Society of PetroleumEngineers

SPE 27907

Evaluation of the Pseudosteady-StateSystems in Various Flow GeometriesM.B. [ssaka and A.K. Ambastha, U. of Alberta

SPE Members

Method for Composite

Copyright ! 994, Society.1 Petroleum Englneers, 1..

TM% paper was prepared for presentation at the western Regicral Meeting held in Long Beach, Celiforn(a, U.S.A., 2*25 March 1894,

This paper was ,detied .t.r presentation by a. SPE Prwram committee following rwew of information contained In % .~r.~ ..bmf!?d b? fh. ..m.KSl. c..f..t. ~ [h. PaP.,,as preyled, have not be.. reviewed by the society of Petroleum Engineers and are s.bjec! 10 correcu.a” by the author(s). The tialerlal, as pmeenl%d, does n.! neesssarily reflectany Pomtion of lhe So.cletY of Pelroleum E.g!naers, [1s ofltcers or members, Papers premnted al SPE meeting. am subject to pu+lice.!!on revreu by Eailoriti Cammit!aes.{ the societyof Petroleum Engineers. Perrniwo” 10 COPYis restricted to a. abstract of not nmrelhan 300 words. !I!.suat(ons 1!M%not be cnpied. The abstract should ..n!ah ccm@c...ackcwled9mententof where and by whom lh. P.PW !. Pmse”ted. Write Ubra,ieJ, SPE, P.O. BOX 833838, Richards.”. TX 75083.3836, USA. Telex 165245 SPEUT.

ABSTRACT

The pseudosteady state method as applied to radial compositesystems, k been used to estimate swept region volumes for thermalrecovery projects with a good degree” of success. However, theincreasing scope and complexity of thermal recovery projects, aswell as, the general heterogeneity of” petrnIeum reservoirs, havenecessitated the consideration of different flow geomenks otherthan radial. Using analytical solutions this study seeks to evaluatethe applicability of, the pseudosteady state melhod m compositesystems in radial, linear, eUiptical and spherical flow geometies.

To compare the solutions for the different flow geometries, lhedimensionless pressure and time values have been normalized toaccount for the difftieit deftitions for dmemionless variables ineach system. Results of the study indicate that, for the samemobility and storativity coiithsti between the inner and outerregions, the dimensionless pressure derivatives for the four flowgeometries can be collapsed into one curve during !he period whenthe various systems approximate pseudosteady state flow. Thisobservation confirms” the pseudosteady state method as beingindependent of the regularly-shaped swept region shape.

Results atso show that the pseudosteady state flow period begins atthe same time for atl four flow geometries. However, the time to theend of pseudosteady state vwies. The linear flow geometry showsthe longest pseudosteady state flow period while the spherical flowgeometry shows the shortest. The radial and elliptical flowgeometries show Lhe.mme duration of pseidosteady state flow.

lle conditions under which a pseudosteady state flow period of areasonable duration will occur for each of the four flow geometriesare presented. Times for the start and end of the pseudosteady stateflow period for each flow geometry,. as functions of mobility andstorativity coittras”m,-tie also””r+scussed. This study will help toidenti~ the conditions under which the ~seudosteady state methodcan b used to estimate the swept volume for !hennal recove~

.— . . .. —---- —---- -- ..— —. . . . .

projects undez various resexvoir situations.

INTRODUCTION

The pseudosteady state method derives from the moblliv tidstorativity contrasts betweti the inner and outer regions of acomposi~ reservoir. The medwd was pmtwsed by Egg.em&@ et~,1 ~ ,.sfiak tie “Olwe of tie inner region of a radbl, cOmpOsite

Ieservoir. Eggemchwiler et d. 1 observed that if the mobility andstorativity conmws are large, the inner region coutd behave as aclosed system for n short period of dme after the end of the fxstsemi-log sinight lime. A Cartesian graph of pressure versus timeduring this period may indicate a straight line whose sIope, W, canb-erelated m the swept (iier region) volume, V~, by

r&=J@ , .. .. . . ..._v~

(1)

To determine the swept volume for themmt recovery projects usingwell test amdysis, reservoirs have been idealized es radial compositesystems.. However, @ inq:@ng scope ‘and complexity of thermalrecovery projects, as well a.%the general heterogeneity of petroleumreservoirs, have necessitated the consideration of different flowgeometries, otk than ractkl.

