Ogunsanya bo phd

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
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
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
System (hfD = 0.2, zwD = 0.5, zfD = 0.5, m = 1.0)………………………...…………132
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
Fracture System (hfD = 0.8, zwD = 0.5, zfD = 0.5, m = 1.0)……………….…………135
Fracture System (hfD = 1.0, zwD = 0.5, zfD = 0.5, m = 1.0)………………….………136
Fracture System (LDxf=0.05, zwD=0.5, zfD=0.5, m=1.0)……………………………137
Fracture System (LDxf=0.1, zwD=0.5, zfD=0.5, m=1.0)……………………..………138
Fracture System (LDxf=0.75, zwD=0.5, zfD=0.5, m=1.0)…………………..………140
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
σ
σ
σ
= 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=
∂
∂
=hz
z
p
z
x
0=
∂
∂
=0z
z
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
Dxf
DxfDfDfDDxf
t
0 Dxf
2
D
Dxf
D
x
Dxf
D
x
y
DxfDDDD
t
dt
)t,z,z,h,L(Z
t4
y
exp
t2
x
k
k
erf
t2
x
k
k
erf
k
k
4
)t,z,y,x(p
Dxf
•












 −
•














−
+
+
π
=
∫
(2.1.1)
Where:
[ ])t,z,y,x(pp
qB2.141
kh
)t,z,y,x(p iDxfDDDD −
µ
= (2.1.2)
2
ft
Dxf
xc
kt001056.0
t
µφ
= (2.1.3)
xf
w
D
k
k
x
)xx(2
x
−
= (2.1.4)
yf
w
D
k
k
x
)yy(2
y
−
= (2.1.5)
h
z
zD = (2.1.6)
h
h
h f
fD = (2.1.7)
zf
Dxf
k
k
x
h2
L = (2.1.8)

12
( ) ( ) ( )








πππ




 π
−
π
+
=
∑
∞
=1n
wDDfD2
Dxf
Dxf
22
fD
DxfDfDfDDxf
zncoszncoshn5.0sin
L
tn
exp
n
1
h
0.4
1
)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
DimensionlessPressure,pwD
1.00E-01
1.00E+00
1.00E+01
1.00E+02
1.00E-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
1.00E-01
1.00E+00
1.00E+01
1.00E-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
2
D
Dxf
D
x
Dxf
D
x
y
DxfDDDD
t
dt
t4
y
exp
t2
x
k
k
erf
t2
x
k
k
erf
k
k
4
)t,z,y,x(p
Dxf
∫ 




 −
•














−
+
+
π
=
(2.1.10)
At short time,
β=
−
+
+
Dxf
D
x
Dxf
D
x
t2
x
k
k
erf
t2
x
k
k
erf (2.1.11)
Where






>
=
<
=β
xD
xD
xD
kkxfor0
kkxfor1
kkxfor2
(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
Dxf
t
0 Dxf
2
D
y
DxfDDDD
t
dt
t4
y
exp
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
D
D
Dxf
2
D
Dxf
y
DxfDDDD
hhfor
t2
y
erfcy
2t4
y
expt
k
k
2
)t,z,y,x(p
=
















−
π
−





−π
β
=
(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
Dxf
t
0 Dxf
2
D
yfD
DxfDDDD
t
dt
t4
y
exp
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
D
D
Dxf
2
D
Dxf
yfD
DxfDDDD
hhfor
t2
y
erfcy
2t4
y
expt
k
k
h2
)t,z,y,x(p
>
















−
π
−





−π
β
=
(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
2
D
Dxf
Dx
Dxf
Dx
y
Df
t
dt
t4
y
exp
t2
xkk
erf
t2
xkk
erf
4
kk
p
Dxf
∫ 












 −
•







 −
+
+π
=
(2.1.16)
and
[ ]}
Dxf
Dxf
DxfDfDfDDxf
t
0 Dxf
2
D
Dxf
Dx
Dxf
Dx
y
'
t
dt
1)t,z,z,h,L(Z
t4
y
exp
t2
xkk
erf
t2
xkk
erf
4
kkF
Dxf
−
•









