基于反演基质矿物模量和多约束条件的双相介质AVO正演方法（英文）(1).pdf

395 Manuscript received by the Editor November 18, 2013; revised manuscript received September 29, 2014. *This work was supported by the National Natural Science Foundation of China Grant Nos. 41404101, 41174114, 41274130, and 41404102. 1. Key Lab of Earth Exploration Gassmann, 1951. Two-phase theory is widely used in oil and gas exploration Sa, 2003. Oil and gas reservoirs are mainly divided into sandstone and carbonate reservoirs. The rock skeleton of sandstone reservoirs comprises quartz, feldspar, lithic fragments, cement, and fl uid-fi lled pores. Similarly, carbonate reservoirs consist of limestone, dolomite, and marl, with oil-, gas-, and water-fi lled cracks or cavities. Thus, they can be treated as two-phase media. The model for saturated porous two-phase media proposed by Gassmann 1951 and Biot 1956 laid the theoretical foundation for explaining seismic wave propagation in two-phase media. Plona 1980 observed the existence of slow P waves in experiments and confi rmed the theoretical assumptions of the two-phase medium model. Nur 1989 used the two-phase medium model in seismic exploration. AVO forward modeling in two-phase media is based on the equations derived by Wang 1990, which are similar to the Zoeppritz equations Zoeppritz, 1919. Qiao et al. 1992 discussed plane wave refl ection and transmission in porous media. 396 AVO forwarding modeling in two-phase media Based on the study of Wang 1990, Mou 1996 refi ned the AVO equations in two-phase media. The reflection and transmission equation coefficients for the top interface were derived from seismic exploration data, while experiments confirmed the presence of slow P waves in two-phase media. Based on the results of Mou 1996, Yong et al. 2006 further refi ned the refl ection and transmission equations considering 1 the interface between two-phase media, 2 the interface between an one-phase and two-phase medium at the top boundary, 3 the interface between a two-phase and one-phase medium at the bottom boundary, and 4 the interface between one-phase media in isotropic rocks. AVO forward modeling in two-phase media aims to calculate the two-phase parameters using parameters, such as the rock bulk modulus, P- and S-wave velocity, density, porosity, and fluid type. Unreasonable model values will lead to unreasonable results or errors. For example, if the coupling density is negative and the slow P-wave velocity is greater than the fast P-wave velocity, the corresponding root of the slow P-wave velocity is negative. Typically, P- and S-wave velocity, density, porosity, and fluid types are obtained from laboratory tests or petrophysical logs. Subsequently, the data undergo statistical analysis. Accurate interpretation of the lithology is particularly important as it affects the assumed matrix bulk modulus, which depends on diagenesis, post-diagenesis reconstruction, ation pressure, temperature, and so on. The estimation of the bulk modulus greatly affects the AVO forward modeling of two-phase media. Therefore, Yong 2006 proposed a self-consistent to calculate the equivalent matrix mineral bulk modulus and dry rock skeleton modulus. Unfortunately, the modulus coupling results in constant modulus and the lack of coupling precision hinders computations. We used matrix mineral bulk modulus inversion Lin et al., 2011 to obtain the inverse matrix mineral bulk modulus, and the simplified Xu–White model Keys et al., 2002 to calculate the dry rock skeleton bulk modulus for ellipsoidal pore shape by adjusting the rock equivalent porosity aspect ratio and using physically meaningful multiple constraints. Theory Matrix mineral bulk modulus inversion The Gassmann equations are based on the following assumptions 1 the rock, including the matrix and the skeleton, is macroscopically homogeneous and isotropic; 2 all pores are interconnected or communicating; 3 the pores are filled with fluids; 4 the rock–fluid system is a closed system; and 5 there is no pore fluid–rock interaction that softens or hardens the skeleton. The Gassmann equations Mavko, 1995 are , satdry KKf 1 2 0 2 00 1 , 1 dry dry fl K K f K KKK 2 , satdry 3 where K0, Ksat, Kdry, and Kfl are the bulk moduli of the matrix minerals, the saturated and dry rock skeleton, and fl uid, respectively. In the case of known water saturation Sw, Kfl is calculated by using Wood’s equation. μsat and μdry is the shear modulus of the saturated and dry rock skeleton, respectively. φ is the porosity and f is the fl uid –rock mixing term, also known as the Gassmann fluid factor. For fl uid-saturated porous media, the equations for the P-wave velocity Vp and S-wave velocity Vs are Mavko, 1998 44 33 , satdrydrydry P satsatsat KKf Sf V 4 , dry s sat V 5 where ρsat is the fluid-saturated rock density and drydry KS 3 4 represents the dry skeleton. The P-wave modulus M ρsatVp 2 Mavko, 1998 is introduced and equation 4 is transed to the Gassmann–Biot–Geertsma equation 4 . 