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A PHYSICALLY CONSISTENT SOLUTION FOR DESCRIBING

THE TRANSIENT RESPONSE OF HYDRAULICALLY

FRACTURED AND HORIZONTAL WELLS

by

BABAFEMI OLUWASEGUN OGUNSANYA, B.S., M.S.

A DISSERTATION

IN

PETROLEUM ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Teddy Oetama Chairperson of the Committee

Lloyd Heinze

James Lea

Accepted

John Borrelli Dean of the Graduate School

May, 2005

ii

ACKNOWLEDGEMENTS

Financial support from the Roy Butler Professorship grant at the Petroleum

Engineering Department, Texas Tech University is gratefully acknowledged. Special

thanks to Drs. Teddy P. Oetama, Lloyd R. Heinze, Akanni S. Lawal, and James F. Lea

for their inspiration and support during the course of this work. Special thanks go to my

lovely wife, Temitayo for proof-reading the initial draft of this work.

iii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS…………………….………………………….…….……....ii

ABSTRACT………………….……………….……………………….………….……...vi

LIST OF TABLES………………….………………………………...……….…............vii

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

LIST OF ABBREVIATIONS………………….…………………….……………........xiii

CHAPTER

I. INTRODUCTION……………………….…………………….………………….1

II. CONVENTIONAL TRANSIENT RESPONSE

SOLUTIONS…………………………………………………….………………..8

2.1 Vertical Fracture Model ……………….……………………..….……………9

2.2.1 Asymptotic Forms of the Vertical

Fracture Solution ……………………………………..………...…..……15

2.1.2 Wellbore Boundary Conditions……………………….……..…….20

2.2 Horizontal Fracture Model ………………………………………….….……25

2.2.1 Special Case Approximations……………………………….……..29

2.2.2 Asymptotic Forms of the Horizontal

Fracture Solution…………………………………………….…….……..33

2.3 Horizontal Wells……………………………………………………………..36

iv

2.3.1 Asymptotic Forms of the Horizontal

Well Solution…………………………………………………………….39

2.3.2 Computation of Horizontal Well Response………..………………42

III. MODEL DEVELOPMENT……………………………………………..……….44

3.1 Uniform-Flux Solid Bar Source Solution…………………………..………..44

3.2 Transient-State Behavior of the Solid Bar Source

Solution…………………………………………………….…………….….…...53

3.3 Asymptotic Behavior of the Solid Bar Source Solution………………..……59

IV. APPLICATION OF THE SOLID BAR SOURCE

SOLUTION TO HYDRAULIC FRACTURES AND

LIMITED ENTRY WELLS……………………………………………….……64

4.1 Vertical Fracture System……………………………………….…………….65

4.2 Horizontal Fracture System…………………………………….……………69

4.2.1 Asymptotic Forms of the Horizontal Fracture Solution………………71

4.2.2 Discussion of Horizontal Fracture Pressure Response…………….….73

4.3 Limited Entry Wells…………………………………………………………82

V. APPLICATION OF THE SOLID BAR SOURCE

SOLUTION TO HORIZONTAL WELLS………………………………..……..85

5.1 Mathematical Model…………………………………………………………86

5.2 Asymptotic Forms of the Solid Bar Source

v

Approximation for Horizontal Wells…………………………………………….92

5.3 Computation of Horizontal Wellbore Pressure ………………………….…..93

5.4 Effect of Dimensionless Radius on Horizontal

Well Response………………………………………………………………….101

5.5 Effect of Dimensionless Height on Horizontal

Well Response………………………………………………………………….104

5.6 The Concept of Physically Equivalent Models (PEM)…………………….109

VI. CONCLUSIONS…………………………..………………………….………..117

BIBLIOGRAPHY………………………………………..….………………………….119

APPENDIX

A. APPLICATION OF GREEN’S FUNCTIONS FOR

THE SOLUTION OF BOUNDARY-VALUE PROBLEMS……………………....123

B. HYDRAULIC FRACTURE/HORIZONTAL WELL

TYPE CURVES……………………………………………………………….……131

vi

ABSTRACT

Conventional horizontal well transient response models are generally based on the

line source approximation of the partially penetrating vertical fracture solution1. These

models have three major limitations: (i) it is impossible to compute wellbore pressure

within the source, (ii) it is difficult to conduct a realistic comparison between horizontal

well and vertical fracture transient pressure responses, and (iii) the line source

approximation may not be adequate for reservoirs with thin pay zones. This work

attempts to overcome these limitations by developing a more flexible analytical solution

using the solid bar approximation. A technique that permits the conversion of the

pressure response of any horizontal well system into a physically equivalent vertical

fracture response is also presented.

A new type curve solution is developed for a hydraulically fractured and

horizontal well producing from a solid bar source in an infinite-acting. Analysis of

computed horizontal wellbore pressures reveals that error ranging from 5 to 20%

depending on the value of dimensionless radius (rwD) was introduced by the line source

assumption. The proposed analytical solution reduces to the existing fully/partially

penetrating vertical fracture solution developed by Raghavan et al.1 as the aspect ratio

aspect ratio (m) approached zero (m ≤ 10-4), and to the horizontal fracture solution

developed by Gringarten and Ramey2 as m approaches unity. Our horizontal fracture

solution yields superior early time (tDxf < 10-3) solution and improved computational

vii

efficiency compared to the Gringarten and Ramey’s2 solution, and yields excellent

agreement for tDxf ≥ 10-3.

A dimensionless rate function (β -function) is introduced to convert the pressure

response of a horizontal well into an equivalent vertical fracture response. A step-wise

algorithm for the computation of β -function is developed. This provides an easier way of

representing horizontal wells in numerical reservoir simulation without the rigor of

employing complex formulations for the computation of effective well block radius.

viii

LIST OF TABLES

3.1.1: Dimensionless Pressure, pD for a Reservoir Producing from a Fully Penetrating Solid Bar Source Located at the Center of the Reservoir (Uniform-Flux Case)…………………………...……….57 3.1.2: Dimensionless Derivative, p’D for a Reservoir Producing from a Fully Penetrating Solid Bar Source Located at the Center of the Reservoir (Uniform-Flux Case)…………………………….………58 5.3.1: Influence of Computation Point on pwD for Horizontal Well - Infinite Conductivity Case (LD=0.05, zwD=0.5)……………………………...…..100 5.5.1: Effect of hfD on pwD for Horizontal Well-Infinite Conductivity Case (LD=0.05, rwD=10-4, zwD=0.5)………………………………………………108

ix

LIST OF FIGURES

2.1.1: Front View of Vertical Fracture Model……………………………………...……10

2.1.2: Plan View of Vertical Fracture Model………………………………………….…10

2.1.3: A Typical Vertical Fracture Wellbore Pressure Response Uniform flux Case (LD = 5.0, zD = 0.5, zwD = 0.5)…………………...…13 2.1.4: A Typical Vertical Fracture Wellbore Pressure Response Uniform flux Case (LD = 5.0, zD = 0.5, zwD = 0.5)………………….…..14 2.2.1: Front View Cross-Section of Horizontal Fracture Model………………………………………………………………………….…..26 2.2.2: Plan View Cross-Section of Horizontal Fracture Model………………………………………………………………………….…..26 2.2.3: A Typical Horizontal Fracture Wellbore Pressure Response Uniform flux Case (zD = 0.5, zwD = 0.5)…………………………..……31 2.2.4: A Typical Horizontal Fracture Wellbore Derivative Response Uniform flux Case (zD = 0.5, zwD = 0.5)………………………….…….32 2.3.1: Schematic of the Horizontal Well-Reservoir System……………………………..36

2.3.1: A Typical Horizontal Wellbore Pressure Response – Infinite Conductivity (zD = 0.5, zwD = 0.5)…………………………….……….….38 3.1.1: Cartesian coordinate system (x, y, z) of the Solid Bar Source Reservoir ………………………………………………..….…..45 3.1.2: Front View of the Solid Bar Source Reservoir System…………………………………………………………………………….46 3.1.3: Side View of the Solid Bar Source Reservoir System…………………….………46

3.1.4: Transient Response of a Fully Penetrating Solid Bar Source (Uniform-Flux Case)…………………………………………….………..54 3.1.5: Derivative Response of a Fully Penetrating Solid Bar Source (Uniform-Flux Case)………………………………………………...……55

x

4.1.1: Cartesian coordinate system (x, y, z) of the Vertically Fractured Reservoir ………………………………….….……………..66 4.1.2: Front View of the Vertically Fractured Reservoir………………….……………..67 4.1.3: Side View of the Vertically Fractured Reservoir ………………….….…………..67 4.2.1: Cartesian coordinate system (x, y, z) of the Horizontal Fracture System…………………………………………….…………69 4.2.2: Front View of Horizontal Fracture System………………………………………. 70 4.2.3: Side View of the Horizontal Fracture System………………………….…………70 4.2.4: Comparison of the Horizontal Fracture Solution Using the Solid Bar Source Solution Versus Gringarten et al…………………………………………………………………….76 4.2.5: Horizontal Fracture Type-Curve Solution Using the Solid Bar Source Solution…………………………………..………….77 4.2.6: Semi-Log Plot of Horizontal Fracture Solution Using the Solid Bar Source Solution…………………………….………78 4.2.7: Comparison of Horizontal Slab Source versus Vertical Slab Source Solutions…………………………………………….79 4.2.8: Illustration of the Effect of Dimensionless Height on Horizontal Fracture Pressure Response…………………………..……80 4.2.9: Illustration of the Effect of Dimensionless Height on Horizontal Fracture Derivative Response………………………...……81 5.1.1: Cartesian Coordinate System (x, y, z) of the Horizontal Well System………………………………………………….…..……87 5.1.2: Front View of the Solid Bar Source Reservoir System………………….…..……87

5.1.3: Side View of the Solid Bar Source Reservoir System……………………………88

5.1.4: Illustration of the Pressure Profile in a Horizontal Well…………………….……94

xi

5.3.1: Pressure Response for Horizontal Well - Infinite Conductivity Case (rwD = 10-4)……………………………………………96 5.3.2: Derivative Response for Horizontal Well – Infinite Conductivity Case (rwD = 10-4)……………………………………………97 5.3.3: Pressure Response for Horizontal Well – Infinite Conductivity Case (rwD = 5x10-4)…………………………………………98 5.3.4: Derivative Response for Horizontal Well – Infinite Conductivity Case (rwD = 5x10-4)…………………………………………99 5.4.1: Effect of rwD on the Transient Pressure Behavior of Horizontal Wells-Uniform Flux………………………………………………102 5.4.2: Effect of rwD on the Derivative Response of Horizontal Wells-Uniform Flux……………………………………….…………103 5.5.1: The Effect of Number of Term ‘n’ on the Line Source Approximation As hfD Approaches Zero ……………………..…………106 5.5.2: Effect of hfD on Transient Pressure Behavior of Horizontal Wells-Infinite Conductivity…………………………….……………107 5.6.1: Base Model (Slab Source)…………………………………………..….………..110 5.6.2: Primary Model (Solid bar Source)………………………………….……………110 5.6.1: Log-Log Plot of β -Function vs. tD – Uniform Flux………………….………….115 5.6.2: Composite Plot for a Pair of PEM…………………………………….…………116

B.1: Type-Curve for a Uniform Flux Horizontal Fracture System (hfD = 0.1, zwD = 0.5, zfD = 0.5, m = 1.0)……………………...……………131 B.2: Type-Curve for a Uniform Flux Horizontal Fracture System (hfD = 0.2, zwD = 0.5, zfD = 0.5, m = 1.0)………………………...…………132 B.3: Type-Curve for a Uniform Flux Horizontal Fracture System (hfD = 0.4, zwD = 0.5, zfD = 0.5, m = 1.0)…………………………...………133 B.4: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (hfD = 0.6, zwD = 0.5, zfD = 0.5, m = 1.0)…………………………134

xii

B.5: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (hfD = 0.8, zwD = 0.5, zfD = 0.5, m = 1.0)……………….…………135 B.6: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (hfD = 1.0, zwD = 0.5, zfD = 0.5, m = 1.0)………………….………136 B.7: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (LDxf=0.05, zwD=0.5, zfD=0.5, m=1.0)……………………………137 B.8: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (LDxf=0.1, zwD=0.5, zfD=0.5, m=1.0)……………………..………138 B.9: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (LDxf=0.25, zwD=0.5, zfD=0.5, m=1.0)……………………………139 B.10: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (LDxf=0.75, zwD=0.5, zfD=0.5, m=1.0)…………………..………140 B.11: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (LDxf=1.0, zwD=0.5, zfD=0.5, m=1.0)……………………………141 B.12: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (LDxf=2.5, zwD=0.5, zfD=0.5, m=1.0)……………………………142 B.13: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (LDxf=5.0, zwD=0.5, zfD=0.5, m=1.0)……………………………143 B.14: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (LDxf=10.0, zwD=0.5, zfD=0.5, m=1.0)…………………..………144

xiii

LIST OF ABBREVIATIONS

B = base matrix, defined in Equation 5.6.7

ct = total compressibility, psi-1 [kpa-1]

F’, F,

F1 and F2 = defined in Equations 2.1.15, 2.2.13, 3.3.10 and 3.3.19

respectively

h = reservoir thickness, ft [m]

hfD = dimensionless fracture thickness

hf = fracture thickness, ft [m]

Io = modified Bessel function of the first kind of order zero

k = horizontal permeability, md

kj = permeability in the j-direction, j = x, y, z , md

Ko = modified Bessel function of the second kind of order zero

L = horizontal well length, ft [m]

LD = dimensionless well length, ft [m]

LDrf = dimensionless time based on fracture radius, rf

LDxrf = dimensionless time based on fracture half length, 0.5xf

m = aspect ratio

M = positive integer

n = positive integer

P = primary matrix, defined in Equation 5.6.8

xiv

p = pressure, psia [kpa]

pD = dimensionless pressure

pi = initial reservoir pressure, psia [kpa]

pwD = dimensionless wellbore pressure

)t,r(p DrfD = P-Function in radial coordinate

)t,y,x(P DxfDD = P-Function in Cartesian coordinate

q = flow rate, STB/D [stock-tank m3/d]

r = radial distance, ft [m]

rf = fracture radius, ft [m]

rw = wellbore radius, ft [m]

rwD = dimensionless wellbore radius

Dr = defined in Equations 2.1.25, 3.3.11 and 3.3.20

s = Laplace variable

)t,z,y,x(S DDDD = defined in Equation 5.1.4

t = time, hours or days

Dit = defined in Equation 5.6.4

tD = dimensionless time

tDrf = dimensionless time based on fracture radius, rf

tDxf = dimensionless time based on fracture half length, 0.5xf

wf = fraction half width, ft [m]

x = distance in the x-direction, ft [m]

xD = dimensionless distance in the x-direction

xv

xf = fracture length, ft [m]

y = distance in the y-direction, ft [m]

yD = dimensionless distance in the y-direction

yf = fracture width , ft [m]

z = distance in the z-direction, ft [m]

zD = dimensionless distance in the z-direction

xw, yw, zw = well location in the x, y, and z-directions, respectively, ft

[m]

zwD = dimensionless well location

β = see Equations 2.1.11 and 2.3.9

)t( Dβ = beta-function

ξ = truncation error

jη = diffusivity constant, j = x, y, z

µ = fluid viscosity, cp [mpa.s]

)y,x(

),y,x(

),y,x(

DD2

DD1

DD

σσσ

= defined in Equations 2.2.25, 3.3.9 and 3.3.18, respectively

φ = formation porosity

jθ = weight fraction

)t,z,z,h,L(Z DxfDfDfDDxf = Z -Function

1

CHAPTER 1

INTRODUCTION

Hydraulically fractured wells and horizontal well completions are intended to

provide a larger surface area for fluid withdrawal and thus, improve well productivity.

This increase in well productivity is usually measured in terms of negative skin generated

as a result of a particular completion type. Hydraulic fractures leading to horizontal or

vertical fractures could produce the same negative skin effect as a horizontal well, but

possibly different transient pressure response; hence, having a good understanding of the

transient behavior of hydraulic fractures systems and horizontal well completion is very

vital for accurate interpretation of well test data.

The orientation of hydraulic fractures is dependent on stress distribution. The

orientation of fracture plane should be normal to the direction of minimum stress. Since

most producing formations are deep, the maximum principle stress is proportional to the

overburden load. Thus, vertical fractures are more common than horizontal fractures. The

only difference between a vertical and a horizontal fracture system is the orientation of

the fracture plane; a vertical fracture can be viewed as parallelepiped with zero width,

while a horizontal fracture, as a parallelepiped with zero fracture height. This same

argument can be extended to horizontal well completions; a horizontal wellbore can be

viewed as a parallelepiped with the height and width equal to the wellbore diameter. This

configuration makes a horizontal well completion behavior like a coupled fracture system

made up of both vertical and horizontal fracture systems. Considering the similarity in

2

the physical models, one will expect a single analytical solution can be developed for

hydraulically fractured (vertical and/or horizontal) well and horizontal well completions.

The primary purpose of this work is to present a general analytical solution for describing

the transient pressure behaviors of (i) vertical fracture system, (ii) horizontal fracture

system, and (iii) horizontal well or drainhole. New physical insights of the critical

variables that govern the performance of these completions are also provided.

Until now, different analytical solutions have been developed for vertical and

horizontal fracture systems using different source functions. A vertically fractured well is

viewed as a well producing from a slab source with zero fracture width1, while a

horizontal fracture is viewed a well producing from a solid cylinder source2. This

approach to hydraulic fracture system fails to establish a link between the transient

behaviors of hydraulic fracture systems. Each fracture system is treated as a separate

system producing from a different source. An analytical solution for a well with a single

horizontal, uniform-flux fracture located at the center of a formation with impermeable

upper and lower boundaries in an infinite reservoir system was presented in Ref. 2. The

authors observed that for certain configuration of horizontal fracture system

(dimensionless length, hD > 0.7), the transient pressure response of horizontal fracture is

indistinguishable from that of a vertically fractured well. This observation provided one

of the most compelling evidence of the existence of a gap in the knowledge of fractured

well behavior. In Chapter II of this report, a detailed review of the physical and analytical

models for describing the transient pressure response of vertical fracture, horizontal

3

fracture, and horizontal well will be presented. The aim of this chapter is preparing a

platform upon which the methodology employed in Chapters III to V is based.

Our attempt to eliminate this gap that exist in the correlation of the transient

behavior of hydraulically fractured well and fracture orientation can be resolved if one

examines a more general/flexible physical model. Thus, in Chapter III of this work, a

general and flexible physical model is developed. Any hydraulic fracture system can be

obtained from this proposed physical model by reducing the model into a special case

configuration. Based of the aspect ratio (m) defined as the ratio of fracture width (yf) to

fracture length (xf), three special case configurations were considered in Chapter IV: (i)

vertical fracture system when the m-value is zero, (ii) horizontal fracture system when the

m-value is greater than zero, and (iii) partially penetrating vertical wells or limited entry

wells. This approach combines the vertical and horizontal fracture analytical solutions

into one single solution. The development of a single analytical model for describing the

transient behavior of both vertical and horizontal fractures provides addition knowledge

about the relationship between the two fracture systems. Although, some of the solutions

presented in Chapter III do not directly pertain to horizontal well analysis, Chapter III

provides information and new insights of the variables that govern horizontal well

performance.

