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UNIVERSITY OF CALGARY

Modeling of Matrix-Fracture Interaction for Conventional Fractured Gas

Reservoirs

by

Ehsan Ranjbar

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF CHEMICAL AND PETROLEUM ENGINEERING

CALGARY, ALBERTA

JANUARY, 2014

Ehsan Ranjbar 2014



iv

Dedication

I dedicate this work to my Mother for

her unconditional love and support



v

Table of Contents

Abstract ............................................................................................................................... iiAcknowledgements ............................................................................................................ iiiDedication........................................................................................................................... ivTable of Contents ................................................................................................................ v

List of Tables .................................................................................................................... viii

List of Figures and Illustrations .......................................................................................... ix

CHAPTER ONE: INTRODUCTION ................................................................................. 11.1 Background ............................................................................................................... 11.2 Dual-porosity system ................................................................................................. 11.3 Shape factor concept ................................................................................................. 31.4 Motivations and objectives ........................................................................................ 41.5 Components and outline of this study ....................................................................... 5References ....................................................................................................................... 8

CHAPTER TWO: EFFECT OF FRACTURE PRESSURE DEPLETION REGIMESON THE DUAL-POROSITY SHAPE FACTOR FOR FLOW OFCOMPRESSIBLE FLUIDS IN FRACTURED POROUS MEDIA ........................ 10

Abstract ......................................................................................................................... 102.1 Introduction ............................................................................................................. 122.2 Methodology ........................................................................................................... 16

2.2.1 Constant fracture pressure ............................................................................... 192.2.2 Linearly declining fracture pressure ............................................................... 192.2.3 Exponentially declining fracture pressure ....................................................... 20

2.3 The approximate analytical solutions ...................................................................... 212.3.1 Constant fracture pressure ............................................................................... 22

2.3.2 Linearly declining fracture pressure ............................................................... 232.3.3 Exponentially declining fracture pressure ....................................................... 26

2.4 Model verification ................................................................................................... 292.5 Results ..................................................................................................................... 33

2.5.1 Comparison of model with Warren and Root model ...................................... 332.5.2 Linearly declining fracture pressure ............................................................... 342.5.3 Exponentially declining fracture pressure ....................................................... 35

2.6 Conclusions ............................................................................................................. 38Nomenclature ................................................................................................................ 40References ..................................................................................................................... 42Appendix 2.A: Solution of the gas diffusivity equation for different fracture

depletion conditions .............................................................................................. 46

2.A1: Linearly declining fracture pressure .............................................................. 462.A2: Exponentially declining fracture pressure ...................................................... 49

CHAPTER THREE: ONE DIMENSIONAL MATRIX-FRACTURE TRANSFER INDUAL-POROSITY SYSTEMS WITH VARIABLE BLOCK SIZEDISTRIBUTION ...................................................................................................... 54

Abstract ......................................................................................................................... 54



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3.1 Introduction ............................................................................................................. 553.2 Matrix block size distributions ................................................................................ 58

3.2.1 Uniform or rectangular distribution ................................................................ 593.2.2 Exponential distribution .................................................................................. 593.2.3 Normal or Gaussian distribution ..................................................................... 60

3.2.4 Linear distribution ........................................................................................... 60

3.2.5 Log-normal distribution .................................................................................. 603.3 Equivalent lengths for different matrix block size distributions ............................. 62

3.3.1 Equivalent length concept ............................................................................... 623.3.2 Discrete matrix block size distribution ........................................................... 623.3.3 Continuous matrix block size distribution ...................................................... 63

3.4 Mathematical model for flow of compressible fluid in fractured media withvariable block size distribution .............................................................................. 64

3.5 Slightly compressible fluids .................................................................................... 683.6 Validation ................................................................................................................ 693.7 Results ..................................................................................................................... 73

3.7.1 Rectangular, discrete, normal and log-normal distributions ........................... 74

3.7.2 Linear distribution ........................................................................................... 753.7.3 Exponential distribution .................................................................................. 77

3.8 Discussion ............................................................................................................... 783.9 Conclusions ............................................................................................................. 82Nomenclature ................................................................................................................ 83References ..................................................................................................................... 85Appendix 3.A: Solution of diffusion equation for variable block size distribution ...... 88Appendix 3.B: Derivation of dimensionless rate and dimensionless cumulative

production for different matrix block size distribution ......................................... 90

CHAPTER FOUR: SEMI-ANALYTICAL SOLUTIONS FOR RELEASE OF FLUIDSFROM ROCK MATRIX BLOCKS WITH DIFFERENT SHAPES, SIZES ANDDEPLETION REGIMES.......................................................................................... 94

Abstract ......................................................................................................................... 944.1 Introduction and previous studies............................................................................ 954.2 Approximate analytical solution ............................................................................. 98

4.2.1 Constant fracture pressure ............................................................................. 1034.2.2 Variable fracture pressure ............................................................................. 1054.2.3 Variable block size distribution (multiple blocks) ........................................ 107

4.3 Model verification ................................................................................................. 1124.4 Results ................................................................................................................... 116

4.4.1 Effect of fracture pressure depletion regime ................................................. 1164.4.2 Block size distribution effect ........................................................................ 120

4.4.3 Comparison of different block geometries .................................................... 1224.5 Conclusions ........................................................................................................... 125Nomenclature .............................................................................................................. 126References ................................................................................................................... 128Appendix 4.A: Analytical solution for cylindrical blocks .......................................... 133

4.A1: Constant fracture pressure ............................................................................ 1334.A2: Variable fracture pressure ............................................................................ 136



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4.A3: Variable block size distributions .................................................................. 141Appendix 4.B: Analytical solution for spherical blocks ............................................. 144

4.B1: Constant fracture pressure ............................................................................ 1444.B2: Variable fracture pressure ............................................................................ 1474.B3: Variable Block size distributions ................................................................. 149

CHAPTER FIVE: CONCLUSIONS AND RECOMMENDATIONS............................ 1525.1 Conclusions ........................................................................................................... 152

5.1.1 Fracture pressure boundary condition effect for a slab-shaped block .......... 1525.1.3 Block size distribution effect for slab-shaped blocks ................................... 1535.1.4 Block geometry effect ................................................................................... 153

5.2 Recommendations ................................................................................................. 154

APPENDIX A: COPYRIGHT PERMISSIONS ............................................................. 155



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List of Tables

Table 2.1: Stabilized values of the shape factor and time at which the pressuredisturbance reaches the inner boundary for different depletion regimes in thefracture. ........................................................................................................................ 38

Table 3.1: Observed frequency of matrix block size in the soil column (Gwo et al.,1998) ............................................................................................................................ 63

Table 3.2: Block size distribution for the discrete distributions. ......................................... 72

Table 3.3: Values of dimensionless equivalent length for different matrix block sizedistributions. ................................................................................................................ 81

Table 4.1: Different probability distribution function and their equivalent radius. .......... 111

Table 4.2: Data used for semi-analytical and numerical models....................................... 115

Table 4.3: Stabilized values of the shape factor based on this study and literaturemodels. ....................................................................................................................... 116

Table 4.4: Values of dimensionless equivalent radius for different matrix block sizedistributions. .............................................................................................................. 120



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List of Figures and Illustrations

Figure 1.1: An idealized dual-porosity system (Lemonnier and Bourbiaux, 2010). ............. 2

