SPESPE 12967

The Interpretation of Interference Tests in a ReservoirWith Sealing and Partially Communicating Faultsby G. Stewart, ● Heriot-Watt U.; A. Gupta, Chaflerhouse Petroleum; and P. Westaway,”Schlumberger Technical Services

● SPE Members

This papw was preaentad at the 1984 European Petroleum Conference held m London England, OctoLwr 25-28, 19S4 The MSt9fial IS subJacf to oorrac-tmn by the author Permission to copy IS restricted to an abstract of not more than SDO words Write SPE,S200North Central Expressway, P O Box64706, 0s11ss, Texas, 75206, USA Telex 7S0989 SPEDAL

ABSIWXT vith the Introduction of hi@) resolutionpreeeure gaugee that suoh teohniquettahave beooae

The interference test hae been extensively Ueed feaetble.in Worth Sea reeervoire to aeeese ~ioationbetiaen uelle anU In thie peper the ●ffect of The qwetion of wntimzity between wells .ieveryfault planee on interference teet meponee ie emlohaaeoclated with theana.lyeed.

Ooourrenoe of teotonicIn the firet part ●impl.esgetene of ~nt end faulting. In m ci~tancem

Interjecting an8 parallel eeallng faults are the two eectione of a peraeeble fonaetlon my beanalyno Uelng the *?soa of IRagee. The Uiepl.aoadtoeuohaneatmtt hat both faoewarwntchlng of Observed and model reepmwe to gilm aslj-nt to lmpemeable - ae ehmm In Flgumthe ~ir m=-terekhf~anamtie- la. In thi* ease the fault plane oonetitutee aoarried out using a -hod baeed on *lioit vertioal pemeeblllty barrier, 1.0. the claeeicfunction theory Origlnauy propwed m ~ M.neer Uieoontinuity. lbet IOorthW reeemire

- -. Xntheeeoond ~-ef-ofa are 41- Into Uietinot , Ieolatti fault b-pRrtWIY eeallrq fauXk bebmen the active lall by thle prooea. The firet evMenoe for theana the dmervatlcsms well ie enalyeed using a exietenoe of fault planee %* of ~ Obtaimdtnm-uilaenaionel •lw~ ~cal fma geologloal ml et-me -* *I-C.Sf.aulator. Aaetoflog—log typesxsrvee e- dlmeter aml log 8ata. mamserltmuetbe~eaafunction of~forvarioue valuee ofa etated that there is -1A•• Uncertainty in the

-It t~-~~ltY ~ - *-. faint m mrated fmm thle Infomtlon.mny teotonlc ~ntedonotmeul tin a00mplete throw @ often Pemeame be-, Pertsam

w~* tO ~iffe- f=~-$ -Y - at*W~of=~ —-=oIr heat partially aagacmtt aox9Be the fault plane~ion for Ehe ●imlation of on moovery Uehmmlnrigumlb. mepromemthenarieeee@sa9ee baeeaon?n.8p~ pmoeaeeauohae a8tothe degrwof~oatlon ~ thewater floomng orgaein~eotion rn-- fa!nt plane; thi8 8ituation-11 b raferrd to~-. In this oontext the ●ffect of U Um pertlany ~ioating fault.pe~llty layering on water or ge8breekthrou@l a the infhenoe of horlaontal m Is very Ulffioult to aeae8e - eloglcal

~-~m~ on gravity data Mtether or not a fault is eealing or~ion m -U ~. - aeterairmtion non-seaM.ng. mexwl.e alwaya&ubtaeto theof the ●ffeotlm Wertioal ~xxty of a eatent of the dim ~tarMeven-~he, reoentlybeemetmaled~~ ale perueable beue 9ay be tiyaoent tlm ~tcvQ&(*) * ~lon with Single -u -Z’tioal

P@=-%wtilw ~(z) ~~* ~prooeeeuyhave~ the xuck in theVtoinity of the fault plane Zw9aering lt

in—ferenoe teat with QSatrfh8ted preeeum qxueabla. medegneof ~icatlut can~attm Obeervatias -11. m only be m~ byanalmie of thepneeum~ of Oontimrity~ -Ile Ie aMo kSmwimm of the reeervoir. me -m*Oeemtlal for- proper aeaign ofu~ enmneeriw etudy of the 9hiatle fi-~ published

—to-ueIl ;nter~~ - -~~ teatiag by maalid~) In Hn ~ hat praviouelyIsmchuaesl inthisraepeot. Xn hi@s ~ eealing faulte H to be poetulatd~l~e ~r8 with large -n apecing *~-m—=— depletlon of the-f=——= affeotatiich—m~ at ~rwir e8 ~f====-Y=-~“ =* =:la arr q mz ens at &e O@y aeva~ -he In Mm ~ f-~~. In

tiia~the~ pra89um deoline -Idonxy be wxplainedon the be8is of -ler fault

References and Illustrations at end of paper

411

THE INTERPRETATION OF INTERFERENCE TESTS IN A RESERVOIR WITH

2 SEALING AND PARTIALLY CONNUNICATING FAULTS SPE 12967

blocks than premnaad in -e original geological-1. conversely in other reservoirs theabsence of anticipated areal pressure variationshas lndioated that faults thought to be sealingwere allowing ~oatbn . The determinationof effeotIve vertical pemaabiltty by mataingthe vertical meeeure profile (predioted by theffinvo:r =Xmlaksr ? ak em obeematian USU to a

meamredw’r survey IS w -11 Setab2-.lim?wer, theprewnce orabeenoeof araal

preseeure variations as SVMSWU2 byl?F7 surveysin the develqaent USH.S is an indi~tor ofhorisonta2 ~ication ~u fault @.anes .