The importance issue presented in Chapter V is the extension of the mathematical

model developed for hydraulic fracture systems to horizontal well configuration.

Conventional models for horizontal well test analysis were mostly developed during the

1980s. The rapid increase in the applications of horizontal well technology during this

4

period led to a sudden need for the development of analytical models capable of

evaluating the performance of horizontal wells. Ramey and Clonts3 developed one of the

earliest analytical solutions for horizontal well analysis based on the line source

approximation of the partially penetrating vertical fracture solution. The conventional

models 4-16 assume that a horizontal well may be viewed as a well producing from a line

source in an infinite-acting reservoir system. These models have three major limitations:

(i) it is impossible to compute wellbore pressure within the source, so wellbore pressure

is computed at a finite radius outside the source, (ii) it is difficult to conduct a realistic

comparison between horizontal well and vertical fracture productivities, because,

wellbore pressures are not computed at the same point, (iii) the line source approximation

may not be adequate for reservoirs with thin pay zones.

The increased complexity in the configuration of horizontal well completions and

applications towards the end of the 1980s made us question the validity of the horizontal

well models and the well-test concepts adopted from vertical fracture analogies. In the

beginning of the 1990s a new development in horizontal-well solutions17-27 under more

realistic conditions emerged. As a result, some contemporary models were developed to

eliminate the limitations of the earlier horizontal well models. However, the basic

assumptions and methodology employed in the development of the new solutions have

remained relatively the same as those of the earlier models. Ozkan28 presented one of the

most compelling arguments for the fact that horizontal wells deserve genuine models and

concepts that are robust enough to meet the increasingly challenging task of accurately

evaluating horizontal well performance. Ozkan’s work presented a critique of the

5

conventional and contemporary horizontal well test analysis procedures with the aim of

establishing a set of conditions when the conventional models will not be adequate and

the margin of error associated with these situations. This work attempts to overcome the

basic limitations of the classical horizontal well model by modifying the source function.

A horizontal well is visualized as a well producing from a solid bar source rather that the

line source idealization. The new source function allows the computation of wellbore

pressure within the source itself and not at a finite radius outside the source

In Chapter V, a special case approximation for horizontal well is obtained from

the physical model proposed in Chapter III by assuming that a horizontal wellbore can be

viewed as a parallelepiped with the height and width equal to its wellbore diameter. The

most distinctive flow characteristic of this model is that fluid flows into the wellbore in

both y- and z-directions to produce the well with a constant total rate. This flow

characteristic makes a horizontal well act like a coupled fracture system at early time; the

combination of both horizontal and vertical fracture flow characteristics leads to the

distinctive early time flow behavior of horizontal wells. since conventional horizontal

well models visualize a horizontal well as a well producing from a line source, it is

impossible to compute the pressure drop within the source; hence, wellbore pressure has

to be computed at a finite radius outside the source. Thus, consideration must be given to

the following two factors in the choice of computation point for horizontal wells: (i)

unlike vertically fractured wells, the horizontal well response is a function of rwD.

Therefore ignoring the effect of wellbore radius in vertically fractured wells is

acceptable, since the wellbore radius is significantly smaller than the distance to the

6

closest boundary; this is not the case in horizontal wells. The proximity of the wellbore to

the boundary in the z-direction makes the effect of wellbore radius more critical in

horizontal wells, and (ii) the pressure outside the source is higher than the pressure inside

the source. Therefore, computing the wellbore pressure at a finite radius outside the

source could lead to a significant error depending on the value of rwD. Unlike

conventional horizontal well models, it is possible to compute wellbore pressure response

inside the source using the horizontal well solution developed in Chapter V. However, it

can be readily decided when the line-source assumption for the finite-radius horizontal

well becomes acceptable; at this point the error introduced in the definition of the

wellbore-pressure measurement point would not have a significant impact on the

accuracy of the results.

The later part of Chapter V was devoted to the effect of dimensionless height, hfD

on the transient response horizontal well especially in thin reservoir. The line source

idealization views a horizontal well as a vertical-fracture where the fracture height

approaches zero in the limit of the Z-function. Clonts and Ramey3 were one of the first

authors to impose this limit on the horizontal well solution. A simple numerical

experiment will be conducted using values of hfD that are likely to be encountered in

practice to validate the applicability of the line source assumption to horizontal well

solutions.

Another aspect of horizontal well technology that has evolved dramatically over

the years is the representation of a horizontal well in numerical reservoir simulation. The

challenge in this area is the accurate formulation of the relationship between wellblock

7

and wellbore pressure in numerical simulation of horizontal wells. In 1983 Peaceman29

published a formulation which provided an equation for calculating effective well-block

radius (ro) when the well block is a rectangle and/or the formation is anisotropic. This

equation was initially developed for vertical wells, and later was modified for horizontal

wells by interchanging x∆ and z∆ , as well as kx and ky. Odeh30 proposed an analytical

solution for computing the effective well-block radius using the horizontal IPR earlier

published by Odeh and Babu31. Prior to Odeh’s formulation, no method was available in

the literature to test the applicability of Peaceman's formulation to horizontal wells. Odeh

pointed out that the Peaceman formulation is not always applicable to horizontal well

simply by interchanging the variables; this is due to the fact that horizontal well

configurations almost always violate the assumption of isolated well, where the well

location is sufficiently far from the boundaries. In a later publication, Peaceman32

revisited his previous formulations in order to stress the effects of the inherent

assumptions made on their applicability to horizontal wells. Two major assumptions were

highlighted in his review: (i) uniform grid size, and (ii) the concept of isolated well

location. The range of configurations when the Peaceman’s formulation yields the well

pressure within 10% error relative to Odeh’s formulation was established. Peaceman

pointed out in his discussion of Odeh’s work that his formulated effective well-block

radius should divided by a scaling factor. This notion was also shared by Brigham33. To

compare the pressure response in hydraulically fractured versus horizontal wells; we

introduce the concept of physically equivalent models (PEM), which is explained in

details in Chapter IV. Two models are said to be physically equivalent if both models

8

produce identical transient pressure behaviors under the same reservoir conditions. The

implementation of PEM concept led us to find a combination of dimensionless rates: β -

function, for which a slab source solution produces the same pressure drop as a solid bar

solid source. This provides an easier way of representing horizontal wells in numerical

reservoir simulation without the rigor of employing complex formulations for the

computation of effective wellbore radius.

Although there have been many models developed for analyzing vertical fracture

systems, horizontal fracture systems, and horizontal wells. No single model is capable of

analyzing both vertical and horizontal fracture systems as well as horizontal wells. Hence

the objectives of this research are to:

1. Develop a single analytical model capable of describing the transient response of

the following models

a. Fully/partially penetrating vertical fracture system,

b. Horizontal fracture system,

c. Limited entry well,

d. Horizontal well.

2. Attempt to overcome the limitations of the line source solution by developing a

more robust horizontal well model using the solid bar source solution

3. Develop a technique for converting the transient-response of a horizontal well

into an equivalent vertical fracture response.

4. Develop a technique for comparison of vertical fracture and horizontal well

pressure responses

9

CHAPTER II

CONVENTIONAL TRANSIENT RESPONSE SOLUTIONS

This chapter takes a critical look at both the physical and mathematical model of

hydraulic fracture and horizontal well systems using already developed techniques and

logic. Three major configurations will be examined in the chapter namely:

a. Fully penetrating vertical fracture configuration,

b. Horizontal fracture configuration,

c. Horizontal well configuration.

The main focus of this section is to highlight the pertinent similarities and differences

between the physical and the analytical models of these three configurations as well as to

present many of the solutions that will be use later in Chapters III and IV.

2.1 Vertical Fracture Model

This section presents the physical and the analytical models employed in

development of the vertical fracture solution in Ref. 1. The most pertinent characteristic

of this analytical model lies is that it can easily be reduced to the line source solution for

horizontal wells. Hence, a lot of similarities exist between this solution and the line

source approximation for horizontal wells.

The physical model leading to the development of the vertical fracture solution is

presented in Figures. 2.1.1 and 2.1.2. The most critical assumption in the model is that

10

the fracture thickness is negligible; hence, there is no flow into the fracture in the z-

direction.

Figure 2.1.1: Front View of Vertical Fracture Model

Figure 2.1.2: Plan View of Vertical Fracture Model

zf

0.5xf

hf

0=∂∂

=hzz p

z

x

0=∂∂

=0zz p

h

-xf

y

x

+xf

Infinite Conductivity or

Uniform Flux

11

The general solution for a fully/partially penetration vertical fracture system is

given as follows

}Dxf

DxfDxfDfDfDDxf

t

0 Dxf

2D

Dxf

Dx

Dxf

Dx

y

DxfDDDD

t

dt)t,z,z,h,L(Z

t4

yexp

t2

xk

k

erft2

xk

k

erfk

k

4

)t,z,y,x(p

Dxf

•

−•

−

++

π

=

∫ (2.1.1)

Where:

[ ])t,z,y,x(ppqB2.141

kh)t,z,y,x(p iDxfDDDD −µ

= (2.1.2)

2ft

Dxf xc

kt001056.0t

µφ= (2.1.3)

xf

wD k

k

x

)xx(2x

−= (2.1.4)

yf

wD k

k

x

)yy(2y

−= (2.1.5)

h

zzD = (2.1.6)

h

hh f

fD = (2.1.7)

zf

Dxf k

k

x

h2L = (2.1.8)

12

( ) ( ) ( )

πππ

π−π

+

=

∑∞

=1nwDDfD2

Dxf

Dxf22

fD

DxfDfDfDDxf

zncoszncoshn5.0sinL

tnexp

n

1

h

0.41

)t,z,z,h,L(Z

(2.1.9)

The function )t,z,z,h,L(Z DxfDfDfDDxf , called Z-function, is proportional to the

instantaneous source function for an infinite slab reservoir with impermeable boundaries.

The Z-function accounts for the partial penetration of the slab source. For a fully

penetrating source, the Z-function is unity. Figures 2.1.3 and 2.1.4 illustrate a typical

wellbore pressure response and derivative response, respectively, for a fully/partial

penetrating vertical fracture system.

13

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03

Dimensionless Time, tDxf

Dim

ension

less

Pre

ssur

e, p

wD

1.00E-01

1.00E+00

1.00E+01

1.00E+021.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

hfD=0.1

0.5

1.0

0.2

Figure 2.1.3: A Typical Vertical Fracture Wellbore Pressure Response Uniform flux Case (LD = 5.0, zD = 0.5, zwD = 0.5)

14

1.0E-01

1.0E+00

1.0E+01

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03


Dim

ension

less

Pre

ssur

e, p

wD

1.00E-01

1.00E+00

1.00E+011.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

hfD=0.1

0.5

1.0

0.2

Figure 2.1.4: A Typical Vertical Fracture Wellbore Pressure Response Uniform flux Case (LD = 5.0, zD = 0.5, zwD = 0.5)

15

2.1.1 Asymptotic Forms of the Vertical Fracture Solution

Short- and long-time approximations of Equation 2.1.1 can be derived using

methods similar to those given in Ref. 1. The main goal of obtaining the asymptotic

forms of the vertical fracture solution is relate the behaviors of the physical model to that

of the mathematical model. If the behavior of the mathematical model is consistent with

that of the physical model physical, the analytical solution is said to a physically

consistent solution.

a. Short-Time Approximation:

Assuming a fully penetrating vertical fracture system (h = hf), Equation 2.1.1

becomes

Dxf

Dxf

t

0 Dxf

2D

Dxf

Dx

Dxf

Dx

y

DxfDDDD

t

dt

t4

yexp

t2

xk

k

erft2

xk

k

erfk

k

4

)t,z,y,x(p

Dxf

∫

−•

−

++

π

=

(2.1.10)

At short time,

β=−

++

Dxf

D

x

Dxf

D

x

t2

xk

k

erft2

xk

k

erf (2.1.11)

Where

>

=

<

=β

xD

xD

xD

kkxfor 0

kkxfor 1

kkxfor 2

(2.1.12)

16

Substituting Equation 2.1.11 into Equation 2.1.10 and assuming that Equation

2.1.12 is satisfied we get

Dxf

Dxft

0 Dxf

2D

yDxfDDDD

t

dt

t4

yexp

k

k

4)t,z,y,x(p

Dxf

∫

−πβ= (2.1.13)

Integrating Equation 2.1.13 with respect to tDxf we get

f

Dxf

DD

Dxf

2D

Dxfy

DxfDDDD

hh for t2

yerfcy

2t4

yexpt

k

k

2

)t,z,y,x(p

=

−π−

−πβ

= (2.1.14)

Equation 2.1.14 represents a vertical linear flow into the fracture at early time. For

a fully penetrating vertical fracture system, the duration of the linear flow is

limited by the distance from the pressure point to 0.5xf.

For a partially penetrating vertical fracture system (h > hf), the short-time

approximation for Equation 2.1.9 developed by Gringarten and Ramey2 will be

utilized.

fD

DxfDfDfDDxf h

1)t,z,z,h,L(Z ≈ (2.1.15)

Substituting Equations 2.1.11 and 2.1.15 into Equation 2.1.1, we obtain,

Dxf

Dxft

0 Dxf

2D

yfDDxfDDDD

t

dt

t4

yexp

k

k

h4)t,z,y,x(p

Dxf

∫

−πβ= (2.1.16)

Integrating Equation 2.1.16 with respect to tDxf we get

f

Dxf

DD

Dxf

2D

DxfyfD

DxfDDDD

hhfor t2

yerfcy

2t4

yexpt

k

k

h2

)t,z,y,x(p

>

−π−

−πβ

= (2.1.17)

17

Equation 2.1.17 represents a horizontal linear flow into the partially penetrating

fracture at early time. The duration of which is limited by the distance of the

fracture from the closest upper or lower boundary and the distance from the

pressure point to 0.5xf.

b. Long-Time Approximation:

For an anisotropic reservoir system, Equation 2.1.1 may be expressed as follows,

'DfD Fpp += (2.1.15)

Where:

Dxf

Dxf

t

0 Dxf

2D

Dxf

Dx

Dxf

Dxy

Df

t

dt

t4

yexp

t2

xkkerf

t2

xkkerf

4kk

p

Dxf

∫

−•

−+

+π

=

(2.1.16)

and

[ ]}Dxf

DxfDxfDfDfDDxf

t

0 Dxf

2D

Dxf

Dx

Dxf

Dxy

'

t

dt1)t,z,z,h,L(Z

t4

yexp

t2

xkkerf

t2

xkkerf

4kkF

Dxf

−

•

−•

−+

+π= ∫ (2.1.17)

Substituting Equation 2.1.9 into 2.1.17, we get

( ) ( ) ( )Dxf

Dxf

1nwDDfD2

Dxf

Dxf22

fD

t

0 Dxf

2D

Dxf

Dx

Dxf

Dxy

'

t

dtzncoszncoshn5.0sin

L

tnexp

n

1

h

0.4

t4

yexp

t2

xkkerf

t2

xkkerf

4kkF

Dxf

πππ

π−π

•

−•

−+

+π=

∑

∫∞

=

(2.1.18)

Recall,

18

( )α

α−π

=−

++

∫+

−

dt4

kkxexp

tkk

t2xkk

erft2

xkkerf

1

1 Dxf

2

xD

Dxf

x

Dxf

Dx

Dxf

Dx

(2.1.19)

Substituting Equation 2.1.19 into Equations 2.1.16 and 2.1.18, we get

( )Dxf

t

0

1

1 Dxf

2D

2

xDDf dtd

t4

ykkxexp

4

1p

Dxf

α

−α−= ∫ ∫

+

−

(2.1.20)

and

( ) ( ) ( )

( )Dxf

Dxf

t

0

1

1 Dxf

2D

2

xD

2Dxf

Dxf22

1nwDDfD

fD

'

t

dtd

t4

ykkxexp

L

tnexp

zncoszncoshn5.0sinn

1

h

1F

Dxf α

−α−

π−

ππππ

=

∫ ∫

∑+

−

∞

= (2.1.21)

Revising the integral in Equation 2.1.20, we get

( )α

−α−−= ∫

+

−

dt4

ykkxEi

4

1p

1

1 Dxf

2D

2

xDDf

(2.1.22)

Replacing the Ei(-x) function in the right hand side of Equation 2.1.22 by the

logarithmic approximation suitable for small values of its argument (large time),

then we can write the long time approximation of the fracture solution as

( ) α

+

−α−= ∫

+

−

d80907.0ykkx

tln

4

1p

1

12D

2

xD

DxfDf

(2.1.23)

To evaluate the long time approximation of Equation 2.1.18, we transform

Equation 2.1.18 into Laplace space and find the limit as the Laplace variable (s)

19

tends to zero. Taking the Laplace transform of Equation 2.1.18 with respect to

tDxf, we obtain

( )

( ) ( ) ( ) α

π+ππππ

=

∫∑+

−

∞

=

dL

nsrKzncoszncoshn5.0sin

n

1

hs

2

sF

2Dxf

22

D

1

1

01n

wDDfDfD

'

(2.1.24)

Where:

( ) 2D

2

xDD ykkxr −α−= (2.1.25)

If

2Dxf

2 L/01.0s π≤ (2.1.26)

or

22DxfDxf /L100t π≥ (2.1.27)

We can assume that ( ) ( )2Dxf

222Dxf

22 L/nL/ns π≈π+ and the long time

approximation of Equation 2.1.24 is given by

( ) ( ) ( ) ( ) α

πππππ

= ∫∑+

−

∞

=d

L

nrKzncoszncoshn5.0sin

n

1

hs

2sF

Dxf

D

1

1

01n

wDDfD

fD

' (2.1.28)

Evaluating the inverse Laplace transform of Equation 2.1.28, we obtain the

following expression

( ) ( ) ( ) α

πππππ

= ∫∑+

−

∞

=d

L


n

1

h

2F

Dxf

D

1

1

01n

wDDfD

fD

' (2.1.29)

Using Equation 2.1.23 and 2.1.29, the long time approximation for a

fully/partially penetrating vertical fracture system can be written as

( ) )L,z,z,y,x(F)y,x(80907.2tln5.0)t,L,z,z,y,x(p

DwDDDDDDD

DDwDDDDD

+σ++= (2.1.30)

20

Where

( ) ( )[ ]{( ) ( )[ ]

( )[ ]}x2D

2DDD

2D

2

xDxD

2D

2

xDxDDD

kkyx/y2arctany2

ykkxlnkkx

ykkxlnkkx25.0)y,x(

−+−

+++−

+−−=σ

(2.1.31)

For

( )[ ]

+±

π≥

2D

2

xD

22Dxf

Dxfykkx25

/L100t (2.1.32)

Equation 2.1.30 represents a radial flow into the fracture system after the fracture

linear flow diminishes; the radial flow period is identified by a straight line with a

slope of 1.151 on the log-log plot of pD vs. tDxf. This is consistent with the

behavior of the physical model (i.e. vertically fractured wells)

2.1.2 Wellbore Boundary Conditions

Two major wellbore boundary conditions are considered in development of a

fully/partially penetrating vertical fracture solution namely; uniform-flux and infinite-

conductivity boundary conditions. The solution presented in Section 2.1.1 above assumes

the uniform-flux condition. For the uniform-flux case, the wellbore pressure was

computed at the center of the fracture (0, 0, zwD). For the infinite-conductivity case,

wellbore pressure is computed at the location of the x-coordinate at which the wellbore

pressure drop is the same as the uniform-flux case. This concept was first introduced in

Ref. 34. Gringarten et al.34 noted that once the stabilized flux distribution is attained, then

it is possible to find a point along the x-axis in the uniform-flux system at which the

21

pressure drops in the uniform-flux fracture and the infinite-conductivity fracture are be

the same. This point is usually referred to as the equivalent pressure point and is used to

obtain wellbore pressure of an infinite-conductivity well by using the solution developed

under the uniform-flux assumption. A unique solution for infinite-conductivity case may

be developed by repeating a similar procedure for all time, but Ref. 34 suggests that the

use of the equivalent point obtained during the stabilized flow period for all time would

not introduce a significant error.

