PERFORMANCE EVALUATION OF WELLS WITH...

PERFORMANCE EVALUATION OF WELLS WITH

SINGLE AND MULTIPLE FRACTURES IN TIGHT

FORMATIONS

A Thesis

Submitted to the Faculty of Graduate Studies and Research

In Partial Fulfillment of the Requirements

For the Degree of

Master of Applied Science

In

Petroleum Systems Engineering

University of Regina

By

Feng Zhang

Regina, Saskatchewan

August, 2013

Copyright 2013: F. Zhang

UNIVERSITY OF REGINA

FACULTY OF GRADUATE STUDIES AND RESEARCH

SUPERVISORY AND EXAMINING COMMITTEE

Feng Zhang, candidate for the degree of Master of Applied Science in Petroleum Systems Engineering, has presented a thesis titled, Performance Evaluation of Wells With Single and Multiple Fractures in Tight Formations, in an oral examination held on August 9, 2013. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: Dr. Adisorn Aroonwilas, Industrial Systems Engineering

Supervisor: Dr. Daoyong Yang, Petroleum Systems Engineering

Committee Member: Dr. Farshid Torabi, Petroleum Systems Engineering

Committee Member: Dr. Yongan Gu, Petroleum Systems Engineeing

Chair of Defense: Dr. Kelvin Ng, Environmental Systems Engineering

i

ABSTRACT

Hydraulic fracturing technology together with horizontal wells leads to the

exploration and development of unconventional resources economically feasible. Due to

complex fracture network and geology nature, evaluating performance of fluids flow in

such a hydraulic fracture system is really a challenge. Numerical modeling and

simulation usually requires very fine grid to reduce the truncation error and enhance the

simulation accuracy. Sometimes even with fine grid, the associated computation accuracy

is compromised. On the other hand, numerous studies have been conducted on pressure

transient analysis with analytical and/or numerical methods, though some underlying

mechanisms have not been well understood since fluid flow in such tight formations can

be completely dependent on the fracture network while the matrix only plays a source

role. Therefore, it is of fundamental and practical importance to evaluate performance of

a well with single and multiple fractures in a tight formation.

Theoretically, a semi-analytical method is developed to couple the fluid flow in the

matrix and the fracture so as to improve the accuracy and efficiency of pressure transient

analysis in a tight formation. As for the semi-analytical method, the fracture is firstly

discretized into several small segments. Equations for the matrix and fracture system can

be solved in the Laplace domain with source functions, respectively. Continuity equations

can be obtained along the fracture interface, leading to a linear system. Solving the linear

system, the flux in each segment can be obtained, which is substituted into the first

segment to determine the bottomhole pressure in the Laplace domain. Finally, the

Stephest inverse algorithm can be employed to convert the solutions in the Laplace

domain to those in the real time domain. The widely-accepted point source function is

ii

used to describe the fluid flow in the matrix for a vertical well with a single fracture,

while a novel slab source function in the Laplace domain is developed to achieve the

same purpose for a horizontal well with multiple fractures. Compared to the point source

function, slab source function is more general, which attributes geometry in each

direction and considers the pressure effect inside the source by superposition principle.

The non-Darcy flow effect inside the fracture based on the Barree-Conway model is

solved by introducing the pseudo-time and iterative methods.

The semi-analytical method using source functions has been validated with

numerical simulation method and field case, respectively. Subsequently, two

dimensionless parameters, i.e., relative minimum permeability (kmr) and non-Darcy

number (FND), have been introduced to incorporate the non-Darcy effect on flow

performance for a vertical well with single fracture. Compared with the Forchheimer’s

equation, the Barree-Conway model causes a smaller pressure drop at a higher non-Darcy

number. The fracture conductivity would be underestimated when non-Darcy flow exists

inside the fracture. For a horizontal well with multiple fractures, fracture conductivity,

fracture stages, fracture number, and fracture dimension are discussed under

consideration of both non-Darcy effect and partially penetrating ratio. Compared to the

vertical well with single fracture, non-Darcy effect in a horizontal well with multiple

fractures can cause a larger pressure drop as non-Darcy number is increased and relative

minimum permeability is decreased. The partially penetrating ratio greatly affects the

early well response, which leads to an early radial flow regime when the ratio is set to be

a small value. Finally, the newly proposed method has been successfully extended to

determine the reservoir and fracture parameters in a real field case.

iii

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my academic supervisor, Dr.

Daoyong (Tony) Yang, for providing me the precious opportunity together with his

continuous encouragement and support throughout the course of my graduate studies.

I also wish to thank the following individuals or organizations for their support and

friendship during my MASc studies at the University of Regina:

My past and present research group members: Dr. Huazhou Li, Mr. Sixu Zheng,

Mr. Yin Zhang, Ms. Xiaoli Li, Mr. Deyue Zhou, Mr. Chengyao Song, Ms. Min

Yang , and Ms. Ping Yang;

Mr. John Styles for providing the transient pressure data and valuable technical

discussions;

Natural Sciences and Engineering Research Council (NSERC) of Canada for a

Discovery Grant to Dr. Yang;

Petroleum Technology Research Centre (PTRC) for the innovation fund to Dr.

Yang;

Faculty of Graduate Studies and Research (FGSR) at the University of Regina for

awarding the FGSR Graduate Scholarship (2012 Winter and 2012 Fall); and

Many wonderful friends and families who extended their care, support and

friendship to me during my stay in Regina.

iv

DEDICATION

This thesis is dedicated to my dearest parents, Mrs. Xiuwen Ji and Mr. Youjin

Zhang, for their continuous support and love.

v

TABLE OF CONTENTS

ABSTRACT… .................................................................................................................... i

ACKNOWLEDGEMENTS ............................................................................................ iii

DEDICATION .................................................................................................................. iv

TABLE OF CONTENTS ................................................................................................. v

LIST OF TABLES ......................................................................................................... viii

LIST OF FIGURES ......................................................................................................... ix

NOMENCLATURE ........................................................................................................ xii

CHAPTER 1 INTRODUCTION .................................................................................... 1

1.1 Unconventional Resources ................................................................................. 1

1.2 Pressure Transient Analysis ............................................................................... 2

1.3 Importance of Non-Darcy Flow ......................................................................... 3

1.4 Purpose of This Thesis Study ............................................................................. 5

1.5 Outline of the Thesis .......................................................................................... 5

CHAPTER 2 LITERATURE REVIEW ........................................................................ 7

2.1 Source and Green Function ................................................................................ 8

2.1.1 Point source function .................................................................................. 8

2.1.2 Volumetric source function ........................................................................ 9

2.1.3 Slab source function ................................................................................. 10

2.2 Pressure Transient Analysis ............................................................................. 11

2.2.1 Vertical well.............................................................................................. 12

2.2.2 Horizontal well ......................................................................................... 15

2.3 Non-Darcy Flow ............................................................................................... 17

vi

2.4 Summary .......................................................................................................... 22

CHAPTER 3 DETERMINATION OF FRACTURE CONDUCTIVITY IN TIGHT

FORMATIONS WITH NON-DARCY FLOW BEHAVIOUR .......... 24

3.1 Theoretical Model ............................................................................................ 26

3.1.1 Mathematical formulation ........................................................................ 26

3.1.2 Semi-analytical solution ........................................................................... 29

3.1.3 Continuity conditions ............................................................................... 33

3.2 Model Validation .............................................................................................. 36

3.2.1 Forchheimer equation ............................................................................... 36

3.2.2 Numerical simulation................................................................................ 41

3.3 Sensitivity Analysis .......................................................................................... 41

3.3.1 Relative minimum permeability (kmr) ....................................................... 44

3.3.2 Non-Darcy number (FND) ......................................................................... 46

3.3.3 Comparison ............................................................................................... 46

3.4 Effect of Non-Darcy Flow ................................................................................ 50

3.4.1 Tight oil formations .................................................................................. 53

3.4.2 Tight gas formations ................................................................................. 62

3.5 Case Study ........................................................................................................ 67

3.6 Summary .......................................................................................................... 67

CHAPTER 4 SLAB SOURCE FUNCTION FOR EVALUATING

PERFORMANCE OF A HORIZONTAL WELL WITH MULTIPLE

FRACUTRES .......................................................................................... 70

4.1 Mathematical Formulation ............................................................................... 72

vii

4.1.1 Product principle ....................................................................................... 73

4.1.2 Slab source function ................................................................................. 74

4.2 Linear System ................................................................................................... 79

4.2.1 Constraint equations ................................................................................. 80

4.2.2 Flow equations .......................................................................................... 81

4.2.3 Continuity equations ................................................................................. 82

4.2.4 Wellbore equations ................................................................................... 83

4.3 Model Validation .............................................................................................. 83

4.4 Results and Discussion ..................................................................................... 87

4.4.1 Fracture stages .......................................................................................... 87

4.4.2 Fracture conductivity ................................................................................ 91

4.4.3 Fracture length .......................................................................................... 93

4.4.4 Fracture spacing ........................................................................................ 96

4.4.5 Penetrating ratio ........................................................................................ 96

4.4.6 Non-Darcy effect ...................................................................................... 98

4.5 Case Study ...................................................................................................... 106

4.6 Summary ........................................................................................................ 113

CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS ............................. 114

5.1 Conclusions .................................................................................................... 114

5.2 Recommendations .......................................................................................... 116

REFERENCES .............................................................................................................. 118

viii

LIST OF TABLES

Table 3.1 Basic parameters used for simulating pressure-transient response in a tight gas

formation (Vincent et al., 1999; Settari et al., 2000; Gill et al., 2003) ............ 39

Table 3.2 Basic parameters used for simulating pressure transient response .................. 42

Table 3.3 Basic parameters used for simulating pressure-transient response in a tight oil

formation .......................................................................................................... 51


formation .......................................................................................................... 52

Table 4.1 Basic instantaneous source functions in an infinite slab reservoir (Chen et al.,

1991) ................................................................................................................. 75

Table 4.2 Values of h(sn)/g'(sn) and (sn, n=0,1,...) for instantaneous plane sources in an

infinite slab reservoir (Chen et al., 1991) ......................................................... 76

Table 4.3 Basic parameters used for simulating pressure transient response .................. 84

Table 4.4 Parameters used for describing non-Darcy flow with Barree-Conway model in

multiple fracutures (Lai et al., 2012) .............................................................. 101

Table 4.5 Reservoir and fracture properties for the well in the Bakken formation ........ 107

Table 4.6 Type curve matching results for the buildup test ........................................... 111

ix

LIST OF FIGURES

Figure 3.1 Schematic diagram of a vertical hydraulic fracture in a closed cylindrical

reservoir. ...................................................................................................... 25

Figure 3.2 Schematic of discretization of one fracture in the global fracture subsystem. 31

Figure 3.3 Flowchart of solving the non-linear system. .................................................. 35

Figure 3.4 The coefficient matrix for solving the matrix-hydraulic fracture system in the

single fractured vertical well. ...................................................................... 37

Figure 3.5 Model validation for the Forchheimer equation ............................................. 40

Figure 3.6 (a) Top view of the 3D grid system and (b) Model validation between this

study and CMG simulation results. ............................................................. 43

Figure 3.7 Effect of non-Darcy number FND (a) at FND = 100, and (b) at FND = 5. ......... 45

Figure 3.8 Effect of non-Darcy number FND (a) at kmr = 0.01, and (b) at kmr = 0.10. ...... 47

Figure 3.9 Comparison on dimensionless pressure calculated from the Forchheimer

equation and the Barree-Conway model together with the relative errors. . 48

Figure 3.10 Type curves for non-Darcy effect in a tight oil formation at CfD = 100.0. ... 54

Figure 3.11 Flux distribution from the matrix to the fracture in a tight oil formation at

CfD = 100 and FND =22.001. ......................................................................... 56

Figure 3.12 Flux distribution inside the fracture in a tight oil formation at CfD = 100 and

FND =22.001. ................................................................................................ 57

Figure 3.13 Type curves for non-Darcy effect in a tight oil formation at CfD = 20.0. ..... 59


CfD = 20.0 and FND =22.001. ........................................................................ 60

Figure 3.15 Bilinear flow analysis (a) at CfD = 20.0, and (b) at CfD = 100.0. .................. 61

x

Figure 3.16 Type curves for non-Darcy effect in a tight gas formation at CfD = 100.0. .. 63

Figure 3.17 Flux distribution from the matrix to the fracture in a tight gas formation at

CfD = 100.0 and FND =121.328. .................................................................... 64

Figure 3.18 Type curves for non-Darcy effect in a tight gas formation at CfD = 20.0. .... 65


CfD = 20.0 and FND =121.328. ...................................................................... 66

Figure 3.20 Type curve match for the field data on an oil well in a tight formation ....... 68

Figure 4.1 Schematic diagram of a horizontal well with multi-stage fractures in a box-

shaped reservoir ........................................................................................... 71

Figure 4.2 Schematic of a slab source model ................................................................... 77

Figure 4.3 The coefficient matrix for solving the matrix-hydraulic fracture system ....... 85

Figure 4.4 (a) Top view of the 3D grid system and (b) Cross-section of the 3D grid

system .......................................................................................................... 86

Figure 4.5 Model validation between this study and CMG simulation results ................ 88

Figure 4.6 Comparison between this study and CMG simulation results ........................ 89

Figure 4.7 Effect of fracture stages on the well response with the same fracture length. 90

Figure 4.8 Pressure response together with its derivative for five pairs of fractures under

various fracture conductivities ..................................................................... 92

Figure 4.9 Effect of fracture conductivity on the flow distribution at (a) tD = 1.0E-6, and

(b) tD = 1.0E2 ............................................................................................... 94

Figure 4.10 Effect of fracture length on the well response at four pairs of fractures ...... 95

Figure 4.11 Effect of fracture spacing on the well response at four pairs of fractures .... 97

xi

Figure 4.12 Effect of partially penetrating ratio of three stages at CfD=10 and (a) α>0.5,

and (b) α≤0.5 .............................................................................................. 99

Figure 4.13 Effect of non-Darcy number FND (a) at kmr = 0.01, and (b) at kmr = 0.10 ... 102

Figure 4.14 Effect of non-Darcy number kmr at (a) FND = 1000, and (b) FND = 10 ....... 103

Figure 4.15 Effect of non-Darcy flow on the flow distribution at (a) tD = 1.0E-6, and (b)

tD = 1.0E3 ................................................................................................... 105

Figure 4.16 Pressure buildup test and production history .............................................. 108

Figure 4.17 Flow regime interpretation for the pressure buildup response ................... 109

Figure 4.18 Type curve matching of pressure buildup data ........................................... 112

xii

NOMENCLATURE

Notations

Bg = formation volume factor, bbl/STB or bbl/Mscf

b = coefficient defined in Eq. [3.23a]

CD = dimensionless wellbore storage

Cf = fracture conductivity, mD·ft

Cη = fracture diffusivity, fraction

c = coefficient defined in Eq. [3.23b]

ct = total compressibility, psi-1

d = coefficient defined in Eq. [3.23c]

E = parameter defined in Eq. [2.4]

f1, f2 = two function defined in Eq. [4.2]

Fn, Fm = parameter defined in Eqs. [4.9d] and [4.9e]

,

,

m n

i jF = parameter defined in Eq. [A-11]

