Petroleum Reservoir Engineering by Non-linear Singular Integral Equations

Transcription

1 Mchacal Egrg Rsarch Vol. 1 No. 1; Dcmbr 11 Ptrolm Rsrvor Egrg b No-lar Sglar Itgral Eqatos E. G. Laopolos Itrpapr Rsarch Orgazato 8 Dma Str. Aths GR Grc Rcv: Agst 8 11 Accpt: Sptmbr 1 11 Pblsh: Dcmbr o:1.5539/mr.v11p URL: Abstract For th trmato of th proprts of svral rsrvor matrals wh ol rsrvs ar movg throgh poros ma a w mathmatcal approach s propos. Sch problm s vr mch mportat for ptrolm rsrvor grg. Ths th abov mto problm s rc to th solto of a o-lar sglar tgral qato whch s mrcall valat b sg th Sglar Itgral Oprators Mtho S.I.O.M.. Bo th abov svral proprts ar aalz a vstgat for th poros mm qato f as a Hlmholtz ffrtal qato. Fall a applcato s gv for a wll tstg to b chc wh a htrogos ol rsrvor s movg a poros mm. Hc b sg th S.I.O.M. th th prssr rspos from th wll tst coct th abov htrogos ol rsrvor s mrcall calclat a vstgat. Kwors: Sglar Itgral Oprators Mtho S.I.O.M. No-lar sglar tgral qato Ol rsrvs Poros ma Ptrolm rsrvor grg Hlmholtz ffrtal qato 1. Itrocto Th st of th movmt of ol rsrvs throgh poros ma s vr mch mportat problm o ptrolm rsrvor grg. Thrfor b applg a wll tst aalss th a hstor matchg procss tas plac for th trmato of th proprts of th rsrvor matrals. Th movmt of ol rsrvs throgh poros ma procs both sgl-phas a mltphas flows. Frthrmor f a wll tst s coct th th wll s sbct to a chag of th flow rat a th prssr rspos ca b frthr masr. For th trmato of svral ptrolm rsrvor paramtrs sch as prmablt th mrcal calclatos shol b s as aaltcal soltos most cass ar ot possbl to b rv. Drg th past ars svral varats of th Boar Elmt Mtho wr s for th solto of ptrolm rsrvor grg problms. At th of ght's Laf a Chg 1987 propos a BEM for th solto of sta flows htrogos sols. Drg th sam pro Masawa a Hor 1988 a Nmbr a Tab 1988 appl boar lmts for sta stat problms of straml tracg. Frthrmor Ka a Hor 199 solv trast problms b sg a Laplac spac boar lmt mol for th aalss of wll tsts svral arbtrarl shap rsrvors. Bo th abov Koh a Tab 1993 s boar lmts to scrb th flow aro tortos horzotal wlls for homogos or pcws homogos rsrvors. Sato a Hor ; appl prtrbato boar lmts for th st of htrogos rsrvors. Also El Harro Qazar Wrobl a Chg 1996 propos th s of a trasform form of Darc's law comb wth al rcproct boar lmt mtho to hal htrogt. O th othr ha Ow 1997 appl a Gr lmt mtho to sothrmal flows wth sco orr ractos. Th sam athor Ow O.O ; Ow O.O s a comb mtho of boar lmts togthr wth ft lmts for th st of htrogos rsrvors. Bo th abov Tagb a Ow 1997 appl a trast o-msoal trasport qato b sg a m Gr lmt mtho. Drg th last ars svral o-lar sglar tgral qato mthos wr s sccssfll b Laopolos Sprgr Vrlag for th solto of appl problms of sol mchacs ISSN E-ISSN

