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 ANALYSIS OF PRODUCTION DECLINE CHARACTERISTICS OF A MULTI-STAGE HYDRAULICALLY FRACTURED HORIZONTAL WELL IN A NATURALLY FRACTURED RESERVOIR A Thesis in Energy and Mineral Engineering by Sarath Pavan Ketineni 2012 Sarath Pavan Ketineni Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 2012

2 The thesis of Sarath Pavan Ketineni was reviewed and approved* by the following: Turgay Ertekin Professor of Petroleum and Natural Gas Engineering George E.Timble Chair in Earth and Mineral Sciences Thesis Advisor Luis F. Ayala H. Associate Professor of Petroleum and Natural Gas Engineering John Yilin Wang Assistant Professor of Petroleum and Natural Gas Engineering Yaw D Yeboah Professor of Energy and Mineral Engineering Department Head of Energy and Mineral Engineering *Signatures are on file in the Graduate School

3 iii ABSTRACT Unconventional hydrocarbon reserves exploration has seen a new high in the recent times owing to the decline in production from conventional reserves. For sustained hydrocarbon production, it is essential for the market to shift from conventional to unconventional sources of energy. Shale gas and tight gas reserves are considered unconventional in nature. Innovative technologies like horizontal well drilling and hydraulic fracturing have made the extraction of hydrocarbons from these unconventional resources economically viable. Most shale gas reservoirs are naturally fractured in nature and exhibit dual porosity characteristics. The flow from these tight gas reserves could accurately be modeled as flow around a horizontal well in a naturally fractured formation. Hydraulic fracturing often alters the reservoir parameters around the wellbore, thus, creating a rubble zone (stimulated reservoir volume-srv) having different characteristics when compared to the outer zone. This problem could ideally be approximated as flow around a horizontal wellbore in a composite naturally fractured formation. Elliptical flow regime was long considered a transient regime in between radial and pseudo radial flow, but in case of anisotropic reservoirs and low permeability reservoirs (tight gas and shale gas) the elliptical flow regime extends for a considerably large period of time. A solution to the elliptical flow problem that considers flow into a horizontal wellbore in a truly composite naturally fractured reservoir has been attempted. Mathieu s modified functions were used to solve the elliptical flow problem.

4 iv The generated solution is validated with other existing solutions by collapsing it into simpler forms given in the literature. Forward solutions are generated for various dimensionless parameters.

5 v TABLE OF CONTENTS LIST OF FIGURES....vii LIST OF TABLES...x NOMENCLATURE.xi Chapter 1 Introduction... 1 Chapter 2 Critical Literature Review : Decline Curve Fundamentals : Radial Flow around a Vertical Well : Case1: Homogeneous Reservoir : Case 2: Double Porosity/Naturally Fractured Reservoirs : Case 3: Composite Reservoirs : Case 4: Composite Dual Porosity/Naturally Fractured Reservoirs : Elliptical Flow around a Horizontal Well : Case1: Homogeneous Reservoir : Case 2: Composite Reservoirs : Case 3: Naturally Fractured Reservoir Chapter 3 Mathematical Formulation of the Problem : Governing Equations for Homogeneous Isotropic Reservoir : Warren and Root Double Porosity Model : Coordinate Transformation : Flow Model : Boundary and Initial Conditions of the Model Chapter 4 Analytical Solution : Laplace Transformation : Inner Boundary Conditions : Solution Procedure : Infinite Reservoir case : Finite Outer Boundary Chapter 5 Mathieu Functions : Angular Mathieu Functions : Radial Mathieu Functions : Calculation of Fourier Coefficients and Characteristic Value : Calculation of Characteristic Numbers : Calculation of the Coefficients Chapter 6 Computational Method... 64

6 vi Chapter 7 Results and Discussion : Validation with Data from Literature : Sensitivity Analysis : Forward Solutions : Numerical Example : Numerical Example Chapter 8 Summary and Conclusions Suggestions for Future Studies Appendix A Stehfest Algorithm Appendix B Graphic User Interface Appendix C Transition to Radial Coordinates

7 vii LIST OF FIGURES Figure 2-1: Resource Triangle (Holditch, 2004)... 6 Figure 2-2: Fetkovich Analytical Type Curve (Fekete, 2010)... 9 Figure 2-3: Fetkovich Composite Type Curve (Analytical/Empirical) (Fetkovich, 1987)... 9 Figure 2-4: Idealization of the Heterogeneous porous medium (Warren & Root, 1963) Figure 2-5: Composite radial reservoir (modified from Satman, Eggenschwiler, & Ramey Jr., 1980) Figure 2-6: Schematic of composite naturally fractured Reservoir (modified from Satman A., 1991) Figure 2-7: Elliptical Coordinate system (McLachlan, 1964) Figure 2-8: Genealogy of the Analytical Solutions Figure 3-1: Equivalent representation of the multi-stage hydraulically fractured horizontal well in a naturally fractured reservoir Figure 3-2: Elliptical Coordinate Representation Figure 3-3: Sugar Cube Model Representation (Warren & Root, 1963) Figure 6-1: Work-flow for the computational method Figure 6-2: Work-flow for an outer boundary type specification at each time step Figure 7-1: Comparison of data with Obut & Ertekin(1987) Study Figure 7-2: Comparison of data with Alpheus(2007) study Figure 7-3: Comparison of data with Kuchuk(1979) study Figure 7-4: Comparison of data with Satman(1991) study Figure 7-5: Comparison of data with Eggenschwiler(1980) study Figure 7-6: Comparison of data with Da Prat(1981) study Figure 7-7: Comparison of data with Economides(1979) study Figure 7-8: Sensitivity analysis with respect to mobility ratio (M) Figure 7-9: Sensitivity analysis with respect to diffusivity ratio (ζ) Figure 7-10: Sensitivity analysis with respect to interporosity flow coefficient (λ)... 87

8 viii Figure 7-11: Sensitivity analysis with respect to storativity ratio (ω) of Region Figure 7-12: Sensitivity analysis with respect to interface distance (ξ 0 ) Figure 7-13: Tree of forward solutions generated Figure 7-14: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =2, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = Figure 7-15: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =2, ω 1 =0.1, ω 2 =0.01, λ 1 =λ 2 = Figure 7-16: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =2, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = Figure 7-19: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =2, ω 1 =0.1, ω 2 =0.1, λ 1 =10-5, λ 2 = Figure 7-20: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =0.5, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = Figure 7-22: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =0.5, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = Figure 7-23: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =0.5, ω 1 =0.1, ω 2 =0.01, λ 1 =λ 2 = Figure 7-24: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =0.5, ω 1 =0.1, ω 2 =0.1, λ 1 =10-6, λ 2 = Figure 7-25: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =0.5, ω 1 =0.1, ω 2 =0.1, λ 1 =10-5, λ 2 = Figure 7-26: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =0.5, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = Figure 7-28: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =0.5, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = Figure 7-29: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =0.5, ω 1 =0.1, ω 2 =0.01, λ 1 =λ 2 = Figure 7-30: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =0.5, ω 1 =0.1, ω 2 =0.1, λ 1 =10-6, λ 2 = Figure 7-31: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =0.5, ω 1 =0.1, ω 2 =0.1, λ 1 =10-5, λ 2 =

9 Figure 7-32: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =2, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = Figure 7-33: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =2, ω 1 =0.1, ω 2 =0.01, λ 1 =λ 2 = Figure 7-34: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =2, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = Figure 7-35: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =2, ω 1 =0.1, ω 2 =0.01, λ 1 =λ 2 = Figure 7-36: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =2, ω 1 =0.1, ω 2 =0.1, λ 1 =10-6, λ 2 = Figure 7-37: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =2, ω 1 =0.1, ω 2 =0.1, λ 1 =10-5, λ 2 = Figure 7-38: Snapshot of the CMG IMEX model used for validation Figure 7-39: Screenshot of the CMG IMEX pressure transient data at the end of the run Figure 7-40: Comparative study of results from a commercial simulator CMG IMEX Figure B-1: Snapshot of the GUI Figure B-2: Results generated by GUI for a vertical well producing from a homogeneous reservoir Figure B-3: Snapshot of GUI for horizontal well homogeneous reservoir case Figure B-4: Snapshot of the GUI for horizontal well in a double porosity reservoir case Figure B-5: Snapshot of the GUI for vertical well in a composite reservoir case Figure C-1: Coordinate representation ix

10 x LIST OF TABLES Table 7-1: Reservoir parameters assumed Table 7-2: Comparison of data with Obut & Ertekin(1987) study Table 7-3: Reservoir properties assumed Table 7-4: Comparison of data with Alpheus(2007) study Table 7-5: Reservoir properties assumed Table 7-6: Comparison of data with Kuchuk(1979) study Table 7-7: Reservoir properties assumed Table 7-8: Reservoir properties assumed Table 7-9: Reservoir properties assumed Table 7-10: Reservoir properties assumed Table 7-12: Reservoir properties assigned for the numerical simulator model Table 7-13: Dimensionless parameters for generation of forward solution Table 7-14: Reservoir parameters assumed Table 7-15: Dimensionless parameters for the analytical model Table 7-16: q D vs. t D data generated by analytical solution Table 7-17: q vs. t data generated through analytical solution

11 xi NOMENCLATURE Alphabet a b B B 2n c C 2n Ce 2n ce 2n c f c g c t D E 2n F 2n Fek 2n h h f h ft h ma h mat I r k k f k m K r L l L M m(p) n p p D p fd p i p mad P wd Q q D q Dd q i r r w s Fourier coefficients of order 2n Characteristic value of a Mathieu s function Decline exponential Formation volume factor Fourier coefficients to satisfy the boundary conditions Compressibility Fourier coefficients to satisfy the boundary conditions First solution of Mathieu s modified equation First solution of Mathieu s function Rock compressibility Fluid compressibility Total compressibility Decline rate (1/time) Fourier coefficients to satisfy the boundary conditions Fourier coefficients to satisfy the boundary conditions Second solution of Mathieu s modified equation Height of the reservoir (ft) Height of a fracture (ft) Total height of the fractures (ft) Height of a matrix (ft) Total height of the matrix blocks (ft) Second kind of Bessel s modified I function of order r Permeability of Reservoir (md) Fracture permeability (md) Matrix Permeability (md) Second kind of Bessel s modified K function of order r Laplace transform function Warren and Root characteristic length Characteristic half-length of a horizontal well Mobility ratio Pseudopressure (psi 2 /cp) Number of repetitive blocks in double porosity model by Kazemi & deswaan Pressure (psi) Dimensionless pressure Dimensionless fracture pressure Initial pressure of a reservoir Dimensionless matrix pressure Dimensionless pressure at wellbore Flow rate Dimensionless flow rate Dimensionless flow rate defined by Fetkovich Initial flow rate Radius(ft) Radius of the wellbore (ft) Laplace parameter

12 xii S T t a t D t Dd V v r Skin factor Temperature Pseduotime Dimensionless time Dimensionless time define by Fetkovich Volume of a reservoir Radial velocity defined by Darcy Symbols Viscosity Porosity Density Elliptical wellbore coordinate Angular coordinate Storativity ratio Elliptical interface coordinate Interporosity flow coefficient Diffusivity ratio Shape factor Squareroot of the parametric value Subscript m and ma were used for representing matrix properties, subscript f for denoting fracture properties, 1 and 2 for representing the inner and outer regions of a composite reservoir, respectively.

13 xiii ACKNOWLEDGEMENTS I would like to express my sincere thanks and gratitude to Dr. Turgay Ertekin for being my thesis advisor and my mentor throughout the course of my master s studies. His wide knowledge and his logical way of thinking have been of great value to me. He helped me navigate through all the obstacles with ease and guided me in the right direction from time to time. I would like to thank Dr. Sabih I. Hayek for his directions and guidance in understanding mathematics of the problem better. His extensive discussions around my work and valuable suggestions have been very helpful for this study. I owe my sincere thanks to Professor Yaw D.Yeboah, Department Head of Energy and Mineral Engineering for providing me teaching assistantship through the course of the study. I would like to thank doctoral student Yogesh Bansal for his help throughout the course work during my master s study. I take this opportunity to thank my friends Taha Husain Murtuza, Phani Kiran Pamidimukkala, Phani Bhushan Chintalapati, Vaibhav Rajput for their help and motivation throughout my master s study. Sincere thanks to my friends, Siddarth Sitamraju, Aditya Chowdary and SoumyaDeep Ghosh for their detailed review, constructive criticism, and excellent advice throughout the preparation of the thesis. I owe my loving thanks to my father Rama Rao, mother Prameela, brother Subramanyam, sister VidyaVathi and girlfriend Revathi Dukkipati for their unconditional love and support in every possible way throughout the process of this course, this thesis and beyond.

