τ ), respectively. Then

Transcription

1 Downloaded /0/ to Reditribution ubject to SEG licene or copyright; ee Term of Ue at Quality factor affect channel wave propagation in 3D iotropic vicoelatic medium Hui i*, Univerity of Houton, eimin Zhu, China Univerity of geocience, Guangzhong Ji, Xi an Branch of China Coal Reearch Intitute Summary Due to coal bed eiting in the middle of higher velocity wall rock, eimic wave interferencing with each other generate guide wave propagating within coal bed and urrounding wall rock (iu,994). Three dimenional channel wave propagating within homogeneou iotropic vicoelatic medium containing coal bed i imulated with an eplicit, time-domain, high order taggered finite difference algorithm (Carcione, 998). General tandard linear olid (GSS) ytem, repreenting diperion/attenuation characteritic at both low frequency band and high frequency band, i ued to reflect vicoelatic property. Conidering maive computational time-conuming, a parallel computational implementation, utilizing OpenM trategy, allow invetigation of largecale three dimenional model and/or broadband wave propagation within reaonable eecution time. Introduction Sedimentary rock and coal bed how vicoelatic characteritic rather than elatic characteritic that generally aumed. Theoretical, numerical, laboratory, and field data indicate that eimic wave through a coal bed i ignificantly affected by the preence of vicoelatic property. In particular, complicated coupling mechanim of channel wave between wall rock and coal bed varie with change of quality factor (Q). It i an important phyical parameter for attenuation and diperion of eimic wave. In turn, quality factor (Q) can be ued to eplain the meaurement of channel wave attenuation/diperion. In preent work, a time-domain high-order taggered grid finite difference (FD) algorithm i developed for numerically olving vicoelatic wave equation in three dimenional patial dimenion. Our algorithm i an adaptation of the high order taggered grid velocity-tre approach. Theory and Method According to vicoelatic theory (Mou, 003), tre ( ) and train ( ) relation can be epreed a follow σ =Λ * δε + M* δ ij ij kk ij Stre and velocity ( ) relation can be given: () εij = ( iυj + jυi ) Π M Auming =Λ+, then we known for GSS model p t l Π= π[ ( ) e σ ] θ ( t) σ (3) l t l M = μ[ ( ) e σ ] θ ( t) (4) repectively. In the formulation, π = λ+ μ, λ, μ indicate lame elatic coefficient. θ () t denote memory variable. Π, M depend on relaation contant of -wave p train ( ), repectively. Then ) and S-wave train ( ubtituting equation (3), (4) into (). σ = ( Π M)* υ + M* υ, i = j (5a) ij k k i j σ = M*( υ + υ ) i j ij i j j i (), (5b) Then ubtituting equation (3) and (4) into (5a) and (5b), tre equation can be given by linear approimation σ = { π[ ( )] μ[ ( )]} p ij kυk μ[ ( )] iυj rijl + + i = j σ μ υ υ, (6a) ij = [ ( )]( i j ++ j i ) + rijl i j, (6b) And memory variable equation can be given p { [ ( ) ( rijl = rijl + π μ )] kυk + μ( )] iυj} 0 SEG DOI SEG a Vega 0 Annual Meeting age

2 Quality factor affect channel wave propagation Downloaded /0/ to Reditribution ubject to SEG licene or copyright; ee Term of Ue at l, i = j (7a) [ ( rijl = rijl + μ )( iυj + jυi )], l, i j For particle velocity equation i given i jσij fi (7b) ρυ = + (8) f i denote eternal force, i, j, k =, y, z. i function of -wave velocity, -wave relaation time, and frequency; µ i function of S-wave velocity, S-wave relaation time, and frequency. The relation between relaation time and -wave quality factor ( Q ) and S wave quality factor ( Q ) are given ( σ = + ), l = ω Q Q (9a) p 0, + ω σq ε = ε = ω ωq ω (9b) σ repectively. ω denote domain frequency. When =, the above equation can be deteriorated into tandard linear vicoleatic equation. O(,N) Staggered Grid Finite Difference Scheme p : σ, σ, σ, r, r, r, π, λ, μ, σ, ε, ε yy zz yy zz : V, ρ ; : V, y ρ ; : Vz, ρ : σ, r, σ, r, σ, r μ yz yz σ Figure : Staggered patial torage cheme for the vicoelatic dependent variable The partial differential equation (6a), (6b), (7a), (7b) and (8) are numerically olved with an eplicit, time-domain high order taggered grid finite difference technique. Dependent variable are tored on taggered patial and temporal grid. Figure depict the ditribution of the dependent variable over an elementary cell of the 3D grid. All patial derivative are approimated with high order taggered FD operator proceing N th order (4 th order i ued in thi paper) accuracy in the dicretization interval. Stree are tored on the integer time rater tn, and velocitie are tored on the interlaced half-integer time rater tn+/, where i time tep. Temporal FD operator are centered and have nd order accuracy. Thee FD updating formulate are (only howing one component) σ (, i j, k) = σ (, i j, k) + π(, i j, k) {[ + ( i, j, k)]{ [ v ( i, j, k)] + [ v ( i, j, k)] + [ v ( i, j, k )]}} μ( i, j, k) Δ t[ + ( i, j, k)] { y[ vy(, i j, k)] + z[ vz(, i j, k )]} + [ Rl ( i, j, k) + Rl ( i, j, k)] σz (, i j, k) = σz (, i j, k) + μ(, i j, k) [ + ( i, j, k)]{ [ vz( i, j, k )] + z[ v( i, j, k)]} + [ R ( i, j, k) + R ( i, j, k)] zl zl R i j k R i j k (0) () n+ n l (,, ) = ( + ) {( ) l (,, ) π (, i j, k) (, i j, k){ [ v( i, j, k)] + [ v ( i, j, k)] + [ v ( i, j, k )]} 0 SEG DOI SEG a Vega 0 Annual Meeting age

