Well test analysis on pressure of viscoelastic polymer solution with variable rheological parameters *

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1 Vol., No., 7- ( do:.46/ns..6 Ntul Scence Well test nlyss on pessue of vscoelstc polyme soluton th vble heologcl pmetes * Hongjun Yn #, Wel Yng, Syun Meng, Mng C Key Lbotoy of Enhnced Ol nd Gs Recovey Mnsty of Educton; Nothest Petoleum Unvesty; Dqng, Chn; # Coespondng Autho: ynhj776@6.com Receved 7 Septembe ; evsed 8 Octobe ; ccepted Novembe. ABSTRACT Accodng to the behvo tht the polyme soluton hs both the chctestcs of vscosty nd elstc popetes, the tnsent flo mthemtcl model consdeed the vscoelstcty of the polyme soluton hs been estblshed. The model, n hch the vton of the heologcl pmetes dung the seepge flo hs been lso tken nto consdeton, hs been solved usng fnte-dffeence method. The type cuves hve been plotted. The nfluence of some popetes of polyme soluton ncludng the vscocty, the elstc popetes nd the heologcl pmetes hs been nlyzed. Comped th the cuves of the poe-l flud, t s shon tht the pessue devtve cuve consdeng the elstcty of the polyme soluton upps less t the dl flo egme. Besdes, t ll come don s the vton of the heologcl pmetes, hch s qute dffeent fom the cse egdng them s constnts. Theefoe, n ell test nlyss on pessue of polyme soluton, t s necessy to consde the elstcty nd the vton of the heologcl pmetes. Keyods: Polyme Soluton; Vscoelstc; Relxton Tme; Rheologcl Pmetes. INTRODUCTION The polyme soluton used n the ol feld s typcl knd of non-netonn flud nd ts heologcl popety n the poous med s vey complcted becuse of the effect of she degdton, deconcentton, dsopton nd entpment. * Poject suppoted by the Ntonl Scence Foundton of Chn (Gnt No: 5874 nd by the Helongjng Povncl Scence nd Technology Pln Poject (Gnt No: GZ9A47 nd by the Resech Pogm of Innovton Tem of Scence nd Technology n Enhnced Ol nd Gs Recovey (Gnt No: 9td8. Mny domestc nd foegn schols hve studed ts heologcl popety nd estblshed sevel heologcl model. The ntl eseches on the flo behvo of the non-netonn flud e mnly focus on the poe-l flud. In most ell test nlyss on pessue of polyme soluton, t s lso ssumed tht the polyme soluton s pue vscous flud, only the she vscosty s consdeed nd the heologcl pmetes e lys teted s constnts. But lge numbe of expements hve demonstted tht polyme soluton hs vscoelstc behvo nd the heologcl pmetes chnges n the seepge flo pocess. The vscosty of the polyme soluton ll declne s the sng of the she te t eltvely loe Dcy velocty. In the flo event, the vscosty s the domnnt nfluentl fcto nd the elstc popety cn be neglected. In ddton, the heologcl popety of polyme soluton cn be expessed by usng pseudo-plstc poe-l model. Hoeve, once the Dcy velocty exceeds the ctcl vlue,the vscosty ll se th the ncesng of the she te, the elstc effect ll enhnce gdully nd ts nfluence ll be too sgnfcnt to be gnoed. Refeence [] hs decled tht the effectve vscosty of polyme soluton s composed of the ndvdul contbutons of she nd stn vscosty. The elstcty behvo nd heologcl popety of polyme soluton n poous med unde the condtons of esevo flo te hs been studed n []. A poe-l flud vscoelstc sem-empcl model hch could descbe the vscoelstc effect of polyme soluton n poous med hs been developed n []. The eseches on the vscoelstcty hve pomoted the development of pecolton theoy. Refeence [4] hs conducted numecl smulton study fo mpovement of polyme floodng by vscoelstc effect. Refeence [5] hs studed the flo behvo of vscoelstc flud, poe l flud nd Netonn flud n poe thot by numecl method. Dffeence mthemtcl models of vscous-elstc polyme soluton hve been estblshed n [6] nd [7] fom df- Copyght ScRes. Openly ccessble t