2 Mchacal Egrg Rsarch Vol. 1 No. 1; Dcmbr 11 lastoamcs strctral aalss fl mchacs a aroamcs. Ths th prst rsarch th o-lar sglar tgral qatos wll b s orr to trm th proprts of th rsrvor matrals wh ol rsrvs ar movg throgh poros sols. B sg thrfor th Sglar Itgral Oprators Mtho S.I.O.M. th th prssr rspos from th wll tst coct a htrogos rsrvor wll b compt. Also som proprts of th poros mm qato whch s a Hlmholtz ffrtal qato ar propos a vstgat. Ths basc proprts of th famtal solto wll b aalz a vstgat. Fall a applcato s gv for a wll tstg to b vstgat wh a htrogos ol rsrvor s movg a poros mm. Th ths problm wll b solv b sg th Sglar Itgral Oprators Mtho a so th prssr rspos from th wll tst coct ths htrogos ol rsrvor wll b compt. Hc th o-lar sglar tgral qato mthos whch wr s wth bg sccss for th solto of svral grg problms of fl mchacs hralcs aroamcs sol mchacs lastoamcs a strctral aalss ar frthr t th prst st for th solto of ol rsrvor grg problms. I sch a cas th o-lar sglar tgral qatos ar s for th solto of o of th most mportat a trstg problms for ptrolm grs.. Wll Tst Aalss for Ol Rsrvor Ol wll tst aalss s a of a mportat hstor matchg procss for th trmato of th proprts of rsrvor matrals. Ths rg th movmt of ol rsrvor throgh poros ma th both sgl-phas a mltphas flow occrs. Also wh a ptrolm wll tst s coct th th wll s sbct to a chag of ts flow rat a th rsltg prssr rspos s possbl to b masr. Morovr ths prssr s compar to aaltcal or mrcal mols orr to stmat rsrvor paramtrs sch as prmablt. I gral a ol rsrvor wll tst a sgl-phas rsrvor s calclat b sg th poros mm qato: whch λ ots th prmablt th porost ξ th vscost p th prssr of th rsrvor t th tm a c t th comprssblt. B rplacg varabls as follows: th.1 ca b wrtt as: wth : Hc q..3 s a Hlmholtz ffrtal qato. Bo th abov cosr b th famtal solto of a pot bcas of th sorc pot. Th th famtal solto ca b gv b th followg qato: whch ma b frthr wrtt as: p c a.5b Ths q..5 s th Hlmholtz pottal qato govrg th famtal solto. Cosr frthr b th famtal solto chos so that to forc th Hlmholtz qato trms of th fcto a wa form. Th th wa form of Hlmholtz qato wll b wrtt as followg: 1/ p t 1/ p t 1/ Pblsh b Caaa Ctr of Scc a Ecato 3

3 Mchacal Egrg Rsarch Vol. 1 No. 1; Dcmbr 11 ISSN E-ISSN th solto oma Ω. Also b applg th vrgc thorm oc.6 o obtas a smmtrc wa form:.7 whch ots th otwar ormal vctor of th srfac S. Thrfor th smmtrc wa form th fcto a th famtal solto ar ol rqr to b frst - orr ffrtabl. B applg frthr th vrgc thorm twc.6 w hav:.8 Hc.8 s th asmmtrc wa form a th famtal solto s rqr to b sco - orr ffrtabl. O th othr ha s ot rqr to b ffrtabl th oma Ω. B combg qs.5 a.8 th o obtas:.9 whch ca b frthr wrtt as:.1 whr q ots th pottal grat alog th otwar ormal rcto of th boar srfac:.11 a th rl fcto:.1 B ffrtatg.1 wth rspct to w obta th tgral qato for pottal grats b th followg formla: Famtal Solto's Basc Proprts Bo th abov w rwrt th wa form of.5 govrg th famtal solto as follows: 3.1 whr c ots a costat cosrg as th tst fcto. Also q. 3.1 ca b wrtt as: 3. Frthrmor 3. tas th form: 3.3 B cosrg frthr a arbtrar fcto Ω as th tst fcto th th wa form of.5 wll b wrtt as:. R q q R R q c c 1 1

4 Mchacal Egrg Rsarch Vol. 1 No. 1; Dcmbr 11 Pblsh b Caaa Ctr of Scc a Ecato a also as: 3.5 Fall 3.5 tas th form: 3.6 If approachs th smooth boar th th frst trm 3.6 ma b wrtt as: lm 3.7 th ss of a Cach Prcpal Val CPV tgral. For th rstag of th phscal mag of 3.7 w rwrt 3.3 a 3.6 as: 3.8 a: 3.9 B 3.8 follows that ol a half of th sorc fcto at pot s appl to th oma Ω wh th pot approachs a smooth boar. Also cosr aothr wa form of q.5 b spposg th vctor fctos to b th grats of a arbtrar fcto Ω chos sch a wa that th hav costat vals: for = Th th wa form of q.5 wll b wrtt as: 3.11 B applg frthr th vrgc thorm th q 3.11 tas th form: 3.1 Frthrmor th followg proprt sts: 3.13 B ag qs 3.1 a 3.13 th o obtas: 3.14 whch tas fall th form: 1 CPV CPV 1 CPV 1

