Finite Element Model Simulations Associates With Hydraulic Fracturing

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Finite Element Model Simulationsssociated With Hydraulic racturingSunder H. Advani, Ohio State U.1.K. Lee, Ohio State U.

bstractRecently emphasis has been placed on the developmentand testing of innovative well stimulation techniques forthe recovery of unconventional gas resources. Thedesign of optimal hydraulic fracturing treatments forspecified reservoir conditions requires sophisticatedmodels for predicting the induced fracture geometry andinterpreting governing mechanisms.This paper presents methodology and results pertinentto hydraulic fracture modeling for the U.S. DOE sEastern Gas Shales Program EGSP). The presentedfinite-element model simulations extend availablemodeling efforts and provide a unified framework forevaluation of fracture dimensions and associatedresponses. Examples illustrating the role of multilayering, in-situ stress, joint interaction, and branched cracksare given. Selected comparisons and applications alsoare discussed.IntroductionSelection and design of stimulation treatments for Devonian shale wells has received considerable attention inrecent years. 1 3 The production of natural gas from suchtight eastern petroliferous basins is dependent on the vertical thickness of the organically rich shale matrix, its inherent fracture system density, anisotropy, and extent,and the communication-link characteristics of the induced fracture system s). The investigation of stimulation techniques based on resource characterization,reservoir property evaluation, theoretical and laboratorymodel simulations, and field testing is a logical steptoward the development of commercial technology foroptimizing gas production and related costs.This paper reports formulations, methodology, andresults associated with analytical simulations ofhydraulic fracturing for EGSP. The presented model extends work reported by Perkins and Kern,4 Nordgren, 501977520/82/00048941 00.25

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Geertsma and DeKlerk,6 and Geertsma and Haafkens.The simulations provide a finite-element modelframework for studying vertically induced fractureresponses with the effects of multi layering and in-situstress considered. In this context, Brechtel et al., 8Daneshy,9 Cleary, 10 and Anderson et al. II have donerecent studies addressing specific aspects of this problem. The use of finite-element model techniques forstudying mixed-mode fracture problems encountered indendritic fracturing and vertical fracture/joint interactionalso is illustrated along with application of suitablefailure criteria.Vertical Hydraulic Fracture ModelFormulationsCoupled structural fracture mechanics and fracture fluidresponse models for predicting hydraulically inducedfracture responses have been reported previously. 12.13These simulations incorporate specified reservoir properties, in-situ stress conditions, and stimulation treatmentparameters. One shortcoming of this modeling effort isthat finite-element techniques are used for the structuraland stress intensity simulations, while a finite-differenceapproach is used to evaluate the leakoff and fracturefluid response in the vertical crack. A consistentframework for conducting all simulations using finiteelement modeling is formulated here.The steady-state and transient fracture-width response,governed by the fracture fluid variables, multilayering,and minimum effective horizontal in-situ stress, is determined initially by combining the formulations and solutions presented by Geertsma and Haafkens 7 and Advaniet al. 12 The plane-strain vertical-crack model is illustrated in Fig. 1, with fluid coupling provided by thecrack interface pressure. The one-dimensional widthequation, as applied by Nordgren 5 on the basis of thePerkins and Kern model,4 for a porous-permeableisotropic elastic medium is

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...... A.E,;'''z

J fl f, '1J.....Ea. .

/: ..

Fig. 1-Plane-strain idealization of vertical-crack model.

E128 l-v2) (2H)p. ax at

8C= 7r.Jt-T(X) , ........................... 1)

along with the initial condition W X,O) =0 and the boundary conditionsW(X,t) =0, for x> L(t)

and_(OW4) _ =256p.(1-v 2 )Q

ax X-a 7rEfor a two-sided fracture.The steady-state solution and finite-difference resultsfor the transient problem have been presented by Nordgren 5 and Geertsma and Haafkens. 7 The discretizedfinite-element weak form of the extended version of Eq.1 obtained by the conventional Galerkin approach, 4 is

where

and

Fi(t)=_8C ) --==N=i dx__N_ i I7r .Jt-T(X) 7rH x=o210

where Ni is the selected interpolation function associatedwith the ith node defined by

and commas designate differentiation .For the steady-state problem, the appropriate equationisKi/a)ai +Fi(t)=O, ........................ 3)

withEK

Ij 128 l-v2) (2H)p.