 −
•







 −
+
+π
= ∫
(2.1.17)
Substituting Equation 2.1.9 into 2.1.17, we get
( ) ( ) ( )
Dxf
Dxf
1n
wDDfD2
Dxf
Dxf
22
fD
t
0 Dxf
2
D
Dxf
Dx
Dxf
Dx
y
'
t
dt
zncoszncoshn5.0sin
L
tn
exp
n
1
h
0.4
t4
y
exp
t2
xkk
erf
t2
xkk
erf
4
kkF
Dxf




πππ




 π
−
π




•




 −
•







 −
+
+π
=
∑
∫
∞
=
(2.1.18)
Recall,

18
( ) α







 α−
π
=
−
+
+
∫
+
−
d
t4
kkx
exp
t
kk
t2
xkk
erf
t2
xkk
erf
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
2
D
2
xD
Df dtd
t4
ykkx
exp
4
1
p
Dxf
α







 −α−
= ∫ ∫
+
−
(2.1.20)
and
( ) ( ) ( )
( )
Dxf
Dxf
t
0
1
1 Dxf
2
D
2
xD
2
Dxf
Dxf
22
1n
wDDfD
fD
'
t
dtd
t4
ykkx
exp
L
tn
exp
zncoszncoshn5.0sin
n
1
h
1
F
Dxf
α







 −α−





 π
−
πππ
π
=
∫ ∫
∑
+
−
∞
=
(2.1.21)
Revising the integral in Equation 2.1.20, we get
( ) α







 −α−
−= ∫
+
−
d
t4
ykkx
Ei
4
1
p
1
1 Dxf
2
D
2
xD
Df
(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−
= ∫
+
−
d80907.0
ykkx
t
ln
4
1
p
1
1
2
D
2
xD
Dxf
Df
(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
+πππ
π
=
∫∑
+
−
∞
=
d
L
n
srKzncoszncoshn5.0sin
n
1
hs
2
sF
2
Dxf
22
D
1
1
0
1n
wDDfD
fD
'
(2.1.24)
Where:
( ) 2
D
2
xDD ykkxr −α−= (2.1.25)
If
2
Dxf
2
L/01.0s π≤ (2.1.26)
or
22
DxfDxf /L100t π≥ (2.1.27)
We can assume that ( ) ( )2
Dxf
222
Dxf
22
L/nL/ns π≈π+ and the long time
approximation of Equation 2.1.24 is given by
( ) ( ) ( ) ( ) α




 π
πππ
π
= ∫∑
+
−
∞
=
d
L
n
rKzncoszncoshn5.0sin
n
1
hs
2
sF
Dxf
D
1
1
0
1n
wDDfD
fD
'
(2.1.28)
Evaluating the inverse Laplace transform of Equation 2.1.28, we obtain the
following expression
( ) ( ) ( ) α




 π
πππ
π
= ∫∑
+
−
∞
=
d
L
n
n
1
h
2
F
Dxf
D
1
1
0
1n
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
( ) ( )[ ]{
( ) ( )[ ]
( )[ ]}x
2
D
2
DDD
2
D
2
xDxD
2
D
2
xDxDDD
kkyx/y2arctany2
ykkxlnkkx
ykkxlnkkx25.0)y,x(
−+−
+++−
+−−=σ
(2.1.31)
For
( )[ ]



+±
π
≥
2
D
2
xD
22
Dxf
Dxf
ykkx25
/L100
t (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
2
D
Dxf
D
Dxf
D
Df
t
dt
t4
y
exp
t2
x1
erf
t2
x1
erf
4
p
Dxf
∫ 












 −
•







 −
+
+π
= (2.1.33)
[ ]}
Dxf
Dxf
DxfDfDfDDxf
t
0 Dxf
2
D
Dxf
D
Dxf
D'
t
dt
1)t,z,z,h,L(Z
t4
y
exp
t2
x1
erf
t2
x1
erf
4
F
Dxf
−




•




 −
•







 −
+
+π
= ∫
(2.1.34)
Using the relation given in Equation 2.1.19, Equation 2.1.33 may be expressed as
( )
α




 −α−
−= ∫
+
−
d
t4
yx
Ei
4
1
p
1
1 Dxf
2
D
2
D
Df
(2.1.35)
Using Equation 2.1.19 and 2.1.35, Equation 2.1.15 can be written as

