analytical solutions for a composite, cylindrical reservoir with a power-law permeability...
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Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability
Distribution in the Inner Cylinder
Ryan Sawyer BroussardDepartment of Petroleum
EngineeringTexas A&M University
College Station, TX 77843-3116 (USA)
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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●Problem Statement●Research Objectives●Stimulation Concepts:
— Hydraulic Fracturing— Power-law permeability
●Analytical Model and Solution Derivations:— Dimensionless pressure solution with a constant rate I.B.C— Dimensionless rate solution with a constant pressure I.B.C.
●Presentation and Validation of the Solutions●Power-Law Permeability vs. Multi-Fractured Horizontal
— Simulation Parameters and Gridding— Comparisons— Conclusions
●Summary and Final Conclusions
Outline
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Problem Statement■Multi-stage hydraulic fracturing along a horizontal well is the current
stimulation practice used in low permeability reservoirs
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Problem Statement Cont.■Hydraulic Fracturing Issues:
Provided by: MicrosoftProvided by: Microsoft
(US EIA 2012)
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Problem Statement Cont.
■Proposed Stimulation Techniques:
■We are not proposing a new technique
■We evaluate a stimulation concept:■ Creating an altered permeability zone
■ Permeability decreases from the wellbore following a power-law function
■How does this type of stimulation perform in low permeability reservoirs?
■How does it perform compared to hydraulic fracturing?
(Carter 2009)(Texas Tech University 2011)
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Research Objectives:■Develop an analytical representation of the rate and pressure
behavior for a horizontal well producing in the center of a reservoir with an altered zone characterized by a power-law permeability distribution
■Validate the analytical solutions by comparison to numerical reservoir simulation
■Compare the power-law permeability reservoir (PPR) to a multi-fracture horizontal (MFH) to determine the PPR’s suitability to low permeability reservoirs
MS Thesis Defense— Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Stimulation Concept: Multi-fracture horizontal■Pump large volumes of fluid at high rates and pressure into the formation
■The high pressure breaks down the formation, creating fractures that propagate out into the reservoir
■Direction determined by maximum and minimum stresses created by the surrounding rock
■Process repeated several times along the length of the horizontal wellbore
(Valko: PETE 629 Lectures)
(Freeman 2010)
MS Thesis Defense— Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Stimulation Concept: Power-Law Permeability■A hypothetical stimulation process creates an altered permeability zone
surrounding the horizontal wellbore.
■The permeability within the altered zone follows a power-law function:
n
s
or r
rkrk
MS Thesis Defense— Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Analytical Model●Geometry
■Composite, cylinder consists of two regions:—Inner region is stimulated. Permeability follows a
power-law function. —Outer region is unstimulated and homogeneous.
■Horizontal well is in the center of the cylindrical volume
■Wellbore spans the entire length of the reservoir (i.e. radial flow only)
●Mathematics■Solution obtained in Laplace
Space■ Inverted numerically by Gaver-
Wynn-Rho algorithm (Mathematica; Valko and Abate 2004)
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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●Assumptions:■Slightly compressible liquid■Single-phase Darcy flow■Constant formation porosity and liquid viscosity■Negligible gravity effects
●Governing Equations:■Stimulated Zone:
■Unstimulated Zone:
Analytical Solution Derivation: Dimensionless Pressure
D
D
D
DnD
DDnsD t
p
r
pr
rrr
111
11
D
D
D
DD
DD t
p
r
pr
rr
22
1
wD r
rr
w
ssD r
rr
ppqB
Lkp i
eoD
2
2wt
oD
rc
tkt
sDD rr 1
eDDsD rrr
w
eeD r
rr
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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●Initial and Boundary Conditions
■ Initial Condtions: Uniform pressure at t=0
■Outer Boundary: No flow
■ Inner Boundary: Constant rate
■Region Interface: Continuous pressure across the interface
■Region Interface: Continuous flux across the interface
Analytical Solution Derivation: Dimensionless Pressure
0,, 0201 DtDDDDtDDD trptrp
02
eDrDrD
DD r
pr
nsD
DrD
DD r
r
pr
1
1
sDrDrDsDrDrD pp 21
sDrDrD
D
sDrDrD
Dr
p
r
p
21
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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●General Solutions in the Laplace Domain:■Stimulated Zone: Solution from Bowman (1958) and Mursal (2002)
■Unstimulated Zone: Well known solution (obtained from Van Everdingen and Hurst (1949))