14 Chapter 1 Introduction The demand for the hydrocarbon resources has been ever increasing owing to the population growth and rapid industrialization. Especially North America has seen a substantial growth in share of unconventional resources to the total energy needs in the last two decades (Imad Brohi, Mehran Pooladi-Darvish, & Roberto Aguilera, 2011). The primary reason as per Imad Brohi et al. (2011) is because North America has crossed the peak production from its conventional resources. For sustained hydrocarbon production, it is essential for the market to shift from conventional to unconventional sources of energy. Conventional reservoirs have permeability values in the range of millidarcy to darcy, while unconventional reservoir permeability values lie in the range of microdarcy or lesser (Ozkan, Brown, Raghavan, & Kazemi, 2009). Shale gas and tight gas reservoirs are considered to be unconventional gas resources. Improved technologies like horizontal well drilling and hydraulic fracturing have enabled corporations to extract hydrocarbon reserves economically. Experts believe that natural gas holds the key to the future owing to the volumes of gas reserves found so far and due to the decline in availability of the conventional oil resources. Clonts & Ramey Jr.,(1986) proposed horizontal drilling in reservoirs with low permeability, thin oil columns, or those having small density difference between oil and water columns. Clonts et al. (1986) mentioned productivity index increase and greater sweep efficiencies in enhanced oil recovery as some of the major advantages of drilling the horizontal wellbores. Giger (1984) proved that the horizontal wells seldom have higher productivity than the conventional wells through a field study in France. Horizontal wells help getting better sweep

15 2 efficiencies in reservoirs with thin oil columns or narrow pay zones by increasing the surface area of contact with the hydrocarbon rich zone. Hydraulic fracturing, secondary or tertiary recovery processes like water/steam flooding, in-situ combustion or CO 2 miscible flooding often causes a change in permeability of the region adjacent to the wellbore (Ambastha, 1988). A composite reservoir is defined to have two or more regions, where each region has distinct rock and fluid properties. Several studies have been conducted on geothermal processes, and stimulated reservoirs, where sharp discontinuities were observed in terms of fluid and rock properties across the interface (Satman, 1991). Hydraulically fractured reservoirs are often treated as composite reservoirs with a radial discontinuity owing to finite nature of fracture propagation. Most shale gas reservoirs are naturally fractured in nature and hence exhibit dual porosity characteristics. Naturally fractured reservoirs are heterogeneous porous media comprising of fractures and matrix blocks. The matrix stores the bulk of the fluid/hydrocarbon, but has low permeability, while fractures exhibit high permeability and doesn t contribute much to the volume of reserves (Warren & Root, 1963). Fractures conduct the fluid from matrix into the wellbore, and this is guided by the matrix fluid transport capacity that is referred to as interporosity flow. Various models (de Swaan O., (1976), Kazemi (1969), Warren & Root(1963)) have been suggested in literature for the transfer of fluid steady state, pseudo steady state and unsteady state methods. However Warren & Root s (1963) method is adopted for the current study. Elliptical flow has long been considered as a transient state in between the early transient and pseudo radial flow regimes by most authors (Amini, Ilk, & Blasingame, 2007). Obut & Ertekin (1987), and Kucuk & Brigham(1979) have studied elliptical flow problem and has come up with solutions for various reservoirs. Elliptical flow has been observed in cases of anisotropic reservoirs, horizontal wells, reservoir with elliptical boundaries, and vertical well with horizontal

16 3 fractures. Based on practical experience coupled with analytical/numerical methods confirmed the dominance of elliptical flow regime in low/ultra-low permeability reservoirs (Amini, Ilk, & Blasingame, 2007). Production decline type curves have been introduced to enable engineers estimate the initial volume of oil/hydrocarbon in place and calculate recoverable resources till the abandonment conditions are reached. Several authors Ram G. Agarwal(1999), Palacio & Blasingame(1993), Carter(1985), and Arps(1956) have worked on developing accurate type curves based on numerical/analytical and semi analytical methods. Type curves could be seen as theoretical plots of generated analytical solutions for various characteristic values of permeability, porosity, mobility, diffusivity ratio and storativity ratio at the given inner and outer boundary conditions. These methods range from basic material balances to decline type curve analyses, but has varying limitations based on the cases considered. A rigorous analytical method would predict the decline characteristics more accurately than a heuristic based approach or a numerical model. A single phase slightly compressible model has been assumed for determining the important effects of mobility ratio, diffusivity ratio, storativities and interporosity flow coefficients on the productions rates. For solving the case of a fully compressible fluid flow, a simple transformation into pseudo pressure needs to be adopted. Moreover at extremely high pressures p/μz assumes a constant value because of which the diffusivity equation collapses the compressible fluid case to the slightly compressible fluid case. According to Ehlig-Economides (1979), constant pressure production is a more relevant inner boundary condition than the constant flow rate assumption in the case of low permeability reservoirs. The purpose of the present work is to develop an exact analytical solution for flow at constant bottom-hole pressure into a multi-stage hydraulically fractured horizontal well in a

17 4 naturally fractured reservoir. Forward solutions (dimensionless flow rate vs. dimensionless time) are presented for different values of the storativities, interporosity flow coefficients, mobility, and diffusivity ratios. Chapter 2 presents a brief discussion on the literature reviewed for this study and summarizes the various developments that have been made by several authors leading to this study. Development of the mathematical model along with the assumptions is presented in chapter 3. Chapter 4 discusses the analytical solution technique adopted, followed by Chapter 5 where the relevant Mathieu functions are explained in brief. Chapter 6 outlines the computational approach to generate the solutions through a computer program coded in Matlab *.Chapter 7 presents the results and discussion that include validation of the current model and sensitivity analysis. Chapter 8 presents summary and conclusions that were derived from this study. *MATLAB (matrix laboratory) is a numerical computing environment and fourth- generation programming language. Developed by Math Works, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages, including C, C++, Java, and FORTRAN.

18 Chapter 2 Critical Literature Review Decline curve analysis has generated significant interest in petroleum industry for the past 80 years. Since then many changes have been observed and many authors have come up with new and improved techniques for prediction of hydrocarbon reserves from time to time. Typically, all oil and gas fields based on their geological structure have an upper limit on the hydrocarbons they contain. The size of the reservoir, which can be defined by geological and geophysical methods, gives an estimate of the potential volume of hydrocarbon reserves in the field. The total volume of the oil/gas in a reservoir is commonly referred to as original oil in place (OOIP). Unconventional reserves are expected to play a major role in meeting the ever increasing global energy demands. Unconventional reserves, especially wet shale gas and oil, are currently being pursued aggressively in US and Canada for large scale development (Duong, 2011). Existence of these resources is known for decades, but the difficulties involved in exploration and economics involved in the extraction have discouraged various corporations in the past. In general, these hydrocarbon reserves are in many ways inferior compared to the conventional reserves, but have huge hydrocarbon storage capacities (George D. Vassilellis, 2009). The way unconventional reserves are defined has changed over time. Earlier the reservoirs which were uneconomical to produce were referred as unconventional reservoirs (István Lakatos, 2009). With the advances in technology, the reservoirs which were earlier considered not economical started to become economical enough for production. Therefore a new definition based on geological properties and physical properties of the fluid has been proposed by István Lakatos(2009). According to the new definition, unconventional oil can be categorized as oil shale, tight oil

19 6 reservoirs, heavy oil reservoirs, oil/tar sand and pyro bitumen deposits and unconventional gas reservoirs can be grouped as gas shale, gas sand, tight sand gas, basin concentrated gas accumulation, associated gas, coal bed methane and methane hydrates. Figure 2.1 represents a resource triangle based on the difficulty on extraction and complexity involved as indicated by Holditch et al. (2004) Figure 2-1: Resource Triangle (Holditch, 2004) The emphasis in recent times has shifted to extraction of unconventional resources owing to the volume of resources available.

20 7 2.1: Decline Curve Fundamentals This section of the literature review explains several existing methods for generation of the decline curves for various reservoirs. Arps (1945) laid the foundation of decline curve analysis by proposing simple mathematical curves, i.e. exponential, harmonic or hyperbolic. It served as an effective tool for creating a reasonable outlook of the production from an oil well on the onset of decline. The flow rates are plotted against the cumulative production or time, and on analysis give us an estimate of the ultimate recoveries. A tentative classification of the decline curves, based on their loss ratio can be summarized as: (2.1) ( ) (2.2) Exponential decline occurs when D is constant, if D varies, then the decline could be classified as hyperbolic or harmonic. To incorporate the varying nature of decline parameter, an exponent b is coupled with the equation in the following way, where exponential decline is given by Equation 2.3, (2.3) Hyperbolic decline is given by Equation 2.4, (2.4)

21 8 where D i is the decline at flow rate q i, and b is the exponent that varies from 0 to 1. When b=1, it is called harmonic decline and, when 0 < b < 1 the decline is termed as hyperbolic. b =0 signifies an exponential decline. Arps (1956) later generalized these curves by introducing a dimensionless parameter q Dd =q(t)/q i, and a dimensionless t Dd =D i t. The resulting equation was: (2.5) The decline curves derived by Arps (1956) were empirical in nature, lacked a sound theoretical basis, but could appropriately describe the decline associated to boundary dominated flow regimes of reservoir depletions. Hurst (1943) proposed solutions for steady-state water influx. But it was Fetkovich (1980), who used these modified versions of these solutions for water influx and proposed a relation combining Arps decline curves. Fetkovich (1980) provided the theoretical explanation for the Arps decline curves. He used analytical solutions for predicting the early transient flow regime and the boundary dominated flow is predicted by Arps form of decline. As a result, the type curves of Fetkovich are split into two regions, transient and boundary dominated. He proposed the following relations: [ ( ) ] (2.6) [( ) ] [ ( ) ] (2.7)

22 9 Figure 2-2: Fetkovich Analytical Type Curve (Fekete, 2010) Figure 2-3: Fetkovich Composite Type Curve (Analytical/Empirical) (Fetkovich, 1987)

23 10 Fetkovich et al. (1980 and 1987) proposed curves where the late time data gives an indication of the reserves, which is a function of the external radius, re of the reservoir (M.Ebrahimi, 2010). From Figure 2.3, it appears that the value of b doesn t change with the reservoir in a given decline. In the case of conventional gas reservoirs, hyperbolic rate-decline relation could be used. But, the accurate estimation of b is often difficult when production data from tight gas sand and shale gas wells are assessed using the hyperbolic rate-decline relation, owing to the very long transient period. In practice for the parameter b, values higher than 1.0 are observed before the onset of true boundary-dominated flow. This leads to huge overestimation of the reserves (J.A. Rushing et al.,2007) on assessing the reserves using Arps hyperbolic rate-decline relation. Carter s gas type curves (Carter, 1985) presented gas-production-rate results for finite radial and linear flow systems produced at constant bottom-hole pressure, which when analyzed are helpful to determine the reservoir size and shape. Palacio & Blasingame(1993) have come up with a new algorithm which accounted for the changes in viscosity and compressibility that occur during the reservoir depletion. They used modified time functions in their approach to account for the non-ideality in the assumptions of constant rate and constant bottom-hole pressures. For better predictions of gas flow, Al-Hussainy et al., (1966), and Agarwal, Ram G. et al.,(1979) and have proposed the following modified functions for time and pressure: (2.8) (2.9)

24 11 Subsequently, a large number of modifications were suggested in the way the time was computed, and new iterative techniques were developed in the process of finding an appropriate decline curve for better predictions. Traditional decline curve analysis is largely an empirical procedure used mainly to predict the future production rates and existing reserves, based on the boundary dominated declining rate. Modern type curve analysis, however, are partially (Fetkovich) or fully derived analytically, based on reservoir fluid flow equations and assuming some simplifying conditions. The modern decline curve analysis or rate transient analysis based on analytical methods require the bottom-hole flowing pressure in addition to the flow rate data to estimate the reservoir properties and obtain gas in place (Fetkovich M. (1980), Palacio & Blasingame (1993), Doublet et al.(1994), Ram G. Agarwal(1999), and D. Ilk, (2007)).Such type curves are generally used for predicting reserves and future production rates as well as reservoir parameters (M.Ebrahimi, 2010). Simultaneously, along with type curve generations, several authors have worked on developing exact analytical solutions for the various reservoir problems. This part of the literature review explains the various chronological developments in terms on analytical solutions developed for various types of reservoirs. 2.2: Radial Flow around a Vertical Well One of the most widely known and popular flow regimes observed in depletion of reservoirs is radial flow regime. The boundary dominated condition is often an important case when production data is being analyzed. The continuity equation relating change of pressure with flow rate for a reservoir with radial flow can be written as:

25 12 ( ) (2.10) Assuming Darcy s law to be valid, (laminar flow conditions) (2.11) where k represents permeability of the isotropic medium, r is radius under consideration, t is time under consideration, φ is porosity of the homogeneous system, ρ is density of the fluid, μ is viscosity of the fluid considered. Neglecting the depth gradients, the following can be written: (2.12) Substitution of in the above equation results in ( ( )) (2.13) ( ) Assuming the fluid considered being slightly compressible, the above equation could be simplified to: ( ) (2.14) Compressibility equation states that

26 13 ( ) ( ) (2.15) ( ) (2.16) and assuming the medium to be incompressible yields us the final equation: ( ) (2.17) The above equations are transformed into dimensionless variables p D and t D. Laplace transform is carried out on the equation formed after dimensionless variable transformation (Van Everdingen, 1949)and are subsequently solved with the use of modified Bessel s functions. Once the equation is solved for a p D, Van Everdingen and Hurst (1949) suggested the following way to get constant pressure solution from constant rate solutions. Duhamel s principle states that: (2.18) where is the dimensionless flow rate at the wellbore in Laplace domain, and is the dimensionless pressure at the wellbore in Laplace domain (Rajagopal, 1993). The following are the solutions for infinite acting reservoirs, closed outer boundary and constant pressure outer boundary reservoirs.

