Numerical solution of compressible fluid flow in porous media with boundary element method

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1 Flud Sruur Inraon and Mong Boundary Probls IV 43 Nural soluon o orssbl lud low n orous da wh boundary ln hod R. Jl, L. Šrg & J. Krar Fauly o Cl Engnrng, Unrsy o Marbor, Slona Fauly o Mhanal Engnrng, Unrsy o Marbor, Slona Absra In h rsn wor a ully dlod boundary ln hod nural sh s rsnd or h sulaon o orssbl lud low n orous da wh rsron o h subson lows. h low s odld by ulzng h Brnan ndd Dary onu quaon whh s oonly usd whn s oran o sasy h non-sl boundary ondon on rabl suras ha bound h orous da doan. h gornng quaons ar ransord by usng h loy ory arabls orulaon and hror h ouaon sh s arond no na and n ar. h hod s ald o onsdr buoyany drn low n losd orous ay, drnally had undr larg raur gradns. h rsuls n rs o loy and raur rdsrbuon as wll as h oal ha ransr aross h ay wll b rsnd or drn gornng arars. Kywords: orous da, orssbl lud low, boundary doan ngral hod, boundary ln hod, naural onon. Inroduon Mos o h suds dalng wh ransor hnona n orous da ar basd on rsung h lud s norssbl and sous, whr h ass dnsy s a onsan quany h loy dos no dnd on h ass dnsy and rssur s sly a or n h lnar onu balan quaon. In hs wor, h boundary ln hod, whh has bn sablshd or h sous norssbl lud oon [], s odd and ndd o aur h orssbl lud sa wh rsron o h subson lows. ha ans ha WI ransaons on h Bul Enronn, Vol 9, ISSN (on-ln) do:.495/fsi73 7 WI Prss

2 44 Flud Sruur Inraon and Mong Boundary Probls IV h drn n ass dnsy sgnanly hangs h loy ld bu hr ar no sho was and no suddn shar hangs n h alus o h ld unons. Furhror h rssur s a hrodyna quany whh s raur and ass dnsy dndn. Mahaal orulaon h rsn wor s onrnd wh naural onon n a wo-dnsonal, rangular nlosur wh h ral walls hld a drn raurs and h onnng horzonal walls onsdrd adaba. h nlosur s lld wh orous aral, whh s hoognous and soro, and h lud ha sauras h sold ar and boh ar n loal hrodyna qulbru. h low s assud o b sady, lanar, and orssbl. h gnral s o quaons or onsraon o ass, onu and nrgy ar basd on h Brnan ndd Dary low odl. h quaons gornng h onsraon o ass, onu and nrgy an b wrn as ollows [] φ, () φ φ g φ φ ( ˆ ), K () [ ( ) ( φ )( ) ] φ ( ) φ s λ. (3) φ h arars, usd abo ar: olu-aragd loy, h -h oordna, φ orosy,, dnsy, dna sosy, h rssur gradn, g gray, K rably o orous da, ˆ on o bul sosy, s raur, λ h hral onduy o h orous da gn as λ φλ ( φ) λs, whr λ and λ s ar hral ondus or h lud and sold hass, rsly. Furhror ( ) and ( ) s rrsn ha aay or h lud and sold hass. Inrodung nw arabls φ and φ, and wh h dnon o Sos aral dra o h arabl ( ) as D( ) D ( ) (), onnuy quaon an b wrn as WI ransaons on h Bul Enronn, Vol 9, ISSN (on-ln) 7 WI Prss

3 D D D, (4) whr D rrsn a loal anson ra. Aordng o Sos hyohss h sond sous on an b an as 3 ˆ. Baus o analyal rasons o dlo loy ory orulaon o gornng quaons, h onu q. () s worh wrng n s ndd or [3, 4] K g D D φ ω ω D D 3 4 3, (5) wh h nroduon o h ory or ω, rrsnng h url o h loy or,, ω ω, (6) and hang n nd ha n our as h orgnal ory s rlad by h soalld orssbl ory ω, ( )ω φ ω. I s oran o srss ou ha orosy φ s an o b onsan or nddual subdoan bu hangabl n rs o h whol ouaonal doan. Rrsnng h aral rors as a su o a onsan and arabl ar ( ) λ λ λ,,,, (7) hn h onu and nrgy quaons (3) and (5) an b wrn as K g D D φ ω, (8) S a D D, (9) whr a s hral dusy, and h sudo body or and sudo ha sour S rs nlud h s o arabl aral rors, gn by a ω ω D D 3 4 3, () 7 WI Prss ISSN (on-ln) WI ransaons on h Bul Enronn, Vol 9, Flud Sruur Inraon and Mong Boundary Probls IV 45

