Velocity Projection with Upwind Scheme Based on the Discontinuous Galerkin Methods for the Two Phase Flow Problem

Transcription

1 raoal Joural of Mor Nolar Tory a Applao Publs Ol Ju 5 Rs. p://.srp.org/joural/jma p://x.o.org/.436/jma.5.49 Vloy Projo Up m Bas o Dsouous Galr Mos for To Pas Flo Problm Jagyog Hou Wjg Ya * J C ool of Mamas a ass X a Jaoog Uvrsy X a Ca Cr for Compuaoal Goss X a Jaoog Uvrsy X a Ca Emal: * jgya@mal.xju.u. Rv Mar 5; ap Ju 5; publs 7 Ju 5 Copyrg 5 by auors a f Rsar Publsg. Ts or s ls ur Crav Commos Arbuo raoal Ls (CC BY). p://ravommos.org/lss/by/4./ Absra T up sm s vry mpora umral approxmao of som problms su as ovo oma problm o-pas flo problm a so o. For fraoal flo formulao of o-pas flo problm Paly Dsouous Galr (PDG) mos omb up sm ar usually us o solv pas prssur quao. s as ulss up sm s a o osrao vloy rosruo loal mass bala ao ol xaly. s papr prs a sm of vloy rosruo som H(v) spas osrg up sm oally. Furrmor ffr ays o alula olar offs may av s a sgfa ffs av b vsga by som auors. W propos a algorm o oba a mor ffv a sabl approxmao of offs ur osrao of up sm. yors Vloy Projo Up m Paly Dsouous Galr Mos To Pas Flo Porous Ma. rouo ox of som fls su as molg a smulao of flu flos prolum or grouar rsrvors sus of prosss of smulaous flo of o or mor flu pass a porous mum ar of gra sgfa. s papr osr ass of o-pas flo r flus ar mmsbl. * Corrspog auor. Ho o s papr: Hou J.Y. Ya W.J. a C J. (5) Vloy Projo Up m Bas o Dsouous Galr Mos for To Pas Flo Problm. raoal Joural of Mor Nolar Tory a Applao p://x.o.org/.436/jma.5.49

2 J. Y. Hou al. A larg umbr of mos ar bas o f ffr (FD) f volum (FV) or f lm (FE) mos av b vlop o al o-pas flo problm. As s ll o o mar of umral mos s us up sm s of gra sgfa approxmao of som problms su as ovo oma problm o-pas flo problm a so o. To av sabl umral ompuaos smulao of o-pas flo problm a aura approxmao of flux s o of mos mpora a srabl grs. f us Paly Dsouous Galr (PDG) mos o srz prssur quao l []-[3] bo prssur a saurao quaos ll b srz by PDG mos a a pross of rosruo of vloy s o b o afr prssur quao s solv. [] a avrag oal vloy as pos pross by subsug ps osas of prssur gra a saurao gra o vloy-prssur xprsso rly. Aually su rosru vloy o som lvl blogs o los orr Ravar-Tomas f lm spa. [4] a pos-pross oal vloy s rosru Brzz-Douglas-Mar (BDM) f lm spas. Bu s a gr of polyomal s mor a o a s usg lar approxmao DG mo s o oug o rosru a vloy BDM spa. A mor sabl a aura rosruo as vlop [] vloy rosru from ps lar prssurs oul v blog o frs-orr Ravar-Tomas f lm spa. Hovr all rosruos mo abov osr up sm as basally us srzao of quaos. T propry of loal mass osrvao s rual porous ma flo a raspor problms. T up sm as r ff o loal mass osrvao of rosru vloy. W fou a ulss up sm a paly rms ar us srzao of o-pas flo problm ar osr ogr o vloy rosruo rror of loal mass osrvao ao ra a sasfaory lvl. s papr prs a sm of vloy rosruo som H(v) spas [5] osrg up sm oally. T ffr ays o alula olar offs may av s a sgfa ffs av b vsga by som auors. For approxmao of offs x o us [6] o a a off lm s valua as avrag of up valu o. Ts mprovs sably of umral sm v a xpl sm s us. oras xpl sm srb [] our xpl PDG sm s spal approxmao of offs a o oly g r of xra pals from prssur quao bu also av a robus prforma rogous ma. T rs of arl s orgaz as follos: ao o rouo a oluso v x of s oum o four pars. o s frs par a osss of o subsos rou govrg quaos of o-pas flo problm a orrspog rfa oos ubsos. a. rspvly. T so par o 3 oms four subsos. ubso 3. up avrag approxmaos of offs ar rou. ubsos 3. a 3.3 PDG mos ar us for prssur quao a vloy rosruo s prs rspvly. ubso 3.4 PDG mos ar us for saurao quao. T r par o 4 osss of o subsos. ubsos 4. a 4. rou all possbl projo sms rsp o vloy rosruo a sm ou ay xpl projos. las par o 5 svral umral xampls o msos ar prov.. Problm Mol.. Mamaal Formulao W osr o mmsbl omprssbl flus porous ma a r s o mass rasfr b pass. Varous a alrav mol quaos for o-pas flo problm a b fou rfr [7]. Hr us pas formulao for prmary varabls ar g pas prssur a saurao ( p a ) a abs of gravy a s/sour rm av: ( λ D p λ D p ) + = () p φ + λ fd + ( fu ) = () r oaos a mags of a off a r rlaosps ar f as follos: D s 8