22
( )
( ) ( ) ( ) α









 π
πππ
π
+












+
−α−
=
∑
∫
∞
=
+
−
d
L
n
n
1
h
2
80907.0
ykkx
t
ln
4
1
p
Dxf
D0
1n
wDDfD
fD
1
1
2
D
2
xD
Dxf
D
(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








 π
πππ
π
+












+
−α−
=
∑
∫
∞
=
−
d
L
n
n
1
h
2
80907.0
ykkx
t
ln
4
1
qp
Dxf
D0
1n
wDDfD
fD
M/m
M/1m
2
D
2
xD
Dxf
mD
(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









 π
πππ
π
+












+
−α−
−=
∑
∫
∞
=
−
−−
d
L
n
n
1
h
2
80907.0
ykkx
t
ln
4
1
qp
Dxf
D0
1n
wDDfD
fD
M/m
M/1m
2
D
2
xD
Dxf
mD
(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:
( )
( ) ( ) ( ) 




 π
πππ
π
+








+
−α−
=α
∑
∞
= Dxf
D0
1n
wDDfD
fD
2
D
2
xD
Dxf
L
n
n
1
h
2
80907.0
ykkx
t
ln
4
1
)(f
(2.1.40)
Let
( )
( ) ( ) ( ) α








 π
πππ
π
+












+
−α−
=
∑
∫
∞
=
−
d
L
n
n
1
h
2
80907.0
ykkx
t
ln
4
1
p
Dxf
D0
1n
wDDfD
fD
M/m
M/m
2
D
2
xD
Dxf
Dm
(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
1m
1DmDmmD ppqp (2.1.42)
and
f
M
1m
ffm
q
M
hxq
=∑=
(2.1.43)
or
M
q
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,jt,0,
M2
1j2
xpt,0,
M2
1j2
xp DDwDDDwD =




 +
==




 −
= (2.1.45)
The resulting pressure drop from the total length of the fracture can be expressed
( ) )L,z,z,y,x(F)y,x(80907.2tln5.0
)t,L,z,z,y,x(p
DwDDDDDDD
DDwDDDDD
+σ++
=
(2.1.46)
Where:
( )
( ) ( )





















−
+




 −
−+





−+





 −
++
−








+




 −
−




 −
−−








+




 −
+




 −
++








+





+





+−












+





−





−=σ ∑=
2
D22
2
D
2
D2
2
2
D
2
D
2
2
D
2
D
D
D
2
D
2
DD
2
D
2
DD
2
D
2
DD
M
1m
2
D
2
DDDD1
y
M
1mm4
M
1mm
yx
M
m
yx
M
1mm
yx
M
y2
arctany2
y
M
1m
xln
M
1m
x
y
M
1m
xln
M
1m
x
y
M
m
xln
M
m
x
y
M
m
xln
M
m
x25.0)y,x(
(2.1.47)
and

25
( ) ( ) ( )
( )
( )
α













 π
−




 π
πππ
π
=
∫∫
∑∑
−
−−−
=
∞
=
d
L
n
rK
L
n
rK
zncoszncoshn5.0sin
n
1
h
2
)L,z,z,y,x(F
M/1m
M/1m Dxf
D0
M/m
M/m Dxf
D0
M
1m 1n
wDDfD
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=
∂
∂
=hz
z
p
z
r
0=
∂
∂
=0z
z
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
0
DrfDrfDfDfDDrfDrfDDrfDDD 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(pp
q
hk2
)t,h,z,z,r(p fffi
f
r
DDfDDDD f
−
µ
π
= (2.2.2)

28
2
ft
r
Drf
rc
tk
t
µφ
= (2.2.3)
f
D
r
r
r = (2.2.4)
h
z
zD = (2.2.5)
h
h
h f
fD = (2.2.6)
z
r
f
Drf
k
k
r
h
L = (2.2.7)














−










−
= ∫
1
0
'
D
'
D
Drf
2'
D
Drf
'
DD
o
Drf
Drf
2
D
DrfD drr
t4
r
exp
t2
rr
I
t2
t4
r
exp
)t,r(p (2.2.8)
( ) ( ) ( )








