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ml. ...I Society of Petroleum Engineers I

SPE 28392

Pressure Transient Behavior of a Finite ConductivityInfinite-Acting and Bounded ReservoirsB.D, Poe Jr. and J.L. Elbel, Schlumberger Dowell, and T.A. Blasingame,

SPE Members

Copyright1994, Societyof PetroleumEngineers,Inc.

IFracture in

Texas A&M U.

0

This paperwas preparedfor presentationat the SPEWh AnnualTechnical~nfermce and Exhibitionheld In NewOrleans,LA,U.S.A.,25-2S September1994.

rhis paperwas selecled tor presentationby an SPE ProgramCommitteefollowingre.lew of informationco.tained In 8n abstract submittedby the author(s).Contentsof the Psper,as presented,have notbeen reviewedby iho .SoCletYof PefJoloumEngineersand are mbJectto wrreCtlOnby the author(s).The material.as presented,does not nwessarily refltmtanYpmitionof the Societyof PetroleumEngineer?,IISoff!~rs, Ormem~rs. paPer8P

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2 PRESSURE TRANSIENT BEHAVIOR OF A FINITE CONDUCTIVITY FRACTURE = SPE28392IN INFINITE-ACTING AND BOUNDED RESERVOIRS

The use of an increasing number of elements in the fracture to accuratelymodel the flux dfstribu&m introduces another numerical difficultly. @increasing tie number of fracture elements.used, thedistarmes betwaanconsecutive fracture nodes (midpoints of elements) decreases (tends fezero), resulting inanumerically singular coefficient matrix for tie system.An additional factor hat also must be considered Is that the use of anumerical Iaplace transform inversion procedure will tandtoma~ify tieerrors in the wellbore pressure and flux distribution values obtained.

The use of higher order (linear) basis functions has been appffed bylGkani5 to more accurately modal tie flux distribution in tie fracture.While this technique greatiy improves the accuracy of tie flux cfstributionand wellbore pressure estimates obtahxf, further improvement car! beobtained by using a smwth continuous basis function for tie unknownflux distribution rather than finear flux cfscrete elements. Based en tieresults of research in this subject, it has been found hat a secend orderrational polynomial can be used to amurafely model tie flux distribution ineach segment of tie fracture that exhibits a monotonic flux distribution.Use of a smooth continuous function to approximate tie flux distributionhas been found to diminate the numadcal difficulties commoniyencountered when using discrete uniform or linear flux elements.

The problem U@ is addressed in this paper is the numerical difficultiesand Inaccuracy ckfen encountered when avafuating the Bounday Elementsolutions in the usual manner. In this paper, a c!etiled analysis procadureis presented which results in the anafytic solution of the transient bahaviorof finite-cenductivify vertical fractures that are located in infinite-acting orfinite reservoirs.

Fracture and Reservoir ModetsThe reservoir mcdels that are considered in this study are hose of ainfinite-acting reservoir, a cylindrically bounded reservoir with either aconstant pressure er no flow outer boundary cenditfon, and a dosedrectangulady beunded reservoir. The reservoir is assumed fo be auniform, homogeneous, horizontal slab with thickness h, average fwrosiiy$ directional permeabitities (lG and ~), constant fluid viscosity (p), andsmall and constant system compressibffify (C,).

The reservoir is considered to contain a single fluid, Gravkaticmal effectsare assumed to be negligible and tie upper end lower boundarfas of thereservoir are assumed tD ba impermeable. The reservoir Is assumed tohave a uniform initial pore pr&sure, PI. Dual porosily reservdr behatief isconsidered in the model using the tachnique reported by Gringarten.d

The fracture is assumed to be a uniform, vertical, homogarraeusrectangular sfab of thickness (b,), height (h, ), permeability (k, ), andfracture half-length (X,). The fracture is assumed to be symmetric abcuttie wellbore. No flow boundatfes are assumed at the fracture tips(XO=XW&l). The well is .assumad to be a plane source.

The cmnvenfional definitions of the variables for geometric mean horizonta!permeability, dimensionless time, pressure, fracture height, fracture width,fracture proppant permeabifify, fracture conductivity, hydraulic diffusivity,spatial variables, and dimensionless fracture flux are given by Eqs. 1-12.

,,.=- .. ...... . .. ... .. . .... .... .... . .. .. .. . . . .. . .. . . .. (8)

x~ -; .. .......... ................................................................................. (9)

. .......-..=-.... ............................. ........................................ (lo)YD.L . .Xfr

oT ........ ..%...s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..s..- (11)

gD(XD,S) = @f(xD, s)M ...>............... .......................... ............ .... .. ..- (12)q

Censfant rate production is assumed for tie system, in which heresarvoir fluid Is produced into the fracture, then via tie varticaf fracture bthe wellbore.

