Chapter 6 Conservation Equations for Multiphase- Multicomponent Flow Through Porous Media

Transcription

1 Chaper 6 Conservaon Equaons for Mulphase- Mulcomponen Flow Through Porous Meda The mass conservaon equaons wll appear repeaedly n many dfferen forms when dfferen dsplacemen processes are consdered. The basc mass conservaon prncple s general and wll be derved us once here and specal cases wll be consdered as subses of he general case. Only sohermal processes wll be consdered here. The maeral used here s exraced from he book by Larry W. Lake, Enhanced Ol Recovery, Prence Hall, Ths book s a recommended reference for hs class. The Connuum Assumpon If we were o he vew he porous medum on a lengh scale on he order of he pore sze, he pore space wll appear exremely dealed and chaoc as n he elecron mcrographs we saw earler. Also, he fluds n he pore space wll be dsrbued n pars of he pore space ha depends of he weably and sauraon hsory n addon o he sauraon self. If we hope o model he flow and ranspor of phases and speces on a macro or mega volume of he porous meda, s necessary o average he descrpon of he sysem over a represenave elemenary volume (REV) ha s large compared o he sze of he pores bu small compared o he macroscopc dmensons of he porous medum. The properes of he sysem wll be descrbed by connuum varables such as porosy, permeably, sauraon, relave permeably, and capllary pressure. Local Thermodynamc Equlbrum I he followng s assumed ha speces whn he REV s n local hermodynamc equlbrum. In means ha he chemcal poenal of each speces s he same n all phases whn he REV. There are some cases when hermodynamc equlbraon s no acheved and s necessary o nclude knec or mass ransfer erms o he mass balance. Examples are mneral dssoluon and ransformaon n he presence of alkalne soluon, laboraory expermens n shor cores a hgh raes, and vscous fngerng wh solven floodng. Fluxes n Isohermal Flow Volumerc Flux We derved earler Darcy's law from momenum conservaon for he flow of a sngle phase n a represenave elemenary volume (REV) of he porous meda. We wll now generalze he defnon of Darcy's law for he volumerc flux of phase n a sysem ha can have N p phases. 6-1

2 u ( p ρ g D) = λ k r where he vscosy n now replaced by he relave mobly of phase, kr λr = μ The pressure s also denfed wh a phase. Two phases ha are coexsng a he same place can have dfferen pressures as a resul of he curvaure of he nerface separang he phases. Ths pressure dfference s denfed as he capllary pressure, P cwo and P cgo. P = p p P = p p cwo o w cgo g o The drvng forces for he volumerc flux of mulple phases can now be denfed as due o: (1) pressure graden, (2) capllary pressure graden as a resul of a graden n he nerfacal curvaure whch can resul from a graden n sauraon or pore szes, and (3) body forces or buoyancy ha s dfferen for phases wh dfferen denses. Dspersve Flux Whn a sngle phase here s a dffusve flux due o a concenraon graden ha can be descrbed by Fck's law. In porous meda here also s dsperson due o mxng of fluds passng hrough he mulple flow pahs hrough he porous medum. The dspersve flux can be expressed as follows. N p N = φ ρ S K ω dsperson = 1 where ρ densy of phase ω mass fracon of speces n phase K dsperson ensor of speces n phase The longudnal and ransverse componens (.e., parallel and perpendcular o he velocy vecor) of he dsperson ensor n an soropc medum are as follows. 6-2

