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Use of Pressure Derivativein Well-Test InterpretationDominique Bourdet, * SPE, J.A. Ayoub, SPE, and Y.M. Plrard, * * SPE, Flopetrol.Johnston Schlumberger

Summary. A we ll -t es t i nt er pr et ati on me th od b as ed o n t he a na ly si s o f t he t ime .r at e o f p re ss ur e c ha ng e a n~ t he . a ct ua l p re ss ur ~ r es po ns ei s d is cu ss ed . A d if fe re nt ia ti on a lg or it hm i s p r op os ed , a nd s ev er al f ie ld e xampl es il lu st ra te h ow t he me th od s un ph fi es t he a na ly si s p ro ce ss ,m ak in g in te rp re ta tio n o f w ell te sts e as ie r a nd mo re a cc ur ate .IntroductionTh e i nt er pr et at io n o f p re ss ur e d at a r ec or de d d ur in g a w e ll t es t h asbeen used f or manyyea r s to evaluate reservo ir character i st ics . Sta ti cr es er vo ir p re ss ur e, m ea su re d i? s hu t- in w ells, is . us ed to p r:< lic tr e se rve s i n p lace t hrough ma t er ia l- b al an ce ca lcu la ti on s. T r an s ien t-p re ss ur e a na ly si s p ro vi de s a d es cr ip ti on o f t he r es er vo ir f low in gb eh av io r. Many me th od s h av e b ee n p ro po se d f or i nt er pr eta ti on o ftr an sie nt te sts ,' b ut th e b es t k no wn a nd mo st w id ely u se d is H om -e r's.2 M ore re ce ntly , typ e c urv es, w hic h ind ic ate th e p re ssu rer es po ns e o f f low in g w e ll s u n d er a v ar ie ty o f w e ll a nd r es er vo ir c on -f igura tions , were in troduced .3-8 Compar ison of t rans ien t-p ressurem e as ur em en ts w ith ty pe c ur ve s p ro vid es th e o nly re lia ble m ea nsfo r i den ti fy in g t ha t po r ti on o f t he p r es su r e d a ta t ha t c an be an al yzedby convent io n al s tr a ig h t- li ne an al ys is me thods .R e ce nt ly , t he q ua li ty o f w e ll -t es t i nt er pr et at io ns h as impr ov edc on si de ra bl y b ec au se o f t he a va il ab il ity o f a cc ur at e p re ss ur e d at a( fr om e lec tr on ic p r es su r e g auges) and t he d eve lopmen t o f n ew so ft -w a re f or c omp ut er -a id ed a na ly si s. A n i nc re as in g n umb er o f t he o-r et ic al i nt er pr et at io n mo de ls t ha t a ll ow a mo re d eta il ed d ef in it io no f th e flo w b eh av io r in th e p ro du cin g form ation a re n ow in u se .S u rp ri si ng ly , t he c ommonly u se d a na ly si s t ec hn iq ue s h av e n otf ol low ed t he g en er al p ro gr es s e vi de nt i n h ar dw a re a nd i n i nt er pr e-t at io n mode ls , mak ing t he i nt erp re ta ti on p rocedu r e compl ic a ted andt ime -con suming . Type cu rv e s a r e s een by va ri ou s an al ys ts a s ov er lys impli s ti c o r overly complex , d i ff i cu l t to d i st ingu ish , and lo r cumber-s ome t o u se . Y e t, me re i de nt if ic at io n o f s tr ai gh t l in es o n a p re ss ur e-v s. - ti me g ra ph is a r ul er a pp ro ac h" - co nv en ie nt f or h an d a na ly si sb ut i gn or in g p owe rf ul c omp ut in g f ac il it ie s t ha t a re a va ila bl e. F u r-t he rmore , t he convent io n al s tr a ig h t- li ne an al ys is me thods f a il t o u sea ll th e d ata a va ila ble a nd c an r es ult in s ig nif ic an t e rr or s.W e p ro po se a n in te rp re ta tio n m eth od b as ed o n th e a na ly sis o fthe d eriv ativ e o f pre ssu re w ith re sp ec t to th e a ppro priate tim ef un ct io n- na tu ra l lo ga ri thm o f t ime o r Ho rn er /s up er po si ti on t imef un ctio ns . T his m e th od c on sid er s th e r es po ns e a s a w ho le , fr omv er y- ea rly -t ime d at a t o t he l as t r ec or de d p oi nt , a nd u se s t he t yp e-c ur ve -ma tc hi ng t ec hn iq ue . I t p ro vi de s a d es cr ip ti on o f t he f low b e-h av io r i n t he r es er vo ir , b ut w it h t he l og ar it hm i c d er iv at iv e, i t a l soemphas iz e s t he i nf in it e r ad ia l f low regime. o f p rime i nt er e st i nwe l l -te st in te rp re ta tio n. T he ap pro ach is a n e xte nsio n o f th e H orn erme th od t o a na ly ze t he g lo ba l r es po ns e w it h impr ov ed d ef in it io n.U se o f th e d er iv at iv e o f p re ss ur e v s. t ime i sma th ema ti ca ll y s at is -f yi ng b ec au se t he d er iv at iv e i s d ir ec tl y r ep re se nt ed i n o ne t erm o fth e d if fu siv ity e qu atio n, w hic h is th e g ov er nin g e qu atio n f or th emo de ls o f t ra ns ie nt -p re ss ur e b eh av io r u se d i n w e ll -t es t a na ly si s.Thus , t he d e ri va ti ve r e spons e i smo re s en s it iv e t o sma ll p h enomenao f i nt er e st t ha t a r e i nt eg r at ed and hence d imi ni sh ed by t he p r es su r e-vs .- t ime so lu t ions .One l imi ta tion of thepressure der ivat ive in analys is i s the d i ff i cu l tyi n c ol le cti ng d if fe re nt ia bl e p re ss ur e- tr an si en t d at a. A c cu ra te a ndf re qu en t p re ss ur e me as ur ement s a re r eq ui re d. H oweve r, p re ss ur emeasu r emen t and t he compu te r p ro ces si ng t echno log ie s now avai l-ab le a t we ll si te s a ll ow p res su r e-de r iv a ti ve an al ys is .The p r es su r e-d er iv a ti ve me t hod i s d emon s tr at ed for a homogene -ou s r es e rvo ir and compa red w i th convent io n al i nt erp re ta ti on t ech -n iq ue s. T h e p ra ct ic al a sp ec ts o f d if fe re nt ia ti on o f a ct ua l p re ss ur ed at a a re d is cu ss ed . A p pl ic at io n o f t he d er iv at iv e a na ly si s t o h et er -'Now with Kappa Engirn>ertng."Now a consul t an t .

C o py rl gh t 1 98 9 SocIety 0 1 Pe tr o le um Eng in e e rsSPE F o rm a ti on E va lu at io n. J un e 1 98 9

o ge ne ou s f orma tio ns r ev ea ls th e g oo d d ef in itio n o bta in ed w ithd e ri va ti ve p lo ts , and t he d is ti nc ti on b e tween cu rr en tl y u s ed i nt erp re -ta tio n mo de ls is c le ar ly s ho wn .TranslentPressure Analysis Appliedto Homogeneous ReservoirsConven ti on a l we ll -t es t i nt erp re ta ti on h as fo cu sed on th e homogene-ous reservo ir so lu t ion . The correspond ing pressure-analys is methodsh av e b ee n d is cu ss ed e xt en si ve ly i n t he l it er at ur e a nd a re c ommonl yused.Two comp lemen ta ry a pp ro ac he s a re u se d f or t ra ns ie nt -p re ss ur eanalysis: (1) a g lo ba l a pp ro ac h is u se d t o d ia gn os e t he p re ss ur e b e-h av io r a nd t o i de nt if y th e v ar io us c ha ra ct er is ti c f low r eg ime s, a nd( 2) s pe ci al iz ed a na ly se s, v al id o nl y f or s pe ci fi c f low r eg ime s, a rep er fo rmed o n s el ec te d p or ti on s o f t he p re ss ur e d at a. R e su lt s o f a na l-y se s w ith b oth a pp ro ac he s m u st b e c on sis te nt.D i agnos is o f p re s su r e b ehav io r i s p e rfo rmed by t yp e -cu rv e an al y-s is . F ig . 1 d e s cr ib es a w e ll w it h w e ll bo re s to ra ge a nd s ki n i n a r es er -v oi r w it h h omo ge ne ou s b eh av io r.s Dimension less pressure, PD 'is p lo tte d o n lo g- lo g s ca le v s. d im e nsio nle ss tim e g ro up , tDICD .T he r es ulta nt c ur ve s, c ha ra cte riz ed b y th e d im e ns io nle ss g ro upCDe2S (A ppendix A of R ef. 9), correspond to w ell conditionsr an gin g f ro m d am ag ed w ells to a cid iz ed a nd f ra ctu re d w ells .T wo flo w re gim es o f in tere st c an b e id entifie d in the p ressu rer es po ns e ( Fig . 1 ). A t e ar ly tim e , a ll th e c urv es m e rg e to a n a symp-t ot e o f s lo p e equa l t o un it y, co rr e spond ing t o pu re we ll bo re - st or ag ee ff ec t g iv en b yPD =tDICD .. . .. . .. . . .. . . .. . .. . . .. . . .. .. . . .. . (1)L at er , w h en a ll s to ra ge e ff ec t i s o v e r, t he c on sta nt s an df ac e f low

r ate is e sta blis he d, a nd th e r es ultin g p re ss ur e b eh av io r p ro du ce sth e u su al s tr aig ht lin e o n a se rn ilo g p lo t:PD=0.5[ln(tDICD)+0.80907+1n CDe2S ) (2)