me process of history matching thethree-d~lonal reServoir relator theraferegives ~ help in dsflniw the la-al~ioation across the reeervotr. EUm!ver theareal ~ Variations luhialdevelop duringthe natural reservoir depletion are not largeUlnees ~oation across fault planes is VezyCU indeed. mm~%n~i=deve~t -Us are Integrated thrO@lsimlator history matching absolute pmeeure

~~wtitsi- Oschmuveyisuade with a different gauge. mis MRItS the~ withWiich~ ~ variationOanbe~ed. Aleoeincemw surveys can only beasde in new deve~t -Us ~reinfomtion is only avai2.atJleon a M9itsd basisat specific Iooationsand specif%c tiaes. EanceUiere is a need to dSViaS ~.E2c k-s *&OhCanhusdto assess Continuitybetlaen -118.Xt is Eor tms ~ that transientin~ tasting has been developed. In aclassic well-to-well interference test theElou-rate atantiiveueii ise+hagadaia%~%sprenure reepmeeataneerby eWtinobeervat.imWell is ~ with a sensitive prweure

~.

&lalYs%s of Xnterferenoe !Msts in FaultedWServoirs

there

h cmstant rate inter femnos teetcanbSnalysedbytyps curve matohingonabg+ogplot of ~ versus % -=-w-w * --~(1). Thepmesure ntoh yieldsan estimateof* ~ ~-on ~llity, k, ~ t?melliptical region of immmnoe defined byvel.a(~) while the time uatoh can be used to

dsteraim the ave~ porosity, +.

4

rhe most attractive fozm of interference test isthe pulse test in which the active -11 isUternatively flowed at oonstant rate and~utin. This technique was first prqxxled byJohnson et al(s) and * analyeis of puma testsLn inflnlte-aotlng reservoirs was considered byKamal and Brigham(s) who showed how reservoirparameters, k and @C+, Oould be determined f-the amplitude and .k- %*- Gf the preeeu??puMes at the obeemtion well.

me of the principal advantages of puma testing1s that for short enough periods the system canne -n as Infinite-aot%ng. Rouever In himpermabillty, faulted reservoirs and for pulse2uration ttaes which are practical, say of themder of one hour or more. tnftiite-act%ngbehavlour Oenlmt be ~. For essmple If the2epth of investigation at the end of the frostpulse is greater then the distance to themamst sealing fault the effeot of the no-flow

mtet be mc7x@sd. Benoe it isto ems>.+= +H!e ef-aot of Sealirbg

barriers on the pressure rwponee at the*mtion well. mrefa siaple fault patternsin a sai-infinite xwservoir are easily handledby the ~of~as~~ by-~d7~P these Srel

rhe presmue drmp at the obsemtlon -11 is~iven by a ~tion of the fbral

Equation (2) refers to the Oonstant rateSraudom situatton. Intheoase -re themctive well flow echadule takes the Eo= of amries of oowtant rate periods (step ratemhedule ) the Obsemtion wIl pressure drvp is@7e by:

B1

AP=A[

r (W - qi-a)m m) 1=s

[

Rr mr(t H-T*-i)D,~] . . . . . . (3)

k-i

.SPE 12967 GEORGE STEWART, ATUL GUPTA, PETER WESTAWAY 3

tnwma superposition hae been ueed to allow forthe actlm -11 rate Xmrlatlon incMIUlng perloaeof #hutln (qi = 0). PD[(t - T%-%)o, *J u of

~ - ~~~ ~-- ~------ --the reeervolr ~lm,pandqand thel-tiolm of the no-f- ~ with r’aap=tto * lmns equation (3) Will Pru21ct t?nl&earvation *11 re@on8e In the m-infinite,yatem.

memethodof~ can raauilyhanae the●imple gemetrlea of -m-. P==~ -~k~le faults and these *VC ● ~repreemtaticul of many real ila~ testSltuatione . me problem of * faultet~ing at a general ang3.eehaeananalytical aozutlon am ha6 been ~ byPraead=). Bouew8r tlm~tlonraqMxe6 to

9===——i=e@==~ M lengthyinvolving ~ integration of a Eeeaelfunction eeries , an8tM*caee tbeeti &entinted here . me metho&a of ena.lyai8duscr-inthiapaper can beextanud tothe~intamecthg fault ~ tf raqu2rea.

~ xct2mati.alUBine mlicit PUnctial

=

me we of a mm-linear opt~tion tachniqMto come * inverse probM5 of ~e8tiRation by matching the OMerw!d elmpreuicted preeema mapnaee at the d--atim$mllllea-~to bewarylalraliama—I==’-w - Crrorta praaentin thedata. Auach~ eaelafactoxy ~ ~Uifmpliclt mmction theoryWea Introauoeainto *petrolaaa enginaerlng literature by ~ -DogrL4”) in 1975. 19m~be~t0a

9-1-- c- of *- i-~-$m -~tim of the -f fi.cmnte of parabolicand elliptic partial differential ~ionc fram

~f- ~ ---

Z4t PJ denotethe~ abaermtlm uailpreaaum at a partlcu3ar W t~. Mao letp(t~, a, m ~-==—1=—lw~ *tfme t~ calouxated froa a chosen na~tical-I of the reaervoir~atip~tlm~ ~thocewalmere tobeUtxted . IR * ~AgSS ~ eg Ca9@9em

~am~~~eti E===*reepectlwely. Foraapecifteu vameofu ~&hat there axiataavaZoeof @eoch H?aks

f(a,8) = Pj -p(b~,a,s) - 0 . . . . . (4)