The solutions of aach of the reaarvoir and fracture ffuid flow problems arefully addressed in be Appendices. As an example, the resewoirdimensionless pressure solution for an infinite-acting reaarvoi~z with afinite-mnductivify fracture is gwan by Eq. 13.

.(...?)-:[}(.) p==.[ k 1~ d~~ -------------- (13)Fer a finite-conductivity verticaf fracturs with fracture storage effects, tiewsllbore dimensionless pressure3 is given by Eq. 14.

[ 4F][~ +-J~~or[X~s)cOs~O~~D]Pw(s).~ y ~kmb~h~

t

.......................................... ..... . ........ ..... ............................ . ... .... ..... . (14)

Similarly, for the ~se where ffie fracture storage effects are assumed tebe negligible, the relationship between tie wellbore pressure and thepressure at some spatial position in the fracture is given by Eq. 15.

[

XD~wD(sj- ~D,(X@). o ;XWD- (XD- XWD)J qDt(XD,S)WD

ho(la)

xD

+ J%?D(xDF)~DKvD 1

The analyticsolutions of tie ceupled resarvoir and fracture systems arepresented in Appandix D. The solution of tie singular integral aquationsresulting from the reservoir and fracture flow problems is accomplished byusing mean value fluxes for intervals of tie fracture length Mat have amenotonic flux distribution. For high conductivity fractures, or even formoderate to low conductivity fractures at vary ear~ times, rhe fluxdistribution in each wing of tie fracture is commonly monotcnicalfyincreaifng from the wellbore to tie fracture tips (fn real-time domain). infhk case a single mean value flux interval maybe assigned for each wingbf the fracture. Nota hat in Laplaca spaca the reverse is true, with theLap lace transformed flux distribution decreasing with increasing dW.ncefrom the waflbnre for a monotcnic flux distribution. To avoid confusion inthe development work and discussions that follow, reference tD tie fluxdfs~ibufion will be as it appeas in real apace.

306

.SPE 28392 B.D. POE, JR,, J.L. ELBEL, AND T.A. BLASINGAME 3

At faterfimas for moderate conductivity fractures, or for practically allvalues oftimewiti ve~lowtimensionless wnductivi~ fractures, rhe fluxdistribution in each wing of the fracture.js nat. monotonic. The fluxdk+tribution in this case is characterized by a monotonically decreasingflux distribution from the wellbore to some point in the fracture(L&&&.+1 ) at which tie flux distribution exhibits a minimum, andthen monotonically increases from that spatial pasifion to the fracture tip.Throughout tie development work presented in this paper, tie spatialpasition at wMch the ffux distribution exhibits a minimum in this case hasbean danotadas%=Xw&. Theiwofypesofffux @ribution discussedare illustrated in Fig. 1.

In each of the cases considered in the study, the well is considered to becenfrefly located in the reservoir drainage area. This is done to simply takeadvantsgeof fhesymmefryof the problem. For a uniform, slab, varticalfracture intersecting averfical well (in which the fracture is symmetricabout Me wellbore) that is centrally located in tie reservcb dr.ahage ares,Iheflux distribution in each wing of the fracture is symmetric about tiewellbore. The symmetry of the problem permits a much simplerapplication of the me% value flux technique to tie finite-conductivityfracture problem, and greauy simplifies the required evafuatiinprocedures. .

Both, tie conditions under which a monotonic flux distribu60n will bexhibited ineach wing of Lhe fracture, endtiesnalytfc pressure transientsolutions obtained for swell intersected by a fmite-conductivity verticalfracture with a monotonic or non-monotonic flux distribution are presentedin Appendx D for the various types of raservoir and fracture modelsconsidered Evaluation of the reservoir and fracture pressure transientsolutions using the mean value flux technique presented in this paperresults in expressions for the transient behavior of a wefl intersected by afinite-conductivity vartical fracture fhat are (1) .@ytic SOIUfiOr!Sof hefinite-conductivity fracture problem, and (2) provide a msans of computingthe pressure or rate-transient behavior of a very low-conduc8vly verticalfracture. Each of lhese iseues is addressed in the discussion of theevaluation of the solutions that follows.

Evaiuatlon of Transfent SolutionsE@fuationof the transient bahavfarof finita-omductivi~ fractures usingthe relationships presented in this paper has been found to provide moreaccurate asfimates of the pressure and rate-transient response offractired wells tian can ba obtained using tie dlscretfzad fracturesohstionsof Refs. 11 through 13. Bycomplefinges much of Uleenafysis

and evaluation procedures as possible analytically, before resorting tonuma+ical techniques, the numerical difficulties ancounterad when hyingto evakrate the transient response of low-conductivity fracture3 isefiminafad.

