Adaptive simulations of two-phase flow by discontinuous Galerkin methods

Transcription

1 Comput. Mtods Appl. Mc. ngrg. 196 (2006) lsvir.com/locat/cma Adaptiv simulations of to-pas flo by discontinuous Galrkin mtods W. Klibr, B. Rivièr * Dpartmnt of Matmatics, Univrsity of Pittsburg, 301 Tackray, Pittsburg, PA 15260, USA Rcivd 4 Octobr 2005; rcivd in rvisd form 12 April 2006; accptd 3 May 2006 Abstract In tis papr prsnt and compar primal discontinuous Galrkin formulations of t to-pas flo quations. T tting pas prssur and saturation quations ar dcoupld and solvd squntially. Proposd adaptivity in spac and tim tcniqus yild accurat and fficint solutions. Slop limitrs valid on non-conforming mss ar also prsntd. Numrical xampls of omognous and trognous mdia ar considrd. Ó 2006 lsvir B.V. All rigts rsrvd. Kyords: rror indicators; Discontinuous Galrkin; Adaptiv tim stpping; Slop limitrs; Fiv-spot; NIPG; SIPG; IIPG; OBB 1. Introduction Accurat simulations of multipas procsss ar ssntial in problms rlatd to t nvironmnt and t nrgy. Tr is a nd for discrtization mtods tat prform ll on vry gnral unstructurd grids. Standard mtods suc as t finit diffrnc mtods, finit volums and xpandd mixd finit lmnt fail to captur t flo pnomna in t cas of igly trognous mdia it full prmability tnsors. Rcntly, discontinuous Galrkin (DG) mtods av bn applid to a varity of flo and transport problm [21,22,2,25,24] and du to tir flxibility, ty av bn son to b comptitiv to standard mtods. Furtrmor, DG mtods allo for unstructurd mss and full tnsor cofficints. vn toug t discontinuous finit lmnt mtods ar mor xpnsiv tan t finit diffrnc mtods, oil nginrs ar illing to pay t pric for accuracy and tus avoid costly mistaks [20]. In tis ork, t prssur-saturation formulation (also knon as t squntial formulation) of t to-pas flo * Corrsponding autor. Tl.: ; fax: mail addrss: rivir@mat.pitt.du (B. Rivièr). problm is discrtizd using svral discontinuous Galrkin mtods. Dscription of t squntial modl and otr formulations for to-pas flo can b found in [14,6]. T unknons ar t tting pas prssur and saturation and t quations ar solvd succssivly. On immdiat advantag is t fact tat t difficulty arising from t non-linarity is rmovd by tim-lagging t cofficints. T prssur quation is solvd by t Odn Baumann Babuska (OBB) mtod [18] ras t saturation quation is solvd by itr t OBB, t nonsymmtric intrior pnalty Galrkin mtod (NIPG) [23], t symmtric intrior pnalty Galrkin mtod (SIPG) [27,1] or t incomplt intrior pnalty Galrkin mtod (IIPG) [9,26]. On can not tat all four mtods OBB, NIPG, SIPG and IIPG ar vry similar to ac otr, and can b dscribd by t sam variational formulation it a bilinar form involving constant paramtrs. For instanc, OBB and NIPG only diffr by t addition of a pnalty trm; SIPG and NIPG only diffr by a sign. T objctiv of tis ork is to invstigat adaptiv simulations in tim and spac on unstructurd mss. W formulat rror indicators for t spatial rfinmnt and drfinmnt tcniqus. W also prsnt an algoritm tat allos t tim stp to vary during t simulation. On /$ - s front mattr Ó 2006 lsvir B.V. All rigts rsrvd. doi: /j.cma