. N N dx) T QN i II X j X 7rH x=oand

Fi= 8C f Ni dx,7r J .Jt-T(X)

where Ni is defined by

Eq. 3 is the familiar linear stiffness/force matrix formulation. On the other hand, the nonlinearity inherent inthe stiffness matrix for Eq. 2 requires the solution of anonlinear set of algebraic equations. The isotropicmedium steady-state and transient width profiles aremodified for the multilayered eometry by introducingthe width scaling coefficient. I Subsequent cumulativeleakoff computations are based on the width response.The fracture height and length evaluations require determination of the vertical crack stress intensity factor andapplication of the percentage leakoff volume. 2 The appropriate volume-balance equation for a two-sided fracture with an elliptical cross-section is

7r [d fL ] 1 dL dT- ( 2H ) - J W(x,t)dx +4HC - dt dT v t - TQ2

o 0.................................. (4)

Eqs. 1 and 4, coupling the transient width and length,provide an iterative framework for computing W(x,t) andL(t). Finite-element computations for several cases andcomparisons with the Perkins and Kern 4 and Geertsmaand DeKlerk 6 model results are given by Lee. 15Eq. 2 can be reduced to an approximate set ofnonlinear algebraic equations by dividing the time rangeinto intervals (tn' t n+ I) For each time interval, the timerate of change of the width extended vector may be approximated by the finite ratio ti* =(an+ -a n)/6.t at aparticular time t* where a* z a(t*), an za(,tn),an+1 ='a(tn+I), and 6.t=tn 1 - tn Eq. 2 then becomes

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where

and

The Newton-Raphson method can be used to solve thenonlinear Eq. 5, following formulation of a curtailedTaylor series expansion, at each time step of the computation. Procedures and results for the simultaneouscomputation ofEq. 1 and the integral Eq. 4 are given. 15Vertical Hydraulic Fracture

odel SimulationsFor the steady-state case, the width and correspondingpressure profiles for specified parameters are comparedwith the Perkins and Kern solution 4 in Figs. 2a and 2b,respectivel y. The transient width profile for equalelapsed propagation time intervals is illustrated in Fig.2c. The parameters, selected from Geertsma andHaafkens 7 for model calibration, are: flow rate Q= 10bbl/min 1.6 m 3 /min), injected volume V=200 bbl 318m 3 ), fracture height 2H= 100 ft 30.48 m), fluid losscoefficient C=0.0015 ft/min 1/ 0.000 46 m/min Ij, ,Poisson s ratio 1 =0.20, shear modulus G=EI2 1 +v)=2.6x 10 6 psi 17.94 GPa) , and viscosityfL=36 cp 36 Pa s).Maximum width comparisons using the onedimensional finite element formulations and the Perkinsand Kern 4 and Geertsma and DeKlerk 6 models are illustrated in Fig. 3.The extension of the preceding simulations of thewidth profile to the case of a multilayered formation withdifferent prevailing layer in-situ stresses necessitatesdevelopment of width scaling curves. These curves incorporate layer elastic moduli ratio, vertical fracturepenetration, and in-situ stress differentials across thelayers. Figs. 4A and 4B reveal the developed fracturewidth scaling curves obtained from finite elementsimulations of the plane-strain model Fig. 1). Thepreviously developed isotropic width magnitudes can beconverted to the layered case by multiplication with thepressure and tectonic stress scale factors and appropriatesuperposition.The corresponding fracture height is obtained by useof the stress intensity factor concept. For the layeredcase, fracture penetration, arrest or interfacial propagation can depend on several factors, such as materialproperty ratio, in-situ stress differences, effective bot-tomhole pressure magnitude, and interface friction/slipcharacteristics. Figs. 5A and 5B illustrate the computedstress intensity factors based on an r 1/, singularity and induced by uniform crack pressure and differential tectonicstress. The interface is assumed to be bonded perfectly.The stress intensity trend in Fig. 5A departs considerablyfrom that of Erdogan and Biricikoglu 6 for Hlh> 1since in the latter the crack pressures are scaled to accommodate the imposed displacement compatibility forthe closed form solution. Stress intensity factorAPRIL 1982

::>co~wa:a..


