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Copyright 2007, Society of Petroleum Engineers

This paper was prepared for presentation at the 2007 SPE Hydraulic Fracturing TechnologyConference held in College Station, Texas, U.S.A., 29–31 January 2007.

This paper was selected for presentation by an SPE Program Committee following review ofinformation contained in an abstract submitted by the author(s). Contents of the paper, aspresented, have not been reviewed by the Society of Petroleum Engineers and are subject tocorrection by the author(s). The material, as presented, does not necessarily reflect anyposition of the Society of Petroleum Engineers, its officers, or members. Papers presented atSPE meetings are subject to publication review by Editorial Committees of the Society ofPetroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paperfor commercial purposes without the written consent of the Society of Petroleum Engineers is

prohibited. Permission to reproduce in print is restricted to an abstract of not more than300 words; illustrations may not be copied. The abstract must contain conspicuousacknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O.Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract

It has been long established from well testing concepts that anelliptical flow pattern exists in hydraulically-fractured wells

producing from low permeability (<0.01 md) and ultra-low

(<0.001 md) formations (often referred to as "tight gas" or

"shale gas" sands). Conceptually, the elliptical flow period is

a transitional flow regime that occurs between the end of bilinear and/or formation linear flow and the onset of pseudo-

radial flow. In a practical sense, the elliptical flow regime can

(and does) dominate the well performance for very low permeability reservoir systems.

Elliptical flow (as with pseudo-radial flow) represents a time

period when the reservoir properties begin to dominate the

reservoir performance. Moreover, the duration of the ellipticalflow period may last for many months, perhaps even years,

depending on the reservoir and hydraulic fracture properties.

Consequently, understanding the elliptical flow period and its

impact on well performance is critical for the optimaldevelopment of a tight gas sand reservoir.

This paper presents a series of decline type-curves for a

system consisting of a hydraulic fracture at the center of anelliptical reservoir. The curves are generated from the

pressure solution obtained using an analytical method. The

curves are generated for different values of the fracture

conductivity and as a function of the elliptical boundary

characteristic parameter (ξ 0).

For the low permeability cases where the elliptical flow period

dominates, the elliptical boundary characteristic parameter (ξ 0)is a useful parameter for establishing the optimum well

spacing and the optimal design for the drainage aspect ratio.

We note that the ξ 0-parameter essentially reduces the

optimization process for tight gas reservoirs to a single

parameter that should yield the optimum configuration fordrainage and production performance. Obviously the fracture

conductivity is also a major factor, as is the reservoir

permeability and the fracture half-length — but the ξ 0-parameter is the key to recovery and performance in systems

dominated by elliptical flow behavior.

Introduction

Elliptical flow has long been considered as a transitional flow

pattern that occurs between linear and pseudoradial flow in thetraditional well test analysis of fractured wells.1 Elliptica

flow also occurs also in circular anisotropic reservoirs.

The study of elliptical flow has a considerable history in the

petroleum engineering literature2-10 (as well as other disci-

plines — e.g., water resources and physics). Prats2, 3

studiedthe effect of vertical fracture on the reservoir behavior for

incompressible and compressible fluid cases. Prats proposed

the representation of a fracture with an equivalent effective

well radius (this was a convenience of the times (early

1960's)).

Kuchuk 6 developed the analytical solution for the transien

elliptical flow problem resulting from an infinite conductivity

fracture producing from an elliptical, or an anisotropic-radiareservoir. The behavior of composite, elliptically-shaped

reservoirs has also been studied by Obut, and Ertekin5 and

Stanislav et al 6,7 — assuming a variety of boundary condi-

tions. Perhaps the most "analytical" treatment of the problemwas provided by Riley8 who developed an analytical solution

for the case of a vertical well with a finite conductivity vertica

fracture in an infinite-acting reservoir system.

In addition to attempts to obtain analytical solutions for the

elliptical flow problem, there are numerous studies which

consider numerical modeling of the elliptical flow problem

Hale9 applied type curve solutions (dimensionless plots)

derived from numerical simulation of the elliptical problem as

a mechanism to interpret well tests in tight gas reservoirs

Liao10 developed a general numerical model for the elliptica

flow case which is capable of accounting for the simultaneouseffects of wellbore storage and fracture face skin effects on the

behavior of pressure transient tests.