The pressure tmnsient behavior of composite resemoirs has receivedconsiderable attention in the literature. Most of the studies haveconsidered redal composite resemoirs. fn 1961, Loucks and

Guerrero 2 presented analytical solutions for radial compositereservoirs usirig Lap] ace transformation. carter 3 presentedsolutions for ‘he pressure transient behavior of a closed, radialcomposite Iesewoir with the well ~oducing at a comtant rate. Bixel

and van Poollen 4 considered the effects of radial discontirnuhes m-.

compxwite reservoirs on pressure buildup and drawdown behaviors.They recommended a semi-log type curve matching method to

References and itlnstrationsat end of paper”571

-.

2 Evshmtion of”the Pseudosteady State Methcd for Composite Systems in Various Ffow Geomemies SPE 027907

determine &e di.wsnce to the discontinuity. Eggenschwiler et al.ipresented an amlytic+ solution in bplace space for an infinitelylarge, two-region, radial composite reservoir producing at a

constsnt rate, with wellbore storage and skin ArnLM.wha5 prssentedpressure derivative responses “for tiim: “and three-region, radialmm~site reservoirs with wellbore storage snd skin, as well as, athin &in at the discontinui~. The outer bmmdsry of rhe reservoirwas considered m be either infinite, closed or at a constam pressure.

Noting that composite reservoirs are not necesmrily all radiaL Bixel

et af.d -ted solutions for the buildup and drawdown behavior ofcomposite resetioirs with linear discontin~ty. Ambas.fha sndSage ev7 presented analytical solutions for linear, compositesystems including the effects of a thin skin at the d~ondnuity thatmay be caused by a psrfially-communicsdng fault sepsradng the tworegions.

O bur and f?rrekin 8 snd .Stanishzv et al,9 presented analyticalsolutions for the transient presmre responses of an iti~nite-conductivity vertical tlacture in an elliptical, compxite reservoti.Recemty, StanisIav et al. 10 have. presented presmre derivativeresponses for composite sys~ems in elliptical flow geometry,including the effect-s of wellbore storage_Wd :%. a?.weIl,.W, .a ~skin at the discondnuitj.

Onyckonwu and Home 11 have studied the pressure transientbehavior of composite reservoirs in spherical flow geometry, Theoute~ bounday wss assumed to,be !@nite m :ex[ent, No .welfboresforageor skin effects were “con.iidered.

The preceding ~is.+ssion on cornposite resgvoirs shows thatsignificant studies have ken conducted on each of the variom flowgeomemies.. However, to the best of our knowledge, no .mempt hasbeen made to compare the pmsnne tmmsiem behavim of cmnpasitesystems in tie various flow geometries. Using ansfyficd solutions,this study seeks to compsre the transient pressure behavior ofcomposite reservoirs “i radiat, etlipticat, linear and spherical flowgeometries, and, in psrdmdar, to establiih fhe conditions imderwhich pssudosteady state flow witl occur for each system.

THEORY

In this section, we present the analytical solutions for esck systemin the pmpmsed study. The following s.isumprions hold true for themmpsite systems in all four now geomemiex

1. The fornnilion consists of two discontinuousregions, with homogeneous and isotropicproperties on each side of the discontinuity.

2. “‘The front is of infinitesimal thickess, ~d isconsidered stationary !hroughout the test period.

3. Lsminar flow of a single phase fluid with slight, butconstant, compressibithy occurs in each region.

4. Gravity and capilkity effi-sts are negligible,

Radial Composite System

fn this study, the Ambastha 5 “solut@t k-used as the model for theradiaI composite system, witi some modifications. .Wellbrxestorage and skin at the active welf, in addition to the skin at thediscontinuity are neglscfe~ since the effects .of these parameters onsfl four flow geombtdes” should be the ssrne. A constant flow rate atfhe well is assumed. i%gure 1 shows a schemafic of a two-region,radial composite reservoir.

The diffusivity equations, in dimensionless form, governing fluidflow in a two-region. radial composite reservou .me given by.