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inem)0.200.5)

0.4)0.15Iti

3w (03 )0::: Jt:i 0.10l t ( 02 )

0.05(01 )

Q = 10 bpm (1 .6 m3/min)C = 0.0015 ft/.'iiiin (0.00046 m/.'iiiin)

2H = 100 ft (30.5 m)v = 0.15G = 2.6 x 10 6 psi (1.79265 x 10 10 N/m2)JJ = 36 cP (0.6 x 1 O ~ Pa min)

p : Perkins & Kern's modelC : Combined model

pCG

G: Geertsma & De Klerk's model

0.00 - - - r - - - - - - - - - - . - - - - r - - ~ -o 10 20 30 40 50 60INJECTION TIME ( min)Fig. 3-Comparison of instantaneous maximum fracturewidth.

VZ= I =0 2

WE, 0 32hA,; 0 20.10 0 1.0

H h

Fig. 4A-Average fracture width plots for induced pressures.

E./E, '1.01.5

1.52 0to 3 0

WE, p ..ft P.

~5 2H 2h E, '2 = ', =0.2E0.0 1.0 1 1 1.2 1 3 1.5l i /h

Fig. 48-Average fracture width plots for in-situ stress differentials.

212

U

I1.6

U

1.2

to"o. ,':, \

.- \ \ \0.6

IQ4 - FINITE ELEMENT SOLUTIONS -1 < 0I

EXTRAPOlATED VAlUES II,1: 1 I: 0.2 IO. II

0 2 0 4 0 6 0 8 1.0 1.2 1.4H h

Fig. SA-Stress intensity factor plots for induced pressures.

T ... . aO.20 6 2H1 E,0.5

fb

Fig. 58-Stress intensity factor plots for in-situ stress differentials.



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_r--------------------------------,---,- I: IIO....r.. (8.1 xl 05MPam1/a).. A'.I:"'lIIO .. c.


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I

inem)

0.5(1.2)

0.4

0.8)0.3wa:::::la:: 0.2LL

. 0.4)X


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TABLE 1 STRESS INTENSITY FACTORS FOR DENDRITIC FRACTURE MODELc/b 0.00 0.25

Loading _ _x_ _ _x_Kf/(p,sX syl Ja 0.762 -0.460 -0.320 0.730 -0.465 -0.280 0.672

\ < ~ / ( P , S X S y ) . J a -0.510 -0.030 0.547 -0.592 -0.040 0.630 -0.472K[/(P,SX Sy);{a 1.025 0.210 -1.310 1.471

y. / w t

1x = H1 XH1

Fig 1 Two dimensional dendritic fracture model.

BRNCH CRCKSIGMA g

l J -2500.00C ) -2000.00

-1500.00+ -1000.00X -500.00 x0 0.00 L y+ 500.00lI: 1000.00Y 1500.00) 2000.00lI 2500.00

Fig A-Stress contours dxx induced by fracture fluidpressure p =1,000 psi (6.9 MPa).

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0.50 0.75 1.00_S_X_ _ _X_ _ ___ _X _y_

-0.473 -0.217 0.590 -0.483 -0.120 0.489 -0.488 -0.021-0.026 0.500 -0.300 -0.130 0.360 -0.248 -0.025 0.2570.193 -1.681 1.710 0.150 -1.874 1.839 0.107 -1.966

BRNCH CRCKSIGMA 'l l -2000.00< l -1000.00A 0.00+ 1000.00X 2000.00 3000.00-t ~ o o o o o: 5000.00y 6000.00) 7000.00lIE 8000.00

Fig B-Stress contours ayy induced by fracture fluidpressure p = 1,000 psi (6.9 MPa _

0 .110 J J l t 1 j aHMIN l1f1E z ~ ,---------y.--------. ,--

O .... N 1 T r 1 I I r _NFig 12-Joint interaction at bi-material interface(EiE1 = 1.25) and vertical-crack model.

