Nearly perfectly matched layer method for seismic wave propagation in poroelastic media

Transcription

1 ρ u + ρ w = λ + µ u + µ u + α M w c ρ u + m w + b w = α M u + µ u + M w CANADIAN JOURNAL o EXPLORATION GEOPHYSlCS VOL 37, NO 1 June 01, P -7 Nearly perecly mached layer mehod or sesmc wave propagaon n poroelasc meda Jngy Chen Deparmen o Geoscences, The Unversy o Tulsa, Tulsa, OK, USA ABSTRACT The nearly perecly mached layer NPML mehod has been successully appled o elecromagnec elasodynamc wave equaons ncludng elasc soropc ansoropc meda ypes o absorb spurous arrvals releced rom nroduced boundares n a number o numercal soluon mehods I has been shown o be exremely sasacory or hs purpose In hs paper, a rs-order velocy-sress-pressure sysem wh NPML s obaned rom Bo s equaons A saggered-grd ne-derence mehod wh ourh-order accuracy n space second-order accuracy n me s developed o smulae sesmc wave propagaon n a D ludsauraed poroelasc meda The absorbng perormance o NPML s nvesgaed by waveeld snapshos, waveorm energy decay measuremens n numercal expermens The numercal resuls demonsrae ha NPML suppresses he spurous relecons n a hghly eecve manner Inroducon Elasc wave propagaon n lud-sauraed poroelasc meda s o grea neres n exploraon geophyscs sesmology n parcular Numercal mehods are employed o asss n he beer undersng nerpreaon o observed sesmc characerscs Carcone, 1996 Examples o numercal mehods nclude he ne-derence Zhu McMechan, 1991, ne elemen Panneon Aalla, 1997, specral elemen Morency Tromp, 008 pseudospecral mehods Özdenvar McMechan, 1997 In hs paper, poroelasc wave equaons based on Bo s heory Bo, 196 are reormulaed no a rs-order hyperbolc sysem whch ncludes he sold velocy componen, he lud relave o sold velocy componen, he sold sress componen, he lud pressure The ne derence mehod s a possbly he bes choce due o s comparave smplcy Here, he ne derence mehod wh ourh-order accuracy n space secondorder accuracy n me s developed n a saggered-grd o smulae wave propagaon n poroelasc meda Vreux, 1986; Lever, 1988; Graves, 1996 The smulaon o sesmc wave propagaon n unbounded meda requres ha absorbng boundary condons are ncorporaed o mnmze ougong sesmc waves a he edges o he numercal model Such suaons nclude sponge boundary condons Ceran e al, 1985; Koslo Koslo, 1986 paraxal condons Clayon Engqus, 1977; Reynolds, 1978 However, hese absorbng boundary condons only exhb good behavor or small ncden angles Bérenger 1994 proposed he perecly mached layer PML as an absorbng layer boundary condon or elecromagnec equaons Ths mehod has become wdely used n a varey o wave equaon problems Collno Tsoga, 001; Zeng Lu, 001 In addon, several moded PMLs were nroduced, such as convoluonal perecly mached layer CPML Roden Gedney, 000; Marn e al, 008, sard complex sreched perecly mached layer Chew Weedon, 1994 nearly perecly mached layer NPML Cummer, 003 Compared o oher PMLs, NPML has he eaures o mplemenaon smplcy compuaonal ecency Bérenger, 004 NPML has been urher appled o acousc elasc wave equaons Hu e al, 007; Chen Zhao, 011 Ths paper s he exenson o he NPML sudy by Chen 011 In he rs secon, Bo s equaons are reormulaed no a rsorder hyperbolc sysem n velocy-sress-pressure orm The new equaons wh NPML are subsequenly presened hrough ransormaons beween he me he requency domans The saggered-grd ne-derence mehod s mplemened o smulae sesmc wave propagaon To chec he absorbng perormance o NPML, numercal expermens are carred ou n he second secon The numercal resuls show NPML can be a valuable absorbng layer boundary condon or he suppresson o he arcal relecons rom