The numerical diffiiultfes associated with the discret!zed fracturesolutions are due largely to tie singular nature of tie integral equations ofthe transiant solutions themselves. The singularities in both tie fluxdistribution and the Greenss function of the reservoir ad fractureeolufions cbnotpose a significant problem when heenalytic techniquesemployed in the Appendices of this paper are used.

A comparison of the accuracy obfehred with tie analytic solutions of Wsstudy, with those resu[tingfromth,e numsricel sokttlon of tie discretizedfracfure solutions of Refs. 11 end13-is presented in Fig.2. Thedsviatiocof tiedfscretized fracture solution response at aerfy transient times andIow-conducfivitfesls due to tie fact fhatfhe cfiscretjzed fracmre so[uflonresults in a numerically singular coefficient matrix. This observation isobtined by examination of the condition number of thecoaffkientm afrix,end from the discussion given previously concerning thesingularnaturaof the reservoir and fracture solutions.

The transient behavforof afinita wnducfivify fracture that is centraffylocated in a closed cylindrically bounded resswoir is presenfad in Fig. 3.The transient end pseudosfeacty-srafe behavfor of various drainage arealsxtenfs are presented in this figure to @amonstrate the stabffity of ffia

solution procedures used in this stuctj for both earfy and lata timebehavior.

The corresponding evefuation of me transient behavior of a well,intersected by afinite-conductivity fracture, mat is centrally located in a@ndrically kunded reservoir with a constant pressure outer boundary isgiven in Fig. 4. The early transient end late time steady-state response areboth accurately and readily evaluated using the solutions developed in thi3study.

The pressure transient behavior of a finite-conductivity fracture that iscantrally Iocatad in a ractangukxly boundsd reservoir with no flow outarboundaries is presented in Fig. 5. The transient behavior presented in thisfigure is genarafed by coupfing me early time transient behavior of aninfinite-acting resewoir with the late time behavior of me racfangularlybounded reservoir solution.

The results obtained with me infinite-acting and finite reservoir solutionspresented in this paper are found to be in agreement with both pubfishedresults and finite-difference numerical simulation results for me transientbehavior of finite-conductivity fractures. The transiant solutions presentedin this papsr are analytic and provide a rapid maans of estimating thetransient behavior of ffnife-cxmducfivify fractures.

The prachcat effects of reservoir permeability anisotropy end duet-porosityresewoir behavfor are wall reported in the literature end need not breproduced in this discussion. However, these effects have beanincorporated into tie transient solutions presented in this work so that mesolutions can be readily used for well-test analysis or well performancemcdeffng when necessary.

The general anafytic solution procedure outlined in Appendix D provides atschniqua for evaluating tie wellbore pressure transient behavior of moregsneral types of finife reservoir shapes and boundary cardiffons.n AH oftieresarvoir pressure solutions can be expressed as Fredholm integralequations with limits of integration over tie entire fracture length. lhewellbore pressure transient behavfor of finite-conductivity fractures thatare centrally located in other drahage area shape reservoirs, or inrecfangulerfy bounded resewoirs with mixaci Dfrfchlet and Neumann outerboundary conditions (both lateral and upper and lower boundaries) cantherefore be developed by simply using tie appropriate resavoirdimensionless solution in this general soluton procedure.

CONCLUSIONSThe following conclusions and observations are obtained from the resultsof this study.

.

1. The ameJytic solutions presented in this work provida en accuratemaans of estimating me transient bahavior of finita-conductivityfractures in infinite-acting reservoirs, or centrally located incyfinckically bounded reservoirs with eithar Neumann or Dfrichletouter boundaries, and closed rectangularly bounded reservoirs,

2. The analytic pressure transient solutions reported in Ibis workprovide a more accurate and far more rapid means of estimating meearfy tmnsient behavfor of low-conductivity fractures tian do thedkcretizad fracture semi-anefytic pressure transient solutions.

3. The generaf snalytk sofufion procadure srnployad in thk work canba used to develop tia wellbore transient solutions for more dive+setype of reservoir drtinage area shapes and mixad buund~conditions than those considered in thk work.