(2)

(3)

Afl vsriables used in Eqs. (2) and (3), and subsequent equations are&fmed in the Nomenclature.

Initial cottditIons:

fn Cfimensionk.ssform, the inidat conditions for the two regions are

~(mo). o. (5)

Inner boundary condition:

Since flow is lamhw snd the rate is constant at the inner boundary(well), Darcy’s law is applicable. In dimensionless. form, thiscondition ix

()?EQLb) D=l=-l

(6)

Also, since wellbore storage snd skin effects are neglected,dimensionless wellbore pressure ix

Pwfl = PDl for~.1 (7)

Conditions at the discontinuity:

At rhe discontinuity (RD), it is required thst pressures. as well as,flow rates in regions 1 and 2 be equal. These conditioti arc

PfJl=nx for rD=RD, and (g)

(9)

Outer boundary condition:

The reservoir is considered m be of infiite extemc. llms, the outertmmdary condition is

~(lD+CU, tD)= f). (lo)

A“ solution of Eqs. (2) and (3), subject to the initial snd bundsryconditiom (Eqs. 4 through 10), is csrried out in Laplace space.

Following the approach of Ambastha 5, and dropping out the

SPE 027907 : “’ “’“-”M. B. ISSlkt and A. K. Ambastha 3

wellbore storage and skin terms. the dimensionless wellborepressure in Laplace space is givti h“ tenis @f Bessel functions as

fib(z) =clfoti)”+cz”G(YT) ““...(n)

The constants, Cl and CZ are obtained by solving tie system ofequstions resulting from the use of the bounds.ry conditions.fnversion of Eq. (11 j iita real space is carried out numerically usingthe .SIehfest 12 algorithm.

Elliptical Co”mpbsite System

The model used for tie elliptical composite sy?~m is sfil~ @ hat

of .%mi&w et af. 10, witi wellbore storage and skin effects beingneglected. The reservou consists of two distinct elliptical regions,as shown in Fig. 2. An elliptical swept region is assumed to occur inthe presence of an infinite-conductivity vertical fracture. Thevertically-fractured welf is located at the center of the kmer region,and fully penetrates tie formation.

In dimensionless form, the diffusivity equaiioiis in ‘ellipticalcoordinates for Regioms 1 and 2 arc

and

The initial snd boundary conditions ire simiiar to those for the radialcomposite system,

A solution of Eqs. (i2) and (13) is afso carried out in Laplace spaceusing a separation” of”” variables techniquel 3, with subsequentnum~ic=l i*versioh,12 The diiemimdess weltbore pressure blLaplsce space is gR@i “-mtirms ofhfatieu functions s-x

FtiDe (&,@’ ~~o c~*(~. ‘~) [ C2nCYn(&, ‘a)

+ F2nFekti(6., -cO 1 (14)

The constam CZR and F2n are.Fourier coefficients that are obminedby solving the system. of equations” resulting frOm tie USe.Ofboundary conditions.g

Linear ‘Composite System

The linear composite system solution employed in this study ismodelled after the Ambastha and Sageev 7 solution. The effect ofskin at the discondn’iity is; ~however, neglected. The linearcomposite system is rectangular, and considered to be semi-inFhite,ss shown in FIX. 3.

The dimensionless diffusivity equatioti for tie pressure behavior inRegions 1 and 2 are:

‘=* ~ for l) SXD<aD; and “(15)@D hl

~“=o~ .: ““” ~foraD<XD <M. (16)

Transforming Eqs. (15) and (16) into Lapatce space, using Ihe iihialconditions, leads to a pair of ordinary differential equations, whichcan be solved easily. For the linear system. the dimensionlesswelfimre pressure in Laplace space is given as:

(17)

The constant CI is found by using the bmmdsry conditions.?

