figure 3-9 and 3-10 suggest that infinite-acting GORs associated with radial flow tend to be significantly lower in magnitude and are found much closer to undersaturated GOR values compared to the linear case. Such is the reduced dependency of infinite-acting GOR on prevailing BHP for radial cases that it can also be shown that its value would tend to stabilize under both constant pressure and rate production conditions during infinite-acting flow, though it should be rigorously time-dependent as demonstrated by Equation 3-4. This stabilized GOR figure observed in radial regime under both constant pressure and rate constraints can be seen as the result of a vanishing wellbore radius in the finite wellbore model implemented in the proposed semi-analytical solution (Appendix D): the resulting profiles (p and S o) tend to remain unchanged when the change of time-dependent inner boundary X w ( =ln (r w φ/(kt sp )) approaches zero as t sp. t (days) Figure 3-7 Linear vs. radial infinite-acting GORs for gas condensate example 61

75 GOR (SCF/STB) p (psia) Figure 3-8 Linear vs. radial infinite-acting GORs for volatile oil example t (days) Simulated GOR Predicted GOR Simulated outer boundary Figure 3-9 Predicted vs. simulated GORs for pwf = 1000 psia under pi = 4000 psia of radial regime 62

76 GOR (SCF/STB) p (psia) Undersaturated GOR Stabilized GOR t (days) Simulated GOR Predicted GOR Simulated outer boundary Figure 3-10 Predicted vs. simulated GORs for qgsc = 10 MSCF/D under pi = 3000 psia scenario of radial regime Effect of bottom-hole specification Figures 3-11 and 3-12 explore the dependency of producing GOR with respect to prevailing p wf specifications maintain a constant initial pressure of p i = 4,000 psia. Due to the close correspondence between gas condensate and volatile oil examples shown in previous figures, only gas condensate case is shown in this section. In these figures, both analytical and numerical predictions are shown demonstrating again an excellent agreement. Numerical results are generated using the same model as section (outer boundary is 500 ft) in order to observe GOR behavior after the onset of boundary effects. As shown, the nature of the pressure-saturation relationship is BHP-dependent and this has a significant effect on linear GOR. Figure 3-11 shows that the pressure-saturation relationship inside the reservoir is significantly p wf-dependent, resulting in a changing GOR for the same reservoir, initial pressure, and fluid system (Figure 3-12). Figure 3-11 further shows that a lower BHP specification leads to lower oil saturation values at all locations inside the reservoir. In this figure, oil saturation at p = p i (4,000 psia, right hand side of the figure) is representative of conditions at the outer boundary; and saturations at p = p wf corresponds to conditions at the fracture face. Condensate build-up starts once the fluid crosses the upper dew point of this fluid ( psia). As p wf decreases, condensate revaporization starts to play an important role. At very low p wf specifications (100 psia, in Figure 3-11), the condensate phase completely 63

77 volatilizes around the area near fracture after some degree of build-up inside the reservoir. At this low p wf condition, since there exist a single-phase gas flow at the fracture condition, the produced oil at surface is 100% condensed from the reservoir gas phase produced at the fracture. In Figure 3-12, all cases yield constant GORs (horizontal lines) during infinite-acting conditions regardless of prevailing p wf specification. In this figure, analytical predictions are compared against numerical simulation, corroborating that flat GOR levels are maintained until BDF effects are felt. The location of the flat GOR level is also significantly depend on the p wf specification. For the case of p wf = psia (fluid s dew point), the entire reservoir remains in a single-phase condition. GOR is constant and equal to reciprocal of volatile oil ratio (i.e. 1/R v) at the specified pressure (GOR = 1/R vi = 7,661 SCF/STB) as would be predicted for any single-phase reservoir system. For the case of p wf = 100 psia, since BHP is very low, the condensate build-up revolatilizes completely around the fracture location. In this case, GOR is constant again and equal to the reciprocal of volatile oil ratio (i.e. 1/R v) at the specified bottomhole pressure (GOR= 1/R v = 13,973 SCF/STB at 100 psia). For all other scenarios, condensate saturation at the fracture S o is higher than the critical oil saturation, and GOR is successfully predicted during infinite-acting flow via the proposed solution, reaching levels of 27,551 SCF/STB at p wf = 1000 psia and 13,316 SCF/STB at p wf = 2000 psia 1.7 and 3.6 times larger than would have predicted on the basis of known R v values at initial reservoir conditions. As discussed before, GORs during infinite-acting two-phase flow are significantly larger than GORs exhibited by the same undersaturated single-phase fluid ( undersaturated GOR ) and are not constrained by any of the R v-values of the reservoir fluid PVT data at the corresponding pressure. To better showcase the dependency of infinite-acting GOR on the specified value of constant BHP, Figure 3-13 presents constant-bhp, linear GOR predictions for a variety of BHP specifications. A strong dependency is observed between specified BHP and producing GOR. A clear trend is observed: GOR is shown to increase with decreasing BHP before a maximum value is reached, after which GOR decreases with decreasing pressure within the low-p wf region. A peak GOR is observed around the p wf = 310 psia specification at a value of 33,833 SCF/STB more than four times the GOR associated with the undersaturated fluid. A constant rate specification is, by definition, a condition of variable BHP at the sandface. For comparison purposes, a numerically simulated infinite-acting GOR scenario producing under a constant surface-gas rate (20 MSCF/D) constraint is also plotted against BHP changes in Figure Comparing the two profiles, the GOR under a constant-gas rate scenario similarly tends to increase with decreasing BHP pressure before a maximum value is reached which is consistent with the finding in Figure 3-11 that GOR is p wf-dependent during early-transient flow. Based on these results, it follows that during a variable-p wf 64

78 situation, GOR changes would be highly dependent on the history of the p wf changes during the life of the well. This is an area of future exploration and work is currently underway to capture using a constant-rate similarity-based formulation and superposition principles. Other authors (e.g. Behmanesh et al., 2015b) have used superposition of constant-p wf solutions during early-transient to study variable-p wf effects p wf = 2000 psia p wf = 1000 psia p wf = 100 psia Proposed solution S o pwf =100 psia pwf =1000 psia pwf =2000 psia p (psia) Figure 3-11 Pressure-saturation relationship of gas condensate example under different pwf specifications Figure 3-12 Producing GOR comparison of gas condensate example for different pwf scenarios 65

79 Figure 3-13 Producing GOR changes with pwf for gas condensate example 3.5 Concluding Remarks In this study, we demonstrate that constant GOR is an infinite-acting effect that can be expected in multiphase (gas condensate and volatile oil) linear flow under constant BHP production constraints. The proposed similarity solution makes the prediction of such GOR possible prior to the collection of any field production data. Pressure and saturation profiles are simultaneously solved in terms of similarity variables. Proposed method has been compared and validated against numerical simulation. Some important conclusions are listed as follows: 1. Nonlinear terms in multiphase flow equations can be successfully handled by similarity method coupled Runge-Kutta solver; governing flow equations in terms of black-oil fluid formulations are solved as a system of simultaneous equations for pressure and saturation; 2. For linear flow under a constant-bhp constraint, unique profiles of pressure and saturation can be solved as functions of the Boltzmann variable; the time-invariant fracture condition is obtained from such unique profiles--leading to a constant GOR-- which can be fully predicted prior to production using proposed solution; such constant GOR characteristic is shown to be valid only under infinite-acting flow; 66

80 3. A constant infinite-acting GOR solution is proved to be rigorously valid for infinite-acting linear flow geometries. Infinite-acting GOR in radial systems is shown to be time-dependent but only during very-early times. Radial GORs also stabilize at constant (near-undersaturated) GOR values at times of more practical interest during infinite-acting conditions; 4. Infinite-acting GOR responses under different BHP specifications are demonstrated to be p wf dependent. The proposed solution is able to fully capture this dependency; 5. Infinite-acting GOR values during two-phase flow can be a number of magnitudes larger than GORs exhibited by the same single-phase fluid producing at conditions higher than dew point pressure ( undersaturated GOR ) for linear flow conditions. These producing GOR values are clearly not constrained by the prevailing volatized oil ratios (R v-values) or solution gas ratios (R s- values) at the corresponding sandface pressure. 67

81 4 Similarity-based Study of Flowing and In-Situ Compositions in Multiphase Reservoirs during Early Transient Periods 4.1 Chapter Summary Liquid-rich gases in unconventional reservoir environments can exhibit complex phase and flow behavior due to gas condensation and re-vaporization and differences in phase mobilities that results in compositional variations inside the system. To date, the analysis of in situ and flowing composition variation in unconventional liquid-rich wells has been largely limited to numerical modeling. This work uses an analytical approach to study the in situ and flowing fluid composition of gas condensate wells producing under infinite-acting linear flow a commonly observed flow regime in hydraulically-fractured horizontal wells in unconventional formations. We propose a semi-analytical solution to the governing partial differential equations (PDEs) written in terms a compositional fluid formulation. The proposed solution is developed using Boltzmann s transformation and is validated by both analytical development and numerical simulation data. Results corroborate that when hydraulically-fractured horizontal wells are producing against a constant bottomhole pressure (BHP) constraint, the producing wellbore fluid composition remains constant as long as the system remains infinite acting, leading to a constant producing gas-oil ratio (GOR). This constant wellstream composition is shown to be very different from in situ composition, which varies according to pressure and production condition inside the reservoir. 4.2 Background With an increasing interest and demand for unconventional gas resources, accurate and comprehensive characterization of these systems is critical. In unconventional liquid-rich environments, predicting a producing GOR in advance of drilling is a key factor in the development of the exploration strategy and field planning purposes. Because there can be significant differences between the composition of in-situ fluids compared to what is produced at the surface, a good understanding of in situ and flowing compositions of reservoir fluid is important to the optimization of the production plan and understanding of reservoir performance. The composition of fluid produced from conventional gas condensate wells is expected to vary with production time as a result of the shifting phase equilibrium associated with declining reservoir pressure. This change of wellstream composition results in a typical increase of producing GOR in conventional gas 68

82 condensate wells (McCain, 1999a; 1994b). However, this is not the case in unconventional systems. Prior studies have observed from field examples that even when BHP is below the dew point, producing GOR remains roughly constant in hydraulically fractured wells (Beliveau, 2014; Clarkson and Qanbari, 2015). Studies using finely gridded compositional numerical simulation models show that constant BHP multiphase production in a hydraulically fractured configuration, characterized as a one-dimensional (1D) planar system, can lead to a constant GOR signature (e.g. Whitson and Sunjerga, 2012). A constant GOR is indicative of a constant wellstream composition. Similar observations for gas flow have been made by Li et al. (2015) using the single-phase compositional flow models: constant wellstream producing composition is also observed when when adsorption/desorption is accounted for and as long as flow remains infinite-acting prior to pressure-transients reaching the domain boundary. In prior analytical studies of the multiphase flow effects in unconventional systems, simplified (black-oil) fluid formulations have been used when seeking a solution to the governing PDEs (Chapters 2 and 3). This study proposes a semi-analytical solution based on compositional formulation which solves governing PDEs for pressure and multi-component fluid composition simultaneously using rigorous multicomponent fluid thermodynamics. In particular, we first demonstrate analytically that a constant wellstream composition indeed prevails in a compositional multiphase system from the combined effect of infiniteacting conditions, linear-flow regime, and constant BHP production. This outcome also confirms our previous finding in Chapter 2 and 3 that infinite-acting GOR is constant under constant BHP linear flow through the use of the simplified black-oil formulation. In solving the associated governing compositional equations, we explore two common approaches in this study: 1) solving for pressure and moles per unit reservoir pore volume, and 2) solving for pressure and overall composition. To arrive at the analytical solutions, we apply the similarity method (Boltzmann transformation) to transform the governing system of PDEs and into a set of first-order ODEs that can be straightforwardly solved via Runge-Kutta integration coupled with a shooting method. This methodology has been implemented previously to analyze singlephase gas flow (Chapter 1) and multiphase flow using black-oil formulations (Chapter 2). 4.3 Similarity solution to compositional multiphase flow equations Emphasizing on constant pressure production constraint, this chapter proposes a similarity-based semianalytical solution for multiphase compositional equation under 1-D linear and radial regimes. Considering the resemblance of analytical derivation and solving technique between the two flow regimes- -as shown in black-oil fluid formulation in Chapter 2--this section presents the details for 1-D linear system while radial solution (with synthetic case study) is provided in appendix (Appendix F). 69