NOMENCLATUREB Resewofr fluid formation volume factor, r m31mb, Fracfure width, mb,~ Dimensionless fracture widrhc, ToW system cwmpreseibility of reservoir, I/Pa

I

307

.PRESSURE TRANSIENT BEHAVIOR OF A FINITE CONDUCTIVITY FRACTURE SPE 2S392IN lNFIN~E-ACTING AND BOUNDED RESERVOIRS

Total system compressibility of fracture, I/PaDual porosity reservoir functionReservoir net thickness, mFracture height, mDimensionless fracture heightGeometric mean horfzonfal permeability, mzFracture proppant permeability, mDimensionless fracture permeabilityX direction reservoir permeability, m2Y direction reservoir parmeabiff~, mzFourier series indexPressure, PaDimensionless pressureLaplaca space dirnens~onl&s reservoir pressure

Laplace space dfmensionles3 fracture pressureInitiaf reservoir pressure, PaDimensionless wellbore pressureLaplace space dimensionless wellbore pressureSandface flowing pressure, PaConstant well production rate, mslsDimensionless fluxLaplace space resewoir dimensionless fluxLaplace space fracture dimensionless fluxDimensionless radial spatiaf podionCyffndrical drainage radfus, mDimensionless cylindrical drainage radiusbplace space parameterTime, sDimensionless time, referenced to system characteristic lenghDimensionless time, referenced to drahage areaParameter of integrationX direction spatiaf position, mDimensi0nlr3ss X diraction spatiaf positionX directfon drainage a reaJ extent, mDfmensionleeeXdirection drainagearealextentFractura half-length (system characteristic length), mX dfracfion wellbore spaffd position, mX direction dimensionless wellbore spatial positionY direction epatfal position, mY direction dimensionless spatial positionY dfrmtfon drainage a real extent, mY direction dimensionless drainage a real extentY direction wellbore spatial position, mY direction dimensionless wellbore position

Mean value fiuxpo9ition forinterval Xm

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SPE 28392 B.D. POE, JR., J.L. ELBEL, AND T.A. BLASINGAME 5

14.

15,

Gringarten, A.C.: lnterpretatfon of Tests in f%ured Reservoirs andMultilayered Reservoirs With Double Porosily B%havbr: Theory andPractice, papar SPE 10044 presented at the internationalPetroleum Exhibtion and Technical 6ymposium, Bejing, china,March 1982.

fGkonl, J.: FIux Determination of Finlta-Conductivkv Fracturas UsinaHigher Order Interpolation Functions, paper SPE22658 presanttiat the 66fh Annual Tachnical Cmfarence and Exhibition, Dallas, TX,Oct. 6-9, 1991.

APPENDICESAPPENDfX A Reservoir Praasure DistributionsThe development of the solutions of the pressure transient bahavfor of awell infersectad by asymmetric, rinits-conductivity vertical fracture in ainfinite-acting or finite reservoir is amompfishad by decomposing meproblem into two somper@s; (1 ) the reservoir, fluldflowprob!em, and (2)the fluid flow within the fracture. The resarvoir pressure transient behaviorcreated by the constant rate production vfa a fully penetrating vticalfracture is addressed in Appendix A. The solution of Ure Iranslentbahavior of fluid flowwifhin affnita-conductivity fracfure that indudestiefracture storage effect is considered in Appendix B, and the Wansientsolution of fluid flow in a finite-conductivity fracture L?at neglacfa the fluidaccumulation component is addre&ed in Appandix C. The Wansientsolutions for fhe fracture presented in Appendices Band Cccnsider thefrachire to bearecfangular slab with uniform width (b,), height (h,), andproppant permaabitity (k~).

The continuity conditions that are imposed on Ure &composed reswvoirand fracture fluid flow problems are such that along the fracture face(Y.=Yw), tie resarvoir and fracture solutions and flux distributions aeidentical. These continuity conditions ere expressed in Eqs. A-1 end A-2for the pressure solutions and fluxes, reapecfivefy.

D(xD,%@) =FD,(XD;S) ..........-_._.-~ --:--- (A-1 )

~D(~DsS) = ?D#D,S) . . . .. . . . . . . .. .. . . . . .. . . . . ..J .. . . . . . . . .. . . . . ... . . .. . . .. . .. . (A-2)

The reservoir is assumed to ba a homogenews slab with uniformthidrness (h). Dbactional permeability anisotropy is considered in thereservoir transient solutions that follow, In which he anisotropy effects aredirectty included in the Greens function rather than using a spatialccerdinate kanslation which can skew tie finite reservoir drainage areaahapa. Dual porosi~ reservoir behavtor is considered In tie followingreservoir transient solutions using the classic tima-dependent functions,