Spherical Composite System

ne Onye~nWU and Homl 1 solution for the pressure -viol Of arsss$voir with spherically discontinuous properties is used for thespherical composite- system in this study. Figure 4 shows aschematic of . sphericat composite system. The reservoir modelconsists of two concentric spheres, representing the inner and outerregions of the composite system, with the wellbore at the center.The outsr region is considered to be infiite in extent

The dimensionless diffusivi~ equations for the two regions of !hespherical composite system are:

(19)

A solution for the dimensionless wellbore pressure in Laplace spaceis given ax

%Ds(l) = Alti. +A2d. .. ‘.fi ‘/7

(20)

where the constants Al snd A2 are determined from using the

boun.aary Condltiom.11

COMPARISON OF SOLUTIONS

A computer pro8mm hai been written to generate the wetttmrepressure and pressure derivative r=pmzses in Laplace space for thefour flow geome!xies described previously. Subsequently, theLaplace space wlutions have been inverted into real space using tie

Sfekfest12 algorithm. A comparison of the solutions is undertakenby examining the bshavior of” the dimensionless pressure derivativeresponses. Both the C=t&.iin and the semi-log pressure derivativesare considered.

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4 Evaluation of the Pseudosteady Slste Method for Composite Systems in.Vafious Flow Geomehies SPE-027907

To enable a comparison of the solutions, the dimensionless pressureand tie .aluw have to b. nonnafized .R_account Em the differen~definitions for–some of the .dimensionfess variables in each system.‘fboughno tnecessary,the radial system snlution has been chosen asthere ference for the comparison. fn addition, the tkne values foreach of tie four sohnions have beerno~dizedby ~e area of theinnez region of each composite system. Thus, in each system, the

5’10. Theresults are not affected by the size of the iwer region. .normalizing fsctors for each system are presented in the following.

Radial System

Since the radial system is the reference, the normalizeddimensionless wellbore pressure is equal to the dimensionlesswellbore pressureof rhemdisl system

%DN = Pwm. ‘“’” (21)

The normafizd time coadinate for the radiaf system ix

kl %t _ ~ :, .=:.4 .,.. _.. .... ....tDN =($YCJII? nR2 - tiR~

(22)

E1llpt~cai system

Since the dimensionless .yellbore pressure for the radial andelliptical systems me sqwif, by deftitiom

pwDN=pwW. : ::(23)

The elliptic p.mmieter, ~o, that defines the.size of the inner regionfor the elliptical system is related to the rdlus of the radid systemby ,10

R.$&. ~ ~(24)

Thus, the normalized time coordinate of the elliptical systembecomes: ‘

QN=~~.%@. “’” . . ...(25)

Linear system

Con.siderimgthe definitions of dirnetiionfess. weffbore pressure forthe radid and linear systems, the normalized dimensionless wellborepressure for the linem system becormm

(26>

The normalizing factor f{ the“’tie coordinate oj Ihe lini~ system isdeterminedby equa@ the F.% of .tie,inner region for the r%tiaf ~dlinear systems.

fi2=ab. “C27)

The normsked time coixdiiate then Lxwmex

(28)

Spherical system

Comparing the deftitions of dimensionless wellbore pressme for“the radial snd spherical systems, the nornmlized dimensionlesswelfbore pressure for the sphericsl system becomes

PWDN = !Q2PWDS. (29)

Letting a = R, snd then equating the inmex Iegion volumes of tieradial and spherical systems, we get

b=:$Q (30)

Substituting for hD in Eq. (29), the nommfized dimensionlesswellbore pressure for the spherical system becomes:

fiDN =:$9 %fh.

The normalized time coordinate for the spherical system is:

(31)

(32)

DISCUSS1ON OF RESULTS

As d=wsed by Ambastha 5, in the absence of wellbore storage andskin, the parameters that affect the pressure derivative rqmnse of aninfiitely large, radial composite system are the mobili~ ratio, Mthe diffusivity ratio, to; and the dimensionless distance to fiediscontinuity, RD... .~e norgmlim.tion exercise carried out in the.fieceding=section eliminates the dependence on RD. The remainimg

pammetczs are M snd te. Since, by deftition, M is includ.xt in (e, astomtivity ratio, F, is defined that expresses onfy the conu.asc inporosity-compressibility between the tiff and outer regions. Thestorativky ratio is given by

F=M.~.(0+ “ (33)