Figure 1.2: Road map of this study. ...................................................................................... 7

Figure 2.1: Schematic of the matrix-fracture model. ......................................................... 17

Figure 2.2: Comparison of the matrix-fracture cumulative fluid production obtainedfrom the approximate analytical solution and the numerical model of Eclipse forconstant fracture pressure. ........................................................................................... 30

Figure 2.3: Comparison of the developed shape factor model with literature models forslightly compressible fluid in the case of linearly declining fracture pressure. .......... 30

Figure 2.4: Comparison of the developed shape factor model with literature models forslightly compressible fluid in the case of exponentially declining fracture pressurefor small values of exponent (k=0.0001). ................................................................... 31

Figure 2.5: Comparison of the developed shape factor model with literature models forslightly compressible fluid in the case of exponentially declining fracture pressure(k=0.632). .................................................................................................................... 32

Figure 2.6: Comparison of the developed shape factor model with literature models forslightly compressible fluid in the case of exponentially declining fracture pressure(k=1). ........................................................................................................................... 32

Figure 2.7: Comparison of the developed model with numerical and Warren and Rootmodel. .......................................................................................................................... 33

Figure 2.8: Comparison of the shape factor for linearly declining and constant fracturepressure. ....................................................................................................................... 35

Figure 2.9: Shape factor comparison for different exponents for exponentially decliningfracture pressure. ......................................................................................................... 36

Figure 2.10: Comparison of the dimensionless shape factor for different pressuredepletion regime in the fracture. .................................................................................. 37

Figure 3.1: Different probability density functions. ............................................................ 61

Figure 3.2: Illustration of a matrix-fracture system and its boundary conditions (left)and representation of fractured reservoirs in the case of non-ideal matrix blocksize distribution (right). ............................................................................................... 65

Figure 3.3: Comparison of the presented model with literature models (Chang 1995,Hassanzadeh and Pooladi-Darvish 2006) for slightly compressible fluid and idealblock size distributions. ............................................................................................... 71


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Figure 3.4: Comparison of the presented semi-analytical model with the numericalresults for the first and second case. Continuous lines are predictions by analyticalmodel. Dots and dashes are from numerical simulations. ........................................... 73

Figure 3.5: Dimensionless rate versus dimensionless time for ideal, rectangular,discrete, normal and log-normal distributions whenF

h=0.1. ...................................... 75

Figure 3.6: Comparison of dimensionless rate for linearly increasing (m=100/81,b=35/81), linearly decreasing (m=-100/81, b=145/81) distribution with Fh=0.1and ideal matrix block size distributions. .................................................................... 76

Figure 3.7: Comparison of dimensionless rate for exponential (exponent values of -20,-5, 5and 20) withFh=0.1and ideal block size distributions. ..................................... 77

Figure 3.8: Dimensionless cumulative production versus dimensionless time fordifferent matrix block size distributions with Fh=0.1. Blocks showing thedimensionless equivalent size of the matrix blockLc/Lcmax. ........................................ 79

Figure 3.9: Dimensionless shape factor for different matrix block size distributions.Blocks showing the dimensionless equivalent size of the matrix blockLc/Lcmax. ....... 81

Figure 4.1: Schematic representation of the problem for a cylindrical block. .................. 102

Figure 4.2: Comparison of the presented model with the numerical simulation for 2Dflow (cylindrical block approximation). .................................................................... 114

Figure 4.3: Comparison of the presented model with the numerical simulation for 3Dflow (spherical block approximation). ...................................................................... 114

Figure 4.4: Dimensionless rate versus dimensionless time for different fracturedepletion regimes for a cylindrical block. ................................................................. 118

Figure 4.5: Dimensionless cumulative release versus dimensionless time for differentfracture depletion regimes for a cylindrical block. .................................................... 119

Figure 4.6: Dimensionless cumulative release versus dimensionless time for differentfracture depletion regimes for a spherical block. ...................................................... 119

Figure 4.7: Dimensionless cumulative release versus dimensionless time for differentblock size distribution and cylindrical blocks. .......................................................... 121

Figure 4.8: Dimensionless cumulative release versus dimensionless time for differentblock size distribution and spherical blocks .............................................................. 121

Figure 4.9: Dimensionless cumulative release versus dimensionless time for differentblock geometries ........................................................................................................ 123

Figure 4.10: Normalized cumulative fluid release versus square root of scaled time ....... 124



Chapter One:Introduction

1.1 Background

Large portion of the production of natural gas occurs from fractured formations including

naturally fractured reservoirs (NFR), coal bed methane (CBM), shale and tight fractured

gas reservoirs. A naturally fractured reservoir (NFR) is defined as a reservoir that contains

fractures created by natural processes, such as diastrophism or volume shrinkage. These

natural fractures can have a positive or negative effect on fluid flow (Aguilera, 1995;

Ordonez et al., 2001). NFRs are usually thought to consist of an interconnected fracture

network, which provides the main flow paths (fractures have high permeability and low

storage volume), and the reservoir rock or matrix, which acts as the main source of the

fluid storage (matrix blocks have low permeability and high storage volume) (Beckner,1990). On the other hand in these reservoirs, the major storage for the reservoir fluids is in

the matrix whereas flow primarily occurs in the highly conductive fractures (Chen, 1989).

In such a system, in addition to the intrinsic properties of the matrix and fracture, the

interaction between the matrix and fractures should be modeled accurately.

In general, there are computational challenges in upscaling of fluid flow in fractured

formations. Upscaling of transport and flow parameters for porous media has been

investigated from decades and a range of upscaling techniques have been introduced (Deng

et al., 2010). Flow and transport in fractured porous media are often described using a

dual-porosity model.

1.2 Dual-porosity system

The dual-porosity approach has been used in numerical simulation of groundwater, oil, and

gas flow in fractured porous media. This model assumes that the porous medium includes

two different regions, one related with the macropore or fracture network with high

permeability and the other with a less permeable and more porous system of rock matrix

blocks. Dual-porosity models assume that fluid flow can be described by two equations for

matrix and fractures, which are coupled using a term describing the exchange of fluid

between the two pore regions (Gerke and van Genuchten, 1993). Figure 1.1 shows a

schematic representation of an ideal dual-porosity system.



Chapter 1. Introduction

2

Figure 1.1: An idealized dual-porosity system (Lemonnier and Bourbiaux, 2010).

In the literature, two approaches have been widely used to model dual-porosity systems

mathematically including non-coupled (Chang, 1995; Hassanzadeh and Pooladi-Darvish,

2006, Ranjbar and Hassanzadeh, 2011), and coupled (Hassanzadeh et al., 2009) methods.

The equations that are expressed for coupled dual-porosity models are as follows:

qpk

t

pc f

ff

ff

2

, (1.1)

mmm

mm pk

t

pc 2

, (1.2)

mmm p

V

Akq )(

. (1.3)

Equations (1.1) and (1.2) are used to explain the pressure diffusion of single-phase slightly

compressible fluid in fractured and matrix medium respectively. In Equation (1.1), qis the

source term that stands for the net addition of fluid to the fracture system from the matrix

blocks, per unit of total volume. Equation (1.3) is used to couple Equations (1.1) and (1.2)

and shows the matrix-fracture interflow or matrix-fracture transfer function. In these

equations subscripts f and m stand for fracture and matrix, respectively. The pf and pm

represent the fluid pressure in the fracture and matrix respectively, is the fluid viscosity,

is the porosity of the medium and kis the permeability. Laplacian operator 2 represents




3

the divergence of gradient and can be used for different coordinate system and different

matrix block geometries (Cartesian, Cylindrical or Spherical).