The following procedure will summarize steps taken to obtain the stabilized flux

distribution and the determination of the equivalent pressure point for a fully/partially

penetrating vertical fracture solution.

Recall Equation 2.1.15 and assume kx = ky = k, Equation 2.1.16 and 2.1.17 respectively

become:

Dxf

Dxf

t

0 Dxf

2D

Dxf

D

Dxf

DDf

t

dt

t4

yexp

t2

x1erf

t2

x1erf

4p

Dxf

∫

−•

−++π= (2.1.33)

[ ]}Dxf

DxfDxfDfDfDDxf

t

0 Dxf

2D

Dxf

D

Dxf

D'

t

dt1)t,z,z,h,L(Z

t4

yexp

t2

x1erf

t2

x1erf

4F

Dxf

−

•

−•

−++π= ∫ (2.1.34)

Using the relation given in Equation 2.1.19, Equation 2.1.33 may be expressed as

( )α

−α−−= ∫

+

−

dt4

yxEi

4

1p

1

1 Dxf

2D

2D

Df (2.1.35)

Using Equation 2.1.19 and 2.1.35, Equation 2.1.15 can be written as

22

( )( ) ( ) ( ) α

πππππ

+

+

−α−=

∑

∫

∞

=

+

−

dL


n

1

h

2

80907.0ykkx

tln

4

1p

Dxf

D01n

wDDfD

fD

1

12D

2

xD

DxfD

(2.1.36)

If we divide the half length of the fracture ( )2/xf , into M equal segments, then the

pressure drop due to production from the mth uniform-flux element extending from

( )M2/mxf to ( )M2/x)1m( f− in the interval zero to ( )2/xf is given by:

( )( )

( ) ( ) ( ) α

πππππ

+

+

−α−=

∑

∫

∞

=

−

dL


n

1

h

2

80907.0ykkx

tln

4

1qp

Dxf

D01n

wDDfD

fD

M/m

M/1m2D

2

xD

DxfmD

(2.1.37)

Due to symmetry with respect to the center of the well, we consider another flux element

extending from ( )M2/mxf to ( )M2/x)1m( f− in the interval zero to - ( )2/xf yields a

pressure drop given by:

( )( )

( ) ( ) ( ) α

πππππ

+

+

−α−−=

∑

∫

∞

=

−

−−

dL


n

1

h

2

80907.0ykkx

tln

4

1qp

Dxf

D01n

wDDfD

fD

M/m

M/1m2D

2

xD

DxfmD

(2.1.38)

The pressure drop due to simultaneous production from the mth flux element in the

positive and negative x-direction is then obtained by the principle of superposition in

space. Applying this principle to Equation 2.1.37 and 2.1.38, we get

( )

( )

αα−αα= ∫ ∫

−

−

−−

M/m

M/m

M/1m

M/1m

mD d)(fd)(fqp (2.1.39)

23

Where:

( )( ) ( ) ( )

πππππ

+

+

−α−=α

∑∞

= Dxf

D01n

wDDfD

fD

2D

2

xD

Dxf

L


n

1

h

2

80907.0ykkx

tln

4

1)(f

(2.1.40)

Let

( )

( ) ( ) ( ) α

πππππ

+

+

−α−=

∑

∫

∞

=

−

dL


n

1

h

2

80907.0ykkx

tln

4

1p

Dxf

D01n

wDDfD

fD

M/m

M/m2D

2

xD

DxfDm

(2.1.41)

Considering all the flux elements along the fracture, the resulting pressure drop and the

resulting production rate from the total length of the fracture can be expressed,

respectively, as

( )∑=

−−=M

1m1DmDmmD ppqp (2.1.42)

and

f

M

1m

ffm qM

hxq =∑=

(2.1.43)

or

Mq

hxqM

1m f

ffm =∑=

(2.1.44)

If we now choose qm in Equation 2.1.42 such that pD would be approximately constants

along the surface of the fracture, then Equation 2.1.42 yields the pressure distribution due

to production from an infinite-conductivity vertical fracture system. In order to obtain qm

24

to be used in Equation 2.1.42, impose the wellbore boundary condition along the fracture

surface (yD = 0, zD = zwD) is set as such that the pressure drop measured in the middle of

the mth flux element be equal to that in the middle of (m+1)st flux element, that is:

[ ]1-M1,j t,0,M2

1j2xpt,0,M2

1j2xp DDwDDDwD =

+==

−= (2.1.45)

The resulting pressure drop from the total length of the fracture can be expressed


DwDDDDDDD

DDwDDDDD

+σ++= (2.1.46)

Where:

( )

( ) ( )

−+

−−+

−+

−++

−

+

−−

−−−

+

−+

−++

+

+

+−

+

−

−=σ ∑

=

2D22

2D

2D2

22D

2D

22D

2D

DD

2D

2

DD

2D

2

DD

2D

2

DD

M

1m

2D

2

DDDD1

yM

1mm4M

1mmyxMmyx

M1mmyx

My2arctany2

yM

1mxlnM

1mx

yM

1mxlnM

1mx

yMmxln

Mmx

yMmxln

Mmx25.0)y,x(

(2.1.47)

and

25

( ) ( ) ( )

( )

( )

α

π−

π

ππππ

=

∫∫

∑∑−

−−−

=

∞

=

dL

nrK

L

nrK

zncoszncoshn5.0sinn

1

h

2

)L,z,z,y,x(F

M/1m

M/1m Dxf

D0

M/m

M/m Dxf

D0

M

1m 1nwDDfD

fD

DwDDDD

(2.1.48)

Once the stabilized flux distribution, qm is obtained, the infinite conductivity solution can

be obtained by solving Equation 2.1.42. To find the equivalent pressure point, we

compute the pressure distribution along the surface of the fracture for a uniform flux

fracture system by assuming constant qm. The equivalent pressure point is the point at

which the uniform- flux and the infinite conductivity solutions cause the same pressure

drop. This point was computed by Gringarten et al. in Ref. 34 to be 0.732.

2.2 Horizontal Fracture Model

This section presents the physical and analytical models employed in the

developed of the horizontal fracture solution in Ref. 3. The most pertinent characteristic

of the analytical model lies in its solution can easily be reduced to the solution for limited

entry/partially penetrating wells. Hence, a lot of similarities exist between the solution of

this model and the line source approximation for limited entry/partially penetrating wells.

The physical model leading to the development of the horizontal fracture solution is

presented in Figures 2.2.1 and 2.2.2.

26

Figure 2.2.1: Front View Cross-Section of Horizontal Fracture Model

Figure 2.2.2: Plan View Cross-Section of Horizontal Fracture Model

zf

rf

hf

0=∂∂

=hzz p

z

r

0=∂∂

=0zz p

x +rf -rf

y

θ

rf

27

A cross-section of the idealized horizontal-fracture system is shown in Figures 2.2.1 and

2.2.2. The following assumptions are made:

1. The reservoir is horizontal, homogenous, and has anisotropic radial (kr) and

vertical permeabilities, kz.

2. Infinite-acting reservoir system completely penetrated by a well with radius (rw),

and the effect of rw is neglected, thus line-source solution applies

3. A single, horizontal, symmetrical fracture with radius (rf), and thickness (hf) is

centered at the well and the horizontal plane of symmetry of the fracture is at an

altitude (zf)

4. A single-phase, slightly compressible liquid flows from the reservoir into the

fracture at a constant rate qf , which is uniform over the fracture volume (uniform-

flux case)

5. There is no flow across the upper and lower boundaries of the reservoir, and the

pressure remains unchanged and equals to the initial pressure as the radial

distance (r) approaches infinity

The general solution for a fully/partially penetration horizontal fracture system is given as

follows

∫=Drft

0DrfDrfDfDfDDrfDrfDDrfDDD dt)t,z,z,h,L(Z)t,r(p2)t,z,r(p (2.2.1)

Where:

[ ])t,h,h,z,z,r,r(ppq

hk2)t,h,z,z,r(p fffi

f

rDDfDDDD f

−µ

π= (2.2.2)

28

2ft

rDrf rc

tktµφ

= (2.2.3)

f

D rrr = (2.2.4)

hzzD = (2.2.5)

h

hh f

fD = (2.2.6)

z

r

fDrf k

krhL = (2.2.7)

−

−

= ∫1

0

'D

'D

Drf

2'D

Drf

'DD

oDrf

Drf

2D

DrfD drrt4

rexp

t2

rrI

t2

t4

rexp

)t,r(p (2.2.8)

( ) ( ) ( )

πππ

π−

π+

=

∑∞

=1nwDDfD2

Drf

Drf22

fD

DrfDfDfDDrf

zncoszncoshn5.0sinL

tnexp

n1

h0.41

)t,z,z,h,L(Z (2.2.9)

Equation 2.2.8 is known as the P-function35, this expression is proportional to the

instantaneous source function for a solid cylinder source in an infinite-acting reservoir.

The pressure distribution created by a continuous cylinder source can be obtained by

integrating Equation 2.2.8 with respect to dimensionless time: tDrf and is shown as follow:

∫=Drft

0

DrfDrfDDrfDD dt)t,r(p2)t,r(p (2.2.10)

Equation 2.2.9 is called the Z-function2. This function is proportional to the instantaneous

function for an infinite horizontal slab source in an infinite-acting horizontal slab

reservoir with impermeable boundaries. It accounts for the partial penetration effect of

29

the solid cylinder source in the reservoir. For a fully penetrating solid cylinder source, Z-

function is unity.

2.2.1 Special Case Approximations

Two special case approximations of Equation 2.2.1 were considered by

Gringarten et al.2 namely:

I. Pressure distribution created by a horizontal fracture with zero thickness.

Taking the limit of the Z-function as hfD tends to zero yields the pressure distribution:

∫=Drft

0

DrfDrfDfDDrfDrfDDrfDDD dt)'t,z,z,0,L(Z)'t,r(p2)t,z,r(p (2.2.11)

Where:

( ) ( )

ππ

π−+

=

∑∞

=1nwDD2

Drf

Drf22

DrfDfDDrf

zncoszncosL

tnexp21

)t,z,z,0,L(Z

(2.2.12)

II. Pressure distribution created by a line-source well with partial penetration or limited

entry.

The pressure distribution is obtained from Equation 2.2.1 by taking the limit of the P-

function as rf approaches zero. The resulting expression is as follow (for r ≥ rf):

∫

−

=Drft

0DrfDrfDfDDrf

Drf

Drf

2D

DrfDDD 'dt)'t,z,z,0,L(Z't2

't4rexp

)t,z,r(p (2.2.13)

The horizontal fracture solution given in Equation 2.2.1 can also be written in a similar

way to the vertical fracture solution as follows:

30

)t,z,r()t,r(p)t,z,r(p DrfDDDrfDDrfDDD σ+= (2.2.14)

Where:

[ ]∫ −=σDrft

0DrfDrfDfDfDDrfDrfDDrfDD 'dt1)'t,z,z,h,L(Z)'t,r(p2)t,z,r( (2.2.15)

Equation 2.2.15 is called the “pseudo skin function”. This skin function represents

additional time-dependent pressure drop in a zone of finite radial distance. Figures 2.2.3

and 2.2.4 represent a typical horizontal fracture wellbore pressure response and derivative

response, respectively

31

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

Dimensionless Time, tDrf

Dim

ension

less

Pre

ssur

e, p

wD

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

LDrf=10.0

5.0

3.0

1.0

0.5

0.3

0.05

zD=0.5 zfD=0.5

Figure 2.2.3: A Typical Horizontal Fracture Wellbore Pressure Response Uniform flux Case (zD = 0.5, zwD = 0.5)

32

0.001

0.01

0.1

1

10

1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Der

ivat

ive

Res

pons

e, p

' wD

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

LDrf=10

5.0

3.0

1.0

0.3

0.05

Solid Bar Source Solution

Figure 2.2.4: A Typical Horizontal Fracture Wellbore Derivative Response Uniform flux Case (zD = 0.5, zwD = 0.5)

33

2.2.2 Asymptotic Forms of the Horizontal Fracture Solution


methods similar to those given in Ref. 2

a. Short Time Behavior:

The short-time behavior of the Equation 2.2.1 can be obtained by examining the short-

time behaviors of the P- and Z- functions. The short time behaviors of these functions

were described by Gringarten et al. in Ref 2 and are presented below. The P-function

becomes constant at early time, (± 1 percent) when Equation 2.2.16 and 2.2.17 are

satisfied. This constant is unity for 0 ≤ rD < 1, one half for rD = 1, and zero for rD > 1.

The P-Function is constant, when

( ) 1r ,20

r1t D

2D

Drf ≠−≤ (2.2.16)

Or in terms of real variable,

1r ,10t D4

Drf =π≤ − (2.2.17)

Hence, at early time flow occurs only in the 0 ≤ rD < 1 region and the pressure-drop

function: Equation 2.2.1, becomes

∫=Drft

0DrfDrfDfDfDDrfDrfDDD dt)t,z,z,h,L(Z2)t,z,r(p (2.2.18)

From Equation 2.2.18 we note that the pressure-drop function is independent of rD

at early time and indicates vertical linear flow into the fracture. The early-time

behavior of Equation 2.2.1 depends only on the form of the Z-function. Two cases

of the Z-function were considered in the Ref. 2:

34

I. Horizontal Fracture of Finite Thickness (hfD ≠ 0).

The pressure-drop distribution function above or below the fracture at early time

was shown in Ref. 2 to be equivalent to

δ−π

δ−

δ

δ+

=<≤

Drf

2DDrf

DD

D2D

DrffD

DrfDDD

t4expt

t2erfc

2t

h1

)t,z,1r0(p

(2.2.19)

The variable Dδ represents the dimensionless vertical distance from the pressure

point to the closest (upper or lower) horizontal face of the fracture. Equation 2.2.19

represents vertical linear flow into the fracture with a fracture storage effect caused

by the finite thickness of the fracture. The fracture storage constant is equal to hfD.

On the horizontal fracture faces ( Dδ =0) the pressure drop is one-half within the

fracture at early time. Therefore, at early time the only flow is within the fracture,

and is of a fracture storage type. A unit slope line is obtained when the pressure

drop is plotted against time on log-log coordinates. As time increases, the linear

vertical flow into the fracture become dominate, and a half slope line is obtained on

log-log coordinates. The length of this last straight line is limited by the distance

from the pressure point to the closest upper or lower boundary and the distance

from the pressure point to rf.

II. Plane Horizontal Fracture (hfD=0)

At early time, the pressure drop function can be expressed as follows

( )

−−−

−−

π

=<≤

Drf

DfDfDrf

Drf

2DfDrf

Drf

DrfDDD

t2zz

erfzzLt4zz

exptL2

)t,z,1r0(p (2.2.20)

35

Equation 2.2.20 represents a linear vertical flow without storage in the fracture, and a half

slope line will be obtained on log-log coordinates.

b. Long Time Behavior

The long-time behavior of the Equation 2.2.1 can be obtained by using procedures similar

to that of the short-time behavior. The long-time approximation of Equation 2.2.1 was

obtained in Ref. 2 is as follows:

( ) 0for r r80907.1tln5.0)t,1r0(p f2DDrfDrfDD >−+=<≤ (2.2.21)

and

0for r 80907.0r

tln5.0)t,1r(p f2

D

DrfDrfDD ≥

−+=> (2.2.22)

Equation 2.2.21 and 2.2.22 were obtained by obtaining the long-time approximations of

the P- and Z-functions. At late time the Z-function approaches unity (± 1 percent), when

2Drf2Drf L5t

π≥ (2.2.23)

and the P-function is equivalent to 0.25/tDrf when

( )1r25.12t 2DDrf +≥ (2.2.24)

From Equation 2.2.33, we notice when Equation 2.2.33 is satisfied, the maximum

pseudo-skin from Equation 2.2.16 can be written as:

[ ]∫π

−=π

σ

2Drf2

L5

0

DrfDrfDfDfDDrfDrfD2Drf2DD dt1)'t,z,z,h,L(Z)'t,r(p2)L

5,z,r( (2.2.25)

36

2.3 Horizontal Wells

The horizontal well model studied in this section is illustrated in Figure 2.3.1. The

model development techniques employed in obtaining the horizontal well solution are

very similar to those employed in Section 2.1 above. The most pertinent goal in this

section is the introduction of the line source approximation into the partially penetrating

vertical fracture solution in order to generate the horizontal well solution. Another critical

point is the effect of wellbore radius on horizontal well pressure, which is computed at a

finite radius (rw) outside from the source. A detailed analysis of the effect of computing

the well pressure at the point: yD = rwD, zD = zwD, will be presented in Chapter V of this

dissertation.

Figure: 2.3.1: Schematic of the Horizontal Well-Reservoir System

Zw

L/2

0=∂∂

=hzz p

z

x

0=∂∂

=0zz p

37

The solution for the pressure distribution in the above horizontal well

configuration was developed in Refs. 3 and 36 using the Green’s Function approach37.

The well is assumed to be located at any location (zw) within the vertical interval and is

considered to be a line source. The general solution for this horizontal well configuration

is given as follow

}Dxf

DxfDxfDfDfDDxf

0hfD

t

0 Dxf

2D

Dxf

Dx

Dxf

Dx

y

DxfDDDD

t

dt)t,z,z,h,L(Zlim

t4

yexp

t2

xk

k

erft2

xk

k

erfk

k

4

)t,z,y,x(p

Dxf

→

•

−•

−

++

π

=

∫ (2.3.1)

Where:

))t,h,z,z,y,y,x,x(pp(qB2.141

kh)t,z,z,y,x(p fffiDDDDDD f−

µ= (2.3.2)

2t

D Lckt001056.0t

µφ= (2.3.3)

x

wD k

kL

)xx(2x

−= (2.3.4)

y

wD k

kL

)yy(2y

−= (2.3.5)

h

zzD = (2.3.6)

z

D kk

Lh2L = (2.3.7)

38

( ) ( )

ππ

π−+

=

∑∞

=

→

1nwDD2

D

D22

DDfDfDD0hfD

zncoszncosL

tnexp21

)t,z,z,h,L(Zlim

(2.3.8)

Figure 2.3.1 represents a typical horizontal wellbore pressure response for an infinite

conductivity wellbore boundary condition.