3 drydry MKf 6 We defi ne the Biot coeffi cient β 0 1. dry K K 7 397 Lin et al. Next, parameter Y is defined using the dry rock Poisson’s ratio σdry 314 1. 13 drydry drydry Y K 8 By substituting equations 7 and 8 into equation 6, a quadratic expression of the Gassmann–Biot–Geertsma equation is obtained 20 0 0 0 11 10. fl fl KM YYY KK KM Y KK 9 For known Vp, Vs, φ, Sw, K0, and σdry, equation 9 can be used to compute β. Then, β is substituted into equation 7 to obtain Kdry and equation 2 is used to calculate the Gassmann fl uid factor f. The P-wave impedance ZP and S-wave impedance ZS Mavko, 1998 are expressed by the following equations , PsatP ZV 10a . SsatS ZV 10b By substituting equations 4 and 5 into equation 10, we obtain the new impedance expressions 2222 , PsatPsatsat sat Sf ZVSf 11 2222 . sat SsatSsatsatsat sat ZV 12 The combined P- and S-wave impedance is 22 . PSsatsat ZcZSfc 13 For the value of c that the product of c and μ is equal to the dry skeleton term S, i.e., S cμsat , the Russell fl uid factor Russell et al., 2003 is sat SP cZZ f 22 14 and 2 4 4 33 21 . 12 drydrydry satsatdry dry P Sdry dry KK c V V 15 The fluid term f is obtained using two different s. Simplifi ed Xu–White model Using the dry rock approximation, Keys and Xu 2002 improved the Xu–White model and calculated the dry rock skeleton bulk modulus using the expressions 0 1 , p KK 16a , 1 , 3 liijjl l s c pT 16b where Kφ is the bulk modulus of the dry rock skeleton, which corresponds to Kdry in the Gassmann equations, p depends on α, υc and υs is the volume fraction of clay and sand, respectively, and T is the pore aspect ratio function derived from Eshelby’s tensor Eshelby, 1957; Berryman, 1980. Two-phase medium refl ection and transmission equations The allocation of the underground interface energy is of fundamental importance in seismic wave propagation models, and provides the theoretical support for the AVO s. AVO s based on the one-phase Zoeppritz equations is a simple approximation of complex underground conditions. If the rock reservoir is saturated with fl uids, then it is a two-phase medium. In two-phase media, Biot theory Biot, 1956 predicts the existence of fast and slow P waves. Thus, the corresponding reflection and transmission mechanisms are not the same as in one-phase medium. The use of Zoeppritz equations for one-phase medium to describe the propagation of seismic waves in a two-phase medium is deemed inadequate. Therefore, four simplified reflection and transmission equations Yong, 2006 are chosen for the AVO forward modeling, in this study. In semi-infi nite two-phase media separated by a planar interface, the plane wave Pi amplitude APi, incidence angle αi hits the boundary and produces the following waves 1 refl ected fast P wave P11 with refl ection angle α11; 2 reflected slow P wave P12 with reflection angle 398 AVO forwarding modeling in two-phase media α12 ; 3 refl ected shear wave S1 with refl ection angle β1; 4 transmitted fast P wave P21 with transmission angle α21; 5 transmitted slow P wave P22 with transmission angle α12; and 6 transmitted shear wave S1 with transmission angle β2 . The corresponding refl ection and transmission equation coefficients are RP11, RP12, RPS1, RP21, RP22, RS2. Then, the AVO equation for the internal boundary of the reservoirthe surface between two two-phase media under six boundary conditions is obtained 1112121 222 1112121 222 11 1112121 222 1112121 222 2 1111111111111 coscossincos cossincos, sinsinco ssin sincossin, 2co s PPSP PSi PPSP PSi P RRRT TT RRRT TT R lANm QQm R 12 1 21 22 2 2 1211121211121 111 2 2122212122212 2 2222222222222 222 2 1111 2co s sin2 2cos 2cos sin2 2cos P S P P S ii R lANm QQm R R l N T lANm QQm R T lANm QQm R T l N l ANm Q 1112 121 222 1112 1 11111 1111112112 11121221 22222222 1 1111111212 12 1 , sin2sin2 cos2sin2 sin2cos2 sin2, 1cos1cos 1 PP SP PS ii PP S Qm R R l NR l N R l NT l N T l NT l N l N RmRm R 21 222 1112 2122 1 122121 