FND = non-Darcy number defined in Eq. [3.9c]

h = payzone thickness, ft

hf = fracture height, ft

k = permeability, mD

kapp = apparent permeability, mD

xiii

kd = constant Darcy permeability, mD

kmin = minimum permeability at high rate, mD

kmr = relative minimum permeability, fraction

kr = relative permeability, fraction

L = length, ft

Lr = reference length, ft

NRe = Reynolds number, dimensionless

p = pressure, psi

qf = flux from reservoir to fracture along fracture, STB/(d·ft) or Mscf/(d·ft)

qN = flux at the node, STB/d or Mscf/d

Qsc = surface flow rate, STB/d or Mscf/d

Qmax = maximum flow rate desired, STB/d or Mscf/d

R = residue of function f1 in Laplace domain at pole sn

s = Laplace transform variable

sn = pole of function f1 in Laplace domain

S = parameter defined in Eq. [A-2]

Sf = parameter defined in Eq. [A-1]

Sg = gas saturation, fraction

Sp, Sp(m), Sp(n), Sp(nm) = parameter defined from Eqs. [4.9a] to [4.9d]

xiv

t = time, hr

V = parameter defined in Eq. [3.25]

v = velocity, ft/s

wf =fracture width, ft

x = x coordinate, ft

xf = fracture half length, or half-length of a slab source, ft

xw = well location, ft

y = y coordinate, ft

yf = half-width of a slab source, ft

z = z coordinate, ft

zf = half-height of a slab source, ft

Greek letters

p L = potential gradient, psi/ft

α = penetrating ratio, fraction

αn = parameter defined in Eq. [4.9g]

β = non-Darcy flow coefficient, ft-1

βm = parameter defined in Eq. [4.9h]

η = hydraulic diffusivity

xv

μ = fluid viscosity, cP

ρ = fluid density, lbm/ft3

τ = characteristic length, ft

τD = dimensionless pseudo-time defined in Eq. [3.15]

δ = parameter defined in Eq. [3.4]

= porosity, fraction

Subscript

D = dimensionless

e = external boundary

f = fracture

i = initial condition, or index, ith

node

m = matrix

max = maximum

min = minimum

N = node

w = wellbore

wf = wellbore flowing

xvi

SI Metric Conversion Factors

bbl × 1.589874 E-01 = m3

cp × 1.0* E-03 = Pa·s

ft × 3.048* E-01 = m

mD × 9.869233 E-04 = µm2

lbm × 4.535924 E-01 = kg

psi × 6.894757 E+00 = kPa

*Conversion factor is exact

1

CHAPTER 1 INTRODUCTION

1.1 Unconventional Resources

Unconventional resources (e.g., shale gas, tight gas, tight oil, oilsands, and coalbed

methane) have recently become a significant increasing source of hydrocarbon

production. Physically, unconventional reservoir systems are difficult to produce and

therefore require special completion, stimulation, and production techniques to achieve

economic success. There have encountered numerous technical challenges due to the

inherent low permeability together with complex geological and petrophysical

characteristics.

Canada has extensive exploration and production of unconventional resources.

With the horizontal drilling and multi-stage hydraulic fracturing techniques for low

permeability reservoirs, tight oil production has recently seen a significant increase in the

Bakken formation which extends from North Dakota and Montana in the United States to

Saskatchewan and Manitoba in Canada (Bahrami and Siavoshi, 2013). The typical

Bakken well in Canada produces roughly 210 bbl/d in the first month (NEB, 2012). Till

June 2011, production from the Bakken play reached nearly 400,000 bbl/d in the USA

and 78,000 bbl/d in Saskatchewan and Manitoba of Canada, respectively (USGS, 2012).

Besides the Bakken Formation, tight oil plays highlighted include the Cardium

Formation, Viking Formation, Lower Shaunavon Formation, Montney Formation,

Duvernay Formation, Beaverhill Lake Group, and Lower Amaranth Formation (NEB,

2012). The development of tight oil resources in the Western Canadian Sedimentary

Basin is still in its infancy. In the recent years, the Cardium play in the west-central

2

Alberta has attracted a great deal of attention with an estimated reserves of 305 million

barrels of light oil, while recent drilling activity has shown that the economics, though

attractive, fall short of those in the Bakken play (AGS, 2009).

Much of the past tight gas activity in the Western Canadian Sedimentary Basin has

focused on the plays in the northeast British Columbia (BC), among which the Montney

play is a major discovery (Wood, 2012). Since horizontal drilling with multi-stage

fracturing technology has made some gas wells with production of 5 to 6 MMscf/d, the

Horn River basin continues to move toward the commercial development stage. With

consideration of the uncertainty, the original gas in place is estimated to be 372 to 529

Tscf (NEB, 2012). In eastern Canada, tight gas resources are also discovered in the

Appalachian Basin, Sydney Basin, and Maritimes Basin (USGS, 2012).

1.2 Pressure Transient Analysis

Due to complex fracture networks and geology natures, modeling and simulation of

flow in such a hydraulic fracture system is really a challenge. As for numerical simulation,

it is usually required to reduce the truncation error and enhance the simulation accuracy

by using very fine grid. This often causes large grid numbers and costs huge computer

memory and storage space. Therefore, analytical and semi-analytical methods by the

source approach offer a flexible tool in well performance evaluation (Lin and Zhu, 2012).

In a tight reservoir system, fluid flow behavior near a single fracture can be

described as linear, elliptical, and pseudo-radial flow regimes (Thompson, 1981). An

elliptical solution for the case of an infinite-acting reservoir was proposed for a vertical

well with finite conductivity fracture, though multiple fracture system makes the

3

simulation more complex. An analytical solution was initially proposed by van Kruysdijk

and Dullaert (1989), which introduces the concept “compound linear flow”. It should be

noted that at early production linear flow is dominant, then the flow turns to

perpendicular to the fracture surface until pressure transients of the individual fractures

begin to interfere, leading to a compound linear flow regime (Ilk, 2010). Other analytical

solutions are provided to model the pressure transient behaviour of horizontal wells (Guo

and Evans, 1994; Larsen and Hegre, 1994; Horne and Temeng, 1995). A semi-analytical

method has been developed and applied to model the entire range of flow regimes around

a horizontal well with multiple fractures (Medeiros et al., 2006). In this case, a dual

permeability region surrounding the fractures is assumed to represent complex fractured

region.

1.3 Importance of Non-Darcy Flow

Darcy’s law is considered as the fundamental principle in fluid flow in porous

media. However, the limitations of Darcy’s law constrained to a relatively small velocity

region have long been recognized (Barree and Conway, 2004). Any deviation from the

linear Darcy flow can be considered as non-Darcy flow. Forchheimer (1901) found that

Darcy’s law was inadequate for describing high-velocity gas flow in porous media and

added a new term to account for the additional pressure drop. A constant parameter (i.e.,

β) is proposed to describe the increasing contribution to pressure drop caused by inertial

losses.

Since the β factor was proposed, different mechanisms have been discussed to

explain the non-Darcy flow in the Forchheimer’s model. Since Non-Darcy flow is

4

attributed to turbulence, β is also called the turbulence factor. Most researchers still agree

that Forchheimer non-Darcy flow is not resulted from turbulence other than inertial

effects because non-Darcy flow is found to occur even at a low Reynolds number (Barak,

1987; Ruth and Ma, 1992; Whitaker, 1996).

Many correlations from the experimental data have been proposed to calculate the

inertial factor β (Geertsma, 1974; Frederick and Graves, 1994; Li and Engler, 2001). All

the correlations suggest that inertial factor is highly correlated with permeability: the

lower the permeability is, the larger the inertial factor will be. Although experimental

studies provide a simple and straightforward correlation to obtain the inertial factor value,

field evidences indicate that the β value at the field scale is significantly greater than the

corresponding experimental measurements (Frederick, 1980). It was suggested that the β

value estimated from field observations are much more reliable than those from

experimental measurements (Narayanaswamy et al., 1999).

Even in the Forchheimer model, some data sets could not be described by the

quadratic flow equation (Firoozabadi and Katz, 1979). The Forchheimer plot can become

concave downwards, leading to the observation of a higher apparent permeability than

that predicted by the Forchheimer’s equation at a high velocity (Jones, 1972). A new

model is proposed to replace the inertial factor in the Forchheimer’s model with two new

parameters, i.e., minimum permeability plateau and characteristic length (Barree and

Conway, 2004). According to the experimental measurements, Lai et al. (2012) claimed

that Barree-Conway model is capable of describing non-Darcy flow in a wider velocity

range than the traditional Forchheimer’s model.

5

1.4 Purpose of This Thesis Study

The purpose of this thesis study is to evaluate performance of a vertical well with

single fracture and a horizontal well with multiple fractures in a tight formation. The

primary objectives of this study include:

1) To determine fracture conductivity of a vertical well with single fracture in a

tight formation under non-Darcy effect with the point source function;

2) To develop a novel slab source function in the Laplace domain to accurately

describe the fluid flow in the porous media;

3) To evaluate the non-Darcy flow effects with the Barree-Conway model, and

compare with traditional Forchheimer’s model;

4) To apply the slab source function in the Laplace domain in a horizontal well

with multiple fractures, and then examine effects of fracture conductivity,

fracture dimension, fracture spacing, fracture number, partially penetrating ratio

and non-Darcy flow; and

5) To extend the newly developed techniques for field applications under non-

Darcy effect and subsequently determine the fracture conductivity and non-

Darcy number.

1.5 Outline of the Thesis

This thesis is composed of five chapters. More specifically, Chapter 1 is an

introduction to the thesis topic together with its major research objectives and scope.

Chapter 2 provides an updated literature review on the pressure transient analysis of

6

fractured wells (i.e., single fracture and multiple fractures). It also includes the non-Darcy

flow behaviour in the fracture. Chapter 3 determines the fracture conductivity of a

vertical well with single fracture under non-Darcy effect. Chapter 4 proposes a novel slab

source function and extends its application to describe pressure transient behaviour in a

horizontal well with multiple fractures. Chapter 5 summarizes the major scientific

findings of this study and provides some recommendations for future research.

7

CHAPTER 2 LITERATURE REVIEW

It has long been recognized that unlocking unconventional resources (e.g., tight oil

and shale gas) is uneconomically viable due to the inherent complexity and low

permeability. The recent advances in hydraulic fracturing technology together with

horizontal wells, however, lead to the exploitation and development of unconventional

resources economically feasible. Such a complex system including matrix, horizontal

wells, multiple hydraulic fractures, and natural fractures presents a great challenge to

accurately model and predict its performance, though numerical simulation is a powerful

tool to evaluate the performance of fractured wells in a complex reservoir. In addition, it

is time-consuming for numerical simulation with inherent uncertainties to achieve such a

purpose under conditions of the required ultra-fine grid.

In practice, the source and Green function is widely applied to solve the single-

phase fluid flow in the porous media when the fluid movement is induced from a

complex fractured well system. Especially, point source function over either a line or a

surface has been mostly used for its easy computation. However, limitations have been

noted in the point source function due to the fact that the source dimension and pressure

inside the source have been neglected.

In this chapter, the source and Green functions applied to evaluate performance of

hydraulic fracture systems are firstly reviewed, followed by pressure transient analysis

with single or multiple fractures. Also, non-Darcy effects in the hydraulic fractures are

presented and discussed.

8

2.1 Source and Green Function

Numerous analytical models have been made available to describe the fluid flow in

a tight formation where partial of the respective contribution from matrix, natural and

hydraulic fractures, and horizontal wellbores has been considered. Gringarten and Ramey

(1973) firstly introduced the Green source function to the problem of unsteady-state fluid

flow in such a reservoir. Since a continuous source solution can be obtained by

integrating the response to an instantaneous source solution, the Green function was

introduced under different boundary conditions for a point, line, plane, and slab source,

respectively. The Newman product method under certain initial and boundary conditions

allows the solution of a multi-dimensional problem to be constructed from one-

dimensional solutions (Newman, 1936).

2.1.1 Point source function

Gringarten et al. (1974) applied the point source solution to the unsteady-state

pressure distribution induced by a well with vertical fractures of infinite conductivity.

The fracture is divided into N segments, each of which is assumed to be a uniform flux

source. By using the Green function in the Laplace domain, Cinco-Ley and Samaniego

(1981) developed a vertical fracture model of finite conductivity in an infinite reservoir

so that variable rates, wellbore storage, skin effect, and conductivity can be easily

quantified. A bilinear flow model is then developed to perform pressure transient analysis

for vertical finite conductivity fractures.

9

Babu and Odeh (1988) predicted horizontal well performance in a closed reservoir

under pseudosteady-state condition by integrating the point source function to the line

source function. Ozkan and Raghavan (1991) developed a point source solution in the

Laplace domain together with an extensive library of solutions under various wellbore

configurations, noting that Gringarten and Ramey’s solution becomes limited if one

desires to transform such a solution to the Laplace domain. Due to the fact that the

product solution technique is only applicable in the real time domain, Chen et al. (1991)

developed an approach to derive solutions in the Laplace domain under various wellbore

and boundary conditions by extending the product principle to the Laplace domain.

2.1.2 Volumetric source function

Amini and Valko (2007) distributed the volume source function to evaluate

performance of horizontal wells with multi-stage fractures in a box-shaped reservoir,

making it possible to describe the pressure behaviour inside the source. A source term

was added to the diffusivity equation to calculate the pressure distribution, while the

production rate from a fracture can be calculated. The volumetric source function

provides reliable results in modeling permeability anisotropy, partial penetration of the

source, and additional pressure losses stemming from the particular geometry of the flow

path inside the sources. Zhu et al. (2007) applied the volumetric source model to some

field cases. The volumetric source method was then successfully adjusted to take into

account the non-Darcy flow effect in a horizontal well with multiple fractures by Amini

and Valko (2010), showing that this method is efficient to predict the production

performance and pressure response.

10

2.1.3 Slab source function

After introducing the procedure to develop slab source function with modified

product principle, Chen et al. (1991) applied it to the partially penetrating fracture by

assuming the fracture as a plane source, which is intercepted by two slab sources and one

plane source. Although the fracture is still assumed to be a plane, the slab source function

has been incorporated in a different way such that the point source function is integrated

along a plane.

Ogunsanya et al. (2005) presented a slab source function or the so-called solid bar

source, which was developed in an infinite-acting reservoir with impermeable upper and

lower boundaries. With the assigned dimension in each direction, the bar source function

can be used to analyze pressure transient behaviour of both vertical and horizontal

fracture systems.

Lin and Zhu (2012) developed a slab source method in a closed boundary box-

shaped reservoir to evaluate performance of horizontal wells with and without fractures,

assigning the source to have a three-dimensional (3D) geometry with consideration of the

effect of pressure behaviour inside the sources by using the superposition principle. The

slab source function is derived based on the instantaneous source functions (Gringarten

and Ramey, 1973) with the Newman’s product method. The slab source function is

claimed to be a general source function which is applicable in vertical wells with

fractures, and horizontal wells with and without fractures.

11

2.2 Pressure Transient Analysis

As for an unconventional reservoir, pressure transient analysis for a hydraulically

fractured well plays an important role in not only evaluating well performance, but also

determining its complex reservoir characteristics, including permeability, initial pressure,

wellbore storage, skin factor, fracture dimension, and conductivity. Different flow

regimes that may be present in a well with single or multiple hydraulic fractures are

summarized as follows.