27 : Case1: Homogeneous Reservoir Ehlig-Economides (1979) summarized the solutions obtained for constant pressure, constant rate at well and has developed solutions for constant pressure at the well head. Infinite Acting: ( ) { [ ( ) ( )]} (2.19) Where l is Laplace variable, K i is Bessel s K function of order i and s represents skin factor Closed outer boundary: [ ( ) ( ) ( ) ( )] (2.20) where ; {[ ( ) ( ) ( ) ( )] (2.21) [ ( ) ( ) ( ) ( )]} Constant pressure outer boundary: [ ( ) ( ) ( ) ( )] (2.22) where ;

28 15 {[ ( ) ( ) ( ) ( )] (2.23) [ ( ) ( ) ( ) ( )]} 2.2.2: Case 2: Double Porosity/Naturally Fractured Reservoirs Warren & Root (1963) defined a double porosity model to account for the characteristic behavior of a permeabile medium which contains regions that contribute significantly to the pore volume but negligibly to the flow capacity. They identified two parameters ω and λ which are sufficient to characterize and differentiate a double porosity system from a homogeneous system. Figure 2-4: Idealization of the Heterogeneous porous medium (Warren & Root, 1963) A solution for a vertical well based on Warren and Root s model have successfully been documented by Da Prat et al.(1981). Mavor & Cinco-Ley(1979) were amongst the first ones to the test the validity of the model with practical data and have included skin and wellbore storage effects. Da Prat (1981) provided an extensive decline curve analysis for reservoirs producing at constant pressure in both finite and infinite systems.

29 : Case 3: Composite Reservoirs Satman, Eggenschwiler, & Ramey Jr., (1980) developed a solution for the reservoirs with radial discontinuity when attempting to solve a thermal oil recovery problem. The reservoir properties- permeability, porosity and compressibility for the fluid for two concentric regions namely inner and outer region are assumed different. A solution in Laplacian space is found which is later inverted using a numerical inversion algorithm suggested by Stehfest (1970). They also analyzed wellbore effects and skin effect on the pressure transient to determine the influence they had on the pressure transient curve through the recovery process. Subsequently, Olarewaju & Lee (1987) studied the composite reservoir problem in detail to analyze the effects of phase redistribution on pressure behavior and proposed six flow regimes. Region 2 Region 1 r w r i r e Figure 2-5: Composite radial reservoir (modified from Satman, Eggenschwiler, & Ramey Jr., 1980)

30 : Case 4: Composite Dual Porosity/Naturally Fractured Reservoirs Satman A. (1991) and Olarewaju et al., (1991) simultaneously developed solutions for composite dual porosity reservoirs. Satman (1991) used a double porosity model suggested by Kazemi, (1969) and de Swaan O., (1976). The reservoir model considered by Satman (1991) involved a matrix structure comprising of rectangular slabs as indicated in the figure below. The thickness of the reservoir containing n horizontal fractures was given by: ( ) (2.24) Fracture z=0 z=h ma /2 Matrix Repetitive element h ma r=r w r=r i r=r e Region 1 Region 2 Figure 2-6: Schematic of composite naturally fractured Reservoir (modified from Satman A., 1991) Satman (1991) defined the following dimensionless parameters to solve the problem:

31 18 ( ) (2.25) ( ) (2.26) (2.27) (2.28) (2.29) He defined the dimensionless transfer coefficients and matrix storativities as The diffusivity and mobility ratios were defined in the following way: ( ) (2.30) ( ) ( ) (2.31) ( ) Olarewaju & Lee (1991) analyzed the composite naturally fractured reservoir by analyzing the fractures on the basis of Warren & Root (1963) model of double porosity.

32 19 2.3: Elliptical Flow around a Horizontal Well Elliptical flow model is in general applicable to elliptical shaped reservoirs, fully penetrated hydraulic fractures, naturally fractured reservoirs, reservoirs with pronounced anisotropy and horizontal well problems. Radial flow assumptions have given rise to errors in solving an elliptical reservoir problem in the study of Coats et al., According to the author, the error observed is inversely proportional to the magnitude of time. This indicates that the flow may not be radial from the beginning. The formation anisotropy controls the extent of elliptical flow regime : Case1: Homogeneous Reservoir Prats et al., (1962) used elliptical flow to describe the flow of compressible fluids at constant pressure from a vertically fractured reservoir. They presented the solution by plotting pressure distribution with various values of the ratio. When the ratio is less than one the flow assumes elliptical nature and when it is greater than one, the flow is almost circular. Prats et al., (1962) used Mathieu functions to generate the results. This paper by Prats et al., (1962) is considered to be one of the classic papers written on elliptical flow regimes and formed the basis for further studies. Gringarten et al., (1975) found that earlier studies conducted by Russell & Truitt (1964) are unsuitable for short time analysis. Kuchuk & Brigham (1979) presented an analytical solution for transient flow in elliptical systems. Their solution is applicable to vertically fractured wells, elliptically shaped reservoirs, anisotropic reservoirs and horizontal wells for both the inner boundary conditions constant pressure and constant flow rate at the wellbore. Their work indicated early linear flow which is followed by radial flow with a semi log slope of

33 20 Hale & Evers (1981) successfully showed that elliptical flow problem can properly model linear, radial or transitional flows in between for a vertically fractured well. Conformal mapping technique is used to solve the steady state elliptic flow regime. They used a modified unsteady state solution by incorporating a new variable, radius of investigation that varied with time. They could numerically generate a family of type curves that could quantify the anisotropy of the reservoir from the pressure response of a well. The studies carried out by Okoye (1988) are patterned on the lines of Kuchuk & Brigham (1979) study. In their solution they found the elliptical model solution worked for the transition phase in depletion of a naturally fractured reservoir : Case 2: Composite Reservoirs Obut & Ertekin (1987) generated a composite system solution in elliptical flow geometry. An infinite conducting vertical fracture is analyzed in a composite reservoir. The solution obtained involved Mathieu functions and the Laplace space solutions are inverted using a numerical inversion algorithm suggested by Stehfest(1970). Their solutions when simplified for a homogenous case are found to be in line with results of Kuchuk & Brigham (1979), and also in line with Satman, Eggenschwiler, & Ramey Jr. (1980), work when the problem is converted to radial coordinates.

34 21 Figure 2-7: Elliptical Coordinate system (McLachlan, 1964) However, it was Riley(1991), who presented a detailed study on elliptical flow in a vertical hydraulic fractured well with finite conductivity. Like all other elliptical studies, the reservoir pressure is obtained as a series of Mathieu functions, and fracture pressure a series of cosines. For extremely small values of time, approximate solutions are given and for all other times, exact analytical solutions are presented. Two flow regimes namely linear and bilinear are observed based on the elliptical conductivity.

35 : Case 3: Naturally Fractured Reservoir Several authors have attempted to solve the elliptical flow problem for water injection wells in homogenous systems. In recent times Amini, Ilk, & Blasingame (2007) concluded that elliptical flow is not just a transitional flow regime, but depending on the reservoir and hydraulic fracture properties, the elliptical flow can last for longer periods. They evaluated the decline curve characteristics in hydraulically fractured wells in a tight gas reservoir and found that elliptical flow is a dominant flow regime in low permeability systems. Alpheus & Tiab (2007) developed an exact analytical solution for elliptical flow in a naturally fractured reservoir. Warren & Root (1963) pseudo steady state double porosity model is employed in this study. This study quantified the anisotropy in naturally fractured reservoirs, where the elliptical flow model is extended to hydraulic fractures with infinite conductivity. Imad Brohi et al. (2011) solved a horizontal well problem in composite dual porosity reservoir. They presented a linear composite model using linear dual porosity model for the inner zone and linear single porosity for the outer region. The interface conditions are effectively implemented. They obtained a solution that is continuous with respect to pressure and flux at the interface. Type curves are presented for various cases, and they observed three linear flow periods based on the reservoir parameters assumed. The first linear flow is from the fractures into the wellbore followed by flow from matrix to fracture and the lastly from the outer single porosity region. Traditionally, elliptical flow models have been applied to vertical fractures, horizontal wells, anisotropic reservoirs, naturally fractured reservoirs, composite reservoirs. However there are no existing solutions in literature for flow from a horizontal well in a truly composite naturally fractured reservoir (where both inner and outer region are considered to be dual porosity regions each).

36 23 In this this study, the works of Alpheus & Tiab (2007), and Obut & Ertekin (1987) are used primarily to find an exact analytical solution to a horizontal well producing from a composite reservoir where both inner and outer zones are represented by dual porosity regions. The genealogy of the analytical solutions available including the analytical solution proposed in this thesis (shown in red) is represented below in Figure 2.8. Single Region Single Porosity (Ehlig- Economides,1979) Double Porosity (Da Prat,1981) Vertical well Single Porosity (Eggenschwiler,1980) Composite Region Reservoir Diffusivity Equation Double Porosity (Satman,1991) Single Porosity (Kuchuk,1979) Horizontal well Single Region Composite Region Double Porosity (Alpheus & Tiab,2007) Single Porosity (Obut & Ertekin,1987) Double Porosity (Current study,2012) Figure 2-8: Genealogy of the Analytical Solutions The current solution encompasses all the existing solutions. In other words, the solution proposed in this work can be collapsed to each of the individual works mentioned above by suitable implementation of the necessary conditions.

37 Chapter 3 Mathematical Formulation of the Problem In general, the flow into a vertical well located in porous, isotropic and homogeneous reservoirs is always considered to be radial in nature. But the heterogeneity of the formation and orientation of the well often distort the geometry of the flow. Specifically in the cases where directional permeability exists and the values of the permeability are in the order of micro and nanodarcies, elliptical flow seems to be a more accurate assumption than a radial flow. Elliptical flow occurs in the following two scenarios; first where the permeability distribution shows directional dependence in nature and second where the fluids are flowing into a horizontal well. Multistage hydraulically fractured horizontal wells in naturally fractured reservoirs can be approximated as composite regions with inner region, Region 1 to represent stimulated reservoir volume (SRV with dual porosity characteristics) and the second region, Region 2 to be double porosity (naturally fractured) region with different values of permeability and porosity from those of Region 1 as shown in Figure 3.1. The problem becomes challenging in nature when dealt with composite, naturally fractured reservoir formations. Flow geometry suggests that the flow into the horizontal well is elliptical in nature with confocal ellipses representing the isopotentials and orthogonal hyperbolas representing the streamlines.