πππ




 π
−
π
+
=
∑
∞
=1n
wDDfD2
Drf
Drf
22
fD
DrfDfDfDDrf
zncoszncoshn5.0sin
L
tn
exp
n
1
h
0.4
1
)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
−+
=
∑
∞
=1n
wDD2
Drf
Drf
22
DrfDfDDrf
zncoszncos
L
tn
exp21
)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
0
DrfDrfDfDDrf
Drf
Drf
2
D
DrfDDD 'dt)'t,z,z,0,L(Z
't2
't4
r
exp
)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:
[ ]∫ −=σ
Drft
0
DrfDrfDfDfDDrfDrfDDrfDD '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
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 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
DerivativeResponse,p'wD
1.00E-03
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 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
r1
t D
2
D
Drf ≠
−
≤ (2.2.16)
Or in terms of real variable,
1r,10t D
4
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
0
DrfDrfDfDfDDrfDrfDDD 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
2
DDrf
D
D
D
2
D
Drf
fD
DrfDDD
t4
exp
t
t2
erfc
2
t
h
1
)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
Df
DfDrf
Drf
2
DfDrf
Drf
DrfDDD
t2
zz
erfzzL
t4
zz
exp
t
L2
)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 rr80907.1tln5.0)t,1r0(p f
2
DDrfDrfDD >−+=<≤ (2.2.21)
and
0for r80907.0
r
t
ln5.0)t,1r(p f2
D
Drf
DrfDD ≥





−+=> (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
2
Drf2Drf L
5
t
π
≥ (2.2.23)
and the P-function is equivalent to 0.25/tDrf when
( )1r25.12t 2
DDrf +≥ (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
2
Drf2
L
5
0
DrfDrfDfDfDDrfDrfD
2
Drf2DD 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=
∂
∂
=hz
z
p
z
x
0=
∂
∂
=0z
z
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
Dxf
DxfDfDfDDxf
0hfD
t
0 Dxf
2
D
Dxf
D
x
Dxf
D
x
y
DxfDDDD
t
dt
)t,z,z,h,L(Zlim
t4
y
exp
t2
x
k
k
erf
t2
x
k
k
erf
k
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)
2
t
D
Lc
kt001056.0
t
µφ
= (2.3.3)
x
w
D
k
k
L
)xx(2
x
−
= (2.3.4)
y
w
D
k
k
L
)yy(2
y
−
= (2.3.5)
h
z
zD = (2.3.6)
z
D
k
k
L
h2
L = (2.3.7)

38
( ) ( )








ππ




 π
−+
=
∑
∞
=
→
1n
wDD2
D
D
22
DDfDfDD
0hfD
zncoszncos
L
tn
exp21
)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
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00E+02
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
0.01
0.02
0.1
0.2
1.0
2.0
4.0
LD=10.0
Vertical Fracture
Solution
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
D
x
D
D
x
t2
x
k
k
erf
t2
x
k
k
erf (2.3.9)
Where:






>
=
<
=β
xD
xD
xD
kkxfor0
kkxfor1
kkxfor2
(2.3.10)
Substituting Equation 2.3.9 into Equation 2.3.1 we get
( ) ( )
D
D
1n
wDD2
D
D
22
t
0 D
2
D
y
DDDDD
t
dt
zncoszncos
L
tn
exp21
t4
y
exp
k
k
4
)t,z,y,x(p
Dxf








ππ




 π
−+





 −πβ
=
∑
∫
∞
=
(2.3.11)
Expanding Equation 2.3.11, we get
( ) ( )
D
D
1n
wDD2
D
D
22t
0 D
2
D
y
t
0 D
D
D
2
D
y
DDDDD
t
dt
zncoszncos
L
tn
exp
t4
y
exp
k
k
2
t
dt
t4
y
exp
k
k
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
2
D
y
Df
t
dt
t4
y
exp
k
k
4
p (2.3.14)
and
( ) ( )
D
D
1n
wDD2
D
D
22t
0 D
2
D
y t
dt
zncoszncos
L
tn
exp
t4
y
exp
k
k
2
F
Dxf
∑∫
∞
=
ππ