3 ( K ) ( K ) D α u = + τ φs D α u = + τ φs where D molecular dspersvy of speces n phase τ oruosy α l longudnal dspersvy coeffcen n phase α ransverse dspersvy coeffcen n phase Mass Balance Consder an arbrary, fxed volume V embedded whn a permeable medum hrough whch s flowng =1,2,...,N c chemcal speces dsrbued among =1,2,...,N p phases. The volume V s greaer han equal o he REV bu smaller han or equal o he macroscopc porous medum dmensons. As Fg. 6.1 shows, he surface area of V s made up of elemenal surface areas ΔA from he cener of whch s ponng a un ouward normal vecor n (bold ype wll be used for vecors n he ex). The sum of all he surface elemens ΔA s he oal surface area of V. Ths sum becomes a surface negral as he larges ΔA approaches zero. The mass conservaon equaon for speces n volume V s Fg. 6.1 Arbrary elemen of volume (Lake 1989) rae of Ne rae of Ne rae of accumulaon = ranspored + producon, = 1, 2,..., N c of n V no V of n V The mass conservaon equaon s usually saed as "accumulaon equals ne flux (or npu mnus oupu) plus ne source (or generaon)". These wll be derved n he followng. The accumulaon erm for speces s 6-3

4 rae of Toal mass accumulaon = = WdV of n V V of n V where W s he overall concenraon of n uns of mass of per un bulk volume. Snce V s a fxed volume, = WdV V V W dv The ne flux erm follows from consderng he rae of ranspor across a surface elemen as shown n Fg Le N be he flux vecor of speces evaluaed a he cener of ΔA n uns of mass of per surface area-me. The ranspor of across ΔA s gven by he negave of he normal componen of N Ne rae of ranspor = n N da A of no V Ths equaon can be ransformed no a volume negral over V by applyng he dvergence heorem. Ne rae of ranspor = NdV V of no V The ne rae of producon of n V s Ne rae of producon = R dv V of n V where R s he mass rae of producon of n uns of mass of per un bulk volume-me. Ths erm can accoun for boh producon or generaon (R > ) and desrucon (R < ) of, eher hrough one or more chemcal reacons or hrough physcal sources (wells) n V. Combnng each erm no he mass balance equaons gves, W + N R dv =, = 1,2,... Nc V Bu snce V s arbrary, he negrand mus be zero. 6-4

5 W + N R =, = 1,2,..., Nc Ths equaon s he dfferenal form for he speces conservaon equaon. I apples o any pon whn he macroscopc dmensons of he permeable medum. In he nex secon we gve specfc expressons for he erms n he equaon. Defnons and Consuve Equaons for Isohermal Flow Table 2.2 (Lake 1989) summarzes he equaons needed for a complee descrpon of sohermal, mulphase flow n permeable meda. Column 1 n Table 2-2 gves he dfferenal form of he equaon named n column 2. Column 3 gves he number of scalar equaons represened by he equaon named n column 1. Columns 4 and 5 gve he deny and number of ndependen varables added o he formulaon by he equaon n column 1. N D s he number of spaal dmensons (N D 3). The saonary phase s reaed here as a sngle homogeneous phase hough more han one sold phase can exs (e.g., dssoluon and precpaon of mnerals). A normally subscrped quany (for example ω ) appearng whou subscrps n Table 2-2 ndcaes a relaonshp nvolvng, a mos, all members of he subscrped se. In he lsng of dependen varables, he prmary meda properes, such as he porosy φ, and he permeably k, are gven funcons of poson x whn he porous meda. These funcons are, o a slgh degree, a funcon of he flud pressure because of compacon. The frs four equaons n Table 2-2 are he speces conservaon equaon and defnons for he accumulaon, flux, and source erms n hs equaon. The overall concenraons are formally defned here as an overall mass fracon. When we ge o specal cases, we wll see ha here are convenen ways for defnng he overall concenraon n erms of phase concenraons and pore volume. 6-5

