TY - JOUR
AU1 - Golalzadeh, A, R
AU2 - Javaherian,, A
AU3 - Nabi-Bidhendi,, M
AB - Abstract In this paper, we introduce a formula for extracting Lame's parameters (λ and μ) in a VTI medium. We show the application of the inversion method to this formula for extracting reflection coefficients of P- and S-wave velocities in a VTI medium. Finally, we show the application of this method to a carbonate reservoir in South West Iran. Results of this research indicate that, if anisotropy parameters are used in steps while extracting λ and μ, we can distinguish between reservoir zones with different lithology and fluid content. Lame's parameters, AVO attributes, reflection coefficients, VTI, reservoir characterization, carbonate rocks 1. Introduction Seismic attributes are powerful tools in reservoir characterization, seismic interpretation, monitoring and simulation of hydrocarbon reservoirs since they are sensitive to the desired geologic features and reservoir properties of interest and they allow us to explain the structural or depositional environment. In one of the various classifications of seismic attributes, they are divided into two general categories: prestack and poststack. Very popular prestack attributes are those of AVO, which measure the change in reflection amplitude and phase as a function of source-to-receiver offset. AVO attributes are employed to identify the boundaries of geologic features and the distribution of hydrocarbons. They are used successfully as a direct hydrocarbon indicator. Conventional AVO analysis is based on analytic expressions for P-wave reflection coefficients in isotropic media. Empirical and analytical studies show, however, that the presence of anisotropy can significantly distort conventional AVO analysis (Wright 1987, Banik 1987, Kim et al1993). New attributes that are estimated from prestack seismic data are Lame's parameters (λ and μ). AVO inversion for Lame's parameters can furnish additional insight into the geologic complexity. Conventional methods for extracting Lame's parameters consider relations between changes in seismic amplitude and offset of source and receiver. The first method for extracting λ and μ was introduced by Goodway et al (1997). They showed that these two attributes are powerful indicators for distinguishing shale zones from gas bearing sand zones. In fact, λ is dependent on properties of the fluids contained in the pores, while μ is independent of types of fluids in the pores or their properties. The theory and concept for extracting Lame's parameters are provided by Burianyk (2000), Goodway (2001), Ma (2001) and Dufour et al (2002). The methods of these workers are, however, only effective in an isotropic medium. On the other hand, the properties of anisotropic rocks are important for seismic imaging, seismic interpretation and reservoir characterization. They also affect the quality of prestack seismic analysis, amplitude analysis, velocity analysis and rock property inversion. The theory behind the reflection and transmission of plane waves at an interface between two anisotropic media has been discussed by Musgrave (1970), Henneke (1972), Keith and Crampin (1977), Daley and Hron (1977), Thomsen (1986, 1988, 1995, 2001), Banik (1987), Lynn et al (1995) and Rüger (1997). We introduce here a method for extracting λ and μ in a VTI medium. The inversion technique has been used successfully to test this method on one of the carbonate reservoirs in South West Iran. 2. Methodology The Zoeppritz (1919) equations describe the relationships between incident, reflected and transmitted P- and S-waves on both sides of an interface between isotropic media. These equations and their approximations are often used in AVO modelling and seismic data processing. They provide exact solutions but have complex forms and need simplification for use in industry. Aki and Richards (1980) developed simplified forms of the Zoeppritz equations. If the zero-offset reflectivity is very much lower than unity, the difference between the exact solution and the solution from the Aki–Richards approximation is small. A number of researchers have rearranged the Aki–Richards equations so that they may be used to describe and extract AVO attributes in an isotropic medium. Included in these are Shuey (1985) who studied zero-offset P-reflectivity and gradient, Smith and Gidlow (1987) and Fatti et al (1994) who studied P- and S-velocity reflectivities, Verm and Hilterman (1995) who studied P-reflectivity and Poisson's ratio and Goodway et al (1997), Goodway (2001) and Gray et al (1999) who determined λ and μ. All of these methods are valid, however, only in isotropic media. Rüger (1997) introduced a new formula for considering changes of seismic amplitude with offset