3 J. Y. Hou al. absolu prmably sor a s souous rogous ma; φ os porosy of mum; p s apllary prssur; λ λ a λ ar g og a oal mobly rspvly; f s fraoal flo; a ar gra opraor a vrg opraor rspvly. W ll us Broos-Cory mol [8] rougou s papr som of s offs ar o-lar fuos f blo: p = p θ (3) λ λ ( ) ( ) + θ θ ( ) = (4) µ + 3θ θ = (5) µ λ λ λ = λ + λ f = f λ = λ (6) r = (7) r r r p s ry prssur for g pas o r larg pors ar omplly fll o-g pas; µ a µ ar o-g a g pas vsosy; θ s paramr assoa por sz srbuo; r a r ar rsual saurao. om oaos for ms ar gv blo: Ω s oma; Ω s bouary of oma; s paro of Ω ; s f lm ; s gs of lm ; : = {: all gs } s s of all gs oa ; s s of all ror gs oa. T Equaos () a () ar subj o appropra al a bouary oos o los sysm. Hr gv o fasbl ss of bouary oos: o s Mx-Numa bouary oo as [] ( u λ fd p) u o + = Γ (8) sm λ f D p = g o Γ (9) p N sn = g o Γ () D pd a or s Numa-Drl bouary oo us [9] u = o Γ () pn ( λ ) o u = fd p+ fu = g Γ () p N sn = o Γ (3) r sd = p o Γ (4) r pd u = g o Γ. (5) N pn T ol bouary of porous mum oma Ω s v o r muually sjo pars: flo oflo a ouflo bouars ( Γ Γ o Γ ou ) rspvly. as of Numa-Drl bouary oo Γ pd a Γ sd oupy ouflo bouary Γ pn a Γ sn oupy flo a o-flo bouars su a g N < o flo bouary a g N = o o-flo bouary. as of mx- Numa bouary oo Γ pd oups flo a ouflo bouars Γ pn oups o-flo bouary Γ sm oups flo bouary a Γ sn oups o-flo a ouflo bouars. 9

4 J. Y. Hou al... rfa Coos orr o s barrr ff pomo of o pas flo olar rfa oo suss [] [6] []-[3] ll b rou r. Follog [9] assum a ally fully ar saura oma ( Ω=Ω Ω ) a rfa Γ J b o ffr sas a ol s j from flo par of bouary Γ s Fgur. ao assum a Ω sas oars sa a Ω s f sa. T pross of pomo s srb brfly blo. Frs ol approas maral rfa bu ao pra a bg o aumula. s as oly ar prssur p s ouous o rfa apllary prssur p a saurao ar souous a sasfy: θ p Ω = p p Ω = p = ( r ). T mor a mor ols aumula a rfa a apllary prssur o oars s xs ry prssur of or s ( p Ω p ) ols bg o pra a r f sa. A s m bo p a p ar ouous bu saurao s sll souous a sasfs: p r = ( r r ) θ r. p r r W o a a ral po of saurao a b fou apllary prssur o oars s rass o valu quval o rsol prssur o f s. Ta s ug from p = p av θ * p = ( r r ) + r p. Ts po ll b us o jug r og pas a or ao pra maral rfa. o rfa oos a b rr form blo. For apllary prssur a for g pas saurao p Ω θ * p p p Ω = > * θ θ Ω > * r θ = θ p θ r * ( r r ) r. p r r (6) (7) (8) (9) () Fgur. T rfa (as l) b o subomas ffr ro proprs. 3