83 wells in unconventional reservoirs. This work proposes a similarity-based semi-analytical solution for multiphase compositional equations under 1-D linear flow for constant pressure production and infiniteacting conditions. The simplified 1-D linear system in this work incorporates the following necessary assumptions: 1) the reservoir is undergoing isothermal depletion; 2) the conductivity of the fracture is infinite, and fluid flow can be represented in a 1-D linear geometry; 3) the system of interest is a single porosity/permeability system; that is, we study the flow from the reservoir formation toward the fracture where no natural fracture is present; 4) capillary pressure and pressure-dependent reservoir property (permeability and porosity) effects are not considered. In compositional formulations, mass conservation applied to each mixture component results in the following set of continuity equations for the flow of each i-th component: (ρ g v g y i + ρ o v o x i ) = t [φ(ρ gs g y i + ρ o S o x i )] Introducing Darcy s law for linear regime (v g = kk rg μ g p x, v o = kk ro μ o p x ), Equation 4-1 becomes: x [(ρ k ro k rg g x μ i + ρ o y o μ i ) p g x ] = φ k t (ρ os o x i + ρ g S g y i ); i = 1,2 n c Equation 4-1 which written in terms of simplified notation becomes: Equation 4-2 x (λ p i x ) = φ k W i t ; i = 1,2,, n c where λ i is the mobility term of i-th component defined as: Equation 4-3 λ i = ρ o k ro μ o x i + ρ g k rg μ g y i ; i = 1,2,, n c and W i is the accumulation term of i-th component defined as: Equation 4-4 W i = ρ o S o x i + ρ g S g y i ; i = 1,2,, n c Equation 4-5 In terms of compositional formulation, the entire system is at thermodynamic equilibrium and thus subject to the equal fugacity constraint for i-th component in both phases. To solve the system of governing PDEs, principle unknown must be selected. In compositional modeling studies, some authors choose pressure and 70

84 number of moles per component (i.e. Chang 1990; Collins et al. 1992) and others use pressure and overall composition (i.e. Nghiem et al. 1983; Ayala et al. 2006) as principle unknowns. In this work, we present the analytical solution techniques for both of these options in the subsequent subsections Solving system of ODEs for pressure (p) and moles per unit pore volume (N i ) To select pressure (p) and moles of i-th component per pore volume (N i ) as principle unknowns, we first define moles of i-th component per reservoir pore volume as: N i = ρ ov o x i + ρ g v g y i v o + v g = ρ o S o x i + ρ g S g y i ; (i = 1,2 n c ) Equation 4-6 where V o and V g are the volumes occupied by the reservoir oil and gas phases, respectively, within the total pore volume occupied by reservoir fluids (V t ). Following the definition of N i, the total pore volume occupied by the multicomponent mixture (V t ) is made equal unity, which is also equal to the summation of phases (gas and oil) volumes: V t (p, N 1,, N nc ) = V o (p, N 1,, N nc ) + V g (p, N 1,, N nc ) = 1 ft 3 Equation 4-7 Compositional primary unknowns could be alternatively defined to be pressure and moles of i-th component per unit bulk volume, and the right-hand-side of Equation 7 would become equal to 1/φ (Chang, 1990; Collins et al, 1992). In this study, given the assumption of negligible rock compressibility (φ remains constant), we use moles per unit pore volume for simplicity. By using p and N i as principle unknowns, all remaining terms in Equations 2 and 3 including λ i and W i are evaluated straightforwardly once values of these n c + 1 principle unknowns (p and n c- N i s) become available: a) Overall in-situ composition of the mixture is calculated directly as the ratio of moles of each component to total moles: z i = N i n c i N i ; i = 1,2,, n c Equation 4-8 b) Phase molar densities (ρ o and ρ g ), vapor molar fraction (f ng), and resulting phase compositions (x i, y i ) are obtained via flash calculations at the prevailing pressure, temperature, and overall composition. Germane to compositional simulation is the assumption that oil and gas phases reach thermodynamic equilibrium at all points in the reservoir domain. In this study, the Peng-Robinson equation of state (Peng and Robinson, 1976) is used to model hydrocarbon PVT and fluid equilibria via fugacity calculations. 71

85 c) Fluid saturation is calculated from the volumetric ratio: where f no = 1 - f ng. S o = V o f no = V o + V g f no + f ng (ρ o /ρ g ) Equation 4-9 For early-transient (infinite-acting) flow, the initial and outer boundary conditions are written in terms of these principle unknowns as follows: p(x, t = 0) = p i N i (r, t = 0) = N i,o ; i = 1,2,, n c lim x p = p i Equation 4-10 lim x N i = N i,o ; i = 1,2,, n c Equation 4-11 where N i,o represents the number of moles of i-th component present at initial condition. Since we consider constant BHP production condition at the inner boundary, we also have: p(x = 0, t) = p wf,sp Equation 4-12 Appendix E shows how the application of similarity theory to Equation 4-3 and initial and boundary conditions in Equations 4-10 to 4-12 leads the the following system of ODEs and boundary conditions for this 1-D linear compositional problem in terms of the similarity variable: d dη (λ dp i dη ) = η dw i 2α dη ; i = 1,2,, n c p(η = 0) = p wf,sp Equation 4-13 lim p = p i and lim N i = N i,o ; i = 1,2,, n c η η Equation 4-14 Equation 4-15 In order to solve this system of n c-simultaneous ODEs in terms of the stated n c+1 primary unknowns (p and N i s), Equation 4-13 must be more explicitly written in terms of p- and N i- derivatives. This is accomplished by rewriting mobility λ i (p, N 1,, N nc ) and accumulation W i (p, N 1,, N nc ) derivatives in terms of their total differentials via the expansion: 72

86 n c dm dη = M dp p dη + M dn i N i dη i Equation 4-16 where M is a generic variable representing either λ i or W i. In order to match number of equations (n c- compositional ODEs) to number of unknowns (n c+1 unknowns), the additional closure relationship given by the volume constraint for V t (p, N 1,, N nc ) in Equation 7 is implemented. This volume constraint is written in its differential form as follows: n c dv t dη = V t dp p dη + V t dn i = 0 N i dη i Equation 4-17 It is further noted that the n c-simultaneous ODEs in Equation 13 are second-order because of the presence of d2 p dη2 derivatives. In order to reduce this system of second-order ODEs to a set of first-order ODEs, we rewrite the system in terms of n c+2 dependent variables or functions (f 1, f 2. f nc, f nc+1, f nc+2) which includes the first derivative of pressure as an additional unknown as shown below: f 1 = p f i+1 = N i ; i = 1,2,, n c f nc +2 = dp dη Equation 4-18 By substituting Equations 4-16 and 4-17 into 4-13 and implementing the definitions shown in Equation 4-18, the original system of n c second-order ODEs (Equations 4-13) as well as volume constraint (Equation 17) are reduced to the following system of n c+2 first-order ODEs: 1 λ 1 f N nc +2 + η W N 1 λ nc W nc f N nc +2 + η 1 2 N 1 V t [ N 1 0 λ 1 f N nc +2 + η W 1 nc 2 dn nc λ nc f N nc +2 + η nc 2 V t N nc W nc N nc 0 λ 1 λ nc 0 ] [ df 1 dη df 2 dη df nc +1 dη df nc +2 dη ] = [ η 2α f nc +2 W 1 p λ 1 dp (f n c +2) 2 η W nc 2α p λ n c dp (f n c +2) 2 V t p f n c +2 ] Equation

87 subject to the following boundary conditions, restated from Equations 4-14 and 4-15: f 1 (η = 0) = p wf,sp lim η f 1 = p i Equation 4-20 lim η f i+1 = N i,o ; i = 1,2,, n c lim f n η c +2 = 0 Equation 4-21 This ODE set can be straight forwardly integrated via a fourth-order Runge-Kutta method with adaptive step size (Press et al., 2007) within the 0 < η < solution domain. The application of this technique is detailed in Chapter 2 for the case of multiphase black-oil fluid formulations. Similar to Chapter 2, the problem presented by Equations 4-14 and 4-15 here is a boundary-value-problem (BVP). The Runge-Kutta method marches from one end (either η = 0 or η ) to the other end of the domain but some unknowns are being specified at both ends. In this problem, f 1 is specified at the inner boundary (η = 0, see Equation 4-20) but values of f 1 to f nc+2 are specified at the other end of the domain (η, Equation 21). Also, in actual implementation, the infinite outer boundary η has to be replaced by a finite boundary location (η max ) still able to capture the infinite-acting nature of the problem. In our implementation, this accomplished the shooting method (Press et al., 2007). Starting at a suitable large value of η max, an initial guess for f nc +2 (= dp dη ) is used (small, but not exactly zero given that η max ) to start the process. A suitable solution is reached when the selection of the appropriate combination of η max, f nc +2 at the outer boundary makes the value of f 1 at η = 0 equal to p wf,sp (within tolerance, typically 10-5 psia) as per the constant BHP constraint (Equation 4-20). The secant method is used to carry out this iterative scheme. To find the best suitable large value of η max, an iterative search for the η max that allows the pressure gradient (f nc +2 in Equation 4-18) to vanish at η = η max within a tolerance (i.e. f nc+2 η=η max < 10 9 ). In this study, η f max = 250 proved to be adequate for all cases forthe prescribed n c+2 η=0 tolerances. After solving the proposed system of 1 st -order ODE equations, pressure (p) and moles (N i) profiles are obtained as functions of the similarity variable η. Given that this system of equations admits a similarity solution, unique p and N i profiles are found as functions of the similarity variable η for infinite-acting linear flow. These unique profiles for pressure and composition lead to fixed, unique values of component 74

88 mobilities λ i (p, N 1,, N nc ) at any point η in the domain, including the fracture location (η = 0). Flowing composition at any point η in the domain in thus directly calculated as follows: z i,flow (η) = λ i(η) n c λ i (η) 1 Equation 4-22 Any dependent variable calculated as function of η, including flowing composition, can be readily transformed to functions of time (t) or distance (x) by invoking the definition of η = x φ/(kt). Detailed examples are provided in results section. It is important to note that the wellstream composition (i.e., z i,well or flowing composition at η = 0): z i,well = z i,flow (η = 0) = λ i(η = 0) n c λ i (η = 0) 1 Equation 4-23 is bound to remain constant as long as constant BHP production in 1-D linear systems remains infiniteacting because the values of λ i (p, N 1,, N nc ) are constant at η = 0 because p and N i take unique, invariant values at η = 0. Mobilities at the fracture face control molar flow rates for each i-th component in the wellstream. Application of Darcy s law at η = 0 yields: p n i = 2αx f kh (λ i x ) dp = 2αx f h φk (λ i x=0 dη ) η=0 1 t Equation 4-24 dp Because (λ i ) is single-valued (invariant), Equation 4-24 indicates that the produced molar flow rate dη η=0 of each component yields a linear relationship between rate versus a square-root-time plot. Given the constant producing wellstream composition, flashing producing wellstream fluids at surface conditions will yield a constant producing GOR. Resulting surface-gas and oil rates will also follow a linear relationship when plotted against square-root-of-time, with their ratio (GOR) remaining constant throughout infiniteacting conditions. These observations are fully consistent with prior field and numerical findings of Whitson and Sunjerga (2012), Behmanesh et al. (2013), Beliveau (2014), Tabatabaie (2014), Clarkson and Qanbari (2015), and Behmanesh et al. (2015a, b) that corroborated the constant-producing-gor signature of infinite-acting 1-D linear multiphase systems. This also further corroborates our prior findings (Chapters 2 and 3) where the presence of infinite-acting constant GOR signatures in constant BHP linear multiphase flow was analytically demonstrated using simplified fluid thermodynamics (black-oil fluid formulations). In this study, we now show that the constant GOR behavior is retained using rigorous, multi-component fluid thermodynamics in fully compositional models. 75