2 8 H. J. Yn et l. / Ntul Scence ( 7- feent ngles. The nfluence of dffeent fctos on the pessue of the fomton ne the njecton ell s nlyzed too. But the vton of heologcl behvo s seldom tken nto consdeton nd the consstency coeffcent nd the poe-l ndex ee lys teted s constnts. Besdes, the eseches on the ell test of vscoelstc polyme soluton ee seldom. Theefoe, n ths ppe, the eltonshp beteen elstc vscosty nd she vscosty hs been nssted n ths ppe. In ddton, the vton of the heologcl pmetes n the flo event hs been tken nto consdeton. Wht s moe, the expesson of the ppent vscosty of polyme soluton hch consdeed the vscoelstcty nd the chngng pmetes of polyme soluton hs been developed. Then the non-dmensonl dl nstble flo mthemtcl model hs been estblshed nd solved usng fnte-dffeence methods. The pessue of dffeent fomton ponts ne the bottom of the ell t dffeent tme hs been clculted. Then the ell test nlyss cuves hve been plotted. In the end, the nfluence of dffeent fctos on the cuves hs been nlyzed.. APPARENT VISCOCITY OF POLYMER SOLUTION.. The She Vscosty The vscosty of the poe-l flud cn be descbed s: v n H ( hee s the dus y fom the ellboe, m; γ s the she te, s - ; H ( s the consstency coeffcent, mp s n ; n ( s the poe-l ndex of polyme soluton, dmensonless. Refeence [8] hs studed the vton of heologcl behvo though expements nd developed the bsc models bout the vton of heologcl pmetes long the seepge flo decton. In ths ppe t mens tht the consstency coeffcent nd the poe-l ndex e chngng long the dl decton. Fo the ell hee the polyme soluton s njected nto, the vton of heologcl pmetes cn be expessed s: H H e ( n n ( hee s the dus of the ellboe, m; α s the vton fcto of the consstency coeffcent, dmensonless; β s the vton fcto of the poe-l ndex, dmensonless. Accodng to the eseches n [9], the eltonshp beteen the sheng te nd the seepng te s s follos: n v γ (4 n CK ' hee v s the flo velocty though poous medum, m/s; K s the pemeblty of the fomton, μm ; c s the fcto elted to the totuosty of cplly,.8~.5; Φ s the poosty of the fomton, dmensonless. The flo velocty though poous medum cn be ppoxmtely expessed by usng the follong lne elton: q v (5 h hee q s the njecton te of the polyme soluton, m /d; h s the esevo thckness, m. Then the expesson of sheng te consdeng the vton of heologcl pmetes cn be deved. n q n CK ' πh (6 The poe-l ndex chnges th the dus. Hoeve, the esultng chnges of the she te mke lttle sense to the vscosty of the flud t the sme pont. As esult, the vton of the poe-l ndex cn be gnoed hen clcultng the sheng te. Then the sheng te cn tun to the expesson s follos: n q n CK ' πh (7 hee n( s the poe-l ndex of polyme soluton n the bottom hole, dmensonless. Lettng n q Fs n CK ' πh The sheng te cn be eventully smplfed s: (8 Fs (9 And the fnl expesson of the she vscosty of the polyme soluton consdeng the vton of theologcl pmetes cn be eched... The Elstc Vscosty The eltonshp beteen elstc vscosty nd she vscosty hs been obtned n []. ( e f v hee, θ f s the elxton tme, s. Copyght ScRes. Openly ccessble t