Analytical Solution Derivation: Dimensionless Pressure
2
2
2
2
12
2
2
2
2
2
2
1231 n
srrIc
n
srrIc
r
r
sp
nsD
n
D
n
n
nsD
n
D
n
nnD
nsD
D
DDD rsKcrsIcp 04032
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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●Particular Solution■Stimulated Zone:
■Unstimulated Zone:
■Simplifying Notation:
Analytical Solution Derivation: Dimensionless Pressure
5140031201
51401 bfbfbbfbfb
bfbfc
5140031201
51400
12 1
1
bfbfbbfbfb
bfbfb
bc
6543
5123 ffff
aKfc
6543
5124 ffff
aIfc
2,
2
2
n
n
n
eDsDn
sDnsD raaraaraaaasrasa 0504
21231210 ,,,,,
353433322120 ,,,,, aIbaIbaIbaIbaIbaIb
,, 4151514110151404051100 aKaIaKaIaafaKaIaKaIaaf
,,, 31505141142140514013345212 bbbbaKaIafbbbbaKaIafbbbbaf
4134025111641540451015 , aKbaKbaIbafaKbaKbaIbaf
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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■Dimensionless Variables:
■Inner Boundary: Constant pressure
■Van Everdingen and Hurst (1949) presented a relationship between constant pressure and constant rate solutions
Analytical Solution Derivation: Dimensionless Rate
spsq
DD
1
s
1
2
wieoD ppLk
2
1
iw
iD pp
ppp
1,11 DD tp
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Solution Presentation
● Analytical Model Parameters
400eDr
n 300 200 100 50 20
1000 -1.2110825 -1.303764 -1.5000001 -1.7657738 -2.3058654
500 -1.0895573 -1.1729398 -1.3494849 -1.5885914 -2.0744872
200 -0.92891244 -1 -1.150515 -1.3543665 -1.7686218
100 -0.807388 -0.86917582 -1 -1.1771831 -1.5372436
50 -0.68586456 -0.73835182 -0.84948507 -1 -1.3058653
20 -0.52521819 -0.56541192 -0.65051493 -0.7657756 -1
10 -0.40369398 -0.43458787 -0.5 -0.58859184 -0.76832176
5 -0.28217007 -0.30376361 -0.34948491 -0.41140789 -0.5372436
2 -0.12152103 -0.13082287 -0.15051508 -0.17718366 -0.23137841
r sD
k wD
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Solution Presentation: Dp
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Solution Presentation: )(pd/dr DD
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Solution Presentation: )(pd/dt t DDD
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Solution Presentation: Dq
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Solution Validation: Simulation Parameters and Gridding
Model Parameters k = 100 nd = .04
Le = 5000 ft
re = 100 ft
rw = .25 ft
pi = 3500 psi
pw = 1000 psi
q = 3.067 bbl/d B = 1
ct = 7.58423E-05 1/psi
µ = .493 cp
Radial grid increments = 2 cm.
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Solution Validation: 300) (rp sD D
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Solution Validation: 200) (rp sD D
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Solution Validation: 100) (rp sD D
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Solution Validation: 50) (rp sD D
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Solution Validation: 20) (rp sD D
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Solution Validation: Dq
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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PPR vs. MFH: Simulation Parameters
Reservoir and Completion Properties SI Units Field Units
Reservoir permeability, k o 9.8693x10-20 m2 1.0x10-4 md
Matrix compressibility, c f 1.0x10-9 1/Pa 6.8948x10-6 1/psia
Matrix porosity (at p i ), ϕ i 0.04 0.04
Reservoir height, h or 2*r e 60.96 m 200 ft
Reservoir width, w or 2*r e 60.96 m 200 ft
Reservoir length, L e 1524 m 5000 ft
Wellbore length, L w 1524 m 5000 ft
Wellbore radius, r w 7.62x10-2 m .25 ft
Fracture width, w f 3.048 mm .01 ft
Fracture porosity, ϕ f 0.33 0.33
Initial reservoir pressure, p i 2.4132x107 Pa 3500 psia
Well pressure, p wf 6.8946x106 Pa 1000 psia
Fluid Properties SI Units Field Units
Oil compressibility, c o 1.0x10-8 1/Pa 6.8948x10-5 1/psia
Oil density (at 14.7 psia), ρ atm 696.658 kg/m3 43.5 lbm/ft3
Oil viscosity, µ o 4.93x10-4 Pa·sec .493 cp
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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PPR vs. MFH: MFH Gridding
■Take advantage of MFH symmetry■Simulate stencil
■ Quarter of the reservoir■ Half of a fracture■ xf = hf/2
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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PPR vs. MFH: Comparisons
●xf = 75 ft., wkf = 10 md-ft., FcD = 1333.33
●See evacuation of near fracture, then formation linear flow
●PPR Perm declines quickly, small surface area with high perm
●MFH more favorable in all cases except 25 fracture case
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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PPR vs. MFH: Comparisons
●xf = 75 ft., wkf = 1 md-ft., FcD = 133.33
●MFH early time rates reduced by an order of magnitude
●Extended time to evacuate fracture and near fracture region
●MFH more favorable in all cases except 25 fracture case
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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PPR vs. MFH: Comparisons