Th is r egime, ca ll ed i nf in it e- a ct in g r ad ia l f low , do es no t s how a char -a cte ris tic s ha pe o n lo g- lo g s ca le . T he lo cu s " ap pr ox im ate s ta rt o ft he s em il og s tr ai gh t l in e" t he re fo re has b ee n m ark ed on the typec ur ve o f F ig . 1.The i nte rp re ta ti on p ro ce du re w it h t hi s t yp e c ur vei s i ll us tr at ed w it h a 3 0- ho ur b ui ld up (Ta bl e I ), w h os e d et ail ed i nt er -p re ta tio n w as p rese nte d in R ef. 1 0.T h e f ir st s te p is t o p lo t t he b ui ld up p re ss ur e d if fe re nc e, p(~t)-p(~t=O), v s. t he e la ps ed t ime , ~t, s in ce th e w ell w as c lo se d ( Fig .2). T his p lo t is th en co mp ared w ith the typ e cu rv es: th e lon g u nits lo pe s tr aig ht l in e a t e ar ly t ime s, i nd ic at iv e o f w e ll b or e s to ra ge e f-fe ct, is m atch ed o n th e e arly -tim e a sy mpto te o f th e typ e c urv es.B y mo vin g a lo ng th is 4 5 lin e, th e b est c ur ve m atc h is a ttem pte d.In th is c ase , a ll c urv es a bo ve CDe2S =10 8, in th e d am ag ed w ella re a, m atc h th e d ata e qu ally w ell. T he p os sib le m atc he s a ls o s ho wth at th e lim it " ap pr ox im ate s ta rt o f th e sem ilo g s tr aig ht lin e" hasb ee n a tta ine d a fte r a bo ut 2 3 h ou rs of shu t-in .A s em ilo g a na ly sis is th en p erf orme d o n th e la st 7 h ou rs o f b uild -u p; th e p re ssu re is p lo tte d w ith re sp ec t to th e lo ga rith m o f H or ne rtim e (F ig. 3). A straight line develops at the end of the plot andis u se d in th e c on ve ntio na l w ay to e stim ate /chIp. ( fr om t he s lo pe ),P* ( fr om ex tr apo la ted p r es su r e t o i nf in it e s hut -i n t ime ) , and S (fromth e s tr ai gh t- li ne d is pl ac ement a t 1 h ou r) .T he p en ne ab il it y g ro up /chIp. b ein g f ix ed , th e p re ss ur e m a tc h isk no wn a nd it is p os sib le to a dju st th e ty pe -c ur ve m atc h. T he f in almatch is m ade on CDe2S =4 x 109 R esu lts o f the a na lysis a reg ive n in A pp en dix A .

29 3

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Fig. 1-Wel lbore-storage and skin type curves for a homo-geneous reservoir. 5

29 4

TABLE l-PRESSURE vs. ELAPSED TIME,BUILDUP 2 (continued)Elapsed Pressure Pressure Pressure SuperpositionTime Change Derivative Derivative nme(hours) (psi) L-O.O L-O.l (hours)2.25000 659.71000 162.25279 159.78026 -2.055832.37500 667.19000 149.94510 151.22713 -2.008852.50000 673.44000 139.99198 149.04678 -1.964592.75000 684.65000 140.37167 138.91767 -1.883213.00000 695.11000 138.47093 126.58152 -1.809933.25000 704.06000 113.73940 135.38378 -1.743433.50000 709.80000 135.84157 134.55333 -1.682693.75000 719.50000 148.73954 111.93829 -1.626884.00000 725.97000 106.36197 109.31883 -1.575364.25000 730.20000 63.06567 86.38834 -1.527594.50000 731.95000 40.78765 75.31986 -1.483124.75000 733.70000 56.85782 70.70269 -1.441585.00000 736.45000 79.90940 66.24425 -1.402665.25000 739.69000 87.07801 69.37689 -1.366095.50000 742.64000 74.17246 62.20195 -1.331655.75000 744.70000 72.37656 55.61930 -1.299126.00000 747.19000 70.17938 53.35277 -1.268366.25000 748.94000 33.00165 42.51997 -1.239196.75000 748.02000 21.38772 37.84478 -1.185137.25000 750.78000 52.87202 29.90081 -1.136067.75000 753.01000 42.98848 34.27710 -1.091278.25000 754.52000 41.68190 42.43457 -1.050198.75000 756.27000 40.60712 39.99428 -1.012339.25000 757.51000 33.15981 37.85081 -0.977319.75000 758.52000 40.47033 37.11028 -0.9448010.25000 760.01000 37.30584 36.78727 -0.9145310.75000 760.75000 32.36145 36.18443 -0.8862611;25000 761.76000 33.83440 34.92720 -0.8597911.75000 762.50000 36.69708 35.73146 -0.8349412.25000 763.51000 38.24900 33.19398 -0.8115612.75000 784.25000 36.56733 32.83639 -0.7895313.25000 765.07000 30.36816 33.82743 -0.7687113.75000 765.50000 27.79694 33.49276 -0.7490114.50000 766.50000 32.60263 33.22815 -0.7213715.25000 767.25000 30.24218 32.87881 -0.6957716.00000 767.99000 32.53602 31.30233 -0.6719916.75000 768.74000 34.84104 31.89127 -0.8498317.50000 769.48000 30.88607 32.05207 -0.6291418.25000 769.99000 33.73485 31.28438 -0.6097719.00000 770.73000 27.55673 30.18581 -0.5915819.75000 770.99000 23.34734 30.37291 -0.57448

20.50000 771.49000 40.41950 29.85512 -0.5583621.25000 772.24000 39.06894 29.75090 -0.5431422.25000 772.74000 26.72085 28.67977 -0.5241323.25000 773.22000 21.23086 28.48296 -0.5064324.25000 773.48000 24.65156 28.54120 -0.4899125.25000 773.99000 33.76653 28.71951 -0.4744526.25000 774.49000 25.79328 26.18140 -0.4599527.25000 774.73000 23.98014 31.10344 -0.4463328.50000 775.23000 31.41352 26.52348 -0.43041Flow History

tp hours 15.33q, STB/D 174Well and Reservoir Parameters

B 1.06Ct. pSi-I 4.2x 10-6

h. ft 107c p 0.25p o . cp 2.5r w' ft 0.29

In t hi s ex amp l e. t he an al yzed da ta we re r eco rded du ri ng bu il dup .A s em i lo g s tr ai gh t l in e c ou ld d ev el op ( Fi g. 3 ) b ec au se t he d at a a rec or re ct ed f or b ui ld up e ff ec t w it h t he Ho rn er me th od . A c or re ct io ns ho ul d a ls o b e p er fo rmed f or t he l og -l og a na ly si s b ec au se t he typec urv es o f F ig . 1 a re d esig ned to d esc rib e d ra wd ow ns.F ig . 4 illu str ate s th e p re ss ur e r es po ns e d ur in g a n " id ea l" te st.T h e w e ll , f ir st a t i ni ti al p re ss ur e, Pi. is o pe ne d a nd p ro du ce d a tcons tan t r a te du ri ng tr T he n it is c lo se d f or b uild up , a nd a fte r in -fm ite sh ut-in tim e, th e p re ssu re w ill b e b ack a t Pi ( if t he s ys tembehav io r i s in fin it e-ac t ing) . In t erm s o f p re ss ur e c ha ng e, i t w i ll t he nS PE F orm atio n E va lu at io n, J un e 1 98 9

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. .. . . .

. . . . . . .