5 J. Y. Hou al. Coo () s sam as a srb [] xp a g pas (sa of og pas) s us as saurao varabl. Morovr (9) s oly r for apllary prssur a o for g pas prssur s varabl p s alays ouous problm suss. Nog a f sub-oma Ω as a fr xur a Ω all rlaosp abov a b ra a smlar mar suprsrp a rvrs. 3. Dsr ms 3.. Approxmao of Coffs For approxmao of offs x o us [6]. L o ay offs ag for som propr approxmaos. Frsly rall orgal ay o approxma offs T approa srb [6] s = =. = = 3 ( T ) ( T ) r T os ma ar saurao o g of lm s [6] for mor als. No s x o follog o () () = = 3. (3) r up valu of s avrag o ror g s osr. T quay s all up flux s o rsp o ormal ompo of oal vloy u su a for all + u = + u >. (4) r ormal vor pos from + o. Trougou s papr all offs o lm a g ar alula by up avrag osa a gral avrag osa ar srb (3). 3.. Prssur Approxmao PDG s so apply Paly Dsouous Galr (PDG) mos [4] su as Nosymmr ror Paly Galr (NPG) o prssur Equao (). om oaos for DG mos ar f: + + v : = v + v v : = v v (5) { } [ ] { } [ ] X { L P } v : = v v : = v Ω (6) : = Ω (7) 3

6 J. Y. Hou al. r v ± ar rsros of v o o aja lms ± rspvly. T prssur Equao () srz by PDG ras as follos. + f p X for all v X { λ }[ ] λ D p v D p v + + ΓD + ε + β + + { λ D v } p p [ v] ΓD ΓD p p = λ D v + λ D ΓD + ε λ D vp + p v g v. r r N ΓD ΓD β ΓD [ v] (8) PDG mos ar oly appl o g-pas prssur rm for apllary prssur rm a raoal DG mo up sm s us Vloy Rosruo Afr solvg sr prssur Equao (8) oal vloy ll b rosru los-orr Ravar-Tomas spa ( RT ) frs-orr Ravar-Tomas spa ( RT ) a frs-orr Brzz-Douglas- Mar spa ( BDM ) rspvly rfr o [5] for mor als abou os spas. T ma a of rosruo urr so follos o p [] a ll x o suao a srzao of prssur quao oas a up sm. A propr rosruo of vloy sms from loal mass osrvao la as so follog srpo. Frsly ras varaoal Equao (8) o lm o o pars as follos + + p + u v λ D p v λ D v ε { λ D v } p (9) = + + p β u v = { λ D p }[ v] λ D [ v] + p [ v]. (3) Combg (9) a (3) s asly s a loal mass s osrv ( ) = + = ( + ) u v u v u v q q v (3) r q a q ar s a sour rms ar zros r. Nog a f g blogs o bo a Γ D o rg a s of Equaos (9) a (3) p + s qual o + ± ( p p ) r a sg ± s rm by ro of for xampl sg s posv s our ormal vor rsp o. oly usg (9) a (3) as gr of from for som H(v) spas oal vloy ll b oba as som appropra projos or rpolaos s spas. orr o av a propr rpolao RT RT a BDM spas soul spfy a s of gr of from (DOF) for s H(v) spas a a orrspog s of bass fuos. f l v b ay osa polyomal spa of gr zro P (9) ll vas a (3) ll bom RT spa s DOF s gral of ormal ompo of vloy o a g. Corrspogly s of bass fuos for RT o rfr lm s ˆ ϕ = ( x a) = 3 (3) ˆ r ˆ s ara of rfr lm ˆ a a s o of s vrs. 3