In tie following &velopment work, the solution of tie Fredholm integralequtions of the first kind (resulting from the coupled reservoir andfracture flow problems) are evaluated by appffcation of tie first meanVafue theorem. Numerical solution of thie problem in the past hasgenerally been accompffshsd by dkcrefizing tie fractura length in space,writing an aquaffon for each element of the discretized fracture length,and simultaneously evaluating both we flux distribution and tie wellborepressure. Except for very low dimensionless mnductivily fracturae, or atvw @ trmsient times (~ses where fhe coefficient matrix of the Iinaasystem is numerically singular), the wellbore pressures obtained in thfsmanner em reasonably accurate, Howeva+, the flux distribution obfainadwith the Laplace trsnsfonnad rmavoir and fracture solutions [n thismannar is seldom accurate for low-conductivity rlactures when L?edifferences in the dimensionless pressures we computed< as in rJWsolutions of Refs. 11 and 13.

For tie evaluation procedure where the fluxes are computed by directiysoupfing Lhe reservoir and fracture pressure solutions (as in Refs. 12 and15), the fluxas are generally evaluatad accurately enough. However, lhatnumerical evacuation procedure afso has its own set of numericaldifficulties to contend with.

From real space numerlcaf simulation of the resarvoir and fracture ROWproblems it has baen obsarved hat two general ty~s of fracture fluxdistributions may be exhibited in each wing of the vertical fracture. meflux distribution in. each wing of tie ticture may (1) be morvatonicallyincreasing from tie weilbore to the fractura tips, or (2) may Esmonotonically dacreaslng from the walltmre to seine spatfaJ position in thefracture between the welibore and the fracture tip, at which it exhibits aminimum (XO=Q and then monotonically increasing from IAat point to Uwfracture tip. Due to the singular nature of t?ese integral equations, tiefracture tip (XO=XW~+l) and Lhe wallbore (Xo=XwO) solutions should not kaevaluated direcffy.

Instead, the first mea vafue thaorem can be applied in the solution of tieintegral equations. The mean value theorem, in its most general form,Insures us that for a monotonic flux distribution over a @ven element oftie fracture length [a, ~ in whfch both the flux distribution and tie kernelfunction g(x) are integrable, end g(x) is efways non-negafffe or afwaysnon-pdive, then tiers is at least one number (d~(IX,s)), averaga fluxvahie such that

J3 J

.6 PRESSURE TRANSIENT BEHAVIOR OF A FINITE CONDUCTIVITY FRACTURE SPE 28392IN INFINITE-ACTING AND BOUNDED RESERVOIRS

The flux distribution in each wing of the vertical fracture may & -aWermonotonic, or may axhibit singularities at both the wefibore and-fracturetips. This is due to tie nature of the singular integral equations thatdescribe the transient behavior of tie coupled reservoir and fracturesystem. The conditions under whfchthe fli distribution in each wing ofthe fracture is monotenic or not is deta~mihed by tie direct solution of thecouplad reservoir and fracture flow solutions, At this point it is sufficient fedsvelop the appropriate fracture face (Y.=YWO=O) solufions hat are afunction of the type of flux distribution exhibited atagivanvatue of timeand reservoir properties.

For the case where the flux distribution is not monotonic, at some spatialposition in the fracture betwean tie wellbore and fracture tip (O < XO c 1),be flux distribution exhibita a minimum. This spatial position thus dvidasthe fracture wing into two segmenti, with eacfr segment having amonotonic flux distribution.

Application of tie mean value thaorem to each of the segments of Ulefracture wing (O s XD

.SPE 28392 B.D. POE, JR., J.L ELBEL, AND T.A. BLASINGAME 7

Note hat tie first term on the right-hand side of Eq. A-16 is idanffcal tothat given by Eq. A-8. The finite reservoir solutions that follow can thus berepresented by sn infinita-acting reservoir component and a componentthat is due to the bounded nature of the resewoir.

The fracture face resewoir pressure sofufion given by Eq. A-16 isevafuated in a Ike manner as employed for tAe infinite-acting reservoirsolution prewouafy. For the case where tie flux distribution in aach wingof tie fracture is non-monotonic, the relationship of Eq. A-16 may beexpressed as in Eq. A-17.

.,...,,.*-_., ,-_, ,_.. Q&,,~.__,,,-_--&+H.._J,,=e,=r.zr-z->:.->:...._~,,A:~. (A-17)

From the fracture face reservoir pressure solution given in Eq. A-17, wscan readily evafuate the reservoir solution at tie spaffal positionscorresponding to be XO=w,,X..p, and minimum flux position (~).Thase solutions are exprassed in Eqs. A-18, A-19, and A.20,respectively.