. . ./././

. . . .Fig. 2-A diagnostic tool: log-log plot of buildup data.

ta ke a n infinite s hu t- in t ime t o r ea ch a t.pBU o f t he s ame ampl it ud ea s th e p re ssu re d ro p a t th e e nd o f th e d ra wd ow n t.pDd(tp)' As ar es ul t, d rawdown an d b ui ld up c ur ve s a re n o t i de nt ic al . InF ig . 5 ,th e d otte d lin e c orre sp on ds to a d ra wd ow n ty pe c urv e. A fte r th ew ell is sh ut in a t tp. t h e r e su lt in g bu il dup r e spon se ( th ick l in e ) d ev i-a te s f ro m th e d rawd ow n ty pe c ur ve a nd f la tte ns to w ar d th e s am elevel as the last draw do wn-pressu re change before shut-in,t.pDd(t ). T his d ev ia ti on is m o re p ro no un ce d w he n th e f lo w tim eb ef or e ~ hu t- in is r ela tiv ely s ho rt, a s is o fte n th e c as e w he n e xp lo -r atio n w el ls a re te st ed .Inp rac t ic e , i t i s n o t po ss ib l e t o a s ce r ta in a p e rf e ct ly cons tan t f lowra te du ri ng d r awdown, e s pec ia l ly du ri ng t he i n it ia l i ns tan ts o f f low .Log -l og an a ly s is cons id e rs t h e g lob al r e spons e du ri ng a f low pe ri oda nd t he re fo re d o es n o t a c c ommoda te a ny r at e v ar ia ti on d u ri ng t hep er io d a na ly ze d. A s a r es ult, o nly b uild up s, r ec or de d o n s hu t- inw ells , g en er ally a re s uita ble f or ty pe -c ur ve m a tc hi ng . W h en th ein te rp re ta tio n is p er fo rm e d o n c omp ute r, th e b uild up ty pe c ur veis g e ne ra te d f or t he a ct ua l f low h is to ry b e fo re s hu t- in (mul ti ra tecurves):P D= ~~: [(q i-qi~I)/(qn-I-qn)][P D( ~~: ~ tjD )

-P D( ~~: ~tjD +~tD ) ]+ PD (~ tD )' (3)a nd th e m a tc h is p er fo rm e d o n th e e xa ct th eo re tic al r es po ns e. R e-c en tly p ub lis he d e xample s s ho w th is m a y b ec om e c ru cia l. IIDerivative of PressureF ig . 6 r ep re se nt s t he s ame r es po ns e a s i n F i g . 1 b u t w it h t he s em i lo gs lo pe o f t he d imen si on le ss p re ss ur e r es po ns e o n t he y a xi s, v s. t heu sua l d imen s ion le s s t ime g roup tDICD o n th e x axis. 10 The cu rv e s

Pi

Fig. 4-Pressure hIstory of a sImple drawdownlbulldup test.SPE FonnationEvaluation. June 1989

.000

3 7 ' 1 l O ~i! \I

'.: J . 5 O O \".. ..3250 '-................ . .3000

1 10 102 1Q3 1Q4(tp.,1U/AI

Fig. 3-Homer plot.

a re g en er at ed b y t ak in g t he d er iv at iv e o f t he p re ss ur e w it h r es pe ctto th e n atu ra l lo ga rith m o f tim e.d pD /[ d I n( tD ICD )] = ( tDICDH dpDI[d( tDICD )]} =( tDICD)pb .

.................................... (4)T he first ty pic al r eg im e o bse rv ed o n th e ty pe c urv e o f F ig . I isw ellb ore -sto ra ge e ffe ct. B y c om bin in g E qs. 1 a nd 4 , w e o bta in( tD ICD )p b = tD ICD . . . .. . . .. . .. . .. .. . .. . .. . .. . (5)As f or p re ss ur e, all t he d e ri va ti ve b ehav io r s a r e i den ti ca l a t e a rl ytim e, a nd th e c urv es m erg e o n a sin gle a sy mp to te o f slo pe e qu alt o u n it y.Wh en t he i nf in it e- ac ti ng r ad ia l f low r eg ime h as b ee n r ea ch ed -i .e ., a ft er t he l im i t " ap p ro ximat e s ta rt o f t he s em i lo g s tr ai gh t l in e"( Fig . I )- th e p re ss ur e b eh av io r is d es cr ib ed b y E q. 2 . T he s em ilo gs lo pe i s c o ns ta nt :( tDICD)pb =0.5, (6)

an d all t he d e ri va ti ve c u rv es me rg e t o a s ec on d a sympt ot e, t he o ne -h al f s tr ai gh t l in e. B e ca us e t he i nf in it e- ac ti ng r ad ia l f low p ro du ce sa c ha ra cte ris tic s tr aig ht lin e o n lo g- lo g s ca le , th e d er iv ativ e p lo tc an b e u se d in p la ce o f th e c on ve nti on al s em ilo g p re ss ur e p lo t f ort he a cc ur ate d ete rm in atio n o f /chIp..B etw een the tw o asym ptotes, and depending on the CDe2Sg ro up , e ac h c ur ve s ho ws a s pe cif ic s ha pe mu ch mo re p ro no un ce dt ha n t ha t o f t he u su al p re ss ur e c ur ve s ( Fi g. I ). T h er ef or e, t he d er iv a -t iv e m e th od is p ow e rf ul f or d ia gn os is a nd , in f ac t, it c om b in es o nth e sa me lo g-lo g p lo t th e g lo ba l a pp ro ac h b y ty pe c urv es a nd th ea c cu ra te s pe c ia li ze d a na ly si s o f r ad ia l f low . T h us , t he re i s n o n ee dfo r r e fmemen ts ; t h e ma t ch i s d i r ec t, s imp l if y in g t he an a ly s is p roce ss .P ro vid ed th at th e d ata s ho w w ellb or e s to ra ge a nd in fin it e- ac tin gr ad ia l f low r eg ime s, t he ma tc h i s u n iq ue b ec au se o f u ni qu e b e ha vi ora t b oth e nd s. T he c ur ve m a tc h is o bta in ed b y id en tif ic atio n o f th e

~ drawdow"~ .........../~uild-upLOG AI

Fig. 5-Drawdown and buildup type curves.295

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" " . . ..",... .",...

10 '". , . ,~ ?l l

. , . , 10 " '.F ig . 6 -D er lv atlv e ty pe c ur ve f or h omo ge ne ou s r es er vo ir . 10

c urv e fo llo win g th e d ata a t in te rm ed ia te tim e b etw ee n th e tw oasymptotes.F o r b ui ld up a na ly si s, t he s ame c ur ve s w e re f ou nd t o b e a pp li ca b le( Re f. 1 0) p ro vid ed th at th e d er iv ativ e is ta ke n, n ot w ith r es pe ct ton at ur al l og ar it hm o f t ime , b u t w it h r es pe ct t o n at ur al l og ar it hm o fthe H om er tim e, as m odified by A garw al 12:d p /{ d I n[ tp A tI (tp +At)]} = At[(tp +A t)/tp ](dp ldt). . ..... (7)These bu il dup de ri va ti ve r e spons e s, when p lo tt ed v s . t he ac tu a l s hut -i n t ime , At, m atc h o n th e type curve of F ig . 6. T his behavior isp re se nt w he n th e H om e r m e th od is v alid =- i.e ., th e d rawd ow n h asto r ea ch r ad ia l f lo w b ef or e s hu t- in .F ig. 7 presents the slope of the exam ple H om er plot (F ig. 3),p lo tt ed o n a l og -l og s ca le v s. s hu t- in t ime . T h e ma tc hi ng p ro ce du reon the type curve of Fig . 6 is as follow s.1. T he c on sta nt- de riv ativ e p ar t o f th e d ata p lo t is p la ce d o n th eo ne -h al f s tr ai gh t l in e o f t he t yp e c ur ve . T h e p re ss ur e ma tc h i s t h enf ix ed a cc ur at el y a nd kh l p . i s k nown.2 . T h e d at a p lo t i s d i sp la ce d a lo ng t he o ne -h al f s tr ai gh t l in e u nt ilwe ll bo r e- st or age d a ta ma t ch on t he ea rl y- time un it -s lo p e a symp t ot eo f th e type c ur ve . T he tim e m a tc h is n ow f ix ed , y ie ld in g th e w ell-b or e- st or ag e c on st an t, C .3 . A d ire ct re ad in g te lls w hic h CDe2S c ur ve p ro vid es th e b es tm a tc h b etw ee n th e tw o a sympto te s, g iv in g a cc es s to th e s kin f ac -tor, S .E x pe ri en ce h as s hown t ha t f or p ra ct ic al r ea so ns d is cu ss ed l at erw i th t h e d if fe r en ti at io n o f ac tu a l d a t a , i t i s conveni en t t o ma t ch bo thp re ss ur e a nd p re ss ur e- de ri va ti ve c ur ve s, e ve n t ho ug h i t i s r ed un -dant .IO W ith th e d ou ble m atc h, a h ig he r d eg re e o f c on fid en ce int he r es ul ts i s o bt ai ne d. T o i ll us tr at e t hi s, t he f in al ma tc h o f t he e x-am ple in Table 1 is show n in Fig. 8.T h e s ki n c oe ff ic ie nt i s n o l on ge r p re se nt o n d er iv at iv e r es po ns eswh en t he i nf in it e- ac ti ng r ad ia l f low con fi gu ra ti on i s r ea ch ed ( onth e o ne -h alf lin e), a nd a s a re su lt, S c an be e st ima te d f rom o nl yt he d er iv at iv e CDe2S ma tc h d ur in g t he t ra ns it io n b etwe en t he two

. . . .C o o "a-. . .c

10 . . .. . .. , ~ . , . .li

11>'

F ig. 8 - T he com bined m atch.29 6

. , ., . 10 ,,)lF ig. 7-M atch of the d erivative of actual data.