he runcaed boundares Governng Equaons The heory o sesmc wave propagaon n a poroelasc medum s descrbed by Bo s heory Bo, 196 whch aes no consderaon he energy dsspaon due o he relave moon beween pore lud he sold marx Boh he as slow P wave modes are ncluded n wha ollows In a poroelasc medum, whch s lud sauraed, macroscopcally soropc locally homogeneous, Bo s equaons n a Caresan spaal sysem have he orm ρ u + ρ w = λ + µ u + µ u + α M w ρ c u + m w + b w = α M u + µ u + M w where u s he dsplacemen vecor or he sold, w he dsplacemen vecor or he lud relave o ha or he sold, r s he macroscopc densy o he lud sauraed medum deermned by r= r +1 r s, r r s beng he denses o he lud sold, he porosy, l c he Lamé s parameer or he sauraed marx, m he Lamé s shear modulus parameer or he dry 1 CJEG June 01

2 Jngy Chen porous marx, T he oruosy, a srucural acor oen aen o be equal o 1/1+1/, m he eecve lud densy dened as m=t r /, η he vscosy o he lud, he permeably o he porous medum, b he mobly o he lud dened by b=η/ The eecve value o b s usually speced n he orm, b= η/ Ks K are he bul modul compressbles o he sold lud, Kb he bul modulus compressbly o he dry porous marx, a he poroelasc coecen o eecve sress dened by a =1 K b /K s, M he couplng parameer beween he sold he lud gven by M=[ /K + a /K s ] 1 One can noce ha a porovscoelasc problem s acually consdered because he ncluson o he erm n b nroduces aenuaon no he above dened suaon However, he less complex erm poroelasc wll be reaned n hs paper, wh aenuaon mplc From he denon o he sran energy uncon n porous meda Bo,196, he sress he pore lud pressure P are speced by τ = µe + δ λc e + α M w 3 Implemenaon o NPML The spaal coordnae srechng coecens s1 s are expressed as sℓ =1+d ℓ / w ℓŒ{1,} along he x1 laeral x vercal Caresan drecons Cummer, 003 The d l are he PML decay acors n he respecve drecons, w s he angular requency = 1 Followng Collno Tsoga 001 Komasch Marn 007, he PML decay acors are expressed as dℓ = d 0 xℓ /L N ℓŒ{1,} where L s he wdh o he absorbng layers Fgure d 0 s obaned rom he heorecal relecon coecen R c Komasch Marn, 007 s gven by d 0 = N+1V m logr c /L Here, V m s he maxmum P-wave n he poroelasc meda N s an neger Aer applyng a emporal Fourer ransorm o equaons 7-10, he ollowng equaon se resuls P = α M e M w u = u + u 4 5 e =e +e u = u + u The me dervaves o he dsplacemen can now be wren n erms o he sress pore lud pressure as mρ ρ u ωp = αm v M V mρ ρ ω P = α M v M V 14 = m τ s + ρ bv + ρ P s 15 = m τ s + ρ bv + ρ P s w = ρ τ + ρ b w + ρ P mρ ρ v = m τ + ρ bv + ρ P ω mρ ρ V = ρ τ s + ρbv + ρ P s τ + ρbv + ρ P +δ λc v s + α M V s 17 +δ λc v s + α M V s ω P = α M v s M V s 9 18 Pω = αm v s M V s By nroducng 1 auxlary varables resulng n 1 exra paral derenal equaons, he NPML equaons can be rewren as τ=µ v + v +δλ c v +αmv P = α M v M V ωτ = µ v s + v s mρ ρ V= ρ τ +ρbv+ρ P τ = µ v + v + δ λ c v + α M V ωτ = µ v s + v s mρ ρ V = ρ 16 = ρ τ s + ρ bv + ρ P s 7 8 ω m ρ ρ V mρ ρ v =m τ +ρ bv+ρ P P = αm v M V 13 ω mρ ρ v These equaons can be expressed as a se o rs order paral derenal equaons n he me doman by derenang equaons 1-4 wh respec o me resulng n ωτ = µ v + v + δ λc v + α M V 1 ω mρ ρ v = m τ + ρ b w + ρ P For smplcy, as ambguy s unlely, no ndcaon s gven n he equaon ses wheher he me requency doman s beng consdered Through coordnae srechng n he requency doman Cummer, 003, equaons ae he orm mρ ρ w = ρ τ + ρb w + ρ P mρ ρ u = m τ + ρ b w + ρ P ω m ρ ρ V = ρ τ + ρbv + ρ P 6 11 ωτ= µ v + v +δλ c v +αmv where e s he sran ensor [e = u + u / ] d = 1 or = d = 0 oherwse For a wo dmensonal medum, x 