As for a well with a single hydraulic fracture, we have (Brown et al., 2009; Ozkan

et al., 2009):

1) Bilinear flow regime is observed in a finite conductivity fracture with slope of

1/4 on the pressure derivative;

2) Linear flow regime follows the bilinear flow towards the hydraulic fracture

wings surrounding the hydraulic fractures with slope of 1/2 on the pressure

derivative;

3) Elliptical flow regime develops towards the drainage area around the fractures

with slope of 1/3 approximately on the pressure derivative; and

4) Pseudo-radial flow regime appears at a later time when pressure propagates deep

enough into the reservoir with slope of 0 on the pressure derivative.

Comparing to the single fracture case, reservoir flow regimes in a horizontal well

with multiple fractures are much more complicated (Bahrami and Siavoshi, 2013):

1) Radial-linear flow occurs inside the hydraulic fractures;

2) Bilinear/linear flow regime corresponds to the same flow regimes in the single

fracture case with slope of 1/4 and 1/2 on the pressure derivative;

12

3) Early radial flow regime is developed towards each fracture when the spacing

between each fracture and its half length is suitable;

4) Early-boundary-dominated flow regime appears when the stimulated reservoir

volume (SRV) around each hydraulic fracture is depleted. Drainage area around

each hydraulic fracture interferes with those of other fractures. This process is

similar to the boundary effect with slope in a range of 0.5 and 1.0; and

5) Elliptical flow/pseudo radial flow regime is similar to the corresponding flow

regimes in the single fracture case.

Compound linear flow regime shows that the flow pattern is predominantly normal

to the vertical plane that contains the horizontal well (van Kruysdijk and Dullaert, 1989).

It should be noted that one or more flow regimes may not be present, depending on the

magnitudes of the reservoir and fractures. As for a finite reservoir, a boundary dominated

flow regime will appear where closed boundaries are involved.

2.2.1 Vertical well

Numerous efforts have been made to analyze production and pressure data so as to

obtain well/reservoir parameters for a hydraulically fractured vertical well in a low

permeability reservoir (Thompson, 1981; Amini et al., 2007; Cipolla et al., 2009).

Gringarten et al. (1974) developed an analytical solution to the problem of transient fluid

flow in the vertical wells with fractures. Infinite conductivity and uniform flux vertical

fracture cases were presented based on the Source and Green’s functions proposed by

Gringarten and Ramey (1973).

13

Studies on the effect of conductivity on the pressure response of a fractured well

were initiated by Cinco-Ley et al. (1976). A semi-analytical solution was developed to

tackle the problem of transient flow of fluids towards vertical fractures with finite

conductivity. As such, this method separates the reservoir system into two subsystems

(i.e., the fracture and matrix subsystems) which can be solved analytically by using the

Source and Green’s functions. The fracture and matrix solutions are then coupled

together with continuity conditions between the two media to yield a solution for the flux

distribution along the fracture. Type curves of well pressure versus time as dimensionless

terms were generated for analyzing well test data under different dimensionless

conductivities.

After analyzing pressure data of fractured wells, Cinco-Ley and Samaniego (1978)

proposed a new analytical solution for the transient behaviour of wells under low and

intermediate conductivities at the initial production. This solution includes a bilinear flow

regime in which only the transient effects of the region of the matrix surrounding the

fracture can affect the well response. Then, a new correlation for the finite conductivity

fractures was presented by correlating the early flow solutions into a single envelope

curve.

Raghavan et al. (1978) examined the effect of a partially penetrating fracture on the

transient pressure behaviour of a well. An analytical solution was developed by assuming

uniform flux across the fracture. An approximate solution was developed for evaluating

the transient pressure behaviour of a well that intersects a partially penetrating fracture

with infinite conductivity. Rodriguez and Cinco-Ley (1984) studied the unsteady-state

behaviour of a reservoir with a well intersected by a partially penetrating vertical fracture

14

of finite conductivity. A semi-analytical method similar to the Cinco-Ley et al. (1976)

was then developed with consideration of the partially penetrating effect. The flow

behaviour of the partially penetrating fractures during the initial production is equivalent

to that of the totally penetrating fractures. An early infinite-acting flow period was found

to follow the bilinear/linear flow regime.

Cinco-Ley and Meng (1988) introduced a semi-analytical and simplified fully

analytical model to study the transient behaviour of a well intersected by a vertical

fracture in a dual porosity reservoir, respectively. Tiab (1988) proposed a method known

as the “TDS-Tiab’s Direct Synthesis Technique” for pressure transient analysis instead of

the conventional semi-log and log-log type curve matching methods. Tiab (1993) applied

the TDS method to uniform and vertically fractured wells with finite and infinite

conductivity.

Wattenbarger et al. (1998) proposed an effective fracture network length

accounting for a single vertical fracture and obtained the solutions based on the linear

flow concept. The natural fracture network is added to the linear flow solutions to form a

linear dual porosity solution, stating that the linear flow regime should generally be

considered as the major one for shale gas reservoirs (Bello and Wattenbarger, 2010). The

slope of the straight line in the linear flow regime yields the matrix drainage area,

provided that the matrix permeability is known during the transient linear flow regime.

However, the linear flow trend could be subjective and the conductivity may affect the

production behaviour (Ilk, 2010). Amini et al. (2007) provided an elliptical flow solution

for hydraulically fractured vertical wells, while the application of elliptical flow type

curves is presented elsewhere (Ilk et al., 2007).

15

2.2.2 Horizontal well

As for a horizontal well with multiple fractures in a tight formation, it becomes

much more difficult to evaluate its performance with huge uncertainties due to the lack of

fundamental understanding (Bello and Wattenbarger, 2010). van Kruysdijk and Dullaert

(1989) firstly provided an analytical solution to present a compound linear flow regime at

late times. Guo et al. (1994) developed an analytical solution without accounting for the

interference among fractures.

Leif and Hegre (1994) provided a comprehensive investigation of the pressure

transient behaviour of horizontal wells with both single and multiple vertical fractures,

either longitudinal or transverse. In this case, fracture radial, radial-linear, formation

linear and pseudo-radial flow can be observed. After examining the pressure response of

horizontal wells with single and multiple fractures in homogenous systems, Kuchuk and

Habusky (1994) noted that wellbore storage effect cannot be neglected in the low

conductivity fractures.

Horne and Temeng (1995) presented an analytical model for evaluating inflow

performance and transient behaviour of a horizontal well with multiple transverse

fractures. Such pressure transient solutions were firstly developed for a uniform flux

single fracture, and then superposition principle was applied to generate final solutions

for multiple fractures.

Raghavan (1997) developed an analytical solution to discern the response of

horizontal wells with multiple fractures. Based on the analytical model, three significant

flow regimes can be observed: 1) Early flow regime in which the system behaves like the

16

one with n-layers; 2) Intermediate flow regime in which interference between fractures is

reflected; and 3) Late flow regime in which the system behaves like a single fracture.

After carefully examining over 2000 fracture treatments data, Wright et al. (1998)

found that fractures are almost never perfectly vertical. The fractures usually dip 5 to 15

degrees from the vertical and very few fractures dip less than two degrees off the vertical.

Wan and Aziz (1999) developed a general solution for horizontal wells with

multiple fractures. Four flow regimes can be accordingly identified, i.e., the early linear,

transient, late linear, and late radial flow.

Zerzar and Bettam (2003) presented a combined method to couple the boundary

element method and Laplace transformation for the horizontal wells with multiple

fractures. This method includes seven flow regimes, i.e., bilinear, linear, elliptical, radial,

pseudo-radial, second linear, and pseudo-steady flow. Also, Al-Kobaisi and Ozkan (2004)

proposed a hybrid numerical-analytical model for evaluating the pressure transient

response, while effect of the rectangular and elliptical geometry of fractures on pressure

response was examined.

Medeiros et al. (2006) proposed a semi-analytical method to obtain pressure

transient solutions for a heterogeneous system, providing flexibility to model various well

responses in tight reservoirs. Amini and Valko (2007) presented a semi-analytical

solution by using the volume source function to predict the performance of a horizontal

well with multiple transverse fractures in a bounded reservoir.

Ozkan et al. (2009) and Brown et al. (2009) respectively presented a trilinear flow

solution in a dual porosity reservoir to analyze the pressure transient behaviour of

horizontal wells with multiple fractures. Such analytical solutions in the outer reservoir,

17

inner reservoir, and fractures are coupled together on the interfaces between the regions.

Meyer et al. (2010) provided an approximate analytical solution for finite conductivity

multiple fractures including fracture interference.

Al-Rbeawi and Tiab (2012) examined effects of the partially penetrating ratio and

inclined angle for horizontal wells with multiple fractures. An early radial flow regime

can be observed to represent the radial flow around each fracture for scenarios with a

small penetrating ratio. The TDS method was subsequently applied to estimate the

formation and fracture parameters.

2.3 Non-Darcy Flow

Darcy’s law (1856) has laid the foundation for describing the fluid flow in porous

media. As such, the pressure gradient can be related to the fluid viscosity μ and

superficial velocity v in a porous media with constant permeability k, i.e.,

p v

L k

[2.1]

Forchheimer (1901) observed the deviation from Darcy’s law at high flow rates. In

addition to introducing a new parameter called inertial factor, a quadratic flow term was

added to the Darcy’s linear form to account for the deviation from the linearity of the

Darcy’s law. This equation is now commonly referred to as the Forchheimer’s equation,

2+p v

vL k

[2.2]

where β is the inertial flow parameter, and ρ is the density of the fluid.

18

Even in one of the Forchheimer’s publications in 1901, some datasets could not be

described by the quadratic flow equation. Jones (1972) observed that there exists a large

deviation from the linear Forchheimer plot for core samples with a large value of the

inertial factor. The Forchheimer plot becomes concave downward, while the apparent

permeability is calculated to be higher than those predicted by the Forchheimer’s

equation at a high velocity. In spite of its limitations, numerous efforts have been made to

evaluate non-Darcy flow behaviour based on the Forchheimer equation (Andrade et al.,

1997; Hill et al., 2001; Stanley and Andrade, 2001).

Based on the extensive experimental results and field data, Barree and Conway

(2004) proposed a new model by replacing the inertial factor in the Forchheimer’s model

with two new parameters, i.e., minimum permeability plateau and characteristic length. In

this case, the absolute permeability in the Darcy’s law is replaced by the apparent

permeability,

app

p v

L k

[2.3]

where kapp is the apparent permeability which is defined as follows,

min

min

Re1

dapp E

k kk k

N

[2.4]

Due to the introduction of minimum permeability plateau (kmin), apparent

permeability becomes constant at both low (i.e., kd) and high Reynolds number (NRe),

which implies the plateau behaviour. The plateau behaviour at high Reynolds number is

partially validated by Lai (2010). The exponential coefficient E describes the overall

heterogeneity of the test samples. The better the sorting is, the closer the value

19

approaches to 1.0. For single-sieve proppant, the value of E is equal to unity (Barree and

Conway, 2004). In equation [2.4], NRe can be written as follows,

ReN

[2.5]

where τ is the characteristic length. Therefore, coupling Equations [2.3]-[2.5], the Barree-

Conway model can be written as follows,

minmin

1

d

p v

k kxk

v

[2.6]

According to the experimental measurements, Lai et al. (2012) claimed that Barree-

Conway model is capable of describing non-Darcy flow in a much wider velocity range

than that of the traditional Forchheimer’s model.

In a tight formation, long horizontal wells that have been drilled and massively

fractured make the production economically viable (Manrique et al., 2010). In practice,

the designed hydraulic fractures do not usually result in the expected well performance

due potentially to the fracture cleanup, multi-phase flow, non-Darcy flow, and

penetrating ratio. It has been well-accepted that fracture conductivity and half length are

generally underestimated in the presence of non-Darcy flow (Holditch and Morse, 1976;

Guppy et al., 1982a; b; Vincent et al., 1999; Umnuayponwiwat et al., 2000; Gil et al.,

2003). Few attempts have been made to examine the effects of non-Darcy flow and

penetrating ratio on the performance of horizontal wells with multiple fractures.

Therefore, it is of fundamental and practical importance to accurately quantify non-Darcy

20

effect and penetrating ratios with consideration of the appropriate fracture dimension and

its conductivity.

Either the Forchherimer’s equation or the Barree-Conway model can be used to

examine the effects on the transient pressure behaviour, leading to a partial differential

equation with strong non-linearity, which is difficult to be solved analytically. In general,

steady-state non-Darcy flow is assumed to occur in the fracture, whereas unsteady-state

Darcy flow is assumed in the matrix (Guppy et al., 1982a; b; Umnuayponwiwat et al.,

2000).

After analyzing the transient behaviour of various conductivities with a finite-

difference simulator, Holditch and Morse (1976) concluded that non-Darcy flow in the

fracture reduced the apparent conductivity. Guppy et al. (1982a) proposed a

comprehensive semi-analytical model of non-Darcy flow in wells with finite-conductivity

fractures, assuming that there exists steady-state non-Darcy flow in the fracture and

unsteady-state Darcy flow in the matrix. This facilitates displaying changes in the flux

distribution along the fracture, while non-Darcy flow in a fracture leads to the apparent

fracture conductivity significantly less than its true value. Subsequently,

Umnuayponwiwat et al. (2000) used a similar model to investigate the flux distribution

along the fracture and interaction between the fracture and matrix and found that an

increase in the fracture flow would accelerate the non-Darcy flow effects.

Due to its difference from Darcy’s condition, buildup tests could not be analyzed as

superposition of the drawdown tests for the nonlinearity of a non-Darcy problem (Guppy

et al., 1982b). Although numerical simulation is the primary choice by discretizing both

space and time (Holditch and Morse, 1976; Belhaj et al., 2003; Lai et al., 2012), the

21

semi-analytical method (Guppy et al., 1982a; b; Zeng and Zhao, 2010) has also seen its

extensive applications where only fracture is divided into number of segments and the

solution in each segment can be obtained analytically.

By examining non-Darcy effects in the proppant packs with laboratory tests,

Martins et al. (1990) discovered that pressure loss behaviour occurs at a high flow rate.

Vincent et al. (1999) pointed out that increasing fracture width or placing high

permeability proppant could increase conductivity. Smith et al. (2004) found that non-

Darcy flow should not be simply considered as the reduction of the true conductivity,

while Guppy’s correlations would overestimate the non-Darcy flow effects.

As for non-Darcy effects at a low flow rate, Miskimins et al. (2005) concluded that

non-Darcy effects would reduce the flow capacity by 5-30% in a tight gas reservoir with

relatively low permeability (0.01 mD). By incorporating the unsteady-state solutions both

in the fracture and matrix, Zeng and Zhao (2010) pointed out that such a solution can be

especially applied to the tight formations. The Barree-Conway model has been employed

to describe the pressure transient behaviour of non-Darcy flow in hydraulically fractured

wells with the finite difference method (Al-Otaibi and Wu, 2011).

Few attempts have been made to describe non-Darcy flow in a horizontal well with

multiple fractures. Amini and Valko (2007) incorporated the non-Darcy flow effect

obtained from the volume source function. With consideration of the three-dimensional

fracture geometry, Lin and Zhu (2012) developed a slab source method to evaluate

performance of horizontal wells with or without fractures. The non-Darcy flow is solved

with an iteration method by comparing the difference between apparent permeability and

absolute permeability. However, such a solution is based on the real time domain, making

22

it difficult to incorporate viable flow rates and conductivities usually obtained from the

Laplace domain solution. Also, Wu et al. (2013) adopted numerical simulation to address

non-Darcy effect in an unconventional gas reservoir.