89 4.3.2 Solving system of ODEs for pressure (p) and overall molar composition (z i ) Alternatively, pressure (p) and overall molar composition (z i ) (i = 1,2,, n c 1) can be used as principle unknowns in order to solve the system of compositional equations represented by Equations 2 and 3 (Nghiem et al. 1983; Ayala et al. 2006). By using p and z i (i = 1,2,, n c 1) as principle unknowns, all remaining terms in Equations 2 and 3 λ i and W i, for example can be evaluated as a function of the n c principal unknowns: a) Molar fraction of n c-th component is calculated using the composition constraint z nc = 1 n c 1 1 z i ; b) Phase molar densities (ρ o and ρ g ), vapor molar equilibrium fraction (f ng), and resulting phase compositions ( x i, y i ) are obtained via thermodynamic equilibrium calculations at prevailing pressure, temperature, and overall composition using the Peng-Robinson equation of state. c) Fluid saturation is calculated from the volume ratio shown in Equation 4-9. For this new set of principle unknowns, the similarity transformation in Appendix E remains the same for the ODE system (Equation E-8) but initial and boundary conditions in Equations E-9 and E-10 are rewritten as: p(η = 0) = p wf,sp lim p = p i and lim z i = z i,o ; i = 1,2,, n c 1 η η where z i,o represents the molar fraction of the i-th component at initial conditions. Equation 4-25 Equation 4-26 When p and z i are used as principle unknowns, we end up with a system of n c ODEs (Equation A-8) and n c unknowns [p and z i; i = 1,2,, n c 1]. This system of ODEs is now explicitly written in terms of p- and z i- derivatives by rewriting mobility λ i (p, z 1,, z nc 1) and accumulation W i (p, z 1,, z nc 1) derivatives via the expansion: n c 1 dm dη = M dp p dη + M dz i z i dη i=1 Equation 4-27 where M is a generic variable representing either λ i or W i. The resulting set of ODEs is second order, but it can be reduced to a first-order set of ODEs by defining the following functions (f 1, f 2. f nc+1): f 1 = p 76

90 f i+1 = z i ; i = 1,2,, n c 1 f nc +1 = dp dη Equation 4-28 The substitution of these definitions (Equation 4-28) and derivative expansions (Equation 4-27) into the system of n c- second order compositional ODEs (Equation A-8) leads to the following set of n c- first-order compositional ODEs: 1 λ 1 f z nc +1 + η W z 1 λ nc f [ z nc +1 + η 1 2 W nc z 1 λ n c 0 λ 1 f z nc +1 + η nc 1 2 z nc 1 f nc +1 + η 2 W 1 z nc 1 W nc z nc 1 0 λ 1 λ nc ] [ df 1 dη df 2 dη df nc +1 dη ] = η 2α η [ 2α f nc +1 W 1 p λ 1 dp (f n c +1) 2 W nc p λ n c dp (f n c +1) 2 ] Equation 4-29 In Equation 4-29, care should be taken to change the composition of n c-th component when evaluating the partial derivatives with respect to compositional changes of other components (z i, i = 1,2,, n c 1) so that the overall composition constraint is preserved (e.g., λ i (z 1 + ε) =λ i (p, z 1 + ε, z 2, z 3 z nc 1, z nc ε), as in Ayala et al., 2006). Implementation of the Runge-Kutta integration coupled with the shooting method for the solution of this system of n c+1 1 st -order ODEs in Equation 29, subject to the boundary conditions in Equations 4-25 and 4-26, follows the same protocol detailed in the preceding section and thus it is not repeated here. It should also be noted that unique profiles of p and z i are now directly obtained by solving system of equations for pressure and overall compositions. Again, these unique profiles of pressure and overall composition lead to fixed, unique values of component mobilities λ i (p, z 1,, z nc 1) at all points in the domain, including the fracture location (η = 0). As a result, wellstream or flowing composition at fracture conditions become time-invariant (constant) leading to a constant GOR signature and a linear relationship between molar producing rates and square-root-of-time discussed in the preceding section. 4.4 Synthetic case studies In this section, the performance of the proposed analytical similarity solution for compositional multiphase flow equations is investigated and compared against predictions from a commercial compositional 77

91 simulator CMG-GEM (CMG 2012). The Peng-Robinson equation of state (Peng and Robinson, 1976) is used in both cases to model PVT and phase equilibrium. A five-component gas condensate fluid (Table 4-1) is used, and the detailed reservoir and fluid properties are shown in Table 4-2. Figure 4-1 and 4-2 present associated phase envelope and relative permeability data for this system. In the following sections, results from the proposed analytical model are generated from the p and N i s and p and z i s approaches discussed previously, both yielding identical results. Table 4-1 Compositional data for synthetic gas condensate example Component Molar fraction, z i Critical pressure, P c (psia) Critical temperature, T c (R) Acentric factor, ω Molar weight, MW (lb/lbmol) Critical volume (ft 3 /lbmass) BIP with CO 2 CO C C C C Table 4-2 Reservoir and Fluid Properties Absolute Permeability, k(md) 0.01 Porosity, φ(fraction) 0.03 Rock Compressibility, c R(1/psi) 0 Fracture Half Length, x f (ft) 125 Reservoir thickness, h(ft) 50 Wellbore radius, r w (ft) 0.25 Dew Point Pressure, p dew(psia) 2876 Specified Bottomhole pressure, p wf,ssp (psia) 1000 Initial pressure, p i (psia) 2880 Temperature, T(F) 250 Maximum CVD Liquid Dropout, fraction 14.5% 78

92 Figure 4-1 Phase envelope of 5-component gas condensate fluid Figure 4-2 Relative permeability curves Model Validation For commercial numerical model, the reservoir outer boundary is placed at 10,000 ft to minimize boundary effects in the early-time data. Figures 4-3 to 4-5 display the primary unknowns of pressure (p) and in-situ overall composition (z i), respectively, predicted by the proposed semi-analytical methodology (continuous line) and numerical simulation (markers). In these profiles, the semi-analytical predictions are compared to the numerical simulation predictions using p and composition vs. time data collected at arbitrary fixed locations (25 and 80 feet away from the wellbore), as well as at arbitrary fixed times (30 th and 100 th day). It is clear from the simulation results that pressure/composition changes at any location and at any time fall 79

93 on top of unique profiles when expressed in terms of the similarity variable η. The agreement between the predicted profiles and the numerical simulation is excellent. It should be noted that the semi-analytical solutions for the two options (solving for p and N i, and solving for p and z i) yield identical results. Figure 4-6 highlights the unique S o vs. pressure relationship that results from Figures 4-3 to 4-5. As revealed, unique pressure and composition profiles as functions of similarity variable η (Figures 4-3 to 4-5) lead to fixed values of pressure and saturation at any point η in the domain, including at the fracture location (η = 0). At that location, it becomes clear from this solution that pressure and in-situ composition values are bound to remain constant as long as infinite-acting constant BHP conditions prevail. Therefore, the producing wellstream composition must remain constant and time-independent in these systems, as predicted by proposed the semi-analytical method and shown in Figure 4-7 as per Equation This constant wellstream composition figure leads to a constant producing GOR, which is displayed in Figure 4-8. It is noted that the value of this constant GOR is around six times higher than undersaturated GOR, from the comparisons in shown in Figure 4-8. Molar flow rates for each component (Equation 4-24) present in the surface produced gas and oil, respectively, are obtained via flash calculation of the wellstream composition at surface conditions are displayed in Figures 4-9 and Following a linear relationship with respect to square-root-time, all flow rates are accurately predicted by the proposed analytical solution (shown in solid lines). Full agreement with finely-gridded reservoir compositional simulator predictions for all these figures highlight the reliability of the proposed model. Figure 4-3 Pressure profiles of validation example 80

94 Figure 4-4 In-situ composition profiles of validation example Figure 4-5 In situ composition profiles of validation example 81

95 Figure 4-6 Pressure-saturation profile of validation example Figure 4-7 Producing wellstream composition history 82

96 Figure 4-8 Producing GOR comparisons Figure 4-9 Molar rate of each component in surface-gas 83

97 Figure 4-10 Molar rate of each component in surface-oil Flowing vs. In situ composition In a multiphase flow system, differences in vapor and liquid phase mobility results in compositional variations between flowing and in situ compositions. Via the proposed semi-analytical solution, Figures 4-11 and 4-12 display the associated flowing and in-situ composition variations for the case under study. In both figures, the composition of each component is normalized with respect to its initial value (z i,o) to highlight their differences: z i flow = z i flow z i,o z i in situ = z i z i,o Equation 4-30 In Figure 4-11, it is clear that lighter components (C 1, C 2 and CO 2) are preferentially recovered from this system as evidenced by the increase in their concentration within the flowing mixture (z i flow > 1) near the fracture. As a result, their in-situ compositions decrease as fluid moves towards the fracture (η = 0, where z i in situ < 1 for these lighter components). Conversely, Figure 4-12 shows that heavier components (C 5 84

98 Normalized flowing and or in-situ compositions Normalized flowing and or in-situ compositions and C 7+) tend to become more concentrated inside the reservoir as fluid moves out (z i in situ > 1, with z c7+ in situ 3). It is noted that component C 1 is being removed the most when compared to all others (flowing composition increased by roughly 8% at the fracture site, or z c7+ in situ 1.08 in Figure 4-11). Figure 4-12 shows, under the initial-to-bhp drop of 1880 psia, the heaviest component C 7+ is recovered the least among all components inside the reservoir due to its higher concentration in least mobile oil phase, with its in-situ composition being increased more than three-fold around fracture conditions (z c7+ in situ 3 in Figure 4-12). Flowing In-situ Figure 4-11 Flowing and in-situ composition comparison In-situ Flowing Figure 4-12 Flowing and in-situ composition comparison 85

99 4.4.3 Effect of initial and bottomhole condition on composition variation Figures 4-13 to 4-18 further investigate the effect of that initial-to-bhp drawdown has on flowing and in situ compositions. Two scenarios are investigated: a) Figures 4-13 to 4-15 present the effect of changes in initial pressure while BHP is held at p wf = 1000 psia; b) Figures 4-16 to 4-18 illustrate the effect of changes in prevailing p wf specifications on compositions while initial pressure is held p i = 4,000 psia. It is noted that Figures 4-13 to 4-18 present similar trends as those observed in the previous section, thus only only C 1 (lightest) and C 7+ (heaviest) compositional variations are shown in this section. Other components are hence omitted to simplify the presentation. For the effect of changes in initial pressure, composition variation in Figures 4-13 (for C 1) and 4-14 (for C 7+) is that flowing (wellstream) fluid composition becomes richer (i.e., more C 7+ and less C 1) as initial pressure increases. This results in richer wellstreams with lower GOR signatures at the surface. As initial reservoir pressure increases against the same p wf, initial drawdown is significantly larger leading to more extensive fracture depletion and higher concentrations of heavier components around the fracture due to increased condensation and reduced relative mobility (increased z c7+ in situ around the fracture). The increased condensation and accumulation of heavies around the fracture is also evidenced by the higher oil saturations around the fracture displayed in Figure 4-15 for the highest p i s. The higher oil saturation translates to higher oil phase mobilities, relative to multiphase cases when oil is immobile, leading to lower flowing (and producing) GORs. The p wf-drawdown effect on composition is presented in Figures 4-16 and For the case of the lowest p wf specification (100 psia), it is observed that in-situ and flowing composition paths converge near the fracture region because near-fracture condensate has been fully volatilized (i.e, oil saturation at fracture conditions goes to zero, see Figure 4-18) and the flowing fluid becomes in-situ single-phase gas. This is the direct consequence of the condensation re-vaporization process illustrated in the S o-p path of Figure 4-18 for near-fracture conditions at p wf = 100 psia. The sharp decrease in oil saturation near the fracture leads to vaporization of all heavies near the fracture and a sharp decrease of in-situ C 7+ normalized concentration as the fracture is approached. This revaporization occurs because the pressure path crosses the lower dew point of the in-situ fluid (~300 psia). For p wf conditions higher than lower dew-point, Figures 4-16 to 4-18 show that higher drawdown due to lower p wf s at fixed p i would lead to leaner wellstream fluids (i.e., with more C 1 and less C 7+), lower oil phase saturations (Figure 4-18), and thus increased flowing (and producing) GORs. 86

100 Flowing In-situ Figure 4-13 Effect of initial pressure on the flowing and in-situ composition of C1 In-situ Flowing Figure 4-14 Effect of initial pressure on the flowing and in-situ composition of C1 87

101 Figure 4-15 Effect of initial pressure on saturation-pressure relationship Flowing In-situ Figure 4-16 Effect of drawdown on the flowing and in-situ composition of C1 88

102 In-situ Flowing Figure 4-17 Effect of drawdown on the flowing and in-situ composition of C7+ Figure 4-18 Effect of drawdown on saturation-pressure relationship 89

