Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics PRESSURE BUILDUP I is difficul o kee he rae consan in a roducing well. This is no an issue in a buildu es since he well is closed. The rae is zero. This es may be conduced any ime. The disadvanage is ha he well has o be closed for a eriod. Since he well is closed, i will no generae income during his eriod. Hence he shuin ime should be as shor as ossible. Procedure 1. Produce he well a a consan (sabilized) rae. A ime close he well. 2. Measure he las flowing ressure which we call wf and he shuin ressure ws. 3. Make inerreaion. Figure 1: Schemaic of an ideal buildu es In he above figure, and denoe roducion ime and shuin ime resecively. Infinie-acing reservoir For a new well, he ressure wave associaed wih he flow eriod may no have reached he ouer boundary. Then he following equaion alies: 1
ws B 1.15 = i log + This is he Horner equaion. + The above equaion shows u as a sraigh line on a ws vs. log lo. Deerminaion of ermeabiliy Noe ha he rae rofile of Fig. 1 is idealized. Insananeous shuin is no ossible. There will always be some aferflow (see wellbore sorage). As a consequence he measured ressure will no obey he Horner equaion iniially. Figure 2: Schemaic of a Horner lo of a well wih aferflow and skin. Hr is used o denoe he Horner raio: Hr = + Observe ha he shuin ime,, increases o he lef in he Horner lo, Fig. 2. The Horner raio Hr will decrease as increases.. The sloe of he sraigh line is given by: m = B1.15 Hence he ermeabiliy may be deermined from he following equaion: Tom Aage Jelmer PRESSURE BUILDUP 2
k = B 1.15 2πmh The sloe is also defined by wo oins on he sraigh line m = log Hr 1 1 2 log Hr 2 which simlifies o: when Hr 1 and Hr 2 are one decade aar. m = 1 2 Deerminaion of he reservoir iniial reservoir ressure The Horner equaion may be wrien: Noe ha: ws = i - B 1.15 log Hr ws = i for Hr = 1 The Horner raio will aroach 1 for infinie shuin ime. Hence he iniial reservoir ressure may be obained by exraolaing he sraigh line back o Hr = 1. The echnique is illusraed in Fig.2. Deerminaion of he skin facor The skin is no included in he Horner equaion. To involve his arameer, he las flowing ressure wf is subraced from boh sides of he Horner equaion. The las flowing ressure is given by he drawdown equaion. On he righ hand side of he Horner equaion we subrac he mahemaical model and on he lef hand side he observed ressure. The resul is: B 1.15 = kh log 2π + k log ϕµ c r ws wf 0.351+ 0. 87 2 w Usually he shuin ime is small in comarison wih he roducion ime. Hence S + Then he above equaion will simlify since he roducion ime disaears. Tom Aage Jelmer PRESSURE BUILDUP 3
The modified Horner equaion may be solved for he skin facor once he shuin ime is secified. The radiional choice is = 1h. This choice leads o: S = 1.15 ws = 1h m wf k log 2 ϕµ c r w 3.91 Observaions: 1. The skin facor is conrolled by he disance = ws = 1h - wf 2. The skin facor S will increase wih increasing difference. 3. The measured wellbore ressure a 1 hour may no be on he sraigh Horner line. Then he line is exraolaed unil i inersecs he Hr = 1h verical line. This is illusraed in he below figure. The Horner raio a 1 hour may be comued from Hr = 1h = + 1 Figure 3: The skin deends on he disance. Bounded reservoir Sooner or laer he ressure wave associaed wih he flow eriod will hi he ouer boundary. Suose his is of no-flow ye. If he well is closed during seudo-seady flow, hen he ressure will build u owards he average ressure raher han he iniial ressure. This is illusraed below. Tom Aage Jelmer PRESSURE BUILDUP 4
Figure 4: Pressure versus disance, seudo-seady flow The effec of he ouer boundary aears a he lae ar of he Horner lo while he early ar essenially remains unchanged. The boundary effec will show u as a break off from he sraigh line. Figure 5: Horner lo, bounded reservoir The Horner equaion for he sraigh line secion may be wrien: ws B1.15 + = * log where * is he inersecion wih he Hr =1 axis. The inersecion has been called he false ressure. I has no hysical inerreaion bu i is relaed o he average ressure, Tom Aage Jelmer PRESSURE BUILDUP 5
. The sraigh line on he horner lo may be used o deermine he ermeabiliy and skin facor as discussed reviously. The rocedure will no be reeaed here. Deerminaion of he average ressure Mahews, Brons and Hazebroek resened chars ha relae he false ressure o he average ressure for various geomeries. Figure 6: Schemaic of a Maews,Brons and Hazebroek lo Index D is used o denoe dimensionless variables. The average ressure may be calculaed as follows: 1. Obain he sloe m and he false ressure * from he Horner lo. 2. Esimae he shae and size of he drainage area. 3. Comue he dimensionless roducion ime from he formula: k DA = ϕµ c A 4. Look u he MBH-curve ha corresonds o he esimaed shae of he drainage area, and find P DMBH. 5. Calculae he average ressure from he formula: m DMBH = * 2.303 The difficul ar in his calculaion rocedure is oin 2. Esimaion of he size and shae of he drainage area is beyond he scoe of hese noes. The average ressure is used in maerial balance calculaions. Also i is used o calculae he flow efficiency, FE. wf FE = wf S Tom Aage Jelmer PRESSURE BUILDUP 6
wf is he las flowing ressure. For seudo-seady flow, he difference wf is indeenden of ime. This condiion leads o a consan value of he flow efficiency. Oherwise i will deend on ime. * wf S Someimes he flow efficiency is aroximaed by FE =. ( * wf ) The resul is no as accurae bu easier o obain. Horner ime As menioned reviously i is difficul o kee he flow rae consan for any lengh of ime. The rae may have flucuaed significanly during he roducion eriod. Horner roosed he following correcion: N = qlast where N is he cumulaive roducion since he las major shuin eriod and q LAST is he las sabilized rae. Tom Aage Jelmer PRESSURE BUILDUP 7