Ambastha 5 presented a sensitivity study for the effect of M and F onthe dbneniionless semi-log pressure derivative respomi for radial,composite systems. To compare the pressure derivative resFcmsesfor the four flow geometries in W“ stidy, ibe vahes of M and F havebeen fixed. Figure 5 shows a log-log graph of normalizeddimensionless semi-log pressure derivative versus norti~zed timefor” the four systems. The puameters M snd F are 100 and 1000,respectively. The graph shows three clearly defied flow regimes foreach system. These flow regim.e$ describe the flow behavior, due mthe inner region, the discontinuity, and the outer region of theWrious composite systems. The transition between the inner andouter region flow behavior is chmacterized. .by unit slope lines ofvqtig Imgti for @ fow flow geometries. A unit slope line, aftezthe inner region flow behavior, on a semi-log pressore derivativegraph indicates pseudosteady state behavior. Figure 5 shows that,for the same mobility and storativity rstim, the dimensionless semi-

574

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SPE 027907—

.” “M. B. Issti . .. . and A.. .K.. A.mbastha

log pressure derivatives for the four flow geometries are identicalduring ps.udosteady state flow for iach system. This observationconf%ms the pseudosteady state method as being independent of tieshap+ of the regularly-shaped swept (inner) regions.

Figute 5 atso shows that piadosteady state flow starts at the sametime “for all flow systems. H~wever, the time to the end of thepsetidosteady state varies for different geodetical shapes. Thetinear flow system shows the longest pseudosteady state perio~w.biIe the sphticd flow syst~ shows the shortest. The radisi andelliptical flow systems show the ssme duration of psiudosteady stateflow. Except for the inner region flow behavior, othm pressurederivative characteristics of the rachl and elliptical systems seemidentical.

The responses from” the four flow geometries are also compared inFig. 6, which shows a graph of normalized Cartesian pressurederivative versus iti”atized time for”the same parameters as in Fig.5. Once again; three clearly defined flow r~gimes are shown fm eachflow geometry. In Fig. 6, the uansitionregion between the innerand outer flow regimes is chsmctcrized by horizontal limes (zero-slope) of varying lerigths. A constant Cartesian pressure derivative,or a horizontal line on a log-log graph of Cartesian pressurederivative versus time, is indicative of pseudosteady state flow.When the responses are normalized by the area of the inner region,as in rhis study, !hen the constant Cariiiisn pressure derivativeduring pseudosteady state flow should be %. Figure 6 shows that thelinear flow system “exhibits the longest pseudokteady state flowperiod, while the spherical flow system gives the shortest. Theradial and elliptical flow systems show the same duration ofpseudosteady state flow period.

A detailed comps.risti of the responses from the radd and ellipticalflow systems is shown h .Fig.7. The figure shows a log-log graphof normalized dimensionless semi-log pressure derivative versusnormalized time for” the two systems, with mobility and stormivityratios as cross pmrneters. Mobility ratio is vari”d from 1 to 1W“,while the storativitj ratio ranges from 1 to 1000. The effecrs ofmobility and storativity ratios on the semi-log pressure derivativeresponses for the radial and elliptical composite systems have beendiscussed separately in Refs. 5 and 10, respectively. Here, we seekto compare the respmses for the two systems. Figure 7 coiu%ms theobservation made in Figs. 5 and 6 that”the responses for the radialand elliptical systems sre identical, except for the inner regioneffects. Thus, for sfl practical purp.wes, thermal welf test data forfractured wells can be analyzed using a radial composite modelsolution witftii “tie context of automated (or automatic) type-curvematching, as long as esrly-dme linear flow data. if observed. due tothe presence of the fracme sre excluded from the analysis.

TfME CRITERIA FOR PSEUDO.STEADY STATE FLOW

TO be able to use the pseudosteady state method to determine theswept volume for thqmsl recoveg projtits, there should be a meansof clmasing the correct pseudosteady state Caztesian line. For a willlocated in the center of a closed drainage region of any regula shapepwudosteady state behavior maybe ob~.gy~ when tDA >0.1. where

tDA is based on the drainage mea. ‘4 For a composite systeiI, then,pseudosteady state behavior should also be observed for tDA >0.1(tDA based on the area of the inner region), if the mobility and

. . . . .storativity ratios tie large enough for the inner region toapproximate a clo:ed system. Figures 5 snd 6 show thatpseudosteady stste behavior ii ihdeed”observed for. tDN >0.1 for allfour flow geometi:ii; ”: Recall that the normalized dimensionlesstime, tD N, is based on the area of .Jtte inner region for each

5

mmposite system geometry. Simcc the duration of the pseudosteadystate period vaies for the different flow ge=ameuies,correlations aresought for the time to the end of pseudosteady state flow.

