Exact elastic impedance in orthorhombic media

Transcription

1 Exact elastic impedance in orthorhombic media F. Zhang (hina University of Petroleum), X.Y. Li (hina University of Petroleum, British Geological Survey) SUMMARY onventional elastic/ray impedance approximations are derived as a scalar to heuristically present seismic reflectivity based on the assumptions of weak impedance, isotropic media, or weak anisotropic media. In this paper, we derive the exact elastic impedance tensors of qp- and qs-waves for orthorhombic media based on the stress-velocity law. They represent the unique mechanical properties of the medium, thus are called elastic mechanical impedance (EMI). The impedance tensor can be reduced for an isotropic medium with a vertical symmetry axis (VTI) or a horizontal axis (HTI). Because no assumptions are made during the derivation, the new impedance shows good accuracy at large angle and can be used to characterize unconventional reservoirs with strong anisotropy, such as shale gas and coal-bed methane reservoirs. An approximation of P-wave EMI is also discussed for a TI medium. It is expressed by vertical velocities and thus is applicable to seismic inversion and interpretation. Application tests on real log data and seismic data show the robust interpretation capability of EMI in lithology characterization compared with conventional impedances.

2 σ σ zz xz Introduction Impedance is an intrinsic physical property of a subsurface medium. Seismic elastic impedance is widely used to characterize the reflection and transmission of seismic waves. onventional elastic/ray impedance approximations are derived as a scalar to heuristically present the seismic reflectivity based on varied assumptions, for example weak impedance contrast, isotropic media, or weakly anisotropic media (onnolly, 1999; Wang, 003; Santos and Tygel, 004; VerWest, 004; Rowbotham et al., 003; Martins, 006). In fact, seismic elastic impedance represents a measure of the amount of resistance of the subsurface medium to particle motion. Therefore, in this sense, we derive the exact elastic impedance tensors of qp- and qs-wavesfor orthorhombic media based on the stress-velocity law. They represent the unique mechanical properties of medium, thus are called elastic mechanical impedance (EMI) in this paper. The impedance tensor can be reduced for an isotropic medium with a vertical symmetry axis (VTI) or a horizontal axis (HTI). Elastic mechanical impedance in orthorhombic media Orthorhombic anisotropy resulting from a combination of thin layers and aligned fractures is one of the most common forms of anisotropy in sedimentary basins (Bush and rampin, 1991). onsidering wave propagation in the vertical plane through a subsurface medium with orthorhombic planes of symmetry, the velocity field of qp- (quasi-p) and qs- (quasi-sv) plane waves can be written v z cosθ cosθ = -iω i p rp is rs (1) v x where vz and v x refers to the velocity in the direction of z and x; i p, r p, is and rs refers to the displacement of incident P-wave, reflected P-wave, incident S-wave and reflected S-wave, respectively; the polarization directions are denoted as θ (qp-wave) and φ (qs-wave). Based on Hooke s law, the traction components are expressed in displacement by p = iω where q cosθ i + p ( p cosθ + q ) ( p cosθ + q ) p q cosθ r p 33qβ p + + p 33qβ i s r s qβ p qβ p ( ) ( ) () p = sin θ = β is the horizontal slowness, q = cosθ and q β = cosφ β is respective vertical slowness of qp- and qs-wave; and β is the phase velocity of the qp- and qs-wave, respectively; θ and φ refers to the propagation directions, with respective deviations ζ P and ζ S from the polarization directions sin θ = ( 1+ ζ P ) and sin ϕ = ( 1 + ζ S ) i. (3) Because the impedance that a given medium presents to a given motion is a measure of the amount of resistance to particle motion, impedance in elasticity is a ratio of stress to particle velocity (Aki and Richards, 1980). Therefore the elastic impedance in the incident medium with orthorhombic symmetry is written: I zz-z I xz-z σ = zz vz σ xz vz I = I zz-x I xz-x σzz v x σ xz v x (4) The impedance of reflected waves has the same scale as incident waves but negative sign, which refers to an opposite direction of the medium resistance. In this paper, we discuss the scalar ratio of the medium resistance, so the impedanceis definedusingthe absolute value of each component.the components of the above equation cover all cases of stress-velocity relationship, therefore, the impedance is called elastic mechnical impedance (EMI) in this paper. The impedance propertiesof qpand qs-waves are different. The impedance tensor in terms of qp-wave and qs-wave displacement is respectively