2.4 Summary

Over the past decades, due to its easy computation, the point source function

integrated over a line or a surface has been mostly used to perform the pressure transient

analysis associated with single phase and slightly compressible fluids. Both the

volumetric source function and slab source function have been advanced to include

effects of the source dimension and pressure behaviour inside the sources, though they

are developed in the real time domain. In particular, with consideration of the physical

fracture geometry, the slab source function in the Laplace domain has not been made

available.

Comparing to the real time domain solution, the solution in terms of Laplace

variables can be more easily used to solve the practical problems, particularly those

involving variable rates, skin effect, wellbore storage, and variable conductivities.

Furthermore, it is possible to extend the time range because convolution in the time

domain is reduced to multiplication in the Laplace domain.

Although non-Darcy flow inside a single vertical fracture has been extensively

studied, few attempts have been made to evaluate the performance of a horizontal well

with multiple fractures under non-Darcy effect with the recently proposed Barree-

Conway model. It is of practical and fundamental importance to not only develop a novel

23

slab source function in the Laplace domain, but also evaluate performance of a horizontal

well with multiple fractures in a tight formation under non-Darcy flow effect.

24

CHAPTER 3 DETERMINATION OF FRACTURE

CONDUCTIVITY IN TIGHT

FORMATIONS WITH NON-DARCY FLOW

BEHAVIOUR

Numerous efforts have been made to describe the pressure transient behaviour in

the vertical well with single fracture (Cinco-Ley et al., 1976; Raghavan et al., 1978;

Wattenbarger et al., 1998). Non-Darcy effects inside the fracture have been previously

analyzed based on traditional Forchheimer’s equation (Guppy et al., 1982a; b;

Umnuayponwiwat et al., 2000). In this chapter, non-Darcy effect is incorporated using

the semi-analytical method with newly proposed Barree-Conway model (2004). In this

study, non-Darcy effect is determined by introducing two dimensionless parameters (i.e.,

kmr and FND) based on the Barree-Conway model (Zhang and Yang, 2012a; b; 2013a; b).

Figure 3.1 shows a vertical fracture that is fully penetrated along the well in a

closed box-shaped reservoir. The assumptions are made as follows:

1) The reservoir is assumed to be homogeneous and bounded by upper and lower

impermeable layers. The fluid is considered as slightly compressible, and flow

in the formation is considered as Darcy’s flow, while the non-Darcy flow in the

fracture can be described with the Barree-Conway model (2004).

2) The matrix porosity, m , compressibility, ctm, permeability, km, thickness, h,

reservoir width xe, and length ye are set to be constant. The half length and

width of the vertical fracture is denoted by xf and wf, respectively. The fracture

porosity, f , compressibility, ctf, and permeability, kf are also assumed to be

constant.

25

Figure 3.1 Schematic diagram of a vertical hydraulic fracture in a closed cylindrical

reservoir

h

, ,f t f f

c k

, ,m t m m

c k

fx

z

y

x

h

26

3.1 Theoretical Model

3.1.1 Mathematical formulation

According to the Barree-Conway model (2004), the flow equation in the fracture is

expressed is expressed as follows by dividing kd in the denominator of Equation [2.6],

1

1

mrf mr

p v

x

kk k

v

[3.1]

where is fluid viscosity, v is superficial velocity, is fluid density, mrk is minimum

permeability relative to Darcy permeability, and is the characteristic length.

According to the Darcy’s flow model, the flow equation in the matrix can be

expressed as follows,

m

p v

x k

[3.2]

The fracture is considered as a plane source with non-uniform flux distribution

from the matrix to the fracture, i.e., ,fq x t in 3 /m s m . Based on Equation [3.1], the

equation for fluid flow in the fracture can be written as follows,

f f f f

f tf

f

k p q pc

x x w h t

[3.3]

where

1

1

mrmr

N

f

kk

q

w h

[3.4]

where qN is the flux inside the fracture. The boundary conditions can be written as

follows,

27

02

f f f sc g

x

k hw p Q B

x

[3.5a]

0

f

f

x x

p

x

[3.5b]

As for the pressure drop in the formation, it can be expressed by,

2 2

2 2

mm tm

k p p pc

x y t

[3.6]

with boundary conditions,

0

,mf f f

y

k h pq x x x

y

[3.7a]

, , ip x y t p [3.7b]

The initial condition for both fracture and formation is,

, , 0 ip x y t p [3.7c]

For the convenience of analysis, the following definitions are made to convert the

mathematical models into dimensionless forms,

m i

D

k h p pp

Q

,

m i f

fD

k h p pp

Q

[3.8]

2

mD

m tm f

k tt

c x [3.9a]

D

f

xx

x , D

f

yy

x [3.9b]

sc g

ND

f

Q BF

w h

[3.9c]

28

f f

fD

sc g

q xq

Q B , N

ND

sc g

qq

Q B [3.9d]

f f

fD

m f

k wC

k x [3.9e]

=f m tm

f tf m

k cC

c k

[3.9f]

Equations [3-5b] can then be rewritten as follows,

1fD fD fD

D D fD D

p q p

x x C C t

[3.10]

1

1

mrmr

ND ND

kk

F q

[3.11a]

0

1 1

2D

fD

D fDx

p

x C

[3.11b]

1

0

D

fD

D x

p

x

[3.11c]

Similarly, Equations [6-7b] can be rewritten as follows,

2 2

2 2

D D D

D D D

p p p

x y t

[3.12]

0

, 1 1DfD D

D y

pq x

y

[3.13a]

, , 0D D D Dp x y t [3.13b]

The initial condition can be expressed with dimensionless term,

=0, =0, 0D D D Dp x y t [3.13c]

29

3.1.2 Semi-analytical solution

The semi-analytical method applied to solve the hydraulic fracture system has been

widely discussed in the literature (Guppy et al. 1982a; Umnuayponwiwat et al. 2000;

Zeng and Zhao, 2010). In general, the whole system is divided into matrix system and

fracture system, each of which is solved separately. The fracture system is firstly

discretized into several segments in each of which the flux, qfD, can be approximated to

its average value within the segment. With such an approximation, equations for the

matrix and fracture system can be solved in the Laplace domain in real time, respectively.

Then their solutions are coupled along the fracture interface, leading to a linear system.

Solving the linear system, the flux in each segment can be obtained. Such obtained flux

can be substituted into the first segment to determine the bottomhole pressure in the

Laplace domain in pseudo-time. Finally, the Stehfest inverse algorithm (Stehfest, 1970)

can be employed to convert the bottomhole pressure in the Laplace domain in real time.

Compared to the traditional methods and numerical simulation, the newly developed

technique based on the Laplace domain is not only easily to incorporate the skin factor,

flow rate, wellbore storage, and conductivity in a unified and consistent framework, but

also efficient and accurate in evaluating the petrophysical parameters together with

reservoir performance for a complex reservoir.

Matrix subsystem: The solution to the matrix subsystem can be directly obtained

by applying the point source solution in the Laplace domain to a closed cylindrical

reservoir (Ozkan and Raghavan, 1991), yielding,

21 2

0-1= , -D fD D Dp q x u K u x x y dx

[3.14]

30

where K0 is a zero modified Bessel function of the second type, while it has a singularity

at xD=x. Van Kruysdijk (1988) proposed a method for integration with the singularity.

The modified Bessel function K0(x) can be divided into two parts, -ln(x) and K0(x)+ ln(x).

The singularity in the first part can be integrated analytically.

Fracture subsystem: Because ,D Dx t is a function of both time and space, the

mathematical model for the fracture subsystem is a strong non-linear equation (Equation

[3.15]). To solve such a non-linear equation, the fracture subsystem is firstly discretized

into several segments within each of which the mathematical model can be solved

analytically (see Figure 3.2). Zeng and Zhao (2010) provided a detailed derivation to the

solution in the fracture subsystem.

An average i Dt is employed to simplify the ,D Dx t distribution in each

segment. A pseudo-time is defined based on the average i Dt , i.e.,

0

Dt

Di i d [3.15]

The flux, ,fD D Dq x t , can be approximated to its average value in each fracture

segment i (xDi-1 ≤ xD ≤ xDi),

,fD D D fDi Dq x t q t [3.16]

The mathematical model in segment i (xDi-1 ≤ xD ≤ xDi) is expressed as follows,

2

2

1fD fD

fDi D

D i fD Di

p pq t

x C C

[3.17]

with boundary conditions at xDi-1 and xDi，

31

Figure 3.2 Schematic of discretization of one fracture in the global fracture subsystem

q

Ni x

y

qfDi

qND1

qfD1

qND2

qfD2

…….

…….

qNDi

…….

…….

z

32

11 DiDi

fD ND

D fD xx

p q

x C

,

DiDi

fD ND

D fD xx

p q

x C

[3.18]

The solution to the aforementioned equations can be obtained with the source

function, which can be formulated as follows,

* * *

1 1, , , , ,fDi D i NDi s D Di i NDi s D Di i fDi

fD i

Cp x s q p x x s q p x x s q

C u

[3.19]

where,

1

* * *1= , = , =

D i D i

fDiND NDNDi NDi fDi

ix x

qq qq q q

[3.20]

For no-flow boundary segments, 0 ≤ x'D ≤ ΔxD, with a source at x'D0, van Kruysdijk

(1988) provided the solution:

0

0 0

0

0

2cosh1, , exp

2 exp 2

2coshexp

exp 2

D D i

s D D i D D i

i D i

D D i

D D i

D i

x x s Cp x x s x x s C

u C x s C

x x s Cx x s C

x s C

[3.21]

When the source is located at x'D0 = xDi-1 and x'D0 = xDi, the final solution can be

obtained with superposition principle as follows,

* * *1,

ifD D i i D NDi i D NDi i fDip x s b x q c x q d q [3.22]

where

33

1

1

1 2

2

2cosh1

1

D D i i

D i D i i

D Di i x x s C

i Dx x s C

fD i

x x s Cb x e

C s C e

[3.23a]

1

2

2

2cosh1

1

D i D i

D i D i i

Di D i x x s C

i Dx x s C

fD i

x x s Cc x e

C s C e

[3.23b]

ifD i

Cd

C s

[3.23c]

3.1.3 Continuity conditions

There are two continuity conditions at the interface between any two adjacent

segments in the fracture and boundary between the matrix and the fracture: pressure

continuity and flow rate continuity. Since the solution of the fracture system is based on

the pseudo-time, pressure or flux of any adjacent grids cannot be directly equalized at the

interface in this study. Therefore, a new continuity condition based on the Stehfest

algorithm for the inverse Laplace transformation is proposed as follows. As such, the

solution can be obtained in the Laplace domain, incorporating variable flow rates, skin

effect, wellbore storage, and variable conductivities more easily.

Since the pressure in the real-time domain should be same along the boundary

between the fracture and matrix, we have,

( ) ( )D Di f D Di mp t p t [3.24a]

34

The solution for the fracture system is based on the pseudo-time Laplace

transformation, while the solution for the matrix system is based on the real-time Laplace

transformation. Then Equation [3.24a] can be rewritten as follows,

( ) ( )D i f D Di mp p t [3.24b]

Using the Stehfest algorithm for the inverse Laplace transformation, we have,

1 1

ln 2 ln 2 ln 2 ln 2N N

j jj j

i i D D mf

V p j V p jt t

[3.25]

Accordingly, the pressure continuity condition for the fracture and matrix system

can be obtained as follows,

1 ln 2 1 ln 2=

i i D D mf

p j p jt t

[3.26a]

Similarly, the pressure and flux continuity condition for the node between two

adjacent segments are respectively expressed as follows,

+1 +1

1 ln 2 1 ln 2=

i i i if f

p j p j

[3.26b]

+1 +1

1 ln 2 1 ln 2=

i i i if f

q j q j

[3.26c]

An iterative approach is used here to assume Darcy flow initially occurring in the

fracture, while the non-linear system is transformed to a linear system. The flux at each

node can be re-substituted into the linear system for a new δ. When the flux converges at

each node, the iteration process (see Figure 3.3) terminates to obtain a stable flux

distribution. The linear system can then be solved to obtain the flux from the matrix to the

35

Figure 3.3 Flowchart of solving the non-linear system

qD, PD

Solution of each fracture segment in

the fracture subsystem

Solution of each segment in the

formation subsystem

=0

DAq B

- <new

1

1

mrmr

ND D

kk

F q

,D D newq q

+t t t

YES

Assume non-Darcy Parameter

PPparameter

36

fracture and the flow rate in the fracture at each node. Figure 3.4 presents the format of

coefficient matrix for vertical well with single fracture.

The flux from the matrix to the fracture and the flow rate inside the fracture at each

node are obtained by solving this linear equation system. Substituting the flux at Segment

#1 and flow rate at Node #1 into the original equation for Segment #1 yields the

bottomhole pressure in the Laplace domain in pseudo-time. Subsequently the bottomhole

pressure in the real-time domain can be obtained by using the inverse Laplace algorithm

as proposed by Stehfest (1970).

The solution can also be used for a real gas flow, provided that pseudo-pressure

transformation has been made (Lee and Holditch, 1982; Spivey, 1984). The solution in

terms of pseudo-pressure is defined by,

0

p pm p dp

u p Z p

[3.27]

Since all computations are based on the Laplace transform, wellbore storage can be

easily added with the following well-known identity,

21

wDwD

D wD

pp

u C p

[3.28]

3.2 Model Validation

3.2.1 Forchheimer equation

The newly developed model can be simplified to the Forchheimer equation since it

is the most widely used flow model for describing the non-Darcy flow behavior (Al-

Otaibi and Wu, 2011).

37

Figure 3.4 The coefficient matrix for solving the matrix-hydraulic fracture system in the

single fractured vertical well

38

The common factor between the Barree and Conway model and the Forchheimer

equation is the apparent permeability of non-Darcy flow, which is defined by Equations

[3.29a] and [3.29b], respectively,

1

1

mr

app mr

kk k k

v

[3.29a]

1

app

kk

k v

[3.29b]

where kapp is the apparent permeability, and β is the inertial flow parameter.

When the apparent permeability of two models is set to be the same, the inertial

flow parameter β is solved as follows,

1 mr

mr

k

k k v

[3.30]

where fluid superficial velocity, v, can be approximated using a constant production rate

and cross-sectional area of flow in a well. Equation [3.30] can be used to correlate the

Forchheimer equation and the Barree-Conway model. According to the parameters listed

in Table 3.1 (Vincent et al., 1999; Settari et al., 2000; Gill et al., 2003), the tight gas

reservoir is assumed to be no boundary effects. Figure 3.5 shows an excellent agreement

in the transient stage between the Forchheimer equation simplified from the newly

developed model and the original Barree-Conway model. There is a very minor mismatch

when entering the bilinear period because the flow rate will change significantly at the

initial stage which will then cause a large difference. When reaching the pseudo-radial

flow period, the flow rate will be stabilized and the difference will disappear.