a sympto tic r eg im e s. T his p ro pe rty o f th e d er iv ativ e p re se nts , ins ome c as es , i nt er es ti ng f ea tu re s f or t he i nt er pr et er . F o r e x amp le ,w h en a w e ll i s t e st ed b ef or e a nd a ft er s timul at io n, i f t h e w e ll t re at -ment h as n ot mo di fi ed t he c ha ra ct er is ti cs o f t he p ro du ci ng z on es ,d er iv at iv e b eh av io rs r ec or de d d ur in g b ot h t es ts s ho ul d ma tc h e x-a ct ly wh en t he d at a c ur ve s a re f re e o f a ny w e ll bo re -s to ra ge e ff ec t.Th e l imi ted i nf lu ence o f t he s k in co ef fi ci en t on de ri va ti ve r e spons e swill be o f i n te r es t f o r t he an a ly s is o f h e te rog eneous fo rma t io n s andf or t he i de nt if ic at io n o f b ou nd ar y e ff ec ts . A s d is cu ss ed l at er , t het rad it io n al f low regimes p roduce a ch ar act er is ti c s h ap e much fa st erth an o n th e u su al p re ss ur e c ur ve s.O th er a pp lic atio ns o f th e d er iv ativ e o f p re ss ur e h av e b ee n p ro -p os ed f or o bs er va ti on w e ll s 13 a nd f ra ct ur ed w e ll s 14 th at u se th ed er iv at iv e o f p re ss ur e w it h r es pe ct t o e la ps ed t ime . In t he me t hodp r es ent ed h e re , i t is p ref e rab le t o cons id e r t he d e ri v at iv e a s t he s emi-l og ( or Homer /s up er po si ti on ) s lo pe f or t he f ol low in g r ea so ns :1 . T h e s er ni Io g d er iv at iv e emph a si ze s t he i nf in it e- ac ti ng r ad ia lf low r eg ime o f p rime i nt er es t i n w e ll -t es t i nt er pr et at io n.2 . W h en th e d er iv ativ e is c on sid er ed a s th e s lo pe o f th e s em ilo go r t he s up er po si ti on p lo t, b ot h t he p re ss ur e c ha ng e a nd t he p re ss ur ed er iv ativ e a re m a de d im e ns io nle ss b y u se o f th e s am e g ro up (kh /

141.2qBp. i n u s ua l o il fi el d un i ts ), mak ing t he doub le ma t ch p r ac ti ca l.3 . T h e d er iv at iv e w it h r es pe ct t o t h e Homer /s up er po si ti on f un c-t io n conve r ts bu il dup anal ys is t o t ha t o f d r awdown, s imp l if y in g t heanalys is p rocess .4 . Bu il du p a na ly si s r ev ea ls a n a dd it io na l a dv an ta ge i n t h e u se o fth e Homer/ superpos i tion der ivat ive of pressure: the resul ting curvesa re n ei th er c omp re ss ed o n t he t ime a xi s, a s f or t ra di ti on al H omer /s upe rpo s it io n an al ys is , n o r on t he p r es su r e ax is , a s fo r bu il dup p r es -s ur e t yp e c ur ve s. T h e d e ri va ti ve d is pl ay s t he f ul l ampl it ud e o f t hes ig na l a nd t he re fo re impr ov es t he s en si ti vi ty o f t he a na ly si s p lo ts .5 . T h e n oi se a pp ar en t in t he d e ri va ti ve d a ta can be r edu ced whent he s upe rpo s it io n func ti o n i s u s ed becau se t he s lo p e ( and t h e d e ri va -tiv e) w ill n ot te nd to wa rd z er o d ur in g th e in fin ite -a ctin g p er io d .

L (2)

.A X I t : .X2,.

F ig . 9 -D lf fe re ntla tlo n a lg or ith m u sin g t hr ee p oin ts .

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~ r - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - '10 . .~

~ , . . ......t . . . . .~_.-., . .-. :,_".,.......,. ~.-==... . .10 ." . , . . . . . . . . . . " ". . . .

Fig. 10- Test of the differentiation algorithm on noisy data.

Differentiation AlgorithmT he m ai n c on ce rn w he n a ctu al d ata a re b ein g d if fe re nt ia te d i s toimprove the s ignal -t o -no is e r at io . Some no is e will a lways be p re sen tb ec au se o f g au ge r es ol ut io n , e le ct ro ni c c ir cu it ry , v ib ra ti on s, e tc .D i ff er en ti at io n i s d if fi cu lt , i f n ot i nc on cl us iv e, f or t he r el at iv el yh ig h n oise lev el ass ociated w ith a lo w s am plin g ra te. T his is fre-q ue nt ly t he case w i th m e ch an ic al g au g es . wh ic h a ls o p ro d uc e n oi seo n b oth p res su re an d tim e ax es.S ev er al a pp ro ac he s f or d if fe re nt ia ti ng d at a h av e b ee n t ri ed (Ap-p en dix B ). B ec au se t he c or re ct r es ul t is n ot k nown w he n w or kin gw it h a ct ua l d at a, m odi fi ed type c urv es w ere u sed to e valu ate th ediffe rent methods . A r andom no ise bo th p r opo rt ional t o an d indepen-d en t o f t he a mp li tu de o f th e P D s ig na l w as a dd ed to th e type curve,a nd th e n um be r o f p oi nt s g en era ti ng t he P D c ur ve w as r ed uc ed b ya ra nd om s am pl in g p ro ce ss .P re fe rr ed A l go ri thm . Th e a lg or it hm p re se nt ed h er e i s s im p le , a nd- i s t he b es t a da pt ed t o t es t i nt er pr et at io n n ee ds . T h is d if fe re nt ia ti ona lg or it hm re pro du ce s th e te st ty pe c ur ve o ve r th e c om ple te t im ei nte rv al b et te r th an o th ers . It u se s o ne p oi nt b ef ore a nd o ne p oin ta ft er t he po in t o f i nt er es t, i, ca lculates the cor responding der iva t ives ,an d p la ces th eir w eig hted m ean at th e p oin t co ns id ered (F ig . 9).

( d p / d X) ; = = [ ( L \ J J , / A X1)A X 2+ ( L \ J J z / A X z )A X ,J /(A X , + A X z ),. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

where 1==pointbefore i, 2==po in t a f te r, a nd X= t ime func ti on (In IUf or d rawdown, modi fi ed Home r, o r s up er po si ti on t im e s e xp re ss edin n at ura l lo ga rit hm fo r b uil du ps ). T he s up er po si ti on fu nc ti on i sw ri tt en a s1/(qn-qn-l{ :~: (q;-q;_l)ln( :~: Atj +IU ) ] +In(At).

..................... (9)When c on se cu tiv e p oi nt s a re u se d fo r t he c alc ul at io ns o f Eq. 8,th e d er iv at iv e c urv e is fre qu en tly s ca tte re d a nd c an no t b e u se d fo ra na ly si s. T h is i s t ru e wh en t he p re ss ur e p oi nt s are r ec or de d a t h ig hs am pli ng ra te , s uc h a s w ith e le ct ro ni c g au ge s (r ea di ng s e ve ry fe ws ec on ds ) a nd wh en t he p re ss ur e v ar ia ti on s b ec om e c lo se t o t he reso-l ut io n o f t he s en so r. N o is e e ff ec ts a re r ed u ce d b y c ho os in g t he p oi nt sw he re th e d eri va ti ve i s c alc ula te d s uffi ci en tl y d is ta nt fr om P oi nt

i. T hi s is e ffi cie nt i n r em ov in g th e n oi se b ec au se it in cre as es t hep re ss ur e v ar ia tio ns c on sid ere d. I f th ey b ec om e to o d is ta nt , h ow -e ve r, th e s ha pe o f th e o ri gin al type c ur ve w il l b e d is to rt ed . T h er e-fo re, a co mp ro mis e m us t b e m ad e b etw een th e sm oo th ness o f th ed er iv at iv e a nd th e p os si ble d is to rti on o f th e p re ss ure re sp on se .T he m in im um d is ta nc e c on si de re d b etw ee n t he a bs ci ss a o f th ep oi nt s a nd t ha t o f P oi nt i,L, is e xp re ss ed i n te rm s o f t h e ti me f un c-ti on . T he d iffe re nt ia tio n a lg ori thm s ele cts P oi nts 1 a nd 2 a s b ein gth e first o nes s uch th at A Xl,2 >L ( Fig . 9 ).B ec au se o f t he c om pre ss io n e ffe ct a t l at e ti me s o n th e s em il ogs ca le (m ore p ro no un ce d o n H om er a nd s up erp os iti on p lo ts w he nbuildups are c on si de re d) , t he smoo th in g e ff ec t o f a g iv en L valueS PE F orma ti on E va lu at io n, J un e 1 98 9

-1 2 3. . .