5 Mchacal Egrg Rsarch Vol. 1 No. 1; Dcmbr Aalss b No-lar Sglar Itgral Eqatos Frthrmor th poros mm qato.1 wll b wrtt aothr form orr a sglar tgral qato rprstato to b applcabl: B applg frthr th Gr Elmt Mtho th q 4.1 rcs to th solto of a o-lar sglar tgral qato: whch: I orr th o-lar sglar tgral qato 4. to b mrcall valat th th Sglar Itgral Oprators Mtho S.I.O.M. wll b s. Ths th o-lar sglar tgral qato 4. s appromat b th formla: M l r r 1 p p p r p l r r l p t whr M ots th total mbr of lmts. Bo th abov w troc th followg fctos scrbg th prssr at a pot a lmt trms of th oal prssrs: p N p 4.5 B rplacg 4.5 th q 4.4 tas th form: whr: M 1 t Rt A p p r ct p l p c t B q l r r p t p l r r p 1 p l r r l p t A C l l p l c t D l 1 pl t l r r B l r r N N N l N l Cl l r r D l l r r N N l l ISSN E-ISSN

6 Mchacal Egrg Rsarch Vol. 1 No. 1; Dcmbr Wll Tstgs Applcatos Htrogos Rsrvors Th prvos mto thor wll b appl to th trmato of a wll tstg whch wll b chc a htrogos rsrvor wth a prmablt varg from 1 md to 3 md 1mDarc 1-1 m = 1μm. Hc b sg th Sglar Itgral Oprators Mtho S.I.O.M. as scrb th prvos paragraphs th t has b ffct th comptato of th prssr rspos from th wll tst coct th abov htrogos rsrvor. Frst of all th prssrs wr compt varato wth th tm. Ths Tabl 1 shows th prssr rspos wth rspct to th tm. Bo th abov th prssr rvatvs wr compt wth rspct to th tm as show Tabl. Sch rvatvs ar vr mch mportat of th wll tstgs trprtato as ths ar som stct shaps a spcall th charactrstcs of crta rsrvor fatrs. Th comptatoal rslts of th prssrs a th prssr rvatvs ar compar to th aaltcal soltos of th sam wll tstg problm f th rsrvor was homogos wth prmablt qal to 5 md. Ths th aaltcal rslts ar show Tabl 1 for th prssrs a Tabl for th prssr rvatvs corrspogl. From th abov Tabls t ca b s that thr s vr small ffrc btw th comptatoal rslts a th aaltcal soltos for both th prssrs a th prssr rvatvs. O th othr ha th abov mto small ffrc ca b pla bcas of th ffsv atr of th prssr trasport mchasm. Fall sam rslts ar show corrspogl Fgrs 1 a a thr-msoal form Fgrs 1a a a. 5. Coclsos I th prst vstgato a mathmatcal mol has b prst as a attmpt to trm th proprts of th rsrvor matrals. Ths th st of th movmt of ol rsrvs throgh poros ma s vr mportat for ptrolm rsrvor grs. Th abov mto problm was rc to th solto of a o-lar sglar tgral qato whch was mrcall valat b sg th Sglar Itgral Oprators Mtho S.I.O.M.. Frthrmor svral mportat proprts of th poros mm qato whch s a Hlmholtz ffrtal qato wr aalz a vstgat. Ths th famtal solto of th poros mm qato was propos a st. Also som basc proprts of th famtal solto wr frthr vstgat. Ths ar vr mportat orr th bhavor of th o-lar sglar tgral qato to b wll rstoo. A applcato was fall gv for a wll tstg to b chc wh a htrogos ol rsrvor s movg a poros sol. Th abov problm was solv b sg th Sglar Itgral Oprators Mtho a ths th prssr rspos from th wll tst coct th abov htrogos ol rsrvor was compt. Both th prssrs a th prssr rvatvs wr compt a ths vals wr compar to th aaltcal soltos of th sam wll tstg problm f th rsrvor was homogos wth a ma prmablt. Ovr th last ars o-lar sglar tgral qato mthos hav b s wth a bg sccss for th solto of svral mportat grg problms of strctral aalss lastoamcs hralcs fl mchacs a aroamcs. For th mrcal valato of th o-lar sglar tgral qatos of th abov problms wr s svral aspcts of th Sglar Itgral Oprators Mtho S.I.O.M.. Ths th prst rsarch sch mthos wr t for th solto of ol rsrvs problms ptrolm rsrvor grg. Rfrcs El Harro K. Oazar D. Wrobl L. C. & Chg A. H. D Global trpolato fcto bas DRBEM appl to Darc's flow htrogos ma. Egg Aal. Bo. Elm Ka J. A. & Hor R. N Prssr-trast aalss of arbtrarl shap rsrvors wth th boar lmt mtho. SPE Form. Eval Koh L.S. & Tab D A boar lmt algorthm for mollg 3D horzotal wlls problms sg D grs. SPE Ptrol. Comptr Cof. Nw Orlas LA pp Laopolos E. G No-lar tgro-ffrtal qatos s orthotropc sphrcal shll aalss. Mch. Rs. Comm Laopolos E. G No-lar tgro-ffrtal qatos sawch plats strss aalss. Mch. Rs. Comm Pblsh b Caaa Ctr of Scc a Ecato 7