 π
−




 −πβ
=
(2.3.15)
Integrating Equation 2.3.14 with respect to tD, we get
t2
y
erfcy
2t4
y
expt
k
k
2
p
D
D
D
D
2
D
D
y
Df
















−
π
−





−π
β
= (2.3.16)
Transforming Equation 2.3.15 into Laplace space, we get
( ) ( ) ( )
( ) ( )[ ]
( )[ ]ξπ+
ξ






ξ
π+
−ξ−
ππ
βπ
=
∫
∑
∞
∞
=
2
D
22
0
2
D
2
D
22
1n
wDD
L/ns
d
4
yL/ns
expexp
zncoszncos
s2
sF
(2.3.17)
Integrating Equation 2.3.17 with respect to ξ, we get
( )
( ) ( )
( )[ ]
( )[ ]( )2
D
22
D
1n
2
D
22
wDD
L/nsyexp
L/ns
zncoszncos
s2
sF π+−
π+
ππβπ
= ∑
∞
=
(2.3.18)
Using the procedure presented in Ref. 36 we can recast Equation 2.3.18 as
( ) ( )[ ]{
( )[ ]} ( )syexp
s4
syLn2zzK
syLn2zzK
s4
L
sF
D23
2
D
2
D
2
wDD0
n
2
D
2
D
2
wDD0
D
−
βπ
−+−++
+−−
β
= ∑
+∞
−∞= (2.3.19)
For large s
( )[ ] ( )[ ]syLzzKsyLn2zzK 2
D
2
D
2
wDD0
2
D
2
D
2
wDD0 +−<<+−± (2.3.20)

41
Thus, equation 2.3.19 becomes
( )[ ] ( )syexp
s4
syLzzK
s4
L
)s(F D23
2
D
2
D
2
wDD0
D
−
βπ
−+−
β
= (2.3.21)
Inverting Equation 2.3.21 back into real space, we get
( )
















−
π
−





−π
β
−




 +−
−
β
=
D
D
D
D
2
D
D
y
D
2
D
2
D
2
wDDD
t2
y
erfcy
2t4
y
expt
k
k
2
t4
yLzz
Ei
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
2
D
2
D
2
wDDD
DDDDD
t4
yLzz
Ei
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:
( ) )L,z,z,y,x(F)y,x(80907.2tln5.0
)t,L,z,z,y,x(p
DwDDDDDDD
DDwDDDDD
+σ++
=
(2.2.24)
Where:

42
( ) ( )[ ]{
( ) ( )[ ]
( )[ ]}x
2
D
2
DDD
2
D
2
xDxD
2
D
2
xDxD
DD
kkyx/y2arctany2
ykkxlnkkx
ykkxlnkkx25.0
)y,x(
−+−
+++−
+−−
=σ
(2.2.25)
and
( ) ( ) ( ) α




 π
πππ
=
∫∑
+
−
∞
=
d
L
n
)L,z,z,y,x(F
D
D
1
1
0
1n
wDDfD
DwDDDD
(2.2.26)
When
( )[ ]



+±
π
≥
2
D
2
xD
22
Dxf
D
ykkx25
/L100
t (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=
∂
∂
=hz
z
p
0=
∂
∂
=0z
z
p
yf
hf
zw
kz
ky
0=
∂
∂
=hz
z
p
0=
∂
∂
=0z
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−τ
φ
=∆ ∫ ∫ ddMt,M,MG,Mq
c
1
t,Mp ww
t
0
Dw
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−τ
φ
=∆
t
0
f dt,z,y,xSq
c
1
t,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
( ) ( )∫ ττ
φ
=∆
t
0
fff
d,z,y,xS
hycx
q
t,z,y,xp (3.1.5)
Where: q = total flow rate, and
fff
f
hyx
q
)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
−+
=
t2
xx2/x
erf
t2
xx2/x
erf
2
1
)t,x(S
x
wf
x
wf
(3.1.7)
2. Infinite slab source in y-direction and infinite reservoir
( ) ( )