%.e. at t29e tj the ~ preemra is axaotly

e9=l ——PraU=a Pramct@ by the *1.lbranyepeclfieU valueof awtthlncertain

~---~~ of s--Cetiafiae (4) Oa be folad by ● ~ona3root fillmng a3gor2tlN such U ~ haltingoraqaaci-aeuton~. mia~oanbe-P-- * auaeaalvewa2ue80fa AIM*peireof valmofaenUp~ eat3.afy(4)mbeplot~on a~ofaveraue nae~inPlgure 3. Thle ~1 relatim bebmen a~ B at time t~ may he -Man ~Mcelly Aas

a- 9j(8? . . . . . . . . . . ...(5)

me impliclt function theorem 14+liaa that thereexlsta a function 93(B) such that a = 9j(S)which eat%mfiea Pj = p(tj, 9j(B), B). Thefunction *(p ) mmat be generated mmerically.euPPoeethat thetruevam- Oftheurknowparametem are a and b reapectivelm then a =gj(b) rapreeentaa point (a,b) Coaeuhem on thegra@iof avemuai3fortm tjaaahumonPlgum 3.

If the time tj la now alloueilto take UtfferentmMtmfmctf.On q;13)0an m~f=each time. meee functlonem Uiatinc’tbut thepoint (a,b) repreaentlngthetruevah?ae -be~ to all, i.e. the functione 93(P) fordifferent time mluea Intersectat (a.b ). Rencet?leeatimatea of the —~ pamneter8 canbe fu.md w datezml.ning t?= fun-lone 9*(s)wrreapoauing tocelected ti-cty, 3 =1 .-. .L end obaemhg the tntemecmm at (a,b)> this1s illustrated in PI- 3.

3#4uation t3)uasuaeu top’mdkct thepreaaum~atanobaerwation wel13000ftfmentilwe Wll ~ Pa3allel faulte m ftapart, Zhe aotlve wall m florid for 24 houmat33,000Sm/D,-infer 24houreaEdttmnflommdegain forafurther24 Roursat the aemrate. Zhe fomtion ~zueabfUty, k, an@pomelty, *, - 4000 m and 0.2 racpectively.me fielu geontry m ilmatratad in Pigum 4Aam2 other ~-aragivenl.n Teblel.lmrty ~ nue lmre --m anu themictell~reegmee techcnminm gum*J @m eapactd 0ec2natory ~ incMBMng 8t-lag ieobeerweu. mepmaaurw MlaaeaR*r - ~s mfm~ *~(-fault8) t8ehomin Plgu395aml ltccnbe~tRat the no-f- ~ea Rave afEecteu bomthe alltuae AIM dtape of * 3wapnce.

i THE INTERPRETATION OF INTERFERENCE TESTS IN A RESERVOIR WITH SPE 12967SEALING AND PARTIALLY COMMUNICATING FAULTS

SPE 12967 GEORGS S1’EWART,ATUL GUPTA, PETER WSSTAWAY 5

Ct - 5.% ● 0.54 x 10-S P9i-;

‘f●ffectlwe ~~**

FE=—=of fault ~

ti/ftSf Wlutllof f&llt aula

4

lhetm~ oftha mouaaamthe~ofno fault Wi6n kf is equal to the fomtionpmueablllty on e%thar slda of the fault and theoaw utnra the fault 18 eeaMng, %.e. kf ~ham Tf - sem.

i

.— —--- . .. - -- ”-... ,---- ,., --, CD- t 70G7c THF INTERPRETATIONOF INTERFERENCETESI’SLN i-ll’WS!LKVUlm w ~-l-.m

-SU .&. wrw ---- -.. ——. — .—.

SEALING AND PARTIALLY COMMUNICATING FAULTS

meemmtion lmll With aPartially~%cating Fault

men th reaulte of the simulation 3une are

~~inafi~itrn found that athirdu~leaa group, *, defined ae?

ef k

m efficient to al- for the effect of theSeaiPeY-mme barrier. Thus the ~tlal=~~PH--X-PO-BCOU~- ~x-a~ of PO m- tZ#q2 for varioua vawee of~, tlw amenelonleeB fault Ooneuctlvity.

various einationa of k~ tf, LAO ~ kyielding the eama * value had iUentioalrapmeea in tem9 of pD anU tti~z.

Bance a eerlee of uimulatlonmm correapmdingto ~ taking the values of O.001* o.o~r 0.05.0.1 - 1.0 =- Oarrieaout and the reaulta~aealog-loggraph of~veraue w%’foreaoh ~,theaemehcun in Figure 18.Aleo ahoun 18 the exponential mkagral aowtionfor the no fault case, %.e. ADO = LO. It Can=seen that for vaMe8 of the Uiaanaionleae faultconductivity Mea than unitY the obaemtionlmll ~ Uiffere ~feluy frua the nofault 0a8e. l?lepreamre lsattematad and the

-ofw-m~=== -%o~mailer.

Yroa the point of view of ~er emtmtlonthtelq-logp btoanbe used form curvematching of obaervatkn well data when theaottw well ie fl.ouadat oomtant rats. mprinciple three ~w could be dete~fmaeuoh type ourvematchtng, Wiz.a

-f-the ~ --

-k - frum the t- ma-Tf - f= the matched c=— ~r -1-.

Fiouever, ae2na2ingthe forntion tobe the~on both 8idee of the fault. the permeability kcan be Ueteralind by analyels of the active -Umsponaa. zn thiaI oam only horizontal movum+ntCan beaadeuurtng thematchf.ng Pruoe8e anu thefault amUuctivity - m f= Wia muchgreater precision.