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7 Connuy Equaon If we nser Eqs hrough no 2.19 we arrve a Np Np φ ρ Sω ( 1 φ) ρsωs ( ρ ωu φ ρ ω ) + + S K = 1 = 1 N p = 1 ( 1 φ) = φ Sr +, =1, 2,..., N c r s Ths s he overall mass conservaon equaon for speces. We sum over he N c componens o oban he equaon of connuy or conservaon of oal mass. Np Np φ ρ S ( 1 φ) ρs ρu + + = 1 1 = = Concenraon Varables The concenraons were expressed here as a mass fracon. Ths choce of concenraon s he mos general. When we examne specal cases we wll fnd oher concenraon varables o be more convenen. In he case of ncompressble, mmscble fluds he only changng parameer s he phase sauraons. In he case of on exchange, he concenraon may be expressed n moles or equvalens per un aqueous volume. Tracer quanes may be expressed as pars per mllon, ppm. Ol feld pracce expresses he quany of ol as sock ank barrel (STB) and he quany of gas as housand cubc fee of gas a sandard condons (Mcf) 6-7

8 Specal Cases We wll now consder specal cases of he general mass conservaon equaons derved earler. One Dmensonal Dsplacemen - Fraconal Flow Formulaon The classcal equaon for wo phase dsplacemen s he case of one dmensonal, homogeneous, ncompressble, mmscble, sohermal, dsplacemen wh no capllary pressure (P c =), sorpon (W s =), reacon (R =), or nerphase mass ransfer (ω =δ ). The mass ransfer equaons reduce o, S φ u + =, = 1,2,..., N x P The connuy equaon and he sauraon defnon reduces o, u = x where N P u= u = 1 where u s he oal volumerc flux. The connuy equaon reduces he number of ndependen speces conservaon equaons by one. The pressure graden can be elmnaed from Darcy's law by expressng he fluxes n erms of he fraconal flow funcon. f N p u λr kg snα = = 1 N λrk ( ρ ρk) P u u k = 1 λ where k = 1 dd snα = dx z rk where α s he dp angle,.e., he nclnaon of he drecon of he dsplacemen relave o he horzon, measured posve n he couner clockwse drecon. The flux expressed as a funcon of he fraconal flow can be subsued no he conservaon equaon and he oal flux facored ou o gve he conservaon equaon n erms of he fraconal flow. 6-8

9 S u f + =, = 1,2,..., N φ x P The nal condons s usually a unform sauraon wh only one phase moble. The boundary condons are a specfed oal flux and fraconal flow a he nle end. There s no down sream boundary condons. Buckley and Levere (1942) frs solved hs equaon for wo-phase flow, and he waer flood or gas flood calculaons usng hs equaon s called he Buckley-Levere heory. Ths heory has been exended o much more complex sysems (Larson 1982; Pope 198). Assgnmen 6.1 Buckley-Levere Equaon Derve he precedng equaons (ncludng he fraconal flow funcon) sarng from he equaons n Table 2.2. Sae your assumpons a each sep. Mul - Dmensonal, Immscble, Incompressble Dsplacemen The fraconal flow formulaon s no useful n muldmensonal dsplacemen because he oal volumerc flux s no ndependen of poson as n one dmensonal dsplacemen. However, he equaons sll smplfy n he case of mmscble, ncompressble dsplacemen n muldmensons. Raher han dervng he connuy equaon as a condon of conservaon of oal mass, he densy can be facored ou of he speces conservaon equaons and he sum of he equaons s hen a saemen of he conservaon phase volumes. Snce he sum of he sauraons s equal o uny and hus ndependen of me, he connuy equaon reduces o he saemen ha he dvergence of he oal flux s zero. N P u = = 1 When Darcy's law s subsued no he above equaon he resul s an ellpc paral dfferenal equaon for pressure of one of he phases. The pressures of he oher phases are relaed hrough he capllary pressure relaons. The phase sauraons can be calculaed from he N p -1 speces conservaon equaons. A condon for he exsence of a soluon o he sysem of equaons s ha he ne sum of he volumerc sources and snks mus be zero f here s no flow ou of he boundares of he sysem. 6-9