Based on practical experience as well as numerical/analytica

models, it is apparent that elliptical flow behavior dominates

the performance of fractured wells in low/ultra-low permeabil

ity gas reservoir systems. The primary objective of this pape

is to develop and validate a series of "type curve" solutions fora system consisting of a hydraulically fractured well in a

bounded elliptical shape reservoir. The "type curves" are pre

SPE 106308

Evaluation of the Elliptical Flow Period for Hydraulically-Fractured Wells in Tight GasSands — Theoretical Aspects and Practical ConsiderationsS. Amini, D. Ilk, and T. A. Blasingame, SPE, Texas A&M University

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2 S. Amini, D. Ilk, and T. A. Blasingame SPE 106308

sented (and implemented) as a diagnostic tool — which isused to assess the flow behavior based on production data.

Type curves are presented for different values of the fracture

conductivity ( F E = 1, 10, 100, and 1000 (very low to very high

fracture conductivity cases)), as well as a function of theelliptical boundary characteristic parameter (ξ 0).

Development of the Model

Elliptical flow is considered to be the governing flow regimefor low permeability gas reservoirs — and, as such, weassume that the reservoir has some sort of an elliptical outer

(or perhaps, inner) boundary. Cases such as production from

an elliptical wellbore, an elliptical fracture, or a circular wellbore in an anisotropic reservoir system can be considered to be

examples of an elliptical inner boundary. An elliptic reservoir

surrounded by an (elliptic) aquifer is a relevant example of an

elliptical outer boundary — as is also the case of a very low permeability gas reservoir that achieves only a limited drain-

age pattern (usually modeled by an ellipse).

Our model analytical consists of a hydraulically fractured wellin the center of a reservoir with a closed elliptical boundary.

The schematic illustration for an elliptical reservoir with ahydraulic fracture is shown in Fig. 1.

Figure 1 — Schematic of the elliptical reservoir model.

The following assumptions are made in order to develop an

analytical solution for this case:

The reservoir is assumed to be a single-layer system that isisotropic, horizontal, and of uniform thickness with con-

stant reservoir characteristics. Production is obtained from a hydraulic fracture which

intersects the wellbore. The fracture is assumed to haveelliptical shape — but we can also assume that the fractureis very narrow compared to the length (and area) of the

fracture, and in such cases, we assume a zero-width frac-ture and the fracture transforms into a line whose length isequal to the focal length of the elliptic system.

The elliptical outer boundary is assumed to have a focal

length equal to the fracture length Fracture properties are assumed constant — and we have

considered the cases of both infinite and finite conductivityvertical fractures.

Wellbore storage and skin effects are neglected for the sakeof simplicity (as we are focusing on production data analy-

sis, this issue not significant) The flowing fluid is assumed to be a single-phase, slightly

compressible fluid (although rigorous transformations can be used for gas flow cases). Fluid flow in the formation and in fracture is laminar, and

Darcy's law is presumed valid.

The analytical solutions for the case of an elliptical fractureand outer boundary have been discussed in prior literature2-4,8

However, we provide an appropriate discussion of the compu

tational issues encountered in the solution of the finite and

infinite fracture conductivity cases in Appendix A.

We also introduce a parameter called the elliptical boundary

characteristic parameter (ξ 0) — which is a variable that cor-

relates all the aspects of the drainage area (e.g., the drainage

aspect ratio (a/b) and penetration ratio based on the fracture

half-length ( x f /a). The ξ 0-parameter helps us to characterize

the drainage area using a single dimensionless parametersimilar in concept and application to the dimensionless radius

for the circular bounded reservoir (r eD = r e/ x f ) as introduced by

Pratikno and Blasingame.13 In Appendix B we provide usefu

orientation and detail regarding the correlation between theelliptical boundary characteristic parameter (ξ 0) and the

drainage area size and shape.

Elliptical Flow Model — Type Curves

Given a volumetric reservoir acting under pseudosteady-state

flow conditions (any well/reservoir configuration), we can

write the following material balance/flow relation for thiscase:

DA Dpss pss D t b p π 2, += ...................................................... (1

Where b Dpss is the "pseudosteady-state constant" for a particu-

lar well/reservoir configuration. We can calculate a series o

pressure solutions as functions of dimensionless time for

different values of the fracture conductivity and the elliptica

boundary characteristic parameter (ξ 0) in order to estimate theb Dpss-parameter for different configurations. Cartesian-style

plots of p D,pss versus t DA are used to establish the b Dpss

parameter values. and a sampling of b Dpss-values is presentedin Table 1 for the cases of F E =1, 10, 100, and 1000. The pro

cess of estimating the b Dpss-parameter is discussed in detail inref. 13

Table 1 — Values of bDpss from elliptical flow solution.