1,x, he summaon convenon has e = e + e ω mρ ρ V = ρ τ +ρbv+ρ P τ = µe +δλce +αm w P = α M e M w ω mρ ρ v = m τ + ρ bv + ρ P ω mρ ρ v = m τ +ρbv +ρ P 10 where v = u v = w Equaons 7-10 orm a se o rsorder hyperbolc paral derenal equaons n me or v, V,, P To reerae, v s he dsplacemen velocy vecor n he sold V he dsplacemen velocy vecor o he lud relave o ha n he sold, s he sress P he pore lud pressure ω m ρ ρ v = m τ + ρ bv + ρ P 19 ωmρρ v =m τ +ρbv+ρ P ω m ρ ρ V = ρ τ + ρbv + ρ P 0 ωτ = µ v + v + δ λc v + α M V 1 ω mρ ρ V = ρ τ +ρbv+ρ P ωτ =µ v + v +δλ c v +αm V ωp = αm v M V CJEG 3 June 01 ω P = α M v M V

3 mρ ρ v = m τ + ρ bv + ρ P mρ ρ V = ρ τ + ρbv + ρ P τ = µ v + v + δ λ v + α M V c P = α M v M V ξ + d ξ = ξ ξ { τ, v, V, P} ξ { τ, v, V, P } Nearly perecly mached layer mehod or sesmc wave propagaon n poroelasc meda The orgnal sae varables x x Œ{,v,V,P} are requred o be changed o he auxlary varables x ~ l, where x ~ l = xs l = x/1+d l /w, l Œ{1,} Aer applyng an nverse Fourer ransorm reurnng o he me doman, a se o paral derenal equaons are obaned as 3 mρ ρ v = m τ + ρ bv + ρ P mρ ρ V = ρ τ + ρbv + ρ P τ = µ v + v + δ λ v + α M V c P = α M v M V subec o ξ + d ξ = ξ { } ξ τ, v, V, P ξ { τ, v, V, P } where 7 Equaons 3-7 show ha he reormulaed wave equaons wh NPML preserve he orm o he orgnal wave equaons 7-10 Equaons 3-7 can be dscrezed by a saggeredgrd ne derence mehod Vreux, 1986; Lever, 1988; Graves, 1996 The posons o dscree sresses, lud pressure velocy componens are ndcaed n Fgure 1 Fgure 1 D schemac or he saggered spaal ne derence grd Fgure 3 The snapshos case 1 or he wave propagaon a 10ms 00ms a b represen x 1 componen o relave lud velocy o sold velocy by usng CPML NPML layers, respecvely Fgure The D gas-sauraed model wh our PML layers The blac cross ndcaes he source; he blac-lled rangle ndcaes recever; he dashed lnes represen PML layers Fgure 4 The sesmograms wh recordng lengh o ms a b represen x 1 componen o relave lud velocy o sold velocy a recevers R 1 R, respecvely CJEG 4 June 01

4 Jngy Chen Numercal Tes Case 1: Hgh requency source A D gas-sauraed ssone model dened schemacally n Fgure s used n he numercal ess All requred physcal parameers o he poroelasc model used n Sheen e al 006 are gven n Table 1 The model s dscrezed on a D N x1 N x = spaal grd A vercal pressure source wh me dependence speced by a Rcer wavele wh a predomnan requency 45Hz s locaed a grd 100, 1 The spaal grd spacng s 08mm he me samplng nerval s 0ms The wdh o PML layers s 10 grd pons The mplemenaon o saggered-grd ne-derence sases he Couran-Fredrchs- Lewy sably condon Couran e al, 198 Two recevers are placed a he grd pons 70, 1 80, 80 whch are close o he PML layers The purpose o usng he wo recevers s o chec he NPML wors well a grazng ncdences or long dsances o propagaon In he modelng es, a convoluonal perecly mached layer CPML s mplemened o chec he absorbng ably o NPML As nroduced n Komasch Marn 007, he dampng parameers, N=, R c =0001, a max =p 0 x 1 =x =1 are used n he numercal mplemenaon o he CPML mehod, whle N= R c = 0001 were seleced or mplemenaon n he NPML mehod, where 0 s predomnan requency o source, x 1 x are real 1 Fgure 3 llusraes he waveeld snapshos or wave propagaon a 10ms 00ms, where a b represen he snapshos o he x 1 componen o relave lud velocy o sold velocy by usng CPML NPML layers, respecvely The slow P-wave P s can be observed arrvng aer he as P S wave phases P S I s clear ha he NPML CPML mehods dsplay smlar absorbng perormances Fgure 4 shows ms sesmograms 10,000 me seps, where a b represen he relave lud velocy o sold velocy a