2 W. Klibr, B. Rivièr / Comput. Mtods Appl. Mc. ngrg. 196 (2006) of t main difficultis is t dvlopmnt of slop limitrs tat ould andl mss it anging nods. W propos a limiting tcniqu basd on t on introducd by Durlofsky t al. [10] for conforming mss. By t us of adaptivity, significantly rduc t computational cost il kping t accuracy. To our knoldg, tr is littl ork in t litratur on applications of DG mtods to to-pas flo. In [19,4], simulations r prformd on uniformly rfind mss and it a constant tim stp. In [3], DG is applid to a total prssur-saturation formulation. In [17], DG and mixd finit lmnts ar coupld. Mor rcntly, in [13], a comprssibl air atr to-pas flo problm is numrically solvd on uniform mss using t NIPG/OBB/SIPG mtod and a local discontinuous Galrkin (LDG) [8] discrtization for t saturation quation. In tis cas, a Kircoff transformation is rquird to obtain a diffusiv flux from t prvious tim stp. T saturation quation is solvd xplicitly in tim, ic is computationally appaling; ovr tis rducd cost is compnsatd by t introduction of an additional unknon, intrinsic to t LDG formulation. Finally, in [11], fully coupld DG formulations ar considrd and in tis cas slop limitrs ar not ndd vn for ig ordr of approximation. Hovr, t solution of t fully coupld DG formulations rquir t construction of a Jacobian matrix at ac tim stp for t Nton Rapson mtod. T plan of t papr is as follos. In t nxt sction, prsnt t quations dscribing t to-pas flo problm. Sction 3 contains t discrt scms and notation. T adaptiv stratgy in spac and tim, as ll as t slop limiting tcniqu, ar dscribd in Sction 4. Numrical xampls ar givn in Sction 5. Som conclusions follo. 2. Modl problm T matmatical formulation of to-pas flo in a porous mdium in R 2 consists of a coupld systm of non-linar partial diffrntial quations. T pass considrd r ar a tting pas (suc as atr) and a non-tting pas (suc as oil). For ac pas, t consrvation of mass and a gnralizd Darcy s la ar obtaind. Undr t assumption of incomprssibility, a prssursaturation formulation is drivd, for ic t primary variabls ar t prssur and t saturation of t tting pas dnotd by p and s : rðk t Krp Þ¼rðk o Krp c Þ; ð1þ oð/s Þ k o k k þr Krp ot k c ¼ r u t : ð2þ t k t T cofficints in qs. (1) and (2) ar dfind blo: K is t prmability tnsor and is spatially dpndnt; for trognous mdia, K is discontinuous. T cofficint / dnots t porosity of t mdium. k t = k o + k is t total mobility, tat is, t sum of t mobility of t non-tting pas and t mobility of t tting pas. Mobilitis ar functions tat dpnd on t fluid viscositis l and l o and on t ffctiv tting pas saturation s. T ffctiv saturation dpnds on t rsidual tting pas and non-tting pas saturations s r and s ro as follos: s s r s ¼ : 1:0 s r s ro T mobilitis ar tn givn by t Brooks Cory modl [5]: k ðs Þ¼ 1 s 4 l ; k oðs Þ¼ 1 ð1 s Þ 2 ð1 s 2 l Þ: o T diffrnc of t prssurs of t to pass p c = p n p is t capillary prssur. From t Brooks Cory modl, it dpnds on t ffctiv saturation and a constant ntry prssur p d : p c ðs Þ¼p pd ffiffiffiffi : s From tis quation, s tat p 0 c ðs Þ < 0 and ill rit: rp c ¼ jp 0 c jrs. u t = u o + u is t total vlocity, tat is t sum of t to pass vlocitis. ac pas vlocity is givn as u d ¼ Kk d rp d ; d ¼ o; : Lt n dnot t outard normal to o. W associat to qs. (1) and (2) svral boundary conditions, by first dcomposing t boundary of t porous mdium o into disjoint parts: o ¼ C p1 [ C p2 ¼ C s1 [ C s2 ; C p1 \ C p2 ¼ C s1 \ C s2 ¼;: T boundary conditions for (1) ar of Diriclt and Numann typ: p ¼ p dir ; on C p1 ; ð3þ Kk t rp n ¼ 0; on C p2 : ð4þ T boundary conditions for (2) ar of Robin and Numann typ: s u t þ K k ok p 0 c k rs n ¼ s in u t n; on C s1 ; ð5þ t K k ok p 0 c k rs n ¼ 0; on C s2 : ð6þ t 3. Scm In tis sction, first stablis som notation for t tmporal and spatial discrtization and prsnt our numrical scm. Lt 0 = t 0 < t 1 < < t N = T b a subdivision of t tim intrval (0, T). For any function v tat dpnds on tim and spac, introduc t notation v i = v(t i,æ) for i =0,...,N. W also dfin t tim stp Dt i = t i+1 t i.

3 406 W. Klibr, B. Rivièr / Comput. Mtods Appl. Mc. ngrg. 196 (2006) T domain is subdividd into triangular lmnts tat form a ms. Bcaus of t rfinmnts and drfinmnts, t ms cangs at vry tim stp. Lt us dnot by i ¼fg t ms at tim ti+1. Lt i b t maximum diamtr of t lmnts. Lt C i b t union of t opn sts tat coincid it intrior dgs of lmnts of i. Lt dnot a sgmnt of C i sard by to triangls k and l of i (k > l); associat it, onc and for all, a unit normal vctor n dirctd from k to l and dfin formally t jump and avrag of a function on by ½Š ¼ðj kþj ðj lþj ; fg ¼ 1 2 ðj kþj þ 1 2 ðj lþj : If is adjacnt to o, tn t jump and t avrag of on coincid it t trac of on and t normal vctor n coincids it t outard normal n. T quantity jj dnots t lngt of. For ac intgr r, dfin a finit lmnt subspac of discontinuous picis polynomials: D r ð i Þ¼fv : vj 2 P rðþ 8 2 i g; r P r () is a discrt spac containing t st of polynomials of total dgr lss tan or qual to r on. W ill approximat t tting pas prssur and saturation by discontinuous polynomials of ordr r p and r s, rspctivly. W no driv t variational formulation for t topas flo problm, by considring t prssur q. (1) and t saturation q. (2) sparatly T prssur quation W rrit (1) by dfining v ¼ Kk o rp c ¼ Kk o jp 0 c jrs : r ðkk t rp Þ ¼ r v: ð7þ Multiplying (7) by a tst function v 2 D rp, and using Grn s formula on on lmnt yilds: 2 b 2 1 b b 3 Fig. 1. Slop limiting on non-conforming mss. Fig. 2. Rfinmnt of a triangular lmnt. Fig. 3. Fiv-ll xampl: coars ms at initial tim and adaptiv mss obtaind at 15 and 45 days.