A coupled dual-porosity model can be formulated by assuming that at each point, there is a

matrix block with specified shape. Inside each block the fluid pressure pmis time and space

dependent (Equation (1.2)). Matrix-fracture interflow term qdoes not appear explicitly in

Equation (1.2). This interflow is assumed to be distributed through the fracture media as a

source/sink term; the interflow enters the matrix blocks only at their boundaries

(Zimmerman et al., 1992).

Non-coupled approach is an alternative method which is used to determine the matrix-

fracture transfer rate for fractured reservoirs. In this approach the pressure diffusion is

solved in the matrix block and fracture is used as a boundary for the matrix system. Thesolution of matrix diffusion equation is used to determine the matrix-fracture transfer

function. Equation (1.4) represents the pressure diffusion in the matrix block

mmm

mm pk

t

pc 2

. (1.4)

The fracture pressure is assumed as a boundary condition for this partial differential

equation (PDE) to determine the matrix-fracture transfer function. As we will discuss later

this fracture pressure can be a constant or vary with time based on a pre-specified time

function.

It should be mentioned that Equation (1.4) is used for a slightly compressible fluid and is a

linear partial differential equation and this equation can be solved by common methods like

Laplace transform or separation of variables.

1.3 Shape factor concept

Currently, the dual-porosity approach is one of the computationally efficient and widely

used methods to model fluid flow in fractured reservoirs. In this approach, the matrix andfracture are separated into two different media, each with its own properties. A transfer

function has been used to represent the matrix-fracture interaction and govern the mass

transfer between the matrix blocks and the fractures (Barrenblat et al., 1960). The rate of

mass transferred from the matrix to the fracture is directly proportional to the shape factor.

For modeling of naturally fractured reservoirs, an accurate value of the shape factor is




4

required to account for both the transient and pseudo-steady state behaviour of the matrix-

fracture interaction and also the geometry of the matrix-fracture system.

In the dual-porosity models and pseudo-steady state conditions the blocks are treated as a

lumped system. In this approach the matrix-fracture transfer function is expressed as

follows:

)( fmm pp

kq

. (1.5)

In this equation is called shape factor. This parameter is a function of fracture spacing

and geometry of the blocks and has the dimension of reciprocal of area.

1.4 Motivations and objectives

Study of gas flow in fractured porous media is important in a variety of engineering fields.

In hydrology there exist a large number of mathematical models (numerical, analytical or

semi-analytical) to simulate the flow of compressible fluids in underground environments

and structured soils (You et al., 2011). Flow of compressible and slightly compressible

fluids (water or oil) in fractured reservoirs has been studied extensively with applications

in prediction of production rates and well testing. Therefore, the flow of compressible

fluids like gases and air in fractured porous media is important in hydrological,

environmental and petroleum engineering.

Flow and transport in fractured porous media are often described using a dual-porosity

model. It is emphasized that it is not practical to model a large scale fractured reservoir

based on a fine grid approach due to the requirement of large computational time. The

presented semi-analytical models in this study which are based on a non-coupled approach

can be incorporated into numerical models for accurate modeling of the amount of

transferred fluids between matrix and fractures using available dual-porosity formulation.

In other words, this study is important to reduce the computational time for large scalesimulations of gas flow in fractured porous media and can be nested in a numerical model

to resolve subgridblock scale flows.

The developed model in this study can be used to simulate single-phase compressible fluid

flow in the fractured porous media. The presented models can handle the variable fracture

pressure and variable block size distribution for different block geometries. It also can be




5

used for slightly compressible fluid without requiring numerical inversion or infinite series

calculation, in an efficient manner.

1.5 Components and outline of this study

This is a paper-based thesis with each chapter published in a peer reviewed journal.

Chapters are written so that they can be followed independently where each chapter

consists of separate abstract, introduction and previous studies, materials and

methodologies, verification, results, conclusions, nomenclatures, references and

appendices.

In Chapter two we investigate the effect of pressure decline in the fracture on the matrix-

fracture transfer shape factor for a compressible fluid. We study the influence of the

fracture pressure (as a boundary condition for the matrix block) on the shape factor for

flow of a compressible fluid in a dual-porosity model, which has not been investigated in

the former works. To determine the value of the shape factor, diffusivity equation for gas

flow, which is a nonlinear PDE, is solved using the combination of the heat integral

method, the method of moments and Duhamels theorem and this solution is used to

evaluate the shape factor.

Chapter three represents a new semi-analytical approach to consider the effect of variable

matrix block size distribution on flow of a compressible fluid in dual-porosity media. In

addition, due to the analytical nature of the proposed model, it can be used to model flow

of slightly compressible fluids in fractured media. The proposed model is a new semi-

analytical approach that can handle the variable matrix block size distribution for the flow

of both compressible and slightly compressible fluids in the fractured media. In addition to

the continuous block size distribution, the proposed model is also capable to model matrix-

fracture transfer when there is discrete block size distribution using structural information

of a porous medium.Finally in Chapter four a new semi-analytical model for different matrix block geometries

(cylindrical and spherical) for flow of compressible and slightly compressible fluids in

fractured porous media is developed. This approximation is used to derive the matrix-

fracture fluid transfer for 2D flow or slab-shaped blocks surrounded by two sets of

fractures and 3D flow or slab-shaped blocks surrounded by three sets of fractures. The




6

model can handle various block size distributions and different pressure regimes in the

fracture in the case of different geometries. At the end in Chapter five, conclusions of the

thesis are presented leading to the recommendations for future works. Figure 1.2 shows the

road map of the study.




7

Figure 1.2: Road map of this study.

Slab-Shaped Matrix Block Cylindrical & Spherical Matrix Block

Matrix-Fracture Transfer Function for

Fractured Gas Reservoirs

Fine Grid Numerical Simulation and Comparison with

Literature Model for Validation (Chapters two to four)

Variable Fracture Pressure

(Chapter two)

Variable Block Size

Distribution (Chapter three)

Constant Fracture Pressure

(Chapter four)

Variable Fracture Pressure

(Chapter four)

Variable Block Size

Distribution (Chapter four)




8

References

Aguilera, R. (1995).Naturally fractured reservoirs.Tulsa, Oklahoma: PennWell Press.

Barenblatt, G.E. Zheltov, I.P. Kochina, I.N. (1960). Basic concepts in the theory of seepage

of homogeneous liquids in fissured rocks (strata).J. Appl. Math. Mech.,20, 852-864.

Beckner, B.L. (1990). Improved modeling of imbibition matrix/fracture fluid transfer in

double porosity simulators.PhD dissertation, Stanford University.

Chang, M.M. (1995). Analytical solution to single and two-phase flow problems of

naturally fractured reservoirs: theoretical shape factor and transfer functions. PhD

dissertation, University of Tulsa.

Chen, Z.X. (1989). Transient flow of slightly compressible fluids through double-porosity,

double-permeability systems-A state-of-the-art review. Transp. Porous Med.,4,147-84.