3 p tan θ EMI = 33q p + tan θ 33 q ( p + q tan θ ) 13 p q β 33q β ( + p) SEMI = tanϕ tanϕ (5) ( p tanθ + q ) p 33q β tanϕ ( q β + p tanϕ ) No assumptions are made in the derivation of equation (5), therefore they have high accuracy in the cases of large angle and strong anisotropy. Besides, becausethe plane stress and velocity is discussed in this paper, the EMIs depend on only four of nine elastic moduli: 11, 33,, and 13.In order to discusseach component in equation (5) more easily, we derive the isotropic impedance matrix as follows: ρ0γ cosθ ρβ p 0 EMI = ρβ0 q ρβ0γ SEM I = (6) ρ0γ ρβ0 q ρβ0 p ρβ0γ cosφ where Γ = [ 1 ( β0 0 )] sin θ, 0 and β0 is the phase velocity of the qp- and qs-wave, respectively. In equations (6), we can see that the impedances EMI zz-z and EMI zz-x are functions of acoustic impedance, while EMI xz-z and EMI xz-x are functions of shear impedance. For a given angle, the components of SEMI (equation 6) depend only on shear impedance. The SEMI s component SEMI zz-z and SEMI zz-x corresponds to EMI xz-x and EMI xz-z, respectively. EMI zz-z agrees with the P-wave alone impedance in Morozov (010), it shows similar form as the approximations of ray impedance, while the converted S-wave is included in the derivation of ray impedance. Equation (5) is generalized EMI and SEMI in the orthorhombic medium, therefore they share similar properties of the corresponding isotropic elastic impedances. Table 1 Elastic parameters of the single-interface model. In order to evaluate the interpretation ability of the components of EMIs, we construct a single interface model using the log data of a tight-sand gas reservoir (Table 1). The overburden shale is transversely isotropic and has vertical symmetry axis, and the lower tight sand is isotropic. The velocity anisotropy of the shale layer is strong (Thomsen s parameters ε, δ>0.). Figure 1(a) shows the curves of four EMI components of qp-wave within the phase angle range from 0 o to 90 o. The corresponding isotropic impedances are compared with each other in Figure 1(b). In the isotropic case, the impedance component EMI zz-z and EMI zz-x has infinite value at 90 o and 0 o, respectively; while EMI xz-z and EMI xz-x show obvious symmetric variation within the angle range of 0 o -90 o. However, the exact anisotropic impedances, especially EMI zz-z and EMI xz-z, show considerablyy different performances. The difference between the EMI curves increases with phase angle, thus EMI zz-z and EMI xz-z provide generally good discrimination of sand from shale at large angle. (a) (b) Figure 1(a) Anisotropic EMI components of qp-wave. (b )Isotropic EMI components of P-wave. The overburden shale layer (black) has strong anisotropy, and the lower sand layer is isotropic. Figure EMI zz-z of P-wave and its approximation.

4 Elastic mechanical impedance in TI media Transverse isotropyis the most commonly consideration of anisotropy in seismic exploration. A sequence of thin layers results a medium that has a vertical symmetry axis (VTI), while a medium containing aligned vertical fractures gives rise to a horizontal symmetry axis (HTI) (Bush and rampin, 1991).The EMI and SEMI in equation (5) can be expressed explicitly by means of the phase angle and five elastic moduli. In order to make them applicable to seismic inversion and quantitative interpretation, the approximations of impedances can be derived by using Thomsen s parameters (Thomsen, 1996).Taking the VTI medium for example, EMI s component of σ zz / vz is written: 33 A1 A ρ0 EMI zz-z { 1 [ ( 0 / β0 ) δ ] sin θ} { 1 [ ( 0 / β0 ) δ ] sin θ} (7) acosθ cosθ 4 where A1 1( 1 + δ sin θ cos θ + ε sin θ), A 1/( 1 ζ P tan θ ), δ and ε are anisotropy parameters, ζ P ( 1 ε )( β0 0 )[ δ cos θ ( 1 sin θ ) + ε sin θ cos θ ]. Equation (7) is expressed by means of vertical velocities and anisotropy parameters, therefore is applicable to seismic inversion and interpretation. Though the anisotropy is strong, equation (7) has high accuracy at large angle (Figure ). Applications to real log data and seismic data We first use a real log data for the tight-sand gas reservoir to evaluate the interpretation capability of EMI zz-z. Then the anisotropic EMI zz-z and isotropic EMI zz-z of the real seismic data are estimated based on the inverted AI section. The models of anisotropy parameters are estimated by means of the calibration of a rock physical model (Guo and Li, 01) and logs. ross-plots of conventional elastic impedance (EI), ray impedance (RI), isotropic EMI zz-z and anisotropic EMI zz-z at 50 o versus acoustic impedance (AI) are plotted for comparison of lithology discrimination. The gamma ray log is used for lithology identification.the quadratic discrimination analysis is applied to quantify the performance of the cross-plots using different elastic impedances. ross-plots using EI, RI and isotropic EMI zz-z have relatively high misclassification error rates. RI and isotropic EMI zz-z give a similar characterization of lithology. A clear view of sand/shale can also be found in the cross-plot of anisotropic EMI zz-z (Figure 3d)), where the misclassification error rate is only.94%. (a) (b) (c) (d) Figure 3(a)-(d) ross-plots of isotropic EI, RI, isotropic and anisotropic EMI zz-z with AI. The misclassification errors of (a)-(d) are 19.54%, 18.69%, 18.7% and.94%, respectively. AI is inverted using the seismic data from the tight sand reservoir, and isotropic EMI zz-z and anisotropic EMI zz-z sections. orresponding well logs and a related horizon are plotted on the inverted sections in Figure 4.Thestudy area is in Western Sichuan basin, southwest hina, with the tight-sand gas deposits buried at around 5000m in depth. ompared with inverted AI, gas-bearing sand layers in the isotropic EMI zz-z section show slightly clearer contrast from surrounding rocks (higher impedance regions at well location), and therefore can be more easily identified. The contrast between sand to shale is enhanced when we estimate the corresponding anisotropic EMI zz-z using equation (7). Regions with higher impedance values are much more distinct in Figure 4d.