38 25 Elliptical outer boundary Hydraulic fractures Elliptical outer boundary Rubble zone Natural fractures Multi-stage hydraulically fractured horizontal well in an elliptical naturally fractured reservoir Equivalent representation of a horizontal well in a composite naturally fractured elliptical reservoir Figure 3-1: Equivalent representation of the multi-stage hydraulically fractured horizontal well in a naturally fractured reservoir The following are the assumptions made in development of this model: 1. The reservoir fluid is slightly compressible with single phase flowing in matrix and fracture 2. The reservoir is composite in nature with distinct values of porosity, permeability, compressibility, viscosity in each of the zones. 3. Each zone is naturally fractured. 4. The flow from matrix to fracture is assumed to be in unsteady state as per Warren and Root s model of double porosity (other assumptions for double porosity model are explained later in this chapter)

39 26 5. The horizontal well of length L completely penetrates the formation with a radius r w.is placed on the midplane of the formation. 6. Radius of the well is negligible compared to the length of the well (r w <<L for the flow to be elliptical in nature) 7. Pressure approaches initial pressure in the limit radial distance approaches to infinity. 8. The innermost ellipse collapses to the wellbore itself with 9. The flow is laminar, hence Darcy s law is applicable 10. There is no flow into the ends of the horizontal well 11. Reservoir has uniform thickness h 12. Adsorption and other alternate phenomena of flow generation are not considered. 13. Skin and wellbore storage effects are not considered. Schematic representation of the system under consideration is indicated below. ξ w Wellbore location (degenerate ellipse) Region 2 Region 1 ξ e External boundary ξ Interface location Figure 3-2: Elliptical Coordinate Representation

40 27 3.1: Governing Equations for Homogeneous Isotropic Reservoir Derivation of the flow equation: (3.1) ( ) ( ) assuming fluid flow is slightly compressible, the above equation becomes: (3.2) ( ) ( ) ( ) ( ) The definition of the compressibility is stated as: ( ) (3.3) where, V = m/ρ. This when substituted in Equation 3.3 results in the following equation, ( ) (3.4) assuming the values of ( ) ( ) are negligible, the following equation can be written: ( ) ( ) (3.5)

41 28 For homogeneous and isotropic systems with no source term, and with a single phase incompressible flow, the viscosity doesn t change with position, so the equation collapses to (3.6) The above Equation 3.6 is well known diffusivity equation in 2- dimensional Cartesian coordinates. This is applicable to high pressure gas wells as well owing to the fact that p/μz is constant over the range of pressures considered for production. 3.2: Warren and Root Double Porosity Model The figure below indicates the double porosity (naturally fractured) reservoir model as defined by the (Warren and Root 1963) Figure 3-3: Sugar Cube Model Representation (Warren & Root, 1963)

42 29 Many models have been proposed for the fluid transfer from matrix to fractures; amongst them the two of the following models are generally used. First one considers transient flow from matrix to fractures, and the second one assumes pseudo steady state flow from matrix to fracture. Pseudo steady state flow model suggested by Warren and Root (1963) is being used in the analysis. The assumptions that are made in the double porosity model are: 1. Pseudo steady state matrix flow. 2. Flow is two dimensional. 3. Initial pressure of the reservoir is same throughout the reservoir at time, t=0. 4. The matrix comprises of a set of porous rock systems that are not connected to each other, have a low transmissibility, and high storage capacity. 5. The fracture system interconnects the porous media and has low storage capacity, high transmissibility. 6. Fractures are fed by matrix, and the fractures transport the fluid to the well (i.e. matrix does not provide fluid directly to the well) (Gerami, Pooladi and Darvish 2007). The governing equations for the double porosity model can be written as: (3.7) and the mathematical representation of the transfer of the fluid from matrix to fracture can be represented as: ( ) (3.8a) where, subscript m denotes the properties of matrix and f denotes the properties of the fracture

43 30 (3.8b) The following dimensionless quantities are defined to transform the Equations 3.8a and 3.7 into dimensionless form (for constant sand-face pressure specification): (3.9) (3.10) (3.11) (3.12) For pressure transient analysis, the dimensionless quantities are defined in the following way (for constant flow rate specification): (3.13) (3.14) (3.15)

44 31 (3.16) The main purpose in using dimensionless groups is to categorize important groups that govern the equation being solved and to simplify the algebraic expressions. They may be developed from the boundary conditions that are imposed, and eventually simplify the resultant expression in the process of attaining a solution. Using the dimensionless quantities for the constant bottom-hole pressure specification, the Equation 3.7 transforms into: ( ) ( ) (3.17) where; (3.18) ( ) (3.19) (3.20) The x and y coordinates could then be transformed to x and y as per the transformation indicated below to remove the anisotropy (3.21)

45 Once transformation suggested in Equation 3.21 is implemented in Equation 3.17, then the Equation 3.21 transforms to: 32 ( ) ( ) (3.22) 3.3: Coordinate Transformation In order to solve the flow equation in elliptical coordinates, one needs to switch over from Cartesian to elliptical coordinates. Amongst the different ways that exist in literature, the following transformation is employed: Let (3.23) (3.24) By definition, the partial derivatives can be written in the following way: (3.25) On the other hand, letting (3.26) The above transformation gives us the following: (3.27)

46 33 From Equation 3.27, one can write (3.28) Rearranging the above equations, the following is obtained: (3.29) ( ) (3.30) Using the equalities from Equation 3.29 (3.31) From Equation 3.30, the following can be written: ( ) (3.32) it follows that: (3.33) [ ] ( ) (3.34) Substituting Equation 3.34 in Equation 3.33 and replacing the p and in the resultant expression will yield:

47 34 ( ) (3.35a) Using the above Equation 3.35a and substituting in the diffusivity Equation 3.22 yields: ( ) ( ) ( ) (3.35b) The above Equation 3.35b could be simplified to: { } (3.36) Now the Equation 3.36 represents the elliptical form of Equation 3.7 Since the definition of these dimensionless groups incorporates all the important variables, it is often a practice that these dimensionless groups are eventually plotted in the final solutions (i.e. p D vs. t D or q D vs. t D ). The above Equation 3.36 is the most general form of dimensionless equation representing elliptical flow in a naturally fractured reservoir. 3.4: Flow Model The equations describing the two composite regions could be written by analogy. The definitions for the dimensionless parameters remain the same, however the equations need to be tailored specific to the region. For the Region 1, the Equation 3.37 can be written with a subscript 1: { } (3.37)

48 here the parameters ω 1 and λ 1 represent storativity ratio and interporosity transfer coefficient specific to Region 1. They are defined as: 35 (3.38) (3.39) This equation is valid in the region (starting from wellbore till the interface) The interface is signified by. Here an assumption is made that β value remains constant in both the regions, For Region 2, the equations could be written analogously as: { } 3.40) The parameters ω 2 and λ 2 correspond to the storativity ratio and interporosity transfer coefficient of the Region 2. They are defined as: (3.41) (3.42) (3.43)

49 36 ( ) (3.44) ( ) where ζ represents the diffusivity ratio and M represents mobility ratio. The equation for Region 2 is valid from Thus Equations 3.37 and 3.40 together represent the elliptical reservoir model in dimensionless coordinates. 3.4: Boundary and Initial Conditions of the Model There are two fundamental boundary conditions that could possibly be specified at the wellbore, namely 1. Constant flow-rate specification (Neumann type boundary condition); (3.45) (3.46) 2. Constant bottom-hole-pressure specification (Dirichlet type boundary condition); (3.47) where

50 37 (3.48) The reservoir is initially assumed to be at a constant pressure p i all throughout the medium. This condition when translated into dimensionless pressure will yield a zero initial pressure condition which can be represented as: (3.49) (3.50) At the interface ( ) the solution in pressure must be continuous in nature. The flow entering from one side of the interface has to be the same on the other side of the interface. So it should satisfy the following constraints (continuity requirement): (3.51) (3.52) At the outer boundary of the system, the dimensionless pressure drop at any time is zero. (3.53) Analytical solutions for the above dimensionless equations for Neumann and Dirichlet type of boundary conditions are explained in detail in the next chapter. Solutions are generated for constant bottom-hole pressure case as it is the relevant boundary condition for generation of decline curves.

51 Chapter 4 Analytical Solution Several authors have tried to solve the diffusivity equation with different inner and outer boundary conditions. One should take care in choosing the appropriate boundary conditions and type of solutions they arrive at. The solutions must be consistent with the engineering aspects of the problem and should preserve the continuity in final solutions. Amongst the various models available, the two most important are transient flow model and steady state flow model. Unsteady state/transient model is considered in the current study. In this mathematical model, skin effects and wellbore storage effects are neglected. The properties of the matrix and the fracture are assumed to be constant through the depletion process. Steady state flow models could be solved easily using conformal mapping. But for an unsteady state model, a more rigorous approach is required. Van Everdingen and Hurst (1949) were amongst the first ones to solve the unsteady state problem with a proper imposition of the boundary conditions. The inner boundary conditions at the wellbore and the interface conditions need to be handled with utmost care. For decline curve analysis, a constant bottom-hole pressure assumption is considered in solving the equations. The solutions for the Region 1 are given explicitly, while the solutions for the outer region are written by analogy as it involves a similar development.

52 39 4.1: Laplace Transformation There are many methods of arriving at a solution for the above equations with the specified boundary conditions. However, a useful method for solving this dimensionless equation is utilizing the Laplace transform. Laplace transform enables us to remove the time dependency of the problem with the usage of inner and outer boundary conditions. If p(t) is pressure at any point in the reservoir and a function dependent on time, then the Laplace transformation is expressed in the following way: (4.1) By using the above transformation, the Equations 3.37 (Region1) and 3.40 (Region 2) are transformed into Laplace space. { } (3.37) { } (3.40) The Laplacian variable is denoted by s, multiplying both sides of the Equation 3.37 by and integrating between 0 to infinity, one obtains: (4.2) [ { }] Since p Df1 and p Dm1 are dependent on time and space, the above transformation will automatically remove the dependency on time and leave the variables p Df1 and p Dm1 as a function of Laplacian space alone.

53 40 ( ) (4.3) ( ) (4.4) { } (4.5) { } (4.6) Substituting the above expressions in the Equation 3.37 will yield: { } (4.7) and the matrix transfer equation is reduced to the following equation on Laplace transformation: ( ) (4.8) The above Equation 4.8 is rewritten as (4.9) substitution for Equation 4.9 in Equation 4.7 results in { } (4.10) To simplify the above equation, the following subsitution can be made:

54 41 (4.11) substituting Equation 4.11 in Equation 4.10 will yield: (4.12) where following way:, similarly the equation for Region 2 can be written by analogy in the (4.13) where,. Along with the above transformations, the boundary conditions and interface conditions needs to be transferred to Laplace domain. The boundary conditions can appropriately be transformed in the following way: 4.2: Inner Boundary Conditions 1. Constant Rate specification (Neumann type boundary condition); (4.14) 2. Constant Pressure specification (Dirichlet type boundary condition); (4.15)

55 42 External boundary condition (4.16) Interface conditions: 1. Continuity of pressure : (4.17) 2. Continuity of flux (4.18) 4.3: Solution Procedure The method of separation of variables is used in the solving the Equations 4.12 and 4.13: (4.19) In the above representation, is assumed to be only dependent on and is only dependent on. So, one can write the following partial differential equations from Equation 4.19: (4.20)

56 43 (4.21) Substitution of Equations 4.21 and 4.20 in Equation 4.12 yields: (4.22) The above Equation 4.22 could be rewritten as: (4.23) Since X and Y are independent of each other, the above Equation 4.23 could be written in the following way: (4.24) Assuming that, and the above equations could be rewritten as: (4.25) (4.26) The above equations are called Mathieu s differential equations, and the solutions for these equations are given in terms on Mathieu functions. McLachlan (1964) introduced Mathieu functions to solve problems in elliptical coordinates. Equation 4.26 governs the flow in angular direction and is commonly referred to as angular Mathieu function. Equation 4.25 is usually

57 44 referred to as associated Mathieu equation. As this equation governs the flow in radial direction, it is commonly referred to as radial Mathieu function. A detailed discussion on Mathieu s functions is presented in Appendix A. The solutions sought should satisfy the following conditions for it to be able to represent our system in an appropriate way: 1. The solution must be periodic in ξ and η, with a period of. 2. The solution must be symmetrical by ξ and η axes. 3. Pressure must be continuous in crossing the interfocal line orthogonally (4.27) 4. Pressure gradient should be continuous for the same condition (4.28) 5. The solution must be bounded. According to McLachlan (1964), the following pairs of Mathieu functions satisfy all of the above conditions: These functions are required to satisfy the above mentioned conditions. ; This combination is periodic with a period and also so Equation 4.27 is satisfied. By principle of orthogonality of

58 Mathieu functions (explained later in the chapter 4),, so the Equation 4.28 is satisfied as well. 45 A similar check is performed for this case as well and it is determined that this solution is also suitable to the problem in hand. The individual solutions proposed for X ( and Y ( are represented as : (4.29) (4.30) So, the solution for the pressure of fracture in the Region 1 can be represented as a linear combination of the solutions presented above: (4.31) here C 2n and F 2n are individual Fourier coefficients that are used to satisfy the inner and outer boundary conditions. The interface conditions are applied to obtain their values. For the second region the solution that may satisfy the given constraints would be slightly different from the first region solution. The Fourier coefficients that may be used to satisfy the boundary conditions would be different. The solution for the second region can hence be written as:

59 46 (4.32) Where B 2n and E 2n are obtained by the application of the interface conditions 4.3.1: Infinite Reservoir case Application of the Boundary Conditions: The exact solutions for the given problem are found out by application of boundary conditions on the general solutions for Regions 1 and 2: External boundary condition: (Boundedness condition) (4.33) The above condition could be satisfied if and only if B 2n =0, as Ce 2n approaches infinite value when.this condition modifies the solution for the second region as the following: (4.34) Dirichlet type of boundary condition (Equation 4.15) is applied on Region 1 solution (Equation 4.31), and yields: (4.35)

60 47 The above equation needs to be simplified further before one could calculate the coefficients. This could be done by using the orthogonality properties of Mathieu functions (discussed in Chapter 5): Multiply on both sides with and integrate with respect to in the limits from 0 to. (4.36) From orthogonality properties of Mathieu functions, the following relation can be written: { (4.37) and the integral on the left hand side is evaluated, which results in (4.38) Here 2n superscript represents the order of the Fourier coefficient at which it is evaluated. This is not to be confused with the exponent. Therefore substituting Equations 4.38 and 4.37 in Equation 4.36 results in the following expression: (4.39) The above equation could be represented as (following Obut and Ertekin,1987):