Deng, H. Dai, Z. Wolfsberg, A. Lu, Z. Ye, M. Reimus, P. (2010). Upscaling of reactive

mass transport in fractured rocks with multimodal reactive mineral facies. Water Resour.

Res.,46.

Gerke, H.H. van Genuchten, M.Th. (1993). A dual-porosity model for simulating the

preferential movement of water and solutes in structured porous media, Water Resour.

Res., 29(2), 305-319

Hassanzadeh, H. Pooladi-Darvish, M. (2006). Effects of fracture boundary conditions onmatrix-fracture transfer shape factor. Transp. Porous Med.,64, 51-71.

Hassanzadeh, H. Pooladi-Darvish, M. Atabay, S. (2009). Shape factor in the drawdown

solution for well testing of dual-porosity systems.Adv. Water Res., 32, 1652-1663.

Lemonnier, P. Bourbiaux, B. (2010). Simulation of Naturally Fractured Reservoirs. State

of the Art. Part 2: Matrix-Fracture Transfers and Typical Features of Numerical Studies,

Oil & Gas Science and Technology Rev. IFP, 65 (2), 263-286.

Ordonez, A. Penuela, G. Idrobo, E.A. Medina C.E. (2001). Recent advances in naturally

fractured reservoir modeling. CT&F Ciencia, Tecnologia y Futuro.,2, 51-64.

Ranjbar, E. Hassanzadeh, H. (2011). Matrix-fracture transfer shape factor for modeling

flow of a compressible fluid in dual-porosity media.Adv. Water Res.,34(5), 62739.

You, K. Zhan, H. Li, J. (2011). Analysis of models for induced gas flow in the unsaturated

zone, Water Resour. Res., 47,W04515, doi: 10.1029/2010WR009985.




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Zimmerman, R.W. Chen, G. Bodvarsson, S. (1992). A dual-porosity reservoir model with

an improved coupling term. Seventeenth Workshop on Geothermal Reservoir

Engineering, Stanford.



Chapter Two:Effect of Fracture Pressure Depletion Regimes on the Dual-Porosity

Shape Factor for Flow of Compressible Fluids in Fractured Porous Media1

Abstract

A precise value of thematrix-fracture transfer shape factor is essential for modeling fluid

flow in fractured porous media by a dual-porosity approach. The slightly compressible

fluid shape factor has been widely investigated in the literature. In a recent study, we have

developed a transfer function for flow of a compressible fluid using a constant fracture

pressure boundary condition (Ranjbar and Hassanzadeh, 20112). However, for a

compressible fluid, the consequence of a pressure depletion boundary condition on the

shape factor has not been investigated in the previous studies. The main purpose of this

chapter is, therefore, to investigate the effect of the fracture pressure3depletion regime4on

the shape factor for single-phase flow of a compressible fluid. In the current study, a model

for evaluation of the shape factor is derived using solutions of a nonlinear diffusivity

equation subject to different pressure depletion regimes. A combination of the heat integral

method, the method of moments and Duhamels theorem is used to solve this nonlinear

equation. The developed solution is validated by fine-grid numerical simulations. The

presented model can recover the shape factor of slightly compressible fluids reported in the

1This chapter is an exact copy of: Ranjbar, E. Hassanzadeh, H. Chen, Z. (2011). Effect of Fracture PressureDepletion Regimes on the Dual-Porosity Shape Factor for Flow of Compressible Fluids in Fractured PorousMedia,Advances in Water Resources, Vol. 34 (12), Page: 1681-1693.

2The focus of our previous study (Ranjbar, E. Hassanzadeh, H. (2011), Matrix-fracture transfer shape factorfor modeling flow of a compressible fluid in dual-porosity media,Advances in Water Resources, 34(5), page627-639) was to find the shape factor for the single-phase flow of compressible fluids (gases) in fractured

porous media for the case of constant fracture pressure. In this study (Ranjbar and Hassanzadeh, 2011), atheoretical analysis of the constant fracture pressure shape factor for the flow of a compressible fluid in

fractured porous media was presented. The presented semi-analytical solution for constant fracture pressurewas validated with fine-grid numerical simulations. In this chapter we further develop our previous study toconsider the effect of pressure variation in the fracture on the matrix-fracture shape factor.

3It is worth noting that the fracture pressure in this thesis is different than the hydraulic fracture pressure andit implies the fluid pressure inside the fracture or the boundary condition imposed on the matrix block.

4In this thesis the fracture depletion regime implies the pressure variations in the fracture which acts as aboundary condition for the matrix block.



Chapter 2. Effect of fracture pressure depletion regimes on the dual-porosity

11

literature. This study demonstrates that in the case of a single-phase flow of compressible

fluid, the shape factor is a function of the imposed boundary condition in the fracture and

its variability with time. It is shown that such dependence can be described by an

exponentially declining fracture pressure with different decline exponents. These findings

improve our understanding of fluid flow in fractured porous media.




12

2.1 Introduction

In general, there are computational challenges in upscaling of fluid flow in fractured

formations. Upscaling of transport and flow parameters for porous media has beeninvestigated from decades and a range of upscaling techniques have been introduced (Deng

et al., 2010). Large portion of the produced natural gas occurs from fractured formations

including naturally fractured reservoirs (NFR), coal bed methane (CBM) and tight

fractured gas reservoirs. In these reservoirs, the major storage for the reservoir fluids is in

the matrix whereas flow primarily occurs in the highly conductive fractures (Chen, 1989).

Warren and Root (1963) established the dual-porosity model for modeling of a slightly

compressible fluid flow in the naturally fractured reservoirs. In the dual-porosity approach,

a fractured reservoir is divided into two media with completely different properties:

fracture and matrix. The fracture network supplies the main flow paths and the reservoir

rock or matrix acts as the major source of the fluid storage (Beckner, 1990). On the other

hand most of the fluid storage is in the matrix and fluid flows through the fractures as the

main channel. Therefore, an improved dual-porosity model should be able to accurately

account for the fracture and matrix interaction.

A dual-porosity model, which is an effective and broadly used approach for modeling and

upscaling of fluid flow in the fractured porous media, assumes that two distinct types of

porosity coexist in a representative rock volume (Chen, 1989; Cihan and Tyner, 2010; Di

Donato and Blunt, 2004; Liu and Chen, 1990; Zimmerman et al., 1996). In general,

fracture has a low storage capacity and high transmissivity and the adjacent rock matrix

has a high storage capacity and a relatively low transmissivity (Kazemi and Gilman, 1993).

Defining the transfer shape factor that accounts for the interaction among the matrix and

fracture is a great challenge in dual-porosity upscaling. In dual-porosity models the matrix-

fracture interaction is modeled through a shape factor. An equivalent fracture permeability,matrix-permeability, matrix-fracture transfer coefficient (shape factor) and saturation

functions (for multiphase flow) are essential parameters for the dual-porosity approach.

Studies have been conducted in the past to determine the transfer shape factor for slightly

compressible fluids in the fractured reservoirs (Bourbiaux et al., 1999; Coats, 1989;




13

Kazemi et al., 1976; Quintard and Whitaker, 1996; Quintard and Whitaker, 1996; Quintard

and Whitaker, 1998; Thomas et al., 1983; Ueda et al., 1989).