Ambastha5 hss presented correlations for the time to the end ofpseudosteady state flow, based on mobility and storativi~ ratios, forrsdisl com~site system.. To establish the duration of pseudosteadystate flow, Ref. 5 uses the criteria that the Csrtesian pressurederivsdve be within 2% or 5% of 2x. For tie 2“% criterion, Ref. 5notes that pseudosteady state flow is likely to appear for csses withmobilhy-storativity product, MF 2 Id snd M 210, if pseudosteady

state flow is required to last up to tDA = 0.2. If the 5% criterion is

used, then pseudostesdy stste behsvior is likely for ~es with MF 2103 ~d M > IQ for “pse;dosteady state kastingdtl tDA “= 0.2.

Noting from the previous discussion that the pseudosteady state flowbehavior of the elliptical system is identical to the radial system, forthe elliptical system, pseudosteady state flow lasting up to tDN =0.2 occurs fti the same condhions as the radiaf system.

por the linear system, normalized dimensiordess time, tDN, for theend of pseudostesdy stste @od was calculated based on deviationsof 2% and 5’% from 2K. Dam for selected cases of mobility sndstorativity ratios are presented in Table 1. Correlations for the timeto the end of pseudosteady state flow based on data from Table 1 meshown in Fig. 8. For the criterion of 2%” deviation froni 2x, a“pseudoskady state period lasting till tDN = 02 is likely for cases ofMF 2103 and M 21O. If the 5% criterion is reed, then a

pseatdostsady state period lssdng up to tDN = 0.2 is likely for cases

of MF21@md M210.

For the spherical comprzsite system, Table 2 presents selected dataused 10 develop the correlations for the tie end to pseudosteady statebehavior. The correlations are shown in Fig. 9, for stmativily ratiosof 100 and 1000. Pseudosteady stste behavior was not observed forthe storativily ratio Of-10. Figure 9 shows that for tie criterion of2% deviation front “the Cartesian pressure derivative of 2x, apseudosteady state pmiod IsSting up to tDN = 0.2 is Iiiely for cases

of MF 2 l@, M 2100 and F 2100. If the 570 deviation ti”terion isused, then the required duration of pseudostesdy state is likely for

csse.sof MF21@, M2100and F2100.

The conditions for the time to the end of pseudosteady state flow forthe vsrious flow geometries are compared in Table 3, forpseudosteady state flow lasting till tDN = 0.2. This duration ofpseudosteady state behavior is considered reasonably long for aproper snslysis of the pressure data. For typical reservoir and fluidparameters of a steam hjection process (k=200 md, P=O.01 CP,

ct=o.L34 psi-1, 0 =0.2 discondhuity radius =102 ft), tDN = 0.2converts to a pseudosteady state period of aLxw 5 hours. Table 3.shows that the conditions for the occurrence of pseudostesdy statediffer for the various flow geomekies. The spherical flow geomeuyrequires the highest mobility and stmativity contrasts, while thelinear flow system requires the least.

CONCLUSIONS

1. Norr&lizing factors .tiat” msble comptison of solutions “for”compite systems in radisl, elliptical, linear and sphericalflow geomeuies have been presented.

575

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2.

3.

4.

5.

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Evsfuation of the Pssudosteady State Method for Composite Systems in Various Flow Geometries SPE 027!?07

Time ‘iiteria for the start and end of pseudosteady sWe flowpericd for each flow geometry, as fimctions of mobility and.storatw,ty ratios. have been developed m ensble the correctchoice of pseudosteady state Csrtesisn straight line.

For the sane mobility and stomtivity ratios, the linear flowgeomeky results in tie longest pseudosteady stste period, thespherical flow system yid~i the shortest pseudosteady statepdod. The fidisl and elliptical flow geometries show thesame duration of pseudosteady state.

The pressure derivative responses for composite systems inmdial snd elliptical flow geometries me identicaf, except forthe early-time inner mgxoti behavior.