4 Kk t rp rv ðkk t rp n Þv o ¼ v rv ðv n Þv; o r n is t outard normal to. Summing ovr all t lmnts in i and using t fact tat p and v ar smoot noug, namly [p ] = 0, [Kk t $p Æ n ] = 0 and [v Æ n ]=0, av Kk t rp rv fkk t rp n g½vš 2 i þ 2C i ¼ 2 i 2C i [o fkk t rv n g½p Š v rv 2C [o W. Klibr, B. Rivièr / Comput. Mtods Appl. Mc. ngrg. 196 (2006) v n ½vŠ: Making us of t boundary conditions (3) and (4), obtain 2 i þ Kk t rp rv 2C i [C p1 ¼ 2 i þ 2C p1 2C i [C p1 fkk t rv n g½p Š v rv 2C i [o ðkk t rv nþp dir : 3.2. T saturation quation fkk t rp n g½vš v n ½vŠ Similarly, dfin t auxiliary vctor f ¼ k k t u t. Tn, (2) can b rrittn as oð/s Þ r ot K k ok k t jp 0 c j rs ¼ rf: As for t prssur quation, multiply by a tst function z 2 D rs ovr on lmnt in i, sum ovr all lmnts, and ð8þ ð9þ Fig. 4. Fiv-ll xampl: tr-dimnsional prssur contours at 15, 30, 45 and 52.5 days.

5 408 W. Klibr, B. Rivièr / Comput. Mtods Appl. Mc. ngrg. 196 (2006) us t rgularity of s and f. W finally obtain aftr som algbraic manipulation: oð/s Þ z þ K k ok jp 0 c ot 2 i k jrs rz t 2C i [o þ 2C i ¼ 2 i K k ok k t j p 0 c jrs n K k ok jp 0 c k jrz n t f rz 2C i [o ½s Š ½zŠ f n ½zŠ: Making us of t boundary conditions (5), (6) and t continuity of prssur, av: oð/s Þ z þ K k ok jp 0 c ot 2 i k jrs rz t s u t n z K k ok jp 0 c 2C s1 2C i k jrs n ½zŠ t þ K k ok jp 0 c 2C i k jrz n ½s Šþ r ½s Š½zŠ t jj 2C i ¼ 2 i 2C i f rz 2C i [o k k t fkk t rz n g½p Š f n ½zŠ 2C s1 s in u t n z Kk rz n ðp p dir Þ: ð10þ C p1 T quation abov is paramtrizd by t cofficints 2 { 1,0,1} and r P 0. For a positiv pnalty valu r, t coic = 1 yilds t SIPG mtod, t coic = 0 yilds t IIPG mtod and t coic = 1 t NIPG mtod. If r =0 and = 1, obtain t OBB mtod T discrt scm W discrtiz t tim drivativ by finit diffrnc, ic yilds t backard ulr scm. T initial approximations P 0, S0 ar simply obtaind by a L2 projction of t initial data p (t = 0) and s (t = 0). Basd on (8) and (10), formulat t folloing numrical mtod: givn ðp i ; Si Þ2D r p D rs, find ðp iþ1 ; Siþ1 Þ2D r p D rs suc tat for all ðv; zþ 2D rp D rs : Fig. 5. Fiv-ll xampl: tr-dimnsional saturation contours at 15, 30, 45 and 52.5 days.

6 W. Klibr, B. Rivièr / Comput. Mtods Appl. Mc. ngrg. 196 (2006) ¼ v i rv PRODUCTION v i" n ½vŠ 2 i 2C i [o þ ðkk t ðs i Þrv nþp dir ð11þ 2C p1 INJCTION 2 i þ Kk t ðs i iþ1 ÞrP fkk t ðs i iþ1 ÞrP n g½vš rv 2C i [C p1 fkk t ðs i Þrv n g½p iþ1 Š Fig. 6. Domain and coars ms for quartr-fiv spot. 2C i [C p1 and / Dt i Siþ1 z þ 2 i S iþ1 U i t n z 2C s1 K k oðs i Þk ðs i Þ 2C i k t ðs i Þ þ 2C i þ r ½S i jj Š½zŠ 2C i / ¼ Dt i Si z þ 2 i 2C i [C p1 K k oðs i Þk ðs i Þ k t ðs i Þ jp 0 c ðsi ÞjrSiþ1 rz jp 0 c ðsi ÞjrSiþ1 n K k oðs i Þk ðs i Þ k t ðs i Þ jp 0 c ðsi Þjrz n f i rz f i" n ½zŠ 2C s1 s in U i t n ½zŠ ½S iþ1 Š Fig. 7. To-dimnsional prssur contours at 7.5, 15, 22.5 and 30 days.