39


formation (Vincent et al., 1999; Settari et al., 2000; Gill et al., 2003)

Parameter Value Unit

Reservoir and

Fluid Data

Reservoir Pressure, pi 3000 psi

Reservoir Permeability, km 0.1 mD

Reservoir Porosity, 0.1 fraction

Reservoir Temperature, T 190 oF

Payzone Thickness, h 60 ft

Total Compressibility, ct 1.0×10-5

psi-1

Initial Viscosity, µ 0.018 cP

Density, ρ 0.0625 lbm/ft3

Production Rate, q 10 MMscf/d

Hydraulic

Fracture Data

Fracture Half Length, xf 800 ft

Fracture Width, wf 0.02 ft

Fracture Permeability, kf 180000 mD

Characteristic Length, τ 2.89×104 ft

-1

Relative Minimum Permeability, kmr 0.01 fraction

Equivalent β (Forchheimer) 1.96×105 ft

-1

40

Figure 3.5 Model validation for the Forchheimer equation

41

3.2.2 Numerical simulation

A numerical simulation model is also used for validation. The numerical model is

built by using a reservoir simulator (IMEX, 2010, Computer Modeling Group Ltd.). The

three-dimensional (3D) grid system is 20×20×1 with cell dimension of 20×20×20 ft. The

single cell at the fracture location was locally refined by using the keyword

“Hydraulically Fractured Wells”. Number of refined blocks in each direction is 7×7×1.

Non-Darcy effect was modeled by using the keyword “Non-Darcy Option”.

As for the numerical simulation, non-Darcy flow is described by using the

Geertsma correlation (1974) which is commonly accepted for the unconsolidated

materials,

0.5 5.5

48511=

rg gk k S

[3.31]

The model was built as a tight gas reservoir with porosity of 15% and permeability

of 0.1 mD. Figure 3.6a shows a top view of the 3D grid system, while input parameters

are listed in Table 3.2 (Zhang and Yang, 2013b). As can be seen in Figure 3.6b, there

exists a good agreement between this study and the results from CMG simulation.

3.3 Sensitivity Analysis

Unlike traditional models based on the Forchheimer equation, the non-Darcy effect

is determined by using two dimensionless parameters (i.e., kmr and FND) based on the

Barree-Conway model. Accordingly, effects of these two parameters on pressure

response are to be discussed here since effects of wellbore storage (CD), fracture

42

Table 3.2 Basic parameters used for simulating pressure transient response


Reservoir and Fluid Data


Reservoir Porosity, 0.15



psi-1

Viscosity, μ 0.026 cP


Production Rate, q 0.1 Mscf/d

Initial Reservoir Pressure, pi 5000 psi

Hydraulic Fracture Data




Equivalent β (Forchheimer) 2.332×107 1/ft

Relative Minimum Permeability, kmr 0.1

Characteristic Length, τ 32.742 1/ft

43

(a)

(b)

Figure 3.6 (a) Top view of the 3D grid system and (b) Model validation between this

study and CMG simulation results

44

conductivity (CfD), and fracture diffusivity (Cη) have been extensively analyzed and

discussed (Cinco-Ley. et al., 1978; Guppy et al., 1982a; Lee and Holditch, 1982).

3.3.1 Relative minimum permeability (kmr)

In the Barree-Conway model, mrk is firstly introduced which is caused by

streamlining or flow heterogeneity. The drawdown type curves for kmr =0.01, 0.10, and

0.50 at FND =100 are depicted in Figure 3.7a. The typical one-fourth slope at early times

of both pressure and its derivative curves are exhibited to show a bilinear flow region for

non-Darcy effect. Then a linear flow with a slope of 0.5 is followed at the end of the

bilinear flow period. Finally, a pseudo-radial flow is attained as the flux in the fracture

has been stabilized. As the value of kmr decreases, the pressure drop and its derivative

curves increase due to strong non-Darcy effect. It is found that non-Darcy effect becomes

significant when kmr becomes less than 0.50. As kmr approaches to 1.00, however, non-

Darcy effect tends to disappear and the pressure response is found to be very similar to

the Darcy’s case.

Figure 3.7b presents the type curves for kmr =0.01, 0.10, and 0.50 at FND =5. When

FND is set to be a smaller value, the pressure and its derivative curves also show a similar

trend to the case at a higher FND value. However, non-Darcy effect becomes less

significant compared with that at a high FND value. This means that FND imposes a

smaller impact on kmr as it is decreased. When FND is set to be a small value, non-Darcy

effect will not affect the pressure response significantly even at a very small kmr value.

45

(a)

(b)

Figure 3.7 Effect of non-Darcy number FND (a) at FND = 100, and (b) at FND = 5

46

3.3.2 Non-Darcy number (FND)

Figures 3.8a and 3.8b present drawdown type curves for FND =1, 10, and 100 at kmr

=0.01 and 0.10, respectively. As shown in Figure 3.8a, the pressure drop and its

derivative curves increase as the non-Darcy number FND increases. This is attributed to

the fact that non-Darcy effect is increased as FND is increased. Obviously, as relative

minimum permeability kmr is decreased, non-Darcy effect becomes much stronger.

Meanwhile, non-Darcy number FND becomes more sensitive than that at a larger kmr. This

is because, as mrk is increased to one, the Barree-Conway model can be converted into

Darcy’s model so that the flow will approach to Darcy flow condition. As can be seen in

Figures 3.7a and 3.8a, the non-Darcy effect will become more evident at a smaller

relative minimum permeability (kmr<0.5) and a larger non-Darcy number (FND>1).

Comparing Figure 3.7b with Figure 3.8b, it can be found that both relative minimum

permeability kmr and non-Darcy number FND can affect the pressure response. However,

non-Darcy number FND that usually results in a large pressure drop at a large kmr is more

sensitive than the relative minimum permeability kmr. Similarly, Figure 3.8 shows that

pressure drop and its derivative curves increase as the non-Darcy number is increased.

3.3.3 Comparison

The relative error caused by Forchheimer equation with respect to the Barree-

Conway model is presented in Figure 3.9. The rate-dependent parameter is examined

from 5 to 100 as kmr is set to be 0.01. It is found that, when FND or DNDq is set as 100, the

relative error can reach to 30.9%, indicating that the Forchheimer equation can

47

(a)

(b)

Figure 3.8 Effect of non-Darcy number FND (a) at kmr = 0.01, and (b) at kmr = 0.10

48

Figure 3.9 Comparison on dimensionless pressure calculated from the Forchheimer

equation and the Barree-Conway model together with the relative errors

49

overestimate the dimensionless pressure by 30.9% compared to the Barree-Conway

model. As FND is decreased, the relative error is decreased along a straight line. When

FND approaches zero, the flow becomes Darcy’s flow with its relative error approaching

to zero. If a threshold value of the relative error is set to be 0.2, the corresponding FND is

found to be 65.

In the non-Darcy flow based on the Forchheimer`s equation, the non-Darcy effect is

determined by dimensionless flow rate ( DNDq ) defined as follows (Guppy et al., 1982a; b;

Umnuayponwiwat et al., 2000),

sc g

DNDf

Q Bq

w h

[3.37]

where Bg is gas formation volume factor, and Qsc is the production rate at field unit. It

should be noted that both dimensionless flow rate and non-Darcy number defined in this

study are dependent on flow rate. When these two rate-dependent parameters are set as

the same value, the only difference between the Forchheimer equation and Barree-

Conway model is the usage of the relative minimum permeability (kmr), which is a

parameter revealing the plateau behaviour at a high flow rate. The physical meaning can

be explained by using the streamline theory (Barree and Conway, 2004).







50



found to be 65.









found to be 65.

3.4 Effect of Non-Darcy Flow

Because of its low production rate in a tight formation, non-Darcy effect on

transient pressure behaviour has not been considered as of yet. To respectively analyze

and discuss the effect of non-Darcy flow on flow performance in a tight oil and gas

formation, a basic data set (Cinco-Ley et al., 1978; Guppy et al., 1982a;

Umnuayponwiwat et al., 2000) used in this study are tabulated in Tables 3.3 and 3.4,

respectively. It is assumed that fracture diffusivity (Cη) and relative minimum

permeability ( mrk ) are to be 105 and 0.01, respectively.

51

Table 3.3 Basic parameters used for simulating pressure-transient response in a tight oil

formation






Total Compressibility, ct 18×10-6

psi-1


Density, ρ 60 lbm/ft3

Production Rate, q 195 STB/d




Fracture Permeability, kf 10000, 2000 mD

Fracture Conductivity, CfD 100, 20

52


formation





Reservoir Temperature, T 710 oR



psi-1



Production Rate, q 1 MMscf/d




Fracture Permeability, kf 10000, 2000 mD

Fracture Conductivity, CfD 100, 20

53

3.4.1 Tight oil formations

Although the effect of non-Darcy flow on the transient pressure responses of

hydraulically fractured wells has been examined, few attempts have been made in the

tight oil formations at a low production rate. In the tight oil formation example, the

reservoir and fluid data are sourced from Cinco-Ley et al (1978). As for a tight oil

formation with km =5.00 mD, it is assumed that there is a relatively high-conductivity

fracture with CfD =100.0. As shown in Equation [3.9c], non-Darcy number (FND) is a rate-

dependent factor. Therefore, oil production rate is assumed to be constant (195 STB/D)

so that non-Darcy number (FND) is only a function of the characteristic length ( ). Barree

and Conway (2004) suggested that characteristic length ( ) is related to the particle size

or size distribution of the porous media. In this study, the value of characteristic length

( ) is set according to the examples provided by Al-Otabi and Wu (2011). As can be

seen in Figure 3.10, when is increased from 100 to 10000 ft-1

, the value of FND is set to

be 22.001, 2.200 and 0.220, while responses for two Darcy flow scenarios with CfD =47.9

and CfD =100.0 are included.

Comparing the response between non-Darcy flow and Darcy flow (CfD =100.0),

non-Darcy flow is found to induce an additional pressure drop. As characteristic length

( ) is decreased, non-Darcy number (FND) is increased, leading to a strong non-Darcy

flow effect. As observed by Guppy et al. (1982a) and Umnuayponwiwat et al. (2000), the

additional pressure drop due to non-Darcy effect can also be expressed by a reduction in

the apparent conductivity of the fracture. In Figure 3.10, the Darcy’s scenario of CfD

=47.9 can be matched with the non-Darcy scenario of CfD =100.0 and FND =2.200 all the

time. The traditional analysis method can result in more than 50% error in the estimation

54

Figure 3.10 Type curves for non-Darcy effect in a tight oil formation at CfD = 100.0

55

of conductivity. This is the reason why the type curves that can be matched with the

conventional Darcy’s model would underestimate fracture conductivity if the non-Darcy

effect exists (Guppy et al., 1982; Umnuayponwiwat et al., 2000). Therefore, the fracture

half-length together with other fracture parameters cannot be accurately determined if the

apparent fracture conductivity is evaluated by using the conventional method.

To better understand the non-Darcy effect, Figure 3.11 presents the flux distribution

from the matrix to the fracture as a function of time at FND =22.001. In this case with

strong non-Darcy effect, it is found that the fluid enters from the heel of the fracture at

the very beginning, resulting in a large pressure drop until the pseudo-radial flow is

established. As time proceeds, the flux decreases at the heel of the fracture but increases

at its toe. This is ascribed to the fact that strong non-Darcy effect causes resistance in the

fracture, forcing the fluids to largely enter from the heel at the beginning of well

production. When pseudo-radial flow is established, the flux distribution stabilizes and

remains unchanged. Since the resistance force becomes smaller, most of the fluids enter

the fracture at its toe, causing a smaller pressure drop. It should be noted that the

additional pressure drop caused by non-Darcy effect is rate-dependent and that the flux

distribution along the fracture is a function of time until the pseudo-radial flow is initiated.

To perform further illustration, flux distribution inside the fracture is plotted in

Figure 3.12. At the beginning of well production, the flux inside the fracture generally

decreases at each node, though it is decreased sharply in the half of the fracture close to

the wellbore. This is because most of fluids entering the wellbore from the formation to

the fracture occur at its toe.

56


CfD = 100 and FND =22.001.

57

Figure 3.12 Flux distribution inside the fracture in a tight oil formation at CfD = 100 and

FND =22.001

58

For the purpose of comparison, a relatively low-conductivity fracture is considered

with CfD =20.0 in the same formation by changing the fracture permeability to a smaller

value. As depicted in Figure 3.13, non-Darcy number (FND) is still the same as that in a

high-conductivity fracture. The transient pressure and its derivative curves show a similar

trend to those in a high-conductivity fracture except that a larger pressure drop can be

obtained at the early stages. This is ascribed to the fact that a low fracture conductivity

can also cause an additional pressure drop which is similar to that of the non-Darcy effect.

Figure 3.14 presents flux distribution along the fracture with a low conductivity. As

shown in Figure 3.14, the flux at the early times is larger than that in the high-

conductivity fracture at the same location, directly leading to a larger pressure drop. In

the late stage, the flux distribution is similar to that of a high conductivity scenario,

indicating that the fluid entering the fracture becomes less at its heel but more at its toe.

In practice, bilinear flow analysis proposed by Cinco-Ley and Samaniego-V (1981)

is widely used to estimate the fracture conductivity. It is worthwhile, however, to

examine how non-Darcy behaviour will affect bilinear flow behaviour

(Umnuayponwiwat et al., 2000). As shown in Figure 3.15a, the pressure change is

increased following a straight line as an increase in the fourth of time at a low

conductivity. This is because bilinear flow exists at the beginning of the production. At

the same fourth of time, the pressure change is reduced as characteristic length increases.

This means that non-Darcy flow leads to an increase in the pseudo-pressure drop.

Similarly, Figure 3.15b shows the same trend for a scenario with a higher conductivity.

Obviously, there exists non-Darcy flow behaviour in a tight oil formation even at a low

production rate.

59

Figure 3.13 Type curves for non-Darcy effect in a tight oil formation at CfD = 20.0

60


CfD = 20.0 and FND =22.001

61

(a)

(b)

Figure 3.15 Bilinear flow analysis (a) at CfD = 20.0, and (b) at CfD = 100.0

62

3.4.2 Tight gas formations

The non-Darcy effect has been widely discussed and analyzed in the gas formations

(Martins et al., 1990; Vincent et al., 1999; Smith et al., 2004). The gas production rate is

set to be constant (1 MMscf). In the tight gas formation, the value of FND is set to be

121.328, 24.266 and 12.133 as τ is increased from 1000 to 10000 ft-1

, while responses for

two Darcy flow scenarios with CfD =9.6 and CfD =100.0 are included.

Figure 3.16 presents the type curves for non-Darcy effect in a tight gas formation

with high fracture conductivity. As can be seen, there is a large additional pseudo-

pressure drop at the early stages. Comparing the curve of FND =121.328 with Darcy’s

condition, it is found the pressure drop is larger than that of the tight oil formation. This

is because the non-Darcy number (FND) can be set to be a larger value for a higher gas

production rate. It is also found that the Darcy’s scenario of CfD =9.6 can be matched with

the non-Darcy scenario of CfD =100.0 and FND =24.266. Comparing to the case in tight oil

formation, the traditional method will cause a larger error in the estimation of

conductivity because non-Darcy effect is usually larger in the tight gas formation. Figure

3.17 shows the flux distribution from the matrix to the fracture at a high fracture

conductivity. As shown in Figure 3.17, the trend is very similar to that in the tight oil

formation; however, there is more fluid entering into fracture from the matrix, finally

resulting in a larger pseudo-pressure drop.