61 48 (4.40) where, ( ) (4.41) ( ) (4.42) By imposing the interface condition Equation 4.17 on Equation 4.32 and Equation 4.34, one can obtain: (4.43) where: (4.44) (4.45) (4.46) and by imposing the interface condition Equation 4.18 on Equation 4.32 and Equation 4.34, one can obtain: (4.47) where:

62 49 (4.48) (4.49) (4.50) So, the following system of equations is developed: (4.51) [ ] [ ] [ ] The system of equations from Equation 4.51 is solved to obtain the values of the Fourier coefficients. The volumetric flow entering the internal boundary from the reservoir is written as: ( ) (4.52) In terms of dimensionless coordinates, the Equation 4.52 can be written as: ( ) (4.53) To obtain the dimensionless flow rate in Laplace space, one needs to take derivate of with respect to and evaluate the value of derivative at wellbore. ( ) (4.54)

63 50 Substituting Equation 4.54 in Equation 4.53 results in: { } (4.55) From McLachlan (1964), the following property is considered: (4.56) Substituting Equation 4.56 in Equation 4.55 yields: { } (4.57) Using Duhamel s principle, the solution for pressure at the wellbore is found out in the following way (Van Everdingen,1949); (4.58) Hence, the solution for the constant pressure boundary condition and constant rate condition are related in the form of Equation The solution for pressure at wellbore for constant rate specification is written as: { } (4.59) The above solutions in Equations 4.57 and 4.59 needs to be transformed back to time domain. This can be achieved numerically using (Stehfest 1970) algorithm. The details of Stehfest algorithm are explained in the Appendix A. Caution needs to be exercised when using the

64 Equation 4.59 for calculation of the dimensionless pressure drop, as this expression is only valid at the wellbore : Finite Outer Boundary Boundary conditions: The exact solution for the pressure is found out by application of appropriate boundary conditions at inner boundary and outer boundary. The interface conditions enable us to calculate the Fourier coefficients. External boundary condition: (in Laplace space) 1. No flow outer boundary condition (4.60) 2. Constant pressure outer boundary specification : (4.61) The inner boundary conditions remain unchanged. They are the same as Equations 4.14 through Equation The solutions for the dimensionless pressure of Region 1 and Region 2 are expressed as: (4.62) (4.63)

65 52 The outer boundary is signified by. The developments from Equations 4.39 through Equation 4.43 remain valid for this case as well. From there on, the equations will have an additional Fourier coefficient in the terms of B 2n. The constant B 2n need not be equalized to zero in the case of a finite reservoir. The imposition of interface condition (Equation 4.17) on Equation 4.62 and 4.63 yields: (4.64) where, (4.65) (4.66) (4.67) (4.68) and the interface condition (Equation 4.18) when applied on Equations 4.62 and 4.63 yield: (4.69) (4.70) (4.71) (4.72)

66 53 1. No flow external boundary: a. The external boundary condition (Equation 4.60) when implemented on Equation 4.63 yields: (4.73) where, (4.74) (4.75) 2. Constant pressure external boundary: a. The external boundary (Equation 4.61) when implemented on Equation 4.63 results in: (4.76) where, (4.77) (4.78) Therefore the following system of equations is solved for finding out the Fourier coefficients :

67 54 (4.79) [ ] [ ] [ ] Equations 4.52 through Equation 4.59 are valid developments for this case as well. The coefficients alone would change in presence of a finite outer boundary, so the dimensionless flow rate in Laplace space could be calculated with Equation 4.57 and dimensionless wellbore pressure in Laplace space could be found out using Equation These dimensionless flow rates and pressure drops needs to be transformed from Laplace space to time space in order to calculate. The numerical inversion is done using Stehfest algorithm. Transition to radial coordinates is presented in the Appendix C

68 Chapter 5 Mathieu Functions This chapter presents a brief discussion on Mathieu functions that are relevant to this study. The first ever usage of Mathieu functions dates back to 1868, when Emile Leonard Mathieu attempted to solve the stretched membrane problem having an elliptical boundary. In order to determine the vibrational modes of the membrane, the following problem was solved with the use of Mathieu functions. (5.1) The two-dimensional equation presented in Equation 5.1 was transformed to elliptical (confocal) coordinates, then using separation of variables the problem was converted to two ordinary differential equations. Assuming h to be the semi-interfocal distance, and a an arbitrary separation constant, the equations take the following form: (5.2) (5.3) (5.4) In Mathieu s problem, a and q were real.

69 56 It may be observed that Equation 5.4 is obtained from Equation 5.3 by replacing z with +/- iz, and vice versa. Hence Equations 5.3 and 5.4 are called as Mathieu and modified Mathieu equations. For the elliptical membrane problem, the appropriate solutions of the Equation 5.3 are referred to as ordinary Mathieu functions, being periodic in z with a period. As a consequence, a has special values called characteristic numbers. The corresponding solutions for Equation 5.4 for the same characteristic value a are called modified Mathieu functions. Second set of solutions that could be obtained are nonperiodic in nature. Since those are not relevant to the problem at hand, they are not being discussed here. If one writes [ ] for z in Equation 5.2, then the resultant expression is: (5.5) The solutions of the above Equation 5.5 are generally represented in a series of cosines with a periodicity of, and with q being negative. (5.6) { 5.1: Angular Mathieu Functions (5.7) Comparing Equations 5.7 and 5.6, and by analogy, the following solutions can be written:

70 57 (5.8) { This is referred to as angular function, since the function only varies with the angular coordinate. Equation 5.3 is taken and z is replaced by iz, then it transforms into the following equation: (5.9) The first solution of the above Equation 5.9 could be written as: { (5.10) The second solution of the above Equation 5.10 can be obtained by replacing z with[ ]. [ ] (5.11) 5.2: Radial Mathieu Functions (5.12) Equation 5.9 is comparable to that of Equation 5.12.

71 Hence the solutions obtained in Equations 5.10 and 5.11 are applicable to Equation So the solution by analogy can be written as: 58 { (5.13) [ ] (5.14) The first and second solutions given by Equations 5.13 and 5.14 are computationally demanding and often encounter convergence problems. Hence an alternate form of these equations is given in terms of Bessel s modified functions. As per the definitions outlined by McLachlan (1964), the solutions represented by Equations5.13 and 5.14 can alternately be calculated as: ( ) (5.15) { where, ( ) (5.16)

72 59 (5.17) The alternate solution for Equation 5.14 can be written in terms on Bessel s modified functions in the following way: ( ) (5.18) { I r ( ) represents Bessel s I function of order r and argument. K r ( ) represents Bessel s K function of order r and argument. Of all the above representations, the formulas involving product of Bessel functions in the Equations 5.15 and 5.18 are found out to be superior over other solutions in terms of convergence, and computational speed. From the Equations 5.15 and 5.18, the derivatives of these radial Mathieu functions can be computed as: (5.19)

73 60 (5.20) The derivatives are computed using Equation 5.19 and 5.20 respectively as and when required. It is evident from the above formulae that all of them require a value for Fourier coefficients for the functions to be evaluated. The following section presents a discussion on the ways to calculate the Fourier coefficients. 5.3: Calculation of Fourier Coefficients and Characteristic Value There are predominantly two methods available in the literature, first one computes the Fourier coefficient values through continuous fraction method iteratively, and the second one employs matrix methods. In the iterative scheme, value of the characteristic number is calculated iteratively, which proves to be a setback in terms of accuracy of the calculation of values of a. The matrix method is used for the calculation of constants in our problem : Calculation of Characteristic Numbers Substituting the series of cosines obtained from Equation 5.6 into Equation 5.5 yields the following recurrence relations when r is equated to 0, 1, 2,

74 61 (5.21) The resultant matrix that would be formed can be indicated as: (5.22) [ ] [ ] The above equation suggests that this could happen only if the values of a are chosen such that determinant goes to zero. With a trial and error procedure, the values of a could be computed until a good agreement is obtained, or alternatively the eigenvalues of the following matrix could be found out: (Alpheus and Tiab 2007) Upper Diagonal: a i,i-1 = q for i=1 to m-1 Middle Diagonal: a i,i =(2i-2) 2 for i=2 to m, Lower Diagonal: a i,i-1= q for i=3 to m, a 21 =q. For an example 4 4 matrix where m=4 will give us:

75 62 (5.23) [ ] The above matrix equation could be written as M A=Aa. Where matrix A is the eigen vector and the diagonal matrix a is the eigen value : Calculation of the Coefficients To compute A 2r, one needs to obtain v 2r ratio of two consecutive successive fourier coefficients which is given by: (5.24) For calculating the first coefficient v o, the following formula is used; (5.25) Normalization of the above Fourier coefficients could be done in the following way: [ ] [ ] (5.26) Equation 5.26 gives us:

76 63 ( ) ( ) ( ) (5.27) Using Equations 5.24 and 5.25 the ratios obtained are substituted in Equation 5.27, and the value of could be found out. Rest of the values for the Fourier coefficients could be found out using back substitution method through Equation 5.24.

77 Chapter 6 Computational Method This chapter presents a brief discussion on the computational aspects of the problem solved in Chapter 4, and presents flowcharts for generation of decline curve plots. Riley (1991) in his research concluded that the computation of Mathieu functions are not as demanding as the literature suggests. The current study confirms the observations of Riley (1991). In order to obtain type curves for various values of dimensionless parameters (interporosity transfer coefficients, storativity ratios, mobility ratios and diffusivity ratios), a computer program was coded. This computer code consists of a main program written in Matlab (R2011a), which transfers the control to various subroutines for necessary calculations like eigenvalue generation, Fourier coefficients calculations, values of various Mathieu functions, derivatives of the modified Bessel functions, numerical Laplacian inversion using Stehfest algorithm, and finally returns the value of flow rates/pressure drop at wellbore at each time interval specified. 1. Laplacian parameter is calculated at each time step and corresponding arguments for the Mathieu functions are computed. 2. Eigenvalues and eigenvectors for the given argument are found out, after which Fourier coefficients are computed. 3. Angular Mathieu functions are computed at required arguments orders and stored in workspace. 4. Radial Mathieu functions of first kind and second kind are calculated for different orders, and given arguments are calculated and stored.

78 65 5. The infinite summation loop for the computation of dimensionless flow rate in Laplace space is terminated when the increments < The calculated Qxd (dimensionless flow rate in Laplace space) is transferred back to time space using numerical Laplace inversion as per Stehfest Algorithm. 7. The values of q D vs. t D of three cases (Infinite acting, finite no flow outer boundary, finite constant pressure outer boundary) are then plotted as output for each corresponding set of inputs. The subroutines used in this computer program are: Main Program : Compdualpor.m : This the main script, where the calculation process starts. This program serves as the controller of the total code, transfers the values to respective subroutines, computes the necessary outputs and gets terminated as per the time limit specified by the user. The numerical inversion/stehfest algorithm is implemented in this main program itself. Function Infinite_acting.m: This subroutine calculates the qxd (dimensionless flow rate in Laplace space) at each parametric value of time for the infinite acting Reservoir case. It calculates these values by invoking the subroutines that calculate Fourier coefficients, Mathieu functions, and their derivatives Function finite_noflow.m: This subroutine calculates the qxd (dimensionless flow rate in Laplace space) at each parametric value of time for the no flow outer boundary condition. It calculates these values by invoking the subroutines that calculate Fourier coefficients, Mathieu functions, and their derivatives

79 66 Function finite_constp.m: This subroutine calculates the qxd (dimensionless flow rate in Laplace space) at each parametric value of time for the constant pressure outer boundary condition. It calculates these values by invoking the subroutines that calculate Fourier coefficients, Mathieu functions, and their derivatives Function fcoef1.m The above subroutine calculates the Fourier coefficients at a given parametric value and order. The calculations here closely follow the procedure outlined in Chapter 5. The output of this subroutine is the set of Fourier coefficients at the given integral order m. Eigen vectors are found out after assembling the tri-diagonal matrix. The normalized eigenvectors are then displayed as the outputs. Function cen1.m This function computes the value of angular Mathieu function at a given order (m), parametric value (q) and angular coordinate η. The computation is performed employing the technique outlined in Equation 5.8 by using a series summation of cosines. This function is specifically designed to handle negative arguments (q). Function besseli.m This subroutine is used to calculate the value of modified Bessel I function at the given argument and order. This is an inbuilt function in Matlab which is called as and when required. Function besselk.m This subroutine is used to calculate the value of modified Bessel K function at the given argument and order. This is an inbuilt function in Matlab which is called as and when required.