recevers R 1 R, respecvely The agreemen beween Fgure 5 Decay o he oal energy case 1 n he doman whou he PML layers The sold lne ndcaes he energy compuaon wh CPML, he dashed lne denoes he energy compuaon wh NPML Fgure 7 The sesmograms wh record lengh o s a b represen he x 1 componen o relave lud velocy o sold velocy a recevers R 1 R, respecvely Fgure 6 The snapshos case or he wave propagaon a 01s 03s a b represen he x 1 componen o relave lud velocy o sold velocy by usng CPML NPML layers, respecvely Fgure 8 Decay o he oal energy wh he me 0 s n he doman whou he PML layers CJEG 5 June 01

5 Nearly perecly mached layer mehod or sesmc wave propagaon n poroelasc meda CPML sold gray lnes NPML shor dashed blac lnes s very good For urher comparson, he decay o energy wh me s also suded Fgure 5 represens he me decay o oal energy n he man doman whou PML layers whch s nroduced by Marn e al 008 Theorecally, no energy s remaned n he model aer 058ms because waves have le he man doman All he energy ha remans s consdered spurous energy One can observe ha he oal energy compued wh NPML decreases gradually, whereas he oal energy compued wh CPML shows oscllaory behavor Addonally, one can noce ha he oal energy compued wh NPML decreases aser han ha o CPML Case : Low requency source In case, he predomnan requency o source, spaal grd spacng me samplng nerval are changed o 45Hz, m 0ms Fgure 6 llusraes he waveeld snapshos or he wave propagaon a 01s 03s, where a b represen he snapshos o he x 1 componen o relave lud velocy o sold velocy by usng CPML NPML layers, respecvely In hs es, he slow P wave dsappears due o s hgh duson Ths observaon s conssen wh he descrpon n Sheen e al 006 Fgure 7 llusraes he s sesmograms 10,000 me seps, where a b represen x 1 componen o relave lud velocy o sold velocy a recevers R 1 R, respecvely Fgure 7 shows ha he agreemen beween CPML sold gray lnes NPML shor dashed blac lnes s agan very good Theorecally, no energy remans n he model aer 074s One can observe ha he oal energy compued wh NPML CPML decreases connuously n Fgure 8 However, he oal energy compued wh NPML decreases aser han ha o CPML To chec he sably o numercal smulaon a longer me, boh CPML NPML codes were rerun o 0s 100,000 me seps Fgure 9b shows he close-up o he energy decay curve beween 18s 0s One can observe ha he oal energy compued wh NPML decreases aser han ha o CPML, he oal energy compued wh CPML shows oscllaory behavor Fgure 9 Conclusons In hs paper, nearly perecly mached layer NPML absorbng boundary condon s appled o he D gas-sauraed poroelasc model A grea advanage o he NPML s ha he reormulaed wave equaons preserve he orm o he orgnal non-boundary wave equaons The numercal resuls show ha he arcal relecons rom he model edges are eecvely suppressed In addon, no nsably s observed durng numercal smulaon usng boh low hgh domnan requences o sources eg 45Hz 45Hz, whch means he numercal mehod wh NPML s hghly ecen even or longer me smulaon Acnowledgemens The auhor wshes o han The Unversy o Tulsa or s nd suppor o hs research The open-source codes provded by Dr Dmr Komasch are used n hs sudy The commens suggesons o Edor Larry Lnes Dr PF Daley sgncanly mproved hs manuscrp Fgure 9 a Decay o he oal energy wh he me 0-0s n he doman whou he PML layers, b he close-up o he energy decay curve beween 18s 0s Reerences Bérenger, J, 1994 A perecly mached layer or he absorpon o elecromagnec waves Journal o Compuaonal Physcs, 114, Bérenger, J, 004 On he relecon rom Cummer s nearly perecly mached layer A perecly mached layer or he absorpon o elecromagnec waves IEEE Mcrowave Wreless