ξ 0 F E = 1 F E = 10 F E = 100 F E = 1000

0.25 0.8481 0.2150 0.1306 0.1220

0.50 0.9902 0.3337 0.2396 0.2298

0.75 1.1671 0.4609 0.3540 0.3426

1.00 1.3627 0.6109 0.4936 0.4812

1.25 1.5733 0.7880 0.6634 0.6501

1.50 1.7963 0.9884 0.8591 0.8453

1.75 2.0293 1.2067 1.0743 1.0602

2.00 2.2682 1.4363 1.3021 1.2877

3.00 3.2529 2.4084 2.2716 2.2570

4.00 4.2503 3.4040 3.2669 3.2522

5.00 5.2486 4.4021 4.2649 4.2502

Using the data in Table 1 and a non-linear regression methodwe can estimate the b Dpss-parameter as a function of the

fracture conductivity and the elliptical boundary characteristic

parameter as follows:

754772.0

16703.00794849.000146.1 00

−+

−+= −

B

A

ueb Dpssξ

ξ ................ (3

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SPE 106308 Evaluation of the Elliptical Flow Period for Hydraulically-Fractured Wells in Tight Gas Sands — 3

Theoretical Aspects and Practical Considerations

Where the auxiliary functions are:

)ln( E F u = ........................................................................ (4)

45

34

2321 uauauauaa A ++++= .................................... (5)

45

34

2321 ububububb B ++++= ..................................... (6)

Where the coefficients are given as:

a1 = -4.7468 b1 = -2.4941

a2 = 36.2492 b2 = 21.6755

a3 = 55.0998 b3 = 41.0303

a4 = -3.98311 b4 = -10.4793

a5 = 6.07102 b5 = 5.6108

The b Dpss results for our work and Eq. 3 (the correlation

equation) are presented graphically in Fig. 2.

Figure 2 — bDpss results and the correlating relation for thecase of a well with a finite conductivity verticalfracture in an elliptically bounded reservoir.

This work pertaining to the b Dpss-parameter is directed

primarily at establishing a "correlation" for those interested in

developing "Fetkovich-style" type curves which are based on

the so-called decline variables. For our work, we intend todemonstrate the elliptical boundary model as a diagnostic to

establish the elliptical flow regime using production data — as

such, for convenience (and simplicity) we utilize type curvesolutions in terms of the equivalent constant rate case in

"decline" form (i.e., q D and auxiliary functions q Di and q Did

versus t DA).

Application of the Elliptical Flow Model — IllustrativeExamples

In this section we provide diagnostic examples to demon-strate/illustrate value of the elliptical boundary model. The

purpose of this exercise is not to obtain "answers" (although

several analyses are performed), but rather, to illustrate the

quality matches obtained using the elliptical boundary model.

Example 1: East Texas (US ) — Tight Gas ( good production)

This case is taken from ref. 13, and all of the relevant data for

this case can be found in that reference. The diagnostic plots

for this case are shown in Figs. 3-5. The operator used

"better-than-average" data acquisition practices, and theresults confirm the validity of the elliptical flow model.

Figure 3 — Example 1: Time-Pressure-Rate (TPR) history plot

Shows very good correlation of rate and pressuredata, indicates likelihood of good analysis.

Figure 4 — Example 1: Log-log (rate function) diagnostic plot.Excellent data functions — strong indication offractured well regimes tending to boundary flow.

Figure 5 — Example 1: "Type curve" plot. Extraordinarymatch of data and elliptical boundary model,match confirms concept.

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Example 2: North Texas (US ) — Very Tight Gas This case has reasonably good correlation of time-rate-

pressure data as shown in Fig. 6. All analyses for this case

yield a permeability of < 0.001 md. The log-log diagnostic

and type curve matches are shown in Figs. 7 and 8 (respect-ively) — elliptical reservoir match is reasonable (Fig. 8).

Figure 6 — Example 2: Time-Pressure-Rate (TPR) history plot.Only limited well history was provided, but thedata do appear reasonably correlated.

Figure 7 — Example 2: Log-log (rate function) diagnostic plot.Data functions suggest a high to very high con-ductivity vertical fracture.

Figure 8 — Example 2: "Type curve" plot. Good to possiblyexcellent match of data functions with ellipticalboundary model. Transition flow regime is ap-parent at late times.