Figure 3.18 shows the type curves for non-Darcy effect in a tight gas formation

with a low fracture conductivity. Compared with high-conductivity fracture scenario, the

bilinear flow is more significant than that in a high conductivity fracture mainly because

of low conductivity other than the non-Darcy effect. Figure 3.19 presents the flux

63

Figure 3.16 Type curves for non-Darcy effect in a tight gas formation at CfD = 100.0

64


CfD = 100.0 and FND =121.328

65

Figure 3.18 Type curves for non-Darcy effect in a tight gas formation at CfD = 20.0

66


CfD = 20.0 and FND =121.328

67

3.5 Case Study

The case study is a drawdown test on an oil well which was first reported by Cinco

et al. (1978). Figure 20 shows the type-curves match between the field data on the oil

well in a tight formation and the solution of the semi-analytical model in this study. Two

cases of Darcy and non-Darcy flow are considered to examine the non-Darcy effect on oil

wells. As shown in Figure 20, there exists an excellent agreement between the field

measurements and solutions of the proposed model in this study for both Darcy and non-

Darcy flow cases. As for the Darcy’s case, the dimensionless fracture conductivity CfD is

calculated to be 6.267, which is very close to the original result of 2π. One of the matches

for the non-Darcy case corresponds to CfD =6.274, FND =0.47, kmr =0.12. This result is

consistent with the previous analysis that the non-Darcy effect in a tight oil formation is

usually very small. It should be noted that it is possible to obtain multiple combinations

of Darcy and non-Darcy parameters by adjusting some parameters. Therefore, it is

recommended that at least two flow tests be required to accurately determine the fracture

conductivity, relative minimum permeability and non-Darcy number.

3.6 Summary

The pressure transient behaviour has been analyzed in a vertical well with single

fracture. Five flow regimes in a finite conductivity fracture can be identified with the

point source function. Firstly a bilinear appears at the initial production followed by the

formation linear flow, elliptical flow, pseudo-radial flow, and eventually ended with

boundary dominated flow. Based on Barree-Conway model, Non-Darcy flow effect

68

Figure 3.20 Type curve match for the field data on an oil well in a tight formation

69

inside the fracture is dominated by relative minimum permeability (kmr) and non-Darcy

number (FND). It is found that non-Darcy number (FND) is more sensitive than relative

minimum permeability (kmr). The non-Darcy effect with high non-Darcy number (FND) is

significant even at a small relative minimum permeability (kmr). Both non-Darcy effect in

tight gas formations and tight oil formations are illustrated in this chapter. The results

show that non-Darcy effect is more significant in tight gas formations than that in tight oil

formations.

70

CHAPTER 4 SLAB SOURCE FUNCTION FOR

EVALUATING PERFORMANCE OF A

HORIZONTAL WELL WITH MULTIPLE

FRACUTRES

The pressure transient behaviour is much more complex in a horizontal well with

multiple fractures compared to that of a vertical well with single fracture. In this chapter,

a novel slab source function is firstly developed in the Laplace domain in a box-shaped

reservoir with a closed boundary. When solving the horizontal well and multiple fractures

system in the semi-analytical method, more fracture segments are involved by using the

superposition principle, leading to a large and complex linear system. Effects of fracture

stages, fracture dimension, and fracture conductivity are firstly examined and discussed,

followed by partially penetrating effect. Finally, non-Darcy flow effect is extended from

the single fracture to the horizontal well with multiple fractures based on the Barree-

Conway model.

In this study, a horizontal well with multi-stage fractures is considered in a box-

shaped reservoir (see Figure 4.1). The assumptions are made as follows:

1) The matrix is homogeneous;

2) The formation has rectangular shape with no flow outer boundaries;

3) The horizontal wellbore is parallel to the reservoir boundary;

4) The transverse hydraulic fractures have a finite conductivity;

5) The vertical fractures partially penetrate the formation;

6) The multiple transverse fractures are assumed to be identical;

71

Figure 4.1 Schematic diagram of a horizontal well with multi-stage fractures in a box-

shaped reservoir

72

7) Non-Darcy flow is assumed to be in the hydraulic fractures, while Darcy`s

flow happened in the matrix;

8) The formation fluid can only enter the wellbore through the fractures at the

open ports in the liner or through perforated intervals, as the case may be; and

9) The model is derived for single-phase oil and gas flow, while pseudo-pressure

and time transformations could be used for gas flow.

4.1 Mathematical Formulation

The source and Green’s function has been applied to describe fluid flow in porous

media for decades. Instantaneous Green’s functions in both real time domain and Laplace

domain have been provided to include geometries of the source and different boundary

conditions (Gringarten and Ramey, 1973; Ozkan and Raghavan, 1991). The advantages

of solutions in terms of the Laplace variable have been widely accepted to solve problems

associated with variable rates, wellbore storage, skin effects and conductivity. A problem

arises when applying the Newman’s product principle in the Laplace domain, i.e.,

1 2, , = , ,f x y t f x t f y t [4.1a]

1 2 1 2, , = , , , ,f x y t f x t f y t f x t f y t L L L L [4.1b]

It can be noted that the solution to a 2D problem, , ,f x y t , can be expressed by a

product form of two 1D solutions (i.e., 1 ,f x t and 2 ,f y t ) in the real time domain but

not applicable in the Laplace domain. Chen and Raghavan (1991) proposed the following

method to extend the following product principle to the Laplace domain.

73

4.1.1 Product principle

As noted by LePage (1980), the Laplace transform of the product of two functions

can be written as follows,

[4.2]

where Br is the Bromwich path to the right of all singular points of f1(p). Chen (1990)

shows that if 1f s has only simple poles, the integral of Equation [4.2] is reduced to

1 2 2n nn

f t f t R s f s s L [4.3]

where nR s is the residue of 1f s at pole sn. In particular, if 1f s is of the fractional

form of h s g s with simple poles, sn, arising solely from the zeros of g s and

0nh s , then the residue of sn in Equation [4.3] can be simplified as .

Then, Equation [4.3] can be rewritten as follows,

[4.4]

The first term on the right side of Equation [4.4] accounts for the possibility that

sn=0 (s0) is a pole. The basic instantaneous source functions and the values of

n nh s g s for instantaneous plane sources in infinite slab reservoirs are listed in

Tables 4.1 and 4.2, respectively (Chen et al., 1991).

1 2 1 2

1=

2 Br

f t f t f p f s p dpi

L

n nh s g s

0

1 2 2 21

0

n

nn

n

h s h sf t f t f s f s s

g s g s

L

74

4.1.2 Slab source function

The slab source functions in an infinite reservoir can be obtained by simply

multiplying the corresponding infinite plane source functions in Table 4.1 by a term of

source length as defined as follows (Chen, 1990),

2

,f x

f w f w f

x

sh x sF x s x x x x x

s

， [4.5]

where xw is assumed at the source center with half fracture length xf. The poles and

residues of the slab sources can be obtained by multiplying the corresponding entries in

Table 4.2 by a term defined as follows,

, 2 0f n fF x s x n ， [4.6a]

2sin

, 1f n x

f n

n x

x sF x s n

s

， [4.6b]

A slab source in a closed rectangular reservoir can be written by multiplying three

1D instantaneous source functions in x-, y- and z- direction (see Figure 4.2), respectively.

, , , , , , , , , , , , , , , , , ,w w w f f f w f w f w fS x y z x y z x y z t VII x x x t VII y y y t VII z z z t

[4.7]

where

, , ,w fVII x x x t is the slab source (prescribed flux) in x- direction

, , ,w fVII y y y t is the slab source (prescribed flux) in y- direction

, , ,w fVII z z z t is the slab source (prescribed flux) in z- direction

75

Table 4.1 Basic instantaneous source functions in an infinite slab reservoir (Chen et al.,

1991)

x=0 xw x

e Source Type Function Number

Instantaneous Source Functions For x>xw

(interchange x and xw for x<x

w)

Prescribed

Flux

VII(x)

Prescribed

Pressure

VIII(x)

Mixed

Boundaries

IX(x)

cosh cosh

sinh

w x e x

x x e x

x s x x s

s x s

sinh sinh

sinh

w x e x

x x e x

x s x x s

s x s

cosh sinh

cosh

w x e x

x x e x

x s x x s

s x s

76

Table 4.2 Values of h(sn)/g'(sn) and (sn, n=0,1,...) for instantaneous plane sources in an

infinite slab reservoir (Chen et al., 1991)

x=0 xw

xe Source Type

Function

Number

h(sn)/g'(sn) sn n=1, 2, ... n=0 n=1, 2, …

Prescribed

Flux

VII(x)

Prescribed

Pressure

VIII(x)

Mixed

Boundaries

IX(x)

1

ex

2cos cosw

e e e

n nx x

x x x

2

x

e

n

x

02

sin sinwe e e

n nx x

x x x

0 -0.5 -0.52

cos coswe e e

n nx x

x x x

2

-0.5x

e

n

x

2

x

e

n

x

77

Figure 4.2 Schematic of a slab source model

xe

ye

ze

xf

yf

zf

x

z y

78

The pressure drop induced by the source at (xw, yw, zw) can be expressed by (Zhang

et al., 2013):

1

1

1

, , ,

1, , , , , , , , ,

8

, , ,2

2 cos cos , , , ,

2 cos cos , , , ,

4

w w w f f f

t f f f

p w f

t e e f

n f e w w f np nn

e e

m f e w w f mp mm

e e

n f en

P x y z s

q sS x y z x y z x y z s

c x y z

q sS y y y s

c x z y

n nF x x x x S y y y s

x x

m mF z z z z S y y y s

z z

F x x

1

cos cos

cos cos , , , , ,

w

e e

m f e w w f n mp nmm

e e

n nx x

x x


z z

[4.8]

where

2

, , ,f w e

p w f

e

sh y s ch y s ch y y sS y y y s

s sh y s

[4.9a]

2, , , ,

f n w n e n

w f np n

n e n


s sh y s

[4.9b]

2, , , ,

f m w m e m

w f mp m

m e m


s sh y s

[4.9c]

79

2, , , , ,

f n m w n m

w f n mp nm

n m

e n m

e n m

sh y s ch y sS y y y s

s

ch y y s

sh y s

[4.9d]

sin=

f e

n f e

f e

n x xF x x

n x x

[4.9e]

sin=

f e

m f e

f e

m z zF z z

m z z

[4.9f]

2

n en x [4.9g]

2

m em z [4.9h]

The detailed derivation in Equation [4.8] is presented in Appendix, while the

computation problem is avoided by Ozkan and Raghavan (1991). The slab source

function is used to solve the horizontal well with multiple fractures instead of point

source function (Equation [3.14]).

4.2 Linear System

With the similar semi-analytical method presented in Chapter 3, As for each

discretized segment, uniform flux solution of the slab source function in Equation [4.8] is

80

used to describe fluid flow in the matrix. Flow in the fracture is considered as 1D flow

model using Equation [3.22], assuming that pressure drop is evenly distributed in the

horizontal wellbore. Therefore, these two types of flow are then coupled by using the

equations provided below. The fracture number in the reservoir is set as N, each of which

is divided into M segments. Segment #1 represents the heel of the fracture, while

Segment #m represents the toe. Single fracture is considered when N = 1, while multiple

fractures are assumed when N > 1.

Assuming that no flow enters the toe of the fracture, the model contains 4N×M

unknowns:

1) Pressure at matrix nodes: PD1,1, PD1,2, …, PD1,M; PD2,1, PD2,2, …, PD2,M; …; PDN,1,

PDN,2, …, PDN,M;

2) Pressure at fracture nodes: PfD1,1, PfD1,2, …, PfD1,M; PfD2,1, PfD2,2, …, PfD2,M; …;

PfDN,1, PfDN,2, …, PfDN,M;

3) Flux from matrix to fracture nodes: qfD1,1, qfD1,2, …, qfD1,M; qfD2,1, qfD2,2, …,

qfD2,M; …; qfDN,1, qfDN,2, …, qfDN,M; and

4) Flux in the fracture node: qND1,1, qND1,2, …, qND1,M ; qND2,1, qND2,2, …, qND2,M; …;

qNDN,1, qNDN,2, …, qNDN,M.

4.2.1 Constraint equations

Two constraint conditions are usually used for pressure transient analysis. As for

fixed production and bottomhole pressure control, we respectively have:

max,1

1

=N

NDii

Qq

s

[4.10]

81

and

,min

1,1

wf

ND

pp

s [4.11]

where Qmax is the maximum flow rate and pwf, min is the minimum bottomhole pressure.

One equation is obtained from the constraint equation. Because the solution is developed

in the Laplace domain, the constraints can be changed at any time without superposition

in time.

4.2.2 Flow equations

By the superposition principle, the pressure drop and flux at each source location as

a result of each segment produces at a constant rate is written as follows,

,

, , , , , ,, , , m n

Di j Dm n Dm n Dm n fDi j i jp x y z t q F [4.12]

where ,

,

m n

i jF is the coefficient of single source solution of Segment #j in Fracture #i

relative to Segment #m in Fracture #n. For the entire fracture, a set of linear equations can

be obtained as follows,

1,1 1,1 1,1

1,1 1,1 1,2 1,2 , , 1,1

1,2 1,2 1,2

1,1 1,1 1,2 1,2 , , 1,2

, , ,

1,1 1,1 1,2 1,2 , , ,

+ ...

+ ...

+ ...

fD fD fDN M N M D

fD fD fDN M N M D

N M N M N M

fD fD fDN M N M DN M

q F q F q F P

q F q F q F P

q F q F q F P

[4.13]

A total of N×M equations can be obtained.

With the solution in the fracture subsystem, another set of linear equations are

written in the first fracture as follows,

82

1 1,1 1 1,2 1 1,1 1,1

2 1,2 2 1,3 2 1,2 1,2

1, 1 1, 1,M 1,

+c

+c

+c

ND ND fD fD

ND ND fD fD

M ND M M ND M M fD fD M

b q q d q P

b q q d q P

b q q d q P

[4.14]

As there are N fractures, there will be a number of N×M equations.

4.2.3 Continuity equations

As discussed in Chapter 3, the solution for the matrix subsystem is based on the

real-time Laplace transformation, while the solution for the fracture subsystem is resulted

from the pseudo-time Laplace transformation.

The pressure continuity condition for the fracture and matrix subsystems can be

obtained as follows,

, ,

1 ln 2 1 ln 2=fD D

i j i j D D

p j p jt t

[4.15a]

where τi,j is the pseudo-time at segment i of fracture j. Thus, N×M equations can be

obtained for the N fractures.

Similarly, the pressure continuity condition for the node between any two adjacent

segments is expressed as follows,

, , +1, +1,

1 ln 2 1 ln 2=fD fD

i j i j i j i j

p j p j

[4.15b]

N fractures have N × (M-1) interfaces, so there will be N × (M-1) equations.

83

4.2.4 Wellbore equations

Because no pressure drop is assumed in the horizontal wellbore, frictional,

acceleration and radial inflow effects are not considered in this study. The pressure along

the wellbore should be the same, we have,

1,1 2,1 ,1...fD fD fDNp p p [4.16]

A total of N-1 equations can be written for interfaces between wellbore and

fractures. Therefore, 4N×M equations can be obtained, corresponding to the same number

unknown variables. With the similar iterative method used in Chapter 3 for non-Darcy

flow, a matrix different from single fracture case is shown in Figure 4.3.

4.3 Model Validation

Numerical simulation models were used for validation. The numerical models were

built by using a reservoir simulator (IMEX, 2010, Computer Modelling Group Ltd.). The

global grid system is 27×27×5, and the cell dimensions are 20×20×10 ft. The single cell

at the fracture location was locally refined by using the keyword “Hydraulically

Fractured Wells”. Number of refined blocks in each direction is 5×5×3. Figures 4.4a and

b show a top view and cross-section of the global grid system. Three fractures are evenly

distributed in a horizontal well, where only the perforations in the fractures are open to

make the fluid flow from the fracture to the wellbore. The basic input parameters are

listed in Table 4.3 (Vincent et al., 1999; Gil et al., 2003).

