Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder Ryan Sawyer Broussard Department

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)

[email protected]

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



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Problem Statement■Multi-stage hydraulic fracturing along a horizontal well is the current

stimulation practice used in low permeability reservoirs



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Problem Statement Cont.■Hydraulic Fracturing Issues:

Provided by: MicrosoftProvided by: Microsoft

(US EIA 2012)



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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)



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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


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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)



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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



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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)



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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



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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


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



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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))


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



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●Particular Solution■Stimulated Zone:

■Unstimulated Zone:

■Simplifying Notation:


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



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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

qq

2

1

iw

iD pp

ppp

1,11 DD tp



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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



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Solution Presentation: Dp



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Solution Presentation: )(pd/dr DD



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Solution Presentation: )(pd/dt t DDD



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Solution Presentation: Dq



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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.



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Solution Validation: 300) (rp sD D



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Solution Validation: Dq



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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



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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



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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



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●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



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●xf = 75 ft., wkf = 0.1 md-ft., FcD = 13.33

●PPR compares well with MFH, even slightly better



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●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



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●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



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●xf = 50 ft., wkf = 0.1 md-ft., FcD = 20

●PPR performs better than the MFH



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●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



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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.



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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

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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



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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).

Documents

Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder Ryan Sawyer Broussard Department