Example 3: North Texas (US ) — Tight Gas

This is a "sister" case to Example 2 — but appears to be lesswell-stimulated ( F E ). The data are reasonably well-correlated

(Fig. 9) and the log-log plot (Fig. 10) clearly suggests a low

conductivity vertical fracture. The data match (Fig. 11) isvery good, indicating validity in the elliptical reservoir model.

Figure 9 — Example 3: Time-Pressure-Rate (TPR) history plotData correlation is fair — most significant featuresdo correlate, but not all features are related.

Figure 10 — Example 3: Log-log (rate function) diagnostic plot.Data functions confirm low conductivity verticalfracture, reasonably good correlation of data.

Figure 11 — Example 3: "Type curve" plot. As noted above,good correlation with low conductivity verticalfracture case. Minor concern regarding dataartifacts at late times (good data interpretation).

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Example 4: Mexico — Very Tight Gas (long production)

First and foremost, this reservoir has a permeability of < 0.001

md (multiple analyses), and the well is the only well in the

field. The long production history and high quality data yield

"near textbook" quality diagnostic plots (Figs. 12-15). Fig. 15 confirms validity of the elliptical boundary model.

Figure 12 — Example 4: Time-Pressure-Rate (TPR) history plot— excellent rate performance. Constant ptf → pwf is reasonable considering location/production.

Figure 13 — Example 4: Log-log (rate function) diagnostic plot.Extraordinary character in data (1/2 slope trend),very good diagnostic trends.

Figure 14 — Example 4: "Type curve" plot. Excellent to out-standing match of the data functions — strongconfirmation of the elliptical boundary model.

As a closure in this section we present the matching results fo

this work, as well as the "average" analysis results for these

examples (Tables 2a and 2b). Perhaps the most importan

outcome of this work is not any specific result, model, etc. —

but rather, the proof-of-concept for application of the elliptica boundary model. This observation of the usefulness of the

elliptical boundary model as a diagnostic can (and should)

lead to better production data analysis — including better"model-based" estimates of fluids in-place.

Table 2a — Matching results from interpretation/analysis.

Example

t mba /t DA (days)

(q g /Δ p p)/q D (MSCFD/psi) F E ξ 0

1 228 1.35 1 0.50

2 7200 0.052 100 0.25

3 4000 0.145 10 0.75

4 190,000 0.098 100 0.25

Table 2b — "Average" analysis results.

Example

k(md)

x f (ft)

G

(BSCF)

1 0.006 300 1.62 0.0007 250 0.5

3 0.003 400 0.9

4 0.001 825 23.0

Summary and Conclusions

Summary: Simply put, the development of the elliptica

boundary model for production data diagnostics and analysiis long overdue. It is often argued that the same concept can

be applied using a rectangular or circular reservoir model as a

surrogate for the elliptical boundary model. Given theexamples reviewed in this work, it is difficult to justify a

"surrogate" for the elliptical boundary model which appears

(by far) to be the best model for evaluating fractured wells in

low to ultra-low permeability gas reservoirs.

Conclusions:1. The elliptical boundary model is an essential tool for the

diagnosis and analysis of production data from hydrauli-cally fractured wells in low to ultra-low permeability gas

reservoirs.

2. The diagnostic matches of the production data presented inthis work are excellent — despite the fact that the quality ofthese data, while reasonable, is certainly not comparable to

the quality of data that can be acquired in the market ( e.g.,continuous downhole pressure and surface rate measure-ments). In practice, the elliptical boundary model should

be appropriate for the analysis/diagnosis of production data

for virtually all cases of hydraulically fractured wells intight gas reservoirs.

Recommendations/Comment : We note that the solution for the

elliptical boundary model (Appendix A) is extremely tedious

— at present, the solution is not-suited for dynamic analysishistory matching (i.e., computational times for this solution

can take days on a personal computer). The only practica

mechanisms for application of this solution are the preparation

of data tables (for interpolation) and the generation of an ap propriate suite of "type curves" (see Appendix C for example

type curves).