Componens Leers, 14, Bo, MA, 196 Mechancs deormaon acousc propagaon n porous meda Journal o Appled Physcs, 33, Carcone, J M, 1996 Wave propagaon n ansoropc, sauraed porous meda: plane wave heory numercal smulaon The Journal o he Acouscal Socey o Amerca, 99, Ceran, C, Koslo, D, Koslo, R Reshe, M, 1985 A nonrelecng boundary condon or dscree acousc elasc wave equaons Geophyscs, 50, Chen, J, 011 The applcaon o he nearly perecly mached layer o numercal modelng n poroelasc meda Exped Absrac, 81s SEG Annual Inernaonal Meeng, San Anono, Chen, J Zhao, J, 011 Applcaon o he nearly perecly mached layer o sesmc-wave propagaon modelng n elasc ansoropc meda Bullen o he Sesmologcal Socey o Amerca, 101, CJEG 6 June 01

6 Jngy Chen Chew, WC Weedon, WH, 1994 A 3-D perecly mached medum rom moded Maxwell s equaons wh sreched coordnae Mcrowave Opcal Technology Leers, 7, Clayon, R Engqus, B, 1977 Absorbng boundary condons or acousc elasc wave equaons Bullen o he Sesmologcal Socey o Amerca, 67, Collno, F Tsoga, C, 001 Applcaon o he PML absorbng layer model o he lnear elasodynamc problem n ansoropc heerogeneous meda Geophyscs, 66, Couran, R, Fredrchs, KO Lewy, H, 198 Über de parellen Derenzenglechungen der mahemaschen Phys Mahemasche Annalen, 100, 3-74 Cummer, S A, 003 A smple, nearly perecly mached layer or general elecromagnec meda IEEE Mcrowave Wreless Componens Leers, 13, Graves, R W, 1996 Smulang sesmc wave propagaon n 3D elasc meda usng saggered-grd ne-derences Bullen o he Sesmologcal Socey o Amerca, 86, Hu, W, Abubaar, A Habashy, T M, 007 Applcaon o he nearly perecly mached layer n acousc wave modelng Geophyscs, 7, Komasch, D Marn, R, 007 An unspl convoluonal perecly mached layer mproved a grazng ncdence or he sesmc wave equaon Geophyscs, 7, SM155-SM167 Koslo, R Koslo, D, 1986 Absorbng boundary or wave propagaon problems Journal o Compuaonal Physcs, 63, Lever, A R, 1988 Fourh-order ne-derence P-SV sesmograms Geophyscs, 53, Marn, R, Komasch, D Ezzan, A, 008 An unspl convoluonal perecly mached layer mproved a grazng ncdence or sesmc wave propagaon n poroelasc meda Geophyscs, 73, T51- T61 Morency, C Tromp, J, 008 Specral-elemen smulaons o wave propagaon n porous meda Geophyscal Journal Inernaona, l75, Özdenvar, T McMechan, G A, 1997 Algorhms or saggered-grd compuaons or poroelasc, elasc, acousc, scalar wave equaons Geophyscal Prospecng, 45, Panneon, R Aalla, N, 1997 An ecen ne elemen scheme or solvng he hree-dmensonal poroelascy problem n acouscs The Journal o he Acouscal Socey o Amerca, 101, Reynolds, AC, 1978 Boundary condons or he numercal soluon o wave propagaon problems Geophyscs, 43, Roden, J A Gedney, S D, 000 Convoluonal PML CPML: an ecen FDTD mplemenaon o he CFS-PML or arbrary meda Mcrowave Opcal Technology Leers, 7, Sheen, DH, Tuncay, K, Baag, CE Oroleva, P J, 006 Parallel mplemenaon o a velocy-sress saggered-grd ne-derence mehod or -D poroelasc wave propagaon Compuers & Geoscences, 3, Vreux, J, 1986 P-SV wave propagaon n heerogeneous meda: velocy-sress ne-derence mehod Geophyscs, 51, Zeng, YQ Lu, QH, 001 A saggered-grd ne-derence mehod wh perecly mached layers or poroelasc wave equaons The Journal o he Acouscal Socey o Amerca, 109, Zhu, X McMechan, GA, 1991 Fne-derence modelng o he sesmc response o lud sauraed, porous, elasc meda usng Bo heory Geophyscs, 56, Gas-sauraed Ssone r g/m r g/m m g/m l c GPa 0530 m GPa 1855 M GPa b Pa s/m a Table 1 Physcal properes o he gas-sauraed ssone Sheen e al, 006 Correspondence o: J Chen, Deparmen o Geoscences, The Unversy o Tulsa, Tulsa, OK, USA; E-mal: ngy-chen@uulsaedu CJEG 7 June 01