................. ........... .. .. .... ... .... . ... .......................... . ... .. (A:ZO)

The corresponding fracture face resa+voir dimensionless vressureSOlti(On, evaluated at ~p for a monotonic flux distribution in e~ch wingof the f7actgre Is given by Eq. A-21. Note hat the minimum Laplaca spaceflux position is at the fracture tip for the monotonic fluxdistributioncme. -.

?D (B. $)~p

r

(1+13) ;~

J

(1-@@& (B,o,s)=

2$- }()* x + ~(u,d ~

................... ................ . . . . . ... . . ..... ... . .. ..T.... .... .. .... ...... . (A-21)

Cyllndrlcal ReservoIr-OlrlchIet Outer Boundary CondItfcmThe reservoir pressure distribution created by the constant rateproduction of a finite-conductivity fracture hat is cemtrally Iooated in a@ndri@ly ~unti reservoir with a constant pra.ssura outer boundary isgivan by Eq. A-22.

r ,, ..- --

. . .. .

...... ..... ..... . ..... . . .. . .. ... .. . . .... .... ......... ... ...... (A-22)

~nce again the solution given in Eq. A-22 consists of a mixed ccerdinatesystem. The relationship between fhe retiangul~ coordinate system ofthe fracture and the cylindrical coordinate system of the drainage area hasbean given by Eq. A-15.

.

The fracture face reservoir pressure solution for the case of a nOn-monotonlc flux distribution in each wing of tie fracture is given by Eq.A-23. The ffux distribution is symmetric about the wellbore and hafracture is cegmfly located in tie reservoir.

Fow.0.)-;j?.(x..5)[K~lx..xol~@]+(lxo+xDl~@]popo

~ ..l~;]+j.[lxo+xolgG)~oc*f,o(xo.s, 1. /,..

311

.. ...... ....... .... . ... . . ... .. .. ...... ......... ... . .. ..... .... .._ (A~23)

.8 PRESSURE TRANSIENT BEHAVIOR OF A FINITE CONDUCTfVITV FRACTUREIN INFINITE-ACTING AND BOUNDED RESERVOIRS

SPE 28392

Employing fhe mean value theorem and the solution pmceckre descfibedpreviously, it can be readily shown hat the XD=CX,XU=p, and minimum fluxspatial position solutions are given by Eqs. A-24, A-25, and A-26,respectively.

........................... ...... ... .. ..... ... .... . ... . .. ....... .... . . .... .. .. ....= (A-24)

........ .. .... ....... . . .. .. ... . ... .. ........ . ....= ... .. ...=.... ......>....- (A-26)

The corresponding fracture face reservoir pressure solutions for amonotenic flux distribution in each wing of the fracture, for U% %=pspatial position, k gven by Eq. A-27. Once agan in this case tieminimum bplace space flux position is at the fracture tip.

Rectangular Reservoir-Neumann Outer Boundary CondlflmrThe reservoir pressure distribution created in a rectangularly bounded,closed reservoir by a finke-ccmducfivify fracture that is produced atconstant rate is given by Eq. A-26.

.......... ................... ... . ... . .. . ... . ..... .... .....................................- (A-28)

For a rectangularly bounded reservoir in which the well is cenkally locatedin tie reservoir and tie fracture has symmetric property distributionsabout tie wellbore, tie flux dk4ribuIJon is also symmetric about thewailbore. Thus Eq. A-28 can be exprassad in a much simpler form for tiefracture face solution, given by Eq. A-29.

.......................... ..... .... . .. .... .... ....... . .. ........ ...... ........ .. ............ (A-29)

For he case of a non-monotonic flux distribution in each wing of tiefracture, the application of tie evaluation procedures described previouslyresults in the following expression for tie fracture face reservoir pressuredistribution.

.

.SPE 28392 B.D. POE, JR., J.L. ELBEL, AND T.A. BLASINC3AME9

""'""""'"""'''" "---..-. L--.--'.. -... ---... -.~J-S:...-.S:---- ....-J.G.=..L.. (A-3o)

The %=X.O+W %=XW+13, snd minimum flux spatial position evaluationsof Eq. A-3o are presentad in Eqs. A-31, A-32, and A-33, respectively.

x+a@=*~+]-@~&h@-Fl

................. ...... ..... ..*.. .-,..................... ...... . .......... ... . ... ..... . (A-32)

... ............ ... . ... . .,. .-. ..... .. .... ..... ........ ... .... (A-53)

-. -.

For tie csse of a monotonic flux dlstribufion in each wing of tie fracture,lhe solufion at &=XW+p is given by Eq. A-34. The minimum Iaplacsspace flux for this case occurs at the frscture tip.