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...... mpJeA.... .:amp,. 8

~ 2 r - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ - - - - - - - - - - - - - - ~.l.~2S ~- - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~

Fig. 12-Derlvative type curve for double-porosity reservoir (pseudosteady-state blocks tofissure fiOW}.17

10

When t he c om p le te r ec omm en de d p ro ce du re i s u se d, t he d is to r-t io n s p ro d uc ed b y t he d if fe re n ti at io n a lg o ri thm p re se n te d h er e a rep ra ct ic al ly i nd ep en de nt o f t he p oi nt d en si ty i n t h e c ur ve . T he s am ee ff ec t i s e x p ec te d t o b e p ro d uc ed o n b o th d at a a nd t he o re ti ca l c ur ve s,a s o p po se d t o t he a lg o ri thm s t ha t u s e a ll t he p o in ts p re se nt i n a g iv ent im e i nte rv al f or smo ot hi ng .Some o f the i rr egu la r it ie s obser ved in the de ri vat iv e behav ior we r ef ou n d t o b e p ar t o f t he r es er vo ir r es po n se . F o r e x amp le , o sc il la ti on so f p re ss ur e c au s ed b y t id a l e ff ec ts a re emph a si ze d b y t he d er iv at iv ea t la te tim e, w he n th e s ig na l is b are ly c ha ng in g.A n ot he r a dv an ta ge i s t ha t t he d eri va tiv e w o ul d s ti ll g iv e r es ul tsw hen the last flo win g p ressu re is m issin g, as w hen the gau ge isru n a fte r sh ut-in o r in so me c as es o f c ha ng in g wellbore storage.The p(~t=O) p oi nt i s n ot n ee de d t o p ro du ce t he d er iv at iv e c urv e;th us, p ro vid ed th at en ou gh d ata are a va ila ble , a u niq ue m atc h isp os si bl e a nd t he s ki n i s a cc es si bl e. T h e d er iv at iv e p lo ts a ls o t en dt o c omp e ns at e s ta rt in g -t im e e rr or s e n co u nt er ed wh en s h ut -i n t im ei s n o t a cc ur at e e n ou g h c ompa re d w i th t he p re ss ur e- ga u ge s amp li ngfre qu en cy . In a dd itio n, fo r g as w ells , th e d iffe ren tia l o f th e re algas potentia l m(p) 15 re pl ac es t he c al cu la ti on o f a n i nt eg ra l b y t ha to f a p ro du ct.

dm(p) /d In(.;;lt)= {2p /[JL(p )z(p )]}M Ap '. . (10)Application to Heterogeneous Reservoir BathingRe ce n t t he o re ti ca l d ev el opm en ts a n d r el at ed p u bl ic at io n s d emon -s t ra te a gener a l o i l i ndus t ry in te r es t i n thebehav io r o f he te r ogeneousformations. In f ac t, it i s o u r e xp er ie nc e, b as ed o n a v er y l ar ge n um -b er o f w ell te sts , th at in s om e are as, u p to 30 or 40% o f t he w el lss h ow a h et er og e ne o us b eh av io r. T h is i s e v id e nt wh en h ig h -a cc u ra cyp re ss ur e d a ta , h ig h -d e fi ni ti on a n al ys is t ec hn iq u es s u ch a s p lo ts o fthe d er iv at iv e o f p re ss u re , a n d c ompu te r- ai de d i nt er pr et at io n a reu s ed . T h e c omb in e d r ec en t p ro g re ss i n d at a a cq u is it io n , d a ta p ro c-e ss in g, a nd c om p ut in g t ec hn iq ue s o ff er s n ew p ro sp ec ts f or t he i n-t er pr et at io n o f w e ll -t es t d at a. Mu c h mo re i nf orm at io n i s p u l le d o u to f t he we ll d u ri ng t od a y's t es ts . I nt er pr et at io n s h ou ld m a ke f ul l u seo f a ll d at a a va il ab le f or a na ly si s.F ig . 1 1 p re se nts a ty pic al d ra wd ow n lo g-lo g p lo t o f IIp vs. ~ .F ou r d ifferen t tim e p erio ds can b e iden tified in th e p ressu reresponse.1. Th e wellbore-storage e ff ec t i s a lw ay s t he fi rs t f lo w r eg im e t oappear.2 . E v id e nc e o f w e ll a nd r es er vo ir h et er og en e it ie s t he n m a y f ol low .Su ch b eh av io r m ay be a resu lt of th e effects of a fractu red w ell,a p a rt ia ll y p e ne tr at in g we ll , a f is su re d f orm at io n , o r a mu lt il ay er edreservoir .3 . A f te r s om e p ro d uc ti on t im e , th e system s ta r ts t o e x hi bi t a r a di a lf lo w b eh av io r, r ep re se nt in g a n e qu iv al en t h om o ge ne ou s s ys te mc om p os ed o f a ll p ro du ci ng e le me nts .4 . B ou nd ary e ffec ts m ay o ccu r at la te tim e.298

T hu s, m a ny t yp es o f f lo w r eg im es c an a pp ea r b ef or e ( an d a ft er )th e a ctu al s em ilo g s traig ht lin e d ev elo ps , an d th ey fo llo w a v erys tr ic t c h ro n ol og y i n t h e p re ss u re r es p on s e. On ly a g lo b al d ia g no s is ,w i th i de nt if ic at io n o f al ] s u cc es si ve r eg im e s p re se n t, w i ll i nd ic at ee xac t ly when conven ti ona l a naly si s , l ik e the sem ilog p lo t t e chn ique ,i s j u st if ie d. F u rt he rm o re , t he o th e r c ha ra ct er is ti c r eg im e s c an bea na ly ze d t o p ro v id e mu ch mo re t ha n j u st kh, S, andp* , a s i ll u st ra t edbelow.DoublePoroslty ModelsO ne freq uen tly en cou ntered ty pe of h etero gen eo us respo nseis d ou ble-p oro sity b eh av io r, w hic h is p ro du ced b y fis su re d re s-erv oirs. T wo m odels o f d ou ble-p oro sity b eh av io rs hav e b eenstudied 16-18.21: o n e a ss ume s pseudosteady-state in te rporosity f low;the o the r a s sume s t ra n si e nt i n te r po r os it y f low . Bo th mode ls are con-s id er ed h er e, a nd t he a dv an ta ge o f t he d er iv at iv e p re se nt at io n i nd is ti ng u is hi ng v a ri ou s t yp es o f h et er og en e ou s r es po n se s i s s hown .Pseudosteady-State In terp oro sity F lo w M od el. T he p lo ts o f th es em ilo g s lo pe s o f e xa mp le s are p re se nted in F ig . 12.17 F or E x-a mp le A , w=1.0, >..e-2S=3x 10-4, (CDe2Sk+ma= 10-1; a n d f orE xa mp le B , w=O.I, > . . e -2S=10-7, (CDe2 )/+ma=104 Duringt he h omog en eo u s r eg im e s, t he r es po n se f ol low s a d er iv at iv e CDe2Sc ur ve , wh er ea s a t t ra ns it io n t im e , t he f la tt en in g o f t he p re ss ur e b e-h av io r i s c ha n ge d i nt o a v er y c h ar ac te ri st ic d ro p o f t he d er iv at iv e.T he t ra ns it io n r eg im e i s n ow d es cr ib ed b y t wo f am il ie s o f c ur ve s( Ap pe nd ix A o f R ef . 9) : e a rl y t ra n s it ion i s def ined by the d imens ion -l e ss g r oup (>"CD/+ma)/[w(I-w)] a nd l at e t ra n si ti on b y (>..CD/+ma)1(1-w). If the storage effect is presen t at the start o f th e tran si-t io n ( Ex am p le B ), th e r es po ns e d ev ia te s f rom t he c or re sp on di ng(>"CD/+ma)/[w(I-w)] curve (1.11 x 10-2 i n t hi s c as e) , b u t s to ra geb ei ng o ve r a t l at e t ra ns it io n t im es , t he m at ch o n (>"CD/+ma)/(I-w)(1.1lXIO-3) i s g oo d.T he d ou bl e- po ro si ty m o de l i ll us tr at es t he g ai n i n s en si ti vi ty o fth e d eri va tiv e a pp ro ac h. T h e f la tt en in g o f t he p re ss ure re sp on sed u ri ng t ra ns it io n i s g en er al ly d if fi cu lt t o i de n ti fy o n a l og -l og s ca le .Inm a ny c as es , a s em il og s ca le h as t o b e u se d f or r efi ni ng t he p re s-s ur e c u rv e m a tc h . W i th t he d e ri va ti ve p lo t, t he h et er og en e ou s n at ur eo f th e r es po ns e i s o b v io us , e li mi na tin g t he n ee d f or a ny f ur th er p lo tf o r a d ju s tmen ts .Table 2 c on ta in s f ie ld d at a fr om a p re ss ur e b ui ld up r ec ord ed ina f is su re d f orm at io n . T h e d e ri va ti ve o f p r e ss ur e s ug g es ts t he h e te ro -g en eo us b eh av io r, a nd th e c om b in ed l og -l og p lo t o f p re ss ur e a ndd e ri va ti ve ( Fi g. 1 3 ) i s m a tc h ed a ga in st t he d u al -p o ro si ty t yp e c ur veo f R ef. 1 7. T he b uild up c urv e s ho wn w as g en era te d w ith th e flo wh is to ry b ef or e s hu t- in (m ul ti ra te c ur ve ). T h e d if fe re nt ia l w a s t ak ena s th e s lo pe o f t he s up er po si ti on p lo t. T h e d ou bl e- po ro si ty m o de lu se d p ro vid es a fa ir ly g oo d d es cr ip ti on o f t h e r es po ns e, d es pi te t hed is cr ep an cy d ur in g p ar t o f th e t ra ns it io n. R es ul ts o f a na ly si s a rep re se nted in A pp en dix C .