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Nomenclature

Field Variablesct = Total system compressibility, psi-1 h = Pay thickness, ft

k = Permeability, md

k f = Fracture permeability, mdk R = Reservoir permeability, md

p = Pressure, psia

p f = Fracture pressure, psia pi = Initial pressure, psia

p p = Pseudopressure function, psia

p R = Reservoir pressure, psia pwf = Flowing bottomhole pressure, psia

q = Flowrate, STB/D

q g = Gas flowrate, MSCF/D

r w = Wellbore radius, ftt = Time, hr

x f = Fracture half-length, ft

Dimensionless Variablesb Dpss = Dimensionless pseudosteady-state constant

F E = Elliptical fracture conductivity

D p = Laplace transform of dimensionless pressure

p D = Dimensionless pressure

p Dd = Dimensionless pressure derivative

q D = Dimensionless flowrate

q Di = Dimensionless rate integral

q Did = Dimensionless rate integral derivative

t D = Dimensionless time (wellbore radius)t DA = Dimensionless time (drainage area)t Dxf = Dimensionless time (fracture half-length)

Mathematical Functions and Variablesnr A2

2 = Mathieu function Fourier coefficients

a = Long axis of the elliptical systemb = Short axis of the elliptical system

Bn = Coefficient in solution seriesce2n = π -periodic angular Mathieu functionC e2n = Radial Mathieu function

Dn = Coefficient in solution series

F ek2n = Radial Mathieu function

F ey2n = Radial Matthieu function

H = Angular function I 2r = Bessel function of order r r

K 2r = Bessel function of order r m = Positive integern = Positive integerr = Positive integer

R2n = Coefficient in series solution s = Laplace domain parameter

X = Radial function x = Cartesian coordinate

y = Cartesian coordinateY 2r = Bessel function of order r

Greek Symbols

α = regression coefficient

β = Coefficient in fracture pressure series

δ = Dirac delta function

ε n = 1+ Kronecker delta function

φ = porosity, fraction

γ = coefficient in reservoir pressure seriesη = Angular elliptical coordinate

κ D = Diffusivity ratio, dimensionless

μ = Viscosity, cp

Ω = Kernel of sum equation

ξ = Radial elliptical coordinate

ξ 0 = Elliptical boundary characteristic variable

Subscript D = Dimensionless f = Fracturei = Integral function or initial value

id = Integral derivative function pss = Pseudosteady-state

m = Positive integern = Positive integer

r = Positive integer R = Reservoir

Gas Pseudofunctions:

dp z

p

p p

z p

basei

ii p

i

μ

μ

∫=

dt pc p

t ct

g g gi gia

)()(

1

0 μ

μ ∫=

dt pc p

t qt

t q

ct

g g

gi gi gasmba

)()(

)(

0)(

,μ

μ

∫=

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References

1- Lee, J. and Wattenbarger, R.A.: Gas Reservoir Eng-

ineering, Textbook Series, SPE, Richardson, Texas (1996)

5,139-1402- Prats, M.: "Effect of Vertical Fractures on Reservoir Beha-

vior— Incompressible Fluid Case," SPEJ (June 1961),

105-18; Trans., AIME, 222 3- Prats, M., Hazebroek, P., Strickler, W.R.: "Effect of Verti-

cal Fractures on Reservoir Behavior— Compressible FluidCase," SPEJ (June 1962), 87-94; Trans., AIME, 225

4- Kolchak, F., and Brigham, W.E.: "Transient Flow in Ellip-

tical Systems," SPEJ (December 1979),401-10, Trans.,AIME, 267

5- Obut, S.T., and Ertekin, T.: "A Composite System Solution

in Elliptic Flow Geometry," SPEFE (September 1987),227-38

6- Stanislav, J.F., Easwaran, C.V., and Kokal, S.L.: "Analyti-

cal Solutions for Vertical Fractures in a Composite Sys-

tem," J. Cdn. Pet. Tech. (September-October 1987), 51-6

7- Stanislav, J.F., Easwaran, C.V., and Kokal, S.L.: "Ellipti-cal Flow in Composite Reservoirs," J. Cdn. Pet. Tech.

(December 1992), 47-50

8- Riley, M.F.: Finite Conductivity Fractures in EllipticalCoordinates, Ph.D. Dissertation, Stanford U., Stanford,

California (1991)

9- Hale, B.W.: Elliptical Flow Systems in Vertically Frac-

tured Gas Wells, M.S. Thesis, U. of Wyoming, Laramie,Wyoming (1991).

10- Liao, Y.: Well Production Performance and Well Test

Analysis for Hydraulically Fractured Wells, Ph.D. Dis-

sertation, Texas A&M U., College Station, Texas 1993).