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10 PRESSURE TRANSIENT BEHAVIOR OF A FINITE CONDUCTIVITY FRACTURE SPE 28392IN INFINITE-ACTING AND BOUNDED RESERVOIRS

(5)=%43--(%+)J%H%FPWHl-l

=.:z...==a.-a .. .... (A-34)....... .......... ......... . . .. ....... . ..... ... ... .... .... .... ........

APPENDIX B Fracture Solutlcm Wltfr Fracture StorageThe Ir.amsient solution of the fracture fluid flow problem in which tiefracture storage effect issonsidered is given byEq. B-1. It is assumedthat Me fracture is a rectangular slab of uniform fracture geometry,proppant conductivity, snd is symmetric shout the wellbore. Theseassumptions can be relsxsd 1s without loss of genersfity of L?e solutionprocedure employed in this paper. However, for the sake of sfsrity snd tomore easily demonskate the solution prosedure used, lhe assumption ofuniform and symmetric fracture properties is made.

Noting tie continuity relationship expressed in Eq. A-2, the subscript cmtie fracture fluxes sre dropped in the development work that fellows. Theseine type of evaluation procedure as used for the reservoir pressuredistributions is also sppffed for the fracture dimensionless pressuresolution.

For the case where the flux distribution is not monotonic in each wins ofthe fracture, tie wellbore dimensionless pressure solution gtven in Eq.B-2 csn be expressed more simply in the form given in Eq. B-3 In Eq.@-3it is assumed that the wellbore is the origin of the system.

1[nstanh ~WD1

_@(a,s{&J[J-q]-cosh ~ ~ +1 ............. ...... (B-3)

K

@slm 11insin E, &Irr (r]Cdg:tanh ~?faThe fracture solution for tie spafisl position XC=CL is obfsined from Eq.B-1 In a fike manner and is gtien by Eq. S-4.

............ .................. ............ ....7.. -._fi_fkz-k z.. . . ... . . . . .. . . . .. . . . (B-I)

The fracture dimensionless pressure solution at the wsflkore i: resdiyobtainad from Eq. B-1, and Is given by Eq. B-2.

- -ED(a={=................ .... .. ..... . .. .. ... .. ... ..... . . .....................................

..(B4)

The fracture sofution for Ihe spatia.f position X.=p is presented in Eq. B-5.

314

.SPE 28392 B.D. POE, JR., J.L ELBE~ AND T.A. BLASINGAME 11

lL J

.

r

......................... ............. .... . . . . . .. . .... . . ... . ... . . ... ....._ (B-5)

The fracture pressure solution evaJuated at fhe minimum flux spatialposition is given by Eq. 56

............................... . .......................... . . ......=A...* . . . .. (B-6)

For tie case where tie flux distribution is monotonic in each wing of befracture, the minimum Laplaca space flux spatial position is at the fracturetip and the corresponding pressure at XD=~ Is givan by Eq. B-7.

315

. . .. ..- (B-7)

Thecorresponding wellbore pressure foramonotonic flux dstribution isgiven by Eq. B-6.

r 1

......................... ..=. . . . . ... . . ...... ......................................... (B-6)

APPENDIX C Fracture Solution With Negligible Fracture StorageThe transient solution of the fracturs fluid flow problem in which thefracture.etorage effect is small and can be considered negligible Las alsobeen addressed in fiefs. 11 and 13. For Mecasewhere the fractureha6uniform fracture geometry and proppant conductivity, and is symmetrfcakout the wellbore, the relationship between tie wellbore pressure and thepressure at some position XO in tie fracture is givan by Eq. C-1. Asdemonstrated in Ref. 13, the assumption of uniform fracture propertiescan be readily rdaxed if desired.

,&mhm[(~~:~.)J:moJ:6DfpPW(S)-FD,(XD,S) .~................ .............. ......... .......... .. .... .. ............ . .. . . .. .. . (c-1)

Note that tie itsrated integral appearing in Eq. C-1 can be readyexpressed intermsof simple integrals, asin Eq. C-2.

[y.J:?Df~:s.p:=(x.-xw)j;:,.,(xD,s)wD-~xD@m(xD,s%.