SPE Formation Evalua tion. June 1989

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T ABLE 2 -P RE SSURE CHANG E VS. ELAPSED TIMEpeA t = 0) = 7 ,2 48 p sl g ( co nti nu ed )E la ps ed T ime(hours)un

14 514 614 714 814 915 015 115 215 3

P res su r e Change558.70559.49560.22560.96561.61562.29562.89563.57565.28

31.64132.50733.37134.23635.10135.96636.83137.80040.424

Flow His to ryRun12345

Duration Flow Rates(hours) (STB/D)3.0000 3945.01.5000 1265.01.7500 1470.06.7500 880.0042.000 0.00000

Well and Rese rvo ir P"r"m"'''rcBCt. psi-1h. Itc f>1 1 - , cpr.; It

1.35x 10-6200.081.30.29

~ r - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,

. : - . . . . . . . '.~

. ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~. '" 10'F ig. 1 3-C om blne d m atc h o f do ub le-p oro sity d ata.

T ra ns ie nt I nte rp or os ity F lo w Mo de l. T he d er iv ativ e re sp on se o fth e tra nsie nt inte rp oro sity flo w so lu tio ns is differe nt fro m thep s eudos te ady -s ta te cu rv e s: a s fo r p r es su r e, t he d e ri va ti ve P ' transi-t io n c ur ve s a re o bt ain ed b y d is pl ac in g h omo ge ne ou s CDe2S curvesb y a f ac to r o f tw o a lo ng p re ss ur e a nd tim e a xe s. 1 9 Th e th eo re tic als e rn il og s tr a ig h t l in e o f t he t ran si ti on r eg ime ( sl op e one-h al f o f t ru es emil og ) i s t hen ch anged i nt o a cons tan t- d er iv a ti ve 0.25 lin e. O nd rawdown re spons es , t he t ran si en t model do es no t s how a de ri va ti vepo in t b e low 0.25 d urin g tr an sitio n, a s illu str ate d in F ig . 1 4.F ig . 1 4 p r e se nt s f or t he s ame p ar ame te rs (CDe2S)f +ma=I, P ' =10-4, and w=10-2, the d ra wd ow n re sp on se p ro duc ed b y tw oma tr ix g eomet ri es , s la bs a nd s ph er es . T ho ug h t he p re ss ur e c ur ve sloo k id entic al, th e d eriv ativ es a re d iffe re nt; th e sp he re m od elr es po ns e d oe s n ot r ea ch th e 0.25 s tr aig ht lin e b ut r em a in s a bo veit, whe re as th e sla b c ur ve is ta ng en t to it b ut d oe s n ot fo llo w it f ora s ig n if ic ant du ra ti on .Fo r bu il dup s, t he d e ri va ti ve du ri ng t ran si ti on r egime may exhi bi ta low er value, dow n to 0.20, if th e p rev io us d ra wd ow n h as n otr ea ch ed to ta l s ys tem f lo w a t s hu t-in tim e . 1 9 S im ila r d is to rtio nsh av e b ee n o bse rv ed w ith a p se ud os te ad y- sta te m o de l. 1 7T he d is cu ss io n o f th e d er iv ativ e c ur ve s o f F ig . 1 4 illu str ate th ed ra wba ck o f the m etho d tha t c on sists o f d ra wing in te rm ed iates tr ai gh t l in es o n p re ss ur e p lo ts . I t i s a l wa ys p os si bl e t o f in d s ev er alr ea so na bl y s tr ai gh t p or ti on s o n a s ta nd ar d p re ss ur e c ur ve p lo tt edon any s cal e. Th is do es no t mean t ha t an an al ys is o f t he i nt ermedi at e-s tr ai gh t- li ne c ha ra ct er is ti cs i s j us ti fie d. T h e d er iv ati ve a pp ro ac h30 0

~ r - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,'"

. , . . 10 10' ." 10< ""Fig. 14-E ffect of block geom etry on double-porosity re-spo nses (tra nsie nt b lo ck s to fissure flo w).

p ro du ce s a z oom e ff ec t o n s ma ll p re ss ure c ha ng es a nd th er ef or ecan a s ce r ta in t he p re s en ce o f " st ra ig h t- li ne b ehav io r" and a ls o g iv ea cc ur at el y i ts t ime l im it s.T he s el ec ti on o f t he b es t s ol ut io n b etwe en t he d ou bl e- po ro si tymodels, pseudosteady-state or t rans ien t in terporosi ty f low, i s gener-a ll y s tr ai gh tf o rwa rd ; w i th t he pseudosteady-state mode l, t he d ro pof der ivative dur ing t rans it ion i s a funct ion of the t rans it ion durat ion .Long t ran si ti on r egimes , co rr e spond ing t o sma ll ( . o J values , p roduce(Fig. 13) de ri va ti ve l ev e ls much sma ll er t han t he p r ac ti ca l 0.25 limito f t he t ra ns ie nt s ol uti on . A n ambi gu ity m ig ht o cc ur wh en t he t ra n-s it io n r eg ime is o f s ho rt d ur at io n. I n s uc h c as es , pseudosteady-statecu rv e s (gen er at ed w i th a l arg e w value) can produce s imilar t rans ien ts ol ut io ns , g en er at ed w ith a smal le r w v alu e ( on th e o rd er o f 10-2or 10-3 o r le ss ). K nowle dg e o f th e r es er vo ir g eo lo gy w ill h elpd ec id e b etwe en t he d if fe re nt f is su re s to ra ge f ig ur es .Cinco-Ley et al . 20 h av e s hown t ha t t he p se ud os te ad y- st at e b e-h avi or can be de ri ved f rom the t ran si en t i nt erporo si ty f low sol ut io nby addi ng a sk in e f fe c t on t he s ur face o f t he ma tr ix b lo cks . I t ju s ti fi esa posteriori th e u se of tw o a pp are ntly diffe re nt m ode ls fo r th ed e sc ri pt io n o f f is su r ed - fo rma ti on r e spons e s. Howeve r, C i nco -Leyet al. 's th eo ry su gg ests th at th e p aram eters o btain ed fro m th epseudosteady-state hypo th e si s ( a ls o ca ll ed r es tr ic ted i nt erporo si tyf low ), a s d ef in ed o ri gin al ly b y Wa rr en a nd Ro ot ,l 6 a re n ot a lw a ysapp li cab le . In par ti cular, A s hou ld i nco rpo ra te t he ma tr ix s k in fac-tor.2l ( In fo rma tio n o n o th er w e ll a nd r es er vo ir c on fi gu ra ti on s a ndo th er a pp lic atio ns o f th e d er iv ativ e o f p re ss ur e a re p re se nte d inRef. 22.)

",.

ConclusionsT ra ns ie nt t es t i nt er pr eta ti on t ec hn iq ue s h av e b ee n r ed uc ed t o th ei den ti fi ca ti on o f ch ar act er is ti c r egimes t ha t p roduce a s tr a ig h t l in ewh en th e p re ss ur e is p lo tt ed v s. ti me o n v ar io us s ca le s: r ad ia l f lowwith p v s. 1 0g (A t) , w e ll bo re s to ra ge a nd p se ud os te ad y- st at e w it hAp vs . At, lin ea r f lo w w ith Ap vs . --.f i t, e tc . W ith m od em c om -p utin g f ac ilitie s, th er e is n o r ea so n to lim it th e p re ss ur e a na ly sisto th ose r estr ic te d p or tio ns o f th e d ata d urin g w hic h a d eriv ativ eis c on st an t. S uc h t yp es o f d at a, c or re sp on di ng to p ur e s pe ci fi c r e-g im es , a re o fte n a bs en t.The me thod p re s en ted i n t hi s p ape r cons id e rs cons tan t d e ri va ti ve sa nd c ha ng es o f s lo pe w it h a h ig h d ef in iti on . T he se tr an si ti on al b e-h avi or s a r e i gno red on convent io n al s tr ai gh t- li ne p lo ts and a re o ft enf ea tu re le s s on l og - lo g p r es su r e-vs .- time g raphs . A d iagno s is i s p e r-f ormed , w it h imp ro ve d s en sit iv it y, o n t he g lo ba l r es po ns e; t he v ar -i ou s f low reg imes a re i den ti fi ed , a cco rd ing t o a l og ica l ch ronol ogy .New anal yt ic a l s o lu ti on s a r e n eed ed for g ene r al r e se rvo ir model -i ng t o i nt eg ra te ch ar act er is ti cs n egl ec ted i n t rad it io n al s imp li fi edsolutions.T he c on clu sio ns a re a s f ollo ws :I . Th e de ri va ti ve appro ach improves t he d e fi ni ti on o f t he an al ys isp lo ts a nd t he re fo re th e q ua li ty o f t he i nt er pr et at io n.2 . T h e d if fe re nt ia tio n o f a ctu al d at a h as t o b e c on du ct ed w it h c ar eto r em o ve n oise w ith ou t a ff ec tin g th e s ig na l. T he d eriv ativ e a p-pro ach d oes n ot p ro du ce e rro rs or n oise b ut on ly re vea ls th em .3 . The i nt erp re ta ti on o f p re s su r e d e ri va ti ve i s a s in g le -pl ot p roce -d ur e. I f e no ug h d ata a re a va ila ble , p re ss ur e a nd tim e m a tc he s a re