η
−−
+
η
−+
=
t2
yy2/y
er
t2
yy2/y
erf
2
1
)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
ff
2
z
22
f
f
h
z
ncos
h
z
ncos
h
h
n5.0sin
h
tn
exp
n
1
h
h4
1
h
h
)t,z(S
(3.1.9)
Where:
c
kj
j
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(pp
q2.141
kh
)t,h,z,z,y,x(p
ffffi
DDfDDDDD f
−
µ
=
(3.1.10)
Dimensionless time:
2
ft
Dxf
xc
kt001056.0
t
µφ
= (3.1.11)
Dimensionless distance in the x-direction:
xf
w
D
k
k
x
)xx(2
x
−
= (3.1.12)
Dimensionless distance in the y-direction:
yf
w
D
k
k
x
)yy(2
y
−
= (3.1.13)
Dimensionless distance in the z-direction:

50
h
z
zD = (3.1.14)
Dimensionless reservoir height:
h
h
h f
fD = (3.1.15)
Dimensionless source half-length:
zf
Dxf
k
k
x
h2
L = (3.1.16)
Aspect ratio:
f
f
x
y
m = (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
D
x
Dxf
D
x
DxfD
t2
x
k
k
erf
t2
x
k
k
erf
2
1
)t,x(S (3.1.18)
4. Infinite slab source in y-direction and infinite reservoir

51














+
+
+
=
Dxf
D
y
Dxf
D
y
DxfD
t2
y
k
k
m
erf
t2
y
k
k
m
erf
2
1
)t,y(S (3.1.19)
5. Infinite slab source in z-direction and infinite reservoir








πππ




 π
−
π
+
=
∑
∞
=1n
DwDfD2
Dxf
Dxf
22
fD
fDDxfD
zncoszncoshn5.0sin
L
tn
exp
n
1
h
4
1
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:
Dxf
1n
wDDfD2
Dxf
Dxf
22
fD
t
0 Dxf
D
y
Dxf
D
y
Dxf
D
x
Dxf
D
x
DxfDDDD
dtzncoszncoshn5.0sin
L
tn
exp
n
1
h
4
1
t2
y
k
k
m
erf
t2
y
k
k
m
erf
t2
x
k
k
erf
t2
x
k
k
erf
m8
)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
D
y
Dxf
D
y
Dxf
D
x
Dxf
D
x
DxfDD
t2
y
k
k
m
erf
t2
y
k
k
m
erf
t2
x
k
k
erf
t2
x
k
k
erf
m4
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
+
=
∑
∞
=1n
wDDfD2
Dxf
Dxf
22
fD
DxfDfDfDDxf
zncoszncoshn5.0sin
L
tn
exp
n
1
h
0.4
1
)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
0
DxfDxfwDDfDDxfDxfDD
DxfDDDD
dt)t,z,z,h,L(Z)t,y,x(P
2
)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
2
t
)t(ln
)t,z,y,x(p
DxfwDDfDDxfDxfDD
Dxf
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
DimensionlessPressure,pD
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
DimensionlessPressureDerivative,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 DxfDD
β
= (3.3.1)
Where:







>>
==
<<
=β
yDxD
yDxD
yDxD
kkmyandkkxfor0
kkmyandkkxfor2
kkmyandkkxfor4
(3.3.2)
Substituting Equation 3.3.1 into Equation 3.1.24, we have
∫
πβ
=
Dxft
0
DxfDxfwDDfDDxfDxfDDDD 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 hfD
2
.
For hfD → 0, Equation 3.3.3 becomes:

60















 −−
−




 −−
π
πβ
−
=
D
DfDf
D
2
DfDD
DxfDDDD
t2
zz
erf
2
zz
t4
)zz(
exp
t
m4
L
)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
2
DDrf
D
D
D
2
D
Drf
fD
DxfDDDD
t4
exp
t
t2
erfc
2
t
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
2
Drf2Drf L
5
t
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
−
+
−
∫
d
kkkkmryx
kkmry2
tan
kkmry2
kkmykkxlnkkx
kkmykkxlnkkx
8
1
)y,x(
x
2
ywDD
2
D
ywDD1
ywDD
2
yD
2
xDxD
2
yD
2
xD
1
1
xD
DD1
(3.3.9)
and
( )
( ) ( ) ( ) ( )∑ ∫ ∫
∞
=
+
−
+
−
ωαππππ
π
=
1n
1
1
1
1
DD0wDDfD
fD
DwDDDD1
ddnLrKzncoszncoshn5.0sin
n
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
2
D
y
DxfDD
t
dt
t4
y
exp
k
k
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
Dxf
DxfDfDfDDxf
Dxf
2
D
y
DxfDD
t
dt
)t,z,z,h,L(Z
t4
y
exp
k
k
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
−