7 410 W. Klibr, B. Rivièr / Comput. Mtods Appl. Mc. ngrg. 196 (2006) C i 2C p1 k ðs i Þ k t ðs i Þ fkk tðs i Þrz n g½p i Š Kk ðs i Þrz n ðp i p dirþ; ð12þ r U i t, fi and vi ar t approximats of ui t ; fi and v i. U i t ¼ Kk ðs i ÞrP i Kk oðs i Þðp0 c ðsi ÞrSi þrp i Þ; v i ¼ Kk oðs i Þjp0 c ðsi ÞjrSi ; f i ¼ k ðs i Þ k t ðs i Þ fu i t g: Bcaus of t discontinuous approximations, tr ar to valus for t functions v i and f i on an intrior dg. Ts quantitis ar tn rplacd by t upind numrical fluxs v i" and f i". Upinding is don it rspct to t normal componnt of t avrag of t total vlocity U t ( 8 ¼ o k \ o l ; ðk > lþ; 8; " ¼ j if fu i k t gn P 0; j l if fu i t gn < 0: From t drivations in Sctions 3.1 and 3.2, obtain t consistncy of t scm (11) and (12). Lmma 1. If (p,s ) is a solution of (1), (2), tn (p,s )is also a solution of (11) and (12) Local mass balanc Lt us fix an lmnt and a tst function v 2 D rp tat vaniss outsid of. For simplicity, assum tat is an intrior lmnt in. T prssur quation (11) bcoms: Kk t ðs i iþ1 ÞrP rv fkk t ðs i iþ1 ÞrP n gv o 1 þ o 2 Kk tðs i Þrv n ½P iþ1 Š ¼ v i rv v i" n v: o If in addition, lt v to b qual to on ovr, obtain t local mass proprty satisfid by t approximations: fkk t ðs i iþ1 ÞrP n gv v i" n v ¼ 0: o 3.5. Slop limiting Approximations of ig ordr yild ovrsoot and undrsoot in t nigborood of t front of t injctd pas. Slop limitrs ar t appropriat tools for dcrasing t local oscillations [7,15]. To our knoldg tr is o Fig. 8. To-dimnsional saturation contours at 7.5, 15, 22.5 and 30 days.

8 W. Klibr, B. Rivièr / Comput. Mtods Appl. Mc. ngrg. 196 (2006) no analysis availabl for slop limitrs in 2D and 3D, vn on a conforming ms. W ar also not aar of limitrs tat ould andl non-conforming mss. In tis sction, propos a limiting tcniqu tat can andl mss it anging nods. Tis procdur is succssfully tstd for our to-pas flo problm. W apply t limiting tcniqu to t approximations P iþ1 tim stp t i+1. and S iþ1 aftr ac In at follos, say tat on lmnt is activ if it blongs to t ms i, i.., if it is usd in t computation of (11) and (12). T lmnt can bcom inactiv if it is rfind and tus its cildrn ar cratd and activatd. T limiting procss consists of to stps. First, loop troug all t activ lmnts starting from t oldst gnration to t youngst (in gnral tis ould man tat t ordr is in dcrasing siz). For xampl, Fig. 1 sos an xampl of fiv lmnts of diffrnt gnration: if G dnots t gnration of t lmnts 0 and 1, tn lmnts 3 and 4 ar of youngr gnration G + 1 and lmnt 2 is of oldr gnration G 1. Tus, it is assumd tat t limiting procss as bn alrady applid to 2. (1) Nigbor avrags: W first comput t avrag saturation for t lmnt to b limitd and all nigboring lmnts as follos. Lt S 0 dnot t avrag saturation ovr 0 and lt S j dnot a function associatd to ac sid j 2 {1,2,3} of 0. For 0 and t nigbors of t sam gnration, av t usual avraging oprator: S 0 ¼ Að 0 Þ; S 1 ¼ Að 1 Þ; r AðÞ ¼ 1 jj S iþ1 : To comput S 2 corrsponding to t sid 2 of 0 and t lmnt 2 tat is of oldr gnration, first locat t barycntr b 2 of an imaginary cild C 2 of t sam gnration of 0 (s dasd lins in Fig. 1). W tn st S 2 ¼ S iþ1 j 2 ðb 2 Þ: W not tat S 2 ¼ Að C 2 Þ. T smallr lmnts 3 and 4 blong to a parnt (s dottd lins in Fig. 1). If dnot by F ~ 1 ;...; F ~ 4 t cildrn of, can rit 4 S 3 ¼ 1 BðF ~ l 4 Þ; l¼1 r t function B is dfind rcursivly as (using t notation F l for t lt cild of ): 8 1 S i jj ; if activ; >< BðÞ ¼ 1 4 >: BðF l 4 Þ; otris: l¼1 Fig. 9. To-dimnsional prssur contours at 7.5, 15, 22.5 and 30 days obtaind on uniformly rfind mss.