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f ix ed , s o an al ys is i s f a st e r. Th is i s impor tan t f o r r e al -t ime i nt erp re -t a tion dur ing wel l -t est moni to r ing . Quick decis ions dur ing tests saver ig t ime .Nomenclature

B = F VF, R B/S TB [res m3/ st oc k- ta nk m 3]cI = t ot al comp re s si b il it y, p s i-I [ kP a - 1C = we ll bo re -s to ra ge c on st an t, b bl /p si [m3lkPa)CD = d imen si onl es s s to r ag e cons tan tFy = r a ti o o f t ot al v o lume o f one po rou s s y st em to bulk volumeh = fo rmation th ickness , ft [m ]k = p ermea bi li ty , mdKo = mo dif ie d B es se l f un ctio n, s ec on d k in d, z er o o rd erKI = modi fi ed Be ss el f un ct io n, s ec on d k in d, f ir st o rd erL = d imen s ion le s s d i st an ce on X ax is o f s emil og an al ys is p l otm = a bs ol ut e v al ue o f s em i lo g s tr ai gh t- li ne s lo pe ,ps i /cycle [kPalcycle]m(p) = r ea l g as p ote ntia l, p si2 /c p [ kP a2/Pa's]

P = p re ss ur e, p si [ kP a]P * = ex t rapo la ted pressurePD = dimens ion less pressureP D = Lap lace t ran sfo rmed d imen s ion le s s p r es su r ePi = i ni ti al r es er vo ir p re ss ur e, p si [ kP a ]Ap = pressure change, psi [kPa) .q = f lo w r ate , S TB /D [ sto ck -ta nk m 3/d ]rw = well bor e r adi us , ft [m ]s = L apl ac e s pa ce v ar ia bl e c or re sp on di ng t o tDICDs' = L apl ac e s pa ce v ar ia bl e c or re sp on di ng t o toS = v an E ve rd in ge n a nd H u rs t s kin f ac to rso = d imen s ion le s s t imetp = p roduct io n t imeA t = e la ps ed t ime , h ou rsX = t ime func ti ona = b lo ck s ha p e p ar ame te r, ft -2 [m -2]{i' = o'[(CDe2S)/+ma)IAe-2Sl'= e xp on en ti al o f E ule r c on sta nt ( - 1 .7 8)o ' = b lo ck sh ap e f act or (1 .8914 fo r s l ab ma t ri x b lo cks , 1 .0508f or s ph er ic al ma tr ix b lo ck s)o = a ng le b etw ee n two i nt er se ct in g s ea li ng f au lt sI( = r a ti o o f p e rmeab il it y- th i ckn es s p roduc tsA = p s eudos te ady- st at e i nt erporo si ty f low pa rame te r ,

ar~(kma/k/)P = v is co si ty ,c p [ Pa -s ]t/> = p or os ity o f o ne s ys temw = s tora tiv ity ra tio , (t />Vc,)//[(t />Vc,)/+(t/>Vc,)mal=

[(CDe2S)/+mal/(CDe2s)/SubscriptsD = dimensionlessf = fissuref +rna = t o ta l s y st emi= p o in t o f i nt er es trna = matrixM = match

AcknowledgmentsWe a re g ra te fu l t o t he mana g ement o f F lo pe tr ol -J oh ns to n f or p er -m issio n to p ub lish th is p ap er. A pp re cia tio n is a lso e xte nd ed toA . A la go a, B . B uc ha na n, G . C la rk , M . C olv in , V . K nia ze ff, a ndA . T e ng ir se n k f or t he ir a ss is ta nc e d ur in g t hi s s tu d y. Inparticular,T . Wh it tl e' s ( now w it h SS I) p a rt ic ip at io n i n t hi s w o rk a n d h is v er yuseful com ments before finalization of the m ethod are ac-knowledged.References1. Ea rl ou g he r, R .C . J r. : Adva n c es i n We ll T e st A n a ly s is , Monog ra p h Se r-i es, S PE , R ic ha rd so n, T X ( 19 77 ) S.2 . H o rn er , D .R .: " Pr es su re B u il d-Up i nW e l ls ," Proc. , Th ir d Wo r ld P et .C o ng ., E .J . B ri ll , L ei de n ( 19 51 ) 11, 503.3 . R am e y, H J. J r.: " Sh or t- Tim e We ll T es t D at a I nt er pr et at io n i n t h e P re s-e nc e o f S ki n E ff ec t a nd We ll bo re S to ra ge ," JP T ( Ja n . 1 970 ) 97 -10 4;Trans. , A IM E , 2 49 .S PE F orma ti on E va lu at io n, J un e 1 98 9

4 . A ga rw al, R .G ., A l- Hu ss ain y, R ., a nd R am ey , H .J. Jr .: "An Investi-g a ti on o f We l lb o re S to r ag e and S ki n E ff ec t i n U n st ea dy L iq ui d F low :I . Ana ly t ic a l T reatmen t, " SPEJ ( S ep t. 1 970 ) 2 79- 90; Tr an s . AIME,249.5 . G r in ga rt en , A .C . e t a l. : "A C omp ar is on B et we en D if fe re nt S ki n a ndWe ll bo re S to ra ge T yp e C u rv es f or E ar ly -T im e T ra ns ie nt A n al ys is ,"p a pe r SPE 8205 p re se nt ed a t t h e 1979SPE Annua l Technical Conferencea nd E xh ib it io n, L as V eg as , S ep t. 2 3- 26 .6 . G r in g ar te n, A .C. : "R es er v oi r L im i tT e s ti ng f or F r ac tu r ed We ll s, " paperSPE 7452 p re se nt ed at t he 1978 SPE Annua l T e ch n ic a l C onf er en c e andE xh ib it io n, H o us to n, O ct . 1 -3 .7 . C in co -L ey . H . and Saman ie g o-V ., F .: "E ff ec t o f We ll bo re S to r ag e andDama ge o n t he T ra ns ie nt P re ss ur e B eh av io r o f V er ti ca ll y F ra ct ur edW e ll s," p ap er S PE 6 75 2 p re se nt ed a t t h e 1 97 7 S PE An nu al T ec hn ic alC on fe re nc e a nd E xh ib it io n, D en ve r, O ct . 9 -1 2.8 . B ou rd et , D . a nd G r in ga rt en , A .C .: " D et erm in at io n o f F is su re V o lum eand B loc k S i ze i n F r ac tu re d R e se rv o ir s b y T ype -Cu rv e Anal ys is , .. paperSPE 9293 p r es en te d a t t he 1980 SPE Annua l T e ch n ic al C onf er en c e andE xh ib it io n. D al la s, S ep t. 2 1- 24 .9 . B our d et , . 0 . Ay ou b, J .A ., a nd P ir ar d, Y .M .: "U se o f P re ss ur e D er iv a-tiv e in W ell T est I nte rp re ta tio n," p ap er S PE 1 27 77 p re se nte d at the1 98 4 C al if or ni a R eg io na l M e et in g, L on g B ea ch , A p ri l 1 1- 13 .1 0. B ou rd et , D . et aI.: " A N ew S et o f T y pe C ur ve s S im plif ie s W ell T estAnalysis," World Oil (M ay 1 98 3) 9 5- 10 6.1 1. J ai n, A . and Ayoub , J .: " P re ss ur e Bu il d-Up inGas-Lift OilWel ls , Fa lahF ie ld , O f fs ho r e Dub ai ," JP T (M ar ch 1 98 4) 4 66 -7 4.1 2. A g arwa l, R .G .: " A N ew Me th od t oA c c ou nt f or P ro du ci ng T im e E ff ec tW h en D raw dow n T yp e C u rv es a re U se d t o A n al yz e P re ss ur e B ui ld upa nd O th er T es t D ata ." p ap er S PE 9 28 9 p re se nte d a t t he 1 98 0 S P E A n-n ua l T ec hn ic al C o nf er en ce a nd E xh ib it io n, D al la s, S ep t. 2 1- 24 .1 3. T iab , D . an d K um ar, A .: "A pp licatio n o f the pI , Func ti on t o I n te r-fe rence Ana lys is , " JP T ( Au g. 1 98 0) 1 46 5- 70 .1 4 . P u ti ga i, S .R . and T ia b, D .: " Pr es su re D e ri va ti ve T yp e C u rv es f or V er -t ic a lly Fractu red Wel ls ," SPEFE(Mar ch 1988) 156-58 ; Trans . , AIME,285.IS . A l-H ussain y, R ., R am ey, H .J. Jr . an d C raw fo rd, P.B .: "T he F lo wo f R eal G ases T hrou gh P oro us M ed ia," JP T (M ay 1 96 6) 6 24 -3 6;Trans. , A IM E, 2 37 .1 6. W a rr en , J .E . a nd R oo t, P J. : " Be ha vi or o f N a tu ra ll y F ra ct ur ed R es er -voirs," SPEl ( Se pt . 1 96 3) 2 45 -5 5; Trans. , A IM E , 2 28 ,1 7. B ou rd et , D . et al.: " In te rp r et in g We ll T e st s i n F r ac tu re d R e se rv o ir s, "World Oil ( Oc t. 1 98 3) 7 7- 87 .1 8. C in co -L ey , H ., S am a ni eg o-V. , F . a nd K u cu k, F .: " Th e P re ss ur eT ra n-s ie nt B e ha v io r f o r Na tu r al ly F r ac tu r ed R e se rv o ir s w i th Mu lt ip le B l oc kS iz e," p ap er S PE 1 41 68 p re se nt ed a t t h e 1 98 5 S PE An nu al T ec hn ic alC on fe re nc e a nd E xh ib it io n, L as V eg as , S ep t. 2 2- 25 .1 9. B ou rd et , D . et al.: "N ew T yp e C ur ve s A id A n al ys is o f F is su re d Z on eWe ll T e st s, " World Oil ( Ap ri l 1 98 4) 1 11 -2 4.2 0 . G r in g ar te n, A .C. : " I nt er pr et at io n o f T e st s i n F is su r ed a nd Mu lt il ay er edR e se rv o ir s w i th Doubl e-Po ro si ty B e ha v io r: T h eo r y a n d P ra ct ic e. " JP T( Ap ri l 1 98 4) 5 49 -6 4.2 1. d e S wa an , A .: " In flu en ce o f S ha pe a nd S kin o f M atr ix -R oc k B lo ck so n P re ss ure T ra nsi en ts in F ra ctu re d R es erv oir s," p ap er S PE 1 56 37p re se nt ed a t t he 1 98 6 S PE An nu al T ec hn ic al C on fe re nc e a nd E xh ib i-tion , N ew O rlean s, O ct. 5 -8.2 2. B ou rd et, D ., A yo ub, J.A ., an d Pirard , Y .M .: " Su pp lem en t to S PE12777 , "U se o f P re ss ur e De ri va ti ve inWel l-Test In te rpre ta t ion ," paperS PE 1 92 15 a va ila ble f ro m S PE B oo k O rd er , R ic ha rd so n, T X.2 3. C in co -L ey , H ., S am a ni eg o-V. , F ., an d Vi tu ra t, D . : "Pr es su r e T ra n si en tAna lys is fo r High -Pe rmeab il it y Reservo ir s ," paper SPE 14314pre sen teda t t he 1 98 5 S PE An nu al T ec hn ic al C on fe re nc e a nd E xh ib it io n, L as V e -g as , S ep t. 2 2- 25 .