103 4.4.4 Compositional vs. black-oil fluid formulation As demonstrated by Chapter 2, multiphase flow governing equations written in terms of black-oil formulation can be solved using Boltzmann transformation coupled Runge-Kutta numerical integration. In this chapter, we have proved that this semi-analytical approach can also be applied to solve the system of equations written for i-th component that follows a fully compositional formulation. This section compares the results using the proposed semi-analytical method for fully compositional (a five-component equation of state) and two-component (oil and gas pseudo-components) black-oil formulation. We use the base case (five-component gas condensate fluid) presented in Tables 2-1 and 2-2. Black-oil properties (B o, B g, R s and R v) are generated using the Peng-Robinson EOS and following a constant-volume-depletion process. The results pressure and saturation as functions of the Boltzmann variable are presented in Figures 4-19 and An excellent match is found in pressure profiles (Figure 4-19), and a slight disparity between saturation profiles can be observed. The solved saturation by semi-analytical compositional solution is found to be higher than that of black-oil formulation; in particular, the difference of solved saturation at the point of fracture ( η = 0) may result in a slight difference between GOR predictions using the two formulations. Specifically, the relative difference is calculated to be approximately 3%: producing GOR using compositional solution is SCF/STB, and that from the black oil solution SCF/STB, which is hardly noticeable in the flow rates plot shown in Figure 4-21, when comparing the black-oil semianalytical solution with the numerically-simulated results using fully compositional simulation package (CMG-GEM). After comparing the solution performance, we further flashed the solved composition profile of the fivecomponent fluid at each pressure level inside the reservoir so as to track the black-oil PVT properties (details provided in Appendix M) and then compared them with respect to interpolated value using solved pressure profiles from the black-oil semi-analytical solution. An excellent match can be found in Figures 4-22 and 4-23, which confirms the validity of using black-oil formulation as a fine approximate model for describing the multiphase flow in a gas condensate system. 90

104 Compositional solution Black-oil solution Figure 4-19 Pressure profiles solved by proposed fully compositional and black oil solutions Compositional solution Black-oil solution Figure 4-20 Saturation-pressure relationship predicted by fully compositional and black oil solutions 91

105 Compositional solution Black-oil solution Figure 4-21 Surface-gas and oil flow rates predictions by fully compositional and black oil solutions Black-oil PVT table Figure 4-22 Converted compositional solution to black oil properties compared with black oil solution 92

106 Black-oil PVT table Figure 4-23 Converted compositional solution to black oil properties compared with black oil solution 4.5 Concluding Remarks In this chapter, we propose a similarity-based semi-analytical solution for the multiphase compositional equations in 1-D linear and radial flow systems. Through the use of Boltzmann transformation coupled Runge-Kutta numerical integration, the proposed semi-analytical approach successfully captures the composition variation inside the reservoir with very close alignment with finely-gridded numerical simulation data. We demonstrate that for linear flow under a constant-bhp constraint, unique profiles of pressure and composition can be solved as functions of the Boltzmann variable for the governing PDEs written in term of compositional fluid formulation. A time-invariant fracture condition results from these unique pressure and in-situ composition profiles. Using the semi-analytical method, we further studied the effect of initial and bottomhole pressure on the pressure-composition path, including both in situ and flowing composition variation. We demonstrated that a constant wellstream composition indeed prevails in a compositional multiphase system subject to infinite-acting, linear-flow, and constant BHP production conditions leading to constant GORs. It is shown that in these environments: 1) lighter components are preferentially recovered as fluid moves towards the fracture while heavier components become concentrated at near fracture in-situ conditions; 2) Increased reservoir initial pressures (p i s) under fixed 93

107 BHP, for the same fluid s upper dew-point pressure, results in richer wellstream fluids and lower producing GORs; and 3) For bottomhole pressure conditions higher than lower dew-point, higher drawdowns due to lower p wf s at fixed p i result in leaner wellstream fluids and higher producing GORs. The last section of this chapter compares the results of the proposed semi-analytical solution in an application involving both black-oil and fully compositional formulation and demonstrates that the black-oil formulation is an adequate approach for modeling the multiphase flow, evidenced in a good match with results from a fully compositional solution. 94

108 5 Similarity-based Analytical Analysis of Capillary Pressure Effects on Recovery from Unconventional Reservoirs 5.1 Chapter Summary Small pores in unconventional formations typically in the order of nanometers significantly increase the role that capillary pressure plays on production and recovery performance from a hydrocarbon asset. In this chapter, a semi-analytical method is developed for solving the governing equations applicable for multiphase flow in unconventional systems, highlighting the effect of capillary pressure not only on fluid flow as an additional gradient but also on fluid properties in the context of a black-oil formulation approach. To arrive at the analytical solutions, we apply similarity method (Boltzmann transformation) to the governing system of PDEs, and the resulting system of ODEs is solved simultaneously for pressure and saturation via shooting method coupled Runge-Kutta numerical integration. The validity of the series of proposed similarity-based semi-analytical solutions is verified by matching against numerically simulated data. The chapter is divided in two parts: 1) traditional consideration of capillary pressure as a function dependent on saturation alone; 2) interfacial-tension (IFT)-dependent capillary pressure that affects phase behavior. From the results obtained in both sections, we observe that Pc has significant impact on pressure and saturation profiles when it is included in the fluid property calculations, while the effect as an additional pressure gradient on flux terms could be rather limited even negligible. 5.2 Background The smallest pore sizes typically found in the organic matter and clays of shale gas reservoirs are often between 2 to 50 nm in diameter (mesopore) or smaller than 2 nm (micropore), while the largest pore sizes may exceed 50 nm in diameter (macropore) (Clarkson et al., 2012; Li et al., 2014b). Capillary pressure effect on phase behavior, a factor not considered in the traditional modeling in conventional formations, is exemplified in these small pores found within elements of shale gas and tight oil reservoirs. The traditional treatment of capillary pressure in the governing flow equations applicable to conventional hydrocarbon reservoirs is commonly focused on its effect as an additional pressure gradient in the flux term. In many recent studies regarding unconventional systems, capillary pressure has been shown to have a significant impact on hydrocarbon phase behavior, which leads to a change in the production rate and the producing GOR. These investigations have involved numerical modeling (Nojabaei et al. 2013; Wang et 95

109 al., 2013; Rezaveisi et al., 2015; Alharthy et al., 2016) and experimental studies (Brusilovsky, 1992; Firoozabadi, 1999). They concluded that when the surface curvature is increased in small pores, the bubble point pressure decreases and the dew point pressure increases; thus, in porous media the bubble point and dew point are first achieved in the larger and smaller pores, respectively. In 2015, Rezaveisi incorporated the capillary pressure effect in a University of Texas at Austin in-house simulator (UTcomp). Nojabaei et al. (2015) studied this same effect using a compositional-extended black-oil approach. In state-of-the-art production data analysis (PDA) techniques for evaluating multiphase flow, capillary pressure is typically neglected in order to enable analytical analysis. In all the previous chapters that discuss multiphase system, capillary pressure is always neglected when writing the governing PDEs. However, not accounting for the significantly amplified role of capillary pressure in unconventional formations may generate misleading analysis results during production forecasts and reserve estimations. This work proposes an alternative analytical treatment that can fully capture this effect, providing a potentially more accurate and reliable PDA technique that is eminently applicable to the analysis of unconventional multiphase systems. In this study, we apply the similarity-based semi-analytical approach to different conditions regarding the role of capillary force in fluid behavior. Focusing on constant-pressure production under a linear regime, the following simplifications are employed in order to arrive at the proposed semianalytical solution: 1) the conductivity of fractures is infinite, and fluid flow can be represented as onedimensional linear flow; 2) the flow to the hydraulic fracture comes from a matrix without the presence of natural fractures and thus is represented as single porosity/permeability; 3) pressure-dependent reservoir properties (permeability and porosity) are not considered; 4) the reservoir is under isothermal depletion. 5) water is considered at irreducible saturation and thus assumed not impact behavior of hydrocarbon systems. In the first section of this chapter, we addressed the traditional consideration by treating capillary pressure as a saturation-dependent only function that does not depend on IFT. This effect is referred as IFTindependent capillary effect. In this section, IFT-independent capillary pressure gradient and property effects are included; reference pressure selection is also addressed. In the second section, we explore Pc effect under the environment of unconventional system where capillary pressure is considered to have an important effect on both phase behavior and fluid properties, and thus is treated as a function of not only saturation but IFT. In the last part of the second section, we proposed a recommended procedure for analytical modeling of amplified capillary pressure effect in unconventional systems based on the observations from both sections. 96

110 5.3 IFT-independent capillary pressure effects Theoretically, gas-oil capillary pressure (p cgo) in porous media is considered to be proportional to IFT as given by Young-Laplace equation, which can be further reduced as following if the radii of the interface curvature are equal (in a capillary tube or circular pore, typically): p g p o = p cgo = 2σ R i Equation 5-1 where σ is interfacial tension, R i is the radii of spherical interface, and could be related to the radius of the capillary tube (r c) by contact angle between two phases: cos θ = r c R i Equation 5-2 In porous rocks, the radii of the interface curvature are functions of saturation, wettability, pore geometry and other rock properties (Tiab and Donaldson, 2011). Thus for a certain type of rock, capillary pressure is usually determined experimentally as function of saturation for the given rock sample. In the traditional manner of modeling, the dependency of capillary pressure on IFT, however, only applies when laboratory capillary pressure curves measured at room conditions are later scaled to reservoir conditions by multiplying with the ratio of reservoir-to-laboratory IFT (Christoffersen and Whitson, 1995). That is to say, interfacial-tension is traditionally assumed to be fixed and capillary pressure is thus considered dependent on saturation alone. This section embraces this assumption by treating Pc as a IFT-independent function. While doing so, fixed IFT is a poor assumption in gas/oil systems where mass transfer between phases causes substantial composition variation. This limitation will be removed in section 5.4 by incorporating the dependency of IFT in gas/oil capillary pressure as well as its effect on phase behavior. In this section, two types of IFT-independent Pc effects are discussed: gradient effect as an additional pressure gradient appears in the velocity term in the governing flow equations; and property effect when evaluating phase properties under the presence of capillary pressure. Notably, for traditional multiphase flow using compositional formulation, capillary pressure is always ignored in flash calculations. For blackoil formulation approach, the fluid properties are evaluated using a single PVT table generated from experimental data or tuned equation-of-state (EOS) without the consideration of capillary pressure. Thus in this section, the Pc property effect refers to the condition that gas phase properties are evaluated by reading the single PVT table at p g while oil phase properties are read at p o, which is smaller than p g by a value of p cgo that is saturation-dependent. It should be noted that, this method of capturing Pc property effect by the use of a single PVT table that assumes the absence of capillary pressure is thermodynamically inconsistent 97

111 because p cgo is supposed to be considered while generating the PVT properties. IFT-dependent Pc effect studied in section 5.4 will address this issue. Using the black-oil fluid formulation approach, the governing partial differential equations of surface-gas and oil components in a 1-D linear system when capillary pressure is present are written as: x [ k rg μ g B g p g x + R s k ro p o μ o B o x ] = φ k t (S g S o + R B s ) g B o x [R v k rg μ g B g p g x + k ro μ o B o p o x ] = φ k t (S o B o + R v S g B g ) Equation 5-3 Using simplified notation: Equation 5-4 λ gg = k rg μ g B g k ro λ go = R s μ o B o k rg λ og = R v μ g B g λ oo = k ro μ o B o W gsc = S g B g + R s S o B o W osc = S o B o + R v S g B g Equations 5-3 and 5-4 (respectively) are transformed as follows: Equation 5-5 x (λ p g gg x + λ p o go x ) = φ W gsc k t The initial and outer boundary conditions are: x (λ p g og x + λ p o oo x ) = φ W gsc k t Equation 5-6 Equation

112 p ref (x, t = 0) = p i and S o (x, t = 0) = S o,i lim p ref = p i and lim S o = S oi x x Considering constant bottomhole pressure specification, inner boundary condition is: Equation 5-8 Equation 5-9 p ref (0, t) = p wf Equation 5-10 where p ref refers to reference pressure. Due to the effect of capillary pressure, the reference phase pressure must be selected before the development of the solution technique. The details regarding reference pressure selection under different fluid types (oil or gas dominate system) will be further explained section In the case studies presented in this work, the initial reservoir condition is assumed to be a single phase where the pressure is above saturation pressure; thus S o,i equals zero for the gas condensate cases. For volatile oil cases, S g is chosen as primary unknown in this work, and the initial value S g,i is also equal to zero for undersaturated fluid. To transform governing PDEs (Equation 5-6 and 5-7) to ordinary differential form, we then apply the Boltzmann variable in the form η = x φ. The resulting ODEs are: kt d dη (λ dp g gg dη + λ dp o go dη ) = η dw gsc 2 dη and boundary conditions become: d dη (λ dp g og dη + λ dp o oo dη ) = η dw gsc 2 dη Equation 5-11 Equation 5-12 lim p ref = p i and lim S o = S o,i η η Equation 5-13 p ref (η = 0) = p wf,sp Equation 5-14 This section (section 5.3) takes gas condensate flow as example, and the analytical development for volatile oil case is presented in Appendix H. For a gas-dominate system, it is natural to select the primary phase pressure--gas phase pressure--as the reference pressure, after which the oil phase pressure is correspondingly calculated as: 99