........ ............ .......... .... .... .. .. .. . . ... ...... .. ......... ........ ........... (c-2)

For a finite-conductivity fracture with negligible fracture storage effects, anadditional relationship is also available for tie evaluation of tie fracturepressures. This is the continuity relationship for the production of fluid intotie fracture and tie tmoduction at tie well. For neolidble fluidaccumulation effects in &e fracture, the sum of the fluxes ;;ross. eachelsment of tie fracture length must equaJ the well production rate. Thisrelationship is given by Eq. G3, ~

xwD+l-J 9Df(xD#s)Lw=; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .(C-3)Xwo-iForasymmefric flux distribution eboutlhe well, tie relationship maybeexpressed asin Eq. c-4. Note that fhesubscripts on fhe fluxes have bssndropped in Eq. C-4, duetotie confinuify relationship of Eq. A-2. OnceWaIn the spatial positions dsnotedby sand p correspond to the meanvakre fluxas for the fracture segments % < %s b + ~ md fin + ~s~ s X.. +1, respectively.

t6D(a.s)+(l.t)60(P. s)=+ ........... .................. ....... ......................... . (c-4)

I

I

.12 PRESSURE TRANSIENT BEHAVIOR OF A FINITE CONDUCTIVITY FRACTURE SPE 28392IN lNFINITE-ACTtNG AND BOUNDED RESERVOIRS

Employing the evacuation techniques described earfier in Appendix A, besolution of the pressure differences for the %=XwCJ+a+XS=~~+13, endminimum flux spatial pesifions for the case where tie flux distribution ineach wing oftiefractwa is not monotonic, are given fzy Eqs. G5, G6,and G7, respectively for cases whsre the weilfmre is the origin of tiesystem

F@(s) -FD,(a,s)=~[ .1

~kmb(Dhfa ++{(

..,

SPE 28392 E.D. POE, JR., J.L. ELBEL, AND T.A. BL4SINGAME 13

The transient wellbore prassure behavior of a well intersected by a finite-wnductivity fracture in a closed cyfinc+kally bounded reservoir isevafuated in a hke manner. The wellbore pressure for a monotonic fluxdistribution (~=1 ) is given by Eq. D-5

... .. (D-5)

Foranon-monotonic flux distribution in the fractura (~cl), the wdlbmepressure is given by Eq. D-6 and the mean value flux of the IntewafXWsX,~+~ is given by Eq. D-1 7,

........... .. . .. ... . . . . .. .. ......... .............................. (D-7jThe wailbore transient behavior of a well intersected by a finite-conductivily fracture that is centrally located in a cylinokically boundedreservoir with a constant pressure outer bounda~ can dso be evefuefadin tie mannar previously discussed. The wellbore pressure for amonotonic ffux distribution (g=l) Is given by Eq. D-8.

I FKO(reD$@ (I+L3)~mJ J c(l-P ~mx+ x+ lo(reDJ@ ~L@)* ~ww 1]................................................................................ (D-8)For a non~monotonlc flux ckstribution in the fracture (gel ), the wellboredimensionless pressure is evaluated with Eq, D-9 and the mean value fluxof the interval Xw&&%Dfi is given by Eq. D-1 O.

317

...

14 PRESSURE TRANSIENT BEHAVIOR OF A FINITE CONDUCTIVITY FRACTUREIN INFINITE-ACTING AND BOUNDED RESERVOIRS

-- (D.9j.. . .... .- . ... .. . . ..... -.. -.. ...-. .. . . . ..... .. . . . . . . . . . . . . . . .

pqGJg%- ,().-

. . 1

,,...._.................._-A. .. .s&-+=.~-*-=.:.T= .. ..*_ .. . ..... . (D-10j

For a well intersected by a finite-cenductiti~ fracture in a dosadrectangularly bounded reservoir, Ihe We time (tm20 .01) wellbore kansientbehavior can be evaluated in a similar manner,

The wsflbers dimensionless pressure behavfor for a monoten~ fluxdistribution (g=l ) is given by Eq. D-11.

SPE 28392

........... ...._ ............. .. . . .. .... ..... ... ........... ..... .......... . .._-. - (D.1

..

SPE 28392 B.D. POE, JR., J.L. ELBEL, AND T.A. BiASINGAME 15

........................ .... . ...............=s....... .. .. .. . ..ti.. (D-12)A similar evaluation fachnique has afso been employed for tie finite-conducfivity fracture solutions that Include tie fracture storage effects.Due 10 tie length of he resulting expressions, fhese solutions areprovided in the supplement to this technisal paper.

100

CfD10 O.O1-OiscreteElement

0.1 D,screte0,01

1 ,~

0.1 ,1

10

0.011

0.001 I Ile.om 1s.005 O.OLW1 Owl 0.01 0.1 1 10

tDFrg. 2- Comparison of continuous flux distribution model

with discrete uniform flux element m~el.infinite-acting reservoir.

1,000

/21

0.031 ~ Jle-0061m-orr50.00010.001 0.01 0.1 1 10 lW l,WLI 10,OWtD

Fig.3 - Transient behavior of a finite-conductivi~ fracturein a closed cylindrically bounded reservoir.