BEM COMPARATIVE ANALYSIS OF DYNAMIC RESPONSES OF POROVISCOELASTIC MEDIA USING VISCOELASTIC MODEL

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1 Proceedg o he 7h Ieraoa Coerece o Mechac ad Maera Deg Abuera/Poruga -5 Jue 7. Edor J.. Sva ome ad S.A. Megud. Pub. INEI/EUP 7 PAPER RE: 6957 BEM COMPARATIVE ANALYSIS O DYNAMIC RESPONSES O POROVISCOELASTIC MEDIA USIN VISCOELASTIC MODEL Leod Igumov Akekadr Ipaov * Vaery Novkov earch Iue or Mechac Lobachevky Sae Uvery o Nzh Novgorod Rua * Ema: paov@mech.u.ru ABSTRACT The pree paper dedcaed o umerca ovg o hree dmeoa boudary-vaue probem porovcoeac ormuao. The porovcoeac ormuao reaed a a combao o Bo heory o poroeacy ad eac-vcoeac correpodece prcpe. Such vcoeac mode a Kev-Vog Sadard ear od ad a mode wh weaky guar kere are codered. Boudary eeme mehod BEM ad boudary egra equao BIE mehod are apped o oba Lapace doma ouo o hreedmeoa boudary-vaue probem. Moded Durb agorhm o umerca vero o Lapace raorm ued o perorm ouo me doma. A probem o a porovcoeac cube wh a cavy ubeced o a orma era preure codered. The comparo o dyamc repoe whe porovcoeac maera decrbed by dere vcoeac mode preeed. Vcoy parameer uece o dyamc repoe o dpaceme ad pore preure uded. Keyword: porovcoeacy poroeacy vcoeacy boudary eeme mehod BEM boudary egra equao BIE. INTRODUCTION Wave propagao dpered meda a grea ere o may dcpe. Mechac o advaced maera uch a poro- vco- or porovcoeac maera reeva o uch dcpe a geophyc geo- ad bomechac emoogy phyca chemry peroeum egeerg ec. The mpemeao o he od vcoeac eec he heory o poroeacy wa r roduced by M. Bo Bo 956. The vcoeac deormao od keeo o poroeac meda may be accoued hrough he ue o correpodece prcpe Cree 974. M. Shaz Shaz preeed a aayca ouo he Lapace doma ad aayca me-doma ouo whou coderg vcou coupg eec ad he goverg equao wa exeded or auraed porovcoeac meda by ug he Kev-Vog mode ad obaed a aayca ouo he Lapace doma or he D probem Schaz & Cheg. Becaue he compexy o he era vcoy ad mechaca coupg porou meda mo rae repoe probem ca oy be oved va umerca mehod. Caca ormuao or BIE mehod wh her dcrezed reazao ad radoa BEM are ucceu approache or ovg hree-dmeoa probem Bazheov 8. Exeo o BIE mehod ad BEM o aoropc probe ad probem where maera mode are dere rom eac eed deveopme o peca ew cheme. Nowaday here are wo udameay dere approache or corucg uch umerca cheme: drec ovg -69-

2 Topc-A: Compuaoa Mechac me doma Nard. 98 or ovg Lapace or ourer doma Crue 968. Uoruaey hree-dmeoa cae o poroeacy o away pobe o coruc requred udamea ouo me coderg porou maera oy pobe Lapace or ourer doma Shaz. PROBLEM ORMULATION We coder a homogeeou od Ω hree-dmeoa pace R ad Γ Ω he boudary o Ω. Sod Ω ake a oropc porovcoeac. Bac poroeac maera a wo-phae maera cog o a eac keeo ad comprebe ud or ga er. Porou maera o a voume V ca be coruced a oow: V V V he ummary pore voume ad where V he oa voume V he voume o he keeo. I aumed ha er ca opey eep hrough he pore ad a coed pore are aumed a a par o he keeo. The a correpodece prcpe apped o he keeo o we exed poroeac ormuao o porovcoeacy. Coderg a boudary-vaue probem or Bo mode o uy auraed poroeac couum Lapace doma erm o 4 ukow dpaceme u ad pore preure p he e o derea equao ake he oowg orm Shaz : u K u β ρ p ψ β p φ p ψ β u R u x u~ x ~ V ρβρ u a x Ω u u u p u x Γ u σ x Γ q u σ where Γ ad Γ deoe boudare or boudary codo o ad d kd repecvey K are eac modu φ V / V poroy a are buk body orce κρ φ β φ κ ρ φρ K ψ ad a K R K K φ K K φk K K K are coa reecg eraco bewee keeo ad er κ permeaby. urher ρ ρ φ φρ a buk dey ρ ρ a ρ are od appare ma dey ad er dey repecvey K K are eac buk modu o he keeo ad er repecvey. Appare ma dey a ρ Cφρ wa roduced by Bo o decrbe dyamc eraco bewee ud ad keeo. C a acor depedg o he pore geomery ad excao requecy. Porovcoeac ouo obaed rom poroeac ouo by mea o he eacvcoeac correpodece prcpe apped o keeo coa K ad Lapace doma. orm o uco K ad are deped o choe vcoeac mode. -7-