9 412 W. Klibr, B. Rivièr / Comput. Mtods Appl. Mc. ngrg. 196 (2006) If t dg j is a boundary dg, tn S j is dfind according to t boundary conditions. S j ¼ s in on C s1 ; S j ¼ S 0 on C s2 : (2) Tst: W tn comput t saturation S iþ1 j 0 ðm j Þ valuatd at t midpoint m j of ac dg j and cck tat tis valu is btn S j and S 0. W stop r if t tst is succssful, otris continu to stp 3. (3) Construction of tr linars: Basd on t tcniqu by Durlofsky t al. [10], construct tr linars using t points b j and t avrags S j. For instanc, if rit t linars as L j ðx; yþ ¼a j 0 þ aj 1 x þ aj 2y, for j 2 {1,2,3}, ty ar uniquly dtrmind by L j ðb 0 Þ¼S 0 and L j ðb l Þ¼S l ; for l 6¼ j: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W tn rank t linars by dcrasing ða j 1 Þ2 þða j 2 Þ2 and cck tat for t valus of t linars valuatd at t midpoint m l, L j ðm l Þ, is btn S l and S 0 for 1 P l P 3. If non of t constructd linars satisfy t tst tn t slop is rducd to 0. Scond, loop troug all lmnts and cck tat tir slops ar not too larg in t uclidan norm. If it is largr tan a cut-off valu (st up by usr), scal it by t ratio cut-off/norm. 4. Adaptivity stratgy In tis sction, dfin t rror indicators and prsnt t adaptivity in spac and tim tcniqus. Ty ar basd on t a postriori rror stimats obtaind for a linar convction diffusion tim-dpndnt problm [12], tat as som similarity it t saturation quation. Hovr, tr is no rigorous matmatical proof for our coupld systm of quations and t rror stimators of [12] ar usd r as rror indicators in t adaptivity algoritm rror indicators W dfin t folloing quantitis: R vol ¼ / Dt i ðsiþ1 Si Þþr K k nðs i Þk ðs i Þ k t ðs i Þ jp 0 c ðsi ÞjrSi k ðs i r Þ k t ðs i Þ U i t ; R 1 ¼½S iþ1 Š; R 2 ¼ K k ðs i Þk nðs i Þ k t ðs i Þ jp 0 c ðsi ÞjrSiþ1 n þfu i t gn½siþ1 Š; Fig. 10. To-dimnsional saturation contours at 7.5, 15, 22.5 and 30 days obtaind on uniformly rfind mss.

10 W. Klibr, B. Rivièr / Comput. Mtods Appl. Mc. ngrg. 196 (2006) S 0.6 P Fig. 11. Saturation (lft) and prssur (rigt) fronts along t diagonal lin x = y at 7.5, 15, 22.5 and 30 days. T solid lin corrsponds to a uniform ms rfinmnt (3) and t dasd lin to an adaptivly rfind ms. Tabl 1 Numbr of dgrs of frdom for adaptiv and non-adaptiv simulations in t cas (,r) = (1,0) t (days) DOFS prss DOFS sat AMR UNI AMR UNI R s1 ¼ s in U i t n Siþ1 U i t n þ K k nðs i Þk ðs i Þ k t ðs i Þ jp 0 c ðsi ÞjrSiþ1 n; R s2 ¼ K k nðs i Þk ðs i Þ k t ðs i Þ jp 0 c ðsi ÞjrSiþ1 n: Tn t rror indicator g computd on ac lmnt is g ¼ 4 kr volk 2 0; þ þ 2ono ð 3 kr 2k 2 0; þð þ 1ÞkR 1 k 2 0; Þ 3 kr s1k 2 0; þ 3 kr s2k 2 0; 2o\C s1 2o\C , ,672 s , , , , , ,672 Tabl 2 Total numbr of dgrs of frdom for adaptiv simulations for all mtods! 1=2 ; t (days) OBB NIPG NIPG SIPG SIPG IIPG IIPG r =0 r =1 r =10 5 r =1 r =10 5 r =1 r = S 0.6 P Fig. 12. Saturation (lft) and prssur (rigt) fronts along t diagonal lin x = y at 15 and 30 days. Solid lins corrspond to mtods NIPG, SIPG, IIPG it r =10 5 and OBB. Dasd lins corrspond to mtods NIPG, SIPG and IIPG it r =1.

11 414 W. Klibr, B. Rivièr / Comput. Mtods Appl. Mc. ngrg. 196 (2006) r is t diamtr of t lmnt and ¼ maxð k ; l Þ if t dg is sard by lmnts k and l. Not tat t notation ono mans tat t dgs ar intrior dgs only Adaptiv ms rfinmnt tcniqu Lt us assum tat t solution P i and Si av bn obtaind at t it tim stp. W comput t rror indicator g for ac activ lmnt. Tn, first rfin t appropriat lmnts and scond apply t drfinmnt tcniqu. Rfinmnt: W rfin ac lmnt os rror indicator is gratr tan a trsold valu g R. Not tat tis trsold valu can b a prcntag of t maximum of t rror indicators. Fig. 2 sos o on lmnt (also calld parnt) is rfind into four smallr lmnts (also calld cildrn). Drfinmnt: W considr a parnt lmnt for drfinmnt if (A) all of its cildrn ar activ, (B) t rror indicator of ac of its cildrn is lss tan a trsold valu g D, and (C) t lmnt as not rfind during t currnt tim stp. For ac parnt lmnt mting ts rquirmnts, an L 2 projction is prformd to rtriv t dgrs of frdom of t parnt. Bfor actually doing t drfinmnt, cck tat t parnt rror indicator is lss tan g R. If it is, tn drfin. If it is not, do not drfin Adaptiv tim stpping tcniqu For tim stratgy, allo t tim stp to vary during t simulation. W uniformly divid t simulation intrval (0,T) into ol stps of lngt Dt i. At t start of ac ol stp, try to comput t saturation for tim t i + Dt i, r t i is t currnt tim. If t rsulting saturation function is satisfactory, tn rcord it, calculat t n prssur function, and procd to t nxt ol stp. On t otr and, if t rsulting saturation function is unsatisfactory, tn discard it and subdivid t ol stp into to alf stps. W tn comput t saturation for t tim at ic t first alf stp nds. If t rsult is accptabl, procd to t scond alf stp, and if its rsult is also accptabl, tn continu on to t nxt ol stp. If on of t alf stps dos not yild satisfactory rsults, tn divid it into quartr stps, procding in t sam mannr as bfor, it t xcption tat accpt t rsults of t quartr stps rgardlss of o Fig. 13. To-dimnsional prssur contours at 7.5, 15, 22.5 and 30 days on an inomognous mdium.