7 Mchacal Egrg Rsarch Vol. 1 No. 1; Dcmbr 11 Laopolos E. G No-lar sglar tgral comptatoal aalss for sta flow problms. Rw. Erg Laopolos E. G No-lar sglar tgral rprstato for sta vsc flowfls of -D arfols. Mch. Rs. Comm Laopolos E. G No-lar sglar tgral rprstato aalss for vsc flowfls of sta arfols. It. J. No-L. Mch Laopolos E. G.. No-lar mltmsoal sglar tgral qatos -msoal fl mchacs aalss. It. J. No-L. Mch Laopolos E. G.. Sglar Itgral Eqatos Lar a No-Lar Thor a ts Applcatos Scc a Egrg. Sprgr Vrlag Nw Yor Brl. Laopolos E. G. 3. No-lar two-msoal aroamcs b mltmsoal sglar tgral comptatoal aalss. Forsch. Ig Laopolos E. G. 5. No-lar sglar tgral qatos lastoamcs b sg Hlbrt trasformatos. Nol. Aal. Ral Worl Appl Laopolos E. G. & Zss V. A.. No-lar ft-part sglar tgral qatos arsg two-msoal fl mchacs. Nol. Aal. Th. Mth. Appl Laf. O. E. & Chg A. H-D A prtrbato boar lmt co for sta stat growatr flow htrogos aqfrs. Watr Rsor. Rs Masawa J. & Hor R. N Applcato of th boar tgral mtho to mmscbl splacmt problms. SPE Rsrv. Egg Nmbr D. T. & Tab D A mprov straml gratg tchq that ss th boar tgral lmt mtho. SPE Rsrv. Egg Ow O. O A Gr lmt tratmt of sothrmal flow wth sco orr racto. It. Comm. Hat Mass Trasfr Ow O. O A boar lmt - ft lmt qatos solto to flow htrogos poros ma. Tras. Poros Ma Ow O. O Boar tgral procrs for satrat flow problms. Tras. Poros Ma Sato K. & Hor R. N Prtrbato boar lmt mtho for htrogos rsrvors: Part 1 - Sta - stat flow problms. SPE Form. Eval Sato K. & Hor R. N Prtrbato boar lmt mtho for htrogos rsrvors: Part - Trast flow problms. SPE Form. Eval Tagb A. E. & Ow O. O Trast 1D trasport qato smlat b a m Gr lmt formlato. It. J. Nmr. Mth. Egg ISSN E-ISSN

8 Mchacal Egrg Rsarch Vol. 1 No. 1; Dcmbr 11 Tabl 1. Tm hors Prssr ps S.I.O.M. Prssr ps Aaltcal Tabl. Tm hors Prssr Drvatv ps S.I.O.M. Prssr Drvatv ps Aaltcal Pblsh b Caaa Ctr of Scc a Ecato 9

9 Mchacal Egrg Rsarch Vol. 1 No. 1; Dcmbr 11 Fgr 1. Prssr Rspos for Wll Tst Htrogos Rsrvor Fgr 1a. 3-D Dstrbto of Prssr Rspos for Wll Tst Htrogos Rsrvor 1 ISSN E-ISSN

10 Mchacal Egrg Rsarch Vol. 1 No. 1; Dcmbr 11 Fgr. Prssr Drvatv for Wll Tst Htrogos Rsrvor Fgr a. 3-D Dstrbto of Prssr Drvatv for Wll Tst Htrogos Rsrvor Pblsh b Caaa Ctr of Scc a Ecato 11