3 H. J. Yn et l. / Ntul Scence ( The Appent Vscosty Thee e both the sheng defomton nd the elstc defomton hen the polyme soluton seepng n the poous medum becuse of the contnuous contcton nd spedng of the unne. As esult, the ppent vscosty s composed of the she vscosty (μ v nd the elstc vscosty (μ e. e v f v ( Accodng to the expessons of the she te nd the she vscosty, the fnl expesson of the ppent vscosty cn be obtned. ff s H e n n Fs. MATHETICAL MODEL AND ITS SOLUTION ( The mthemtcl model fo tnsent flo of vscoelstc polyme soluton s deved th the follong equtons. The ptl dffeentl equton fo tnsent flo of vscoelstc polyme soluton: Intl condton: Inne boundy condton: p C p μ K L t t Oute boundy condton: ( p p (4 K p q μ π h p e (5 (6 Defne the dmensonless vble s follos: Dmensonless dus D (7 Dmensonless pessue Dmensonless tme p πkh q B p D * t D * C t p (8 Kt (9 Whee μ * s the chctestc vscosty, tht s the ppent vscosty of the polyme soluton t the bottom of the ellboe. hee n n A C * ( n s A H F ( n f s C H F ( Accodng to the dmensonless vble bove, the seepng model of the polyme soluton consdeng the vton of the heologcl pmetes s estblshed: * p D pd D D D D td p p D td * pd D D D D D D De ( The dffeence equton t the pont (, j cn be estblshed by usng Implct Dffeence Method. p b p cp d N j j j,,, p,,,, N p, p d j j p p, N hee x x D D D D b t D D x ' c x x D D D D j pd j d t x n x e ( e n n A C n n d n n A C n And Δx = ln( e / /N, hch s the gd spcng; e s the dus of the extenl boundy, m; s the dus of the ellboe, m; s the numbe of the node; N s the gd numbe; p e s the supply boundy pessue, P. Copyght ScRes. Openly ccessble t

4 H. J. Yn et l. / Ntul Scence ( 7- Then the tdgonl coeffcent mtx equton of the dmensonless mthemtcl model s bult up s: pd d b c p D d b c pd d..... p DN pe The equton hs been solved. The pessue nd ts devtve vlue t the bottom of the ellboe t dffeent tme hve been clculted. 4. TYPE CURFE OF VISCOELASTIC POLYMER SOLUTION Accodng to the soluton of the mthemtcl model, the type cuve hs been dn, s my be seen n Fgue. As shon n Fgue, hen consdeng the vton of the heologcl pmetes nd the elstcty of the polyme soluton, the chctestc of the type cuve s s follos: The pessue cuve nd the pessue devtve cuve both chnge long the stght lne th the slope of t the pue ellboe stoge phse. In tnston secton, the pessue cuve flttens nd the pessue devtve cuve ppes to be tnspotton hump. The pessue devtve cuve goes up fte the tnston egme. Ths vton s compehensve cton of dsopton, she, nd elstc defomton. The pessue devtve cuve goes don once the nfluence of the chngng heologcl pmetes become obvously. When the effects of the closed oute boundy ply ole, the pessue cuve nd the pessue devtve cuve both go up. 5. ANALYSIS OF INFLUENTIAL FACTORS Some pmetes hch could nfluence the chctestcs of the type cuve hve been nlyzed. These pmetes nclude the elxton tme, the vton fcto of the consstency coeffcent nd the vton fcto of the poe-l ndex, s ell s the consstency coeffcent nd the poe-l ndex of the vscoelstc polyme solutons. The nfluence of elxton tme s nlyzed on the bss of Fgue. As the elxton tme nceses, the pessue nd pessue devtve vlues ncese fte the pue ellboe stoge phse, the hump of the pessue devtve