4 46 Flud Sruur Inraon and Mong Boundary Probls IV S λ D( ) D [( φ )( ) ] s D D. () 3 Nural hod h nural hod hosn or hs nsgaon s h Boundary Doan Ingral Mhod basd on h lassal Boundary Eln Mhod. h na s gn by loy or Posson s quaon ω D, () rrsnng h ass onsraon quaon () and h ory dnon (6), rssng h oably and rsron ondons bwn loy, ory, and ass dnsy ld unons. o alra h onrgn and h sably o h ould loy, ory and rssur ouaonal ra sh, h als ransn aroah ay b usd or quaon () [5]. By addng h aral auulaon r an b wrn n s arabol duson or α ω D, (3) whr α s h rlaaon arar onrollng h duson and auulaon rosss. I s obous ha h gornng loy quaon () s aly sasd only a h sady sa o h aral ransn ( ), whn h als dra anshs. For h wo-dnsonal lan low h ory or has us on oonn rndular o h lan o h low,.g. ω (,, ω ), hror h quaon (3) rdus o α ω D. (4) h ory ns s gn by h ory ransor quaon oband as a url o h onu quaon (8) n h or Dω ω g D ν ω ω D φ ω. K (5) WI ransaons on h Bul Enronn, Vol 9, ISSN (on-ln) 7 WI Prss

5 Flud Sruur Inraon and Mong Boundary Probls IV 47 h ory ransor quaon (5) s a hghly nonlnar aral drnal quaon du o h rodus o loy and ory hang n nd ha h loy s naally dndn on ory. For h wo-dnsonal lan low quaon (5) s rdud o a salar quaon or ory ω Dω ω ν D g φ ω, (6) K rdung also h sudo body or r o ω ω 4 D D 3 a. (7) 3 Du o h buoyany or and arabl aral rory rs, ang as addonal raur and rssur dndn ory sour rs, h ory ransor quaon s ould o h nrgy and rssur quaons, ang h nural rodur ry sr. In h orssbl lud dynas h rssur s a hrodyna quany whh s raur and ass dnsy dndn. Wrng h onu quaon (8) or h rssur gradn w ha ω φ a g. (8) K o dr h rssur quaon, dndng on nown ld and aral unons, h drgn o quaon (8) should b alulad, rsulng n h ll Posson rssur quaon. (9) As abo wh loy or quaon, h als rssur ransn r ay b addd o oban als arabol duson quaon α. () Equaons (9), (4), (6) and () rrsn h ladng non-lnar s o quaons o whh h wghd rsduals hnqu o h BDIM has o b ald [6]. Eah o hs quaons an b wrn as n h ollowng gnral drnal onsraon quaon WI ransaons on h Bul Enronn, Vol 9, ISSN (on-ln) 7 WI Prss

6 48 Flud Sruur Inraon and Mong Boundary Probls IV ℵ [ ] b u, () whr h lnar drnal oraor ℵ [ u ] wll b arabol duson drnal oraor o h or ℵ [ u ] u a u, () wh a qual o: α or nas and rssur, ν and a λ or ory and raur ns, rsly. As h ouaonal rsuls o h rsn wor ar ld o h wodnsonal as, all h subsqun ngral quaons wll onsqunly b wrn or h as o lanar gory only. Ingral rrsnaon o h na quaon s gn wh h ollowng quaon u ( ξ) ( ξ, ) α d d α ud d n n Γ Γ bu d d u d, Γ Γ F, F (3) ' ω whr D b α. h ory ransor quaon s gn wh h ngral rrsnaon u ω ( ξ) ω ( ξ, ) νω d d ν u d d n n Γ Γ b u d d ω u d, Γ Γ F, F (4) φ K whr b ( ω g ) ω Alyng slar rodur o h ha ransor quaon (9), w ha. u ( ξ) ( ξ, ) a d d a u d d n n Γ Γ bu d d u d, Γ Γ F, F (5) WI ransaons on h Bul Enronn, Vol 9, ISSN (on-ln) 7 WI Prss