3 Quality factor affect channel wave propagation Downloaded /0/ to Reditribution ubject to SEG licene or copyright; ee Term of Ue at μ( i, j, k) (, i j, k) { [ v (, i j, k)] + [ v (, i j, k )]}} () R i j k R i j k n+ n zl (,, ) = ( + ) {( ) zl (,, ) μ(, i j, k) (, i j, k){ [ vz(, i j, k )] + [ v ( i, j, k)]} z v ρ( i, j, k) + n n + + { [ σ ( i, j, k)] y[ σy ( i, j, k)] + n + + n + [ σ ( i, j, k )] + f ( i, j, k)} n+ + n + ( i, j, k) = v ( i, j, k) + ) + z z (3) + (4) For the above formulate, denote time interval, upercript n+ and n- denote n+/ and n-/, repectively. i differential operator given N + ( [ u( )] = Ci Δ N ) { u( + iδ) u[ ( i ) Δ]} i= N ( N ) [ u( )] = Ci { u[ + ( i ) Δ] u( iδ)} Δ i= (5) Synthetic Data Eample Timelice in figure illutrate 3D vicoelatic channel wave propagation in the model (00m*00m*00m) coniting of coal bed (thickne i 0m, from z=45m to z=55m). A point eploion ource activated by a Ricker wavelet (domain frequency 0Hz) i located at coal bed (5m,00m,50m). Fine patial (=y=m, z=0.5m) and temporal (0.m) gridding are required for repreentation of channel wave, without numerical diperion. A parallel algorithm implementation, utilizing OpenM trategy, allow thi large model to be invetigated in reaonable eecution time. hyical parameter and quality factor ditribution are diplayed in Table and Table, repectively. In the table, Qp_w and Q_w denote p-wave quality factor of wall rock a nd -wave quality facto r of wall rock, repectively. Qp_ c and Q_c denote p-wave quality factor of coal and - wave quality factor of coal, re pectively. -wave m/ Table : trata phyical parameter S-wave De nity thick lithology m/ (g/cm3) m Sandtone Coal bed Sandtone Table : Quality factor ditribution Model Qp_w Q_w Qp_c Q_c Dicuion of reult/concluion Figure how V component (left column) and Vz component (right column) naphot in the 45m. For the V component, channel wave, concentrating motly eimic wave energy, propagate in the coal bed. Only a little energy till leak into wall rock. e energy tranmiion happen a the quality factor (both wall rock and coal bed) increae. But intenity of hear wave of wall rock lightly increae when quality factor increaing. For the channel wave, the energy eem remain table a quality factor increae. For the Vz component, although channel wave concentrate motly eimic wave energy, partly energy tranmitting into wall rock propagate a hear wave velocity of wall rock. A quality factor increae, the intenity of compreional wave of wall rock decreae dramatically, but for hear wave of wall rock, it till keep almot contant. Conidering of contrat of intenity of wall rock and coal bed, the V component ha higher channel wave tranmiion wave ratio, and the ratio keep relatively table while quality factor increae/decreae. In turn, V component may be more advantage for channel wave eploration. 0 SEG DOI SEG a Vega 0 Annual Meeting age 3

4 Quality factor affect channel wave propagation Downloaded /0/ to Reditribution ubject to SEG licene or copyright; ee Term of Ue at Model V 45m Model V45m Model 3 V 45m Model 4 V 45m Figure : V and Vy component naphot at 45m Mode Vz 45m Mode Vz 45m Mode 3 Vz 45m Mode 4 Vz 45m 0 SEG DOI SEG a Vega 0 Annual Meeting age 4

5 EDITED REFERENCES Downloaded /0/ to Reditribution ubject to SEG licene or copyright; ee Term of Ue at Note: Thi reference lit i a copy-edited verion of the reference lit ubmitted by the author. Reference lit for the 0 SEG Technical rogram Epanded Abtract have been copy edited o that reference provided with the online metadata for each paper will achieve a high degree of linking to cited ource that appear on the Web. REFERENCES Carcione, J., 998, Vico-acoutic wave propagation imulation in the earth: Geophyic, 53, iu, T. F., D. M. an, 994, In-eam eimic eploration: China Univerity of Mining and Technology re. Mou, Y. G., 005, Seimic numerical modeling for 3D comple media: etroleum Indutry re. 0 SEG DOI SEG a Vega 0 Annual Meeting age 5