3 Proceedg o he 7h Ieraoa Coerece o Mechac ad Maera Deg -7- LAPLACE TRANSORM INVERSION The porovcoeac ouo me doma obaed wh he hep o he Lapace raorm vero. Modcao o a Durb agorhm Durb 974 ued or umerca vero o Lapace raorm. Sugge beg a Lapace raorm parameer he: d Im co d e. > Dvde he egrao erva [ R] o ma ub-pace ad deoe he ode wh.... The he egra ad become Zhao 4: d _ 4 d e Im co. > 5 I each ub-pace o [ ] we approxmae ad Im wh ear uco repecvey: Im where Im Im. So he ear approxmao uage ead o oowg ormuae: 6 co co co co e 7 Wh repec o ad he a orm o 7 :

4 Topc-A: Compuaoa Mechac e co co α ormuae have he ame preco order a he rapezoda rue amey O max. BOUNDARY ELEMENT APPROACH Iegra repreeao ad BIE wh egra Lapace raorm are ued. udamea ad guar ouo are codered erm o guary oao. Numerca cheme baed o he ree-be-somgaa ormua. Boudary-vaue probem ca be reduced o he boudary egra equao BIE yem a oow Bazheov 8 Shaz : ~ α Ω u x k ~ Γ ~ ~ T x y u y T x y u x U ~ x y y k x Γ T T q u u u u p where U x T x - udamea ad guar ouo T x guare. k k ~ dγ coa he oaed Boudary urace o our homogeeou od dcrezed by quadraguar ad raguar eeme ad raguar eeme are aumed a guar quadraguar eeme. We ue reerece eeme: quare ξ ξ [ ] ξ ad rage ξ ξ ξ ξ ad each boudary eeme mapped o a reerece oe by he oowg ormua: y ξ 8 N ξ y β k where - oca ode umber eeme k β k - goba ode umber N ξ - hape uco. odhey dpaceme-re mxed mode perormed. To dcreze he boudary urace egh-ode bquadrac quadraera eeme are ued geerazed dpaceme ad raco - are approxmaed by ear ad coa hape uco repecvey. Subeque appyg o coocao mehod ead o he yem o ear equao. A he coocao ode we ake he approxmao ode o boudary uco. or each homogeeou doma o he od ear equao wre geerazed dpaceme approxmao ode are a oow Ugodchkov 986: N 4 N N 4 αω m k χ k k k u A u B k k k N N N. D k u χ k u χ k Lear equao wre geerazed raco approxmao ode ake oowg orm: αω u 8 N 4 N N 4 m k χ k k k A u B k k k N N N D k u χ k u χ k -7-

5 Proceedg o he 7h Ieraoa Coerece o Mechac ad Maera Deg To boudary eeme wh o coac correpod he r erm boh equao o ay he coac codo bewee eeme k ad k - ecod oe. ormuae o accoued coece A B ad D are: A k m k m k k [ R ξ T x y ξ δχ k мt x y ξ ] J ξ dξd ξ B k U m k x y ξ J ξ dξ dξ k D k m k χχ k R ξ U x y ξ J ξ α y ξ dξ dξ where he parameer o Lapace raorm. aua quadraure are ued o cacuae egra o reguar eeme. Bu a eeme coa a guary agorhm o guary avodg or order reducg apped Igumov 4. Whe guary excuded we ue a adapve egrao agorhm. A approprae order o aua quadraure choe rom prmary kow eceary preco mpobe he eeme ubdvded o maer eeme recurvey. k NUMERICAL RESULTS A porovcoeac cube wh de egh L ad coag a pherca cavy o radu R.5 m codered ee g.. The cube camped a he ace x m ad cavy ubeced o a orma era preure: or P P.5 or.5 P 5 Pa. P or > Poroeac maera parameer are: K 8 N / m 6 N / m φ.66 k.9 K.6 N / m m 4 / N Berea adoe. ρ kg / m K 458kg / m ρ 9. N / m x g. - A cube coag a pherca cavy. The BEM meh empoyed co o 4 boudary eeme ad 46 ode. g. -7 how he rae repoe o he dpaceme u ad pore preure p he ceer o upper ace a he oda po wh he coordae m. or porovcoeac aay dere vcoeac mode were codered. Vcoy parameer uece o dyamc repoe uded. -7-