Tbe ~Me In ?igum M axe~rically gmmateu ~ *lutiona WI*the aotl- WIl fl~ at ~ rate. Theu~lonleaa ~, pD, is a funct%on ofdiaenaionleae tlae, ~qjz , anU diameimleee~lvlty, ~. PMcally this ~ -beml*tmH

m= PD(w%zt %0)

In onler to Uevewp ~r ~Interpretation ~itfi~-expreee tm8 functional relation HI= amaerical apprvximation. m achim this eadieolutlon,oorrwpxM mg to a epactfic * vaXue,uaafltm~apalrof~ Polsale in

* m - 109 w%= -Ii- * w%’ -1-‘le. +~ ~ * fo~~--, -----

I09PD = % + ai TS(109tD/rDz)+ az Tz(logt#r&) + . . . .

It wee not poeeible to adequately mpreeent logm by a single pdynaalal over the whole rangeof log t’#z& frum -1 to 3. The aeoenulngooefftcients of these polynomials*. al, a=, .. . ara~lmfunctioneof *and theyinturn wre fittea aa polym6iale of the fozmt

% = boo + %o Tz(%m) + %o T2(%o)

al - box + btt TI(Aoo) + bzz T2(Mo)

416

SPE 12967 GEORGE STEWART, ATUL GUPTA, PETER WESTAWAY 7

Given the matrix of coefficients bij the value (a) A claaslcal nemilog straight line

of~forany valuesof *an13t&r& canbe at early t- of ba8ic slope ml

quickly evaluated . Thie algorithm is very [W/( 4wkh )1 characterizing thecomenimt for ccaputer application and once rwervoir around the uellbom. This

available the Observation -11 pressure correaponUs to the infinteracting period

Corresponding to an? particular time m before the dlacontinuity infMrenceu the

parameter values can be determined. Uote that active well preasura and its ~uration

Ln computer based well teat analysis the El depends on the ~ietance ~.

function is alao generateU frtm a naricalqproxbatlon to the exponential Integral and (b) A period of steeper raeponaa along anthe present 6olution obtained through mmerical aJmoet straight line of slope mz inter-Simulation and surface EMMng ~: mediate b4tween the baelc slope El andequivalent to an analytical solution Ln that the the double slope character%etic of a

* function is repreaenteu ~ an expliclt sealing fault,i.e.mi< m= < 2mi. Theformula in t~& and *. This Polynaial can value of m= and the duration of thisaleo be a%ff~t~ked ko give dp@kD If transition period are depernlent on the-- -? more importantly, the Laplace dimenaionlees fault Oonuuctivity ba6ed

tranaform can be taken to give the mnetant rate on ~, i.e.- = kf~/(@&).solution In Laplace apace frun which it ispossible to generate, for example, the solution (c) At late time there is a third straightfor the Caae where the active mu exhibits line perioU of slope •~ which 18 cloee tomlbre storage. Given the explicit fozm of m: except at small -Wes of ~ leas than~ t@&, ~) the Iaplicit function approach 0.04.can be used to analyae observation WII data intern of Tf, the fault tranamieaibility, and k, In the 8econd period the linear, semix~lethe formation permeability barrier is reatrlcting flow frum the region on

the far side of the fault and the radialActive *II Reeponae with a Partially prmlxwation of the preaawe e%eturbance is~ntcatl ng Fault Rmuified. If the Mrrter m annular in shape

and ~rlcally placed around the active wellIn the caae of a aaal%ng fault the d%atance ~ ther@ial nature of thefti tmuld notbefm the active wan to the Uieoontlnutty can h Uistofieu AM the late time plmeewm behaviourQetetineU fnm the Interaection of first and lmuld reflect the permeability of the Mgion8econa straight 11- on the aemil.og plot. outside the Ulacontinuity.S%ncethe

EoOmver when the~tim WIl ~ ie Virtually barrier is linear thera is a.zwaya -

l~t of t~ -lt ~i~, m- it Uietortion of the -id flow pattern * theis btaaen the active -M and the obaervak~ slope at late t-, m=, is never emactly equal-11, It %8 eaaential to ~= the nature of to 91 (for the can shere the pemeabllities onthe active -11 raapmae lslenthefault ia either side of the fault are equal). For valueapartially ~ting . ~ia information i8 of the dimeneionlae8 fault ~lvity *alao mvailabXe from the 8~ results endthe active ~U behaviour on a -log plot for

greaterthan about0.04 the final slope m= lac3.0eato m*andthepzeaeum ~ ~on

Valuea of ~ of 1.0, 0.1 - 0.01 are ahmnl in the ~ristic 8hape ehafn in rigure 19.Figuree 19, 20- 21 ~imly. ~ the zhe8&pe of themidUle ragion, ma, lsl.eae thanaiatemce to the ~ion well hea no double that of initial or final tiraimt lineInfluence on the active uexl reepmee Uhidl ie periods. m that th~s ~rietic is thecmlyaffacteuby*# hence itie~to inveree of that pertaining to a dual POrwltye!@Soythe~ Ieea ~ivity baaed on system (naturally fractured or ~red~, I.e. ~ . kf ~/(~+) When referring to meemoir ) where tne siope of the iiiterm8S:atethe active lmll . - the S-1atia’1 run8 *raight line ts half that of the early and late

*m”o.3375ti~ x9, 2oend23 time periods.~ to * = 0.387!5,0.03875 m 0.003875re8pectWaly. - no-fault aM eealing fault 7ha ClA93i* ~ Of eat-tx the distance-m~--ew=rw—l = to a 8ealing fault 1s baeed on the intereectlon~-. Ae ~ a trenS frcm no-fault of the fmetstzai@ltune dthe~toeaalingfault~ ia ~m8w etraight Mne of dmuble slope at t- ~ *ichfault Condoctlvity ~. In fact at ~)= %8 -lam to * by the equation~0.001, i.e. ~ = 0.0004 the late time activewell ~ 18 alm3t Im3ietingulsheble fmmtheeealing fault ceee and only an interference