113 p o = p g p cgo (S o ) Equation 5-15 To rewrite the system of differential equations in terms of primary unknowns, we apply the differential form of Equation 5-15: dp o dη = dp g dη p cgo ds o S o dη Equation 5-16 The solution technique used in this work for the boundary-value-problem (Equations 5-13 and 5-14) follows a similar development to that proposed in Chapters 1 to 4 i.e., incorporating a shooting method coupled with Runge-Kutta numerical integration. Following Chapters 1 to 4, a fourth order Runge-Kutta formula combined with an adaptive step-size routine (Press et al., 2007) is be implemented to solve this system of nonlinear ODEs. To solve the problem via Runge-Kutta, the equations are initially reduced to a set of first order ODEs. This is accomplished by rewriting them in terms of first order ODEs for new dependent variables or functions (f 1, f 2 and f 3). Since gas phase pressure (p g) and condensate saturation (S o) are the main unknowns, they are defined as follows: f 1 = p g f 2 = S o Equation 5-17 Because the system of equations are second-order ODEs with respect to pressure, an additional pressurederivative function is defined as: f 3 =dp g /dη Equation 5-18 In this way, the system of 2 nd -order ODEs (Equations 5-9 and 5-10) are reduced to a set of first-order ODEs with respect to f 1, f 2 and f 3. Detailed derivation and 1 st -order ODEs are provided in Appendix H. As a result, Equations H-3 to H-5 (immobile condensate) or Equation H-10, H-14 and H-15 (mobile condensate) provide the system of first order ODEs to be integrated via a fourth-order Runge-Kutta method within the 0 < η < solution domain. The complete profiles within the domain are obtained by marching Runge-Kutta solver from one end (either η = 0 or η ) to the other end of the domain. In this BVP, f 1 is specified at the inner boundary (η = 0, Equation 5-13) and values of f 1 and f 2 are specified at the other end of the domain (η, Equation 5-14). To integrate on this domain, the infinitely large boundary in Equation 5-11 is replaced by a large but necessarily finite outer boundary value η max, which represents the end of the Runge- 100

114 Kutta marching search. As used in previous chapters, η max = 250 is applied and proves to be consistent with the infinite-acting condition. The shooting method (Press et al., 2007) starts with an initial guess for f η max, which is then iteratively calculated until a converged solution is reached as Runge-Kutta integration marches to η = 0 at the value of f 1 in agreement (within a tolerance) with the specified constant BHP constraint. After obtaining the complete profiles for f 1, f 2, and f 3 as functions of η within the entire domain η = 0 to η max, flow rates can be calculated from Darcy s law as: p g q gsc = 2x f hk (λ gg x + λ p o go x ) dp g = 2x f h φk (λ gsc dη λ go x=0 dp cgo dη ) 1 η=0 t p g q osc = 2x f hk (λ og x + λ p o oo x ) dp g = 2x f h φk (λ osc dη λ og x=0 dp cgo dη ) 1 η=0 t Equation 5-19 Equation 5-20 where λ gsc = λ gg + λ go and λ osc = λ og + λ oo, as defined in Appendix H. Notably, in practice, we find the second capillary gradient is found to be negligible because gas pressure gradient at fracture is much larger than capillary gradient ( dp g dη dp cgo dη = p cgo ds o S o η = 0), which is due to a small saturation gradient ds o (in comparison with dp g ). Therefore, the flow rates presented in Equations 5-19 and 5-20 can dη dη be conveniently approximated as: dp g q gsc 2x f h φk (λ gsc dη ) 1 η=0 t dp g q osc 2x f h φk (λ osc dη ) 1 η=0 t Equation 5-21 Producing GOR is calculated as: Equation 5-22 dp g GOR = q λ gsc gsc dη λ go = ( q osc dp g λ osc dη λ go p cgo S o p cgo S o ds o dη ) ds o dη η=0 ( λ gsc λ osc ) η=0 Equation

115 While Equation 5-23 is identical to the GOR expression without the Pc effect (Equation 3-3 in Chapter 3), it is worth noting that capillary force could result in pressure and saturation profiles changes inside reservoir, thus affecting the mobility terms of both phases, λ gsc and λ osc as well as the producing GOR value Synthetic case study: gas condensate For the purposes of this validation, the input reservoir properties displayed in Table 5-1 have been used. The numerical model was constructed on the basis of a one-dimensional logarithmic-grid linear system with one well located at one end of the reservoir domain. The gas condensate case is identical to the synthetic example used in Chapter 2 and 3: a ten-component fluid with a dew point equal to psia at reservoir temperature (250 F) and a 22% maximum CVD liquid dropout volume. Relative permeability is shown in Figure 5-1. A single capillary pressure curve considered to be dependent only on saturation in this section is presented in Figure 5-2. They are generated using van Genuchten (1980) formulation in which tuning parameter is 0.5 and S oc = 0.1. The black-oil properties of this gas condensate fluid are generated using the Peng-Robinson EOS (1976) and are presented in Chapter 2. In the original correlation provided by van Genuchten (1980), p cgo approaches infinite as oil saturation approaches S oc and thus is not suitable for computation in gas condensate system where condensate starts to build up from S o = 0. Thus, we modified the correlation following that proposed by Doughty and Pruess (1992) to obtain a continuous relationship without approaching infinity. Notably, the modification is needed for the computation purpose; the actual behavior of capillary pressure in a gas condensate system before condensed liquid builds up enough to become mobile remains to be studied in future work. Table 5-1 Reservoir and Fluid Properties for gas condensate example Absolute Permeability, k(md) 0.01 Porosity, φ (fraction) 0.03 Rock Compressibility, c R(1/psi) 0 Fracture Half Length, x f (ft) 125 Reservoir thickness, h(ft) 50 Wellbore radius, r w (ft) 0.25 Initial pressure, p i (psia) 4000 Bottom-hole pressure, p wf,sp (psia) 1000 Using the reservoir characterization shown in Table 5-1, performance predictions are obtained for a BHP condition maintained at 1000 psia. For simulation in the numerical model, the reservoir outer boundary is placed at 10,000 feet to minimize boundary effects in the early-time data. In the following figures, we compare the results between the conditions that Pc gradient is present and not present. Simulated pressure and saturation profiles for Pc not present case is taken from p and S o distributions at 100 th day. For the Pc 102

116 present case, in order to validate this study with a widely-used commercial numerical simulator in industry CMG-IMEX (CMG, 2012), which uses oil phase pressure as reference pressure results presented in this subsection are generated by solving governing system of nonlinear ODEs (Equations 5-11 and 5-12) for the oil phase pressure. Detailed development and solution techniques are provided in Appendix I. To be consistent with the simulator, fluid properties for both gas and oil phases are evaluated at reference pressure. However, as previously explained, it is physically reasonable to choose the dominate phase pressure, gas phase pressure, for the gas condensate system. Though theoretically the solved results should be independent of the choice of reference pressure, an inconsistency due to reference pressure selection will have a significant impact on the results because the fluid properties are evaluated at different reference pressures (interpolation of fluid properties in black-oil PVT table enters at different pressures when reference pressure is different). This problem will be further addressed in the next section (5.3.2). Figures 5-3 and 5-4 display pressure and saturation profile predictions, respectively, predicted by the proposed semi-analytical methodology (continuous line) and numerical simulation (markers). As previously discussed, unique pressure and saturation profiles are indeed obtained as a function of the similarity variable. In all figures, semi-analytical predictions are compared against numerical simulation predictions using p o and S o vs. time data collected at arbitrary fixed locations (25 and 100 feet away from the wellbore) as well as at arbitrary fixed times (50 th and 100 th day). Simulation results displayed in markers demonstrate that pressure/saturation changes at any location and at any time fall on top of unique pressure and saturation profiles when expressed in terms of the similarity variable η. The matches between the predicted profiles and numerical simulation results are excellent. Figure 5-5 highlights the unique S o vs. pressure relationship during that is developed during the infinite-acting linear flow under the gradient effect of capillary pressure. As shown, these unique pressure and saturation profiles lead to fixed values of pressure and saturation at any point η in the domain, including the fracture location (η = 0). At that location, it is clear that the values of pressure and saturation remain constant as long as infinite-acting constant BHP conditions prevail. Therefore, as per Equation 5-36, producing GOR must remain constant in these systems when capillary pressure is present; and this constant GOR value compared to simulated value is shown in Figure 5-8. This is an important observation because it was proven in Chapter 3, that constant GOR is expected in linear regime under constant-bhp infinite-acting flow if capillary pressure is neglected. The results shown here further corroborate that even when capillary pressure is present, producing GOR for a multiphase system still remains constant as long as system is under infinite-acting linear flow. Comparing the simulated results for capillary pressure present and not present cases, it can be seen that condensate saturation increases under additional capillary force driven flow. Moreover, the proposed semianalytical solution is proven to successfully capture such an effect. Notably, the plateau-like period of 103

117 pressure profiled in Figure 5-3 is the result of a high p cgo vs. saturation gradient at a low condensate saturation level: dp o dη = dp g dη p cgo S o ds o. Figures 5-6 and 5-7 further compare the gas and oil flow rates dη prediction developed using the proposed methodology with simulated data. Matches between the proposed methodology and CMG-IMEX results are excellent. By combining Figures 5-6 and 5-7 with Figures 5-3 and 5-4, it is evident that both gas and oil rates are increased under a capillary gradient effect. This is because the Pc gradient effect results in a higher pressure gradient around the point of fracture (i.e., f 3 = psi/ft md under Pc and psi/ft md without Pc) as shown in Figure 5-3. However, the oil rate shown in Figure 5-7 evidences a significant increase due to increased oil mobility at the fracture as a result of higher condensate saturation around it, as shown in Figure 5-4. Another direct result of increased oil saturation is the a lower producing GOR value under Pc gradient effect, as shown in Figure 5-8. Although the good agreement observed in Figures 5-3 to 5-8 verifies the validity of the proposed solution in comparison with commercial simulator results, it is worth noting that these data are generated using oil phase pressure as the reference pressure a fact which may not follow reservoir physics, as noted in prior research (e.g. Rezaveisi et al., 2015). The next section further discusses the results using gas phase pressure as the reference pressure and compares the findings with those presented in this case study. Figure 5-1 Relative permeability curves 104

118 Po (psia) Figure 5-2 Saturation-dependent capillary pressure curve (CMG-IMEX w/ Pc) (CMG-IMEX w/ Pc) (CMG-IMEX w/ Pc) (CMG-IMEX w/ Pc) CMG-IMEX w/o Pc Figure 5-3 Pressure profiles under Pc effect (po reference) 105

119 (CMG-IMEX w/ Pc) (CMG-IMEX w/ Pc) (CMG-IMEX w/ Pc) (CMG-IMEX w/ Pc) CMG-IMEX w/o Pc Figure 5-4 Saturation profiles under Pc gradient effect (po reference) (CMG-IMEX w/ Pc) (CMG-IMEX w/ Pc) (CMG-IMEX w/ Pc) (CMG-IMEX w/ Pc) CMG-IMEX w/o Pc Figure 5-5 Saturation-pressure relationship under Pc effect (po reference 106

120 CMG-IMEX w/ Pc CMG-IMEX w/o Pc Figure 5-6 Gas flow rate under Pc gradient effect CMG-IMEX w/ Pc CMG-IMEX w/o Pc Figure 5-7 Oil flow rate under Pc gradient effect 107

121 Figure 5-8 Producing GOR under Pc gradient effect Reference pressure selection As demonstrated in section 5.3.1, the proposed similarity-based solution proves to be able to fully capture the Pc gradient effect by matching perfectly with the commercial numerical simulator CMG-IMEX. The results and matches are achieved by solving the governing PDEs for reference pressure (oil phase pressure) and saturation simultaneously. Although the validity of the methodology is verified, the selection of reference pressure remains to be discussed, because the calculation is physically inconsistent when gas properties are evaluated at oil phase pressure. More importantly, because reference pressure is continuous throughout the reservoir, using oil phase pressure as reference pressure in a gas-dominate system implies that at saturation pressure (dew point), the pressure of the gas phase which should be higher than the oil phase pressure by a value of capillary pressure will experience a discontinuity, as shown by Figure 5-9. Because gas phase is the dominant phase existing both above and below dewpoint, this discontinuous pressure profiles does not follow fluid physics. In short, dominant phase should be continuous and used as reference phase where reference pressure is evaluated. Based on the observation above, we further investigate the solution in which gas phase pressure (p g) is the reference pressure and fluid properties are evaluated at reference pressure (Figures 5-10 and 5-11). From 108