in a VTI medium. Figure 1 illustrates the accuracy of Rüger's equation by comparing the approximate reflection coefficient with the exact VTI solution and the corresponding exact isotropic reflection coefficient. This figure shows that there are significant differences between the isotropic curve and the exact reflection coefficient curve for incident angles greater than 5°. On the other hand, Rüger's equation is a very good approximation of the exact reflection coefficient for incident angles up to 40°. The presence of anisotropy can be seen therefore to distort severely or even reverse the variation of the reflection coefficient with incident angle. Rüger's equation is as follows: 1 where Rp denotes the P-wave reflection coefficient, θ is the incident phase angle (average of θ1 and θ2), V is the vertical P-wave velocity (average of V1 and V2), W is the vertical S-wave velocity (average of W1 and W2), I = ρV is the vertical P-wave seismic impedance (average of I1 and I2), ρ is the density (average of ρ1 and ρ2) and μ = ρW2 is the vertical shear modulus (average of μ1 and μ2). In this equation subscript 1 indicates the upper layer and subscript 2 the lower layer. The differences in anisotropy across the boundary are written as Δδ = (δ2 - δ1) and Δε = (ε2 - ε1), where δ and ε are the anisotropy parameters defined by Thomsen (1986). For extracting Lame's parameters, provided that changes in the elastic properties are small, Rüger's equation is rearranged as follows: 2 3 4 5 6 7 8 9 10 Figure 1 Open in new tabDownload slide Reflection coefficient for a VTI medium (Rüger 1997). The solid line indicates the exact VTI solution. The upper broken line indicates the approximation based on Rüger's equation, and the lower broken line indicates the solution corresponding to the isotropic medium (δ2 = 0; ε2 = 0). Model parameters are Vp01 = 2.9, Vs01 = 1.8, ρ1 = 2.18, ε1 = 0, δ1 = 0,Vp02 = 3.1, Vs02 = 1.85, ρ2 = 2.2, ε2 = 0.1 and δ2 = 0.2. Figure 1 Open in new tabDownload slide Reflection coefficient for a VTI medium (Rüger 1997). The solid line indicates the exact VTI solution. The upper broken line indicates the approximation based on Rüger's equation, and the lower broken line indicates the solution corresponding to the isotropic medium (δ2 = 0; ε2 = 0). Model parameters are Vp01 = 2.9, Vs01 = 1.8, ρ1 = 2.18, ε1 = 0, δ1 = 0,Vp02 = 3.1, Vs02 = 1.85, ρ2 = 2.2, ε2 = 0.1 and δ2 = 0.2. In the above equations, 1/2ΔI/I is the zero-offset P-wave reflectivity and 1/2ΔJ/J is the zero-offset S-wave reflectivity. Spratt et al (1997) introduced equation (11) for calculating θ: 11 where θ denotes the incident angle, Vint is a smoothed version of the interval velocity, Vrms is the root-mean-square velocity,X is the source-to-receiver offset, and T0 is the zero-offset two-way travel time. We have employed this equation for our research. After calculating θ we have five unknowns (1/2 ΔI/I, 1/2 ΔJ/J, Δρ/ρ, Δδ and Δε). To extract reflection coefficients for P- and S-waves and other unknowns, we have applied the inversion method to fit equation (10) for the P-wave reflection amplitudes from CMP gathers of real data. The outputs of the first inversion program are for the zero-offset reflection coefficients of P- and S-waves. For extracting seismic impedances of P- and S-waves we have applied the seismic inversion method. We have then calculated λ and μ for a VTI medium from the seismic impedances of P- and S-waves by Goodway's equations (λρ = I2 - 2J2 and μρ = J2). 3. Application of Lame's parameters in reservoir characterization The reflections recorded in seismic exploration are closely related to subsurface rock properties. The strongest amplitude variation with offsets in the seismic data is often caused by hydrocarbon saturation in the rocks. The essence of the matter for AVO is in the fact that the shear modulus of a rock does not change when the fluid saturation is changed. On the other hand, the bulk modulus changes significantly when the fluid saturation is changed (Gassmann 1951). Since μ has no mechanical dependence on the fluid saturation, it is clear that all the fluid dependence of the bulk modulus resides within λ. Thus, λ is elastically dependent on fluid properties, while μ is not. At low liquid saturations, the bulk modulus of the fluid mixture is dominated by the gas. The effect of the liquid on the bulk modulus is therefore negligible until the porous medium approaches full saturation with either water or oil. As the medium approaches full saturation, the shear modulus μ remains constant, while the bulk modulus (over a very narrow range of saturation values) increases to its full saturation value. Lame's parameter λ of a brine-saturated rock is greater than that of gas-oil-saturated rock (Berryman et