11- McLachlan, N.W.: Theory and Application of Mathieu Functions, Dover Publications, New York City, New York

(1964)12- Stehfest, H.: "Numerical Inversion of Laplace Trans-

forms," Communications of the ACM (January 1970), 13,

No.1, 47-49.(Algorithm 368 with correction (October

1970), 13, No. 10)13- Pratikno, H., Rushing, J.A., and Blasingame, T.A.: "De-

cline Curve Analysis Using Type Curves — Fractured

Wells," paper SPE 84287 presented at the SPE annual

Technical Conference and Exhibition, Denver, Colorado,

5-8 October 2003.

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Appendix A: Development of the Model

For a system containing a single-phase, slightly compressible

fluid, the diffusivity equation is expressed by Eq. A.1

t

p

k

c p t

∂

∂=∇ φμ 2 ............................................................ (A.1)

Where:

p = pressuret = time

k = permeability

f = porosity

m = fluid viscosity

ct = total compressibility

Assuming uniformity in the z -axis, and neglecting any gravity

effects, we can write the diffusivity equation in Cartesiancoordinates as:

t

p

k

c

y

p

x

p t

∂

∂=

∂

∂+

∂

∂ φμ

2

2

2

2

................................................. (A.2)

Eq. A.2 can be transformed into elliptical coordinates using

the following transform correlations:η ξ cos.cosh. f x x = ...................................................... (A.3)

η ξ sin.sinh. f x y = ....................................................... (A.4)

Where x f is defined as the fracture half-length.

The elliptical system is characterized by series of confocal

ellipses (focal radii equal to x f ), which are normal to a family

of hyperbolas of the same focal radii. Fig. A.1 shows a

schematic of elliptical coordinates.

Figure A.1 — Elliptical coordinates.

The diffusivity equation in elliptical coordinates is given as:

t

p

k

xc p p f t

∂

∂−=

∂

∂+

∂

∂)]2cos()2[cosh(

2

2

2

2

2

2

η ξ φμ

η ξ

............ (A.5)

An elliptical system with a boundary at ξ 0 is transformed into

a (ξ 0 × 2p ) rectangle in elliptical coordinates. The fracture

transforms to a line of the width ξ w from 0 to 2p . Fig. A.2

shows how the transformation works. It should be noted thataccording to the symmetry of the problem, we have just shown

the transform in one-quarter of the plane. That is the reason h

is taken from 0 to p/2.

Figure A.2 — Elliptical system after transformation.

Initial Condition and the Boundary Conditions

After defining the governing equation for the model we needto introduce initial and boundary conditions to solve the given

partial differential equation (PDE).

Initial Condition:

We assume a uniform initial pressure ( pi) for reservoir at the

start of production

i p p =)0,,( η ξ ....................................................................(A.6)

Boundary Conditions:

For defining boundary conditions we need orientation from

the geometry of the problem. In the angular direction (h ) weuse the following boundary conditions which are based on the

symmetry of the model:

00 =∂

∂=η

η

p...................................................................(A.7)

0

2

=∂

∂

=π

η η

p.................................................................(A.8

It is obvious that the solution should be π -periodic solution

because of the symmetry in the flow from two sides of thefracture. In the radial direction our first B.C. is given as:

wf pt p =),,0( η ..............................................................(A.9

It should be noted that we have assumed the fracture to be an

infinite-conductivity conduit of zero width lying at x w= 0.

Depending on whether we are dealing with infinite-acting or

finite reservoir case, the second B.C. is defined as a constant

pressure at infinity for the infinite-acting model, and using a

"no-flow" boundary for the bounded case:

i pt p =∞→ξ η ξ ),,( (infinite-acting reservoir) .........(A.10)

00 =

∂

∂

=ξ ξ ξ

p

(no-flow boundary) ..................(A.11

Applying the following dimensionless variables enables us to

transform the diffusivity equation and the boundary and initiaconditions into dimensionless form.