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element method. Table I shows the computed stress intensity factors for the pressure and in-situ stress loadingswith c b=O, 0.25, 0.50, 0.75, and 1.00. The magnitudeof the normalized Mode stress intensity factorsignificantly contributes to the secondary crack propagation. As an example, the analysis in Table Idemonstrates that the secondary crack tip effective stressintensity factor K eff = [ K ~ ) 2 K ~ , ) 2] 1/ exceeds thecorresponding primary tip value K1 for 0::;c b::;0.075with Sx/P=2 and Sv/P= 1. Also, even for c b= 1,secondary crack propagation can be shown for selectedratios of in-situ stresses and crack pressure. Variousmixed-mode fracture propagation criteria are presentedin the subsequent text. Stresses induced by the fracturefluid loading are shown in Figs. l lA and lIB.Another example of mixed-mode conditions resultsfrom the interaction of the induced vertical fracture witha bedding layer interface or joint. Fig. 12 illustrates avertical crack intersecting a horizontal joint at thematerial interface. This bimaterial problem, in theabsence of shear tractions, has been studied by Goreeand Venezia. 2 The shear tractions at the joint interfacecoupled with the interaction of fracture fluid pressureand tectonic stresses produces conditions favorable to interfacial crack propagation. In this context, conditionsfor crack confinement, penetration, or interfacial crackpropagation have been studied experimentally by Teufeland Clark. 22 For an interface crack subjected to uniformshear stress T the computed values of K IT..[; and K /T..[; at the joint tip are -0.198 and 0.593. Thesevalues, when superposed with the in-situ and fracturefluid pressure stress intensity, produce conditions similarto the dendritic fracture model, with c b=O.Mixed-Mode Fracture CriteriaSeveral mixed-mode fracture initiation criteria areavailable in the literature. Ingraffea 23 has presented acomparison of the maximum hoop tensile stress, 24minimum strain eneil Y density, 25 and maximum strainenergy release rate 2 theories. In addition, a fracturecriterion for rock media with crack closure and frictionaleffects has been developed by Advani and Lee. 27 Thesecriteria are reviewed in the following.Maximum Tensile Hoop Stress Theory. 24 In thistheory, the fracture envelope is governed by

O( 203.c o s - K,cos ---K smO =K,c, . . . . (7a)2 2 2where the fracture initial angle, 0 is governed by

K, sin O+K,,(3 cos 0-1)=0 7b)Maximum Strain Energy Density Theory. 25

K1(3 -4v-cosO)(1 cosO+4K,K"sinO[cosO-(l-2v)]+K1,[4(I-v)(I-cosO)+(l +cosO)(3cosO-l)]=4(l-2v)K2,C, (8a)

216

with 0 determined fromK1sinO(2cosO+4v-2 +4K, K" [cos20 - 2v cosO]+K1I(2-4v-6cosO)sinO=0 8b)

and imposition of the condition for stable crack propagation.Maximum Strain Energy Release Rate Theory. 26 Thefracture locus for this theory is defined by

Crack Closure and Frictional Effects Theory. 27 Thistheory, based on the maximum circumferential stress, includes the effects of crack closure and friction. Thefailure threshold is defined by

K I l ) 2 + ~ = 1 .2KI K C . . . . . . . . . . . . . . . . . . . . . . . (10)Although this theory departs considerably from theaforementioned conventional theories, reasonable correlation of this theory with available experimental datafor rock media with shear and compressive loading hasbeen obtained.27 Further controlled experiments onmixed-mode testing under simulated in-situ conditionsare necessary. Fig. 13 illustrates the variation betweenthe theories defined by Eqs. 7 8 9 and 10.

ConclusionsThe theoretical simulations of the induced crack-openingmode and mixed-mode propagation responses provideinterpretive, qualitative, and comparative information onthe governing hydraulic-fracture mechanisms andfracture-geometry prediction. The transient verticalhydraulic fracture formulations and results for crackwidth, height, and length are applicable to layered formulations with differential in-situ stress. Vertical migration of the fracture in the overburden can be minimizedby discrete control of the treatment pressure and/oralteration of the local effective minimum horizontalstress by means of successive pressure drawdownfollowed by sequential fracture propagation. The mixedmode evaluations suggest a possible rationale for the effectiveness of dendritic fracturing for reservoirs with alarge number of pre-existing systematic fractures, jointsystems, and favorable horizontal in-situ stresses withlocal fracture fluid pressure modification.

Several characteristics for the assignment of conventional, foam, dendritic, or explosive treatments haverecently emerged.13.28-30 These factors include consideration of fracture density and extent, shale thickness,relative in-situ stresses, energy assist mechanisms, wellcleanup, shale/fracture fluid interaction, proppant selection, and vertical fracture migration. Preliminary resultsindicate that correlation with the prevailing in-situ stressgradients or isotropy indices are promising diagnostic indicators for fracture-treatment selection and design. Thecomprehensive development of a cost-effective stimulation strategy, however, requires extensive and controlledSOCIETY OF PETROLEUM ENGINEERS JOURNAL