6 Topc-A: Compuaoa Mechac. Kev-Vog mode: K K γ. γ Parameer o Kev-Vog mode: γ 5 ad..5 x poroeac γ γ γ 5 γ u m x - g. - Dpaceme u cae o Kev-Vog mode x -4.5 poroeac γ γ γ 5 γ p N/m x - g. - Pore preure p cae o Kev-Vog mode -74-

7 Proceedg o he 7h Ieraoa Coerece o Mechac ad Maera Deg. Sadard ear od mode: K K β γ β β K. γ K Parameer o adard mode: β 4 γ ad...5 x -6.5 poroeac γ γ γ γ..5 u m x - g. 4 - Dpaceme u cae o adard ear od mode x -4.5 poroeac γ γ γ γ. p N/m x - g. 5 - Pore preure p cae o adard ear od mode -75-

8 Topc-A: Compuaoa Mechac. Mode wh weaky guar kere: K K α h α h Parameer o weaky guar mode: α..5 ad.9 h ad 4. 7 x poroeac α. h α.9 h α.5 h α.5 h 4 u m x - g. 6 - Dpaceme u cae o mode wh weaky guar kere.5 x -4 poroeac α. h α.9 h α.5 h α.5 h 4 p N/m x - g. 7 - Pore preure p cae o mode wh weaky guar kere -76-

9 Proceedg o he 7h Ieraoa Coerece o Mechac ad Maera Deg CONCLUSIONS A probem o a oropc porovcoeac cube wh a cavy ubeced o he era preure codered. u o umerca experme are empoyed. A Lapace doma BEM approach or he rae aay o he hree-dmeoa ear porovcoeac od baed o he correpodece prcpe ha bee preeed. The porovcoeac meda modeg baed o Bo heory o uy auraed poroeac maera combao wh he eac-vcoeac correpodg prcpe. Kev-Vog adard ear od ad weaky guar mode are codered a vcoeac couve mode. Boudary egra equao mehod ad boudary eeme mehod are apped order o ove hree dmeoa boudary-vaue probem. guarzed BIE yem codered. Mxed boudary eeme are ued o perorm he paa dcrezao. Moded Durb agorhm apped o oba ouo me doma. A uece o maera vcoy o rae repoe o dpaceme ad pore preure preeed. Obaed reu or he rae repoe o he dpaceme how he eec o he rao rom aaeou o equbrum vcoeac modu cae o he adard ear od mode uch eec wa earer demoraed Igumov 4. ACKNOWLEDMENTS The work wa acay uppored by Rua Scec oudao gra REERENCES []-Bazheov V.. Igumov L.A. Boudary Iegra Equao & Boudary Eeme Mehod reag he probem o D eaodyamc wh couped ed. PhyMahL Mocow Rua 8 []-Bo M. A. Theory o propagao o eac wave a ud-uuraed porou od. J. Acou. Soc. Am []-Cree R. Iroduco o he heory o vcoeacy. Mr Mocow 974. [4]-Crue T.A. Rzzo.J. A drec ormuao ad umerca ouo o he geera rae eaodyamc probem. Par I. J. Mah. Aa. Appc [5]-Durb. Numerca vero o Lapace raorm: a ece mproveme o Duber ad Abae mehod. The Compuer Joura [6]-Igumov L.A. Lvchuk S.Y. Perov A.N. Beov A.A. Boudary-eeme modeg o -D poroeac ha-pace dyamc. Advaced Maera earch [7]-Igumov L.A. Ipaov A.A. Sabaeva T.A. Maera vcoy eec o dyamc repo vcoeac ad porovcoeac od. Probem o Sregh ad Pacy [8]-Nard D. Brebba C. A. Boudary Eeme Mehod Egeerg C. A. Brebba Ed.. Sprger Ber

10 Topc-A: Compuaoa Mechac [9]-Schaz M. Wave Propagao Vcoeac ad Poroeac Coua. Sprger Ber ermay. []-Schaz M. Cheg A.H.D. Dyamc Aay o a Oe-Dmeoa Porovcoeac Coum J App Mech []-Ugodchkov A.. Huoryak N.M. Boudary eeme mehod deormabe od mechac. Kaza Sae Uvery Kaza Rua 986. []-Zhao X. A ece approach or he umerca vero o Lapace raorm ad appcao dyamc racure aay o a pezoeecrc amae. I. J. Sod Sruc