)(2.141

1 p p

qB

kh p i D −=

μ ...........................................(A.12

t xc

k t

f t

D 20002637.0

φμ = ...........................................(A.13

ξξ 0

ξ w

0

π/2

h = p /2

h = p

h = 3p /2

h = 0

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Solution for the infinite-acting case: (radial flow)

The outer boundary condition (Eq. A.26) implies that a finite

value of pressure occurs as ξ →∞. Among the three aforemen-tioned solutions, only the solution

)4

,(2 s

Fek n −ξ

meets the necessary requirements of the outer boundary

condition. So the radial solution is given as:

)4

,()( 2 s

Fek B X nnn −= ξ ξ ........................................... (A.37)

Combining Eqs. A.32 and A.37 we obtain the preliminarysolution for the dimensionless pressure in the Laplace domain:

)4

,()4

,(),( 220

s Fek

sce B p nn

nn D −−∑=

∞

=ξ η η ξ ............. (A.38)

To fully solve Eq. A.37 we need to find the value of n B s. we

use the remaining unused inner boundary condition (Eq. A.25)

and the orthogonality properties of Mathieu cosine function to

obtain the value for n B . Introducing Eq. A.25 to Eq. A.38:

)4

,0()4

,(1

220

s Fek

sce B

snn

nn −−∑=

∞

=η ........................... (A.39)

Multiplication of both sides of Eq. A.39 by )4

,(2 s

ce m −η and

integrating from (0 to p ) results:

η η η

η η

π

π

d s

ce s

ce s

Fek B

s

d s

ce

mn

nnn

m

)4

,()4

,()4

,0(

)4

,(

20 0

22

02

−−∑ ∫−=

∫ −

∞

=

..................................................................................... (A.40)

From the definition of nce2 function we can see that:

nnn Ad

sce 2

00

2 )1()4

,( π η η π

−=∫ − .................................... (A.41)

0)4

,()4

,(

022 =∫ −−

π η η η d

sce

sce mn (if m ≠ n) ............ (A.42)

2)

4,()

4,(

022

π η η η

π =∫ −− d

sce

sce mn (if m = n) ............ (A.43)

Rearranging Eq. A.40 using Eqs. A.41 to A.43, and solving

for n B , we have:

)4

,0(

2)1(

2

20

s sFek

A B

n

nn

n−

−= ................................................... (A.44)

Finally we have the pressure solution as:

)4

,(

)4

,0(

)4

,(2)1(),( 2

2

2

0

20 s

ce s

Fek

s Fek

s

A p n

n

n

n

nn

D −−

−∑ −

= ∞

=η

ξ η ξ (A.45)

Solution for the case of a bounded reservoir : (radial flow)

In this case, we have two types of Mathieu functions thatsatisfy the outer boundary condition of the problem (Eq

A.21). So the radial solution can be written in the form of a

linear combination of the two functions as:

)4

,()4

,()( 22 s

Ce D s

Fek B X nnnnn −+−= ξ ξ ξ ...............(A.46

Applying the outer boundary condition leads to:

n

n

n

n

nnnn

D s

Fek

sCe

B

sCe D s Fek B

)4

,('

)4

,('

0)4

,(')4

,('

2

2

22

−

−−=

=−+−

ξ

ξ

ξ ξ

.....................(A.47

To calculate the n D coefficients, we utilize the inner boun-

dary condition similar to that for the infinite-acting case:

)4

,(

)4

,0()4

,(')4

,0()4

,('

)4

,()4

,(')4

,()4

,('

2)1(),(

2

202202

202202

0

20

sce

s Fek

sCe

sCe

s Fek

s Fek

sCe

sCe

s Fek

s

A p

n

nnnn

nnnn

n

nn

D

−×

−−−−−

−−−−−×

∑ −

= ∞

=

η

ξ ξ

ξ ξ ξ ξ

η ξ

(A.48

Finite Conductivity Fracture

The case of a well with a finite conductivity fracture becomes

necessary as the pressure drop within the fracture becomessignificant compared to the total pressure drop of the fracture

and reservoir. The case of finite conductivity elliptica

fracture has been studied extensively by Riley8 and this

portion of the work borrows heavily from Riley's work. Thegoverning equation in the fracture can be written as thefollowing in the Laplace domain:

)2

(

))2cos(1(2

2

02

2

π η δ

π

η κ

ξ η ξ

−−=

−−∂

∂+

∂

∂

=

E

f D R

E

f

sF

p s p

F

p

...........(A.49

f Rt

R f t D

k c

k c

)(

)(

φ

φ κ = is called the diffusivity ratio and δ

represents the Dirac Delta function. In practical cases Dκ is a

small number in the order of 10-7

to 10-10

the effect of fracturediffusivity in the governing equation is neglected. Hence, the

final form of the governing equation for the fracture is reduced

to:

)2

(2

02

2π

η δ π

ξ η ξ

−−=∂

∂+

∂

∂

= E

R

E

f

sF

p

F

p...................(A.50

With the boundary conditions:

0

20

=∂

∂=

∂

∂

== π

η η η η

f f p p..............................................(A.51

7/25/2019 [Blasingame] SPE 106308




The solution to Eq. A.50 has been discussed extensively byRiley8 for the case of an infinite-acting reservoir. The solution

is recast slightly to obtain a suitable solution for the bounded

case. In brief, Riley proposed a solution in the form of cosine

series as:

∑= ∞

=0

2 )2cos(),(

r

r f r s p η β η ....................................... (A.52)

Introducing the fracture solution and the reservoir solution fora bounded reservoir into Eq. A.50 we have:

)2

()4

,(.)1(2

)2cos()4()1(

0222

02

2

π η δ

π η γ

η β

−−=∑ −−+

∑ −−

∞

=

∞

=

E nnnn

n

E

r r

r

sF

sce R

F

r r

...................................................................................... (A.53)

Where:

)4

,0()4

,(')4

,0()4

,('

)4

,0(')4

,(')4

,0(')4

,('

202202

202202

2

s Fek sCe sCe s Fek

s Fek

sCe

sCe

s Fek

R

nnnn

nnnn

n

−−−−−

−−−−−=

ξ ξ

ξ ξ .

...................................................................................... (A.54)

Using the definition of the Mathieu function )4

,(2

sce n −η (Eq.

A.33), Riley showed that we can write Mathieu function as a

cosine series or vice versa if we write it in form of the cosineseries we would have:

∑∞

=

=Ω−0

2222

2 22

p r r

r p pr E

sr F

ε β ε β ............................... (A.55)

where,

m

m

m p

mr

pr

r p R A A 2

0

22

22

22

22 ∑

∞

=

=Ω=Ω ................................... (A.56)

and

otherwise1,0if 2 === nn n ε ε .................................. (A.57)

We can solve the system of equations described in Eq. A.55

assuming a finite value for the series (n). It should be noted

that r 2 β oscillates at first and then steadily, but declines very

slowly as r grows. We calculate r 2 β for r values between 0

to n from the system of equation and use regression technique

to obtain the values of r 2 β for r' s greater than n. Our

observation shows that using a regression-based appraoch for

the following model accurately describes the behavior of

r 2 β for r values greater than n:

22r

r

α β = .................................................................. (A.58)

Appendix B: The Elliptical Boundary Characteristic

Variable ( 0)

If we assume the elliptical outer boundary has the same focalength as the hydraulic fracture length, we can write the

following equations correlating all aspects of the drainage area

to a single parameter (ξ 0) (See Fig. B.1 below).

Figure B.1 — Schematic of the elliptical reservoir model

)cosh( 0ξ f xa = .............................................................(B.1)

)sinh( 0ξ f xb = ..............................................................(B.2)

20 )2sinh(

2 f xab Area ξ

π π == .........................................(B.3

)coth(RatioAspectDrainage 0ξ ==b

a..........................(B.4

10 )cosh(RationPenetratio

−== ξ a

x f ............................(B.5

Fixing the boundary characteristic parameter, we can calculate

every other aspects of the size and ratios of the drainage areaas a function of fracture half length ( x f ). Table B.1 shows the

system parameters for some selected values of the elliptica

boundary characteristic variable.

Table B.1 — System ratios and boundary characteristicvariable

4

ξ 0 Aspect Ratio Penetration Ratio

0.25 4.0830 0.9695

0.50 2.1640 0.8868

0.75 1.5744 0.7724

1.00 1.3130 0.6481

1.25 1.1789 0.5295

1.50 1.1048 0.4251

1.75 1.0623 0.3374

2.00 1.0373 0.2658

3.00 1.0050 0.0993

4.00 1.0007 0.03665.00 1.0001 0.0135

Appendix C: Type Curves (qD and tDA format) for theElliptical Boundary Model

The equivalent constant rate format type curves for this work

are shown in q D versus t DA format — see Figs. C.1 to C.4

F E =1, 10, 100, 1000 (respectively).

7/25/2019 [Blasingame] SPE 106308



Figure C.1 — Type curve for a fractured well centered in a closed (homogeneous) elliptical reservoir —FE=1, various 0-values;qD functions versus tDA format.

Figure C.2 — Type curve for a fractured well centered in a closed (homogeneous) elliptical reservoir —FE=10, various 0-values; qD functions versus tDA format.

7/25/2019 [Blasingame] SPE 106308






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