●xf = 75 ft., wkf = 0.1 md-ft., FcD = 13.33
●PPR compares well with MFH, even slightly better
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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PPR vs. MFH: Comparisons
●xf = 50 ft., wkf = 10 md-ft., FcD = 2000
●Reduction in stimulated volume has greatly affected MFH, not so much the PPR
●Now 50 and 25 fracture case produce within the range of PPR
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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PPR vs. MFH: Comparisons
●xf = 50 ft., wkf = 1 md-ft., FcD = 200
●MFH performance from 10 to 1 md-ft. is small
●50 and 25 fracture case produce within the range of PPR
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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PPR vs. MFH: Comparisons
●xf = 50 ft., wkf = 0.1 md-ft., FcD = 20
●PPR performs better than the MFH
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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PPR vs. MFH: Comparisons
●xf = 25 ft., wkf = 10 md-ft., FcD = 4000
●MFH rates dominated by low perm matrix at early times
●Rate decline follows closely to PPR
●PPR performs much better despite infinite conductivity fractures
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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PPR vs. MFH: Conclusions
●The reduction in stimulated volume adversely affects the MFH more than the PPR:—Loss of high conductivity surface area
●The PPR lacks the high permeability surface area that the MFH creates
●Unless the fracture half-length is small or the fracture conductivity low, the PPR will not perform as well as the MFH
●Conditions may exist where achieving high conductivity fractures is difficult. In these situations, the PPR may provide a suitable alternative in ultra-low permeability reservoirs.
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Summary and Conclusions■Introduced a stimulation concept for low perm reservoirs:
■ Altered zone with a power-law permeability distribution
■ Power-law is a “conservative” permeability distribution
■Derived an analytical pressure and rate solutions in the Laplace domain using a radial composite model
■Validated the analytical solutions using numerical simulation
■Compared the PPR stimulation concept to MFH, concluding that:■ The PPR does not perform as well as the MFH unless the fracture surface area is
small and/or the fracture conductivity low
■ The PPR does not provide adequate high permeability rock surface area
■Recommend the PPR when conditions exist that prevent optimal fracture conductivities
n
s
or r
rkrk
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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Recommendations for Future Work■Consider different permeability distributions:
■ Exponential permeability model (Wilson 2003)
■ Inverse-square permeability model (El-Khatib 2009)
■ Linear permeability model
MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M UniversityCollege Station, TX (USA) — 2 October 2012
Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder S
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ReferencesAbate, J. and Valkó, P.P. 2004b. Multi-precision Laplace Transform Inversion. International Journal for Numerical Methods in Engineering. 60: 979-993.
Bowman, F. 1958. Introduction to Bessel Functions, first edition. New York, New York: Dover Publications Inc.
Carter, E.E. 2009. Novel Concepts for Unconventional Gas Development of Gas Resources in Gas Shales, Tight Sands and Coalbeds. RPSEA 07122-7, Carter Technologies Co., Sugar Land, Texas (19 February 2009).
El-Khatib, N.A.F. 2009. Transient Pressure Behavior for a Reservoir With Continuous Permeability Distribution in the Invaded Zone, Paper SPE 120111 presented at the SPE Middle East Oil and Gas Show and Conference, Bahrain, Bahrain, 15-18 March. SPE-120111-MS. http://dx.doi.org/10.2118/120111-MS.
Freeman, C.M. 2010. Study of Flow Regimes in Multiply-Fractured Horizontal Wells in Tight Gas and Shale Gas Reservoir Systems. MS thesis, Texas A&M University, College Station, Texas (May 2010).
Mathematica, version 8.0. 2010. Wolfram Research, Champaign-Urbana, Illinois.
Mursal. 2002. A New Approach For Interpreting a Pressure Transient Test After a Massive Acidizing Treatment. MS thesis, Texas A&M University, College Station, Texas (December 2002).
Texas Tech University. 2011. Dr. M. Rafiqul Awal, http://www.depts.ttu.edu/pe/dept/facstaff/awal/ (accessed 31 October)
van Everdingen, A.F. and Hurst, W. 1949. The Application of the Laplace Transformation to Flow Problems in Reservoirs. J. Pet. Tech. 1 (12): 305-324. SPE-949305-G. http://dx.doi.org/10.2118/949305-G.
Wilson, B. 2003. Modeling of Performance Behavior in Gas Condensate Reservoirs Using a Variable Mobility Concept. MS thesis, Texas A&M University, College Station, Texas (December 2003).