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E -06 1.0E -05 1.0E-04 1.0E-03 1.0E-02 1.0E -01 1.0E+00 1.0E+01 1.0E+02 1.0E+03

Dimensionless Time, tD

Dim

ension

less

Pre

ssur

e, p

wD

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+021.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

0.01

0.02

0.1

0.2

1.0

2.0

4.0

LD=10.0

Vertical FractureSolution

Figure 2.3.1: A Typical Horizontal Wellbore Pressure Response – Infinite Conductivity (zD = 0.5, zwD = 0.5)

39

2.3.1 Asymptotic Forms of the Horizontal Well Solution


methods similar to those given in Ref. 3

a. Short-Time Approximation

At short time,

β=−

++

D

Dx

D

Dx

t2

xkk

erft2

xkk

erf (2.3.9)

Where:

>

=

<

=β

xD

xD

xD

kkxfor 0

kkxfor 1

kkxfor 2

(2.3.10)

Substituting Equation 2.3.9 into Equation 2.3.1 we get

( ) ( )D

D

1nwDD2

D

D22

t

0 D

2D

yDDDDD

tdtzncoszncos

Ltnexp21

t4yexp

kk

4)t,z,y,x(p

Dxf

ππ

π−+

−πβ=

∑

∫∞

=

(2.3.11)

Expanding Equation 2.3.11, we get

( ) ( )D

D

1nwDD2

D

D22t

0 D

2D

y

t

0 D

D

D

2D

yDDDDD

tdt

zncoszncosL

tnexp

t4y

expkk

2

tdt

t4y

expkk

4)t,z,y,x(p

Dxf

Dxf

∑∫

∫∞

=ππ

π−

−πβ+

−πβ= (2.3.12)

Equation 2.3.12 can be written as

Fp)t,z,y,x(p DfDDDDD += (2.3.13)

40

Where

∫

−πβ=Dxft

0 D

D

D

2D

yDf t

dtt4y

expkk

4p (2.3.14)

and

( ) ( )D

D

1nwDD2

D

D22t

0 D

2D

y tdtzncoszncos

Ltnexp

t4yexp

kk

2

FDxf

∑∫∞

=ππ

π−

−πβ

= (2.3.15)

Integrating Equation 2.3.14 with respect to tD, we get

t2

yerfcy

2t4

yexpt

k

k

2p

D

DD

D

2D

Dy

Df

−π−

−πβ= (2.3.16)

Transforming Equation 2.3.15 into Laplace space, we get

( ) ( ) ( )

( ) ( )[ ]( )[ ]ξπ+

ξ

ξ

π+−ξ−

ππβπ=

∫

∑∞

∞

=

2D

220

2D

2D

22

1nwDD

L/nsd

4yL/nsexpexp

zncoszncoss2

sF (2.3.17)

Integrating Equation 2.3.17 with respect to ξ, we get

( ) ( ) ( )( )[ ] ( )[ ]( )2

D22

D1n

2D

22

wDD L/nsyexpL/ns

zncoszncoss2

sF π+−π+

ππβπ= ∑∞

=

(2.3.18)

Using the procedure presented in Ref. 36 we can recast Equation 2.3.18 as

( ) ( )[ ]{

( )[ ]} ( )syexps4

syLn2zzK

syLn2zzKs4

LsF

D232D

2D

2wDD0

n

2D

2D

2wDD0

D

−βπ−+−++

+−−β

= ∑+∞

−∞= (2.3.19)

For large s

( )[ ] ( )[ ]syLzzKsyLn2zzK 2D

2D

2wDD0

2D

2D

2wDD0 +−<<+−± (2.3.20)

41

Thus, equation 2.3.19 becomes

( )[ ] ( )syexps4

syLzzKs4

L)s(F D23

2D

2D

2wDD0

D −βπ−+−β

= (2.3.21)

Inverting Equation 2.3.21 back into real space, we get

( )

−π−

−πβ

−

+−−

β=

D

DD

D

2D

D

y

D

2D

2D

2wDDD

t2

yerfcy

2t4

yexpt

k

k

2

t4

yLzzEi

8

L)s(F

(2.3.22)

Substituting Equations 2.3.16 and 2.3.22 into Equation 2.3.13 we get

( )

+−−

β=

D

2D

2D

2wDDD

DDDDD t4

yLzzEi

8

L)t,z,y,x(p (2.2.23)

Equation 2.2.23 represents early radial flow into the horizontal wellbore; this flow

period is limited by the distance of the location of the wellbore and the closest

upper or lower boundary and the distance from the pressure point to 0.5L.

b. Long-Time Approximation

The long time approximation of the horizontal well solution can be obtained using

techniques similar to that employed in Section 2.1 above.

The long time approximation for a horizontal well is as follows:


DwDDDDDDD

DDwDDDDD

+σ++= (2.2.24)

Where:

42

( ) ( )[ ]{

( ) ( )[ ]( )[ ]}x

2D

2DDD

2D

2

xDxD

2D

2

xDxD

DD

kkyx/y2arctany2

ykkxlnkkx

ykkxlnkkx25.0

)y,x(

−+−

+++−

+−−

=σ

(2.2.25)

and

( ) ( ) ( ) α

ππππ

=

∫∑+

−

∞

=d

L


)L,z,z,y,x(F

D

D

1

1

01n

wDDfD

DwDDDD

(2.2.26)

When

( )[ ]

+±

π≥

2D

2

xD

22Dxf

Dykkx25

/L100t (2.2.27)

2.3.2 Computation of Horizontal Well Response

The fact that wellbore pressure of horizontal well is computed at a finite radius

(rw), has ramification that deserves consideration. The vertical fracture solution given in

Section 2.1 ignores the existence of the wellbore. It is possible to compute the response

for a vertically fractured well at xD = 0, yD = 0, and specify the pressure at this point to be

the wellbore pressure. Mathematically, it implies that it is possible to compute pressures

within the source and that these solutions are bounded at all times. In the horizontal well

case with a line source solution, it is not possible to compute pressure drops inside the

source. Pressure drops have to be computed at some finite radius outside the source.

Thus, consideration must be given to two factors in the analysis of horizontal wellbore

pressure computed using the line source solution: (i) horizontal well response is a strong

43

function of rwD at early time, the pressure computed at a finite radius outside the source is

higher than the pressure computed at the same radius inside the source, (ii) since the

vertical fracture and the horizontal well solutions are not computed at the same point, it is

difficult to conduct a realistic comparison between vertical fracture and horizontal well

pressure responses

44

CHAPTER III

MODEL DEVELOPMENT

In this chapter we develop the general mathematical solution for a well producing

from a solid bar source. This solution is valid for oil reservoirs under some physical and

boundaries conditions given is Section 3.1. The three-dimensional solution for the

transient pressure response of a well producing from a solid bar source is derived from

three one-dimensional instantaneous sources using Green’s functions37 and Newman

product solution38. The solution obtained in the section will provide a platform for the

development of hydraulic fracture (vertical, horizontal and coupled fractures), limited

entry well, and horizontal well solutions in Chapters IV and V.

3.1 Uniform-Flux Solid Bar Source Solution

The mathematical model for developed in this section assumes: Flow of a slightly

compressible fluid in a solid bar source

1. The porous medium is uniform and homogenous

2. Formation has anisotropic properties

3. Pressure is constant everywhere at time t = 0, i.e., ip)0,z,y,x(p = )

4. Pressure gradients are small everywhere and gravity effects are not included

5. A single, horizontal, symmetrical solid bar source of length (xf), width (yf), and

height (hf) is centered at the well.

45

Figure 3.1.1: Cartesian coordinate system (x, y, z) of the Solid Bar Source Reservoir

hf

yf xf

(0, 0, 0)

h zw

No flow Upper Boundary

No flow Lower Boundary

46

Figure 3.1.2: Front View of the Solid Bar Source Reservoir System

Figure 3.1.3: Side View of the Solid Bar Source Reservoir System

0 =∂

∂

=hzz

p

0 =∂

∂

=0zz

p

yf

hf

zw kz ky

0 = ∂

∂

= h z z

p

0 = ∂

∂

= 0 z z

p

xf

zw kz kx

hf

47

As illustrated by the coordinate system in Figures 3.1.1 to 3.1.3, the model

assumes a solid bar placed parallel to the x-axis. It is located at an elevation zw in the

vertical (z) direction, and is parallel to the top and bottom boundaries. The center of the

solid bar source, as shown by the coordinate system in Figure 3.1.1, is located at the

coordinates (xw, yw, zw), while the coordinates (x, y, z) represents any point in the porous

media at which pressure is computed. Note also that, for the coordinate system shown,

the coordinate of the center of the source are (x = 0, y = 0, z = zw).

Assuming the fluid withdrawal rate is uniform over the length of the source, the

pressure drop at any point in the reservoir can be expressed in terms of instantaneous

source functions (see Appendix A) as

( ) ( ) ( ) ττ−τφ

=∆ ∫ ∫ ddMt,M,MG,Mqc

1t,Mp ww

t

0Dw

wf (3.1.1)

Where:

( ) ( )∫ ττ−=Dw

ww ddMt,M,MGt,MS (3.1.2)

For a three-dimensional model with the same coordinate system as Figure 3.1.1, Equation

3.1.1 becomes

( ) ( ) ( )∫ ττ−τφ

=∆t

0 f dt,z,y,xSqc

1t,z,y,xp (3.1.3)

Where: ( )t,z,y,xS is the total source function in the three-dimensional space.

Using Newman product rule 38 this total source function may be defined as the

product of three one-dimensional instantaneous source functions.

( ) ( ) ( ) ( )t,zSt,ySt,xSt,z,y,xS = (3.1.4)

48

We also define qf(t) as the well flow rate per unit volume of the source. Further, we

assume a constant rate distributed uniformly over the length of the source (i.e. a uniform-

flux source boundary condition). Equation (3.1.3) can be expressed as

( ) ( )∫ ττφ

=∆t

0fff

d,z,y,xShycx

qt,z,y,xp (3.1.5)

Where: q = total flow rate, and

fff

f hyxq)t(q = (3.1.6)

As shown in Appendix A, we model the solid bar source reservoir as the intersection of

three one-dimensional instantaneous source2, 37: (i) an infinite slab source in an infinite-

acting reservoir in the x-direction, (ii)an infinite slab source in an infinite-acting reservoir

in the y-direction, and (iii) an infinite plane source in an slab reservoir in the z-direction.

These source functions, which have been derived and tabulated by Gringarten and

Ramey37 can be written as

1. Infinite slab source in x-direction and infinite reservoir

( ) ( )

η−−+

η−+=

t2xx2/xerf

t2xx2/xerf

21)t,x(S

x

wf

x

wf (3.1.7)

2. Infinite slab source in y-direction and infinite reservoir

( ) ( )

η−−+

η−+=

t2yy2/yer

t2yy2/yerf

21)t,y(S

y

wf

y

wf (3.1.8)

3. Infinite plane source in z-direction and slab reservoir

49

πππ

ηπ−π

+

=

∑∞

=1n

ff2

z22

f

f

hzncos

hzncos

hhn5.0sin

htnexp

n1

hh41

hh

)t,z(S

(3.1.9)

Where: c

k jj φµ=η , j = x, y, or z

To facilitate presentation of the solutions for the solid bar source over a wide

range of variables, we also recast the equations in terms of dimensionless parameters.

The following are the definitions of dimensionless parameters, given in Darcy units.

Dimensionless pressure:

[ ])t,h,h,z,z,y,y,x,x(ppq2.141

kh

)t,h,z,z,y,x(p

ffffi

DDfDDDDD f

−µ

= (3.1.10)

Dimensionless time:

2ft

Dxf xckt001056.0t

µφ= (3.1.11)

Dimensionless distance in the x-direction:

xf

wD k

kx

)xx(2x

−= (3.1.12)

Dimensionless distance in the y-direction:

yf

wD k

kx

)yy(2y

−= (3.1.13)

Dimensionless distance in the z-direction:

50

hzzD = (3.1.14)

Dimensionless reservoir height:

h

hh f

fD = (3.1.15)

Dimensionless source half-length:

zf

Dxf kk

xh2L = (3.1.16)

Aspect ratio:

f

f

xym = (3.1.17)

Note that the dimensionless time and dimensionless distances are presented in terms of

source half-length. Furthermore, the permeability anisotropy is included in the definitions

of the dimensionless distances in x- and y-directions, and the dimensionless source half

length.

Substituting the dimensionless variables defined in Equation 3.1.10 through

3.1.17 into Equations 3.1.7 through 3.1.9, we obtain the following expressions:

1. Infinite slab source in x-direction and infinite reservoir

+

++

=Dxf

Dx

Dxf

Dx

DxfD t2

xkk

erft2

xkk

erf21)t,x(S (3.1.18)

4. Infinite slab source in y-direction and infinite reservoir

51

+

++

=Dxf

Dy

Dxf

Dy

DxfD t2

ykkm

erft2

ykkm

erf21)t,y(S (3.1.19)

5. Infinite slab source in z-direction and infinite reservoir

πππ

π−π

+

=

∑∞

=1nDwDfD2

Dxf

Dxf22

fD

fDDxfD

zncoszncoshn5.0sinL

tnexpn1

h41

h)t,z(S

(3.1.20)

Substituting Equation 3.1.4, 3.1.18 through 3.1.20 into Equation 3.1.5, the mathematical

model for a solid bar source reservoir can be written as follow:

Dxf1n

wDDfD2Dxf

Dxf22

fD

t

0 Dxf

Dy

Dxf

Dy

Dxf

Dx

Dxf

Dx

DxfDDDD

dtzncoszncoshn5.0sinL

tnexp

n

1

h

41

t2

yk

km

erft2

yk

km

erft2

xk

k

erft2

xk

k

erfm8

)t,z,y,x(p

Dxf

πππ

π−

π+•

−

+

+

•

−

++

π

=

∑

∫

∞

=

(3.1.21)

The study of the behavior of Equation 3.1.21 is simplified by the introduction of the

following functions which were first introduced in Refs. 2 and 35.

1. The P-function:

−

+

+

•

−

++

=

Dxf

Dy

Dxf

Dy

Dxf

D

x

Dxf

D

x

DxfDD

t2

yk

km

erft2

yk

km

erft2

xk

k

erft2

xk

k

erfm4

1

)t,y,x(P

(3.1.22)

52

The P-function is proportional to the instantaneous source function for a solid bar source

in an infinite-acting reservoir. When the m-value is unity, Equation 3.1.22 indicates

excellent agreement with the P-function developed in Ref. 2 for tDxf ≥ 10-3. For tDxf < 10-3

Equation (3.1.22) yields a better solution. This is due to the fact that at early time, the

modified Bessel Function of the first kind (Io) approaches infinity. An early time

approximation for Io function was used to eliminate this problem in Ref. 2.

2. The Z-function2

πππ

π−π

+

=

∑∞

=1nwDDfD2

Dxf

Dxf22

fD

DxfDfDfDDxf

zncoszncoshn5.0sinL

tnexpn1

h0.41

)t,z,z,h,L(Z (3.1.23)

The Z-function is proportional to the instantaneous source function for an infinite

horizontal slab reservoir with impermeable boundaries, and accounts for the partial

penetration of the solid bar source. For a fully penetrating source the function is unity.

Substituting Equation 3.1.22 and 3.1.23 into Equation 3.1.21, the mathematical

solution for a solid bar source reservoir can be expressed as:

∫

π

=Dxft

0DxfDxfwDDfDDxfDxfDD

DxfDDDD

dt)t,z,z,h,L(Z)t,y,x(P2

)t,z,y,x(p (3.1.24)

From Equation 3.1.24 the pressure derivative function of the solid bar source solution is

given by

)t,z,z,h,L(Z)t,y,x(P

2t

)t(ln)t,z,y,x(p

DxfwDDfDDxfDxfDDDxf

Dxf

DxfDDDD

π

=∂

∂

(3.1.25)

53

3.2 Transient-State Behavior of the Solid Bar Source Solution

Before extending the solid bar solution to hydraulic fracture and horizontal well

solutions, we studies the behavior of the solid bar source solution with the aim of gaining

insight into the sensitivity of this solution to critical parameters as well as gaining deeper

understanding into the computational efficiency of this solution during the early and late

time periods. A general analysis of the asymptotic behavior of this solution will be

included in this chapter; specific cases will be studied for hydraulic fracture systems and

horizontal well configurations in Chapters IV and V, respectively.

To study the influence of the P-function on the solid bar solution, we consider a

fully penetrating solid bar source by setting the Z-function equal to unity. Figures 3.1.4

and 3.1.5 show the transient pressure and derivative response of a well producing from a

fully penetrating solid bar source, respectively. The effect of aspect ratio ‘m’ was

investigated in the plots. In Figures 3.1.4 and 3.1.5 we note that as ‘m’ tends to zero, the

behavior of a fully penetrating solid bar solution is indistinguishable from that of a fully

penetrating slab source solution. This observation is explains why we expect the solid bar

source solution to agreement closely with both the vertical and horizontal fracture

solutions in chapters VI and V.

54

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03


Dim

ension

less

Pre

ssur

e, p

D

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

1.0

0.8

0.6

0.4

0.2

m=0.001

0.1

Figure 3.1.4: Transient Response of a Fully Penetrating Solid Bar Source (Uniform-Flux Case)

55

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03


Dim

ension

less

Pre

ssur

e D

eriv

ativ

e, p

' D

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

1.0

0.8

0.6

0.4

0.2

m=0.001 0.1

Figure 3.1.5: Derivative Response of a Fully Penetrating Solid Bar Source

(Uniform-Flux Case)

56

Tables 3.1.1 and 3.1.2 present the dimensionless pressure, pD and derivative response, p’D

for a reservoir producing at a constant rate from a fully penetrating solid bar source.

The capability of the P-function to model a well producing from both a solid bar

source as well as a slab source gives the solid bar source solution a broad applicability.

The effect of the Z-function on both the short and long time behaviors of the solid bar

source solution is more difficult to achieve. The effects of the Z-function on the

asymptotic behavior of the hydraulic fracture and horizontal well will be demonstrated in

Chapters VI and V.