cuve n tnston secton nceses nd the lg(p D, lg(p D ' t D lg(p D lg(p D ' t D Fgue. The type cuve of vscoelstc polyme soluton. lg(p D,lg(p D ' t D θ f =.s θ f =.5s θ f =s Fgue. The nfluence of the elxton tme. pessue devtve cuve upps less t the dl flo egme. Wht s moe, comped th the cuves smply consdeed the polyme soluton s poe-l flud, t s shon tht the pessue devtve cuve uptuns less t the dl flo egme consdeng the elstcty of the polyme soluton. The lge the elxton tme s, the elstc popety of the polyme soluton s stonge nd the gete the enegy s equed, then the lge the bottom hole pessue s nd the fste t chnges. Theefoe, t s necessy to consde the elstcty n ell test nlyss on pessue of vscoelstc polyme soluton. The nfluence of the consstency coeffcent t the ellboe s nlyzed on the bss of Fgue. It s shon tht the consstency coeffcent of the polyme soluton t the ellboe hs no sgnfcnt effect on the go-up of the pessue devtve cuve. Hoeve, s the consstency coeffcent t the ellboe ncese, the ppent vscosty of the polyme t the ellboe ll ncese, the pessue nd ts devtve vlue of the dl flo egme ll ncese. The nfluence of the poe-l ndex t the ellboe s nlyzed on the bss of Fgue 4. The poe-l ndex t the ellboe mnly nfluences the dl flo egme. As the poe-l ndex t the ellboe deceses, the uptuned level of the pessue devtve cuve nceses. It mens tht the flud s Netonn flud hen the vlue of the poe-l ndex s. The moe the poe-l ndex t the ellboe devted fom the vlue of, the moe obvously of the non-netonn flo chctestc s, the gete the flo esstnce s nd the moe obvously the pessue devtve cuve goes up. Copyght ScRes. Openly ccessble t

5 H. J. Yn et l. / Ntul Scence ( 7- lg(p D,lg(p D ' t D 4 H ( =.6P s n H ( =.4P s n H ( =.P s n Fgue. The nfluence of consstency coeffcent t the ellboe. lg(p D,lg(p D ' t D 4 n ( =.5 n ( =. n ( = Fgue 4. The nfluence of poe-l ndex t the ellboe. The nfluence of vton fcto of the consstency coeffcent s nlyzed on the bss of Fgue 5. As the vton fcto of the consstency coeffcent nceses, the dl flo egme dsppe ele, the pessue devtve cuve goes don ele nd deepe nsted of gong on tunng up. It s mnly fo the follong esons: The smlle the vton fcto of the consstency coeffcent s, the fste the consstency coeffcent chnges nd the fste the ppent vscosty of polyme soluton deceses. In the sme peod of flong tme, the smlle the ppent vscosty of the polyme soluton s, the smlle the flo esstnce s. As esult, the pessue devtve cuve goes don ele, the ecessed pt s de nd the dl flo peod stopped ele. The nfluence of vton fcto of the poe-l ndex s nlyzed on the bss of Fgue 6. As the vton fcto of the poe-l ndex nceses, pessue nd pessue devtve vlues e smlle n the sme peod of flong tme, the pessue devtve cuve upps less obvously t the dl flo egme nd the dl flo peod become shote. Wht s moe, the Concve ppes ele nd deepe. It s mnly fo the follong esons. The gete the vton fcto of the poe-l ndex s, the fste the poe-l ndex chnges, the soone the ppent vscosty of polyme soluton deceses, then the smlle the ppent vscosty s nd the loe the flo esstnce s. It s the esons lg(p D,lg(p D ' t D α =-. α =-.6 α = Fgue 5. The nfluence of the vton fcto of consstency coeffcent. lg(p D,lg(p D ' t D 4 β =.5 β =. β = Fgue 6. The nfluence of vton fcto of the poe-l ndex. bove mke the pessue devtve cuve flls ele. 6. CONCLUSIONS Accodng