Generalized form of reflection coefficients in terms of impedance matrices of qp-qp and qp-qs waves in TI media

Transcription

1 Generalzed form of reflecton coeffcents n terms of mpedance matrces of q-q and q-q waves n TI meda Feng Zhang and Xangyang L CNC Geophyscal KeyLab, Chna Unversty of etroleum, Bejng, Chna ummary Reflecton coeffcents of q- and q- ncdent waves n orthorhombc meda can be explctly expressed by means of mpedance matrces. In ths paper, we extend prevous studes and derve generalzed lnearzed equatons of q-q and q-q waves. Impedance matrces representng complete reflecton propertes of the medum can be nverted based on the smplfed equatons. Besdes, these equatons are reduced for an sotropc medum wth a vertcal symmetry axs (VTI or a horzontal axs (HTI. We dscuss ther accuracy and applcablty by means of three models: two models of VTI/VTI nterfaces wth moderate and strong ansotropy, and a VTI/HTI model. The approxmatons have suffcent accuracy for relatvely strong ansotropy. Therefore they can be appled n the characterzaton of unconventonal resources, for example, shale gas or coalbed methane. 74 th EAGE Conference & Exhbton ncorporatng E EUROEC Copenhagen, Denmark, 4-7 June

2 Introducton AVO analyss and nverson are essental technques n the characterzaton of hydrocarbon reservors. Ther development s based on the plane wave reflecton coeffcents from an nterface between two subsurface meda. The frst analytc soluton of sotropc meda s publshed by Zoepprtz (99. ubsequently, a seres of approxmatons of - and -wave reflecton coeffcents have been studed for AVO analyss and nverson. Analytcal solutons of reflecton coeffcents n ansotropc meda can be found for wave propagaton n symmetry planes, such as by Daley and Hron (977 for transversely sotropc (TI meda wth vertcal symmetry axs, and choenberg and rotazo (99 for symmetrc plane of orthorhombc meda. On the assumpton of weak TI meda, dfferent lnear equatons of q-q and q-q waves have been nvestgated, for example by Ruger (997 and Jlek (. In ths paper, we extend prevous studes and derve a generalzed equatons of q-q and q-q reflecton coeffcents for orthorhombc meda n terms of mpedance matrces. These equatons can be reduced for an sotropc medum wth a vertcal symmetry axs (VTI or a horzontal axs (HTI. We dscuss ther accuracy and applcablty by means of three models: two models of VTI/VTI nterfaces wth both moderate and strong ansotropy, and a VTI/HTI model. The approxmatons have suffcent accuracy for relatvely strong ansotropy. Generalzed form of reflecton coeffcents and nverson n orthorhombc meda Orthorhombc ansotropy results from a combnaton of thn layers and algned fractures. It s one of the most common forms of ansotropy n sedmentary basn (Bush and Crampn, 99. In ths case, we use q to denote the quas-v wave, because the q- and qv-wave are decoupled from the H wave n the symmetry planes. An explct equaton for reflecton coeffcents at a planar nterface separatng two layers wth orthorhombc ansotropy s expressed by ( X X Y Y ( X X Y Y Rqq Rqq R ( + Rqq Rqq where refers to the th medum layer, and X and Y are the frequency ndependent mpedance matrces ntroduced by choenberg and rotazo(99: snθ X ρ β snϕ snθ ρ β snϕ, sn( θ + θ + C55( / cos( ϕ ϕ C55( Y. ( cos( θ θ C33( / + sn( ϕ ϕ C33( snϕ In the above equatons, and β are the phase veloctes of the q- and q-wave, respectvely; ρ ( C33( C3( reduces to the densty ρ for sotropc meda; the polarzaton drectons are denoted as θ (q-wave and ϕ (q-wave, and sn θ ( + ζ ( snθ ; sn ϕ ( + ζ ( snϕ, where C( ( cos ( C( C55( ( δ θ ε ζ C and ( ζ ( δ cos ϕ + ε C cos ϕ /( C cos ϕ 55( (, ( C C ( ( 55( representng the polarzaton devaton from the propagaton drecton θ and ϕ : and δ ( C55 ( + C3( C( / C(, / ( ε ρ C, and ε ρβ / C55(. The determnants of the mpedance matrces are and snθ / Y C55( / sn ϕ β snθ X C33(, / snϕ. In order to evaluate the effect of ansotropy on the AVO response and to nvert for the elastc propertes, we derve a smple expresson for reflecton coeffcents n terms of the frst-order elastc propertes. Equaton ( can be approxmated for meda of small mpedance contrast as 74 th EAGE Conference & Exhbton ncorporatng E EUROEC Copenhagen, Denmark, 4-7 June