57

Table 3.1.1: Dimensionless Pressure, pD for a Reservoir Producing from a Fully Penetrating Solid Bar Source Located at the Center of the Reservoir

(Uniform-Flux Case)

Dimensionless Pressure , pD tDxf m=1.0 m=0.8 m=0.6 m=0.4 m=0.2 m=0.001

1.E-06 1.57E-06 1.96E-06 2.62E-06 3.93E-06 7.85E-06 1.13E-03 1.E-05 1.57E-05 1.96E-05 2.62E-05 3.93E-05 7.85E-05 4.87E-03 1.E-04 1.57E-04 1.96E-04 2.62E-04 3.93E-04 7.85E-04 1.70E-02 1.E-03 1.57E-03 1.96E-03 2.62E-03 3.93E-03 7.85E-03 5.53E-02 1.E-02 1.57E-02 1.96E-02 2.62E-02 3.92E-02 7.41E-02 1.76E-01 1.E-01 1.55E-01 1.91E-01 2.43E-01 3.17E-01 4.20E-01 5.58E-01 1.E+00 8.51E-01 9.42E-01 1.05E+00 1.16E+00 1.30E+00 1.44E+00 1.E+01 1.93E+00 2.04E+00 2.15E+00 2.27E+00 2.41E+00 2.56E+00 2.E+01 2.27E+00 2.38E+00 2.49E+00 2.62E+00 2.75E+00 2.90E+00 3.E+01 2.48E+00 2.58E+00 2.69E+00 2.82E+00 2.96E+00 3.11E+00 4.E+01 2.62E+00 2.72E+00 2.84E+00 2.96E+00 3.10E+00 3.25E+00 5.E+01 2.73E+00 2.83E+00 2.95E+00 3.07E+00 3.21E+00 3.36E+00 6.E+01 2.82E+00 2.93E+00 3.04E+00 3.16E+00 3.30E+00 3.45E+00 7.E+01 2.90E+00 3.00E+00 3.12E+00 3.24E+00 3.38E+00 3.53E+00 8.E+01 2.96E+00 3.07E+00 3.18E+00 3.31E+00 3.45E+00 3.60E+00 9.E+01 3.02E+00 3.13E+00 3.24E+00 3.37E+00 3.50E+00 3.65E+00 1.E+02 3.08E+00 3.18E+00 3.29E+00 3.42E+00 3.56E+00 3.71E+00 2.E+02 3.42E+00 3.53E+00 3.64E+00 3.77E+00 3.90E+00 4.05E+00 3.E+02 3.62E+00 3.73E+00 3.84E+00 3.97E+00 4.11E+00 4.26E+00 4.E+02 3.77E+00 3.87E+00 3.99E+00 4.11E+00 4.25E+00 4.40E+00 5.E+02 3.88E+00 3.98E+00 4.10E+00 4.22E+00 4.36E+00 4.51E+00 6.E+02 3.97E+00 4.08E+00 4.19E+00 4.32E+00 4.45E+00 4.60E+00 7.E+02 4.05E+00 4.15E+00 4.27E+00 4.39E+00 4.53E+00 4.68E+00 8.E+02 4.12E+00 4.22E+00 4.33E+00 4.46E+00 4.60E+00 4.75E+00 9.E+02 4.17E+00 4.28E+00 4.39E+00 4.52E+00 4.66E+00 4.81E+00 1.E+03 4.23E+00 4.33E+00 4.45E+00 4.57E+00 4.71E+00 4.86E+00

58

Table 3.1.2: Dimensionless Derivative, p’D for a Reservoir Producing from a Fully Penetrating Solid Bar Source Located at the Center of the Reservoir

(Uniform-Flux Case)

Dimensionless Pressure Derivative, p’D tDxf m=1.0 m=0.8 m=0.6 m=0.4 m=0.2 m=0.001

1.E-06 1.57E-06 1.96E-06 2.62E-06 3.93E-06 7.85E-06 8.18E-04 1.E-05 1.57E-05 1.96E-05 2.62E-05 3.93E-05 7.85E-05 2.78E-03 1.E-04 1.57E-04 1.96E-04 2.62E-04 3.93E-04 7.85E-04 8.85E-03 1.E-03 1.57E-03 1.96E-03 2.62E-03 3.93E-03 7.85E-03 2.80E-02 1.E-02 1.57E-02 1.96E-02 2.62E-02 3.91E-02 6.62E-02 8.86E-02 1.E-01 1.49E-01 1.77E-01 2.09E-01 2.41E-01 2.64E-01 2.73E-01 1.E+00 4.26E-01 4.38E-01 4.48E-01 4.55E-01 4.60E-01 4.61E-01 1.E+01 4.92E-01 4.93E-01 4.94E-01 4.95E-01 4.96E-01 4.96E-01 2.E+01 4.96E-01 4.97E-01 4.97E-01 4.98E-01 4.98E-01 4.98E-01 3.E+01 4.97E-01 4.98E-01 4.98E-01 4.98E-01 4.99E-01 4.99E-01 4.E+01 4.98E-01 4.98E-01 4.99E-01 4.99E-01 4.99E-01 4.99E-01 5.E+01 4.98E-01 4.99E-01 4.99E-01 4.99E-01 4.99E-01 4.99E-01 6.E+01 4.99E-01 4.99E-01 4.99E-01 4.99E-01 4.99E-01 4.99E-01 7.E+01 4.99E-01 4.99E-01 4.99E-01 4.99E-01 4.99E-01 4.99E-01 8.E+01 4.99E-01 4.99E-01 4.99E-01 4.99E-01 4.99E-01 4.99E-01 9.E+01 4.99E-01 4.99E-01 4.99E-01 4.99E-01 5.00E-01 5.00E-01 1.E+02 4.99E-01 4.99E-01 4.99E-01 5.00E-01 5.00E-01 5.00E-01 2.E+02 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 3.E+02 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 4.E+02 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.E+02 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 6.E+02 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 7.E+02 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 8.E+02 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 9.E+02 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 1.E+03 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01 5.00E-01

59

3.3 Asymptotic Behavior of the Solid Bar Source Solution


methods similar to those present in Chapter II. In Section 3.2, we have established that,

for small m-values, the behavior of the solid bar solution is similar to that of a slab

source. So, the general asymptotic forms of the solid bar source solution presented for the

different cases are as follows:

Case 1: m >> 0

At early time the P-function is constant:

m4

)t,y,x(P DxfDDβ= (3.3.1)

Where:

>>

==

<<

=β

yDxD

yDxD

yDxD

kkmy andkkxfor 0

kkmy andkkxfor 2

kkmy and kkxfor 4

(3.3.2)

Substituting Equation 3.3.1 into Equation 3.1.24, we have

∫πβ=

Dxft

0DxfDxfwDDfDDxfDxfDDDD dt)t,z,z,h,L(Z

m8)t,z,y,x(p (3.3.3)

In Equation 3.3.3 we notice that the early time behavior of the solid bar source depends

only the Z-function. The integral in Equation 3.3.3 may have different forms depending

on the value of hfD2.

For hfD → 0, Equation 3.3.3 becomes:

60

−−−

−−π

πβ−

=

D

DfDf

D

2DfDD

DxfDDDD

t2zz

erf2zz

t4)zz(expt

m4L

)t,z,y,x(p

(3.3.4)

Equation 3.3.4 represents linear vertical flow into the source

For hfD >> 0 Equation 3.3.3 becomes

δ−π

δ−

δ

δ+πβ

=

Drf

2DDrf

DD

D2D

DrffD

DxfDDDD

t4expt

t2erfc

2t

mh8

)t,z,y,x(p

(3.3.5)

Equation 3.3.5 represents a storage dominated flow.

At late time the Z-function approaches unity and the transient behavior of a well

producing from a solid bar source depends only on the P-function.

1)t,z,z,h,L(Z DxfDfDfDDxf = (3.3.6)

for

2Drf2Drf L5t

π≥ (3.3.7)

The long-time approximation of Equation (3.1.24) is given by

( )( )DwDDDD1DD1

DDDDDD

L,z,z,y,xF)y,x(

80907.2tln5.0)t,z,y,x(p

+σ++= (3.3.8)

Where

61

( ){ ( ) ( )[ ]( ) ( ) ( )[ ]( )

( )( )( ) ω

−−+

−

−−

ω++++−

ω−+−−

=σ

−

+

−∫

dkkkkmryx

kkmry2tan

kkmry2

kkmykkxlnkkx

kkmykkxlnkkx8

1

)y,x(

x

2

ywDD2D

ywDD1

ywDD

2

yD

2

xDxD

2

yD

2

xD

1

1

xD

DD1

(3.3.9)

and

( )

( ) ( ) ( ) ( )∑ ∫ ∫∞

=

+

−

+

−

ωαπππππ

=

1n

1

1

1

1

DD0wDDfDfD

DwDDDD1

ddnLrKzncoszncoshn5.0sinn

1

h

1

L,z,z,y,xF

(3.3.10)

Where:

( ) ( ) ω−+α−=

2

yD

2

xDD kkmykkxr (3.3.11)

Case 2: m → 0

As m tends to zero, the early time behavior of the solid bar source solution can be

approximated by that of a slab source, which is shown as follows:

Dxf

Dxf

Dxf

2D

yDxfDD t

dtt4yexp

kk

2)t,y,x(P

−π

β≈ (3.3.12)

Substituting Equation 3.3.12 into Equation 3.1.24, it yields

∫

−πβ

≈Dxft

0 Dxf

DxfDxfDfDfDDxf

Dxf

2D

y

DxfDD

tdt)t,z,z,h,L(Z

t4yexp

kk

4

)t,y,x(P (3.3.13)

62

Following the same procedure highlighted in Section 2.2 of chapter 2, Equation 3.3.13

can be expressed as:

−π−

−πβ

=

D

DD

D

2D

Dy

fD

DxfDDDD

t2y

erfcy2t4

yexptkk

2h

)t,z,y,x(p

(3.3.14)

Where:

>

=

<

=β

xD

xD

xD

kkxfor 0

kkxfor 1

kkxfor 2

(3.3.15)

Equation 3.3.14 represents a linear vertical flow into the source.

At late time the long-time approximation of Equation 3.1.24 is given by:

( )( )DwDDDD2DD2

DDDDDD

L,z,z,y,xF)y,x(

80907.2tln5.0)t,z,y,x(p

+σ++= (3.3.16)

Where

( ) ( )[ ]{( ) ( )[ ]

( )[ ]}x2D

2DD

1D

2D

2

xDxD

2D

2

xDxD

DD2

kkyx/y2tany2

ykkxlnkkx

ykkxlnkkx4

1

)y,x(

−+−

+++−

+−−

=σ

−

(3.3.17)

and

( )

( ) ( ) ( ) ( ) απππππ

=

∫∑+

−

∞

=

dnLrkzncoszncoshn5.0sinn

1

h

2

L,z,z,y,xF1

1

DD01n

wDDfDfD

DwDDDD2

(3.3.18)

As hfD tends to zero, Equation 3.3.18 reduces to:

63

( )

( ) ( ) ( ) απππ

=

∫∑+

−

∞

=

dnLrkzncoszncos

L,z,z,y,xF1

1

DD01n

wDD

DwDDDD2

(3.3.19)

Where:

( )

+α−= 2

D

2

xDD ykkxr (3.3.20)

64

CHAPTER IV

APPLICATION OF THE SOLID BAR SOURCE SOLUTION TO

HYDRAULIC FRACTURES AND LIMITED ENTRY WELLS

Although there have been many analytical studies on pressure-transient behavior

of hydraulic fracture systems, no single analytical solution capable of describing both

vertical and horizontal fracture transient state behaviors has been developed. The purpose

of this work is to develop a general analytical solution that is robust enough to fit this

need.

In this Chapter we present a type curve solution for a well producing from a solid

bar source in an infinite-acting reservoir with impermeable upper and lower boundaries.

Computation of dimensionless pressure reveals that the pressure-transient behavior of any

hydraulic fracture system is governed by two critical parameters: (i) aspect ratio:

ff x/ym = and (ii) dimensionless length: zDxf kk)L/h2(L = . Analysis of a typical

log-log plot of pwD vs. tDxf indicates the existence of four distinct flow periods (i) vertical

linear flow period, (ii) fracture fill-up period causing a typical storage dominated flow,

(iii) transition period, and (iv) radial flow period. As the aspect ratio tends to zero, the

first and second fracture fill-up periods disappear resulting in typical fully/partially

penetrating vertical fracture pressure response.

This analytical solution reduces to the existing fully/partially penetrating vertical

fracture solution developed by Raghavan et al.1 as the aspect ratio tends to zero, and a

horizontal fracture solution is obtained as the aspect ratio tends to unity. This new

65

horizontal fracture solution yields superior early-time (tDxf < 10-3) solution compared with

the existing horizontal fracture solution developed by Gringarten and Ramey2, and

indicates excellent agreement for tDxf > 10-3. Possibility of extending this new solution to

horizontal well analysis is discussed in Chapter V.

4.1 Vertical Fracture System

For very small m-values (xf >> yf) the solid bar source solution reduces to the

fully/partially penetrating vertical fracture solution. Figures 4.1.1 and 4.1.2 illustrates the

vertical fracture model used in this study. The model is physically the same as the model

studied in Ref. 1. As the m-value approaches zero, we have a fully/partially penetrating

slab source (vertical fracture) with zero thickness. From Section 3.3 we see that only the

P-function is affected by this approximation, while the Z-function remains the same.

Figure 4.1.3 illustrates the mode of fluid flow into the vertical fracture system. Note that

flows occurs only in the y-direction ( yqq = ); this is the most distinctive flow

characteristic of vertical fracture systems.

66

Figure 4.1.1: Cartesian coordinate system (x, y, z) of the Vertically Fractured Reservoir

hf

yf

xf

(0, 0, 0)

h zw



67

Figure 4.1.2: Front View of the Vertically Fractured Reservoir

Figure 4.1.3: Side View of the Vertically Fractured Reservoir

0 =∂

∂

=hzz

p

0 =∂

∂

=0zz

p

hf

kz ky

q = qy

0 = ∂

∂

= h z z

p

0 = ∂

∂

= 0 z z

p

xf

zw kz kx

hf

68

The solution for a fully/partially penetrating vertical fracture system can be expressed as

follows:

DxfDxffDDfDDxfDxfDD

t

00m

DxfDDDD

dt)t,z,z,h,L(Z)t,y,x(Plim2

)t,z,y,x(pDxf

∫ →

π

= (4.1.1)

Substituting the limit of the P-function as m approaches zero into Equation 4.1.1, we get:

}Dxf

DxfDxfDfDfDDxf

Dxf

2D

t

0 Dxf

Dx

Dxf

Dx

y

DxfDDDD

tdt

)t,z,z,h,L(Zt4y

exp

t2

xkk

erft2

xkk

erfkk

4

)t,z,y,x(p

Dxf

−

−

++

π

=

∫ (4.1.2)

Equation 4.1.2 is the same as the fully/partially vertical fracture solution developed in

Ref. 1.

The pressure derivation function of the solid bar source solution can be derived from

Equation 4.1.2 and is shown as:

)t,z,z,h,L(Z)t,y,x(Plim

2t

)t(ln)t,z,y,x(p

DxfwDDfDDxfDxfDD0m

Dxf

Dxf

DxfDDDD

→

π

=∂

∂

(4.1.3)

Equation 4.1.2 is exactly the same as the partially penetrating vertical fracture solution

shown in Ref. 1 (See Chapter II, Section 2.1), hence a comparison between these two

models (slab source vs. solid bar source solution) will not be carried out in this Chapter.

69

4.2 Horizontal Fracture System

The mathematical model for a horizontal fracture model can be derived following

steps highlighted in chapter 3. This can be written as follow:

Dxf

t

0DxffDDfDDxf0hDxfDD

DxfDDDD

dt)t,z,z,h,L(Zlim)t,y,x(P2

)t,z,y,x(pDxf

f∫ →

π

= (4.2.1)

Taking the limit of solid bar source solution as hfD approaches zero, we get Equation

4.2.1. In Figure 4.2.3 we can see from the physical model that fluid now flows into the

fracture system only in the vertical direction ( zqq = ); this is typical of horizontal fracture

systems. Equation 4.2.2 describes the transient pressure response of a horizontal fracture

system. When the m-value is unity, Equation 4.2.2 shows excellent agreement with the

horizontal fracture model developed in Ref. 2.

Figure 4.2.1: Cartesian coordinate system (x, y, z) of the Horizontal Fracture System

yf xf

(0,0,0

h zw



70

Figure 4.2.2: Front View of Horizontal Fracture System

Figure 4.2.3: Side View of the Horizontal Fracture System Equation 4.2.1 can be expressed as follows

0 =∂

∂

=hzz

p

0 =∂

∂

=0zz

p

kz ky

q=qz

yf

0 = ∂

∂

= h z z

p

0 = ∂

∂

= 0 z z

p

xf

zw kz kx

71

( ) ( ) Dxf1n

wDD2Dxf

Dxf22

Dxf

Dy

Dxf

Dy

t

0 Dxf

Dx

Dxf

Dx

DxfDDDD

dtzncoszncosL

tnexp21

t2

ykk

m

erft2

ykk

m

erf

t2

xkk

erft2

xkk

erfm8

)t,z,y,x(p

Dxf

ππ

π−+

−

+

+

−

++

π

=

∑

∫

∞

=

(4.2.2)

The derivative response of a horizontal fracture system can be obtained from Equation

4.2.1and is shown as follows:

)t,z,z,h,L(Zlim)t,y,x(P

2t

)t(ln)t,z,y,x(p

DxfwDDfDDxf0hfDDxfDDDxf

Dxf

DxfDDDD

→

π

=∂

∂

(4.2.3)

4.2.1 Asymptotic Forms of the Horizontal Fracture Solution

The early-time pressure distribution function for horizontal fracture system can

obtained by taking the limit of Equation 3.3.3 in Section 3.3 as hfD tends to zero. Recall

Equation 3.3.3 and remember it as:

∫πβ=

Dxft

0DxfDxfwDDfDDxfDxfDDDD dt)t,z,z,h,L(Z

m8)t,z,y,x(p (4.2.1)

For hfD → 0 Equation 4.2.1 becomes:

72

−−−

−−π

πβ−

=

D

DfDf

D

2DfDD

DxfDDDD

t2zz

erf2zz

t4)zz(expt

m4L

)t,z,y,x(p

(4.2.2)

Equation 4.2.2 represents linear vertical flow into the source

The late time pressure distribution function for horizontal fracture system can

obtained from Equation 3.3.8 in Section 3.3. Recall Equation 3.3.8 and remember it as:

( )( )DwDDDD1DD1

DDDDDD

L,z,z,y,xF)y,x(

80907.2tln5.0)t,z,y,x(p

+σ++= (4.2.3)

Where

( ){ ( ) ( )[ ]( ) ( ) ( )[ ]( )

( )( )( ) ω

−−+

−

−−

ω++++−

ω−+−−

=σ

−

+

−∫

dkkkkmryx

kkmry2tan

kkmry2

kkmykkxlnkkx

kkmykkxlnkkx8

1

)y,x(

x

2

ywDD2D

ywDD1

ywDD

2

yD

2

xDxD

2

yD

2

xD

1

1

xD

DD1

(4.2.4)

and

( )

( ) ( ) ( )∑ ∫ ∫∞

=

+

−

+

−

ωαπππ

=

1n

1

1

1

1

DD0wDD

DwDDDD1

ddnLrkzncoszncos

L,z,z,y,xF

(4.2.5)

Where

( ) ( ) ω−+α−=

2

yD

2

xDD kkmykkxr (4.2.6)

73

4.2.2 Discussion of Horizontal Fracture Pressure Response

Analysis of the pressure response of a horizontal fracture system indicates that

this fracture configuration exhibits four distinct flow periods: (i) vertical linear flow

period, (ii) fracture fill-up period causing a typical storage dominated flow, (iii) transition

period, and (iv) radial flow period. This behavior is consistent with the observations of

Gringarten et al.2. To compare the performance of Equation 4.2.1 with the solution in

Ref. 2, we assume equal fracture volumes for both the fracture systems: the horizontal

rectangular slab and the solid cylinder source, using this assumption we obtain the

equivalent dimensionless variables as follows:

f2ffff hrhyx π= (4.3.1)

Substituting ff xym = into Equation 4.3.1, we get

π

=2f2

fmx

r (4.3.2)

Substituting Equation 4.3.2 into the Equations 3.1.11 and 3.1.16, we get:

DxfDrf tm25.0t π= (4.3.3)

and

DxfDrf Lm

5.0L π= (4.3.4)

Here, tDrf and LDrf are equivalent to dimensionless time and dimensionless length defined

in Ref. 2.