80 67 Function dbesseli.m This subroutine is used to calculate the value of derivative of modified Bessel I function at the given argument and order. This uses the inbuilt function besseli in Matlab and then following the formula given below the value is computed. (6.1) Function dbesselk.m This subroutine is used to calculate the value of derivative of modified Bessel K function at the given argument and order. This uses the inbuilt function besselk in Matlab and then following the formula given below the value is computed. (6.2) Function Ce.m This subroutine calculates the value of the first solution of Mathieu s modified equation. This function employs the formula outlined in Equation The Radial function is solved by expressing it as a product of two Bessel I functions. The input arguments include the order (m), radial coordinate (ξ) and the parametric value (q). Computations are performed with the total number of Fourier coefficients available. This subroutine calls for function besseli.m which in turn computes the besseli function value at the given order and argument. The computations are done for the negative arguments. Function Fek.m This subroutine calculates the value of the second solution of Mathieu s modified equation. This function employs the formula outlined in Equation The Radial function is solved by expressing it as a product of two Bessel functions (I and K). The input arguments include the order (m), radial coordinate (ξ) and the parametric value (q). Computations are

81 68 performed with the total number of Fourier coefficients available. This subroutine calls for function besseli.m and besselk.m which in turn computes the values of these modified Bessel functions at the given order and argument. The computations are done for the negative arguments. Function dce.m This subroutine calculates the value of the derivative of first solution of Mathieu s modified equation. This function employs the formula outlined in Equation The Radial function is solved by expressing it as a product of two Bessel I functions. The input arguments include the order (m), radial coordinate (ξ) and the parametric value (q). Computations are performed with the total number of Fourier coefficients available. This subroutine calls for functions besseli.m and dbesseli.m as required at a specific order and argument. The computations are done for the negative arguments. Function dfek.m This subroutine calculates the value of the derivative of second solution of Mathieu s modified equation. This function employs the formula outlined in Equation The Radial function is solved by expressing it as a product of two Bessel functions (I and K). The input arguments include the order (m), radial coordinate (ξ) and the parametric value (q). Computations are performed with the total number of Fourier coefficients available. This subroutine calls for functions besseli.m, dbesseli.m, besselk.m, and dbesselk.m as required at a specific order and argument. The computations are done for the negative arguments. Function linsolve.m This is another inbuilt function in Matlab, which solves the linear equation of the form A X = B. Here A is the coefficient matrix and B the result vector. X is the vector which returns the values of Fourier coefficients C 2n, F 2n, B 2n, E 2n. The maximum computation time is for evaluating the involved subroutines in iterating for the convergence criterion in the infinite_acting.m/ finite_noflow/ finite_constp. The computation

82 time highly depended on the convergence criterion used to stop the iterations at each parametric value for the incremental qxd value in the loop. 69 Compdualpor.m (Controller) User prompts for the input parameters Calculate the modified parameters Initiate the time step calculations at t D =1:10 6 Infinite_acting.m Finite_noflow.m Finite_constp.m Laplace space calculations Stehfest algorithm Calculate q D for corresponding t D Plot q D vs. t D Figure 6-1: Work-flow for the computational method

83 70 Infinite_acting.m/ finite_noflow.m/ finite_constp.m While tolerance >10-5 Calculate the Fourier coefficients Compute matrix A While loop Solve for C 2n,F 2n from AX=B (using linsolve) Calculate qxd(n) qxd=qxd+qxd(n) Tol=qxd(n)/qxd Output=qxd Figure 6-2: Work-flow for an outer boundary type specification at each time step The work flow remains very similar for all the three flow regimes, but the way in which the matrix A is computed differs significantly based on the outer boundary condition specification. The computation of matrix A invokes the usage of subroutines Ce.m, dce.m, Fek.m, dfek.m and other Bessel functions.

84 71 Chapter 7 Results and Discussion The closed form of solutions has been developed for a horizontal well in composite naturally fractured reservoir. Before the generation of final decline type curves, the model was validated with solutions available in the literature. The problem was collapsed to some special cases and the solutions generated by the computer code were validated with the available published data in literature. 7.1: Validation with Data from Literature Case 1: Validation with data from the paper by Obut & Ertekin (1987) Obut & Ertekin (1987) developed closed form solutions for the elliptical flow model in a composite reservoir. If one replaces value of q= s*f(s)/4 to q= s/4 in Equations 4.25 and 4.26, then the composite, naturally fractured reservoir problem collapses to composite single porosity case. Infinite outer boundary is considered for validating the results. Solution presented in the form of equations from Equations 4.33 through Equation 4.59 for the infinite reservoir case is coded by replacing s*f(s) with s. Thus the results obtained from this simulation run should ideally collapse to values generated by Obut & Ertekin (1987). Table 7.2 shows the comparison between the values generated from this model and Obut & Ertekin s (1987) model. The results seem to be in exact agreement as indicated in the Figure 7.1. In the Figure 7.1, the dots represent the data points generated from Obut and Ertekin (1987) model, and the solid line represents the data obtained from the current study.

85 Log10(qD) 72 The following reservoir parameters indicated in Table 7.1 were assumed in generation of the results: Table 7-1: Reservoir parameters assumed Elliptical Wellbore Coordiante,ξ w 0 Elliptical Interface Coordinate,ξ Diffusivity ratio, 1.1 Mobility ratio, 5 Time under consideration,t D Comparison of Data with Tanju Obut & Ertekin's Study qd from Current Study qd from Tanju Obut's Study Log10(tD) Figure 7-1: Comparison of data with Obut & Ertekin(1987) Study

86 73 Table 7-2: Comparison of data with Obut & Ertekin(1987) study t D q D (Tanju Obut et al.) q D (current Study) Error % % % % % % % % % % % % % % % % % % % % % % % % % % % Case 2: Validation with data from the paper by Alpheus & Tiab (2007) One could collapse this model to single region double porosity by assuming same values of ω and λ for the both regions under consideration. Thus the model should collapse to results obtained by model of Alpheus & Tiab (2007). Table 7.4 presents the comparison between the results obtained by the aforementioned model and current model. The current model predicts the values of q D with an approximate error of 10% between the times t D = 4 to At later times, the predictions

87 log10(q D ) 74 seem to be in line with Alpheus s study. This is evident from Figure 7.2, that at late times the solution predicted by the current model lies in exact agreement with the Alpheus s model. The reference model predictions are depicted by blue dots and the data from the current model is plotted as a black solid line. Table 7-3: Reservoir properties assumed Elliptical wellbore coordiante,ξ w 0 Elliptical interface coordinate,ξ Storativity ratio, 0.9 Interporosity transfer coefficient, Time under consideration, t D Comparison with data from Alpheus study q D from Alpheus's study q D from current study log10(t D ) Figure 7-2: Comparison of data with Alpheus(2007) study

88 75 Table 7-4: Comparison of data with Alpheus(2007) study t D q D (Alpheus et al.,) q D (Current Study) Error % % % % % % % % % % % % % % % % % % % % % % % % % % % % Case 3: Validation with data from the paper by Kuchuk (1979) In addition to the aforementioned models, further checks are conducted by collapsing the current model to a homogeneous model. The model could be collapsed to homogeneous case by equating diffusivity and mobility ratios to unity and replacing s*f(s) by s in the problem

89 log10(q D ) 76 formulation. Thus the model should be able to predict the results generated by Kuchuk s(1979) study. Table 7.5 indicates the reservoir properties assumed. Comparative study is presented in Figure 7.3 and Table 7.6. From the results generated, it is evident that the model provides an exact match with the data from literature. Table 7-5: Reservoir properties assumed Elliptical wellbore coordiante,ξ w 0 Elliptical interface coordinate,ξ Diffusivity ratio, 1 Mobility ratio, 1 Time under consideration,t D Comparison with data from Kucuk's study q D from current study q D from Kucuk's study log10(t D ) Figure 7-3: Comparison of data with Kuchuk(1979) study

90 77 Table 7-6: Comparison of data with Kuchuk(1979) study t D q D (kuchuk's study) q D (Current study) error % % % % % % % % % % % % % % % % % % % % % % % % % % % A closer look at these three tables indicates that consistently, better agreements have been obtained for larger time values. Thus it could be concluded with confidence that the model is able

91 78 to predict accurate values for dimensionless flow rates at various t D values for horizontal well configurations: In order to establish the validity of the current model, comparisons are drawn with the well know radial solutions by collapsing the current model to be applicable to radial flow around a vertical well, the following checks are carried out Case 4: Validation with data from the paper by Satman (1991) Satman (1991) published a solution that could predict flow rates from a vertical well in a composite naturally fractured reservoir. For the current study to be applicable to radial coordinates, a validity check with respect to Satman s model is extremely important. The elliptical outer boundary and the interface boundary parametric values were obtained upon coordinate transformation as indicated in chapter 3. The length of the horizontal well was equated to the radius of the wellbore for the current work to collapse to radial coordinates. As indicated in Figure 7.4, a good agreement has been obtained between the solutions proposed by Satman et al.,(1991) and the current work. However there is a minor difference in the predictions of flow rates from Satman s work and the current study, which could be attributed to the difference in the flow models assumed for the transfer of fluid from matrix to fracture. A transient flow model was assumed in Satman s work, whereas the current work assumes a pseudo-steady state model..

92 log10(q D ) Comparison with composite naturally fractured vertical well case 1 = 2 =1 q D from current study q D from Satman's work log10(t D ) Figure 7-4: Comparison of data with Satman(1991) study Case 5: Validation with data from the paper by Eggenschwiler (1980) Similarly, the current work is compared with the data from a composite vertical well solution proposed by Eggenschwiler et al.,(1980). The comparison of the data is indicated in the Figure 7.5, where a close agreement was found between the data generated using Eggenschwiler s solution and reduced solution from the current study. In order to collapse the current work to a composite vertical well solution, the storativity ratio (ω) values of the both the regions are considered at unity and interporosity transfer coefficients (λ) a very small value of These assumptions necessarily force the two regions to act as single porosity regions, hence could retain the composite nature by setting the mobility and diffusivity ratio values different from unity.

93 log10(q D ) 80 In order to compare the current work with Eggenschwiler s study, care was taken to ensure the appropriate translation of the interface diameters to properly depict the elliptical system under consideration. In Figure 7.5, the blue dots depict the data generated from the solution proposed by Eggenschwiler et al.,(1980) and the solid black line plots the data generated by collapsing the current model to a radial system with a vertical well Comparison with composite vertical well solution q D from current study q D from Eggenschwiler's study log10(t D ) Figure 7-5: Comparison of data with Eggenschwiler(1980) study Case 6: Validation with data from the paper by Da Prat (1981) The current work could further be reduced to obtain a solution that could predict the decline characteristics of flow from a vertical well in a naturally fractured reservoir. In order to compare the results generated, data from Da Prat s thesis was considered. The comparison between the data generated could be observed from Figure 7.6. The data shows close agreement throughout

94 log10(q D ) 81 the range of t D considered. The storativity values (ω) for both the regions were assumed an identical value of 0.01, and the interporosity flow coefficient (λ) values for both the regions are set at Comparison with double porosity vertical well solution q D from Da Prat's study q D from Current study Validation with vertical wells case for single region dual porosity case with 1 = 2 =0.01, 1 = 2 =10( - 5) log10(t D ) Figure 7-6: Comparison of data with Da Prat(1981) study Case 7: Validation with data from the paper by Economides (1979) As a final validation check, the current work is collapsed to the simplest case under consideration i.e. radial flow around a vertical well. For comparison and reference, solution suggested by Ehlig- Economides,(1979) has been considered. The results are indicated in Figure 7.7. The storativity ratio (ω) values, diffusivity ratio (, mobility ratio (M) were all assumed to take a value of unity for both the regions to become single porosity and reservoir homogeneous in nature. The interporosity flow coefficient (λ) values were assumed at a very low value of 10-8 which prevents

95 log10(q D ) 82 any possibility of flow from matrix to fracture or vice versa and also enables us to remove the singularity that might arise by assuming a zero value for interporosity flow coefficient (λ) Comparison with homogeneous solution for vertical well ( 1 = 2 =1, 1 = 2 =10( - 8)) q D from Current study q D from Economides study log10(t D ) Figure 7-7: Comparison of data with Economides(1979) study The Figure 7.7 indicates that the current model could be collapsed to as simple a system as a radial flow around a vertical well with appropriate assusmptions. With having compared the data generated from Economides(1979) study and the current work, it clearly indicates that the model developed in the current study could succesfully encompass all the solutions indicated in the Figure 2.8 (Genealogy of the type curves). 7.2: Sensitivity Analysis The following sensitivity analysis was done in order to gain better insight into the capabilities of the solution. The results of these tests are displayed in Figures 7.8 through Figure Critical

96 83 analysis of the following figures should be able to generate some meaningful implications about the sensitivity of the proposed model. In the first check, the values of diffusivity ratio (ζ), interporosity flow coefficient (λ) values are set at constant values of 1, 10-6 respectively. The storativity ratio (ω) of Region 1 is set at 0.01 and Region 2 at The mobility ratio (M) alone was varied from 0.1 to 5.0 and the dimensionless flow rates thus obtained were analyzed. The following Table 7.7 indicates the reservoir properties assumed in the analysis. Table 7-7: Reservoir properties assumed First Check Parameter Symbol Value Elliptical wellbore coordinate ξ w 0 Elliptical interface coordinate ξ Storativity ratio of Region 1 ω Storativity ratio of Region 2 ω Interporosity transfer coefficient of Region 1 λ Interporosity transfer coefficient of Region 2 λ Diffusivity ratio ζ 1 Mobility ratio M 0.1,0.5,2.0,5.0 Dimensionless time considered t D < 10 6