3 - - R X X Y Y (3 In equaton (3, and denotes the average and change of the elastc propertes of the th and the (+th medum layers, respectvely. Inverson of mpedance matrx The mpedance matrces X and Y are functons of elastc parameters. They represent the complete reflecton propertes of the medum. If the ampltude of all types of reflecton from and ncdent waves are recorded, X - X and Y - Y can be nverted from R based on equaton (3. Then X and Y can be solved by the recursve matrx-nverson formula that extends the conventonal recursve equaton: Lnearzaton of Reflecton coeffcents X X X / X, and k k- k k- k k Y Y Y / Y (4 The matrces of mpedance contrast X and Y can be derved by dfferentatng X and Y n each of the relevant varables: d ρ + C33 + C55 + C3 + + β + θ + ϕ + ζ + ζ (5 ρ C33 C55 C3 The reflecton coeffcents of qq- and qq- waves are expressed by means of the average and contrast of the elastc propertes of the adjacent meda: C ρ β ρ β β C β R qq θ + θ θ + + θζ C C + ρ β sn sn tan tan C (6 sn θ C33 β C55 ρ β ρ β sn ϕ R qq sn θ ζ (7 C C C ρ β sn θ mplfed soluton for weak VTI meda Frst, we present the reflecton coeffcents for a VTI medum smplfed from equatons (6 and (7. A VTI medum has a vertcal symmetry axs. The ansotropy results from a sequence of thn layers (Bush and Crampn, 99. For the sotropc meda, ρ ρ,, β β and ζ ζ ; the average elastc constants and ther contrast across the nterface s gven by 33 ρ 55 C 44 ρ β C, C and C 33 / C33 ρ / ρ + ( /, C 44 / C44 ρ / ρ + ( β. Consderng weak VTI ansotropy, Thomsen (986 proposed a set of parameters whch are lnked to practcal sesmc measurements: ε C ( C 33( ( C 3( + C 44( ( C 33( C 44 C 44( + C 3( C 33( ; δ ; γ C 33( C 33( ( C 33( C 44 C 33( and the phase veloctes are expressed as 4 ( + δ sn θ cos θ + ε θ β θ ϕ ζ C ζ 66( C 44( C 44( ( sn ; β + ( ( β ( ( ε δ sn ϕ cos ϕ. (9 ( Other relevant varables can be wrtten as ( ( ( ρ ρ ( ( δ ( ( ε δ sn ϕ cos ϕ, ζ [ δ + ( ε δ sn θ ] cos θ /( β and ζ [ δ + ( ε δ sn ϕ] cos ϕ /( β. Consderng polarzaton devatons are small, mpedance matrces and ther contrast can be expressed as ( ( ( ( ( ( ( ( X X + X, Y Y + Y, X d ( X + d( X, and Y d ( Y + d( Y. ( ( ( Notng that X and Y are small, we have ( ( ( ( ( ( R X [ X + X ] Y [ Y + Y ]. ( (8 74 th EAGE Conference & Exhbton ncorporatng E EUROEC Copenhagen, Denmark, 4-7 June

4 In the case of VTI meda, the mpedance matrces that can be nverted are now ( ( ( ( ( ( ( ( X [ X + X ] and Y [ Y + Y ]. X and Y are the sotropc mpedance matrces snθ ( X and ( ρβ / sn θ ρβ cos ϕ Y ; ( ρ(sn ϕ ρ β sn ϕ snϕ tanθ / snϕ ( + + ϕ βρ + ρ β + ( X sn ϕ( ρ 4ρ β sn (, ζ snθ ζ tan ϕ X, C33 / ( / C33 / C33 ρ β cos ϕ ζ ρ sn ϕ ζ (ρ β cos ϕ + C3 tanϕ C55 / [ tanθ / C55 / [4sn ϕβ ( θ + ϕ β β + ( Y sn ( / C cos ( / / 55 / C55 C55 C and ζ C55 / a cos θ tanθ ζ C sn ϕ Y ζ tan θ ζ snϕ snθ tanθ / ubsttutng these equatons nto equatons (6 and (7 yelds R qq( ( θ R( ( θ + δ sn θ + ε sn θ tan θ (3 4 Rqq( ( θ R( ( θ + snθ / [ δ sn θ + ( ε δ sn θ ] + ( snθ / snϕ / ζ (4 Fgures and compare the exact - and -wave reflecton coeffcents wth the dfferent approxmatons ntroduced above. Two sngle-nterface models wth VTI ansotropy are used for the evaluaton. The frst model has moderate ansotropy (% for -wave velocty and % for -wave velocty, whle the second model has strong anstoropy (6% for -wave velocty and 48% for -wave velocty. For both cases, the generalzed approxmatons of q-q (equaton 6 agree well wth the exact numercal results, even at a large angle of ncdence. The reduced -wave approxmaton shows good accuracy when the angle s smaller than 3 o. A smlar stuaton s found n the case of R qq, except the accuracy of the generalzed approxmaton n Fgure b s relatvely lower. (a (b (c (d Fgure (a - and (b -wave reflecton coeffcents for a VTI/VTI nterface. The Thomsen parameters (ε,δ,and γ for the upper and lower layers are.,. and., and.5,. and., respectvely. (c - and (d -wave reflecton coeffcents for a strong VTI nterface. The Thomsen parameters (ε,δ,and γ for the upper and lower layers are.56, -.5 and.48, and.,. and., respectvely.the exact reflectvtes (sold black are compared wth dfferent approxmatons (equatons (5 and (6 are dotted black curves; equatons (7 and (8 are dotted green curves; equatons (4 and (5 are dotted blue curves AVO response n fractured HTI meda A medum contanng algned vertcal fractures gves rse to effectvely HTI ansotropy. The dervaton of the reflecton coeffcents for weak HTI meda s smlar to that of VTI meda. β R qq ( ( θ ( ( θ δ ε 8 γ R + + sn θ + ε sn θ tan θ (5 R qq ( ( θ R( ( θ + snθ / [ β / γ + ( δ ε sn θ (6 4 + ( β γ θ + ( ε δ θ 4 / sn sn + ( snθ / snϕ / ζ 74 th EAGE Conference & Exhbton ncorporatng E EUROEC Copenhagen, Denmark, 4-7 June