12 satisfactory ty ar. For t numrical simulations in tis papr, a rsulting saturation function as dmd unsatisfactory if t avrag saturation in any lmnt xcdd t pysically prmissibl rang by mor tan 0.01; otris, it as considrd satisfactory. T main purpos of tis tim stpping tcniqu is to spd up computation itout losing accuracy, tus to incras t fficincy of t mtod. 5. Numrical xampls W. Klibr, B. Rivièr / Comput. Mtods Appl. Mc. ngrg. 196 (2006) In t folloing simulations, assum tat t fluid and mdium proprtis ar l o ¼ 0:002 kg=ðmsþ; l ¼ 0:0005 kg=ðmsþ; / ¼ 0:2; s ðt ¼ 0Þ ¼0:2; p ðt ¼ 0Þ ¼3: Pa; s in ¼ 0:95; s r ¼ 0:15; s ro ¼ 0; p d ¼ Pa: T ordrs of approximation ar discontinuous picis linars for t saturation and discontinuous picis quadratics for t prssur. For t adaptiv rfinmnts and drfinmnts, cos g R = and g D = (1/3)g R. T ll-knon fiv-spot problm on omognous and trognous mdia is first considrd, tn simulations it igly varying prmability ar prsntd. Fig. 15. Prmability fild and coars ms: prmability is in it rgions and lsr Fiv-spot on omognous mdium T prmability tnsor is K =10 11 dm 2, r d is t Kronckr dlta tnsor. Fig. 3 sos t coars ms and Fig. 14. To-dimnsional saturation contours at 7.5, 15, 22.5 and 30 days on an inomognous mdium.

13 416 W. Klibr, B. Rivi r / Comput. Mtods Appl. Mc. ngrg. 196 (2006) t domain mbddd into t squar ( 300, 300)2. Four production lls ar locatd at ac cornr of t domain; t ll bor corrsponds to part of Cp1, r assum tat pdir = Pa. An injction ll is locatd in t intrior of t domain; t ll bor corrsponds to t boundary Cs1 and t rmaindr of Cp1 and t prssur is st pdir = Pa. T flo of t pass is tus drivn by t gradint of prssur from t injction ll to t production lls. T simulation is run for 52.5 days it a tim stp varying btn days and days. T paramtrs in (12) ar r = 0 and = 1. Tr dimnsional vis of contours of tting pas prssur and saturation at slctd tims ar son in Figs. 4 and 5. In ordr to bttr analyz tis xampl and bcaus of t symmtry of t problm, r-run t simulations on on quartr of t domain; tis yilds t quartr-fiv spot problm son in Fig. 6. T injction ll is at t lft bottom cornr ras t production ll is at t rigt top cornr. T domain is no mbddd into (0, 300)2. T contours of tting pas prssur and saturation at slctd tims ar son in Figs. 7 and 8. T locally rfind and drfind mss ar also givn on ts figurs. On can conclud tat t proposd rror indicators captur ll t location of t front. As xpctd, t ms is mor rfind in t nigborood of t saturation front. It also appars tat t ms stays rfind at t nigborood of t injction ll bor. W compar t adaptiv rsults it tos obtaind on t coars ms rfind uniformly tr tims. T prssur and saturation contours ar givn in Figs. 9 and 10. Hr, t tim stp varis btn days and days. T contours ar similar to t adaptiv ons. For bttr comparison, so t prssur and saturation profils along t diagonal {(x, y): x = y} (s Fig. 11). Using adaptiv rfinmnt and drfinmnt dcrass significantly t cost of t computation, as son in Tabl 1. T columns for AMR corrspond to adaptivly rfind mss, and t columns for UNI corrspond to uniformly rfind mss. W no compar t saturation and prssur profils obtaind by varying t paramtrs r and. In particular, considr t cass (, r) 2 {(1, 1), (1, 10 5)}, ic yild t NIPG mtod it small and larg pnalty valus; t cass (, r) 2 {( 1, 1), ( 1, 10 5)}, ic yild t SIPG mtod it small and larg pnalty valus; t cass (, r) 2 {(0, 1), (0, 10 5)}, ic yild t IIPG mtod it small and larg pnalty valus and t OBB mtod usd abov. If t pnalty valu is small noug, i.., r = 10 5, Fig. 16. To-dimnsional saturation contours at 17.5, 35, 52.5 and 70 days: ( = 1, r = 0).