10 Mscble Dsplacemen We now rea he sohermal case of many componens flowng smulaneously n a sngle flud phase. Thus only one phase flows regardless of composon, bu boh convecon and dsperson of hese componens mus be ncluded. Mscble processes of neres nclude (1) ruly mscble dsplacemen of ol by a solven; (2) chromaographc processes such as analycal chromaography, separaon chromaography, on exchange process, and adsorpon of chemcals as hey percolae hrough sols or oher permeable meda; (3) leachng processes such as he n su mnng of uranum; and (4) chemcal reacon processes of many ypes n fxed bed reacors. The speces mass conservaon equaon for sngle phase mscble dsplacemen s ( ) φρω ( 1 φ) ρsωs [ ρωu φρ ω ] R + + K = = 1,2,..., N c The second subscrp for he phase number s now superfluous and has been dropped. The auxlary Eqs. (2.2-5), (2.2-6), (2.2-8), and (2.2-1) hrough (2.2-12) are sll needed, bu he ohers are no longer pernen. Eq (2.2-5) or Darcy's law has a consderably smpler form, k u = ( p ρg D ) μ If densy s consan, he pressure and deph gradens can be replaced wh a flow poenal graden. Snce he relave permeably s consan, s lumped wh he absolue permeably. However, he vscosy may be srongly dependen on he composon. If he sauraon of he flowng phase s no uny (.e., here s an mmoble phase) can be lumped wh he porosy. For mscble solvens, he sorpon and reacon erms are zero, gvng ( φρω ) + = = [ ρω u φρk ω ], 1,2,..., A specal one dmensonal case of he above equaon s obaned when he effec of composon and pressure on densy s negleced and he dspersvy s consan. Leng C =ω ρ be he mass concenraon of componen, follows ha N c 2 C C C φ + u = φk, 1,2,..., = N 2 x x C 6-1

11 where K he longudnal dsperson coeffcen, s now a scalar, K D τ α u = + φ Because of he connuy equaon, u s ndependen of x. If D s aken as a consan, he above equaon reduces o he lnear convecve-dffuson equaon. Chromaographc Transpor The equaons for chromaographc ranspor are specal cases of he of he convecve dffuson equaon excep ha he sorpon erms, C s, mus be reaned. The bass for chromaographc ranspor s he separaon of componens due o he dfference n C s. These sorpon reacons may be adsorpon, he exchange of one on for anoher on he saonary subsrae, or precpaon-dssoluon reacons. Dsperson may smear dsplacemen frons and reduce peak concenraons bu wll no aler he relave ranspor of he speces. Thus he dsperson erm wll be negleced so ha he equaons can be nvesgaed wh he nsghful mehods avalable for hyperbolc paral dfferenal equaons, e.g., mehod of characerscs and coherence heory. These frs order paral dfferenal equaons are somemes called he chromaographc equaons. C Cs C φ + ( 1 φ) + u =, = 1,2,..., N x Semmscble, Incompressble, One Dmensonal Dsplacemen wh Ideal Mxng Helfferch (1981) has shown ha he hghly developed heory for chromaographc dsplacemen can be also appled o mulphase, mulcomponen sysems. The componens can paron beween he phase wh known equlbrum relaonshps. However, an assumpon of deal mxng s needed for he oal volumerc flux o be ndependen of poson. Also, he concenraons are expressed as volume fracons. C 6-11

12 1 ρ N C ω =, = 1,2,..., N o ρ = 1 P C ω ρ = ρ o ω = = 1 NC NC C ρ o = 1 = 1 ρ If we assume ncompressble fluds, consan porosy, and deal mxng and dvde he speces conservaon equaon by he respecve pure componen densy, we have NP NP φρ S φ CS + ( 1 φ) Cs + Cu K ω o = = 1 = 1 ρ = 1,2,..., N C If we furher assume he phase densy can be ncluded n he graden, he equaon can be expressed n erms of C. If we also assume ha he dvergence of he sum of he dffusve fluxes s zero, he sum of he speces balances gves N P u = = 1 Ths means ha n one dmenson he oal volumerc flux s ndependen of poson. Wh hese assumpons, he equaons reduce o NP NP NP C φ CS + ( 1 φ) Cs + u fc φsκ = = 1 x = 1 x = 1 x = 1,2,..., N C If we neglec he dsperson erms, hese equaons are mahemacally equvalen o he chromaographc equaons. However, phase sauraons and fraconal flows are now addonal dependen varables. Sngle Phase, Consan Compressbly When only one phase s flowng and only small perurbaons n pressure s consdered, he dependence of denses and porosy can be lnearzed n pressure and analycal soluons can be derved from he resulng lnear 6-12