5 where the Thomas parameters are defned n dfferent way, due to the 9 o rotaton of the symmetry axs. In order to evaluate the effects of fractures on the multcomponent AVO response, we buld a model wth a overburden shale and four types of sand. The shale has weak VTI ansotropy (ε.,δ,and γ.; the gas sand s dscussed n Gassmann (95 and Castagna (993. Ths model has a hgh/low mpedance contrast, and s common for gas reservors (sold curve n Fgure 3. Gas sands wth dfferent fracture densty are smulated by Hudson s equaton (Hudson 98. The fracture-nduced ansotropy reduces the ampltude of the q-q wave, especally for wde ncdent angles. In contrast, the ampltude of the q-q wave ncreases wth ncreasng fracture densty, from o up to nearly 35 o. (a (b Fgure 3 (a - and (b -wave reflecton coeffcents for four VTI/HTI nterfaces. The overburden shale has weak VTI ansotropy (ε.,δ,and γ.. Gas sands wth dfferent fracture densty from 5% to % are smulated. Conclusons We have ntroduced a generalzed form of the q-q and the q-q wave reflecton coeffcents for TI meda n terms of mpedance matrces. The equatons are derved, based on the exact soluton for orthrhombc meda. Reduced equatons for VTI and HTI meda are derved by consderng weak ansotropy. We use three sngle-nterface models to evaluate the equatons. The generalzed equatons have suffcent accuracy even for relatvely strong ansotropy. mplfed equatons for VTI meda show good accuracy when the angle of ncdence s small. We use the reflecton coeffcents for HTI meda to evaluate the effects of fractures on gas sand AVO responses. The ampltude of the q-q wave at wde angle apertures, and that of q-q at small angle apertures, s senstve to the presence of fractures. Acknowlegement Ths study s sponsored by the research funds provded to new recrutments of Chna Unversty of etroleum-bejng (YJRC--3. References Ak, K., and Rchards,. G., 98, Quanttatve sesmology: theory and methods: W.H.Freeman. Bortfeld, R., 96, Approxmaton to the reflecton and transmsson coeffcents of plane longtudnal and transverse waves: Geophyscal rospectng, 9, Bush, I. and Crampn,., 99, ars Basn Vs: case hstory establshng combnaton of fnelayerng (and matrx ansotropy and crack ansotropy from modelng shear wavefelds near pont sngulartes: Geophyscal Journal Internatonal, 7, Hudson, J. A., 98, Wave speeds and attenuaton of elastc waves n materal contanng cracks: Geophyscal Journal of the Royal Astronomcal ocety, 64, Jlek,.,, Converted -wave reflecton coeffcents n weakly ansotropc meda: ure and Appled Geophyscs, 59, Ruger, A., 997, -wave reflecton coeffcents for transversely sotropc models wth vertcal and horzontal axs of symmetry: Geophyscs, 6, choenberg, M. and rotazo J., 99, Zoepprtz ratonalzed and generalzed to ansotropy: Journal of esmc Exploraton,, Thomsen, L.A., 986, Weak elastc ansotropy: Geophyscs, 5, th EAGE Conference & Exhbton ncorporatng E EUROEC Copenhagen, Denmark, 4-7 June