Gladfelter Deconvolution

case. from simultaneously rncasured rkm nholc pressure and flow Thr Cltrdfclfer ( (;larffelter. Tracy. and lf’ilwy] ) dec-nnvnlw rate, psD is alsn called “deccmvolved pressure, influence func[i(]lt rrw( hod has bwn nwd for t}Ic las( thrw decades for dr(er[ion or unif rwpunse bctlts~iur.Once p.~D is obtained. the n]ining the constant rate behavior of the ivell-rcwrvoir sys!em conventional interpretation methods can be used for the deterfrcrnt simultaneously mrwurcd dowrlhole flow rate (afterflow) minaticm of the well-reservoir system and its parameters. and ilrmsure data. The success of the C,ladfeltvr deconvolution met hod dcpmrds crn the brha~irrr of the downhole flmv rate. It The purpose of this work is to study the general behavior of is sliutvn that the method works only if the do Jvnhole flow rate Gladfeltet deconvolution and to explore its application to ravarics Ii]tcarly \vith time. It is also shown t}la( the commonly dial. linear, and spherical well-reservoir systems. trssnlncd first semilug s!raight line due to the Glad felter dm-rrnTheoretical Developments vnlu:ion is instead a twrgcotial line that is parallel tn thr final scmilog straight line. Consider a well-reservoir system (the single phase fluid flow, in a The validitv of the Glarffelter dcconvolu(iw nlethod is investiporous medium which is described by the diffusivity equation) gatcd for different tvellhore georm,tries such as Iirle, c} Iii]drical. initially at constant and uniform pressure. The relationship between downhole flow rate (input) and pressure (output) signals and spherical source \vclls, as sell as fractured ~vells. Some can f-e described as a convolution operation useful asymptotic sulutions arc presented fnr the interpretation of simultaneuuslj rrlcasu red duwnhole flow rate find pressure, Introduction / lD P1.D(frJ) = ~ ‘?b(r)PSD[~D ~)dr In 1955, Gladfeltcr. Tracy, aIId \Yilscyl stated that {11(. recip10 rocal productivity index ,( ~q~~ ). is a linear function of the — / ~ qD(r)Pir(~D r)~r -.$ go(t~) (1) logarithm of I.ime for buildup tcs{s wlwre APU, is the change in sh,]!-ir] prrwurr and q$r is the change in aftcrflo~v rate. The G ladfelt cr. Tracy. and N’iisry] method. \vtlich WX, call W ladfelpruvided that qD(O) is zero, tw dmwn~ olution.has bee], used in well testi]jg since 1’355 for .Qt houg,i, the rmnvolu[ion in(egral gi~en by Eq. 1 can be easily ?he interpretation of simnltanwusll rtlswsured duwrtho l:? pressure and Row rate. Over the jcars t}tere has been a nu,nber ( omputrxi from qD(f D) and p5D(tD). the deconvolution (comof studies on this subject, put ing) of p,s~j from pwIZ and qD becomes an ill-prised problem namely, \\ ’irlcstock and Colpitis:, if qr) has measurement errors’1. Ral~,e\3.4. Fctkt>yich and Vitvrol 5. Ku~huk and Aywtararr6,7, Th[)rllps(m et al.8. Thompson and Reynolds9. and \leunier and The cbjective of the Gladfelter deconvolution is to r~cover p.s3 \l’iltrna], nlo. from rrleasured pwD and qD data as In terms of dimensionless variables. the (;lsidfelter deconvolution can be dehned as determining dimensiorrlcss pressure J)$D CPD(fD) ‘ 6-$ = ~(~) (2) at the sandface, including skin (pD -S) fnr the constant rate