122 Po (psia) the pressure and saturation comparisons between the two scenarios, it is clear that the red dash lines (p g as reference pressure) yield slightly higher condensate saturation levels, but the overall levels are very close to the green dash lines in which capillary pressure is not accounted for in system of equations. This suggests that when capillary pressure is solely considered as an additional pressure gradient driving fluid flow, the effect is rather small (even negligible), especially as seen in the pressure profiles shown in Figure A further comparison with solved profiles using p o as reference pressure (solid lines) in both figures also indicates that, when significant Pc gradient effect is captured by commercial simulator that is biased towards liquid system (initially developed for oil reservoirs) such as CMG-IMEX, the results (e.g. Figures 5-3 to 5-8) could be rather misleading because incorrect reference pressure is used. p g p o p dew p o as reference p (CMG-IMEX w/ Pc) (CMG-IMEX w/ Pc) (CMG-IMEX w/ Pc) (CMG-IMEX w/ Pc) CMG-IMEX w/o Pc Figure 5-9 Pressure comparisons for Pc gradient effect using oil phase pressure as reference in gas condensate system 109

123 Figure 5-10 Pressure profiles for Pc gradient effect using difference reference pressures p (psia) Figure 5-11 Saturation-pressure relationships for Pc gradient effect using difference reference pressures 110

124 5.3.3 Capillary pressure gradient vs. fluid property effects As demonstrated in section , when using the correct reference phase--gas phase pressure (p g)-- in gas condensate system, Pc gradient effect is observed to have very limited influence on results if phase properties are both evaluated at reference pressure (a typical assumption used in commercial simulators). In this subsection, we first investigate the potential reason behind such observation starting from the terms in governing flow equations that differentiate from the equations without capillary pressure (e.g. Equations 2-6 and 2-7) --the flux terms of the surface gas and oil pseudocomponents. dp g F gsc = λ gg dη + λ dp o go dη = λ dp g gsc dη λ dp cgo go dη dp g F osc = λ og dη + λ dp o oo dη = λ dp g osc dη λ dp cgo oo dη Equation 5-24 Equation 5-25 As p cgo being a IFT-independent function that causes additional pressure drop due to saturation gradient, the terms influenced by Pc gradient effect in Equations 5-22 and 5-23 are the additional flux of gas and oil pseudocomponents caused by the secondary phase (oil phase) appearance due to condensation-- λ go dp cgo dη dp and λ cgo oo, given the mobility terms λ dη go and λ oo represent for gas and oil pseudocomponents mobility contributed by reservoir oil phase. Besides component mobilities, these flux terms are also affected by capillary pressure gradient dp cgo. In order to better understand Pc gradient effect, in addition to the set of dη p cgo vs. S o curve used in section (shown as Set A in Figure 5-12), another set of capillary pressure curve (Set B in Figure 5-12) is also implemented and results are compared with Set A in this section. Comparing with the two sets in Figure 5-12, Set B is set to have higher absolute value of p cgo (overall level is above Set A in Figure 5-12) while slope is milder due to smaller capillary pressure gradient dp cgo ds o. Using gas phase pressure as reference pressure, predicted saturation-pressure profiles using proposed semianalytical solution under the two sets of p cgo curves are shown in Figure Comparison with the So-p relationship without Pc effect indicates that Set B yields closer results to no Pc profiles, and thus has less impact on fluid flow behavior. From a further comparison for pressure and capillary pressure displayed Figure 5-14, it is clear that though the absolute value of p cgo of Set A is lower than that of Set B, the steeper slope of dp cgo dη at near dewpoint ( psia) region increases condensate build up, leading to higher oil saturation in Figure A more important observation is that, though Figure 5-14 suggests that the slope of capillary pressure curve plays a role in the S o-p profiles in Figure 5-13, the overall Pc gradient effect 111

125 under both Set A and B curves is very limited compared with no Pc profile in This limited Pc gradient effect, as implied by the additional flux terms ( λ go dp cgo dη dp and λ cgo oo ), is the direct result of the dη combination of low secondary phase (oil phase) mobility and small p cgo gradient. To be more specific, the mobility terms λ go and λ oo are contributed by reservoir oil phase--the second phase much less mobile compared with dominant gas phase. Thus, the values of λ go and λ oo are expected to be much smaller compared with λ gsc and λ osc. The magnitude of λ gsc and λ osc ratios diaplayed in Figures 5-15 and 5-16 λ go λ oo confirms this observation. Secondly, capillary pressure gradient is found to be typically many times smaller than dominant pressure gradient dp g dη, as illustrated by a much milder change in pcgo profiles compared to p g in In particular, this difference between induced pressure drop and capillary pressure drop caused by saturation change could be expected to be significant in the production of unconventional reservoirs where induced pressure drop is usually large in order to improve production, resulting in an rather negligible Pc gradient effect. This observation also confirms the approximation taken by Equations 5-19 to 5-21 for the calculation of production rates and producing GOR. Figure 5-12 Saturation-dependent pcgo curves input 112

126 Figure 5-13 Saturation-pressure relationships under difference capillary pressure curves Figure 5-14 Pressure comparisons under difference capillary pressure curves 113

127 Figure 5-15 Mobility comparisons under different Pc curves Figure 5-16 Mobility comparisons under different Pc curves 114

128 Based on the discussion above, we further investigate the scenario where p g is the reference pressure and fluid properties are evaluated at each phase pressure (R v, B g and µ g at p g; R s, B o and µ o at p o). By doing so, phase fluid properties are affected by capillary pressure and thus referred as Pc property effect. Set A p cgo curve is implemented for demonstration. The results from applying the proposed similarity-based solution method to is represented by black broken lines in Figures 5-17 to 5-18, and also compared with profiles obtained from evaluating phase properties at reference pressure (oil or gas phase). It is clear from these figures that reference pressure selection affects the pressure level at which fluid properties are evaluated, leading to different performance results. The disparity found in these figures is due to the fact that fluid properties are fundamentally evaluated by entering a single PVT table at different pressures, and this will affect the results significantly. This observation will be visualized in a black-oil pseudocomponent PX diagram and explained in details in section Furthermore, an additional group of profiles are included in Figures 5-17 and 5-18 as yellow markers, which are obtained by solving governing system of equations without p cgo gradient terms (λ go dp cgo dη and λ dp cgo oo are neglected, details provided in Appendix J) under dη the condition that phase properties are evaluated at each phase pressure. In other word, Pc gradient effect is neglected in these profiles while property effect is considered. Comparing with the black broken lines where both Pc gradient and property effects are accounted for, yellow markers fall exactly on top of black broken lines. These overlapping profiles suggest that capillary pressure property effect plays a much more significant role in both pressure and saturation profiles, corroborating our previous observation based on Figures 5-13 to 5-16 that Pc gradient effect has very limited impact on multiphase flow behavior. w/ Pc gradient w/ Pc gradient p g reference w/ Pc gradient (fluid phase p) p g reference w/o Pc gradient (fluid phase p) Figure 5-17 Pressure profiles under Pc gradient and property effects (Set A) 115

129 w/ Pc gradient w/ Pc gradient p g reference w/ Pc gradient (fluid phase p) p g reference w/o Pc gradient (fluid phase p) Figure 5-18 Saturation-pressure relationships under IFT-independent Pc gradient and property effects (Set A) Compositional paths in PX diagram for the black-oil pseudocomponent formulation under pc- gradient and property effects As demonstrated in and 5.3.3, saturation-pressure relationships under IFT-independent Pc effect are shown to be affected by both Pc gradient and property effects. In order to better examine and visualize such dependency, we utilize the pressure-composition (PX) diagram for black-oil pseudocomponent, on which pressure-composition paths can be straightforwardly mapped. Similar investigation has been employed before for the cases where Pc is not present (Chapter 2) and the details regarding constructing the dewpoint (ω gg ) and bubblepoint ( ω go ) lines using black-oil properties (R v, B g, R s, B o) have been provided in section Similar to section 5.3.3, the discussion for composition paths in PX diagram under IFT-independent Pc effect is divided to two parts: gradient and property effects. In the scenario of Pc gradient effect, horizontal tie lines are drawn at reference pressure at which both phase properties are evaluated. Thus at each reference pressure level in Figure 5-19, lever rule and volumetric calculation are implemented to construct the saturation contour maps: f mo = ω gg z g ω gg ω go Equation

130 S o = V o 1 = V o + V g 1 + f mg /f mo (ρ o /ρ g ) Equation 5-27 In this way, this series of S o contours does not consider p cgo and thus is identical to that presented in Chapter 2. After mapping in-situ composition paths using inverse lever rule in Figure 5-19, it is clear that the slight difference of S o-p relationships between no Pc and Pc gradient effect (p g as reference) is the result of different pressure gradient when entering two-phase region caused by the additional capillary pressure drop. The results obtained from section that uses oil phase as reference pressure is also included in Figure 5-19; but it is worth addressing that this p o - composition path does not follow fluid physics due to the discontinuity gas phase pressure experiences after condensate appears below dewpoint. In short, the oil phase pressure path should not be continuous starting from above dewpoint at all. Pc property effect is highlighted in Figure As addressed a few times before, a single black-oil PVT table that assumes no p cgo in its preparation is implemented when p cgo is considered a saturation-dependentonly function. Thus, the outlines of PX plot (dewpoint and bubblepoint lines) remain unchanged in Figure For Pc property effect, because phase properties are evaluated at phase pressures that differs by the value of p cgo, saturation contours calculated by Equations 5-26 and 5-27 are also affected by p cgo(s o). To include p cgo in S o contour maps, a simple iterative scheme employing successive substitution procedure is used to find the iso-s o lines. The resulting S o-contours considering p cgo(s o) is displayed as red lines in Figure 5-18; and S o-contours without p cgo (exact same iso-lines presented in Figure 5-19) are shown in grey shades. Comparing these two series of S o-contours, it is clear that the entire S o-contours are shifted to left-handside when p cgo is considered, suggesting that overall oil saturation levels could be lowered. Such shifted iso-s o lines are the result of evaluating phase properties at their phase pressure respectively rather than at reference pressure altogether. To be more specific, due to the pressure difference between phases, the lever rule applied in Figure 5-20 is performed on the projections of z g, ω gg and ω go on x-axis. Therefore, observing the p g and p o entries in Figure 5-20, it is clear that ω go is smaller when evaluated at p o instead of p g while ω gg remains same, resulting in a decreased f mo and increased f mg at certain composition z g. Besides the mass fraction ratio (f mg /f mo ), Equation 5-27 also indicates that saturation values are affected by phase densities. As shown in Figure 5-20 where different phase pressure entries are marked, oil phase density is higher when evaluated at oil phase pressure (which is smaller than p g by the value of p cgo) compared to that under p g, leading to an increased ρ o /ρ g ratio. Therefore, the combination of increased mass fraction ratio (f mg /f mo ) and phase density ratio (ρ o /ρ g ) results in the lowered S o level at certain composition, leading to an overall shifted S o-contours. After identifying the shifted S o-contours, we map the pressurecomposition path for Pc property effect in Figure Though travelling similar paths as that under Pc 117

131 gradient effect, overall saturation level (as shown in Figure 5-18) under Pc property effect is much lower, as expected from the discussion above. w/o Pc P g reference P o reference Reference p Figure 5-19 Composition paths under IFT-independent Pc gradient effect (Set A) p g p o p cgo Figure 5-20 Composition paths under IFT-independent Pc property and gradient effect (Set A) 118

132 p o p g Figure 5-21 Evaluating phase densities under IFT-independent Pc property effect (Set A) 5.4 IFT dependent capillary pressure effects The effect of capillary pressure on the phase behavior of multiphase flow in porous media has been broadly investigated via numerical simulation and experimental studies (Sigmund et.al. 1973; Brusllovsky, 1992; Firoozabadi, 1999; Finncioglu et al., 2012; Nojabaei et al. 2013). These studies show that bubble point pressure decreases and dew point pressure increases when phase behavior is affected by amplified capillary force in small pores. This suppressed bubble point effect is considered to be the reason for commonly observed constant GOR phenomenon in unconventional oil reservoirs (Nojabaei et al. 2013; Khoshghadam, et al., 2015): a reservoir undergoes single-phase flow when BHP drops below the original bubble point while still above the suppressed bubble point. Numerical simulation studies have also been made to capture the amplified capillary forces on phase behavior in unconventional oil and gas systems (Rezaveisi et al., 2015; Nojabaei 2015; Jiang and Younis, 2016). Some authors capture capillary pressure effect in phase behavior by incorporating Pc dependency on interfacial tension (IFT) in flash calculation while adsorption due to molecular interactions with the surface and changes in fluid properties is neglected (i.e. Rezaveisi et al., 2015; Nojabaei 2015); some authors suggest shifting critical properties in confined nanopores due to altered interaction between molecules, and between molecules and surface (i.e. Alharthy et al. 2016). Based on these studies, this section presents an alternative, analytical method to model the capillary effect on 119