Appendix A-Results of Analysis of Data, Table 1D ata a re m atc he d a ga in st th e ty pe c urv e fo r a w ell w ith w ellb ores to ra ge a nd s kin in a r es er vo ir w ith h omo ge ne ou s b eh av io r. T hem a tc h p ar am e te rs a re d ef in ed a s CDe2S =4 x 109, p re ss ur e m a tc h(PD IAp)M=1.79XlO-2 psi-I[0.26xIO-2 kPa-I], a nd tim ematch [(tDICD)/At]M=14.8 hours r+,It follo ws as detailed in R ef. 10: kh=[l41.2qBp(PDIAp)M]=1 ,1 65 r nd -f t [ 35 5 md-rn], C={0.000295(khlp)[lltl(tDICD)M}=9.3x 10-3 bbl/psi [0.21x 10-3 m3/k Pa ), a nd S=[0.5 In( C Oe2S1CD)]=7.7.Appendix B-Summary of the DifferentiationAlgorithms ConsideredTh re e d if fe re nt a pp ro ac h es c an b e u se d . Smo ot hi ng i s a p p li ed e it he ro n p re ss ur e d at a b ef or e d if fe re nti atio n ( Alg or ith m s A a nd B ), o nt he d e ri va ti ve cu rv e (A lgo ri thm B) , o r on a s econd o r t hi rd d e ri va ti ve

30 1

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(Algorithm C), b efo re in te gra tio n o f th e d ata to p ro du ce th e firstd e ri vat iv e r esponse .A lg orith m A fits a p oly no mia l th ro ug h d ata p oin ts a ro un d th ep o in t o f i nt er es t a nd takes t he e xa ct p o ly n om i al d er iv at iv e. T h e u se rc an d efin e th e le ng th o f th e tim e in te rv al an d th e n um be r o f p o in tsf or th e p ol yn om ia l fi t. T he d eg re e o f t h e p o ly nom ia l c ou ld b e v ari edb y modi fy in g t he s o ur ce p ro g ram . A l th o ug h t hi s p ro ce du re smoo th st he d at a b ef or e d if fe re nt ia tio n, i t g en er al ly w o rk s f or a ct ua l d at a,p ro vid ed th at an ad ju stm en t o f th e p oly no mial d eg re e is m ad e tos uit e ac h p ar ti cu la r c as e. C on se qu en tl y, i ts u se i s c um be rs om e. Ina dd itio n, th e s ha pe o f th e o rig in al d eriv ativ e is a ffe cte d.A lg or it hm B u se s a s et o f p ar ab ol as , e ac h d ef in ed b y t hr ee p oi nt so f th e v icin ity o f th e p oin t c on sid ere d. T he trip le ts are c ho se n a sev en ly sp ac ed a s p os sib le . A cco rd in g to th e " qu ality " o f th e d ata,5 o r m ore th an 15 s u rr ou n di ng p o in ts p ar ti ci pa te i n t he c al cu la ti ono f e ac h lo cal d eriv ativ e, w hic h is an av erag e o f th e d eriv ativ e o fthe parabolas used. The smoo th ing is ob ta ined by aver a ging p r es sur edata andlor p re ss ure d eri va ti ve o ve r a g iv en t im e in te rv al . T hi s a l-g o ri thm f ai ls t o r ed u ce t he n o is e e ff ec t s uf fi ci en tl y, e ve n w i th l ar ges mo oth in g, w hich a ffe cts th e o rig in al s hap e o f th e type curve.A l go ri thm C c al cu la te s u p t o t he t hi rd d er iv at iv e f or e ve nl y spacedp oin ts , s mo oth s it, an d th en in te grate s to o btain th e fin al v alu e o ft he f ir st d er iv at iv e. It t en ds to c re at e fa ls e c on tin uo us o sc il la ti on sa t l at e ti me s d uri ng i nf in it e- ac ti ng r ad ia l f lo w.O th er s mo oth in g te ch niq ues h av e b een p ro po sed ,2 3 b ut th e al-g orith m p re sen te d in th is p ap er w as ch os en fo r its s im plicity a ndit s e ff ic ie nc y a t smo ot hi ng d at a w ith l ow d is to rti on e ff ec ts a nd b e-cause it is in dep en dent o f the d en sity of po ints. T he sam e effectca n b e ap plie d to ac tu al d ata an d to th eo re tic al cu rv es.

30 2

Appendix C-Results of Analysis of Data, Table 2Dat a a re m at ch ed a ga in st t he type c ur ve f or a w ell w it h w ell bo res to ra ge a nd s kin i n a r es erv oir w it h d ou bl e- po ro si ty b eh av io r a ndp se ud o st ea d y- st at e i nt er po ro si ty f low . T h e m a tc h p ar am e te rs a redefined as CDe2S=1.1, w=0.015, Ae-2S=4xI0-4 , pressurematch (PDIAp)M =8.72 x 10 -3 psi-I [1.26XIO-3 kPa-I). an dt im e m at ch [(tDICD)/M)M=370 hours r+,I t f o ll ows thatkh=[141.2qBp(PDIAp)M)=1.830 md-ft [558md-m], C={0.OOO295(khlp)[M(tDICD)]M} =0.001 bbllpsi [23x10 -6 m3/kPa]. S=[0.5ln(CDe2SICD)]=-3.6, w=0.015, an d A=2.9x 10-7 Th e e xt ra po la te d r es er vo ir p re ss ur e wa s e va lu at edatp"'=7.843 psig [54.1 MPa].SI Metric Conversion Factors

bb l x 1.589873 E-OI m3cp x 1.0'" E-03 Pa-sft x 3.048* E-Ol mmd x 9.869233 E-04 = pm2ps i x 6.894757 E+OO = kP apsi-I x 1.450377 E-Ol = kPa-1

Convers lon fac tor is exact. SPEFEOriginal SPEmanuscript received for reviewAprit14. 1984.Paperaccepted for publieationApril 28. 1988.Revised manuscr lpl_ Ma rch 7.1989.Paper (SPE 12777 )first presented at the 1984 California Regional Meellng held in Long Beach. April 11-13.SPE 19215. "Supplemant to SPE 12m. Use 01Pressure Oerivativeln WenTestlnterpr ...tatlon," available from SPE Book Order Dept.

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