7 J. Y. Hou al. L v a v b ay fuos spa of polyomal of gr o P (9) a (3) bom DOFs for RT. T orrspog bass fuos for RT spa o rfr lms ar 6x 8xy 8x 8xy x + 6x ˆ ϕ ˆ = ϕ = 6x + y 8xy 8y 4 6xy 4y x + 8y x 4 x x+ ˆ ˆ 8 xy 8 x ϕ3 = ϕ4 = 8xy y 8y 8xy y 8xy 6y x + 4x + y 6xy 8x 6 ˆ ϕ ˆ 5 = ϕ 6 = 8y 4y y 8xy 6y 6x 8xy 6x 8x 6xy 8x ˆ ϕ ˆ 7 = ϕ8 =. 8y 6xy 8y 6y 8xy 6y L v b ay fuos P polyomal spa us bass fuos of BDM a b oba a smlar mar xp a (9) s o us. All DOFs for BDM ar jus f o gs of lm so oly (3) s us o rm bass fuos. T orrspog bass fuos for BDM ar x 6x ˆ ϕ ˆ = ϕ = 6x+ 4y 4 6 6y x 4x 6x ˆ ϕ ˆ 3 = ϕ4 = y 6y 6y x 6x+ y 6 ˆ ϕ ˆ 5 = ϕ6 =. 4y 6y s o a o of DOFs for BDM a RT spas s o uqu for xampl alf-g gral of ormal ompos of vloy s also avalabl a applabl aurao Approxmao T spaal srzao of saurao quao s smlar o a of prssur quao gv (8). T ffuso rm of saurao quao s srz by PDG mos a avv rm s srz by a raoal DG mo usg up sm. A Eulr sm m s us. T saurao Equao () qupp Mx-Numa bouary oos (8)-() oul b r as: + f X for all v X φ p v λ f D v + + p + p + + λ f D [ v] ε λ f D v + + u v+ [ v] β Γ sm φ = v + fu v f u v g v u v. [ ] ΓsN N ΓsM T varaoal form rms of Numa-Drl bouary oos ()-(5) ras: (33) 33

8 J. Y. Hou al. X v X + f for all φ p v λ f D v + + p + + λ f D v ΓD p ε λ f D v v β ΓD Γ D φ = v + ( fu ) v ( fu ) [ v] p gnv ε λ f D v r + rv β ΓN Γ D ΓD p ε λ f D v J + v J β Γ Γ [ ] ΓD [ ] r J( ) s rfa oo of saurao srb (). 4. Fasbl Projos of Dsr rags 4.. DDG Mos om Or Projos T abbrvao DDG mas a DG mos ar us for bo prssur a saurao quaos. For a lar omparso umral xprms ls all possbl a fasbl projos blo. Frsly o RT () (or BDM () ) as vloy spa proj o RT (or BDM) spa by (9) a (3). oly RT () (or BDM () ) mas projo o RT (or BDM) spa osrg up sm bu ou paly rm Trly for RT (or up sm p + λ + λ (35) u v = D p v+ D v + + u v = { λ D p }[ v] λ D [ v]. (36) p BDM ) mas projo osrg paly rm bu ou [ ] (34) + + p u v = λ D p v+ λ D v { λ D v } + ε p + (37) A las for RT (or + + u v = { λ D p }[ v] λ D [ v] + p β + [ v]. p BDM ) mas projo ou osrg bo paly rm a (38) 34

9 J. Y. Hou al. up sm p + λ + λ (39) u v = D p v+ D v + + u v = { λ D p }[ v] λ D [ v]. (4) As a rouo for all DDG mos vloy rv by projo RT (or prsrvs loal mass osrvao propry bs ll so umral xampls. 4.. DDG Mo ou Expl Projos p BDM ) [] vloy s us rly as ombao of gra of soluos a offs as follos + + p u = { λ D p } λ D (4) p + = λ + λ u D p D r avrag of oal vloy s us ror gs. Aloug os us ay projos xplly vlos osru from (4) a (4) ar som of mpl projos o RT spa. T vloy rv from (4) a (4) s los o vloy projo RT spa s osru by (39) a (4). Bu r valu a lm s ffr. Furrmor DDG mo usg vloy rosruo prs s subso as ra ffrs oras o a propos [] ar rfl o asps blo: ) T varaoal form of saurao quao os orpora ay aoal pals from prssur quao. ) T approxmaos of offs ar oally ffr. 5. Numral Exampls s so prs som ompur xprms o xam propos mos o o msoal spas. Bo o bouary oos ffr yps ar us xamao of all mos. ss a osr splam of o-g pas by g pas s smlar o so all quarr-fv spo problm rou []. s 3 osr splam of g pas by o-g pas s us o smula barrr ff [9]. T omas us xprms ar squar ( ) o orrs b u off a for ms us souous problm a small squar ffr ro propry s fx s oma s Fgur (a) a Fgur (b). a s us Nosymmr ror Paly Galr (NPG) mo paly paramrs ε = = a β =. orr o prv osllaos a slop lmr prour srb [5] s us. f osrg mx-numa yp bouary (8)-(9) for saurao quao follog al a bouary oos ar us: (4) = =. (43) =.9 o Γ Γ (44) sm g = m s o Γ Γ Γ (45) N sn o ou 6 r 3.45 Pa o pd p = Γ Γ (46) 35