A precise value of the shape factor is essential to consider transient and pseudo-steady state

performance of the matrix-fracture interaction and also geometry of the matrix-fracture

system. It should be noted that the functionality of the fracture pressure as a boundary to

the matrix blocks may also have a significant effect on the stabilized value of the shape

factor for a slightly compressible fluid (Chang, 1995; Hassanzadeh and Pooladi-Darvish,

2006).

In traditional dual-porosity formulation the flow between the matrix and the fracture is

considered by a transfer function, which acts as a source term in the governing equation for

fluid flow in the fractures. Darcys law is used in this source function over the mean pathbetween the matrix and the adjacent fracture. In the non-coupled dual-porosity formulation,

flow in the fractures acts as a boundary condition for flow in the matrix (Kazemi and

Gilman, 1993). This transfer function and the amount of fluid that is transferred from the

matrix to the fracture are directly proportional to the shape factor. Numerical simulation of

naturally fractured reservoirs using a dual-porosity approach requires a precise value of the

shape factor for the entire period of the production time.

In general, there are two models to consider the matrix and fracture interaction including

pseudo-steady state and transient transfer. The former model ignores the pressure transient

in the matrix while the latter model accounts for the pressure transient in the matrix. The

matrix-fracture shape factor for a slightly compressible fluid can be obtained using the

following equation (Lim and Aziz, 1995):

)( fm

m

m

mm

ppt

p

k

c

, (2.1)

where is the fluid viscosity, cm, m, and km are the total isothermal compressibility,porosity and permeability, respectively, mp shows the average pressure of the matrix

block,fp is the fracture pressure and is the matrix-fracture transfer shape factor with

dimension of L2. In a pseudo-steady state model the matrix blocks are considered as a




14

lumped system with an average pressure, mp , while in the transient model one needs to

find the solution of the pressure diffusivity equation given by:

tpcpk mmmmm

, (2.2)

For a slightly compressible fluid, negligible variation of the fluid viscosity and isothermal

compressibility with the pressure leads to a linear flow equation for the pressure variation

in the matrix. This equation can be solved by common analytical or semi-analytical

methods such as the Laplace transform or separation of variables method (Zimmerman et

al., 1996; Lim and Aziz, 1995; Hassanzadeh et al., 2009; Shan and Pruess, 2005;

Zimmerman et al., 1993).

Determination of the matrix-fracture transfer shape factor for a slightly compressible fluid

based on the pseudo-steady state or transient transfer model has been studied in the past.

Investigators have considered the effect of fracture boundary conditions on the dual-

porosity formulation and shape factor for the slightly compressible fluid (Chang, 1995;

Hassanzadeh and Pooladi-Darvish, 2006; Rangel-German et al., 2010). It has been shown

that the fracture pressure and its variation with time affect the transient and pseudo-steady

state value of the shape factor.

There have been new efforts to determine the shape factors for multi-phase flow andthermal methods in fractured porous media (Rangel-German et al., 2010; Civan and

Rasmussen, 2002; van Heel et al., 2008). There have also been a few reports in the

literature to model dual-porosity systems for compressible fluids with different approaches

than this study (Lu and Connel, 2007; Penuela et al., 2002). A more detailed review of

shape factor developments was discussed elsewhere (Ranjbar and Hassanzadeh, 2011).

Although the dual-porosity approach with the shape factor concept has some limitations, it

has been widely used and well accepted approach in hydrological sciences and petroleum

reservoir modeling. This may be because of its simplicity, computational efficiency and

flexibility in application to various fluid flow and transport problems. In addition, lack of

more advanced and efficient models that can accurately take into account the matrix-

fracture interaction have contributed to extensive use of the dual-porosity models.

Currently, the majority of commercial flow simulators use the dual-porosity approach.




15

However, it should be pointed out that a new line of attack on tackling fluid flow and

transport in fractured rocks has been recently introduced based on discrete fracture network

models (Hoteit and Firoozabadi, 2005). Hoteit and Firoozabadi (2005) presented a discrete

fracture model for single phase flow of compressible fluids in heterogeneous and fractured

media. They developed a numerical model by combining the mixed finite element and the

discontinuous Galerkin methods for multi-component gas flow. Discrete fracture model

also have been used for multiphase flow and water injection in fractured media (Karimi-

Frad and Firoozabadi, 2003; Lemmonnier and Bourbiaux, 2010).

It has been reported in the literature that in the case a slightly compressible fluid the

pseudo-steady state value of the matrix-fracture shape factor is a function of the pressure

decline regime in the fracture. Contrary to the slightly compressible fluid case, thevariation of the isothermal compressibility and viscosity with pressure cannot be ignored

when dealing with a compressible fluid. This leads to a nonlinear PDE. Therefore, the

reported shape factors for slightly compressible fluids cannot be applied for compressible

fluids or their application has not been validated in the previous studies. In a recent study

we derived the matrix-fracture shape factor for a compressible fluid in dual-porosity

systems (Ranjbar and Hassanzadeh, 2011). The effect of fracture pressure decline on the

compressible fluid shape factor has not been reported in the previous studies. In this study,

we further develop our previous study to investigate the effect of pressure decline in the

fracture on the matrix-fracture transfer shape factor for a compressible fluid.

We study the influence of the fracture pressure (as a boundary condition for the matrix

block) on the shape factor for flow of a compressible fluid in a dual-porosity model which

has not been investigated in the former works. To obtain the matrix-fracture shape factor a

nonlinear diffusivity equation is solved using the heat integral method and the method of

moments. To consider the effect of the time variation of the boundary conditions a

modified trial solution (early time) and Duhamels theorem (late time) are used to derive

the early and late time shape factors for the declining fracture pressure cases. The

developed approximate analytical solution is validated by a numerical model (Ranjbar and

Hassanzadeh, 2011). The developed shape factor model can recover predictions from the

shape factor models available in literature for a slightly compressible fluid. This shape




16

factor may find applications in dual-porosity modeling of the conventional and

unconventional naturally fractured gas reservoirs such as coalbed methane and fractured

tight gas reservoirs.

This paper is organized in a manner that follows a methodology for derivation of the shape

factor for compressible fluids. Next solution of the nonlinear diffusivity equation subject to

a declining fracture pressure is obtained using the heat integral and moment methods and

Duhamels theorem. Afterwards model verification and results are discussed followed by

conclusions.

2.2 Methodology

In this section the shape factor for flow of a compressible fluid from a matrix block under

different fracture boundary conditions is derived by taking into account the pressure

dependency of the viscosity and isothermal compressibility. Darcys law for flow of gas in

the porous media is expressed as follows:

dx

dp

B

Akq

g

mgsc

, (2.3)

where A is the cross-section area and Bg is the gas formation volume factor. Using the

definitions of the gas formation volume factor and real gas pseudo-pressure (Ikoku, 1992)

and writing the Darcys law over some characteristics length l leads to the following

equation:

2fm

sc

scmsc

l

A

Tp

Tkq

. (2.4)

As shown in Equation (2.4), lis a length where the matrix pressure is equal to its average

pressure and this length changes with time during transient matrix production. For a

matrix-fracture combination shown in Figure 2.1, Equation (2.4) is multiplied and divided

by the bulk volume of the matrix-block to define the transfer function for compressiblefluids. Using the definition of the shape factor (Equation (2.5)); the final equation for the

matrix-fracture transfer function for compressible fluids (e.g. gases), is expressed as

Equation (2.6) (Ranjbar and Hassanzadeh, 2011):




17

)2/( bVl

A , (2.5)

)(4 fmm

sc

bsc

sc T

k

p

VT

q

, (2.6)

In this equation Tis the absolute temperature, is the shape factor, m shows the average

matrix block pseudo-pressure and f is the fracture pseudo-pressure.