*k t=

teetuoulU indicate that auoh a~llevalof = 1.781

/

. . . . . . . (8)

~cation tistad . *P-\z

A ~~, ~~~~a~ic alnpe to the umi-2og active W1l

In the m of a Partialxg ~ioating fault

~~inan~eethe interjection ~thefir8tti~

= reaerWoir , notably rtirai@tlineei8~nt omLgpamU on theU-lonlesa fault Ooductivity ** me Slopsof theeeconU perloU, mz, ia alaoafumtlonof*. tunceani~ ion~for-

417

THE INTERPRETATIONOF INl?ERI’ERENCETESTS IN A RESERVOIR WITH SPE 12967

SEALING AND PARTIALLY COMMUNICATING FAULTS

active wall responee can ba devlaed IMloh labaed on the detaotion of the ftret end eaoon8straight line periode on a eemll.og plot andgtves both the di~ to km fallt ans its

Woo-sei*.ty . T!l&a .~ that tM

lnftnite-acting flret perioS 18 not entirelymaeked bg mllbore storage and Oan M14entifieel the Slape, -~, of *- ~t -Uof ~ give the formation ~lllty, k.

xnomler todevelopmc!ha method the active-n S-later Outptu for wariouavameeof*Uereanalgeea anlktha ●lopaerl&, mzena=sd8teminad taeinga ~ aqtaame ~. It18 not U%ff%cult to identify the lnfinlte-aotlrqand i~iate etraight lines tit the finalline of ●mpe D~ ia * -11 *flnaa ~ atvaMe8 of ~ greater than about 0.04. Themreeolts are preeented in Figure 22 there theratio R@l (16E*162 ) Is given ae a Shnctmnof~~fort?tepreeentoa8e ~=2.S81~en6the~of Pl@re22 anSF2Pure23abou2dbe ooneide~ to be 2.581 ~ 8inoQ ~ iE notrelevant . mflx Me8 batlnen ma ~~valmofla2 ~lng to no fault (*= =) and a eealing fault (~ = O). The ratioabmx W =s0 wan for ~ketenaea but for8ma21valuea of~ the true valmeofm=lsonlgattalnad at very late times (mot aohleved in the●*xatlon nane ). In a reaX te8t lt %s Mke2ythat only aufflcient date to eeflne theIntermediate region -IO be obtalmd on a~le perioe of tima.

* i~l.ul of * first a eeoonustraight linae denoted + ~ axeo emnedand t?le ratio wtmlJ, ~ t= 18 the%nternction tima for a eeallng fault given by(8),2e~eaafunotmn of~lnrigure23.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

coNcMlsmNs

Xmpllcit fcmct%on theory haa been fbund to be a

--l ~ for anaXy3tng $nterfemnce te8tsand, in ganral, It 18 ~maean

axoellent apprwdi to So:v.ng t!18 znlJar!MProblm of panmater eatlmation -n 8emi20ganalgaie 18 not agplzable . mmathouieMbuetanu haetbeuietinct ~of@.elUing the oonfidanoa Umita for tba eetmted

~-. me application to interfemmtesting hae baen oonaidexwd ham but tt ieavtdent that theteohniqtfeoan alaobeueedtoanalyaa tha aotiwa well preaeurn behaviour andin Particular the late tranaient parlo4 Whenboundarle8 have affected the maeured reqonae.mxemample~ingkh and Shave beaneatermllnd fmm tlla early tima ~ -~liolt function mathotlcan be ured to analgeala- *- eat= %= tam Qf Eault gaoau?triee.

TM xlioit funotion m8tlKM 23 ●imply aaxwmntont way of matolling the preuictad

P~ ~ ofamaYel totheob3ervedpmsrure zxmpnaeamxtheae ahouxdalwawba

* ~=- - ~ of fit.TM main di~ of the tachritqua is thatitleltmitedt o— anknow ~re.~r, :tiepouibXe tooarxy out tbarntohing at various valuea of a thiti unknown~ramlto Qatermine tbe beat overallmat- ~ ~elm30baerv88 reepo---!lm8appmaoho anbaueedfor QXetoalxoufor the ●fw of aotf= -Uwra storaga oninterfe~ testtng. In thie Oaaa theanalytical -Mttion is only aWai2abxe in L3pxaaapaoaansthesteh-mtl- i8requ2.routom- --1 Pr’aalotea~.

!Ma equAva2ent of an analyt%oal eoht20n for tlmpreaeum reqmnea atanobeervaklon WIlontlmfar 82da of ● partially ~loating fault f=● oawtant rateaott=uallbae baenobtatnaUbylnmarxoax mf6iuxaMon . %k3 f!unotson 28~by a polynomial =tlon -*oan be wed for auperpoaition in variabla rata*itmaiaioiona,e.q. pg~~~~w~ *

zap- ~ If rqUir9U. xmpMott -ionthaorgoan be -tom atoll tbe~a?ma--tom -n rae&m8e totht8mdelaulhenoadetemim t.hafaultt Xmnaieatblllty Tf*the~~ -Mebaenuactifkn te(fetexmtna dfmmtlle acti- =11

~.

a,b s tnle VaWaa of ~-*-

mlmatd

*’ u2mene20nlea8 di8tamae frum mot%- or

w -11 to Obeamion *U

%’ Uimansionleea fault CXxMuotvrity

%’ ~ ~-lllw

II 1 forntion w~

k t fozion pazmaablltty

kf 1 fault aona pezu3abizity

.