10 J. Y. Hou al. (a) Fgur. Mss us xprms. (a) quarr-fv spo ms us omogous mum; (b) quarr- fv spo ms us souous ma. 6 r.4 Pa o pd ou p = Γ Γ (47) u = m s o Γ Γ. (48) pn o W Numa-Drl yp bouary ()-(3) s us for saurao quao follog al a bouary oos ar osr: = = (49) u u = Γ Γ (5) m s o sn m s o sn o = Γ Γ (5) = o Γ Γ (5) r sd ou u = Γ Γ (53).5 m s o pn u = Γ Γ (54) m s o pn o 5 r. Pa o pd ou. p = Γ Γ (55) T paramrs lug ro a flu proprs us smulao ar summarz Tabl. 5.. Ts s xam propry of loal mass osrvao la. For s purpos solv so-all quarr-fv spo problm o a omogous mum a umral loal mass of rosru vloy. All projo mos suss abov ll b us a ompar. T oma us xprm s squar ( ) o orrs b u off s Fgur (a). T al a bouary oos (43)-(48) s us. T paramrs rsp o ro propry a Broos-Cory mol ar ls Tabl Ts. Fgur (a) spo a lf boom s flo bouary Γ ouflo bouary Γ ou s loa a rg op orr rs of bouary s oflo bouary Γ o. To ma sur a ar fro says s oma fal m s s o T = 6 s. W us a osa m sp a rao of m sp o spa sp s squar s abou 4.5. W us DDG o o DDG mo ou (b) 36

11 J. Y. Hou al. Tabl. Paramrs us umral smulaos. porosy φ =.4 D = 5.4 ; Ts prmably [m ] [ ] vsosy[g/(ms)] 3 µ =. µ =. 4 rsual saurao =.5 r =. r Broos-Cory 3 p = 3 Pa θ = 3 Ts porosy φ =. D = ; prmably [m ] [ ] vsosy[g/(ms)]. 3 µ = µ = 5. 4 rsual saurao =.5 =. r r Broos-Cory 3 p = 5 Pa θ = Ts 3 porosy φ =. φ =. prmably [m ] D = [ ;] D = [ ;] vsosy[g/(ms)] µ =. µ =. 3 rsual saurao = = = = r r r r Broos-Cory 4 p = Pa 4.5 Pa p = θ = θ = xpl projos. r s o s a sour rms xa loal mass s zro o a lms a s u = r s ouar u ormal vor o. Tus a asly f rrors of loal mass osrvao ur vor orms l a l ar rspvly max u a u. T rrors of loal mass osrvao a sl ms ar ls Tabl a Tabl Ts s s so umral soluos solv by our sm usg projo RT a omogous ma. For rsuls usg projo BDM a RT y ar smlar usg RT so ar om. T ms us s s sam as prvous s.t al a bouary oos (43)-(48) ar us s s. To ma sur a ar fro says s oma fal m s s o T = 8 s. A osa m sp s us a rao of m sp o spa sp s squar s abou 4.5. T paramrs of ro propry a Broos-Cory mol ar ls Tabl Ts. T oours of g pas saurao omogous mum a sl ms ar prs Fgur Ts 3 las s xam our sm a souous ma. W assum a oma us r s ally fully ar saura a rfas b o ffr sas s Fgur 4. Ω s f + sa a Ω s oars sa so ol-rapp pomo ll appar o rfas Γ J s Fgur 4. * + * T ral po (9) s.44 a ol ll pra rfa Γ J. T ms (56) 37

12 J. Y. Hou al. Tabl. Numral rrors of loal mass osrvao s a ms = 4 s a = 8 s. = 4 s = 8 s l l l l DDG RT ( ) RT RT RT BDM ( ) BDM BDM BDM RT ( ) RT RT RT Tabl 3. Numral rrors of loal mass bala s a ms = s a = 6 s. = s = 6 s l l l l DDG RT ( ) RT RT RT BDM ( ) BDM BDM BDM RT ( ) RT RT RT