According to the Warren and Root (1963) dual-porosity model for a slightly compressible

fluid, the interporosity flow rate per unit volume of the rock can be expressed in terms of

the accumulation rate in the matrix as follows:

t

p

cqm

mm

. (2.7)

The interporosity flow rate for compressible fluids can be expressed as follows (Ranjbar

and Hassanzadeh, 2011):

tT

c

p

VTq mmm

sc

bscsc

4. (2.8)

Combination of Equations (2.6) and (2.8) leads to the following equation for single-phase

shape factor of compressible fluids as follows (Ranjbar and Hassanzadeh, 2011):

tkc m

fmm

mm

. (2.9)

Figure 2.1: Schematic of the matrix-fracture model.

x

hm

Lc

Fracture

Matrix




18

There is another alternative to derive the shape factor for compressible fluids by integrating

of the diffusivity equation over the matrix-block volume. This method was used by

Zimmerman et al., (1993) to derive the shape factor for slightly compressible fluids and

leads to Equation (2.1). By integrating of the gas diffusivity equation over half of the

matrix block volume we reach to the following equation,

Adxxx

k

Vtc mm

b

mmm )(

2/

1

. (2.10)

Simplification of Equation (2.10) leads to the following equation:

x

k

V

A

tc mm

b

mmm

2/. (2.11)

Using the Warren and Root (1963) approximation we have,

lx

mfm

. (2.12)

Where l is the characteristics length, which is the distance from the matrix-fracture

boundary where the matrix pressure is equal to its average pressure. By substituting this

equation in Equation (2.11) we reach to the following equation:

)()2/(

mfm

b

mmm

k

Vl

A

tc

. (2.13)

Using the definition of the shape factor given by Equation (2.5) we reach to the following

equation for the shape factor of compressible fluids, which is similar to the equation that

was obtained by Zimmerman et al., (1993) for slightly compressible fluids. It should be

noted that in this equation pseudo-pressure is appeared in the final equation as we are

dealing with compressible fluids.

)(mf

m

m

mm tk

c

. (2.14)

This equation is the same as Equation (2.9). It should be pointed out that for a

compressible fluid the viscosity-isothermal compressibility product is a strong function of

pressure, in Equations (2.9) or (2.14) the solution of the nonlinear gas diffusivity equation

is utilized to determine the shape factor for different pressure regimes in the fracture.




19

2.2.1 Constant fracture pressure

In this case it is assumed that at the matrix-fracture interface, the fracture pressure and

hence the pseudo-pressure is a constant. For this case the dimensionless variables are

defined as follows:

if

imD

, (2.15)

c

DL

xx , (2.16)

mD , (2.17)

2c

DL

tt . (2.18)

In Equation (2.17) the average hydraulic diffusivity, is given by (Ranjbar and

Hassanzadeh, 2011):

f

i

f

i

p

p mifm

m

p

p mm

m

if c

dp

pp

kdp

c

k

pp

11. (2.19)

Using the definition of dimensionless variables (Equations (2.15), (2.17) and (2.18)) in the

shape factor equation (Equation (2.9)), the subsequent equation for the dimensionless

shape factor in the case of the constant fracture pressure is obtained:

D

D

DD

mt

h

1

142 . (2.20)

2.2.2 Linearly declining fracture pressure

For a linearly declining fracture pressure we have the following equation for the fracture

pseudo-pressure as given by:

ttif

1),1( (2.21)

where is a decline constant. For this case the dimensionless pseudo-pressure and the

dimensionless fracture pseudo-pressure are defined as follows:




20

i

miD

, (2.22)

DDDfD ttt

1

,)( (2.23)

where is the dimensionless decline constant and is defined as follows:

Dt

t , (2.24)

Applying the explanation of the dimensionless variables, Equations (2.17), (2.18), (2.22)

and (2.23), in the shape factor equation (Equation (2.9)) leads to the following equation for

the shape factor in the case of the linearly declining fracture pressure:

DD

D

D

D

mt

th

42 . (2.25)

2.2.3 Exponentially declining fracture pressure

For this case the fracture pressure declines exponentially with time according to the

following equation:

)(exp)( tif , (2.26)

where )( tf . For an exponential decline, the dimensionless pseudo-pressure

and the fracture dimensionless pseudo-pressure are defined as follows:

i

imDD t

)( , (2.27)

)(exp1)( DDfD tt , (2.28)

where is the dimensionless decline constant and is defined in Equation (2.24). Using the

definition of dimensionless variables (Equations (2.17), (2.18), (2.27) and (2.28)) in theshape factor equation (Equation (2.9)) leads to the following equation for the shape factor

in the case of the exponentially declining fracture pressure:

))(exp1(

42

DD

D

D

D

mt

th

. (2.29)




21

2.3 The approximate analytical solutions

The compressible fluid diffusivity equation for linear flow can be stated as:

tk

c

xm

m

mmm

2

2

. (2.30)

Strong pressure dependence of the viscosity and isothermal compressibility leads to a

nonlinear partial differential equation (PDE) for compressible fluid flow in fractured

porous media. Solution of this PDE cannot be obtained by common methods like Laplace

transform or separation of variables. Equation (2.30) in term of the matrix hydraulic

diffusivity,m= km/cmmis expressed as follows:

2

2

)(x

pt

mm

m

. (2.31)

In Equation (2.31), hydraulic diffusivity is a space and time dependent parameter. To solve

Equation (2.31), we neglect the isothermal compressibility-viscosity product variation with

space and effect of the space is considered by a correction factor, (Ranjbar and

Hassanzadeh, 2011; Agarwal, 1979). Fine-grid numerical simulations are used to

determine this correction factor. Since the gas compressibility is orders of magnitude larger

than the rock compressibility we ignore the rock compressibility in the solution (Agarwal,

1979; Ikoku, 1992). Therefore, we reach the following PDE with the initial and boundaryconditions.

))((x

txt

mm

m

, (2.32)

imt 0 , (2.33a)

00

xx m , (2.33b)

fmcLx . (2.33c)

In Equations (2.32) and (2.33), is used to correct the effect of space on the hydraulic

diffusivity; Lcis characteristic length of the matrix-block which is half of the matrix-block

thickness (hm). Figure 2.1 illustrates a graphical representation of the matrix-fracture

system.