SPE 12967 GEORGE STEWART, ATUL GUPTA, PETER WESTAWAY 9

I

t

I

:

I

!

f

#

t

!

t

I

1

1

t

I

I

fault zone thickness

distanoa from active -11 to

tabaervationwell

diatanoe from aotive well to fault

slope on Semilog plot

preemlm

dimanslonlees preesum

initial pressure

measured obaervatlon well pressure

at time tj

in ultu well flowrate

wellbore radius

dimensionless radius

interaeotlon of f%rst - second

atralglitltneu on semllog plot

lnterseotion time Corresponding to

sealing fault

fault tranam.imaibility

time of prwaura Observation

parsmatera to be estimated

Vlaooelty

1.

2.

Bmmer, R.E., W2auton, H. aTM Vda, . s.:%nalgtioal Model for Ve*loal Interfersnoa-ts aorQsB 310u-Permeability~“ . PaperSPB 11%S presented at SPB-Ml@ 5* Annua2?all ~ioal conference, San ~,a, Oot 5-0, 19sa .

St~, G. ti Wits, M. 1 ‘WtWInt~tat20n of M8tributsd m~ W?- ~e In Pmduoad Reservoirs” .

PaPer BUR272Pre-adattha mropeanPetmm Confemnoe, London, 25-20 Oot,1902.

3.

4.

5.

6.

7.

8.

9.

10.

Nadir, F.T. I miatle Field Develqmmnt” IPaper BUR 165 presented at the EuropeanPetrolmm Conferenoa, London, 21-24 OCt,1980.

vela, s. and McKinley, R.N. ! “- RrealSeterogeneitlen affeot Pulse Test Reeulta”.~. Pet. Enq. J. (June 1970), 181-191)Trans -, *.

Johnson, C.R., Gr9enlcom, R.rn. arid Woode,E.G. : “Pulse-wmting - A W 3W3XM forDeaoribing R@S.rVOlr Flow PYOpWt%eO BetweenWells- . J.Pet.~oh. (Dec. 1%6), 1594-1604~Trana. RIM?, 237.—

~, n. W Brigham, W.E. I “Pulse Testing

Remponse for Unequal Pulse arM ShutinPeriods-. sec. w. Brig. J., (Ott 1975),----- _399-41UJJ Trana . AiitS,259.—

sarlougMr, R,c. I “Advanoes m -11 metRnalwis” , vol. 5, SPs monograph series(1977), 286-191.

Prasad, R,K. : Wresaure Transient Rnalysiminttle Praaence of * IntersectingBoundaries”. J.Pet.lWtI., (Jan. 1975),SP-P6j Trams ~, 259.—

cannon, J.R. and Dogru, A.If.: -Bstim3t2.onof Pamaability W@ Porosity fmm wall matData”. Paper BPS 5345 presented at the 4sth%mq.,m>. . — ~~lfQ~~g& m@onal meting.SPS~, Ventura, CA, Rpril 2-4, 197S.

Wsstawav, P.: - Ilffeot of a PartiallYsealLng- VQrtioal ?ault on Single tiIalltiple wll ~ Transient -eting”.M.sng. tlwsis , Beriot4tatt Unlvarszty, 1983.

I

TAms I

EASIC PARAMETERS POR SYNTHETIC QmMPIs

Distance from aotive wsii to observation =wsZI,LAO 5000 ft

Interwell mobile phase viscosity, P 0.40 Op

Intemll average %ons thicknSSS. h 100 ft

lbta.1Ooalpreasl.bility,c+ 6.25 X 10-8 ~-~

oil fonaation volume factor, % 1.2

lmll radius, rw 1.0 ft

Avarage permeability, k 4oooti

Avsrage Poroeity, 6 0.20

TABLE 2

cOUPIDRKE L13flT3FOR ESTIBM!FED P~ -TEST CA6E A

~mr samaet.ez F.v!?mge True Lower Upper(psi) Value value Confidence Confidence

Limit Limit

●oel k 4050 4000 3900 4200@ 0.20 0.20 0.19 0.21

●leo k 3800 4000 3200 4400e 0.2!2 0,20 0.19 0.21

TABLE 3

Well radius, rw

Interwell mobile phase Vieoosity, B

Interwell average thioknees

lbtal Ocmprsesibility, Ct

Oil formation volme factor, Q

Rater fomtion voluma factor, ~

average porosity,*

xn~eotion Aate(Hi/D)

35,000e

35,0000

35,0000

35,0000

0.33 ft

0.41 Op

90.0 ft

6.4 X 10-6 PSi-x

1.263

1.020

0.23

T2sls(hr)

12

1224242424

SPE12%.-

. . . . . . . . . .. . . . . . ... . . . . ..,. . . . . . ..-. .. ..“

.

1.

. . .

. .. .

l.. !

i.. .

. ..,

..-

.“ . .

. .

. . . .. . .. . .

‘,.

. . . . .

. . . . . . .. .

. ..- . .

. . .. .

. . . . .. .. . . . . . .

---- . .-.. . . .. . . . .

. .

. .. . .

. . . .

. ..’

. . .

. . .. .

. . . ... .

. .. . .. . . ..-, .1 . . . . - . ‘ .. . .. . . . . ., ..

.“. .

,.

. . . ..,

. . .“ I. . . .

. ..’. .

. ..-.

. . . ‘,. .. . . . . . . . .

. . . .. . . . . . . .. . . .. . . . . .

. .

la -

. .

SEAL ING

. .. . .

..

.

,.

. .

FAULT

. . . . .

. . .