14 W. Klibr, B. Rivièr / Comput. Mtods Appl. Mc. ngrg. 196 (2006) all mtods produc idntical solutions for bot prssur and saturation. Fig. 12 sos t profils obtaind at 15 and 30 days: solid lins corrspond to small pnalty valus or t OBB mtod ras dasd lins corrspond to larg pnalty valus. As xpctd, if t pnalty valu incrass, numrical diffusion addd by t jump trm appars in t solutions. T saturation fronts ar sligtly smard. It is also intrsting to not tat for a fixd pnalty valu, all tr pnalty mtods produc t sam solutions. Finally, giv in Tabl 2 t total numbr of dgrs of frdom for prssur and saturation quations and so tat t numrical cost is comparabl for all mtods Fiv spot on trognous mdium Tis simulation is idntical to t on abov xcpt for t prmability tnsor. Hr, K is discontinuous and is qual to dm 2 in a small subdomain. In t rst of domain, K =10 11 dm 2. W prsnt t contours of t prssur and saturation at diffrnt tims in Figs. 13 and 14 in t cas (,r) = (1, 0). Clarly, t rgion of lo prmability is not invadd by t injctd tting pas. Tis sos tat t scm as vry littl numrical diffusion. It is also intrsting to not tat t proposd mtod allos for an arbitrary numbr of anging nods, itout any spcial car Higly varying prmability fild W considr a squar domain (0,400) 2 it varying prmability as son in Fig. 15. T prmability is Im 2 xcpt in svral small rgions r it is 10 5 tims smallr (s [16]). T simulation is run for 70 days. T tim stp varis btn days and days. T vrtical boundaris corrspond to C p1 r t sam prssur p dir as in t prvious xampls is imposd. T lft vrtical boundary corrsponds to C s1. W first considr t OBB mtod ( =1,r = 0). Saturation contours on adaptivly rfind mss ar son in Fig. 16. T dgrs of frdom ar 8232, 13,281, 17,070 and 19,131 for t rspctiv tims 17.5, 35, 52.5 and 70 days. T figurs so clarly tat tr is vry littl numrical diffusion. For comparison, so t contours obtaind on a uniform ms, ic corrsponds to 38,400 dgrs of frdom (s Fig. 17). T coars ms as bn rfind tic, and tis producs a computational tim of 24 on a singl procssor. For anotr lvl of rfinmnt, t simulation ould run for 1 k. Fig. 17. To-dimnsional saturation contours at 17.5, 35, 52.5 and 70 days on uniform mss: ( = 1, r = 0).

15 418 W. Klibr, B. Rivi r / Comput. Mtods Appl. Mc. ngrg. 196 (2006) Fig. 18. NIPG To-dimnsional saturation contours at 35 days: r = 10 6, r = 10 5 and r = 1. W rpat t xprimnt for t NIPG mtod it tr coics of pnalty r 2 {10 6, 10 5, 1}. T saturation contours at 35 days ar son in Fig. 18. Bcaus of t igly varying prmability fild, t mtod is mor snsitiv to t coic of t pnalty. T valu r = 10 6 yilds a comparabl solution to t OBB mtod; tr is vry lit- tl numrical diffusion. Hovr, for r = 10 5, t tting pas floods t rgion of lor prmability and for r = 1, t mtod is too diffusiv to captur t barrir zons. Similar conclusions can b mad it t otr to mtods. W so t saturation contours for r = 10 6 for IIPG and SIPG in Fig. 19. Fig. 19. To-dimnsional saturation contours at 35 days for r = 10 6: IIPG (lft) and SIPG (rigt).