7 Flud Sruur Inraon and Mong Boundary Probls IV 49 wh b λ [( φ )( ) ] s D( ) D. D Fnally, h rssur quaon ay b rwrn n h ollowng ngral san u ( ξ) ( ξ, ) α d d α u dd n n Γ Γ bu d d u d, Γ Γ F F (6) wh b α. In all ngral quaons h onsan araon o ld unons (, ω,, ) s assud whn h nddual nrn F F, hror h ngrals ay b aluad analyally, U a F F F u u d, Q a n F F u d, Q a u F d. (7) Also n all quaons u s h arabol duson undanal soluon n h or r 4aτ u, (8) 4π a τ whr h ror us o a s as rssd abo and r s h agnud o h or ro h sour on o rrn on. Fnal ngral rrsnaon or loy, ory, raur and rssur ar hror gn by h ollowng quaons ( ξ) ( ξ, ) Q dγ D n U dγ ω nu dγ n Γ Γ Γ ω Q d D Q d u d,, F F (9) WI ransaons on h Bul Enronn, Vol 9, ISSN (on-ln) 7 WI Prss

8 5 Flud Sruur Inraon and Mong Boundary Probls IV ω ( ξ) ω ( ξ, ) ω Q dγ nω g U d Γ n Γ Γ ( ω g ) Q d φ ω U d ω u d, K F, F (3) ξ ξ Q dγ λ U dγ λ Q d ( ) (, ) F λ n λ Γ Γ ( φ)( ) s D( ) λ D (3) D u d, F, F ( ξ ) ( ξ, F ) Q dγ Q d F uf d. (3) whr n, g and Γ ar h noral loy, angnal gray and nonlnar aral sour oonns, rsly. For h nural aroa soluon o h ld unons, h ngral quaons ar urhr wrn n a dsrzd annr n whh h ngrals or h boundary Γ and doan ar aroad by a su o h ngrals or all boundary lns and or all nrnal lls. In suh a way w oban h ar or o h quaons, whh ar sold by oulng n and na quaons, onsdrng h orrsondng boundary and nal ondons. Sn h l s o quaons s wrn sulanously or all boundary and nrnal nods, hs rodur rsuls n a ry larg ully oulad sys ar nlund by duson and onon. h onsqun o hs aroah s a ry sabl and aura nural sh wh subsanal our and ory dands. o ro h onos o h ouaon, h subdoan hnqu s usd, whr h nr soluon doan s arond no subdoans o whh h sa dsrbd nural rodur an b ald [6]. h nal sys o quaons or h nr doan s hn oband by addng h ss o quaons or ah subdoan onsdrng h oably and qulbru ondons bwn hr nras, rsulng n a uh sars sys ar suabl o sol wh ra hnqus. 4 Conluson h boundary ln ngral aroah o h soluon o orssbl lud oon n hrally drn orous ay s rsnd. h drd nural WI ransaons on h Bul Enronn, Vol 9, ISSN (on-ln) 7 WI Prss

9 Flud Sruur Inraon and Mong Boundary Probls IV 5 odl s hararzd by h dooson o low no s nas and ns, a rsul o h loy ory orulaon o h odd (orous) Nar-Sos quaon or a orssbl lud. h dsrbd nural algorh lads o srong oulng bwn loy, ory and ass dnsy lds, rsulng n a sabl nural sh. h alaon o h arabol duson undanal soluon n h draon o nal ngral rrsnaons nsurs an aura ouaon o h low ld arabls. A nural odl basd on h rsnd horal wor s, a hs on, n h has o aluaon and sng, hror h rsuls o s als should b rsnd a h onrn. Rrns [] Jl, R., Šrg, L., Boundary ln hod or naural onon n non-nwonan lud saurad squar orous ay, Eng. Anal. Bound. El., 7, , 3. [] Bar, J., Baha, Y., Inroduon o Modlng o ransor Phnona n Porous Mda, Kluwr Aad. Publ., Dordrh, Boson & London, 99. [3] Šrg, L., Jl, R., Corssbl lud dynas n orous da by h Boundary Eln Mhod (Char 6). Ergng hnologs and hnqus n Porous Mda, ds.: D.B. Ingha, A. Ban, E. Mau, I. Po, Kluwr Aad. Publ., Dordrh, Boson & London, 93-7, 4. [4] Šrg, L., Sa, N., BEM or h wo-dnsonal lan orssbl lud dynas, Eng. Anal. Bound. El., 9,. -7, 5. [5] Šrg, L., R, Z., Boundary-doan ngral hod usng a loy ory orulaon, Eng. Anal. Bound. El., 5, , 995. [6] Hrbrš, M., Šrg, L., Ira hods n solng Nar-Sos quaons by h boundary ln hod, In. J. Nur. Mh. Engng., 39,. 5-39, 996. WI ransaons on h Bul Enronn, Vol 9, ISSN (on-ln) 7 WI Prss