. . . . . . . . . . .. . . . . . . . . .,. . . . . .

. . . . . .. . .,.. . . . .

.-. ..-.. . .

. .,. ,. . . 4

.d . . . .

. . d. .

, . . . . .. . .. .

.., . ..,. . . . . . . .

,. .

R. . . .

. . . . , . .. . ., . . . . .. .,.. . . . . . .. . . . . . . I

. .

.

.“

.

,. ..

.

.

. . .. . .

. . .. .

,,. .. . .

..

. .

. . .. . . . . . . . .. .

b.. . . .

., “. .. . . .,

.. . . . .. . ... . . .

.. . . . “ .

. ,... . .. .

. ... .

.... . . . . .

. . . . . .

. . . . . . . . . . . . .

lb - PART IALLY COMMUNICATING FAULT

Fig.1—Sealing and partially communicating faults.

sf’E12967.-

G ----- ----- -

a

-nm

--jI

I

I

I

I

I

I

I

I

1

a

8

o

al

Q

VI 0●

0

FIG 4a TEST CASE A

PARALLEL FAULTS.

— 3ooo’—

FIG 4b TEST CASE A

PRESSURE-TIME RESPONSE FOR TEST CASE A

3515 1 , IINJECTION

/e e e 3512

t

I

/

~ 3~UJ /=

● ACTIVE WELLo OBSERVATION WELL ~ 3506e IMAGE WELL L

35031- 1/ -~

..~O 6 16 24 32 40 46 56 64

MAXIMUM OF 30 IMAGES CONSIDERED.TIME (HOURS)

Fig. d—hterfemne test Mw-n Wellel fauh~Te@ * A.

FIG 5

INFINITE ACTING PRESSURE RESPONSE

3502.50 1 , I1 , 1 !

INJECTION INJECTION

3502.00-

3500.50-

3500.00 t , , ,

0 8 16 24 32 40 46 56 64 7*TIME (HOURS)

Fig. 5-lntinite-sting ayatem raaponaa.

FIG 6 TEST CASE A

GRAPHS OF PERMEABILITY-POROSITY WITH NO ERROR.

mnn

k TRIJE PERMEABILITY - 400h0p

TRUE POROSITY -0.20

4500

1-i t.s? 4000

J

------

%a

L

3500

-.-13MJU’

/ +~\-

\ L I

1.60 1.80 2.00 2.m 2.40 z.EiOxlO-’

POROSITYFig. 5-k+ diagram for Taal Cane A—no error.

FIG 7 TEST CASE A

PERMEABILITY-POROSITY CURVES WITH*O.1 PSI ERROR.

ox..-

POROSITYFig. 7-k-O diagram for Test Ca&e A—*O. I psi error.

FIG 8 TEST CASE A.

INTERSECTIONS OF PERMEABILITY-POROSITY CURVES WITHfo.1 PSI ERROR.

5000,

I

o 0 CALCULATEOSOIUTION

AvG. PERMEABILITY= 4050m0

E 0 AVG.POROSITY -0.20

~

> ;

: 4000 ---T~~EJ=---”~~zE

‘,mo&0.16 0.20 0.26

POROSITYFig. E-intersections of k+ cuws-Test Case A, *0. I psi error.

FIG 9 TEST CASE A

INTERSECTIONS OF PERMEABILITY-POROSITYCURVES WITH *1.OPSI ERROR.

0

00 CALCULATE SOLUTIONAvG. PEMEABILITY- 38~0

0AVG. POROSITY -0.20

TRUE VALUE*------ ------ --

1

1

~;

<i=-1 0=1al o+1

It

14 o.m~ I

POROSITYFiO. 9—htemeti!ons of k+ curves-TeW C= A i 1.0Pi e~or.

10-1

FIG 10 TEST CASE B

PRESSURE TIME RESPONSE FOR TEST CASE B.

3505 \ , , /INJECTION INJECTION I1 I

TIME (HOURS)Fig. 10—Pressure-time response for Te5t Case B-single f8ult.

FIG 11 TEST CASE B

PERMEABILITY OISTANCE CURVES FOR ERROR OF*O.IM5P51.TIME INTERVAL OF 12HRS.

4200

s~

@nlkSHRS

=~- -.=Jwu&

[ \w

‘1 \ 1

\37rn

IB 7(KI 800 ~ ~~ ;1OO Im ml

OISTANCE (FEETIFig. 11-k-L diagram for Te6t Case B-*0.C6 pi error.

SP E12967

TESTCASE B ?0.4 PSI ERROR

k

(rnd)

5000 r

4000

3000

●

● CALCULATED SOLUT ION

AV . PERM?AB IL ITY = 4310 md

AV . DISTANCE 1335 ft

TRUE WLUE---- ----- --- ..*

8

I

f

:8I

Ikl

IgkIa

..

●●

100 500 1000 1500 2100

L

(ft)

Fig. 12—intersections of k.L curves—Tesl Case B, f 0.4 psi error.

FIG 13 MEASURED PRESSURE USING OUARTZ GAUGE.

2641“ -Q- .

* ----

p40 . .-. -. -...-.. -. e. .-

— .. -.

~ 2639- --- -.. -. .-.

%+ .-.. ... -a

— .-0 .: 2636-

. . ...... -. *

.. .... .. -.

2637 “. --e---.... ....... DATA POINTS (MEASUREDI

2636 - e- .’

~ INJECTION

o SAMPLE OATA FUINTS

2635i,

08 1624324646 M647266M$$ 102

TIME (HOURS)Fig. 13—Field elxample pressure-time response.

FIG 14 CURVES OF WI VS W2.