97 log10(q D ) M=0.1 M=0.5 M=2.0 M= Increasing Mobility Ratio log10(t D ) Figure 7-8: Sensitivity analysis with respect to mobility ratio (M) From Figure 7.8, it can clearly be observed that increase in the mobility ratio (M) shifts the flow rate curves downwards. Mobility ratio is the ratio of, so in here if the viscosity of both the regions is assumed to be constant, then the mobility ratio (M) would represent the permeability ratio alone. So with increasing difference in permeability between Region 1 and Region 2, the composite characteristics begin to predominantly influence the solution. The permeability term in the dimensional transformation could account for this downward trend observed. Second check was implemented by varying diffusivity ratio (ζ) by keeping all other dimensionless variables constant. Mobility ratio (M) was set at 2; interporosity flow coefficient

98 log10(q D ) (λ) values at 10-6, storativity ratios (ω 1 and ω 2 ) at 0.01 and respectively, diffusivity ratio (ζ) values were varied from 0.1 to 5 as indicated in the Table Table 7-8: Reservoir properties assumed Second Check Parameter Symbol Value Elliptical wellbore coordinate ξ w 0 Elliptical interface coordinate ξ Storativity ratio of Region 1 ω Storativity ratio of Region 2 ω Interporosity transfer coefficient of Region 1 λ Interporosity transfer coefficient of Region 2 λ Diffusivity ratio ζ 0.1,0.5,2,5 Mobility ratio M 2 Dimensionless time considered t D < =0.1 =0.5 =1.0 = =0.1 2 = = =10-5 M=1 o = log10(t D ) Figure 7-9: Sensitivity analysis with respect to diffusivity ratio (ζ)

99 86 As the diffusivity ratios were increased, the decline curves shifted downward as indicated in Figure 7.9. Diffusivity ratio (ζ) is, but the mobility ratio (M) is assumed to be unity. Hence diffusivity ratio (ζ) simplifies to. From this definition as diffusivity ratio (ζ) increases, the compressibility of the flow in region two is increased (as porosity doesn t vary significantly across the regions). When compressibility of the inner region is higher, naturally more flow can occur for the same amount of pressure drop. Hence the results generated are in line with the theoretical considerations. With higher values of elliptical interface distance, a significant double dual porosity signature is evident as indicated in Figure 7.9. Third check was performed by varying interporosity flow coefficient (λ) values for the both the regions together. All other dimensionless variables assume a constant value. The value of λ 1 and λ 2 are varied from 10-4 to The properties of reservoir used to perform this check are indicated in Table 7.9 Table 7-9: Reservoir properties assumed Third Check Parameter Symbol Value Elliptical wellbore coordinate ξ w 0 Elliptical interface coordinate ξ Storativity ratio of Region 1 ω Storativity ratio of Region 2 ω Interporosity transfer coefficient of Region 1 λ ,10-5,10-6,10-7 Interporosity transfer coefficient of Region 2 λ Diffusivity ratio ζ 1 Mobility ratio M 1 Dimensionless time considered t D < 10 6

100 log10(q D ) = = = = log10(t D ) Figure 7-10: Sensitivity analysis with respect to interporosity flow coefficient (λ) From Figure 7.10, it is evident that the interporosity flow coefficient (λ) does have a significant effect on the dimensionless flow rates. The onset of second linear flow is determined by the value of λ. As the value gets smaller, the onset of second decline gets delayed as seen from Figure At large t D, flow rates again become independent of the interporosity flow coefficient value considered. At low values of t D, the Laplacian parameter becomes large in number, thereby simplifying the following expression as:. This expression hence is almost independent of λ, and is reflected from the Figure 7.9 at early times.

101 88 Fourth check was conducted by varying storativity ratio (ω 1 ) values of region 1 and keeping the rest of the dimensionless variables constant. Mobility ratios (M), diffusivity ratio (ζ), interporosity flow coefficient (λ), are assumed constant values of 10, 1, and 10-6 respectively. The reservoir properties assumed are tabulated below, and the variation of the flow rates with the storativity ratios is indicated in the Figure 7.11 Table 7-10: Reservoir properties assumed Fourth Check Parameter Symbol Value Elliptical wellbore coordinate ξ w 0 Elliptical interface coordinate ξ Storativity ratio of Region 1 ω 1 0.1,0.01,0.001, Storativity ratio of Region 2 ω Interporosity transfer coefficient of Region 1 λ Interporosity transfer coefficient of Region 2 λ Diffusivity ratio ζ 1 Mobility ratio M 10 Dimensionless time considered t D < 10 8

102 log10(q D ) = = = = Decreasing log10(t D ) Figure 7-11: Sensitivity analysis with respect to storativity ratio (ω) of Region 1 The Figure 7.11 indicates that the curves shift upward with decrease in storativity ratio, however decline faster with decreasing storativity ratio values. The flow rates are higher when the matrix stores the bulk of the fluid and that is evident from the Figure At late times when the fluid from the Region 1 has been withdrawn, Region 2 acts as a primary source of production, because of which the curves coincide at higher t D values. In Figure 7.11, the crossover is a result of higher initial rates and faster decline with high storativity ratio values. This could be explained by the fact that when fractures store bulk of the fluid, the initial flow rates are higher and at the same time, they drain faster. As the storativity value (ω) decreases, the matrix holds bulk of the fluid and drainage process takes higher time, as can be seen from the delay of onset of decline in Figure 7.11.

103 log10(q D ) 90 Table 7-11: Reservoir properties assumed Fifth Check Parameter Symbol Value Elliptical wellbore coordinate ξ w 0 Elliptical interface coordinate ξ 0 3,4,5,6 Storativity ratio of Region 1 ω Storativity ratio of Region 2 ω 2.01 Interporosity transfer coefficient of Region 1 λ Interporosity transfer coefficient of Region 2 λ Diffusivity ratio ζ 1.1 Mobility ratio M 20 Dimensionless time considered t D < q D vs.t D for varying ellitpical interface location =6 =5 =4 = =0.1 2 = = =10-5 M=20 = log10(t D ) Figure 7-12: Sensitivity analysis with respect to interface distance (ξ 0 ) A further check with respect to the interface distance is performed and results indicated an interesting trend. Depending on the interface location, the second declines are branched out at

104 91 varying times, with steeper declines arising from shorter distances. The results are indicated in Figure 7.12 shows the dependency of the interface location on the double dual porosity signature. The flow rates values does not vary significantly with change in storativity ratio of second region. At large times, the asymptotic expression for the Fourier coefficients and resultant Mathieu functions need to be used to avoid oscillatory solutions. At large times, the system of equations that is formed for finding the solution may be ill ranked. To avoid potential divergence problems, it is advised to use the asymptotic solutions for large time values. 7.3: Forward Solutions Forward solutions are generated using a computer code for various case scenarios. It is practically impossible to depict an exhaustive set of type curves to include all possible reservoir configurations. However considerable amount of time has been spent on generating these forward solutions which encompasses a broad range of reservoir spectra. Two mobility ratios (M), two different diffusivity ratios (ζ), two interporosity flow coefficient (λ) values, 4 storativity ratios (ω) are presented with four different interface conditions (ξ 0 ). The smooth behavior of the forward solution indicates that error free interpolations could be made from these curves. As the storativity ratio (ω) values go higher i.e. as storage of fluid in fracture increases, the separation of the curves is pronounced in the early and late times, however during the transition phase the curves come close to each other. Higher storativity ratio (ω) values resulted in lower initial production rates, however took longer time to decline through the transition period. For low storativity ratio (ω) values, sharper transitions are observed, and the interface effect is minimal on these curves, hence they almost overlap on each other.

105 92 The interporosity flow coefficient (λ) values determine the time at which transition occurs, lower the values later the onset of transition is observed. It is worthwhile to note that almost all the curves come really close at the transition phase than at early or late time declines. Higher interporosity flow coefficient (λ) values force early transitions and a pronounced effect of these is evident from the generated forward solutions. Mobility ratio (M) has a peculiar effect on the way the curves behave with change in interface location. When ratios lower than unity is considered, the curves shift upward with increase in the interface distance from the wellbore, whereas ratios higher than 1 force the curves to shift downward with increase in interface distance. The effect of diffusivity ratio (ζ) has been minimal on the behavior of the curves, but with the lower diffusivity ratio (ζ) values (less than 1), the curves come closer than expected and with higher diffusivity ratio (ζ) values (greater than 1), curves are separated. For high values of interporosity flow coefficients (λ > 10-3 ), the results generated are oscillatory in nature towards early times, which in turn could be attributed to the divergent series values by Mathieu functions at high arguments. But this may not be considered a setback as the normal range of values for interporosity flow coefficients (λ) are normally never that high, even in case if the values are higher than 10-3 then conversion from dimensional time to dimensionless time would be higher as a result, low dimensionless times translate to extremely low dimensional times. At small times, the arguments might become large in nature and in turn may result in divergent series for the Mathieu function computations. To avoid the potential divergence problems, asymptotic expressions for computation of Mathieu functions are employed. The following tree summarizes the cases for which forward solutions have been generated.

106 93 M=0.5 M=2 ζ =0.5 ζ =2.0 ω 1 =0.01 ω 1 =0.1 ω 2 =0.001 ω 2 =0.01 ω 2 =0.1 λ 1, λ 2 =10-5 λ 1, λ 2 =10-6 λ 1 =10-5 λ 2 =10-6 λ 1 =10-6 λ 2 =10-5 ξ 0 =0.5 ξ 0 =1.0 ξ 0 =1.5 ξ 0 =2.0 Figure 7-13: Tree of forward solutions generated

107 log10(q D ) log10(q D ) =0.5 =1.0 =1.5 = Increasing log10(t D ) Figure 7-14: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =2, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = =0.5 =1.0 =1.5 = Increasing log10(t D ) Figure 7-15: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =2, ω 1 =0.1, ω 2 =0.01, λ 1 =λ 2 =10-5

108 log10(q D ) log10(q D ) =0.5 =1.0 =1.5 = Increasing log10(t D ) Figure 7-16: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =2, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = =0.5 =1.0 =1.5 = Increasing log10(t D ) Figure 7-17: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =2, ω 1 =0.1,ω 2 =0.01, λ 1 =λ 2 =10-6

109 log10(q D ) log10(q D ) =0.5 =1.0 =1.5 = Increasing log10(t D ) Figure 7-18: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =2, ω 1 =0.1, ω 2 =0.1, λ 1 =10-6,λ 2 = =0.5 =1.0 =1.5 = Increasing log10(t D ) Figure 7-19: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =2, ω 1 =0.1, ω 2 =0.1, λ 1 =10-5, λ 2 =10-6

110 log10(q D ) log10(q D ) =0.5 =1.0 =1.5 = Reducing log10(t D ) Figure 7-20: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =0.5, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = =0.5 =1.0 =1.5 = Increasing log10(t D ) Figure 7-21: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =0.5, ω 1 =0.1, ω 2 =0.01, λ 1 =λ 2 =10-5

111 log10(q D ) log10(q D ) =0.5 =1.0 =1.5 = Increasing log10(t D ) Figure 7-22: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =0.5, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = =0.5 =1.0 =1.5 = Increasing log10(t D ) Figure 7-23: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =0.5, ω 1 =0.1, ω 2 =0.01, λ 1 =λ 2 =10-6

112 log10(q D ) log10(q D ) =0.5 = =1.5 = log10(t D ) Figure 7-24: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =0.5, ω 1 =0.1, ω 2 =0.1, λ 1 =10-6, λ 2 = =0.5 =1.0 =1.5 = Increasing log10(t D ) Figure 7-25: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=2, =0.5, ω 1 =0.1, ω 2 =0.1, λ 1 =10-5, λ 2 =10-6

113 log10(q D ) log10(q D ) =0.5 =1.0 =1.5 = Reducing log10(t D ) Figure 7-26: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =0.5, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = =0.5 =1.0 =1.5 = Reducing log10(t D ) Figure 7-27: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =0.5, ω 1 =0.1, ω 2 =0.01, λ 1 =λ 2 =10-5

114 log10(q D ) log10(q D ) =0.5 =1.0 =1.5 = Reducing log10(t D ) Figure 7-28: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =0.5, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = =0.5 =1.0 =1.5 = Reducing log10(t D ) Figure 7-29: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =0.5, ω 1 =0.1, ω 2 =0.01, λ 1 =λ 2 =10-6

115 log10(q D ) log10(q D ) =0.5 =1.0 =1.5 = Reducing log10(t D ) Figure 7-30: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =0.5, ω 1 =0.1, ω 2 =0.1, λ 1 =10-6, λ 2 = =0.5 =1.0 =1.5 = Reducing log10(t D ) Figure 7-31: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =0.5, ω 1 =0.1, ω 2 =0.1, λ 1 =10-5, λ 2 =10-6