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field testing with supportive laboratory and predictiveanalysis.Nomenclature

a, b ca ili*C

CijE

Fi(f)GhH

K I , KK ILNi

pQ

Sx, SytW x,t)xy)

iJv -a -

arex

r -

Superscriptsi j, k

T =

Subscriptseff

HMAXHMINi j, k

n =OVBD

crack dimensions for dendritic modelwidth vector componenttime derivative of width vectorfluid loss coefficienttransient width coefficientelastic modulusleakoff forcing functionshear modulushalf pay zone heighthalf fracture heightMode I and II stress intensity factors,

respectivelycritical Mode I stress intensity factorfracture half lengthinterpolation functioncrack pressurefracture fluid injection ratehorizontal in-situ stressestimefracture widthhorizontal coordinate in fracture

directiontransverse coordinateangular coordinatefluid effective viscosityPoisson's ratiostress componenthorizontal in-situ stress differentialfluid loss delay timejoint shear stress

indicial componentstranspose

effective magnitudehorizontal maximum valuehorizontal minimum magnitudeindicial componentstime step designatoroverburden magnitude

AcknowledgmentsThe support of the U.S. DOE under Contract No. DEAC 21-79 MC 10514 is gratefully acknowledged. Computational assistance was provided by E.Y. Lee.

References1 Komar, C.A.: Development of a Rationale for StimulationDesign in the Devonian Shale, paper SPE 7166 presented at the

APRIL 1982

2.0

Eq.IO0.5

1.0 Eq.9

0.5 Eqs8 a bEqs 7 a b

o. 0.5 1.0

Fig 13-Comparison of mixed-mode fracture envelopes.

SPE Regional Gas Technology Symposium, Omaha, NB, June7-9, 1978.2. Creman, S.P.: Novel Fracturing Treatments in the DevonianShale, Proc., U.S.DOE First Eastern Gas Shales Symposium,Morgantown, WV (1977) 288-308.3. McKetta, S.F.: Investigation of Hydraulic FracturingTechnology in the Devonian Shale, Pmc., U.S. DOE SecondEastern Gas Shales Symposium, Morgantown, WV (1978) 125-34.4. Perkins, T.K., and Kern, L.R.: Widths of Hydraulic Fractures,1 Pet. Tech. (Sept. 1961) 937-949; Trans., AIME, 222.5. Nordgren, R.P.: Propagation of a Vertical Hydraulic Fracture,Soc. Pet. Eng. 1. (Aug. 1972) 306-314; Trans., AIME, 253.6. Geertsma. J., and DeKlerk, F.: A Rapid Method of PredictingWidth and Extent of Hydraulic Induced Fractures, 1. Pet. Tech.(Dec. 1969) 1571-1581; Trans., AIME, 246.7. Geertsma, J and Haafkens, R.: A Comparison of the Theoriesfor Predicting Width and Extent of Vertical Hydraulically InducedFractures, 1. Energy Resource Tech. (March 1979) 8-19.8. Brechtel, c. Abou-Sayed, A.S., and Clifton, R.J.: In SituStress Determination in the Devonian Shales (IRA McCOY20402) within the Rome Basin. Terra Tek Report TR 76-36. SaltLake City (July 1976).

9. Daneshy, A.A.: Hydraulic Fracture Propagation in Layered Formations. Soc. Pet. Eng. 1. (Feb. 1978) 33-41.10 Cleary, M.P.: Primary Fractures Governing Hydraulic Fracturesin Heterogeneous Stratified Porous Formati ons, paper ASME78-PET-47 presented at the Energy Technology Conference andExhibition, Houston, Nov. 5-9, 1978.11 Anderson, G.D., Hanson. M.E., and Shaffer, R.J.: Theore ticaland Experimental Analyses of Hydraulic Fracturing. Pmc U.S.DOE Third Eastern Gas Shales Symposium, Morgantown, WV(1979) 225-246.12 Advani, S.H., GangaRao, H.V.S., Chang, H.Y., Komar, C.A.,and Khan, S.: Hydr auli c Fracture Modeling for the Eastern GasShales Project, Proc U.S. DOE Second Eastern Gas ShalesSymposium, Morgantown, WV (1978) 1 84-98.13 Advani, S.H., Komar, C.A., Chang, H.Y., and Stonesifer, R.:

Rock Mechanics Aspects of Hydraulic Fracturing in the Devonian Shale. Proc., U of Missouri-Rolla, 21st U.S. Symposiumon Rock Mechanics, The State of the Art in Rock Mechanics.Rolla, MO (1980) 701-709.14 Oden, J.T.: Finite Elements of Non-linear Continua, McGrawHill Book Co. Inc., New York City (1972) 123.15 Lee, E.Y.: One Dimensional Finite Element Modal Simulatorfor Predicting Hydraulic Fracture Widths and Lengths. MSthesis. Ohio State U., Columbus (1981).