22

2.3.1 Constant fracture pressure

A solution for the constant fracture pressure is given in our recent work (Ranjbar and

Hassanzadeh, 2011). Since this solution will be used as a basis for the time-dependent

boundary condition, the final form of the solution is given in the following. Using an

integral method (Finlayson, 1972; Goodman, 1964; Pooladi-Darvish, 1994; Zimmerman

and Bodvarsson, 1989) the early time solution for constant fracture pressure can be found

as follow (Ranjbar and Hassanzadeh, 2011):

1

3

111

31

24

1,)

24

11(

2424

)124(

D

D

DD

D

DDDD

DDD

D tt

x

tt

tx

. (2.34)

1

1

24

1,4

24

D

D

DD

D tt

. (2.35)

where D is the average pseudo-pressure and D1is hydraulic diffusivity of the fracture in

dimensionless formand is expressed as follows:

mffmDmD

ckx /)1(1

. (2.36)

The late time solution can be obtained by the method of moments (Chang, 1995; Ames,

1965; Crank, 1975) as given by (Ranjbar and Hassanzadeh, 2011):

,))exp(686.5)exp(541.0(

24

1))exp(175.6)exp(793.1(

))exp(489.0)exp(252.11(),(

321

1

221

21

DDD

D

DDDD

DDDDD

xtt

txtt

tttx

(2.37)

121 24

1),exp(148.0)exp(790.01

D

DDDD ttt

. (2.38)

where

11 486.2 D , (2.39)12 181.32 D , (2.40)

Substituting early and late time average dimensionless pseudo-pressures (Equations (2.35)

and (2.38)) and their derivatives in Equation (2.20) leads to the following equation for the




23

dimensionless shape factor for flow of a compressible fluid from a matrix block subject to

a constant fracture pressure boundary condition:

11

12

24

1,

1)244

1(

64

D

D

DDDD

D

m ttth

(2.41)

121

2112

24

1,

)exp(148.0)exp(790.0

)exp(763.4)exp(964.14

D

D

DD

DD

D

Dm t

tt

tth

. (2.42)

where parameters and Dwere obtained by matching the early and late time cumulative

production from the matrix to the fracture by a numerical flow simulator (Geo-Quest,

2009).

2.3.2 Linearly declining fracture pressure

For the linearly declining fracture pressure the diffusivity equation and its initial and

boundary conditions are expressed as follows:

))((D

DD

DD

D

xt

xt

, (2.43)

00 DDt , (2.44a)

,00

D

DD

xx (2.44b)

DDfDDD ttx )(1 . (2.44c)

For the early time solution of these equations we assume that it has the following form:

3))(1

11(),(

D

DDDDD

t

xttx

. (2.45)

When the boundary condition changes with time the penetration depth in the heat balance

integral method (HBIM) is found by solving the following ordinary differential equation

(Mitchel and Myers, 2010):

)()(

)1()(

1)()(

2D

DfDDDDfD

D t

tn

n

t

n

tt

dt

d

. (2.46)

In this equation n is the exponent in the trial solution, n=3 for our case and 1 . In

Equation (2.46), can be found from the following equation (Mitchel and Myers, 2010):




24

12

)1(2

n

nnt

fD

D

fD

.(2.47)

Solving Equation (2.46) for a linearly declining fracture pressure leads to the following

penetration depth for this case:

DD t181 . (2.48)

It should be noted that Equation (2.48) is obtained by assuming =0 in Equation (2.46)

(Mitchel and Myers, 2010). If we do not use the assumption of =0Equation (2.46) cannot

be solved analytically. The derivation of this equation is shown in Appendix 2.A1 in more

details. Our numerical results show that we can obtain a more accurate solution if we use

the following equation for the penetration depth:

DD t191 . (2.49)

The early time solution is valid till the penetration depth reaches the inner boundary, so we

can find the time at which the pressure disturbance reaches the boundary ( t*) as follows:

1

**1

9

1910

D

D tt

. (2.50)

Therefore, the early time solution of the partial differential equation (2.43) with the

boundary conditions (2.44) can be expressed as follows:

1

3

19

1,)

9

11(),(

D

D

DD

DDDDD t

t

xttx

. (2.51)

Integrating of Equation (2.51) over the matrix block volume, leads to Equation (2.52) for

the early time average dimensionless pseudo-pressure:

1

2/31

9

1,

4

9

D

DD

D

D ttk

. (2.52)

The time dependence of the boundary condition for the late time solution can beconsidered using Duhamels theorem. When the fracture pseudo-pressure varies with time

(Equation (2.44c)), Duhamels theorem provides the basis to solve the problem with

variable boundary conditions based on the solution provided for the constant fracture

pseudo-pressure. Using Duhamels theorem (Chang, 1995; Ozisik, 1993; Polyanin, 2001)




25

the solution of partial differential Equation (2.43) with the boundary conditions, Equations

(2.44b) and (2.44c) can be expressed as:

dtxt

Dt

DDDfD

D

D

0

),()( . (2.53)

In Equation (2.53), D within the integral is the solution when 1fD and D on the left-

hand side is the solution of PDE (2.43) when the matrix-fracture boundary condition

changes with time.

Using Duhamels theorem leads to the following late time solution for the case of the

linearly declining fracture pressure:

1

232

2

1

32

1

9

1),1))(exp(086.1086.0(

47256.5

)1))(exp(314.3314.2(

55790.0

),(

D

DDDD

DDDDDDD

ttxx

txxttx

(2.54)

Derivation of Equation (2.54) is shown in Appendix 2.A1 in more details. The late time

average matrix block pseudo-pressure for the linearly declining fracture pressure is

obtained as follows:

12

21

1

1

0

9

1),1)(exp(

14229.0)1)(exp(

81416.0

),(

D

DDDD

DDDDD

tttt

dxtx

(2.55)

Using the average pseudo-pressure and its derivative in the shape factor equation (Equation

(2.25)) results in the following equations for the early and late time shape factors in the

case of the linearly declining fracture pressure:

11

12

9

1,

1

)14

9(

9

2

3

D

D

D

D

D

D

D

m tt

t

h

(2.56)

121

2112

9

1,

)1)(exp(00442.0)1)(exp(32750.0

)exp(14229.0)exp(81416.014

D

D

DD

DD

D

Dm t

tt

tth

(2.57)

It should be noted that for the linearly declining fracture pressure, the shape factor for

compressible fluid is not a function of the dimensionless decline constant, ; a similar




26

observation was reported by Hassanzadeh and Pooladi-Darvish (2006) for flow of a

slightly compressible fluid in fractured porous media.

2.3.3 Exponentially declining fracture pressure

For the exponentially declining fracture pressure the solution of the following PDE should

be used in Equation (2.29) to derive the shape factor for this case:

))((D

DD

DD

D

xt

xt

, (2.58)

00 DDt , (2.59a)

,00

D

DD

xx (2.59b)

)exp(1)(1 DDfDDD ttx (2.59c)The following function is assumed for the early time solution which satisfies the outer

boundary condition:

3))(1

11))(exp(1(),(

D

DDDDD

t

xttx

. (2.60)

Solving the ODE of Equation (2.46) leads to the following equation for the penetration

depth in the case of an exponentially declining fracture pressure:

))()exp(1)exp(1

2(121 1D

D

DD

DDt

terftt

t

. (2.61)

The derivation of this equation is illustrated in Appendix 2.A2. It should be noted that in

Equation (2.61), erf(x)is the error function defined as follows:

Dt

y

D dyeterf

0

22)( . (2.62)

Since Equation (2.61) is obtained based on an approximation of =0 in Equation (2.46),

our numerical results show that one can obtain a more accurate solution if we use thefollowing equation for penetration depth:

))(

)exp(1)exp(1

2(96.121 1

D

D

DD

DDt

terf

ttt

. (2.63)




27

It should be noted that the effect of pressure disturbance will reach the inner boundary

when 0 and for the exponential decline we cannot obtain an explicit equation for t*

and t* is determined for any values of k by making Equation (2.63) equal to zero.

