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
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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
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
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
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Naturally Fractured Reservoirs,” paper SPE 18302 presented at the 1988 SPE Annual Technical Conference and Exhibition, Houston, TX, Oct. 2-5, 1988.
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water Drive,” SPERE (Aug. 1990) 375.
120
13. Aguilera, R. and Ng, M.C.: “Transient Pressure Analysis of Horizontal Wells in
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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
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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.
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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
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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.