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16. Erdogan, F., and Biricikoglu, V.: Two Bonded Half Planes witha Crack Going Through the Interface," Int. 1 Eng. Sci. 1973)11, No.7, 745-766.17. Erdogan, F.: "Fractu re Problems in Composite Materials," Eng.Frac. Mech. (1972)4, No.4, 811-840.18 Cremean, S., McKetta, S.F., Owens, G.L., and Smith, E.C.:"Massive Hydraulic Fracturing Experiments of the DevonianShale in Lincoln County, West Virginia," Final Report to U.S.DOE, Contract E(46-1) -8014, Columbia Gas Co. (Jan. 1979).19 Swolfs, H.S., Lingle, R., and Thomas, J.M.: "Determination ofthe Strain Relaxation and their Relation to Subsurface Stresses inthe Devonian Shale, Terra Tek Final Report TR 77-12, Salt

Lake City (Feb. 1977).20. Kiel, O.M.: "Hydraulic Fracturing Process Using ReverseFlow, U.S. Patent No. 3,933,205 (1976).21. Goree, J.G., and Venezia, W.A.: "Bonded Elastic Half-Planeswith an Interface Crack and a Perpendicular Intersecting Crackthat Extends Into the Adjacent Material-I, Int. 1 Eng. Sci.(1977) 15, No. I, 1-17.22. Teufel, L.W., and Clark, J.A.: "Hydraulic Fracture Propagationin Layered Rock: Experimental Studies of Fracture Containment, paper SPE 9878 presented at SPEIDOE Low-PermeabilityGas Symposium, Denver, May 27-2 9, 1981.23. Ingraffea, A.R.: On Discrete Fracture Propagation in RockLoaded in Compression," Proc., U. C. Swansea, First IntI. Conference on Numerical Methods in Fracture Mechanics, Swansea,U.K. (1978) 235-248.24. Erdogan, F., and Sih, G.C.: On the Crack Extension in PlatesUnder In Plane Loading and Transverse Shear," 1 Basic Eng.(Dec. 1963) 519-527.25. Sih, G.C.: "Strain-Energy Density Factor Applied to MixedMode Crack Problems," Int. 1 Frac. (1974) 10, No.3, 305-321.

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26. Hellen, T.K., and Blackburn, W.S.: The Calculation of StressIntensity Factors for Combined Tensile and Shear Loading," Int1 Frac. (Aug. 1975) 605-617.27. Advani, S.H., and Lee, K. Y.: "Thermo-mechanica l FailureCriteria for Rock Media, Proc., U. of Texas, 20th U.S. Symposium on Rock Mechanics, Austin (1979) 19-25.28. Komar, C.A., Yost, A.B., and Sinclair, A.R.: "Practical Aspectsof Foam Fracturing in the Devonian Shale," paper SPE 8345presented at the SPE 54th Annual Technical Conference and Exhibition, Las Vegas, Sept. 23-26, 1979.29. Schmidt, R.A., Warpinski, N.R., and Cooper, T.W.: "In-SituEvaluation of Several Tailored-Pulse Well Shooting Concepts,"paper SPE 8934 presented at SPEIDOE Unconventional GasRecovery Symposium, Pittsburgh, May 18-21, 1980.30. Horton, A. ,: A Comparative Analysis of Stimulations in theEastern Gas Shales, Morgantown Energy Technology CenterReport DOE/METC-145 (1981).

SI Metric Conversion Factorsft x 3.048*in. x 2.54*psi x 6.894 757

*Conversion factor is exact.

E OlE+OOE+03

memPaSPEJ

Original manuscript received in Society 1 Petroleum Engineers office March 171980. Paper accepted for publication Sept. 11, 1981. Revised manuscript receivedDec. 22 1981. Paper SPE 8941) first presented at the 1980 SPE/DOE Symposiumon Unconventional Gas Recovery held in Pittsburgh May 18-21.


Finite Element Model Simulations Associates With Hydraulic Fracturing

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