133 phase behavior in unconventional multiphase systems. Between the two methods, the treatment of amplified capillary force effect on fluid and flow behavior in this work follows the similar development proposed by Nojabaei (2015). It is achieved by modifying the flash calculation in vapor-liquid-equilibrium (VLE) condition to include IFT-dependent gas-oil capillary pressure (p cgo), and thus referred as IFT-dependent Pc effect. In the previous section, we focus on analyzing the traditional way of modeling a capillary pressure effect by implementing it as a IFT-independent effect through the use of a similarity-based solution. The proposed solution shows that restricting the capture of this capillary pressure gradient effect in unconventional systems which experience high capillary pressure due to small pores might lead to rather different results, since the data are highly dependent on the selection of reference pressure and actual pressure at the point at which phase properties are evaluated. The evaluation of phase properties under IFT-independent Pc effect could be problematic given the input black-oil table does not consider capillary pressure. Therefore, in order to modify the proposed methodology in section 5.3 using black-oil formulation for the purpose of capturing the dependency of phase behavior and fluid property on capillary pressure, we apply an extended black-oil model approach to calculate interfacial-tension-dependent p cgo using a process proposed by Nojabaei (2015). As shown by Equation 5-1 in section 5.3, capillary pressure in capillary tubes is simply calculated as the ratio between IFT and the radii of spherical interface that can be straightforwardly related to the radius of the capillary tube by contact angle. Unlike idealized capillary tube model, the radii of the curvature between phase interface in actual pores are affected by many factors related to rock types and properties. As the radii of curvature and contact angle vary from one pore to another, Leverett (1941) defined a dimensionless J function to convert measured capillary pressure data to a universal curve: J(S w ) = P c /(σ cos θ) k/φ Equation 5-28 It assumes that the porous rock can be modelled as a bundle of non-connecting capillary tubes, where k/φ is a characteristic length of the capillaries radii and could be interpreted as an estimate of the mean pore radius. Following the similar analogy, one could write rewrite gas/oil capillary pressure in porous media as: p g p o = p cgo = 2σ r p Equation 5-29 Equation 5-29 is in the similar form of Young-Laplace equation for capillary tube (Equation 5-1), and the interpretation here is that gas/oil capillary pressure p cgo in porous rock could be understood in an analogous 120

134 way as that in capillary tubes. Combining Equations 5-28 and 5-29, r p can be understood as the effective radii of interface curvature in pores that is dependent on saturation and mean pore radius. Assuming k/φ is constant, r p is saturation-dependent; and Equation 5-29 provides an efficient and straightforward way to model the IFT-dependent capillary pressure effect in porous media. In this section (5.4), IFT-dependent p cgo included in the calculation of phase properties is referred as IFT-dependent Pc property effect. In the meantime, the presence of p cgo in the flux terms (i.e. Equation 5-24 and 5-25) is also considered and referred as IFT-dependent Pc gradient effect. Because IFT varies with pressure and composition of the phases and thus not constant anymore, using the Macleod and Sugden correlation (Pedersen, 2007), IFT for a hydrocarbon mixture may be calculated as: n 4 c σ = [ χ i (x i ρ o y i ρ g ) ] i Equation 5-30 where χ i is the parachor of i-th component. In black-oil formulation, reservoir fluid is assumed to consist of two pseudo-components: surface-gas and surface-oil. To calculate the interfacial tension between these two pseudo-components, Equation 5-30 is adjusted as: σ = [χ o (ω oo ρ o ω og ρ g ) + χ g (ω go ρ o ω gg ρ g ) ] 4 Equation 5-31 where ω oo is the mass fraction of the oil component in the reservoir oil phase, ω go is the mass fraction of the gas component in the reservoir oil phase, ω gg is the mass fraction of the gas component in the reservoir oil phase, and ω og is the mass fraction of the oil component in the gas phase; and where ρ o and ρ g are oil and gas phase densities, respectively. To further relate these variables to widely used fluid property functions in black-oil formulations, we follow the definition of ρ g, ρ o, ω go and ω gg in section of Chapter 2. To obtain the values of parachors of surface-gas and oil pseudocomponents χ g and χ o, linear regression is applied matching with interfacial tension calculated by flash calculation using Equation 5-30, as suggested by Nojabaei et al. (2015). In this way, capillary pressure is treated as a function of both interfacial tension σ (which depends on pressure and fluid properties) and effective interface radii r p (which is saturation-dependent). To decouple p cgo with regard to its dependency on saturation, a normalized capillary pressure vs. saturation function (p cgo /σ vs. S g) is used, which could be straightforwardly obtained from the Leverett J-function by combining Equation 5-28 and In this study, the normalized capillary pressure function capillary 121

135 pressure divided by interfacial tension is then combined with Equation 5-31 to calculate capillary pressure between oil and gas phases. Because IFT is dependent on fluid properties which are affected by capillary pressure, the calculation of p cgo is a coupled problem that can be solved iteratively in flash calculation under compositional formulation. For this investigation of a black-oil formulation, multiple PVT tables with respect to different effective radii values (r p) were developed using flash calculation and EOS prior to generating a solution. For consistency purpose, the pressure entries in the PVT table must be consistent with the reference pressure selected to be solved in system of equations. Once the effective interface radii value is determined as a function of saturation, the PVT properties corresponding to this specific r p value can be determined through interpolation. The flow chart for this iterative procedure, implemented in compositionally-extended black-oil simulations that use pressure and composition as primary unknowns, can be found in Nojabaei. In this study, we use pressure and saturation as primary knowns, and a detailed flow chart showcasing the process is shown in Figure Solve for primary unknowns: reference pressure (p g in gas condensate and p g in volatile oil), saturation (S o in gas condensate and S g in volatile oil) Assume a value of effective pore radius r Interpolate the PVT table corresponding to the r for black-oil properties; calculate IFT reff p p o r p p c Interpolate p cgo from normalized capillary pressure curve; calculate r from from IFT obtained in previous step Update r r peff Convergence check Figure 5-22 Flow chart for capillary pressure and fluid property calculation in proposed solution Appendix J provides the detailed derivation of proposed similarity-based semi-analytical solution for the IFT-dependent Pc gradient and property effects. Similar to section 5.3, the governing 2 nd -order ODEs are reduced to a set of 1 st -order nonlinear ODEs to be integrated via Runge-Kutta solver. At each Runge-Kutta integration step, procedures shown by Figure 5-22 is employed to carry out the IFT-dependent Pc property effect. A detailed explanation of the solution technique the coupled shooting method and Runge-Kutta solver is not repeated here because it is identical to that identified in section 5.3. After solving for pressure 122

136 and saturation profiles, the flow rates can be calculated incorporating Darcy s law. For the gas condensate system: p g q gsc = 2x f hk (λ gg x + λ p o go x ) x=0 = 2x f h φk ((λ gsc λ go p cgo p g ) dp g dη λ go p cgo S o ds o dη ) 1 t η=0 p g q osc = 2x f hk (λ og x + λ p o oo x ) x=0 = 2x f h φk ((λ gsc λ oo p cgo p g ) dp g dη λ oo p cgo S o ds o dη ) 1 t η=0 Equation 5-32 For the volatile oil system: Equation 5-33 p g q gsc = 2x f hk (λ gg x + λ p o go x ) x=0 p cgo = 2x f h φk ((λ gsc + λ gg ) dp o p o dη + λ p cgo ds g gg S g dη ) 1 t η=0 p g q osc = 2x f hk (λ og x + λ p o oo x ) x=0 p cgo = 2x f h φk ((λ osc + λ og ) dp o p o dη + λ og p cgo S g ds g dη ) η=0 1 t Equation 5-34 Equation 5-35 Approximately, similar to section 5.3, we may neglect the additional pressure gradient at fracture and producing GOR is calculated as Equation Model Validation: Bakken oil Following the development above, we first validate the proposed semi-analytical approach via an actual field example: Bakken Formation oil. Using the black-oil model, a constant-composition-expansion (CCE) calculation is performed to obtain the PVT properties of the reservoir fluid characterization provided by Nojabaei (2015). For this case study, the reservoir properties are determined to be the same as those shown in Table 5-1, but with the initial pressure (p i) set equal to 5500 psia and the bottomhole specification (p wf) 123

137 set equal to 500 psia, following Nojabei. The original bubble point without any capillary pressure effect is found to be psia. The relative permeability curve and normalized capillary pressure curve are shown in Figures 5-23 and Notably, the interfacial tension between gas and oil pseudo-components in this case study is calculated using modified Macleod and Sugden correlation (Equation 5-31), while multiplied by three times in order to keep consistency with experimental data for Bakken formation, as suggested by Nojabaei. The results under the IFT-independent Pc gradient effect shown in the figures are generated using a single p cgo vs. S g relationship, obtained from Figure 5-24 using a fixed interfacial tension value of 2.37 dyne/cm (the IFT value of original fluid under equilibrium at p o = 2000 psia). In this section, we compare the IFT-dependent Pc effect with two other scenarios using a proposed semianalytical method: 1) no Pc at all; 2) IFT-independent Pc gradient effect where phase properties are evaluated at reference pressure. Figures 5-25 and 5-26 provide the comparisons for pressure vs. similarity variable η and gas saturation vs. pressure (using oil phase pressure as the reference pressure) relationship, respectively. To validate the proposed solution, simulated data generated by an in-house compositional simulator that is capable of capturing the IFT-dependent capillary effect (Siripatrachai, 2016) is shown as discrete points in both figures. Observing the results under different capillary pressure effects, it is evident from Figure 5-26 that the original bubble point is suppressed under the IFT-dependent capillary effect, and that overall saturation is also lowered. Unique profiles in both figures for pressure/saturation changes at a fixed time (100 days) and a fixed location (8 ft from fracture) confirm the validity of the similarity theory when IFT-dependent Pc effect is accounted for. This uniqueness suggests that, when reservoir pressure crosses suppressed bubble point under the IFT-dependent Pc effect, producing GOR still remain constant as long as system is under linear regime and infinite-acting period. The good match between numerically simulated data and solved results by proposed similarity-based method verify the validity of the proposed solution under IFT-dependent capillary effect. In a comparison, the disparity between the IFT-dependent capillary effect and the other two scenarios in both figures is clear. However, the difference between the two scenarios (without Pc and IFT-independent Pc gradient effect) is extremely small, as evidenced wherein the solid line and the dashed line are overlapping. This comparison, following similar observation presented by Rezaveisi et al. (2015), confirms again the observation made in section 5.3 during the discussion of reference pressure using a gas condensate example: the capillary pressure gradient effect is negligible when considered only to influence flow as an additional pressure gradient. The effective interface radii (r p) and capillary pressure profiles solved using the proposed methodology are shown in Figures 5-27 and The range of r p is found to be around 20 nm in this case and is consistent with the findings using numerical modeling as presented by Rezaveisi et al. (2015) and Nojabaei (2015). 124

138 Flow rates comparisons using proposed solution are demonstrated by lines in Figures 5-29 and Numerically simulated data from in-house compositional simulator capturing Pc effect on phase behavior and gradient effects is shown as scatters in these figures. Good matches between black dash lines and scatters in both figures verify the capability of proposed semi-analytical method to capture the effect of capillary pressure on fluid properties. In Figure 5-29, the oil flow rate under IFT-dependent Pc effect is 12.4% higher than under other conditions, a direct result of lower gas saturation around the fracture as shown in Figure The gas rate shown in Figure 5-30 is closer yet lower compared to these measures, with a difference equal to 2.1%. This lower gas rate is due to a combined effect of lower gas saturation (Figure 5-26) and lower pressure gradient around fracture, as shown by the less steep pressure gradient in zoom-in plot in Figure Producing GOR under IFT-dependent Pc effect is calculated to be 1.65 MSCF/STB, while the others yield a higher value 1.89 MSCF/STB--due to higher gas saturation at fracture shown in Figure The GOR comparison is shown in Figure Figure 5-23 Relative permeability curves of the Bakken oil example 125

139 Figure 5-24 Normalized pcgo/ift curve for Bakken oil example IFT- independent Pc gradient IFT- dependent Pc property and gradient Siripatrachai (2016) simulation: IFT- dependent Pc property and gradient (t = 100 D) Siripatrachai (2016) simulation: IFT- dependent Pc property and gradient (x= 8 ft) Figure 5-25 Pressure profile comparisons for under difference Pc effects 126