5 (a) (b) (c) Figure 4(a) Inversion result of acoustic impedance, (b) Isotropic EMI zz-z, (c) Anisotropic EMI zz-z. onclusions In this paper, we discuss the exact elastic mechanical impedance tensors of qp- and qs-wave for orthorhombic media based on the stress-velocity law. Because no assumptions are made in the derivation of impedance tensors, they can be used to characterize a medium s property in the cases of very large angle and strong anisotropy.theoretical analysis shows that components of qp-wave impedance, especially EMI zz-z and EMI xz-z, have different variations from those of isotropic P-wave impedance. EMI zz-z and EMI xz-z provide generally good discrimination of sand from shale at large angle. Anapproximation of anisotropic EMI zz-z is then derived by means of vertical velocities and anisotropy parameters, therefore is applicable to seismic inversion and interpretation. It has high accuracy at large angle when the anisotropy is strong. Real log data test of tight-sand gas reservoir shows the robust interpretation capability in lithology characterization of anisotropic EMI zz-z. In the seismic data application, anisotropic EMI zz-z and isotropic EMI zz-z are estimated from the inverted acoustic impedance using anisotropy parameters models estimated by means of the calibration of rock physical model and logs. The conventional AI inversion results could be improved straightforwardly by means of the estimation of anisotropic EMI zz-z. The gas-bearing sand can be more easily identified in the anisotropic EMI zz-z section, than in AI and isotropic EMI zz-z section. Acknowledgements This study is sponsored by National Natural Science Fund Projects (No and No.U1608), Research Funds Provided to New Recruitments of hina University of Petroleum-Beijing (YJR ), and Science Foundation of hina University of Petroleum-Beijing (YJR ). References Aki, K., and Richards, P. G., 1980, Quantitative seismology: theory and methods: W.H.Freeman. Bush, I. and rampin, S., 1991, Paris Basin VSPs: case history establishing combination of finelayering (and matrix) anisotropy and crack anisotropy from modelling shear wavefields near point singularities: Geophysical Journal International, 107, onnolly, P., 1999, Elastic impedance: The Leading Edge, 18, Guo, Z.Q. and Li, X.Y., 01, Rock Physics Templates for Analysis of Brittleness Index, Mineralogy, and Porosity - A Barnett Shale ase Study: 74th EAGE onference & Exhibition. Martins, J. L., 006, Elastic impedance in weakly anisotropic media: Geophysics, 71, D73 D83. Morozov, I. B., 010, Exact elastic P/SV impedance: Geophysics, 75, 7-. Rowbotham, P., Marion, D., Eden, R., Williamson, P., Lamy, P., and Swaby, P., 003, The implications of anisotropy for seismic impedance inversion: First Break, 1, Santos, L. T., and Tygel, M., 004, Impedance-type approximations of the P-P elastic reflection coefficients: Modeling and AVO inversion: Geophysics, 69, Thomsen, L.A., 1986, Weak elastic anisotropy: Geophysics,51, VerWest, B., 004, Elastic impedance revisited: 66 th onference and Exhibition, EAGE, Extended Abstracts, P34. Wang, Y., 003, Seismic Amplitude Inversion in Reflection Tomography: Elsevier, Amsterdam.