Figure 4.2.4 compares the solution from Equation 4.2.1 with those of Gringarten

et al.2. From this plot we observe an excellent agreement between the two solutions (error

74

< 5% at early time). In terms of computation efficiency, the computation time for

Equation 4.2.1 is about five times faster than the Gringarten et al.2. Another advantage of

Equation 4.2.1 over the Gringarten et al.2 solution is the superior early time performance

of Equation 4.2.1. This is mainly due to the fact at early time the P-function contained in

Equation 4.2.1 is more stable than the P-function in Ref. 35. Figure 4.2.5 illustrates the

type-curve solution obtained from Equation 4.2.1 for a wide range of dimensionless time:

10-6 to 103. The plots indicate for LDrf < 0.05, a vertical linear flow period precedes the

storage-dominated flow period; this characteristic is not visible in the horizontal fracture

type-curve solution presented in Ref. 2.

Depending on the reservoir parameters, a horizontally fractured well may exhibit

early-time pressure behavior that is distinctly different from that of either a vertical

fracture or fully/partially penetrating vertical well characteristics. However, for LDxf ≥

0.75 the behavior of a horizontal fracture is essentially indistinguishable from that of a

vertically fractured reservoir2. In Figure 4.2.7 we show type curve solutions for both

horizontal fracture and fully penetrating vertical fracture solutions for a uniform-flux

boundary condition. On these plots we notice that fully penetrating vertical fracture

solution closely matches that of the horizontal fracture case of LDxf ≈ 2.5. This

observation was also observed by Gringarten et al.2 in Ref. 2

For hfD > 0 the early-time pressure behavior of a uniform-flux horizontal fracture

may exhibit an additional flow period depending on the value of LDxf. Figures 4.2.8 and

4.2.9 illustrate type-curve plots for a uniform-flux horizontal fracture with hfD = 0.001,

we note from this plot that for LDxf > 5 a storage-dominated flow period precedes the

75

vertical linear flow period, this is due to the fact that, for hfd ≥ 0.001, the fracture volume

is significant. At very early time, the flow occurs inside the fracture only. This is not seen

for the case with hfD = 0 since the fracture volume is assumed to be negligible.

76

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02


Dim

ension

less

Pre

ssur

e, p

wD

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

Gringarten et al. (1974)

Solid Bar Solution

LDrf=10.0

5.0

3.0

1.0

0.5

0.3

0.05

hD=0.0 zD=0.5 zfD=0.5 m=1.0

Figure 4.2.4: Comparison of the Horizontal Fracture Solution Using the Solid Bar Source

Solution Versus Gringarten et al.2

77

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

0.05

LDrf=10.0

5.0

3.0

1.0

0.2

0.5

hD=0.0 zD=0.5 zfD=0.5 m=1.0

Figure 4.2.5: Horizontal Fracture Type-Curve Solution Using the Solid Bar Source

Solution

78

0

1

2

3

4

5

6

7

8

9

10

1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

0.00E+00

1.00E+00

2.00E+00

3.00E+00

4.00E+00

5.00E+00

6.00E+00

7.00E+00

8.00E+00

9.00E+00

1.00E+01

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

0.05

LDrf=10.0

hD=0.0 zD=0.5 zfD=0.5 m=1.0

Figure 4.2.6: Semi-Log Plot of Horizontal Fracture Solution Using the Solid Bar Source

Solution

79

0.00001

0.0001

0.001

0.01

0.1

1

10

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

Vertical Fracture Solution

LDxf=10

5.0

2.5

0.75

0.25

0.1

0.05 Solid Bar Source Solution

hD=0.0 zD=0.5 zfD=0.5 m=1.0

Figure 4.2.7: Comparison of Horizontal Slab Source versus Vertical Slab Source

Solutions

80

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

LDxf=10

5.0

2.5

0.75

0.25

0.1

0.05

hD=0.001 zD=0.5 zfD=0.5 m=1.0


Figure 4.2.8: Illustration of the Effect of Dimensionless Height on Horizontal Fracture

Pressure Response

81

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Der

ivat

ive

Res

pons

e, p

' wD

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

LDxf=10

5.0

2.5

0.75

0.25

0.1

0.05

hD=0.001 zD=0.5 zfD=0.5 m=1.0


Figure 4.2.9: Illustration of the Effect of Dimensionless Height on Horizontal Fracture

Derivative Response

82

4.3 Limited Entry Wells

To obtain a solution for limited entry wells or vertical wells with partially

penetration, the limit as xf and yf approaches zero is passed on the source functions S(x, t)

and S(y, t) in Equation 3.1.7 and 3.1.8. The mathematical model for a limited entry well

is therefore written as follows:

( ) ( ) ( ) ( )∫ →→φ=∆

t

0 0y0x

f dtt,zS*t,ySlim*t,xSlimc

qt,z,y,xpff

(4.3.1)

Where

( )

η−−

πη=

→ t4)xx(exp

t21t,xSlim

x

2w

x0x f

(4.3.2)

and

( )

η−−

πη=

→ t4)yy(exp

t21t,ySlim

y

2w

y0yf

(4.3.3)

Substituting Equations 4.3.2 and 4.3.3 into 4.3.1 and the expression for Z-function, we

have:

( )

( ) ( ) ( )∫ ∑

πππ

π−

π+

η

−

πηφ=∆

∞

=

t

01n

wDDfD2Dxf

Dxf22

fDr

2

r

f


tnexp

n1

h0.41

t4rexp

tc4q

t,z,y,xp

(4.3.4) Where:

2w

2w

2 )yy()xx(r −+−= (4.3.5)

83

Equation 4.3.4 represents the pressure distribution for a well producing from a partially

penetrating line source in an isotropic reservoir (kx = ky = kr). Rewriting Equation 4.3.5 in

dimensionless variables, we have:

( )

( ) ( ) ( ) D1n

wDDfD2Drw

Drw22

fD

t

0D

2D

DDDDD


tnexp

n1

h0.41

t4r

expt21t,z,rp D

πππ

π−

π+

−=

∑

∫∞

=

(4.3.6)

Where:

wt

D rckt000216.0t

µφ= (4.3.7)

W

D rrr = (4.3.8)

h

hh f

fD = (4.3.9)

z

r

wDrw k

krhL = (4.3.10)

Equation 4.3.6 may also be rewritten as:

( )

( ) ( ) ( )u

duu4

Lrn

uexpzncoszncoshn5.0sinL

tnexp

n1

h0.4

t4r

Ei21t,z,rp

U

2

Drw

D

1nwDDfD2

Drw

D22

fD

D

2D

DDDD

∫∑∞∞

=

π

−−πππ

π−

π+

−−=

(4.3.11)

84

For

Drw2D L5tπ

≥ (4.3.12)

Where

D

2D

t4r

u = (4.3.13)

Equation 4.3.6 is consistent with the limited entry well solution obtained in Refs. 2 and 39

85

CHAPTER V

APPLICATION OF THE SOLID BAR SOURCE SOLUTION TO

HORIZONTAL WELLS

The conventional models of transient pressure response in horizontal wells are

generally based on the line source approximation of the partially penetrating vertical

fracture solution1. These models have three major limitations: (i) wellbore pressure is

computed at a finite radius outside the source; thus, it is impossible to compute wellbore

pressure within the source, (ii) it also is difficult to conduct a realistic comparison

between horizontal well and vertical fracture pressure responses, because wellbore

pressures are not computed at the same point, and (iii) the line source approximation may

not be adequate for reservoirs with thin pay zones. This work attempts to overcome these

limitations by developing a more flexible analytical solution using the solid bar

approximation. A technique that permits the conversion of the pressure response of any

horizontal well system into the equivalent vertical fracture response is also presented.

In this chapter a new type curve solution is developed for a horizontal well

producing from a solid bar source in an infinite-acting reservoir by means of Newman’s

product solution38. Type-curve plots for the ranges 0.01 ≤ dimensionless length (LD) ≤ 10,

and 10-4 ≤ rwD ≤ 1.0 are presented.

A dimensionless rate function (β -function) is introduced to convert the transient-

response of a horizontal well into an equivalent vertical fracture response. A step-wise

algorithm for the computation of β -function is developed using Duhamel’s principle39.

86

This provides an easier way of representing horizontal wells in numerical reservoir

simulation without the rigor of employing complex formulations for the computation of

effective well block radius.

5.1 Mathematical Model

Figures 5.1.1 through 5.1.3 illustrate the idealization of a horizontal well

producing from a solid bar source. The most distinctive flow characteristic of this model

is that fluid flows into the wellbore both in the y- and z-directions to produce a constant

rate: zy qqq += inside the wellbore. This flow characteristic makes a horizontal well act

like a coupled fracture system at early time; the combination of both horizontal and

vertical fracture flow characteristics leads to the distinctive early time flow behavior of

horizontal wells. As time increases, qz approaches zero, and qy approaches q. In Figure

5.1.1 we see that the ideal source function for a horizontal well system is the solid

cylinder source with radial flow of fluid into the wellbore along the circumference of the

circular wellbore source. However, the solid bar approximation was employed in the

development of Equation 5.1.1 for two reasons: (i) the solid cylinder source solution

contains the Io- function. This function is unstable for small values of tD, and (ii) the solid

cylinder source solution is less computationally efficient than the solid bar solution. No

comparative study was conducted to evaluate the percentage error introduced by the solid

bar approximation in this study.

87

Figure 5.1.1: Cartesian Coordinate System (x, y, z) of the Horizontal Well System

Figure 5.1.2: Front View of the Solid Bar Source Reservoir System

0 = ∂

∂

= h z z

p

0 = ∂

∂

= 0 z z

p

L

zw kz kx

2rw

2rw

2rw L

(0, 0, 0)

h zw



88

Figure 5.1.3: Side View of the Solid Bar Source Reservoir System

The analytical solution for a horizontal well model developed from the solid bar

source solution by replacing the aspect ratio (m) with the dimensionless wellbore radius,

rwD, can be written as follows:

} DDxfDfDfDDxf

D

Dy

wD

D

Dy

wD

t

0 D

Dx

D

Dx

wD

DxfDDDD

dt)t,z,z,h,L(Z

t2

ykkr

erft2

ykkr

erf

t2

xkk

erft2

xkk

erfr8

)t,z,y,x(p

D

−

+

+

−

++

π

=

∫

(5.1.1)

0 =∂

∂

=hzz

p

0 =∂

∂

=0zz

p

2rw

2rw

zw kz ky

89

Note that the Z-function expression in Equation 5.1.1 is quite different from that

of conventional horizontal well solutions, which contain the limit of Z-function as hfD

approaches zero. This approximation is not adequate for thin reservoir systems. A

numerical experiment will be used to demonstrate this claim later in this section.

Substituting the expression for P-function into Equation 5.1.1

∫ =

π

=D

wD

t

0DDfDDfDDxfrmDDD

DDDDD

dt)t,z,z,h,L(Z)t,y,x(P2

)t,z,y,x(p (5.1.2)

The pressure derivative function derived from Equation 5.1.2 is as follows

)t,z,z,h,L(Z)t,y,x(P

2t

)t(ln)t,z,y,x(p

DwDDfDDxfrmDDDD

D

DDDDD

wD=

π

=∂

∂

(5.1.3)

The pressure drop created by a horizontal well can be written as the sum of the pressure

drop (pDSB) caused by the solid bar source plus the pressure drops caused by the partial

penetrating effect (see Equation 5.1.4). The function )t,y,x(p DDDDSB represents the

pressure drop distribution created by an infinite solid bar source. The function

)t,z,y,x(S DDDD represents the “pseudo-skin function,” which is the additional time-

dependent pressure drop caused by a partially penetrating source.

)t,z,y,x(S)t,y,x(p)t,z,y,x(p DDDDDDDDSBDDDDD += (5.1.4)

Where

D

t

0rmDDDDDDDSB dt)t,y,x(P

2)t,y,x(p

D

wD∫ =

π= (5.1.5)

90

and

[ ] DDwDDfDD

t

0rmDDD

DDDD

dt1)t,z,z,h,L(Z)t,y,x(P2

)t,z,y,x(SD

wD−•π

=

∫ =

(5.1.6)

5.2 Asymptotic Forms of the Solid Bar Source Approximation for Horizontal Wells


methods similar to those given in Ref. 3. The new result in this work is the establishment

of effect of rwD on the short- and long time behavior of horizontal wells.

Case 1: Small rwD (rwD ≤ 5 x 10-3)

The short time approximation of Equation 5.1.1 is as follows:

( )

+−−β−

=

D

2DD

2wDD

D

DxfDDDD

t4yL/zzEi

L8

)t,z,y,x(p

(5.2.1)

Where:

>

=

<

=β

xD

xD

xD

kkxfor 0

kkxfor 1

kkxfor 2

(5.2.2)

The long-time approximation of Equation (5.1.1) is as follows:

( )( )DwDDDD1DD1

DDDDDD

L,z,z,y,xF)y,x(80907.2tln5.0)t,z,y,x(p

+σ++= (5.2.3)

Where

91

( ) ( )

( ) ( )( )[ ]}x

2D

2DD

1D

2D

2

xDxD

2D

2

xDxD

DD1

kkyx/y2tany2

ykkxlnkkx

ykkxlnkkx41

)y,x(

−+−

+++−

+−−

=σ

−

(5.2.4)

and

( )

( ) ( ) ( ) ( ) απππππ

=

∫∑+

−

∞

=

dnLrKzncoszncoshn5.0sinn1

h2

L,z,z,y,xF1

1DD0

1nwDDfD

fD

DwDDDD1

(5.2.5)

Where

( )

+α−= 2

D

2

xDD ykkxr (5.2.6)

As hfD approaches zero, Equation 5.2.5 is reduced to the form given in Ref. 2

Case 2: Large rwD (rwD ≥5 x 10-3)

The short-time approximation of Equation 5.1.1 is as follows:

−−−

−−π

πβ−

=

D

DfDf

D

2DfD

wD

D

DxfDDDD

t2

zzerf

2zz

t4)zz(expt

r4L

)t,z,y,x(p

(5.2.7)

Where

>>

==

<<

=β

yDxD

yDxD

yDxD

kkmy andkkxfor 0

kkmy andkkxfor 2

kkmy and kkxfor 4

(5.2.8)

92

Equation 5.2.7 represents linear vertical flow into the wellbore; for this range of rwD, the

transient response of a horizontal well is indistinguishable from that of a horizontal

fracture system (see Equation 4.2.2). The Equation was developed by assuming the limit

of the Z-function as hfD approaches zero. For hfD >> 0 a wellbore-storage dominated flow

period may be seen.2

The long-time approximation of Equation 5.1.1 is given below:

( )( )DwDDDD2DD2

DDDDDD

L,z,z,y,xF)y,x(80907.2tln5.0)t,z,y,x(p

+σ++= (5.2.9)

Where

( ){ ( ) ( )( ) ( ) ( )( )

( )( ) ω

−ω−+

ω−

ω−−

ω++++−

ω−+−−

=σ

−

+

−∫

dkkkkryx

kkry2tan

kkry2

kkrykkxlnkkx

kkrykkxlnkkx81

)y,x(

x

2

ywDD2D

ywDD1

ywDD

2

ywDD

2

xDxD

2

ywDD

2

xD

1

1xD

DD2

(5.2.10)

and

( )

( ) ( ) ( ) ( )∑ ∫ ∫∞

=

+

−

+

−

ωαπππππ

=

1n

1

1

1

1DD0wDDfD

fD

DwDDDD2

ddnLrKzncoszncoshn5.0sinn1

h1

L,z,z,y,xF(5.2.11)

Where

( ) ( ) ω−+α−=

2

ywDD

2

xDD kkrykkxr (5.2.12)

93

5.3 Computation of Horizontal Wellbore Pressure (pwD)

Conventional horizontal well models compute wellbore pressure at a finite

wellbore radius (rw) outside of the source. The choice of computation point has some

ramifications that deserve consideration. The vertical fracture solution ignores the

existence of the wellbore. Mathematically, it is possible to compute pressure within the

source and the solutions are bounded for all time in the case of a vertically fractured well.

Due to the fact that conventional horizontal well models visualize a horizontal well as a

well producing from a line source, it is impossible to compute the pressure drop within

the source; hence, wellbore pressure has to be computed at a finite radius outside the

source. Thus, consideration must be given to the following two factors in the choice of

wellbore-pressure computation point for horizontal wells:

I. Unlike vertically fractured wells, horizontal well response is a function of rwD. The

effect of wellbore radius in vertical fracture solution can be ignored, because the

wellbore is relatively far from the closest outer boundary; this is not the case in

horizontal wells. The proximity of the wellbore to the boundary in the z-direction

makes the effect of wellbore radius more critical in horizontal wells,

II. The pressure outside the source is higher than the pressure inside the source.

Therefore computing the wellbore pressure at a finite radius outside the source

could lead to a significant error depending on the value of rwD. (see Figure 5.1.4)

94

Figure 5.1.4: Illustration of the Pressure Profile in a Horizontal Well

Unlike conventional horizontal well models, it is possible to compute wellbore

pressure response inside the source using the solid bar source solution (Equation 5.1.1).

However, it can be readily decided when the line-source assumption for the finite-radius

horizontal well becomes acceptable; at this point the error introduced in the definition of

the wellbore-pressure computation point would not have a significant impact on the

accuracy of the results. Figures 5.3.1 and 5.3.3 compare the computed wellbore pressure

from Equation 5.1.1 (pwD computed at the point: yD = rwD, zD = zwD inside the source)

versus the wellbore pressure computed using the line source approximation (pwD

computed at the point: yD = rwD, zD = zwD outside the source) for rwD values of 10-4 and

5x10-4 respectively. An increase in rwD can be viewed in two different ways: (i) increase

ppwwff

PPrreessssuurree,, ppssiiaa

DDiissttaannccee ffrroomm ssoouurrccee,, fftt

PPwwff**

EErrrroorr == ((ppwwff** -- ppwwff))

rrww

95

in wellbore diameter with constant wellbore length, indicating that pwD is computed

further away from the source in the case of line source approximation, and (ii) decrease in

wellbore length with constant wellbore diameter, indicating a horizontal drainhole. Table

5.3.1 shows a comparison between the line source solution and the solid bar solution. The

larger the wellbore diameter or the shorter the wellbore length, the higher the error

introduced by the line source approximation. For the values of rwD less than or equal to

10-4 (large wellbore length), the line source approximation yields acceptable results

within 5% error, while for the values of rwD greater than 10-4 (short wellbore length/

horizontal drainhole), the line source approximation does not accurately compute the

horizontal wellbore response. Hence, the solid bar source should be used particularly at

early time (tD < 10-3).