?he conditions under which a psendosteady state period of areasonable duration occurs for each of the flow geometrieshave been established. These criteria will be of helu indetermining when the pseudosteady state method wifi beappropriate for the estimation of swept volumes for thermalrecovery projects under various msavoir situations.

ACKNOWLEDGEMENTS

.Coinpudng facilities for this whrk were provided by the Depamneruof, Mining, Met@rgic~ and. Petrpleurn Engineering at theUniversity of’ .A1berta. Resesrch Assistantship for the first authorwas provided tbro:~ a.r-$h ~ntmct born the AlbeIta Ofl .%andsT.%bnology snd Research Au~orily (~OS~).

NOMENCLATURE

,=

,-

.

.

.

distance to discontinuity fm linear andspherical flow geomeuies, mdimensionless distance to discondnui~” (radialand sphericat). a/rw

‘srear of swept (iier) region, rnzwidh of tines msewoiq mdimensionless width of linear reservoir= b/1modified Mathieu function of fmt kind ofintegerorder

totaf compressibility, Pa-1real, even, pericdc Mdhieu function ofintegerordercommit of the system of equadons for linear flowFourier caeficients of the system ofequsdom for eflipfical flowStOrativiV iatio = (@~ I/(.$cJ2modified Mathieu function of second kind ofintegerorderFourier coefficients of Ihe system of equatiomfor elliptical flowreservoir fhickness, mmodified Bessel function of frit tid of ord~zeromodified Bessel function of second kind

w

m

PiPv/e

AdPwTL%

Pwfk

PWD1

PWDN

Pun

PWD6

FWD

. dimensionkss pressure drop in innei

region = 2tilh@i - pl)/~1 PI: dimmsiOnleW pressure drop in outer

region = kk2h(pi - @/qy2~2= initial reservoir pressme, “Pa= well!mre pressure for dliitic-dl flow geonwmy, ‘

—

—

Pawellborepressure for tinear flow g-eomemy, PayeU@e pressure for radial flow gwmehy, Pawellbore pressure for spherical flow geometry,Padimensionless welfbore pressure (elliptical

ftOW)= 2xk1h(pi - pw.)/~@l “dimensimdess weflbore pressure (linear flow)=

klbh(~l Fwl)/Wl~lldimensionless nonnsfized welftore uresmre&op for any flow gwmeixy (see E&. 21.23,26and31)d@mi0ide5s wellbore pr.es.sure(mdiaf flow)=

2xklh@ - ~)/wI~Idimensionless welfbore pressure (spherical

flOW)=4Zkl,w(pi-pwJ/tW@l -

dimensionless wellbore pressure drop inLaPlace sp.,injtitiori or production rates, Sm3/srsdius or rsdird distsnce for ridial snd sphericalflow geometries, m ‘ ~-dimensionless radius m radisf distance (rtiiatand spherical) = rhwdistance r. discondnuiij for mdiat flowge.mnmy, mdimensio”idess distance tb discondnuiw fOr

radisl flow georne~. R/rwtime.s~easionless time based on area = k~t/

(@%)IAdimemiodess time for elliptical flow= k~t /

(@CC), L2

m . dimikonless time for linear flow.. k, t J

DN =

m=

Ds=

v, =:x

“xD =

of order zero Greek SymbolsPenneabilily, m2time vsriable in Laplsce spacefracture hti-length, m P=

mobility ratio = ~~)1/(Wy)2 ?- —

ps.sure, Pa P=

. .dimensionless nortmdized time for sny flawgeometry (see F@. 22.25, 2S and 32)dmensiordess time for rsdial flow = k~t /

.@Pcd]r~2dimen.hbs time foi spherical flow . k~t /

(Wt)trw2Wepl volume, m3distance in tinear flow geometry, mdnensionks d=tsnce in linem fhwgenmmy, x/1

formation volume factor, m31Sm3

space coordinate inelliptical geanetry

viscosity, Pa-s

k=

576

space cmxdinatc in dliptical gemnefzy

SPE 027907

<w

50

;

Subscripts

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REFERENCES

1.

2..

3.

4.

5.

6..

7.

8.

9.

10.