−π
β
=
D
D
D
D
2
D
D
y
fD
DxfDDDD
t2
y
erfcy
2t4
y
expt
k
k
2
h
)t,z,y,x(p
(3.3.14)
Where:






>
=
<
=β
xD
xD
xD
kkxfor0
kkxfor1
kkxfor2
(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
( ) ( )[ ]{
( ) ( )[ ]
( )[ ]}x
2
D
2
DD
1
D
2
D
2
xDxD
2
D
2
xDxD
DD2
kkyx/y2tany2
ykkxlnkkx
ykkxlnkkx
4
1
)y,x(
−+−
+++−
+−−
=σ
−
(3.3.17)
and
( )
( ) ( ) ( ) ( ) αππππ
π
=
∫∑
+
−
∞
=
dnLrkzncoszncoshn5.0sin
n
1
h
2
L,z,z,y,xF
1
1
DD0
1n
wDDfD
fD
DwDDDD2
(3.3.18)
As hfD tends to zero, Equation 3.3.18 reduces to:

63
( )
( ) ( ) ( ) απππ
=
∫∑
+
−
∞
=
dnLrkzncoszncos
L,z,z,y,xF
1
1
DD0
1n
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=
∂
∂
=hz
z
p
0=
∂
∂
=0z
z
p
hf
kz
ky
q = qy
0=
∂
∂
=hz
z
p
0=
∂
∂
=0z
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
0
0m
DxfDDDD
dt)t,z,z,h,L(Z)t,y,x(Plim
2
)t,z,y,x(p
Dxf
∫ →
π
=
(4.1.1)
Substituting the limit of the P-function as m approaches zero into Equation 4.1.1, we get:
}
Dxf
Dxf
DxfDfDfDDxf
Dxf
2
D
t
0 Dxf
D
x
Dxf
D
x
y
DxfDDDD
t
dt
)t,z,z,h,L(Z
t4
y
exp
t2
x
k
k
erf
t2
x
k
k
erf
k
k
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
2
t
)t(ln
)t,z,y,x(p
DxfwDDfDDxfDxfDD
0m
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
0
DxffDDfDDxf
0h
DxfDD
DxfDDDD
dt)t,z,z,h,L(Zlim)t,y,x(P
2
)t,z,y,x(p
Dxf
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
yfxf
(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=
∂
∂
=hz
z
p
0=
∂
∂
=0z
z
p
kz
ky
q=qz
yf
0=
∂
∂
=hz
z
p
0=
∂
∂
=0z
z
p
xf
zw
kz
kx

71
( ) ( ) Dxf
1n
wDD2
Dxf
Dxf
22
Dxf
D
y
Dxf
D
y
t
0 Dxf
D
x
Dxf
D
x
DxfDDDD
dtzncoszncos
L
tn
exp21
t2
y
k
k
m
erf
t2
y
k
k
m
erf
t2
x
k
k
erf
t2
x
k
k
erf
m8
)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
2
t
)t(ln
)t,z,y,x(p
DxfwDDfDDxf
0hfD
DxfDD
Dxf
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
0
DxfDxfwDDfDDxfDxfDDDD 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
2
DfDD
DxfDDDD
t2
zz
erf
2
zz
t4
)zz(
exp
t
m4
L
)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
−
+
−
∫
d
kkkkmryx
kkmry2
tan
kkmry2
kkmykkxlnkkx
kkmykkxlnkkx
8
1
)y,x(
x
2
ywDD
2
D
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:
f
2
ffff hrhyx π= (4.3.1)
Substituting ff xym = into Equation 4.3.1, we get
π
=
2
f2
f
mx
r (4.3.2)
Substituting Equation 4.3.2 into the Equations 3.1.11 and 3.1.16, we get:
DxfDrf t
m
25.0
t
π
= (4.3.3)
and
DxfDrf L
m
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
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
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
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
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

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