al2000, Assefa et al2003). Lithologic changes may also cause amplitude variations versus offset. The lithology differences are closely linked to the rock property differences. Carbonate rocks are important hydrocarbon reservoir rocks with complex textures and petrophysical properties, mainly resulting from various diagenetic processes (compaction, dissolution, precipitation, cementation, etc). The porosity in carbonate rocks can vary in size from inter-crystalline micro-porosity to vuggy and moldic macro-porosity. These complexities cause the relationships between the petrophysical properties and the seismic properties of limestone to become complicated. Assefa et al (2003) indicated that the bulk and shear moduli of water-saturated limestones decrease as the porosity increases, and the shear modulus of water-saturated rocks is lower, by as much as 2 GPa, than that of dry rocks. They attributed this anomalous behaviour in limestones to the water/matrix interaction at grain contacts. 4. Field example The seismic data presented here are from a carbonate reservoir 3D survey in SW Iran. Figure 2 shows the migrated two-way time map of horizon H2. This horizon indicates the top of the Sarvak formation. The area studied is surrounded by a blue rectangle. True amplitude processing including prestack time migration was performed. We used sonic and density logs from two wells in this field. Only one of the wells (well 1) is located in the AVO analysis area. S-wave logs were derived from the Vs and Vp relationship. These logs were used along with seismic horizons to construct macro-velocity, density and low frequency impedance models. Figure 2 Open in new tabDownload slide The migrated two-way time map of horizon H2. This horizon is shown in figure 3 and indicates the top of the Sarvak formation. The studied area is surrounded by a blue rectangle. Figure 2 Open in new tabDownload slide The migrated two-way time map of horizon H2. This horizon is shown in figure 3 and indicates the top of the Sarvak formation. The studied area is surrounded by a blue rectangle. A package in Matlab software was written for this research. The CMP gather, Vp/Vs ratios and Vrms and Vint from check shots were input into the AVO inversion program. The outputs of this program were P- and S-wave reflection coefficients in a VTI medium. Figure 3 shows both the zero-offset P-wave and zero-offset S-wave reflection coefficients derived from prestack CMP gathers through the AVO inversion method. We focused on two formations with different hydrocarbon potentials. These two formations are located between horizons H1 and H3 (figure 3). Horizon H1 refers to the top of the Ilam formation which is above the Sarvak formation, and horizon H3 is the top of the Kazhdomi formation which is below the Sarvak formation. There are two good porous zones in the carbonate lithology Ilam and Sarvak formations, and a shaly lithology between them. The first porous zone is water saturated. The top of the second good porous zone is hydrocarbon saturated. According to the theory and experimental studies, we expect water-saturated zones to have higher values of λ, and the same or lower values of μ, in comparison with hydrocarbon-saturated zones. Figure 4 shows different logs in well 1 with the locations of hydrocarbon- and water-saturated zones. There are two main formations shown in this figure, namely the Ilam and Sarvak, abbreviated IL and SV, respectively. Figure 3 Open in new tabDownload slide Zero-offset reflection coefficients derived from prestack CMP gathers via the AVO inversion method: (a) for P-waves and (b) for S-waves. Figure 3 Open in new tabDownload slide Zero-offset reflection coefficients derived from prestack CMP gathers via the AVO inversion method: (a) for P-waves and (b) for S-waves. Figure 4 Open in new tabDownload slide Different logs in well 1 with locations of hydrocarbon- and water-saturated zones. There are two main formations shown in this figure: namely, the Ilam and Sarvak abbreviated as IL and SV, respectively. Figure 4 Open in new tabDownload slide Different logs in well 1 with locations of hydrocarbon- and water-saturated zones. There are two main formations shown in this figure: namely, the Ilam and Sarvak abbreviated as IL and SV, respectively. We used the Vanguard seismic inversion industry program for extracting the impedances of both P- and S-waves. Figure 5 shows the zero-offset seismic impedances derived from prestack CMP gathers by the AVO inversion method. Figure 5(a) is for P-waves estimated from figure 3(a), and figure 5(b) is for S-waves estimated from figure 3(b). We used the Goodway et al (1997) equations (λρ = I2 - 2J2 and μρ = J2) and macro-density model from the density log for extracting Lame's