1 22 2 12 2222222 2 22 111 1112 11111112 11 2122 22212222 22 sin1cos 1cos1sin 1cos, P PS i PP PP Tm TmT m ll RQR mRQR m ll TQR mTQR m 1111 1 , i l QR m 17 where A1, N1, Q1, R1, ρ11, ρ221, ρ121, φ1 and A2, N2, Q2, R2, ρ12, ρ222, ρ122, φ2 are two-phase parameters and the porosity of the upper and lower medium, respectively. l11, l12, l1, l21, l22, l2 are the circular wave numbers corresponding to the P11, P12, S1, P21, P22, S2 waves, respectively. m11, m12, m21, m22 are the amplitude ratios of the fl uid and solid, corresponding to P11, P12, P21, P22 waves. The top boundary of the reservoir is the interface between the one-phase and two-phase medium. For a tight cap rock, the porosity is zero. The AVO equations for the top boundary of the reservoir can be obtained by making simplifi cations. Similarly, we can obtain the AVO equations at the bottom boundary of the reservoir and the isotropic AVO equations as well. Self-consistent model Based on Eshelby’s strain energy theory, Berryman 1980 proposed a self-consistent model Berryman, 1980 for the deation of isolated inclusions. It is a more general of the self-consistent approximation for N-phase composites ** 1 0, N i iiSC i x KKp 18 where i refers to the ith material, xi and Ki are the corresponding volume fraction and bulk modulus, and K*SC is the self-consistent bulk modulus, namely, coupling modulus. We also defi ne the coupling precision K ** 1 . N i iiSC i Kx KKp 19 Clearly, as K approaches 0, the coupling precision increases. We consider a two-phase rock and per the following steps inversion of the matrix mineral bulk modulus, solving for the two-phase medium parameters using multiple constraints, and AVO forward modeling of the two-phase medium. Data from laboratory tests or petrophysical logs are used as the model parameters. Inversion of the matrix mineral bulk modulus Based on the parameters Vp, Vs, ρsat, φ and Sw, Ksat and μsat are calculated as 22 4 , 3 satps KVV 20a 399 Lin et al. 2. sats V 20b Then, the corresponding ranges of K0 and σdry are determined. According to equation 7 and β 1–[1–φ][3/1–φ] Krief, 1990, we obtain 3/1 0 1. dry KK 21 Furthermore, we use the relations of the bulk moduli drysat KKK 0 22 to obtain 3/13/1 0 11. satdrysat KKKK 23 Obviously, Eq. 23 shows the upper and lower limits of K0. σdry typically ranges between 0.0 and 0.45. For known Vp, Vs, φ, and Sw and within the range of K0 and σdry , the Gassmann and Russell fl uid factors can be calculated, and the same fl uid term f is obtained with different s. K0 and σdry are continuously adjusted. The difference between the two fl uid factors is minimal at convergence and defines the optimal inversion parameters. If K0 is within the known range of mineral moduli Mavko et al., 1998, then, the calculation results are presumably correct. Solving for two-phase parameters with multiple constraints To calculate the dry rock skeleton bulk modulus Kdry, we need to consider the rock equivalent porosity aspect ratio parameter α with values between 0 and 0.9. For each α value and combined with K0, equation 16 can be used to calculate Kdry. First constraint we assume that the saturated rock bulk modulus Ksat is greater than the dry rock skeleton bulk modulus Kdry. If the saturated rock bulk modulus Ksat is less than the dry rock skeleton bulk modulus Kdry, we readjust α within its range and continue the calculations until the constraints are satisfi ed and Kdry is obtained. To calculate the two-phase medium parameters A, N, Q, and R, we use the equations of Geertsma et al. 1961, 2 0 14 , 1 3 drydry f HK KK 24a 0 1 , 1 f K KK 24b 0 1 . 1 f L KK 24c Then, we calculate the two-phase medium parameters 2, RL 25a .QKR 25b We consider A and N with respect to the Lam coefficients λ and shear modulus μ in one-phase medium, and assume that A/N is approximately equal to λ/μ. Based on the relation among the elastic parameters, we obtain 2 2 2. p s V A NV 26a and .22ANPHQR 26b Thus, we obtain parameters A and N 2 2 , 2 . S P V NPAPN V 27 We then calculate the mass parameters ρ11, ρ22, and ρ12 of the two-phase medium and slow P-wave velocity V2. Based on practical experience, the fast P-wave velocity V1 can be used as the P-wave velocity. For known parameters H and ρsat, Biot 1956 derived the equation 28 of the roots Z1 and Z2 of the fast and slow P-wave velocity, respectively, 2 2112222111212112212 22 112212112212 2 0,ZZ 28 2 , c sat H V 29a 2 1 2 1 , c V Z V 29b 2 2 2 2 . c V Z V 29c 40