96

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+021.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

Line Source Approximation


0.01

0.02

0.1

0.2

1.0

2.0

4.0

LD=10.0


zwD=0.5

rwD=10-4

hfD=0.0

Figure 5.3.1: Pressure Response for Horizontal Well -Infinite Conductivity Case (rwD = 10-4)

97

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Der

ivat

ive

Res

pons

e, p

' wD

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

Line Source Approximation

Solid Bar Source Solution0.01

0.02

0.1

0.2

1.0

2.0

4.0

LD=10.0


zwD=0.5

rwD=10-4

hfD=0.0

Figure 5.3.2: Derivative Response for Horizontal Well -Infinite Conductivity Case (rwD = 10-4)

98

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+021.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

Line Source ApproximationSolid Bar Source SolutionVertical Fracture

Solution

0.01

0.02

0.1

0.2

1.0

2.0

4.0

LD=10.0

zwD=0.5

rwD=5x10-4

hfD=0.0

Figure 5.3.3: Pressure Response for Horizontal Well -Infinite Conductivity Case (rwD = 5x10-4)

99

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Der

ivat

ive

Res

pons

e, p

' wD

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

Line Source ApproximationSolid Bar Source Solution


0.01

0.02

0.1

0.2

1.0

2.0

4.0

LD=10.0

zwD=0.5

rwD=5x10-4

hfD=0.0

Figure 5.3.4: Derivative Response for Horizontal Well -Infinite Conductivity Case

(rwD = 5x10-4)

100

Table 5.3.1: Influence of Computation Point on pwD for Horizontal Well-Infinite Conductivity Case (LD=0.05, zwD=0.5)

rwD=10-4 rwD=5x10-4

tD Line Source

Solid Bar Source

% Error

Line Source

Solid Bar Source

% Error

10-6 0.13540 0.14378 5.83 0.05642 0.07083 20.35 10-5 0.19290 0.20127 4.16 0.11261 0.12660 11.05 10-4 0.25050 0.25883 3.22 0.17003 0.18398 7.58 10-3 0.30930 0.31765 2.63 0.22883 0.24278 5.74 10-2 0.43000 0.43831 1.90 0.34949 0.36343 3.84 10-1 0.74800 0.75639 1.11 0.66757 0.68151 2.05

1 1.45700 1.46495 0.54 1.37613 1.39008 1.00 10 2.52100 2.52925 0.33 2.44042 2.45438 0.57

100 3.66200 3.67089 0.24 3.58206 3.59602 0.39 1000 4.81300 4.82122 0.17 4.73238 4.74635 0.29

101

5.4 Effect of Dimensionless Wellbore Radius on Horizontal Well Response

Computations indicate that for rwD ≤ 10-4 the transient-response of a horizontal

well is identical to that of a partially penetrating vertical fracture system, and for rwD >

0.01 the transient response of a horizontal well is indistinguishable from that of a

horizontal fracture system. Analysis of computed dimensionless pressure reveals that the

pressure-transient behavior of any horizontal well system is governed by two critical

parameters: (i) rwD and (ii) LD. Figures 5.4.1 and 5.4.2 show the transition from a partially

penetrating vertical fracture transient pressure behavior to a typical horizontal fracture

transient pressure response as dimensionless wellbore radius increases from zero to unity.

This demonstrates the robust nature of the solid bar solution. The solid bar solution can

be used to analyze the transient pressure behavior of both hydraulic fracture

(vertical/horizontal) systems and horizontal well/drainhole. A significant disagreement

was observed between the horizontal fracture solution developed in Ref. 2 and the solid

bar solution evaluated at rwD = 1.0 for values of tD < 10-3. This is due to the following

reasons: (i) the solution presented in Ref. 2 was developed for a solid cylinder source,

while the solution presented in this work is developed for a solid bar source (ii) the solid

cylinder source solution contains the Io function, which tends to infinity for small values

of tD, and (iii) the integral term in the solid cylinder source solution causes the truncation

error to grow, especially during the early time period. For the values of tD > 10-3 an

excellent overall agreement was observed between the solid bar source solution and the

solid cylinder source solution (< 0.5% difference).

102

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

Vertical Fracture Solution (Ref. 1)

Horizontal Fracture Solution (Ref. 2)

rwD=1.0-4

0.0125

0.125

0.25

0.5

1.0

rwD=2rw/L

zwD=0.5hfD=0.0LD=0.05

Figure 5.4.1: Effect of rwD on the Transient Pressure Behavior of Horizontal Wells-Uniform Flux

103

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03Dimensionless Time, tD

Der

ivat

ive

Res

pons

e, p

' wD

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

rwD=1.0-4

0.01250.125

0.25

0.5

1.0

rwD=2rw/L

zwD=0.5hfD=0.0LD=0.05

Figure 5.4.2: Effect of rwD on the Derivative Response of Horizontal Wells-Uniform Flux

104

5.5 Effect of Dimensionless Height on Horizontal Well Response

The line source idealization views a horizontal well as a vertical-fracture where

the fracture height approaches zero in the limit of the Z-function. Clonts and Ramey3

were one of the first authors to impose this limit on the horizontal well solution. To pass

this limit on the Z-function, they argued that the value of )sin(Ψ can be approximated by

Equation 5.5.1 when the value of hfD approaches zero.

)sin(Ψ ≈ Ψ (5.5.1)

Where

fDhn5.0 π=Ψ (5.5.2)

for n = 1, 2, 3…∞

To validate the applicability of Equation (5.5.1) to horizontal well solutions, a

simple numerical experiment was conducted using values of hfD that are likely to be

encountered in practice. In Figure 5.5.1 we note that as ‘n’ increases, there is a significant

difference between the values of )sin(Ψ and Ψ for hfD = 0.015. This trend is expected

when we consider the fact that Equation 5.5.1 only holds for small values of Ψ .

However, since the value of ‘n’ is increasing Ψ begins to grow and Equation 5.5.1 may

no longer hold. This observation has a significant impact on the method used in

computing the horizontal well response, particularly for the following cases: (i) for thin

reservoirs the value of hfD could be as high as 0.025, and (ii) at early time (tD = 10-6) the

number of terms ‘n’ required for the Z-function to converge ranges from 100 to 15,000

for the range of LD - values investigated in this work. Figure 5.5.2 and Table 5.5.1

illustrates the effect of hfD on the computed wellbore response of a horizontal well, where

105

the line source solution is represented by that of zero-hfD. Considering this results, we

note a significant difference between the line source approximation and the solid bar

source solution during the early time period (tD ≤ 10-2). From this observation, we can

conclude that the line source approximation tends to overestimate the partial penetration

effect of a horizontal well.

106

0.0

0.5

1.0

1.5

0 5 10 15 20 25 30 35 40 45 50

n

Sin(Ψ

) or Ψ

0

0.5

1

1.5

Sin(Ψ) Ψ

hfD=0.015

hfD=0.005

Ψ=sin(nπhfD)

Figure 5.5.1: The Effect of Number of Term ‘n’ on the Line Source Approximation As hfD Approaches Zero

107

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

hfD=0.0 (Line Source Solution)

0.010.015


zwD=0.5

rwD=10-4

LD=0.05

Figure 5.5.2: Effect of hfD on Transient Pressure Behavior of Horizontal Wells-Infinite Conductivity

108

Table 5.5.1: Effect of hfD on pwD for Horizontal Well-Infinite Conductivity Case (LD=0.05, rwD=10-4, zwD=0.5)

hfD tD

0.0 0.010 0.015 10-6 0.07189 0.05669 0.0505010-5 0.10064 0.08538 0.0791210-4 0.12942 0.11416 0.1078910-3 0.16964 0.15438 0.1481110-2 0.29006 0.27479 0.2685210-1 0.60813 0.59287 0.58660

1 1.31670 1.30144 1.2951710 2.38100 2.36574 2.35947

100 3.52264 3.50738 3.501111000 4.67297 4.65770 4.65144

109

5.6 The Concept of Physically Equivalent Models (PEM)

To compare hydraulically fractured well versus horizontal well productivities, we

introduce the concept of physically equivalent model (PEM). Two models are said to be

physically equivalent if both models produce identical transient pressure behaviors under

the same reservoir conditions. The goal of this concept is to find a dimensionless rate

function (β -function) for which a slab source (Figure 5.6.1) yields the same pressure

drop as that of a solid bar source (Figure 5.6.2). Suppose a step-wise continuous function:

)t( Dβ exists, such that Equation 5.6.1 is true for all time.

110

Figure 5.6.1: Base Model (Slab Source)

Figure 5.6.2: Primary Model (Solid bar Source)

wf

hf

zf

00==∂∂

∂∂

==hhzzzz

pp

00==∂∂

∂∂

==00 zzzz

pp

z

y

zf

hf

z

y

00==∂∂

∂∂

==hhzzzz

pp

00== ∂∂

∂∂

==00 zzzz

pp

111

∫ ττ−τβ=Dt

0D

'DBaseDimPrD d)t(P)()t(P … (5.6.1)

Integrating by part, Equation 5.6.1 may be written as

( ) ττ−τβ+β= ∫+ d)t(p

dd)t(pt)t(p DDBase

t

0DDBase0DDimPrD

D

(5.6.2)

Where:

)t(p DimPrD = constant rate solution for the primary model (Solid bar source).

)t(p DDBase = constant rate solution for the base model (Slab source).

Expressing Equation 35 in discrete form such that ...n 1, 0,j ,tt....tt 1DjDj1D0D =<<< +

and 0t D0 ≥ :

( ) ( ) ( )[ ] ξ+−β−β+β

=

∑−

=++

1n

0i1DiDDBaseDi1DiDDBase0D

DimPrD

)tt(ptt)t(pt

)t(p (5.6.3)

Where

1DijDijDi t)1(tt +θ−+θ= (5.6.4)

Where: 10 j ≤θ≤

The choice of jθ determines the stability and accuracy of the resulting numerical scheme.

Like most schemes, the value of ( )Dtβ at 0t D = is required. Raghavan40 suggested that,

if this value is not known, it is best to assume that 0j =θ or 1DiDi tt += another way of

overcoming this problem is to set the initial value ( ) 0t 0D =β + .

112

For small time intervals, the truncation error: ( )ξ approaches zero ( )0 →ξ , and

( ) ( ) ( )[ ]∑−

=++ −β−β+β

≈1n

0i1DiDDBaseDi1DiDDBase0D

DimPrD

)tt(ptt)t(pt

)t(p (5.6.5)

The system of equations in the Equation 5.6.5 can be expressed in matrix as:

PBx = (5.6.6)

Where:

mxmmmm1mm3m2m1

1m1m1m31m21m1

332313

2212

11

bb...bbb0b...bbb........................0...0bbb00...0bb00...00b

B

=

−

−−−−−

(5.6.7)

xm1m1

1m,1

13

12

11

PP

.

.

.PPP

P

=

−

xm1m

1m

3

2

1

.

.

.FunctionX

ββ

βββ

=−β=

−

(5.6.8)

Where:

)tt(pb 1DnDmDBasenm −−= (5.6.9)

)tt(pP 1DnDmimPrDnm −−= (5.6.10)

113

( ) ( )[ ] [ ]mm

m11DmDmm b

SUMptt −=β−β=β − (5.6.11)

Where:

m,1m1mm33m22m11 b...bbbSUM −−β++β+β+β= (5.6.12)

The structure of the base matrix - B may vary depending on the deconvolution

scheme employed in solving Equation 5.6.5. The structure of this matrix affects the

accuracy and stability of Equation 5.6.6. For the deconvolution scheme employed in this

work, the matrix - B is a lower diagonal matrix. This type of matrix structure makes it

easy to solve Equation 5.6.6.

From a computational point of view, it is recommended that small time intervals

be chosen during the early time period, with increasing step size during the medium and

late time periods. Stability consideration dictates this choice. It is desired that the error

does not propagate without bound. A stability criterion (Equation 5.6.12) developed in

Ref. 40 indicates that Equation 5.6.6 is unconditionally stable if β -function is a

monotonically decreasing function (see Equation 5.6.12 for 0 ≤ ℓ ≤ n-1).

1)t(

)t()t(0D

D1D ≤β

β−β + ll (5.6.12)

Figure 5.6.1 presents a log-log plot of β -function vs. tD for a set of physically equivalent

models (Base Model: fully penetrating vertical fracture, Primary Model: horizontal well);

the shape of the computed β -function depends on the deconvolution scheme employed in

solving Equation 5.6.6. The base model corresponds to the curve of β -function = 1.0. In

Figure 5.6.1 we note that the maximum truncation error occurs at the first step in each log

114

cycle. This is due to the logarithm increase in partition step size. For time intervals where

( ) ( )[ ]Di1Di tt β−β + approaches zero, the truncation error is negligible. Figure 5.6.2

illustrates a composite plot for a pair of PEM. This plot shows a combination of β -

function for which the transient pressure response of a fully penetrating vertical fracture

is exactly identical to that of horizontal well with rwD = 10-4. This procedure permits the

representation of a horizontal well by an equivalent vertically-fractured well in numerical

simulation studies.

115

0.01

0.1

1

10

100

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02


β-Fun

ctio

n

0.01

0.1

1

10

1001.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

rwD=10-4

0.0125

0.125

0.25

0.50

1.0

Base Model: Fully Penetrating Vertical FracturerwD=0.0, zwD=0.5, zD=0.5, hfD=1.0, LD=0.05

Primary Model: Horizontal Well/FracturezwD=0.5, zD=0.5, hfD=0.0, LD=0.05

Base Model

0.005

Figure 5.6.1: Log-Log Plot of β -Function vs. tD – Uniform Flux41

116

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02


Dim

ension

less

Pre

ssur

e, p

wD

0.1

1

10

1001.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

β-Fun

ctio

n

Primary ModelBase ModelBase Model w/Beta-FunctionBeta-Function

Base Model: Fully Penetrating Vertical FracturerwD=0.0, zwD=0.5, zD=0.5, hfD=1.0, LD=0.05

Primary Model: Horizontal Well

rwD=10-4, zwD=0.5, zD=0.5, hfD=0.0,

Figure 5.6.2: Composite Plot for a Pair of PEM41

117

CHAPTER VI

CONCLUSION

Having presented the problems, objectives, and results of investigations in the

previous chapters, we arrived at the following conclusion:

1. A new horizontal well solution capable of computing wellbore pressure response

inside the source is developed. This solution views a horizontal well as a well

producing from a partially penetrating solid bar source.

2. Analysis of the solid bar source solution reveals, for rwD ≤ 10-4 (long wellbore length)

the line source approximation yields acceptable results within 5% error. For rwD > 10-

4 (short wellbore length/ horizontal drainhole) the error introduced by the line source

approximation in the computed horizontal wellbore response is about 10 to 20%.

Hence, the solid bar source should be used for this range of rwD.

3. The pressure transient behavior of any horizontal well system is governed by two

critical parameters: (i) rwD and (ii) LD. For rwD ≤ 5x10-4 the transient-response of a

horizontal well is identical to that of a partially penetrating vertical fracture system

with zero fracture height. While, for rwD ≥ 5x10-3 the transient response of a

horizontal well is indistinguishable from that of a horizontal fracture system.

4. For a thin reservoir system (hfD > 0.005) the line source approximation tends to

overestimate the partial penetration effect of a horizontal well. The effect of hfD

should be accounted for in this case.

118

5. The β - function technique can be used to convert the transient-response of a

horizontal well into an equivalent vertical fracture response. This procedure permits

the representation of a horizontal well by an equivalent vertically fractured well in

numerical simulation studies.

6. This analytical solution reduces to the existing fully/partially penetrating vertical

fracture solution developed by Raghavan et al1. as the aspect ratio tends to zero, and a

horizontal fracture solution is obtained as the aspect ratio tends to unity. This new

horizontal fracture solution yields superior early time (tDxf < 10-3) solution compared

with the existing horizontal fracture solution developed by Gringarten and Ramey2,

and exhibits excellent agreement for tDxf > 10-3.

119

BIBLIOGRAPHY

1. Raghavan, R., Thomas, G.W., and Uraiet, A.: “Vertical Fracture Height: Effect on Transient Flow Behavior”, paper SPE 6016 presented at the SPE-AIME 51st Annual Fall Meeting, New Orleans, Oct.3-6, 1976.

2. Gringarten, A. C. and Ramey, H. J.: “Unsteady State Pressure Distribution Created by

a Well With a Single Horizontal Fracture, Partial Penetration, or Restricted Entry”, Soc. Pet. Eng. J. (August, 1974) 413-426, Trans., AIME, Vol. 257.

3. Clonts, M.D. and Ramey, H.J., Jr.: “Pressure Transient Analysis for Wells with

Horizontal Drainholes,” paper SPE 15116 presented at the 1986 California Regional Meeting, Oakland, CA, April 2-4.

4. Daviau, F., Mouronval, G., and Curutchet, P.: “Pressure Analysis for Horizontal

Wells,” paper SPE 14251 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Sept 22-25.

5. Goode, P.A., and Thambynayagam, R.K.M.: “Pressure Drawdown and Buildup

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SPEFE (Dec. 1989) 567. 7. Carvalho, R.S. and Rosa, A.J.: “A Mathematical Model for Pressure Evaluation in an

Infinite-Conductivity Horizontal Well,” SPEFE (Dec. 1989) 567. 8. Odeh, A. S. and Babu, D.K.: “Transient Flow Behavior of Horizontal Wells:

Pressure Drawdown and Buildup Analysis,” SPEFE (March 1990) 7. 9. Carvalho, R.S. and Rosa, A.J.: “Transient Pressure Behavior for Horizontal Wells in

Naturally Fractured Reservoirs,” paper SPE 18302 presented at the 1988 SPE Annual Technical Conference and Exhibition, Houston, TX, Oct. 2-5, 1988.

10. Kuchuk, F. J., Goode, P.A., Brice, B.W., Sherard, D.W., and Thambynayagam,

R.K.M.: “Pressure-Transient Analysis for Horizontal Wells,” JPT (Aug. 1990) 974. 11. Kuchuk, F. J., Goode, P.A., Wilkinson, D.J., and Thambynayagam, R.K.M.:

“Pressure-Transient Behavior of Horizontal Wells with and without Gas Cap or Aquifer,” SPEFE (March 1991) 86.

12. Ozkan, E. and Raghavan, R.: “Performance of Horizontal Wells Subject to Bottom-

water Drive,” SPERE (Aug. 1990) 375.

120

13. Aguilera, R. and Ng, M.C.: “Transient Pressure Analysis of Horizontal Wells in

Anisotropic Naturally Fractured Reservoirs,” SPEFE (March 1991) 95. 14. Kuchuk, F. J.: “Well Testing and Interpretation for Horizontal Wells,” JPT (Jan.