13 J. Y. Hou al. Fgur 3. T oours of g pas saurao omogous mum a sl ms Ts. Fgur 4. Dsouous quarr-fv spo problm. us s s s Fgur (b). T al a bouary oos (49)-(55) ar us s s. T al a bouary oos (49)-(55) ar us s s. To ma sur a ar fro says s oma fal m s s o T = s. A osa m sp s us a rao of m sp o spa sp s squar s abou 4.5. T paramrs of ro propry a Broos-Cory mol ar ls Tabl Ts 3. W ol flos from oars sa o f sa jo of ol from flo bouary Γ mor a mor ol approas a aumulas a fro of rfa of f sa. W aumulao ras a ral po a s apllary prssur a oars s of rfa s grar a a f s aumula ol ll pra rfa a r f sa ara. By oras 39

14 J. Y. Hou al. Fgur 5. T oours of g pas saurao souous ma a sl ms Ts 3. rvrs ro ol mmaly pra rfa a s ol-rapp pomo ll o app f ol flos from f sa o oars sa. T oours of g pas saurao souous ma a sl ms ar prs Fgur Coluso T vlos rosru from projos RT BDM a RT ar mu br o prsrv loal mass osrvao propry a ors. Ta s vloy rosruo projo a osrs bo up sm a paly rm a bs prsrv loal mass osrvao propry. T approxmao of off (3) s vry ssal o sably of all DDG mos. sa of (3) f approxmao of off () s us varaoal form of saurao quao as o orpora aoal pals from prssur quao; ors sm ll b usabl. Fug T or s suppor by Naoal Naural Fouao of Ca (No.3788 a No.4467). Rfrs [] Er A. Mozolvs. a u L. () Dsouous Galr Approxmao of To-Pas Flos Hro- 4

15 J. Y. Hou al. gous Porous Ma Dsouous Capllary Prssurs. Compur Mos Appl Mas a Egrg p://x.o.org/.6/j.ma.9..4 [] lbr W. a Rvèr B. (6) Aapv mulaos of To-Pas Flo by Dsouous Galr Mos. Compur Mos Appl Mas a Egrg p://x.o.org/.6/j.ma [3] Mozolvs. a u L. (3) Numral mulao of To-Pas mmsbl omprssbl Flos Hrogous Porous Ma Capllary Barrrs. Joural of Compuaoal a Appl Mamas 4-7. p://x.o.org/.6/j.am [4] Basa P. a Rvèr B. (3) uprovrg a H(v) Projo for Dsouous Galr Mos. raoal Joural for Numral Mos Flus p://x.o.org/./fl.56 [5] Brzz F. a For M. (99) Mx a ybr f lm mos. prgr N Yor. p://x.o.org/.7/ [6] Nayagum D. äfr G. a Mosé R. (4) Mollg To-Pas omprssbl Flo Porous Ma Usg Mx Hybr a Dsouous F Elms. Compuaoal Goss p://x.o.org/.3/b:comg [7] C Z.X. Hua G.R. a Ma Y.L. (6) Compuaoal Mos for Mulpas Flos Porous Ma. AM Plalpa. p://x.o.org/.37/ [8] Broos R.H. a Cory A.T. (964) Hyraul Proprs of Porous Ma. Colorao a Uvrsy For Colls. [9] Fuč R. a Myša J. () Mx-Hybr F Elm Mo for Mollg To-Pas Flo Porous Ma. Joural of Ma-for-usry [] Ho H. a Froozaba A. (8) Numral Molg of To-Pas Flo Hrogous Prmabl Ma Dffr Capllary Prssurs. Avas War Rsours p://x.o.org/.6/j.avars [] Eéry G. Eymar R. a Ml A. (6) Numral Approxmao of a To-Pas Flo Problm a Porous Mum Dsouous Capllary Fors. AM Joural o Numral Aalyss p://x.o.org/.37/46936 [] Caès C. (9) F Volum m for To-Pas Flo Hrogous Porous Ma volvg Capllary Prssur Dsous. EAM: Mamaal Mollg a Numral Aalyss p://x.o.org/.5/ma/93 [3] Caès C. Gallouë T. a Porra A. (9) To-Pas Flos volvg Capllary Barrrs Hrogous Porous Ma. rfas a Fr Bouars p://x.o.org/.47/fb/ [4] Rvèr B. (8) Dsouous Galr Mos for olvg Ellp a Parabol Equaos: Tory a mplmao. AM Plalpa. p://x.o.org/.37/ [5] Cav G. a Jaffré J. (986) Mamaal Mols a F Elms for Rsrvor mulao. us Mamas a s Applaos. Nor-Holla Amsram. 4