116 log10(q D ) log10(q D ) =0.5 =1.0 =1.5 = Reducing log10(t D ) Figure 7-32: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =2, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = =0.5 =1.0 =1.5 = Reducing log10(t D ) Figure 7-33: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =2, ω 1 =0.1, ω 2 =0.01, λ 1 =λ 2 =10-5

117 log10(q D ) log10(q D ) =0.5 =1.0 =1.5 = Reducing log10(t D ) Figure 7-34: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =2, ω 1 =0.01, ω 2 =0.001, λ 1 =λ 2 = =0.5 =1.0 =1.5 = Reducing log10(t D ) Figure 7-35: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =2, ω 1 =0.1, ω 2 =0.01, λ 1 =λ 2 =10-6

118 log10(q D ) log10(q D ) =0.5 =1.0 =1.5 = Reducing log10(t D ) Figure 7-36: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =2, ω 1 =0.1, ω 2 =0.1, λ 1 =10-6, λ 2 = =0.5 =1.0 =1.5 = Reducing log10(t D ) Figure 7-37: q D vs. t D for a horizontal well in a composite naturally fractured formation with M=0.5, =2, ω 1 =0.1, ω 2 =0.1, λ 1 =10-5, λ 2 =10-6

119 : Numerical Example 1 The following example was considered to validate the type curve data with data from a commercial numerical simulator CMG IMEX*: The sample data set that has been used in CMG IMEX simulations is Table 7-12: Reservoir properties assigned for the numerical simulator model Fracture permeability of Region 1, k f1 (md) 10 Matrix permeability of Region 1, k m1 (md) 5.68E-07 Viscosity of Region 1, μ 1 (cp) Fracture porosity of Region 1, φ f Matrix porosity of Region 1, φ m Matrix compressibility of Region 1, c m1 Fracture compressibility of Region 1, c f1 1.00E E-06 Fracture spacing of Region 1, W f1 1 Fracture permeability of Region 2, k f Matrix permeability of Region 2, k m2 Viscosity of Region 2, μ E E-02 Fracture porosity of Region 2, φ f Matrix porosity of Region 2, φ m Matrix compressibility of Region 2, c m2 Fracture compressibility of Region 2, c f2 1.00E E-06 Fracture spacing of Region 2, W f Length of the horizontal well, L (ft) 1770 Wellbore radius, r w (ft) 0.25

120 107 An inner crushed zone is simulated by changing the properties (fracture spacing, fracture permeability, matrix and fracture porosity) of grid blocks surrounding the wellbore such that, the inner zone represents the stimulated reservoir volume. As it could be observed from Figure 7.38, the inner zone takes the shape of an ellipse. The screen shot of the CMG IMEX model is attached below for visualization of the model. Figure 7-38: Snapshot of the CMG IMEX model used for validation. The above data set was randomly assigned values, and corresponding dimensionless values were found and fed to the analytical solution to generate an appropriate type curve.

121 108 The corresponding dimensionless values are tabulated in Table 7.13: Table 7-13: Dimensionless parameters for generation of forward solution Storativity ratio of Region 1, ω Storativity ratio of Region 2, ω Interporosity flow coefficient of Region 1, λ Interporosity flow coefficient of Region 2, λ Diffusivity ratio, ζ 1000 Mobility ratio, M 1000 Necessary corrections are made to account for the change in viscosity, volume formation factor, and effective permeability to the oil flow in the process of translating the dimensional production to dimensionless production for comparison. The assumptions made in the model are: 1. Infinite acting outer boundary 2. Inner region and outer region are two distinct double porosity regions The flow rate with time data obtained from the commercial simulator is converted into dimensionless flow rate vs. dimensionless time data using the manipulations indicated by Equations 3.12 and 3.11 respectively. The numerical simulator was run for ten years to compare the data with the current work.

122 109 Figure 7-39: Screenshot of the CMG IMEX pressure transient data at the end of the run Dimensionless q D vs. t D values are compared from the numerical simulator CMG IMEX and the current work. The Figure 7.39 indicates that the infinite outer boundary specification is valid throughout the run. The screenshot displays the extent to which pressure transients have travelled by the end of the run time. Owing to the high interporosity flow coefficient, a distinct double porosity signature could not evidently be observed in the Figure 7.40.

123 110 Results are indicated in the following Figure B3: 2.5 q D vs. t D - Comparative Study q D t D CMG IMEX Data Current study Figure 7-40: Comparative study of results from a commercial simulator CMG IMEX As it can be observed from the above data for dimensionless times (t D ) up to 160, a very good agreement has been obtained. Thus it could be concluded with certainty that the solutions developed in the current work are valid. *CMG IMEX is a fully featured three-phase, four component black oil reservoir simulator for modeling primary depletion and secondary recovery processes in conventional oil and gas reservoirs. IMEX also models pseudo-miscible and polymer injection in conventional oil reservoirs, and primary depletion of gas condensate reservoirs, as well as the behavior of naturally or hydraulically fractured reservoirs.

124 : Numerical Example 2 This section presents an example and its solution by explaining the usage of forward solutions developed in this research. The properties of the composite naturally fractured system are given as: Region 1: Table 7-14: Reservoir parameters assumed k f1 200 md k f2 10 md k m md k m md μ 1 1 cp μ 2 1 cp φ f φ f φ m φ m c m c m c f c f W f1 1 ft W f ft Height, h (ft) 50 Volume formation factor, B(Res bbl/stb) 1.1 Initial pressure, p i (psia) 5000 Bottom-hole pressure, p sf (psia) 1000 Half Length of the horizontal well, L (ft) 1500 Volume formation factor, B 1.1

125 112 Table 7-15: Dimensionless parameters for the analytical model Storativity ratio of Region 1, ω Interporosity flow coefficient of Region 1, λ E-02 Storativity ratio of Region 2, ω Interporosity flow coefficient of Region 2, λ E-02 Elliptical horizontal wellbore coordinate, ξ w 0 Diffusivity ratio, ζ 20 Mobility ration, M 20 Elliptical interface coordinate, ξ The problem is to predict the flow rates for a given interface location and other dimensionless parameters presented in Table The dimensionless time and dimensionless flow rate conversion are outlined below q D vs. t D data is generated as following from the analytical solution :

126 113 Table 7-16: q D vs. t D data generated by analytical solution q D Given the dimensionless parameters and few basic assumptions of constant viscosity and constant volume formation factor would enable us with generating the flow rate for the system specified in Table The flow rate data for above system could be translated using the equations for t D and q D : t D

127 114 Table 7-17: q vs. t data generated through analytical solution q ( in STB/day) t( in hours) In order to achieve better results, one may use a data interpolation table for the viscosity and volume formation factor against pressure to improve the accuracy of the predictions. In that case at every data point the translation from dimensionless to dimensional parameters would depend on the calculated viscosity and calculated volume formation factor.

128 Chapter 8 Summary and Conclusions Decline rate predictions for a horizontal well that is subject to multi-stage hydraulic fracturing in a double porosity reservoir is extremely important in evaluating the effectiveness of the fracture job and economics involved in production from the given resource. One of the important concerns for a project engineer is to design an optimal fracturing job that results in the required production. In order to design an efficient process, the permeability increase, extent of fracture propagation, change in the physical properties of the inner zone needs to be established. An analytical solution that associates the change in physical properties to the increase in production greatly helps the engineer to predict accurate decline rates. Therefore, a significant amount of research has been done on this problem and various solutions were proposed. The problem of multi-stage hydraulically fractured horizontal well in a naturally fractured reservoir could approximately be modeled as a composite naturally fractured reservoir comprising of two double porosity regions. The problem of a horizontal well production from a truly composite double porosity region in elliptical geometry has been less explored. Directional permeability, presence of vertical fractures leads the flow geometry to be elliptical in nature. New solutions were developed for constant pressure and constant rate inner boundary conditions. Owing to the large number of dimensionless variables, rate decline curves could not be presented as type curves. Instead forward solutions were generated for various scenarios of mobility ratios, diffusivity ratios, interporosity flow coefficients, and storativity ratios.

129 116 The presented tree of forward solutions in Figure 7.13 is not an exhaustive case scenario. The most likely values that storativity and interporosity flow coefficient formations may assume are considered. The curves generated indicate that the effect of boundary is minimal in the range of mobility and diffusivity ratios assumed. The flow rate curves shift upward with increasing interface location from the wellbore when the mobility ratios of greater than 1 are assumed. The flow rate curves shift downward with increase in interface distance from the wellbore when the mobility ratios of less than 1 are assumed. Forward solutions generated indicate that the value of interporosity flow coefficient has a huge effect on the onset of second decline arising from drainage of matrix. The solutions developed above are applicable for constant bottom-hole pressure assumption, the solutions for the dimensionless pressure at wellbore could be found out using Van Everdingen s (1949) method of superimposition. The solutions developed validated against the existing solutions which formed the subsets of the current solution. A good agreement was obtained in each of the cases indicating the versatility of the current solution for its applicability over the broad range of reservoir conditons. The model achieved excellent results in the range of values considered for generating these results. At low times, for large mobility ratios and small interporosity flow coefficient values, the solutions seem to oscillate. The users of this study need to exercise caution in using these values for their future study. Necessary modificiations need to be made to include small time and long time asymptotic solutions. Numerical examples demonstrating the usage of the generated solutions are presented in sections 7.4 and 7.5. Section 7.4 presents a comparison of the data generated through current work and that of a numerical simulator. Section 7.5 presents an example demonstrating the usage of forward solutions in predicting the flow rates with time.

130 Suggestions for Future Studies Transient flow model may be assumed for the matrix transfer. It is found on close examination that the solutions for the second region are orthogonal, only when diffusivity ratios close to 1 are assumed. To improve the solution, Riley s(1991) assumptions may be used. In addtion to the current pseudo steady state flow assumption from matrix to fracture, other flow mechanisms namely adsorption and other non-darcian flows that might arise due to concentration difference could be modeled. Thus a multimechanistic model that accounts for the pore size distribution, concentration differences, dynamic slippage, and sorption could be attempted. Well-bore storage effects and the skin effects could be modeled for better predictions. Grouping the dimensionless parameters to obtain type curve solutions could be explored.

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135 122 Appendix A Stehfest Algorithm The solutions obtained in the Laplace space involve complex mathematical relations and cannot easily be transferred back to the real time space. Computation of infinite integrals would take significant amount of computation time. So the solutions are kept in Laplace space and numerically they are inverted back to time domain using an algorithm proposed by Stehfest. If a Laplace transform P(s) is given in a form of a real procedure. Numerical Laplace inversion produces an approximate value F (t) at a real time space. F(t) is evaluated in the following way: ( ) Here N must be even. As it can be seen that V i depends on N alone, in case of evaluation at different times with same N array V needs to be evaluated only once. This procedure is given by Stehfest. The following is the Gaver Stehfest Algorithm method of implementation. The constants V i in the series are evaluated by the following formula: [ ] ( ) The calculation method is based on a probability density function and is bound to give rise to some errors in inverting oscillating functions. For monotonically decreasing and increasing functions, Gaver Stehfest algorithm predicts accurate enough values. The optimal value of N is found to be 12. For most the simulation runs, this value has been used. Higher N sometimes has given risen to oscillations in the problem. So optimal values found were N=12, 14, 16. All these three values seem to working well with the data sets examined in generating forward solution.

136 Appendix B Graphic User Interface A Graphic User Interface (GUI) has been developed to facilitate generation of type curves in a user friendly manner. This feature enables user to enter the reservoir parameters and generate a forward solution/type curve of desired nature. As indicated in the GUI Snapshot below, user can select from the following 8 options: 1. Well type Horizontal or Vertical 2. Reservoir type Homogeneous, Composite, Dual porosity, Composite dual porosity Depending on the type of well and reservoir chosen, the input parameters need to be accordingly fed to the GUI. Sample snapshot of the vertical well, homogenous reservoir case is indicated below:

137 Figure B-1: Snapshot of the GUI User need to choose the radio button Horizontal or Vertical for the selection of well type and scroll down to select from the available reservoir types. Once the options are chosen, a set of blank input parameters are displayed in which, user needs to feed the appropriate input values. Then a pushbutton simulate is used to run the program and populate the results table and the plot. Here an example case could be loaded for each one of the cases to automatically populate the input values for an example case. Reset counter could be used to erase all the input values, and start the program again.

138 125 A working GUI on generation of decline curves for horizontal case can be seen as below Figure B-2: Results generated by GUI for a vertical well producing from a homogeneous reservoir For varying well type and varying reservoir types, different scenarios are presented. The below attached screen shot is the input layout for the horizontal well configuration

139 Figure B-3: Snapshot of GUI for horizontal well homogeneous reservoir case 126

140 Figure B-4: Snapshot of the GUI for horizontal well in a double porosity reservoir case 127

141 Figure B-5: Snapshot of the GUI for vertical well in a composite reservoir case 128