M. B. Issaka and A. K. Ambastha 7

. cawrdinmeof tier bounday= , ~dinat. of Kicon.tinviy”or .fwt. !OCSti.~T

= diff%sivitj ratio= (k/$@I / (k/@pcJ2= porosity, fmction ~ ““

. dimensionless

. elliptical flow gmmeDy= flowing or fraimre= initial- limpr flow geomeby—. radial flow geomeay- spherical flow geome~—= total. wateror wellbore. int.irfaceor flc-ad front= region 1 (inner)=.. region 2 (outer)”

=.. pridic

Eggenschwiler, M., Satman, A., and Ramey, H. J.Jr.:mIIIteqXemtimof JnJ~tion Well_R~j_we Transi~t Data .jnThermal Oil RecovefY,” paper SPE 8908 presefifed at theCalifornia Regional Meeting of SPE, f-m Angeles, CA (April9-11, 1980).

bucks, T. L. and Guerreio, E. T.:” Pressure Drop in aComposite Reservoir.” .SOC.PeJ. Eng. J. (Sept- 1961) 170-76.

Carter, R. D.:” P&sure” Behavior of a Limited CuculerComposite R-ok.” SOC.Pet. E.g. J. (Dec. 1966) 328-34.

Bixel, H. C. and van Poollen. H. K.: “pressure Drawdown andBuiJdup in the Presence of RadM Dk-andnuities.” SOC. Pet.Eng. J. (Sept. 1967) 301-09.

AmbasOia, A. K.:’’Pressure Transi.+u Analysis for CompositeReservoirs.” Ph.D. “Thesis, Stanford Universiw, Stanford, CA(Oct. 1988) 193 PP.

Bixel, H. C., L#rkin. B. K, and van PwUen, H. K.: ‘Effect ofLineaI D~continuities on pressure Buildup and DrawdownBehavior,” J. Pet. Tech. (Aug. 1963) 885-95.

Ambastha. A. K. and Sageev, A.: “Linear Wata Jntlux of anInfinite Aquifer Through a Patinlly C&iummicatin8 Fault,”Prcce?dings of the 12th Geo@nml Res. Engg. Workshop,Sanford, CA (Jan. 20-22, 198.7).

Obuf, S. T. ad Ertekin, T.:”A Composite System Solution inElliptical Flow Genmitfy,” SPEF~ (SepL” 1987) 227-238.

Sianidav, J. F.. b.SW~llL C. V. and Kokal. S. L.:” AlldYdCdSolution for Verticaf Fractures in . “Composite System,” J.Cdn. Pet. Tech. (Sept.-Dolt: 1987) 51-56J . .

Stanislav, J. F., Easwaran, C. V. and Kokal, S. L.:” EllipticalFIOW in Conmesite Reservoirs.’” J. Can. Pet. Tech. (Dec.1992) 47-50. “

11.

12.

1.3.

14.

*

577

Onyekonwu, M. O. and Home, R. N.:” Pressure Respnse of aReservoir With Spherically Dk.mndnuous Properties,” J. Pet.Teck. (Nov. 1983) 2127-34.

Stehfest, H.:” Numerical Inversion of Laplace Trtisfonn:Algorithm 368-(d5); Conwnunication of ACM (Jan. 1970) 47-49.

McLachlan. N. W.: Theory and Application of MathieuFunctions, Oxford University Press, London (1947).

E8rlou8her, R. C.. Jr.: Advances in WeU Test Analysis,Mono~aph Volume 5 of the Henry L Doherty Series, Societyof Pemoleum Enginwm of AIME, New York (1977).

T.IA. 1 ‘1’hm ,. (he end .{ Dssude+lad” <.(? MXY.W —*”. totk k,Mr rtgkl or. Ii.., .-ac -.

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0.112 0.1510.143 02740.164 0.3s80,208 0.48,03& pg

1.1!?0 3.0401.s41 4.316

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8 Evalum.ion of the Pscudostcady State Mctimd for Composite Systems in Various Flow Geometries SPE 027907

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Fig. 2-

_“””””””-&i““h@ Zl@. M>lCO. F>103 IAT>l.?, M>1D3, F>1M

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SPE 027907 ~ ““” ‘1 ‘“B. kS& and A. K Ambastha 9

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Mobiliw Ratio

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104

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579