13 parabolc paral dfferenal equaons. These soluons are used o nerpre ransen pressure analyss for reservor characerzaon. The compressbly of a phase s defned as he relave change n flud volume per un change n pressure. 1 v c = v p 1 ρ = ρ p The formaon compressbly s defned as he relave change n porosy per un change n pressure. c f 1 φ = φ p A change n porosy wh reducon n pressure s usually assocaed wh compacon of he bulk volume. Raher han rgorously defnng a coordnae sysem ha follows he compacon, we wll us assume ha he rock mass per un volume remans consan as he porosy changes. I s assumed ha he changes n sauraon wh change n pressure can be negleced. The small changes n sauraon ha occur wh compacon and dfference n he phase compressbles may be negleced for small perurbaons n pressure bu dssoluon or resoluon of gas can no be negleced. The accumulaon erm of he connuy equaon wll be expressed as a funcon of pressure. p N N P N P φ ρs = φ ρcs + cfφ ρs = 1 = 1 = 1 p There wll only one erm n he dvergence of he connuy equaon, he flux of he one phase ha s flowng. Denoe he phase ndex of hs phase as 1. Assume ha he change n densy of hs phase can be negleced n he dvergence. The connuy equaon hen reduces o he followng equaon. φ ρ cs ρ S p + + u1 = ρ ρ NP NP cf = 1 1 = 1 1 The facor nsde he bracke s he oal compressbly of he formaon and s fluds. I s ofen presened whou he rao of denses. Ths rao resuls from he rgorous dervaon. 6-13

14 c ρ cs = + ρ S NP NP cf = 1 ρ1 = 1 ρ1 I s convenen o express he pressures relave o a daum,.e., express as a flow poenal. ( ) Φ= p ρ g D D o where D o s a daum deph o whch all pressures wll be referred. In he followng equaon he symbol p wll be used n place of Φ. Assume ha he permeably and vscosy are consan and can be facored ou he dvergence. p φ c o kr1k η= φμ c o r1 2 p= k k μ 1 p 2 = p η where Ths equaon s he hea or dffuson equaon and many classcal soluons exs. The parameer η s called he hydraulc dffusvy. 6-14

15 Overall Maeral Balance An overall mass balance s useful because (1) summarzes he behavor of he sysem wh a few "lumped" parameers, and (2) he fne dfference soluon of he dfferenal equaons should be conssen wh he overall maeral balance. The overall mass balance s done by negrang he dfferenal speces balance (Eq n Table 2-2) over he volume of he sysem and from an nal me o a me. I s assumed ha here are no reacons,.e., R =. The negral of he accumulaon erm s as follows. W ' W ' dv d = V d V = V W () W ( ) The negral of he dvergence erm s deermned from he dvergence heorem for fluxes a boundares and sum of he snks and sources. NdVd= n NdA+ ( q qi) d ' P V A ( ) = qb + qp qi d ( Q Q Q ) = + B P I where q s a mass flow rae and Q s he cumulave flux, producon or necon. The saemen of he overall mass balance of speces s now () ( ) = ( ) V W W Q Q Q I B P These overall maeral balance equaons are very useful n esmang he volume of a reservor and/or he parameers for he nflux from an aqufer. Also, hese maeral balances should always be used o valdae numercal smulaon resuls. ' ' 6-15