84

Table 4.3 Basic parameters used for simulating pressure transient response


Reservoir and Fluid

Data

Reservoir Pressure, pi 3000 psia

Reservoir Temperature, T 190 ºF

Matrix Permeability, km 0.1 mD



Initial Gas Viscosity, μi 0.018 cP


psi-1

Gas Density, ρ 8.23 lbm/ft3

Gas Molecular Weight, M 17.382 lbm/lb-mol

Production Rate, q 1.0 MMscf/d

Hydraulic Fracture

Data




85

Figure 4.3 The coefficient matrix for solving the matrix-hydraulic fracture system

1 2 3 … M-2 M-1 M

1

2

3

M-2

M-1

M

*

* * *

… …* * *

* * *

* * *

* *

…

……

*

** *

* *

* *

* *

* *

……

* *

1 2 3 … M-2 M-1 M … 1 2 3 … M-2 M-1 M … 1 2 3 … M-2 M-1 M …

1

2

3

M-2

M-1

M

……

* *

* * *

…* * *

* * *

* * *

…

* *

1

2

3

M-2

M-1

M

1

2

3

M-2

M-1

M

……

** *

* *

* *

* *

* *

……

……

* *

* *

* *…

…

* *

* *

*

* *

* *

* *…

…

* *

* *

*

* * * * * * * * * * * **… …

* * * * * * * * * * * *** * * * * * * * * * * **

* * * * * * * * * * * *** * * * * * * * * * * **

* * * * * * * * * * * **

* * * * * * * * * * * **

* * * * * * * * * * * **

* * * * * * * * * * * **

* * * * * * * * * * * *** * * * * * * * * * * **

* * * * * * * * * * * **

……

……

… …

… …

… …… …

… …

… …

… …

… …

… …

… …

… …

1 2 3 … M-2 M-1 M

1

2

3

M-2

M-1

M

*

* * *

… …* * *

* * *

* * *

* *

…

……

*

** *

* *

* *

* *

* *

……

* *

1 2 3 … M-2 M-1 M … 1 2 3 … M-2 M-1 M … 1 2 3 … M-2 M-1 M …

1

2

3

M-2

M-1

M

……

* *

* * *

…* * *

* * *

* * *

…

* *

1

2

3

M-2

M-1

M

1

2

3

M-2

M-1

M

……

** *

* *

* *

* *

* *

……

……

* *

* *

* *…

…

* *

* *

*

* *

* *

* *…

…

* *

* *

*

* * * * * * * * * * * **… …

* * * * * * * * * * * *** * * * * * * * * * * **

* * * * * * * * * * * *** * * * * * * * * * * **

* * * * * * * * * * * **

* * * * * * * * * * * **

* * * * * * * * * * * **

* * * * * * * * * * * **

* * * * * * * * * * * *** * * * * * * * * * * **

* * * * * * * * * * * **

……

……

… …

… …

… …… …

… …

… …

… …

… …

… …

… …

… …

86

(a)

(b)

Figure 4.4 (a) Top view of the 3D grid system and (b) Cross-section of the 3D grid

system

87

As for the numerical simulation, non-Darcy flow is described by using the

Geertsma correlation (1974) which is commonly accepted for the unconsolidated

materials. Inertial factor β is calculated to be 6.86×107 ft

-1. The common factor between

the Barree-Conway model and the Forchheimer equation is the apparent permeability of

non-Darcy flow, which is defined by Equations [3.37a] and [3.37b], respectively. When

kmr is set to be 5×10-9

, τ is calculated to be 2.133×10-3

ft. As can be seen in Figure 4.5,

there exists an excellent agreement between this work and the CMG simulation result,

showing that the slab source method produces reliable transient pressure results.

The newly slab source function is also validated with widely used point source

function. In the slab source function, the width of fracture is incorporated in a single

vertical fracture case. Figure 4.6 presents the comparison result between the point source

and slab source function. It is found that minor difference exists in the initial production

because the fracture width is very small. However, the development of slab source

function is still meaningful for easy computation, which is not only applicable in the

single vertical fracture, but in the horizontal fracture, multiple fractures and horizontal

well.

4.4 Results and Discussion

4.4.1 Fracture stages

Figure 4.7 presents the pressure response and its derivative curves under different

fracture stages with the same fracture length for each stage. Six flow regimes can be

observed in a multiple-fracture system.

88

Figure 4.5 Model validation between this study and CMG simulation results

89

Figure 4.6 Comparison between this study and CMG simulation results

90

Figure 4.7 Effect of fracture stages on the well response with the same fracture length

91

Firstly, a bilinear/linear flow appears at the initial production followed by the early-

radial flow, compound-linear flow, elliptical flow, pseudo-radial flow, and eventually

ended with boundary flow. As can be seen, the pressure drop decreases as the fracture

stage number is increased. This is due to the fact that more stages lead to a larger fracture

volume and a smaller fracture spacing as well. This results in a longer bilinear/linear flow,

though number of fracture stages is less. As the number of fracture stages increases the

interference among these hydraulic fractures becomes more evident, especially between

the dimensionless time of 0.3-20×10-3

.

In a tight oil or shale gas reservoir, the permeability is often found to be in the

micro-Darcy to nano-Darcy range, so it will take a long time (i.e., measured in decades)

to reach the pseudo-steady state flow (Ozkan et al., 2009). In Figure 4.7, the slope of both

pressure and derivative curves is approaching unity when tD = 2.0×10-3

in the case of 40

fracture stages. This means that the pressure and its derivative curve bend towards to the

pseudo-steady flow. This phenomenon can be explained by the occurrence of the

stimulated reservoir volume (SRV) (Mayerhofer et al., 2010). The fractured well can

only produce from the volume trapped between fractures.

4.4.2 Fracture conductivity

Figure 4.8 presents drawdown type curves for fracture conductivity CfD = 1, 10, 30

and 50 with five pairs of fractures. Fracture conductivity is found to mainly influence

early-stage bilinear/linear flow regime. As fracture conductivity CfD is increased, the

pressure drop and its derivative value become smaller. When fracture conductivity is set

to be large enough, the linear flow regime is followed by the bilinear flow regime.

92

Figure 4.8 Pressure response together with its derivative for five pairs of fractures under

various fracture conductivities

93

To better understand the conductivity effect, Figure 4.9a shows the flux distribution

at the initial production tD = 10-6

when fracture conductivity increased from 1 to 50. The

flux at the heel of the fracture is increased as the conductivity is decreased. Figure 4.9b

illustrates the flux distribution at tD = 102, which allows the flux distribution to be

stabilized. As can be seen in Figure 4.9b, at a small fracture conductivity CfD = 1, there

are still much fluid entering from the heel of the fracture, while, at large conductivity CfD

= 50, most fluid enters the toe of the fracture. This is ascribed to the fact that a smaller

fracture conductivity induces a larger resistance in the fracture, forcing the fluids to enter

the portions of the matrix progressively distant from the fracture heel with time, and

ultimately culminates in production entering the fracture from its toe. When pseudo-

radial flow is established, the flux distribution stabilizes and remains unchanged. Since

the resistance force becomes smaller, most of the fluids enter the fracture at its toe,

resulting in a smaller pressure drop.

4.4.3 Fracture length

Figure 4.10 illustrates effect of fracture length at four pairs of fractures. All the

fractures with different length firstly show bilinear/linear pressure drop followed by a

gradual change that leads to an elliptical flow regime with a slope that is larger than half

but less than 1. As can be seen, the pressure drop increases as the fracture length is

increased, while the linear flow regime is gradually changed to the bilinear regime. It is

found that the longer the fracture is, the longer the duration of the elliptical flow regime

will be.

94

(a)

(b)

Figure 4.9 Effect of fracture conductivity on the flow distribution at (a) tD = 1.0E-6, and

(b) tD = 1.0E2

95

Figure 4.10 Effect of fracture length on the well response at four pairs of fractures

96

4.4.4 Fracture spacing

Different from single fracture, fracturing spacing existing in multiple fractures has

significant effect in the pressure transient behaviour. Figure 4.11 presents the effect of

fracture spacing at four pairs of fractures. The boundaries are set to be constant, and the

horizontal wellbore is assumed to place in the same location. The only varying parameter

is fracture spacing which is illustrated as dimensionless form. As shown in Figure 4.11,

the fracture spacing significantly influences the intermediate radial flow regime. It is

found that the intermediate radial flow regime cannot be observed when fracture spacing

is set as a small value (DD = 3). As fracture spacing increased, the intermediate flow

regime becomes more evident at larger value (DD = 8).

As can be seen, when boundary conditions are kept unchanged, the fracture spacing

also influences the pseudo-radial flow regime. At a large value (DD = 8), the pseudo-

radial flow regime almost disappears. This can be explained that larger fracturing spacing

makes the fracture more approaching to the boundary, which means the pseudo-steady

state flow directly occurs after the elliptical flow regime.

4.4.5 Penetrating ratio

Although numerous efforts have been made to study pressure transient behaviour of

hydraulic fractured wells, there are few studies pertaining to effects of the partially

penetrating fractures (Raghavan et al., 1978; Rodriguez et al., 1984; Alpheous and Tiab,

2008). Typically, it is preferred that the fracture height be equal to the formation

thickness, where fully-penetrating fractures are produced. However, a fully penetrating

97

Figure 4.11 Effect of fracture spacing on the well response at four pairs of fractures

98

fracture may lead to an early water breakthrough in a reservoir with bottom water, or

unanticipated water production in a reservoir with an incompetent top or bottom seal that

has an adjacent permeable water-bearing formation.

Figure 4.12a depicts drawdown type curves of three vertical fractures at fracture

conductivity CfD = 10 when penetrating ratio α = 0.6, 0.7, 0.8, 0.9 and 1.0. Penetrating

ratio α = 0.5 can be used as the threshold value to classify the pressure response (see

Figure 4.12b). The pressure behaviour tends to be similar to that of the fully penetrating

fractures when other factors such as fracture stages, spacing and dimensions remain

unchanged. The first linear flow regime appears at the initial production followed by a

short transition regime.

Figure 4.12b plots the drawdown type curves of three vertical fractures at fracture

conductivity CfD = 10 when penetrating ratio α = 0.1, 0.2, 0.3, 0.4, and 0.5. It is observed

that the pressure drop becomes larger as the penetrating ratio is decreased. The first linear

flow regime becomes shorter as the penetrating ratio is decreased. Meanwhile, an early

radial flow regime instead of the transition regime occurs.

4.4.6 Non-Darcy effect

The non-Darcy effect in the single fracture has been discussed in Chapter 3. It has

been noted that the non-Darcy effect is dominated by both relative minimum permeability

(kmr) and non-Darcy number (FND).

99

(a)

(b)

Figure 4.12 Effect of partially penetrating ratio of three stages at CfD=10 and (a) α>0.5,

and (b) α≤0.5

100

Table 4.4 presents the basic parameters used in the experiments by Lai et al. (2012).

The flow rate ranges from 707.2 Sm3/d to 283168.2 Sm

3/d, covering the most possible

field production rates. Since the non-Darcy number FND is calculated to be 2.7-1091.4

with Equation [16b], it is set to be 1, 10, 100 and 1000 to cover the most possible

scenarios from low to high flow rates.

Figure 4.13a depicts drawdown type curves for FND = 1, 10, 100 and 1000 at kmr =

0.01 with four pairs of fractures, respectively. The trend is very similar to that in the

single fracture. As FND is increased, the pressure drop and its derivative curves increase

due to strong non-Darcy effect. It is found that non-Darcy effect becomes significant

when FND becomes larger than 10. As FND approaches to zero, however, non-Darcy effect

tends to disappear and the pressure response is found to be very similar to that of the

Darcy’s case.

Figure 4.13b plots drawdown type curves for FND = 1, 10, 100 and 1000 at kmr =

0.10, respectively. The pressure drop together with its derivative is becoming much

smaller at kmr = 0.10 than that at kmr = 0.01. Comparing to the larger kmr case, there still

exists a difference in pressure drop even in the pseudo-radial flow regime at the smaller

kmr = 0.01. Figures 4.14a and b show drawdown type curves for kmr = 0.01, 0.05 and 0.10

at FND = 1000 and FND = 10, respectively. As shown in Figure 4.14a, the pressure drop

and its derivative curves increase when kmr is decreased. However, as shown in Figure

4.14b, the pressure drop and its derivative curves are almost the same when kmr is

decreased. The non-Darcy effect becomes insignificant at a smaller non-Darcy number

(FND = 10) when kmr is decreased.

101

Table 4.4 Parameters used for describing non-Darcy flow with Barree-Conway model in

multiple fractures (Lai et al., 2012)


Darcy permeability, kd 10.0 Darcy

Minimum permeability, kmin 0.1, 0.5, 1.0 Darcy

Minimum relative Permeability, kmr 0.01, 0.05, 0.10, fraction

Viscosity, μ 1 cP


Characteristic length, τ 3000-30000 1/ft

102

(a)

(b)

Figure 4.13 Effect of non-Darcy number FND (a) at kmr = 0.01, and (b) at kmr = 0.10

103

(a)

(b)

Figure 4.14 Effect of non-Darcy number kmr at (a) FND = 1000, and (b) FND = 10

104

As can be seen in Figures 4.13 and 4.14, the non-Darcy flow behaviour in multiple

fractures is similar as that of single fracture. Non-Darcy effect will become more evident

at a smaller relative minimum permeability (kmr<0.05) and a larger non-Darcy number

(FND>10). In addition, the non-Darcy number (FND) is more sensitive than the relative

minimum permeability (kmr), resulting in a larger pressure drop even at a larger kmr. The

major difference is that non-Darcy flow can cause a larger pressure drop in the single

fracture than that in the multiple fractures. This is ascribed to the fact that the fixed

production rate is assumed to be unity in all cases, causing the flux in each fracture is

smaller than that of single fracture. Then, a smaller flux can result in a smaller non-Darcy

number (FND), and eventually lead to smaller pressure drop.

To better understand the non-Darcy effect, Figure 4.15a presents the flux

distribution at the initial production tD = 10-6

. As shown in Figure 4.15a, most fluids enter

from the heel of the fracture at the initial production. Non-Darcy effect forces more fluid

entering from the heel of the fracture than that of Darcy flow.

Figure 4.15b illustrates the flux distribution from the matrix to the fracture at tD =

103, which allows the flux distribution to be stabilized. As can be seen in Figure 4.15b, as

for Darcy flow, most of the flow occurs at the toe of the fracture. However, as for non-

Darcy flow, the flux from the heel increases, whereas the flux from the toe decreases with

an increase in FND and a decrease in kmr. At kmr = 0.01 and FND = 1000, the flux at the

heel of the fracture is even larger than that at its toe. This is attributed to the fact that non-

Darcy flow increases the flow resistance in the fracture, forcing more fluid to enter the

fracture from the area close to the wellbore. This is also the reason why the effective

105

(a)

(b)

Figure 4.15 Effect of non-Darcy flow on the flow distribution at (a) tD = 1.0E-6, and (b)

tD = 1.0E3

106

fracture length will be shorter than its original design due to the non-Darcy flow in the

fracture.

4.5 Case Study

The Bakken formation is located in the Williston Basin covering parts of Montana,

North Dakota, and South Saskatchewan. A tremendous hydrocarbon is discovered in this

formation and production increases rapidly with technology advancement in horizontal

drilling and massively fracturing. This formation is composed of three distinct geological

layers: Upper shale, Middle Bakken and Lower shale. The Middle Bakken is a low

porosity and low permeability reservoir composed of mixed clastic and carbonate

sediments primarily deposited in a shallow marine environment (Luo et al., 2010).