16 W. Klibr, B. Rivièr / Comput. Mtods Appl. Mc. ngrg. 196 (2006) Conclusions Tis papr prsnt adaptivity tcniqus in spac and tim. W so tat t adaptiv simulations ar mor fficint tan t simulations obtaind on uniform mss and it constant tim stp. A slop limiting tcniqu for mss it svral anging nods pr fac is dfind. Numrical xprimnts so robustnss of t proposd DG scms on trognous mdia. Comparisons btn SIPG, NIPG, IIPG and OBB mtods ar prformd: if t pnalty valu is small noug, t rsulting numrical solutions ar vry similar. Incrasing t pnalty valu introducs numrical diffusion in t approximations, in particular if t prmability fild igly varis in spac. Svral futur xtnsions ar currntly undr invstigation: for instanc, plan to xtnd our computational rsults to tr-dimnsional problms using unstructurd ttradral mss. W ar also invstigating t ffcts of p-adaptivity on t accuracy and t computational fficincy of t scm. Acknoldgmnt T first autor is partially supportd by a CRDF grant from t Univrsity of Pittsburg and by a Bracknridg fllosip. T scond autor is supportd by a National Scinc Foundation grant DMS Rfrncs [1] D.N. Arnold, An intrior pnalty finit lmnt mtod it discontinuous lmnts, SIAM J. Numr. Anal. 19 (1982) [2] P. Bastian, Hig ordr discontinuous Galrkin mtods for flo and transport in porous mdia, Callngs in Scintific Computing CISC 2002, vol. 35, [3] P. Bastian, Discontinuous Galrkin mtods for to-pas flo in porous mdia, Also tcnical rport, , IWR (SFB 359), Hidlbrg Univrsity, submittd for publication. [4] B. Rivièr, Numrical study of a discontinuous Galrkin mtod for incomprssibl to-pas flo, in: Procdings of CCOMAS, [5] R.H. Brooks, A.T. Cory, Hydraulic proprtis of porous mdia, Hydrol. Pap. 3 (1964). [6] G. Cavnt, J. Jaffré, Matmatical Modls and Finit lmnts for Rsrvoir Simulation, Nort-Holland, [7] B. Cockburn, C.-W. Su, TVB Rung Kutta local projction discontinuous Galrkin finit lmnt mtod for consrvativ las II: gnral framork, Mat. Comput. 52 (1989) [8] B. Cockburn, C.-W. Su, T local discontinuous Galrkin mtod for tim-dpndnt convction diffusion systms, SIAM J. Numr. Anal. 35 (1998) [9] C. Dason, S. Sun, M.F. Wlr, Compatibl algoritms for coupld flo and transport, Comput. Mt. Appl. Mc. ngrg. 193 (2004) [10] L.J. Durlofsky, B. ngquist, S. Osr, Triangl basd adaptiv stncils for t solution of yprbolic consrvation las, J. Comput. Pys. 98 (1992) [11] Y. pstyn, B. Rivièr, Fully implicit discontinuous Galrkin scms for multipas flo, Appl. Numr. Mat., in prss. [12] A. rn, J. Proft, A postriori discontinuous Galrkin rror stimats for transint convction diffusion quations, Appl. Mat. Ltt. 18 (7) (2005) [13] O. slingr, Discontinuous Galrkin finit lmnt mtods applid to to-pas, air-atr flo problms, P.D. tsis, T Univrsity of Txas at Austin, [14] R. Hlmig, Multipas Flo and Transport Procsss in t Subsurfac, Springr, [15] H. Hotit, P. Ackrr, R. Mosé, J. rl, B. Pilipp, N todimnsional slop limitrs for discontinuous Galrkin mtods on arbitrary mss, Int. J. Numr. Mt. ngrg. 61 (2004) [16] L.J. Durlofsky, Accuracy of mixd and control volum finit lmnt approximations to Darcy vlocity and rlatd quantitis, Watr Rsour. Rs. 30 (4) (1994) [17] D. Nayagum, G. Scäfr, R. Mosé, Modlling to-pas incomprssibl flo in porous mdia using mixd ybrid and discontinuous finit lmnts, Comput. Gosci. 8 (1) (2004) [18] J.T. Odn, I. Babus ka, C.. Baumann, A discontinuous p finit lmnt mtod for diffusion problms, J. Comput. Pys. 146 (1998) [19] B. Rivièr, T dgimps modl in ipars: Discontinuous Galrkin for to-pas flo intgratd in a rsrvoir simulator framork, Tcnical Rport 02-29, Txas Institut for Computational and Applid Matmatics, [20] B. Rivièr,. Jnkins, In pursuit of bttr modls and simulations, oil industry looks to t mat scincs, SIAM Ns (January/Fbruary) (2002). [21] B. Rivièr, M.F. Wlr, Discontinuous Galrkin mtods for flo and transport problms in porous mdia, Commun. Numr. Mt. ngrg. 18 (2002) [22] B. Rivièr, M.F. Wlr, Non conforming mtods for transport it nonlinar raction, in: Procdings of an AMS-IMS-SIAM Joint Summr Rsarc Confrnc on Fluid Flo and Transport in Porous Mdia: Matmatical and Numrical Tratmnt, [23] B. Rivièr, M.F. Wlr, V. Girault, Improvd nrgy stimats for intrior pnalty, constraind and discontinuous Galrkin mtods for lliptic problms. Part I, Comput. Gosci. 3 (1999) [24] S. Sun, B. Rivièr, M.F. Wlr, A combind mixd finit lmnt and discontinuous Galrkin mtod for miscibl displacmnt problms porous mdia, in: Procdings of Intrnational Symposium on Computational and Applid PDs, 2002, pp [25] S. Sun, M.F. Wlr, Symmtric and non-symmtric discontinuous Galrkin mtods for ractiv transport in porous mdia, SIAM J. Numr. Anal. 43 (1) (2005) [26] S. Sun, M.F. Wlr, Symmtric and nonsymmtric discontinuous Galrkin mtods for ractiv transport in porous mdia, SIAM J. Numr. Anal. 43 (1) (2005) [27] M.F. Wlr, An lliptic collocation-finit lmnt mtod it intrior pnaltis, SIAM J. Numr. Anal. 15 (1) (1978)