Therefore, the early time solution of Equations (2.58) and (2.59) can be expressed as

follows:

*3

1

,))(

)exp(1)exp(1

2(96.12

11(

))exp(1(),(

tt

t

terf

ttt

x

ttx

D

D

D

DD

DD

D

DDDD

(2.64)

Integrating of Equation (2.64) over the matrix block volume, leads to Equation (2.65) forthe early time average dimensionless pseudo-pressure:

*1 ,)

)(2(96.12

4

)exp(1tt

t

terft

tD

D

D

DD

D

D

(2.65)

Duhamels theorem and the verified solution of the constant fracture pressure boundary

condition lead to the following equation for the late time dimensionless pseudo-pressure in

the case of the exponentially declining fracture pressure:

),exp(

)086.1086.0)()exp()exp(

)exp(1(48743.1

)314.3314.2)()exp()exp(

)exp(1(48743.0

)086.1086.0)(exp())exp()exp(

)exp(1(48743.1

)314.3314.2)(exp())exp()exp(

)exp(1(48743.0

)exp(1),(

32**

2

*

32**

1

*

*322**

2

*

321**

1

*

D

DD

DD

DDDD

DDD

DDDD

t

xxtt

t

xxtt

t

ttxxttt

t

xxttt

t

ttx

(2.66)

More details about the derivation of Equation (2.66) are discussed in Appendix 2.A2.

Integrating over the matrix-block bulk volume results in the following equations for the

average dimensionless pseudo-pressure in the case of the exponentially declining fracture

pressure:




28

),exp())exp()exp(

)exp(1(03867.0)

)exp()exp(

)exp(1(71132.0

)exp())exp()exp(

)exp(1(03867.0

)exp())exp()exp(

)exp(1(71132.0)exp(1),(

**2

*

**1

*

*2

**2

*

1**1

*

D

DD

DDDDD

ttt

t

tt

t

ttttt

t

ttt

tttx

(2.67)

Using the average pseudo-pressure and its derivative in Equation (2.29) leads to the

following equations for the early and late time shape factors in the case of the

exponentially declining fracture pressure:

*

1

2/3

11

2 ,

))exp(1())(

2(96.124

)exp(1

))(

2(2

))(

2())(2

)()exp((

))exp(1(96.12))(2(96.12)exp(12

)exp(

1

tt

tt

terft

t

t

terft

t

terf

t

terf

t

tt

tt

terftt

t

h

D

D

D

D

DD

D

D

D

D

D

D

D

D

D

DD

DD

D

DDD

D

D

D

m

*

2**2

*

1**1

*

**2

*

**1

*

2**2

*2

1**1

*1

**2

*

**1

*

2 ,

)exp())exp()exp(

)exp(1(03867.0)exp()

)exp()exp(

)exp(1(71132.0

)exp())exp()exp(

)exp(1(03867.0)

)exp()exp(

)exp(1(71132.0

))exp())exp()exp(

))exp(1((03867.0)exp()

)exp()exp(

))exp(1((71132.0

))exp()exp(

)exp(1(03867.0)

)exp()exp(

)exp(1(71132.01)exp(

4

tt

ttt

tt

tt

t

ttt

t

tt

t

ttt

tt

tt

t

tt

t

tt

tt

h

D

DD

D

DD

D

D

m




29

Based on Equations (2.68) and (2.69), for an exponentially declining fracture pressure the

shape factor for a compressible fluid is a function of the decline exponent, k. Similar

observations have been made in the previous studies for a slightly compressible fluid

(Chang, 1995; Hassanzadeh and Pooladi-Darvish, 2006).

2.4 Model verification

The developed shape factor was validated by a fine-grid single porosity model (Eclipse

100). The total cumulative production from the matrix to the fracture based on the

simulator was used to find the correction factor () and D and to validate the presented

model. Figure 2.2 shows the matrix-fracture cumulative fluid production versus time for a

case with =0.7, T=93.3C and pressure drawdown of 45 to 22.5 MPa. The values obtained

for the correction factor (), the matching parameter (D), the average hydraulic diffusivity

( ) and the dimensionless fracture hydraulic diffusivity (D1) are 0.730, 0.3127, 0.03457

and 0.3691, respectively. In the model verification studies, a slab-shaped matrix-block with

thickness (hm) of 4m, permeability of 1mD, and porosity of 0.1are considered. We use the

same reservoir data and parameters throughout this paper. As illustrated in Figure 2.2 the

approximate analytical model based on this study is in a good agreement with the fine grid

numerical simulation. More details about the numerical simulations and more validation

cases are discussed elsewhere (Ranjbar and Hassanzadeh, 2011).

The developed model with 11 DD must reproduce the shape factor for a slightly

compressible fluid. For additional validation of the developed model, the shape factor

derived in this study is evaluated with the shape factor of the slightly compressible fluid.

Outcomes show that the models developed for different boundary conditions can reproduce

the slightly compressible fluid shape factor for the complete period of time.

Figure 2.3 compares the developed shape factor model for a slightly compressible fluid (

11 DD ) and models available in the literature (Chang, 1995; Hassanzadeh and

Pooladi-Darvish, 2006) when the fracture pressure declines linearly with time. According

to this figure the presented model shows an acceptable match with other models.




30

Time (sec)

0 500 1000 1500 2000

CumulativeFluidProduction(Sm

3)

0

100

200

300

400

500

600

700

Numerical

Compressible Fluid Model

Figure 2.2: Comparison of the matrix-fracture cumulative fluid production obtained from the

approximate analytical solution and the numerical model of Eclipse for constant fracture pressure.

Dimensionless Time

0.001 0.01 0.1 1 10 100

DimensionlessShapefactor(h

m2)

1

10

100

1000


Hassanzadeh & Pooladi-Darvish Model

Chang Model

Figure 2.3: Comparison of the developed shape factor model with literature models for slightly

compressible fluid in the case of linearly declining fracture pressure.




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Figures 2.4, 2.5 and 2.6 demonstrate the comparisons between the presented shape factor

models in this study with the literature models for a slightly compressible fluid when the

fracture pressure declines exponentially with time for different values of the decline

exponent. These figures demonstrate that the presented model can reproduce the slightly

compressible fluid shape factor with an acceptable accuracy.

Dimensionless Time

0.001 0.01 0.1 1 10 100

DimensionlessShapeFac

tor(h

m2)

1

10

100

1000

Compressible Fluid ModelHassanzadeh & Pooladi-Darvish Model


compressible fluid in the case of exponentially declining fracture pressure for small values of exponent

(k=0.0001).




32

Dimensionless Time

0.001 0.01 0.1 1 10 100

DimensionlessShapeFactor(

hm2

)

1

10

100

1000


Chang Model


compressible fluid in the case of exponentially declining fracture pressure (k=0.632).

Dimensionless Time

0.001 0.01 0.1 1 10 100

DimensionlessShapeFactor(h

m2)

1

10

100

1000


Hassanzadeh & Pooladi-Darvish Model


compressible fluid in the case of exponentially declining fracture pressure (k=1).




33

2.5 Results

In this section a comparison of the developed model with the Warren and Root model

(1963) is presented and then the behaviour of the shape factor for different fracture

pressure depletion regimes for flow of a compressible fluid through dual-porosity media i

Modeling of Matrix-Fracture Interaction for Conventional Fractured Gas -...

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