1995) 36. 15. Du, K., and Stewart, G.: “Transient Pressure Response of Horizontal Wells in

Layered and Naturally Fractured Reservoirs with Dual-Porosity Behavior,” paper SPE 24682 presented at the 1992 SPE Annual Technical Conference and Exhibition, Washington, DC, Oct. 4-7, 1992.

16. Kuchuk, F. J. and Habashy, T.: “Pressure Behavior of Horizontal Wells in Multilayer

Reservoirs with Crossflow,” SPEFE (March 1996) 55. 17. Ozkan, E., Sarica, C., Haciislamoglu, M., Raghavan, R.: “Effect of Conductivity on

Horizontal-Well Pressure-Behavior,” SPE Advanced Technology Series, Vol. 3, No. 1 (March 1995) 85.

18. Suzuki, K.: “Influence of Wellbore Hydraulics on Horizontal Well Pressure Transient

Behavior,” SPEFE (Sept. 97) 175. 19. Guo, G., and Evans, R.D.: “Pressure-Transient Behavior and Inflow Performance of

Horizontal Wells Intersecting Discrete Fractures,” paper SPE 26446 presented at the 1993 SPE Annual Technical Conference and Exhibition, Houston, TX, Oct. 3-6, 1993.

20. Larsen, L., and Hegre, T.M.: “Pressure Transient Analysis of Multifractured

Horizontal Wells,” paper SPE 28389 presented at the 1994 SPE Annual Technical Conference and Exhibition, New Orleans, LA, Sept. 25-28, 1994.

21. Guo, G., Evans, R.D., and Chang. M.M.: “Pressure-Transient Behavior for a

Horizontal Well Intersecting Multiple Random Discrete Fractures,” paper SPE 28390 presented at the 1994 SPE Annual Technical Conference and Exhibition, New Orleans, LA, Sept. 25-28, 1994.

22. Horne, R.N., and Temeng, K.O.: “Relative Productivities and Pressure Transient

Modeling of Horizontal Wells with Multiple Fractures,” paper SPE 29891 presented at the SPE Middle East Oil Show, Bahrain, March 11-14, 1995.

23. Raghavan, R., Chen, C.C., and Agarwal, B.: “An Analysis of Horizontal Wells

Intercepted by Multiple Fractures,” SPEJ (Sept. 1997) 235.

121

24. Chen, C.C. and Raghavan, R.: “A multiply-Fractured Horizontal Well in a Rectangular Drainage Region,” SPEJ (Dec. 1997) 455.

25. Frick, T.P., Brand, C.W., Schlager, B., and Economides, M.J.: “Horizontal Well

Testing of Isolated Segments,” SPEJ (Sept. 1996) 261. 26. Frick, T.P. and Economides, M.J.: “Horizontal Well Damage Characterization and

Removal,” SPEPF (Feb. 1993) 15. 27. Ozkan, E. and Raghavan, R.: “Estimation of Formation Damage in Horizontal

Wells,” paper SPE 37511 presented at the Production Operations Symposium, Oklahoma City, OK, March 9-11, 1997.

28. Orkan, E.: “Analysis of Horizontal-Well Responses: Contemporary vs.

Conventional,” paper SPE 52199 presented at the SPE Mid-Continent Operation Symposium, Oklahoma City, OK, March 28-31, 1999.

29. Peaceman, D.W.: “Interpretation of Well-Block Pressures in Numerical Simulation

with Non-square Grid Blocks and Anisotropic Permeability” SPEJ (June 1983) 531-43.

30. Badu, D. K., Odeh A. S., Al-Khalifa, A.J., and McCann, R.C.: “The Relation

Between Well block and Wellbore Pressures in Numerical Simulation of Horizontal Wells” SPERE (August 1991) 324-28.

31. Badu, D.K., and Odeh, A.S.: “Productivity of a Horizontal Well” SPERE (Nov. 1989)

417-21 32. Peaceman, D. W.: “Representation of a Horizontal Well in Numerical Reservoir

simulation” paper SPE 21271 presented at the SPE Symposium on Reservoir Simulation, Anaheim, CA, February 17-20, 1993.

33. Brigham, W.E.: “Discussion of Productivity of a Horizontal Well” SPERE (May

1990) 245-5. 34. Gringarten, A. C. and Ramey, H. J.: “Unsteady-State Pressure Distribution Created

by a Well with a Single Infinite-Conductivity Vertical Fracture,” paper 4051 presented at the 47th Annual Technical Conference and Exhibition (ATCE), San Antonio, October 8 -11, 1972.

35. Gringarten, A. C. and Ramey, H. J.: “Application of P-Function to Heat Conduction

and Fluid Flow Problems,” paper 3817, submitted to SPE-AIME for publication.

122

36. Orkan, E., Raghavan, R., and Joshi, S.D., “Horizontal-Well Pressure Analysis,” SPEFE (December 1989) 567- 75.

37. Gringarten, A. C. and Ramey, H. J.: “The Use of Source and Green’s Functions in

Solving Unsteady-Flow Problems in Reservoirs,” Soc. Pet. Eng. J. (Oct. 1973) 285 – 296; Trans., AIME, Vol. 225

38. Newman, A, B.: “Heating and Cooling Rectangular and Cylindrical Solids,” Ind. and

Eng. Chem. (1936) Vol. 28, 545. 39. Hantush, M. S.: Nonsteady Flow to a Well Partially Penetrating an Infinite Leaky

Aquifer,” Proceedings Iraq Scientific Society (1957) 40. Raghavan, R.: Well Test Analysis, PTR Prentice-Hall, Inc. New Jersey pp 274-282

(1993). 41. Ogunsanya, B.O.: “A Physically Consistent Solution for Describing the Transient

Response of Horizontal Wells,” SPE94331, presented at the SPE Production Operation Symposium, Oklahoma City, OK, April 17-19, 2005.

123

APPENDIX A

APPLICATION OF GREEN’S FUNCTIONS AND THE NEWMAN PRODUCTION

SOLUTION FOR THE SOLUTION OF BOUNDARY-VALUE PROBLEMS

A.1 Green’s Function Formulation

The use of Green’s function for the solution of partial differential equations is

derived in detailed in many references. Specifically, the application to the solution of heat

conduction problems is described by Carslaw and Jaeger [1959] and Ozisik [1968]. In

this section, we illustrate the use of Green’s functions for the solution of thee-

dimensional boundary-value problem with non-homogeneous boundary conditions.

Following the work of Ozisik [1968], we consider the boundary-value problem

for a three-dimensional bounded region R, which is initially as pressure (pi), for t = 0.

For time t > 0, there is an active source or sink, q , within the region. The partial

differential equation governing the flow of a slightly compressible fluid within the

region, R, is the diffusivity equation,

( ) ,t

pct,mqp

kt

2

∂∂Φ=+∇

µ in the region R for t > 0 (A.1)

where m is a vector representing three-dimensional space.

For this illustration, we will consider general boundary conditions, such as the

general boundary conditions of the third kind

),t,r(fBp

A iii

i =+η∂

∂ on the boundary or surface s, for t > 0 (A.2)

124

While the initial condition is

ip)t,m(p = , in the region R for t = 0 (A.3)

Definitions of terms in Equation (A.1-3) include

2∇ = three-dimensional Laplace operator =

∂∂

∂∂

∂∂

2

2

2

2

2

2

z,

y,

x

P = pressure in the three-dimensional space, )t,m(p at any time t

m = three-dimensional space variable

q = source or sink term as a function of position and time

i

p

η∂∂ = outward-drawn normal to the boundary surface, si

Ai, Bi = arbitrary constants

k = permeability of the porous medium

Φ = porosity of porous medium

ct = total system compressible

µ = fluid viscosity

For mathematical convenience, if we divide both sides of Equation (A.1) by µk , then

we can rewrite it as

( )t

p1t,mq

kp2

∂∂

η=µ+∇ (A.4)

Where η is the diffusivity defined by

125

tc

k

µΦ=η

An auxiliary homogeneous partial deferential equation can be written in terms of

the Green’s function, G, as

( ) ( )t

G1t'mm

QG2

∂∂

η=τ−δ−δ

η+∇ , in the Region R for t > 0 (A.5)

The homogeneous boundary condition of the third kind is

0GBG

A ii

i =+η∂

∂ , on the boundary si for t > τ (A.6)

While the associated initial condition is

G = 0, in the region R for t < τ (A.7)

Definitions of terms in Equations (A.5-7) are as follow:

G = Green’s function for the boundary-value problem given by Equations (A.1 -3)

that describes the pressure at m at any time t due to an instantaneous point heat

source, Q of strength unity. Note that G satisfies the homogeneous boundary

condition of the third kind given by Equation (A.2)

Q = instantaneous point source/sink term of strength unity = ( ) 1t,mqk

=ηµ

( )'mm −δ = three-dimensional Dirac delta function for the space variable, i.e., for

the Cartesian coordinate system

( )τ−δ t = Dirac delta function for the time variable

τ = time variable

126

For our example, the instantaneous Green’s function, ( )τ−t,'m,mG , for the

domain or region R, with respect to the diffusivity equation (Equation A.1) is the

pressure that would be generated at the point m at the time t by an instantaneous

but fictitious source or sink of strength unity located at the point m and activated

at the time t < τ. This function must also satisfy the boundary and initial

conditions defined by Equations (A.6) and (A.7), respectively.

According to Gringarten and Ramey [1973], the instantaneous Green’s

function is a two-point function having the following properties:

1. ( )τ−t,'m,mG is a solution to the adjoint or auxiliary diffusivity

equation (Equation A.5). If L{u} represents the differential form of the

diffusivity equation, then the adjoint differential form, L’{u}, is

defined by the requirements that the expression uL{v} – vL’{u} be

integrable. For our example, L is the operator defined by (A.4)

.t

12

δδ

η−∇

and the adjoint operator, L’ is

δδ

η+∇

t

12 , τ < t

2. It is symmetrical in the two point m and m’

3. ( )τ−t,'m,mG is a delta function that vanishes at all points inside the

boundary of si, as t → τ, except at the point , where it become infinite

so that for any continuous function f(m).

127

( ) ( ) ( )'mf'dmt,'m,mG'mflimis

t=τ−∫τ→

Moreover, from the definition of unit strength instantaneous source,

( )τ−t,'m,mG also satisfies

( )∫ = 1'dmt,'m,mG , t ≥ 0

4. If the pressure is prescribed on the outer boundary si of the domain R,

then the Green’s function vanishes when m is on the boundary, si (i.e.,

Green’s function of the first kind). It the flux is prescribed on si, the

Green’s function normal derivative vanished when r is on the

boundary, si (i.e., Green’s function of the second kind). Of the domain

is infinite in extend Green’s function is aero when r is at infinity

To determine the solution to Equation (A.1) in terms of the Green’s function,

( )τ−t,'m,mG , we first rewrite both Equations (A.1 and A.5) in terms of time variable,

τ, and a point, m’ which is different from the point m in the three-dimensional space as

follows:

( )τ∂

∂η

=τµ+∇ p1,'mq

kp2 , in the region R for t < τ (A.8)

( ) ( )τ∂

∂η

−=−τδ−δη

+∇ G1tm'm

1G2 , in the region R for t < τ (A.9)

Where 2∇ is the three-dimensional Laplacian operator with respect to the space variable

m. The minus sign on the right side of Equation (A.9) is necessary because the Green’s

function depends on time t as a function of t - τ.

128

( ) ( ) ( ) ( ) ( )τ∂

∂η

=−τδ−δη

−τµ+∇−∇ Gp1ptm'm

1G,'mq

kGppG 22 (A.10)

Integrating Equation (A. 10) with respect to the space variable, m, over the region R and

with respect to the time variable, τ, over the interval τ = 0 to τ = t gives

( ) ( ) ( ) ( )

( )∫∫

∫ ∫∫ ∫∫ ∫=τ

=τ

=τ

=τ

=τ

=τ

=τ

=τ

ττ∂

∂η

=

−τδ−δτη

−ττµ+∇−∇τ

t

0R

t

0 R

t

0 R

t

0 R

22

dGp

'dm1

'pdmtm'md1

'Gdm,'mqdk

'dmGppGd

(A.11)

Apply Green’s Theorem [Davis and Snider, 1969] that defines the relationship between

volume and surface integrals, the volume in the first term on the left side of Equation

(A.11) and the evaluation over the volume R can be rewritten in terms of a surface

integral over si as

( ) dsn

Gp

n

pG'dmGppG

is iiR

22 ∫∫

∂∂−

∂∂=∇−∇ (A.12)

The integral involving the delta functions appearing in the third term on the left

side of Equation (A.11) can be evaluated using the properties of the delta function. By

definition, the Dirac function is

( )

=≠

=−δba,1

ba,0ba

Therefore, we can write

( ) ( ) ( ) ( )τ=τ−δτ−τδ∫ ∫=τ

=τ

,mp'pdm,'mpm'mdtt

0 R

(A.13)

Finally, the integral on the right side of Equation (A.11) can be evaluated as

129

( ) [ ] ( ) ( ) ( ) ( ) ( ) i

t

0

t

0

p0G0p0GtptGdGp Gp =τ−==τ=τ−=τ=τ==ττ∂

∂ =τ

=τ

=τ

=τ∫ (A.14)

which makes use of the initial conditions given by Equations (A.3) and (A.7).

Substitution of Equations (A.12), (A.13), and (A.14) into (A.11) and rearranging

yields the following equations for ( )t,mp in terms of Green’s function (G).

dSn

Gp

n

pGd'Gdm)t,'m(qd

k'dm)0(Gp)t,m(p

is ii

t

0R

t

0R

i ∫∫∫∫∫

∂∂−

∂∂τη+τηµ+=τ=

=τ

=τ

=τ

=τ

(A.15)

Each of the terms in Equation (A.15) has a physical significance. The first term

represents the effects of the initial pressure distribution in the system. We can express the

pressure change at any point in space and time, ( )t,mp∆ as the difference between the

initial pressure condition in Equation (A.14) and the ( )t,mp defined by Equation (A.15)

as

( ) ∫ −=τ=∆R

i )t,m(p'dm)0(Gpt,mp (A.16)

The third term or the right side of Equation (A.15), which represent the effects of

the boundary condition functions, become zero for all boundary conditions considered in

the dissertation. As shown by Gringarten and Ramey [1973], the integral in the third

terms becomes zero either when the computational domain, R, is infinite or if the domain

in finite and when the outer boundary conditions is that flux or zero pressure for all

values of m and all times. Therefore, under these conditions,

0dSn

Gp

n

pGd

is ii

t

0

=

∂∂−

∂∂τη ∫∫

=τ

=τ

130

Substituting Equation (A.16) into Equation (A.15) and allowing the third terms to

become zero results in

∫∫∫∫ τΦ

=τηµ=∆R

t

0tR

t

0

'Gdm)t,m(qdc

1'Gdm)t,m(qd

k)t,m(p (A.17)

Assuming the fluid withdrawal is uniform over the source volume (i.e. a uniform-

flux source), Equation (A.17) can be rewritten as

∫ ττ−τΦ

=∆t

0t

d)t,m(S)(qc

1)t,m(p (A.18)

Where:

∫=R

'dm)t,m(G)t,m(S

is defined as the instantaneous uniform-flux source function. A continuous source

function is obtained by integrating the right side of equation (A.19) with respect to time.

Moreover, (A.18) forms the basis for development of the hydraulic fracture and

horizontal well solutions represented in the dissertation.

131

APPENDIX B

HYDRAULIC FRACTURE/HORIZONTAL WELL TYPE CURVES

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

LDxf=10

5.0

2.5

0.75

0.25

0.1

0.05

Figure B.1: Type-Curve for a Uniform Flux Horizontal Fracture System

(hfD = 0.1, zwD = 0.5, zfD = 0.5, m = 1.0)

132

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03Dimensionless Time, tDxf

Dim

ension

less

Pre

ssur

e, p

wD

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

0.05

0.1

0.25

0.75

2.5

5.0

LDxf=10.0


(hfD = 0.2, zwD = 0.5, zfD = 0.5, m = 1.0)

133

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

LDxf=0.05 LDxf=0.1 LDxf=0.25 LDxf=0.75 LDxf=2.5 LDxf=5.0 LDxf=10.0


(hfD = 0.4, zwD = 0.5, zfD = 0.5, m = 1.0)

134

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

Der

ivat

ive

Res

pons

e, p

' wD

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03


Figure B.4: Composite Type-Curve for a Uniform Flux Horizontal Fracture System

(hfD = 0.6, zwD = 0.5, zfD = 0.5, m = 1.0)

135

pD,pD' Vs. tDxf for a Uniform Flux Horizontal Fracture System(hD=0.8, zwD=0.5, zfD=0.5, m=1.0)

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

Der

ivat

ive Res

pons

e, p

' wD

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03


Figure B.5: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (hfD = 0.8, zwD = 0.5, zfD = 0.5, m = 1.0)

136

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03


Dim

ension

less

Pre

ssur

e, p

wD

Der

ivat

ive

Res

pons

e, p

' wD

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03


Figure B.6: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (hfD = 1.0, zwD = 0.5, zfD = 0.5, m = 1.0)

137

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

Der

ivat

ive

Res

pons

e, p

'wD

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03

hD=0.0 hD=0.2 hD=0.4 hD=0.6 hD=0.8 hD=1.0


(LDxf=0.05, zwD=0.5, zfD=0.5, m=1.0)

138

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

Der

ivat

ive

Res

pons

e, p

' wD

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

hD=0.0 hD=0.2 hD=0.4 hD=0.6 hD=0.8 hD=1.0

Figure B.8: Composite Type-Curve for a Uniform Flux Horizontal Fracture System (LDxf=0.1, zwD=0.5, zfD=0.5, m=1.0)

139

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

Der

ivat

ive

Res

pons

e, p

' wD

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

hD=0.0 hD=0.2 hD=0.4 hD=0.6 hD=0.8 hD=1.0


(LDxf=0.25, zwD=0.5, zfD=0.5, m=1.0)

140

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

Der

ivat

ive

Res

pons

e, p

' wD

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

hD=0.0 hD=0.2 hD=0.4 hD=0.6 hD=0.8 hD=1.0


(LDxf=0.75, zwD=0.5, zfD=0.5, m=1.0)

141

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

Der

ivat

ive

Res

pons

e, p

' wD

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

hD=0.0 hD=0.2 hD=0.4 hD=0.6 hD=0.8 hD=1.0


(LDxf=1.0, zwD=0.5, zfD=0.5, m=1.0)

142

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

Der

ivat

ive

Res

pons

e, p

' wD

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

hD=0.0 hD=0.2 hD=0.4 hD=0.6 hD=0.8 hD=1.0


(LDxf=2.5, zwD=0.5, zfD=0.5, m=1.0)

143

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E -01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

Der

ivat

ive

Res

pons

e, p

' wD

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

hD=0.0 hD=0.2 hD=0.4 hD=0.6 hD=0.8 hD=1.0


144

1.0E -06

1.0E -05

1.0E -04

1.0E -03

1.0E -02

1.0E -01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03


Dim

ension

less

Pre

ssur

e, p

wD

Der

ivat

ive

Res

pons

e, p

' wD

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+011.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

hD=0.0 hD=0.2 hD=0.4 hD=0.6 hD=0.8 hD=1.0


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