12 , 1-

10-g=g

~8 -y0

:6 -

4 .0.60 0.70 O.IBO 0.90 1.00 1.10 1.20 1.30

W2X10-6(MD.FTICP).

Fig. 14--[kf(4wcJ]-(kW) diagram ~r field examPle.

FIG 15 INTERSECTIONS OF W1-W2 CURVES.

0 0

CALCULATEDSOMJTIONWI (kl+~Ct)-7.M:KIF MO.PSIICP

o0

W2[khlPB9.70x 105MO.FTICPo

000

0 0

00 8°

0 00

,$: 1 , 1 t

.70 .80 .90 1.00 1.10 1.20 1.30

WZX 1o-6 (MC1.FTICfl.

Fig.15—lntwsetimns of [W@t)Hkhk4 cum-—field example

FIG 16 INTERFERENCETEST-EEST MATCH FOR WI AND W2.

2641- &●...; i!. -*

pll - :* r -o. -*.,-.. -— ● .. .2

g12639-~2. *- ●

“.: s“. a.

%

-.eo

J~ 2636-

. ... ~ .....

- 4-!**.?9’s . -“‘lb -., m. .. . .. . MEASURED DATA

2637- .; -~-% - ● BEST FIT.. WI. M@t.7.EWX lCFMD.PSIICP.

Zm ~.” W2-khl# -9.70x 1(P MO.F71CP.

EZl INJECTION.-*

TIME (tlOURSl

Fig. 16-Comparison of observed and predicted pressur.3time responses.

FIG 17: RESERVOIRGRID.

RESERVOIR IS SYMMETRICALABOUT THE AXIS OF THE WELLS.

~ ‘“

‘i1

32mm

mom

2oaTlciuw

65 z20

II

ALTERED REGIONx

I

rA ACTWE WELL COORDINATES(11,1Il.

YO-1 OIK$HWATIONWELL COORDINATES(16,11},

OBSERVATIONWELL COORDINATES(15,11)

OBSERVATIONWELL CUOROINATES[7,111

Fig,. 17—Carlesian grid for the simulation of a partially communicating fault.

Fig. 18—Type curves for obsavation well response across a linear Parlially communicating fault.

FIG 19: ORAWOOWN AT THE ACTIVE WELL WITH, FAULT AO-l.O.

REAL TIME, OELTA T, HOURS.

~2M 00.37 1.22 4.06 13.46 44.70146.41492.751635.965431.6516033 59874

ml4160.0 -

‘.4060.0- ~.,

‘.& ‘\‘,@ 3960.0- ‘.‘. /z ‘.= 366D.o-

‘. m3‘.‘..

3760.0t

SEALING FAULT AT 300 (‘‘ ‘.

(lflJlfBIE SLOPE”). ‘\ . i

3660.01 I .I ,

1

-1.0 0.201.40 2.60 3.60 5.MI 6.20 7.40 6.60 9.60 11.00LITT.

Fig. 19—Active well resW* fOr Aw=l JJ:

FIG 20 DRAWOOWN AT THE ACTIVE WELL WITH FAULT &O.01.FIG 22 EMPIRICAL RELATIONSHIP FOR SEMI-LOG ORAWOOWN

SLOPES IN THE PRESENCE OF A NON- SEALING FAULT.

REM TIME, DELTA T, HOURS.

0.37 1.22 4.06 13A6 44.70 146.41492.751M.Wml.651~ X$74

r<” ~

4m”0 --J%..

4160.0 -’>,.:.... . ----

-.. .~ 4LM0.O ---- tjO~ULT I“UNITSLOPH

~ 3960.0--.

.m2 -.------.

.‘woo .

~.*

In T.Fig. 20-Active wetl respon5e for Aco=0.01

FIG 21 ORAWOOWN ATTHEACTIVEWELLWITHFAULTAD=O.lO.

REALTIME,OELTAT,HOURS.

0.371.22 4.0613.4644.70148.41492.751635.905431.6518033590744260.0

r“”< :

41q.O mlf q. .

\

‘. -. -..=~ 4060.0

NO FAULT~UNIT SLOPE”I-.. .\ fm \ ●m2 -.. %

---~@ ~.o

x ;,*.“e ----

\ ., -. ----n ‘\ \

3660.0 SEALINO FAULT AT ~ ~’ ‘ >(OOUBLE SLOPH.

\\ ,m

3760.0t

-x\.

. .1

3660.o~,’\

, , , I I

-1.~ 0.20 1.40 2.60 3.00 5.IHI 6.20 7.40 8.60 9.00 11.(NIb T.

Fig. 21 —Active well response for Am= 0.1.

m3iiil

A.

0.001 0.01 0.1 1.02.0

h ~

2.0-. \ \ \ mz

\\ /ml\

mo >’+

1.5ml ‘, - 1.5 !!2

\ ml\\\\

‘.‘m,.

-.---0--------- -

1.0 1.0-4 -3 -2 -1 0 1

LOG A.

Fig. 22—Slope ratio m */m I as a function of AN= 0.2875 Aw

FIG 23 IEMPIRICAL RELATIONSHIP FOR SEMI-LOG DRAWDOWN DATA IIN‘THEPRESENCE OF A NON-SEALING FAULT.

0.01b

0.001 0.1 1.0 101.0

1~

1.0

‘.‘- \

~p. \ \

1

]hb~~ \ h

●

Txo\ Txo

0.5&p ‘\ 0.$b% ‘,

\\ \ \

I\\

‘.‘ ..

-..

1 I 1 , J------0-4 -3 -2 -1 0 1°

LOG A.

Fig. 23 —lnterWCtion time ratio t,lt,o as a function of Am= 0.3875 AW.