Figure 4.16 presents the pressure buildup test and production history from a well in

the Bakken formation of South Saskatchewan. As shown in Figure 4.16, the top part

shows the pressure data reported form an automated acoustic well sounder equipment,

while the bottom part illustrates the well production history used in the analysis of the

pressure data. The well was hydraulically fractured with 15 stages and then commenced

its production on December 20, 2010. The well was shut in for a buildup test on January

27, 2011 and terminated on February 28, 2011 after 769.5 hours of buildup time.

The reservoir and fracture properties are listed in Table 4.5. It has been noted that

non-Darcy flow during the producing period also affects the buildup test response

(Umnuayponwiwat et al., 2000). The pressure data was well matched by using the newly

formulated model with single phase and multiple transverse fractures (see Figure 4.17). It

is shown from the derivative curve that there exists a wellbore storage effects after the

107

Table 4.5 Reservoir and fracture properties for the well in the Bakken formation


Payzone Thickness, h 5.9 ft


Specific Gravity 0.81 fraction


Reservoir Temperature, T 176 ºF

Bottomhole pressure, pw 498.19-2249.97 psi

Wellbore Radius, rw 0.23 ft

Well Length, L 3564 ft

Number of Fractures 15

108

Figure 4.16 Pressure buildup test and production history

109

Figure 4.17 Flow regime interpretation for the pressure buildup response

110

first 1.5 hours. Due to the wellbore storage, the bilinear flow and the early radial flow

cannot be observed. Then a linear flow regime with half slope develops, defining the

formation of the linear flow regime. This is followed by a short elliptical flow which is

not obviously. Eventually, the pseudo-radial flow with constant derivative value is found

in the late stage.

Figure 4.18 presents the measured and simulated pressure buildup data, while the

matching parameters are tabulated in Table 4.6. Due to wellbore storage effect, the

penetrating ratio cannot be determined in the pressure response. Therefore, the fracture

height is assumed to be equal to the formation thickness as the net payzone is only 5.9 ft

thick. Because two new parameters are introduced in this study, it is difficult to obtain the

unique match result by type curve matching. If non-Darcy flow is not considered in the

match, the dimensionless conductivity (CfD) was found to be 20.76. When non-Darcy

effect is incorporated, one of the matches corresponds to the type curve for CfD = 23.52,

kmr = 0.087 and FND = 8.32. The other match is with type curve for CfD = 23.52, kmr =

0.524 and FND = 73.55. Due to the fact that production of this well is not high (83.98

STB/d), the first case of CfD = 23.52, kmr = 0.087 and FND = 8.32 is considered as the

most reasonable matching result. Then the characteristic length (τ) is calculated to be

241.12 1/ft. There exists an excellent agreement between the measured and simulated

pressure for the buildup test as well.

111

Table 4.6 Type curve matching results for the buildup test


Wellbore Storage, C 0.12 bbl/psi

Skin Factor, s 0.21


Fracture Height, hf 5.9 ft

Fracture Conductivity, Cf 5642 mD·ft

Matrix Permeability, km 0.872 mD

Relative Minimum Permeability, kmr 0.087 fraction

Characteristic Length, τ 241.12 ft-1

112

Figure 4.18 Type curve matching of pressure buildup data

113

4.6 Summary

As for a horizontal well with multiple fractures, it is found that the interference

becomes more evident as fracture stages increase. The longer the fracture is, the longer

the duration of the elliptical flow regime will be. Fracture conductivity is found to mainly

affect the early-stage bilinear/linear flow regime. Two types of pressure response can be

classified based on the threshold penetrating ratio α = 0.5. In the case of a large

penetrating ratio, the pressure response is similar to that of the fully penetrating case.

When penetrating ratio is smaller than 0.5, the first linear flow regime occurs followed by

an early radial flow regime. At the same fixed production rate, non-Darcy effect in the

single fractured vertical well can cause a larger pressure drop than that of the horizontal

well with multiple fractures under the same conditions.

114

CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS

5.1 Conclusions

In this study, the point source function was firstly applied in the single vertical

fracture well by developing a semi-analytical method to solve the single fracture problem

with consideration of the finite conductivity and non-Darcy effect in the fracture. In the

semi-analytical method, equations for the matrix and fracture subsystems can be solved in

the Laplace domain with source functions, respectively. Then, their solutions are coupled

along the fracture interface, leading to a linear system. Solving the linear system, the flux

in each segment can be obtained, which is substituted into the first segment to determine

the bottomhole pressure in the Laplace domain. Finally, the Stephest inverse algorithm

can be employed to convert the solutions in the Laplace domain to that in the real time

domain. The non-Darcy flow effect inside the fracture can be solved by converting the

real time to the pseudo-time. Continuity conditions are modified based on the pseudo-

time, and the solutions under non-Darcy flow effect can be obtained by iterative methods.

Due to the difficulty for evaluating performance of horizontal wells with multiple

fractures, a novel slab source function was then developed to describe the fluid flow in

the porous media. The slab source function attributes geometry to sources and considers

the pressure effect inside sources by using the superposition principle over multiple

sources. Flow effect inside fractures can be examined by dividing the fracture into several

segments, each of which can be treated as a slab source. Such a newly developed

technique has been validated with numerical simulation method, and then applied to

115

describe the pressure transient behaviour under various fracture conductivities, fracture

half-lengths and fracture in a field case.

The major conclusions that can be drawn from this thesis study are summarized as

follows:

1) Five flow regimes can be observed in a finite conductivity fracture with the

point source function. More specifically, a bilinear appears at the initial

production followed by the formation linear flow, elliptical flow, pseudo-radial

flow, and eventually ended with boundary dominated flow. The slab source

function in the Laplace domain is developed successfully to describe pressure

transient behaviour in a complex reservoir and well system;

2) The non-Darcy flow effect inside the hydraulic fractures is examined based on

the Barree-Conway model. Comparing to traditional Forchheimer’s equation,

non-Darcy flow effect is dominated by relative minimum permeability (kmr)

and non-Darcy number (FND). Non-Darcy number (FND) is more sensitive than

relative minimum permeability (kmr). The non-Darcy effect with high non-

Darcy number (FND) is significant even at a small relative minimum

permeability (kmr);

3) At lower non-Darcy number (FND), the results are almost the same between

Forchheimer’s equation and Barree-Conway model. At a higher non-Darcy

number (FND), the Forchheimer’s equation can overestimate pressure drop by

30.9% compared to the Barree-Conway model;

4) Non-Darcy effect in a tight gas reservoir is found to be more significant than

that in a tight oil reservoir. At the same fixed production rate, non-Darcy effect

116

in a vertical well with single fracture results in a higher pressure drop than that

in a horizontal well with multiple fractures;

5) Fracture conductivity mainly affects the early-stage bilinear/linear flow regime.

A smaller fracture conductivity causes a larger resistance in the fracture,

forcing the fluids to largely enter the toe at the beginning of production;

6) As the number of the fracture stages increases, the interference among those

hydraulic fractures becomes more evident, especially between the

dimensionless times of 0.3×10-3

and 20.0×10-3

; and

7) Two types of pressure response can be classified based on the threshold

penetrating ratio α = 0.5. In the case of a large penetrating ratio, the pressure

response is similar to that of the fully penetrating case. When penetrating ratio

is smaller than 0.5, the first linear flow regime occurs followed by an early

radial flow regime.

5.2 Recommendations

Based on this thesis study, the following recommendations for future studies are

made:

1) The slab source function can be extended to describe fluid flow in more

complex cases by incorporating natural fractures, horizontal wellbore

contribution, and stimulated reservoir volume (SRV);

2) Experiments should be conducted to quantify the ranges of minimum

permeability plateau for different proppant and fracturing fluids;

117

3) Non-Darcy effect based on the Barree-Conway model should be examined on

more field cases, especially on fractured gas wells in tight gas and/or shale gas

reservoirs; and

4) A comprehensive two-phase flow model should be developed to evaluate well

performance under non-Darcy flow inside the hydraulic fractures.

118

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130

APPENDIX: DERIVATION OF THE SLAB SOURCE SOLUTION

IN THE LAPLACE DOMAIN

Newman’s product solution method is firstly used by Gringarten and Ramey (1973)

in the application of source and Green’s functions for multi-dimensional flow through

porous media. This method, however, is only applicable in the real (spatial and temporal)

domain. For the development of solution in the Laplace domain, Chen et al. (1991)

proposed a method to extend the product principle to the Laplace domain. The detailed

derivation of this method can be found elsewhere (Chen et al., 1991). In this study, the

slab source function is developed by following the same procedure.

A slab can be obtained by considering the intersection of a slab source in a linear

reservoir in the x- and y- directions and a slab source in a linear reservoir in the z-

direction. The required instantaneous source function can be written as follows,

, , , , , , , , , , , , , , , , , ,w w w f f f w f w f w fS x y z x y z x y z t VII x x x t VII y y y t VII z z z t

[4.7]

where

, , ,w fVII x x x t is the slab source (prescribed flux) in x- direction

, , ,w fVII y y y t is the slab source (prescribed flux) in y- direction

, , ,w fVII z z z t is the slab source (prescribed flux) in z- direction

Equation 4.7 can be solved in two steps:

, , , , , , , , , , , ,f w w f f w f w fS x y x y x y t VII x x x t VII y y y t [A-1]

131

, , , , , , , , , , , , , , , , , ,w w w f f f f w w f f w fS x y z x y z x y z t S x y x y x y t VII z z z t [A-2]

Taking the Laplace transform of Equation [A-2], its product source solution can be

expressed as follows,

1

, , ,, , , , , , , , , , , , , , ,

, , ,

, , ,, , , , , ,

, , ,

w f o

w w w f f f f w w f f

w f oVII

w f m

f w w f f mm

w f mVII

h z z z sS x y z x y z x y z s S x y x y x y s

g z z z s

h z z z sS x y x y x y s s

g z z z s

[A-3]

Equation [A-3] can be written as follows,

1

, , , , , , , , ,

2, , , , , ,

2 cos cos , , , , , , , ,

w w w f f f

f

f w w f f

e

n f e w w w f f n mf nmm

e e

S x y z x y z x y z s

zS x y x y x y s

z

n nF z z z z S x y x y x y s

z z

[A-4]

where

1

, , , , , ,

2, , , 2 cos cos , , , ,

f w w f f

f

p w f n f e w w f np nn

e e e

S x y x y x y s

x n nS y y y s F x x x x S y y y s

x x x

[A-5a]

1

, , , , , , , ,

2, , , ,

2 cos cos , , , , ,

w w f f n mf nm

f

w f mp m

e

n f e w w f n mp nmn

e e

S x y x y x y s

xS y y y s

x


x x

[A-5b]

132

Using Equations [A-5a] and [A-5b], Equation [A-4] can be rewritten as follows,

1

1

1

, , , , , , , , ,

4, , , 2 cos cos , , , ,

2 cos cos , , , ,

4 cos cos

w w w f f f

f f

p w f n f e w w f np nn

e e e e

m f e w w f mp mm

e e

n f e wn

e e

S x y z x y z x y z s

x z n nS y y y s F x x x x S y y y s

x z x x


z z

n nF x x x x

x x

1

cos cos , , , , ,m f e w w f n mp nmm

e e


z z

[A-6]

Since slab source is assumed, the source-shape-dependent rate in terms of

withdrawal rate is

ˆ8 f f f

qq

x y z [A-7]

The pressure-rate-source function in the Laplace domain is described by Equations

[A-6] and [A-7],

133

1

1

1

, , ,

1, , , , , , , , ,

8

, , ,2

2 cos cos , , , ,

2 cos cos , , , ,

4

w w w f f f

t f f f

p w f

t e e f

n f e w w f np nn

e e

m f e w w f mp mm

e e

n f en

P x y z s

q sS x y z x y z x y z s

c x y z

q sS y y y s

c x z y


x x


z z

F x x

1

cos cos

cos cos , , , , ,

w

e e

m f e w w f n mp nmm

e e

n nx x

x x


z z

[4.8]

where

2

, , ,f w e

p w f

e


s sh y s

[4.9a]

2, , , ,

f n w n e n

w f np n

n e n


s sh y s

[4.9b]

2, , , ,

f m w m e m

w f mp n

m e m


s sh y s

[4.9c]

134

2, , , , ,

f n m w n m

w f n mp nm

n m

e n m

e n m


s

ch y y s

sh y s

[4.9d]

sin=

f e

n f e

f e

n x xF x x

n x x

[4.9e]

sin=

f e

m f e

f e

m z zF z z

m z z

[4.9f]

2

n en x [4.9g]

2

m em z [4.9h]

Equation [4.8] is based on the coordinates defined in Table 3.1 and 3.2. Equations

[4.8] to [4.9h] can be non-dimensionalized as,

1

1

1

, , ,

, , ,

2 cos cos , , , ,

2 cos cos , , , ,

4 cos

D D D D D

fD

fD p D wD fD D

eD fD

n fD eD wD D D wD fD nD Dp nn

eD eD

m fD eD wD D D wD fD mD Dp mm

eD eD

n fD eDn

P x y z s

xq s S y y y s

x y


x x


z z

nF x x

1

cos

cos cos , , , , ,

wD D

eD eD

m fD eD wD D D wD fD nD mD Dp nmm

eD eD

nx x

x x


z z

[A-8]

where

135

2

, , ,fD D wD D eD D D

p D wD fD D

D eD D


s sh y s

[A-9a]

2, , , ,

fD D nD wD D nD

D wD fD nD Dp n

D nD

eD D D nD

eD D nD


s

ch y y s

sh y s

[A-9b]

2, , , ,

fD D mD wD D mD

D wD fD mD Dp m

D mD

eD D D mD

eD D mD


s

ch y y s

sh y s

[A-9c]

2, , , , ,

fD D nD mD wD D nD mD

D wD fD nD mD Dp nm

D nD mD

eD D D nD mD

eD D nD mD


s

ch y y s

sh y s

[A-9d]

sin=

fD eD

n fD eD

fD eD

n x xF x x

n x x

[A-9e]

sin=

fD eD

m fD eD

fD eD

m z zF z z

m z z

[A-9f]

2

n eDn x [A-9g]

2

m eDm z [A-9h]

136

For the convenience of superposition principle, Equation [A-8] can be written as

follows,

, , , , ,

,

,, , ,i j m n m n m n i j

m n

D D D D D fD i jP x y z s q s F [A-10]

where,

, ,

, , , ,

, , , ,

,

,

1

1

, , ,

2 cos cos , , , ,

2 cos cos , , , ,

m n i j

i j m n m n i j

i j m n m n i j

fDm n

i j p D wD fD D

eD fD

n fD eD wD D D wD fD nD Dp nn

eD eD

m fD eD wD D D wD fD mD Dp mm

eD eD

xF S y y y s

x y


x x


z z

, ,

, , , ,

1

1

4 cos cos

cos cos , , , , ,

i j m n

i j m n m n i j

n fD eD wD Dn

eD eD

m fD eD wD D D wD fD nD mD Dp nmm

eD eD

n nF x x x x

x x


z z

[A-11]

,

,

m n

i jF is the coefficient of single slab source solution of Segment #i in Fracture #j

relative to Segment #m in Fracture #n.

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