to the behvo tht the polyme soluton hs vscoelstcty nd ts heologcl pmetes e not constnts long the seepge flo decton, the tnsent flo mthemtcl model hs been estblshed. It hs been found tht the type cuve s dffeent fom the cuves thout consdeng the vscoelstcty nd the heologcl pmetes vton. Comped th the cuves smply consdeed the polyme soluton s poe-l flud, the pessue cuve of vscoelstc polyme soluton s hghe nd the pessue devtve cuve uptuns less n the dl flo egme. The gete the elxton tme s, the gete the elstc vscosty of polyme soluton s nd then the gete the seepge esstnce the flud encounteed s. It mens tht the enegy equed n the seepge flo s hghe. So the njecton pessue equed s hghe nd the pessue t the othe pont of the fomton s hghe. The smlle the poe-l ndex of polyme soluton n the bottom hole s, the moe seously the non-netonn behvo s. Hence, t the sme te, the ppent vscosty of the polyme soluton ne the bottom hole s smlle. Ths ll mke the pessue chnges moe slo- Copyght ScRes. Openly ccessble t

6 H. J. Yn et l. / Ntul Scence ( 7- ly nd mke the pessue devtve cuve uptuns moe t the dl flo egme. The consstency coeffcent of polyme soluton n the bottom hole hs no sgnfcnt nfluence on the uptuned degee of the pessue devtve cuve. It mnly hs mpct on the vlue of the pessue nd the pessue devtve. The lge the consstency coeffcent of polyme soluton n the bottom hole s, the lge the vlue of the pessue nd the pessue devtve s. Becuse of the vton of the heologcl pmetes, the ppent vscosty of polyme soluton educes contnuously n the flo pocess, hch mkes the pessue devtve cuve declnes fte the uptuned secton n the dl flo nd mke concve secton ppes befoe the uptuned secton cused by the closed oute boundy. The moe seously the heologcl pmetes chnges, the moe obvously the concve ll be. REFERENCES [] Rnjb, M., Rupp, J., Pusch, G. nd Meyn, R. (99 Quntfcton nd optmzton of vscoelstc effects of polyme solutons fo enhnced ol ecovey. Socety of Plstcs Engnees, 454, 5-5. [] Wng, W.Y. (994 Vscoelstcty nd heologcl popety of polyme soluton n poous med. Jounl of Jnghn Petoleum Insttute, 6, (n Chnese. [] Zhng, Y.L., L, C.H., Wng, X.M. nd Hu, J.B. (994 The vscoelstc effects of polyme solutons n poous med. Jounl of Dqng Petoleum Insttute, 8, 9-4 (n Chnese. [4] L, H., Cheng L.S. nd Zhng S.Y. ( Numecl smulton study fo mpovement of polyme floodng by vscoelstc effect. Petoleum Exploton nd Development, 9, 9-9 (n Chnese. [5] Zhng, L.J., Yue, X.A., Lu, Z.C. nd Hou, J.R. (5 Pecolton mechnsm of polyme soluton though poous med. Jounl of Unvesty Petoleum (Ntul Scence Edton, 9, 5-55 (n Chnese. [6] Yn, H.J., Fu, C.Q. nd Lv, Y.P. (4 An unstedy seepge flo model of vscoelstc polyme soluton. Jounl of Hydodynmcs, 6, 9-5 (n Englsh [7] Co, R.Y., Cheng, L.S., Ho B.Y., Go, H.H. nd Yo, D.W. (7 Mthemtcl model of vscous-elstc polyme soluton seepge. Jounl of X n Shyou Unvesty (Ntul Scence Edton,, 7-9 (n Chnese. [8] X, H.F., Yue, X.A., Co, G.S. nd Zhng, S.F. ( Rheologcl behvo of polyme soluton n the couse of seepge flo. Jounl of Dqng Petoleum Insttute, 4, 6-9 (n Chnese. [9] Wng, X.H. nd Zho, G.P. (998 She te of poel flud though poous med. Xnjng Petoleum Geology, 9, -4 (n Chnese. [] X, H.F. ( The theoy nd pplcton of polyme soluton th vsco elstc behvo. Petoleum Industy Pess, Bejng, 97- (n Chnese. Copyght ScRes. Openly ccessble t