140 Siripatrachai (2016) simulation: equilibrium Siripatrachai (2016) simulation: equilibrium Suppressed bubblepoint Original bubblepoint Figure 5-26 Solved pressure-saturation relationship under difference Pc effects Figure 5-27 Gas/oil capillary pressure profile obtained using proposed semi-analytical method 127

141 r p q osc (STB/D) Figure 5-28 Effective interface radii profile obtained using proposed semi-analytical method gradient and property effects Siripatrachai (2016) simulation: gradient and property effects t(days) Figure 5-29 Oil production rate comparisons for under different Pc effects 128

142 P c flow & fluid property effects P c flow & fluid property effects Figure 5-30 Gas production rate comparisons for under different Pc effects Figure 5-31 GOR comparisons for under different Pc effects 129

143 5.4.2 Synthetic case study: gas condensate After validating against simulation results for Bakken oil example, in this section, we apply the proposed solution to further investigate the production and recovery behavior of gas-dominant system under IFTdependent Pc effect. The ten-component synthetic gas condensate fluid used in section 5.3 is discussed here. Following similar procedure for oil system in section 5.4.1, a constant-volume-depletion (CVD) calculation is first to performed to obtain the black-oil PVT properties of the ten-component gas condensate characterization under a series of r p conditions. The original dewpoint without considering IFT-dependent p cgo in phase equilibrium is found to be psia. The normalized capillary pressure curve shown in Figure 5-32 (p cgo /σ vs. S g ) is taken from Set B shown in Figure 5-12 and rescaled by fixed IFT value of σ = 10 dyne/cm. In this case study, two sets of relative permeability curves are implemented and displayed in Figure They are generated using van Genuchten correlation (tuning parameter = 0.8) for two critical oil saturation values: S oc = 0.1 (as identical to that used in section 5.3) and S oc = 0.3. Using these fluid characterization and rock and fluid properties as inputs, three scenarios are studied here: p i = 4000 psia and S oc = 0.1 (Case A); p i = 4000 psia and S oc = 0.3(Case B); p i = 3000 psia and S oc = 0.3 (Case C). Botthomhole pressure (p wf) is specified at 1000 psia for all cases. Solving system of equations for reference pressure (p g) and saturation (S o), results generated using proposed solution under IFT-dependent Pc effect is compared with that when Pc is not accounted for all three scenarios. Figures 5-35 to 5-37 provide the comparison in terms of pressure and saturation profiles as wells production rates for Case A. From the saturation-pressure relationships shown in Figure 5-36, it is evident that dewpoint is increased under IFT-dependent Pc effect, whereas the difference between increased and original dewpoint is much smaller (five times smaller) than that observed in oil case (Figure 5-26). In the meantime, the overall oil saturation level is also elevated under IFT-dependent Pc effect, leading to an increased condensate build-up at fracture (p = p wf). Since this higher oil saturation at fracture is much higher than S oc (= 0.1), Case A result suggests that IFT-dependent Pc effect could leads to in an increased oil phase relative permeability and decreased gas phase relative permeability at fracture, leading to a higher oil rates while lower gas rates as shown in Figures 5-36 and 5-37, as consistent with observation by Jiang and Younis (2016). Figures 5-38 and 5-39 present the solved profiles for gas/oil capillary pressure (p cgo) and effect interface radii (r p). As expected, unlike IFTindependent Pc functions, Figures 5-35 and 5-38 suggest that p cgo vs. S o profile does not follow a monotonic relationship due to its dependency on IFT, as IFT is lowest near saturation pressure. In Figure 5-29, the calculated r p values are found to be within the range of 2 nm to 5 nm. Since the range of r p is implicitly constrained by normalized gas/oil capillary pressure (p cgo/σ) input, the smaller r p values in Figure 5-29 compared with Figure 5-28 is the direct result of small IFT values found in gas condensate fluid that is much closer to critical point than Bakken oil. Nojabaei (2015) states that this IFT-dependent Pc treatment in flash calculation neglects adsorption effect and thus could be considered satisfactory if pore 130

144 sizes are greater than 10 nm and molecules are small ( nm). For pore sizes are smaller than 10 nm, critical properties of the hydrocarbon components need to modified to model the phase behavior in flash calculations (Luchao et al., 2013). Moreover, flash calculation and equation-of-state would become not applicable in extremely tiny pores because of rock and fluid interactions, and microscopic approach like molecular dynamic simulation may be required. These observations being said, we are not claiming pore size can be calculated as r p, which are found to be between 2 nm and 5nm in this case. Because input p cgo/σ function is dependent on the properties of the porous rock, using p cgo/σ in the same scale as that in Bekkn (Figure 5-24), we are trying to showcase the counterpart of the IFT-dependent Pc effect in gas-dominant system of oil reservoirs (i.e. Bakken) via a synthetic case. Though reducing the p cgo/σ magnitude will lead to higher r p values that might be more consistent with the assumptions of IFT-dependent Pc approach, the trend of the results (increased dewpoint, higher oil saturation) would not be changed while the difference might be less obvious. Compared to Case A where condensate saturation is a few times higher than S oc, Case B reduces the mobility of condensate by using a higher S oc (= 0.3) value to see its influence on fluid recovery. While oil saturation is clearly increased under IFT-dependent Pc effect, both S o profiles shown in Figure 5-40 are found to be in less mobile region as S o being slightly higher than S oc. Thus, the change of relative permeability reflected on oil phase mobility does not have as much impact as Case A, leading to a very close production rate comparison in Figure For the gas production, because gas phase is in highly mobile region, the increase of S o restricts the gas phase mobility, resulting in lower gas rate in both cases (Figure 5-39 and 5-41). In the last example (Case C), Figures 5-42 to 5-45 investigates the scenario where condensate is immobile. Overall S o level is lowered due to a lower initial pressure (p i = 3000 psia). This dependency of saturation level on initial pressure (effect of degree of undersaturation) has been discussed in details in Chapter 2. Like Case A and B, condensate build-up is increased under IFT-dependent Pc effect as shown by Figure Saturation profiles under the two conditions (without Pc and IFT-dependent Pc effect) can be both considered condensate immobile. Being immobile, increased S o under IFT-dependent Pc effect does not impact oil phase mobility at fracture (p = p wf) and surface oil production is solely coming from condensation of reservoir gas phase. Therefore, oil rate is observed to decrease under IFT-dependent Pc effect, as the direct consequence of IFT-dependent Pc affecting fluid property in the way that less oil is dissolved in gas phase (as also mentioned by Nojabaei, 2015). The similar observation would not apply to IFT-independent Pc property effect discussed in section 5.3 because the dependency of black-oil fluid property on capillary pressure is not included, and thus volatized-oil-gas ratio (R v) remains unchanged under IFT-independent Pc property effect. 131

145 It is also worth addressing that this decreased oil production due to IFT-dependent Pc effect is consistent with that reported by Rezaveisi et al. (2015), while being contrary to that observed in Case A and Jiang and Yonis (2016). Using the three cases to study sensitivity of recovery on oil phase mobility, we are able to explain the different trends observed by numerical simulation studies regarding oil production under IFTdependent Pc effect in a gas condensate system. The observations from Case A to Case C show that the oil saturation is always increased under IFT-dependent Pc effect, but the production rate as well as producing GOR will be highly-dependent on the mobility of oil phase. Figure 5-32 Normalized pcgo/ift curve for synthetic gas condensate example krg and kro S g Figure 5-33 Two sets of relative permeability curves used in synthetic case study 132

146 w/o Pc IFT-dependent Pc property and gradient Figure 5-34 Pressure profiles for gas condensate case A (pi = 4000 psia, Soc = 0.1) w/o Pc IFT-dependent Pc property and gradient S oc = 0.1 Figure 5-35 Saturation-pressure relationships for gas condensate case A (pi = 4000 psia, Soc = 0.1) 133

147 w/o Pc IFT-dependent Pc property and gradient Figure 5-36 Gas rate comparison for gas condensate case A (pi = 4000 psia, Soc = 0.1) w/o Pc IFT-dependent Pc property and gradient Figure 5-37 Oil rate comparison for gas condensate case A (pi = 4000 psia, Soc = 0.1) 134

148 Figure 5-38 Gas/oil capillary pressure profile for gas condensate case A (pi = 4000 psia, Soc = 0.1) Figure 5-39 Effective radii profiles for gas condensate case A (pi = 4000 psia, Soc = 0.1) 135

149 S oc = 0.3 w/o Pc IFT-dependent Pc property and gradient Figure 5-40 Pressure-saturation relationship comparison for gas condensate case B (pi = 4000 psia, Soc = 0.3) w/o Pc IFT-dependent Pc property and gradient Figure 5-41 Gas rate comparison for gas condensate case B (pi = 4000 psia, Soc = 0.3) 136

150 w/o Pc IFT-dependent Pc property and gradient Figure 5-42 Oil rate comparison for gas condensate case B (pi = 4000 psia, Soc = 0.3) w/o Pc IFT-dependent Pc property and gradient S oc = 0.3 Figure 5-43 Pressure-saturation relationship comparison for gas condensate case C (pi = 3000 psia, Soc = 0.3) 137

151 w/o Pc IFT-dependent Pc property and gradient Figure 5-44 Gas rate comparison for gas condensate case C (pi = 3000 psia, Soc = 0.3) w/o Pc IFT-dependent Pc property and gradient Figure 5-45 Oil rate comparison for gas condensate case C (pi = 3000 psia, Soc = 0.3) 138

152 5.4.3 Recommended semi-analytical solution for capturing amplified capillary pressure effects in unconventional multiphase systems By the use of a semi-analytical approach, the amplified capillary pressure effect typically found in nanopores of unconventional reservoirs is studied in previous subsections. Gas/oil capillary pressure (p cgo) is considered to be a IFT-dependent function that impacts phase behavior, its effect in terms of fluid flow and property on the recovery of oil-dominant or gas-dominant systems are also discussed. The proposed semi-analytical solution, as detailed presented in Appendix K, follows analytical yet rigorous derivation starting from the transformed governing system of ODEs. However, the execution of this rigorous semianalytical solution to be more specific, the simultaneous integration of system of 1 st -order ODEs could be rather troublesome as the evaluation of saturation function (df 2 /dη) involves not only solving a quadratic equation but calculating its secondary derivative (d 2 f 2 /dη 2 ) at one iteration level behind. Observing the governing 2 nd -order ODEs (Equation 5-11 and 5-12), it is clear that all these additional calculation procedure and numerical treatment are due to including the capillary pressure gradient (dp cgo /dη) in the flux terms. Neglecting dp cgo /dη will result in a simplified and approximated system of 1 st -order ODEs with respect to reference pressure and saturation (as shown in Appendix J), and its expression is also identical to the scenario where capillary pressure is not accounted for (Appendix Chapter 2). In the meantime, our observation in section 5.3 and suggests that the IFT-independent Pc gradient effect is extremely limited even negligible when compared with the results when Pc is not present at all. Motivated by both considerations, Figures 5-46 to 5-49 further explore the scenario that p cgo gradient effect is neglected in IFT-dependent Pc effect. This approximate solution follows the same derivation presented in Appendix J, while the evaluation of fluid properties should follow the flow chart shown in Figure Figures 5-46 and 5-47 compare the pressure profiles and saturation-pressure relationships under both IFTindependent and -dependent conditions. Figures 5-48 and 5-48 provide same comparisons for the gas condensate Case C in section In all the figures, the scenario where Pc gradient is neglected under IFT-dependent Pc effect should be highlighted. As expected, it falls exactly on top the results obtained from integrating the rigorously-derived 1 st -order ODEs accounting for both IFT-dependent Pc gradient and property effect. Therefore, it is evident that Pc gradient effect, which is expanded in terms of both pressure and saturation gradient (Equation K-1), is still negligible compared to Pc property effect when p cgo is considered to be IFT-independent and expanded in term of saturation gradient. This is a very important observation for the analytical modeling for capturing IFT-dependent Pc effect in unconventional system because the approximate solution (Appendix J) is proved to efficiently capture the most impactful effect Pc property effect while being easy to implement compared to rigorous solution (Appendix K). 139

153 w/o Pc IFT-independent gradient w/o Pc gradient Pc property IFT-dependent Pc property Pc w/o property Pc gradient IFT-dependent Pc property Pc and property Pc gradient and gradient Figure 5-46 Pressure profiles under Pc effects for Bakken oil fluid property effects flow and fluid property effects S o p o (psia) Figure 5-47 Saturation-pressure relationships under Pc effects for Bakken oil 140