parameters. Figure 6 shows the results of this operation. Figure 6(a) shows the λ parameter and figure 6(b) shows the μ parameter estimated from P- and S- impedances, respectively. As indicated in figure 4, we have a water-saturated zone below horizon H1. We see, therefore, high values of λ with low ones of μ. Below horizon H2, on the other hand, we have a hydrocarbon-saturated zone. Here we see, therefore, low values of λ with constant μ. For a detailed study of the water-saturated and hydrocarbon-saturated zones, we determined the λ/μ attribute. Figure 7 shows the results of this study. Below horizon H1 we see a zone of high λ/μ values, and below horizon H2 we see a zone of low λ/μ values. In this figure, the blue ellipse shows the location of the water-saturated zone, and the green ellipse shows the location of the hydrocarbon zone. Figure 5 Open in new tabDownload slide Estimated seismic impedances: (a) for P-waves estimated from figure 3(a) and (b) for S-waves estimated from figure 3(b). Figure 5 Open in new tabDownload slide Estimated seismic impedances: (a) for P-waves estimated from figure 3(a) and (b) for S-waves estimated from figure 3(b). Figure 6 Open in new tabDownload slide Estimated values for (a) λ and (b) μ from seismic impedances for P- and S-waves, respectively. Figure 6 Open in new tabDownload slide Estimated values for (a) λ and (b) μ from seismic impedances for P- and S-waves, respectively. Figure 7 Open in new tabDownload slide Results for the λ/μ attribute. Below horizon H1 we see a zone of high λ/μ values and below horizon H2 we see a zone of low λ/μ values. The blue ellipse shows the location of the water-saturated zone and the green ellipse shows the location of the hydrocarbon zone. Figure 7 Open in new tabDownload slide Results for the λ/μ attribute. Below horizon H1 we see a zone of high λ/μ values and below horizon H2 we see a zone of low λ/μ values. The blue ellipse shows the location of the water-saturated zone and the green ellipse shows the location of the hydrocarbon zone. For considering lateral changes in lithofacies and fluid content we have applied the integrated amplitude attribute of Lame's parameters on different reservoir zones. The locations of various zones are shown in figure 4. At first we applied this attribute to the water-saturated zone. Figure 8(a) shows the results of this attribute for the λ parameter in the water-saturated zone below horizon H1, as indicated in figure 4. Results of this attribute for the μ parameter in the water-saturated zone below horizon H1 are shown in figure 8(b). We expect water-saturated zones to have high values of λ and constant, low values of μ. The slight decrease in λ to the left is due to the hydrocarbon in the adjacent field. Results successfully show high values of λ and low ones of μ in this water-saturated zone around well 1. We have then applied this attribute to the hydrocarbon-saturated zone. We expect hydrocarbon-saturated zones to have low values of λ and high or constant values of μ. Results successfully show low values of λ and slightly higher values of μ in the hydrocarbon-saturated zone around well 1. Figure 8(c) shows the results of this attribute for the λ parameter below H2, as indicated in figure 4. Results of this attribute for the μ parameter in the hydrocarbon-saturated zone below horizon H2 are shown in figure 8(d). Results successfully show low values of λ and slightly higher values of μ in the hydrocarbon-saturated zone around well 1. There is a water-saturated zone towards the bottom part of the hydrocarbon zone below horizon H2. We have applied the integrated amplitude attribute in this zone. Figure 8(e) shows the results for the λ parameter in this water-saturated zone. Figure 8(f) shows the results of this attribute for the μ parameter in the same water-saturated zone. Results successfully show high values of λ and the same or slightly higher values of μ in this water saturated zone around well 1. Figure 8 Open in new tabDownload slide Results of applying the integrated amplitude attribute: (a) and (b) the λ and μ parameters for the water-saturated zone located below horizon H1, (c) and (d) the λ and μ parameters for the hydrocarbon-saturated zone located below horizon H2, (e) and (f) the λ and μ parameters for the water-saturated zone located towards the bottom part of the hydrocarbon zone below horizon H2. H1 and H2 represent the top of the Ilam and Sarvak formations, respectively. Figure 8 Open in new tabDownload slide Results of applying the integrated amplitude attribute: (a) and (b) the λ and μ parameters for the water-saturated zone located below horizon H1, (c) and (d) the λ and μ parameters for the hydrocarbon-saturated zone located below horizon H2, (e) and (f) the λ and μ parameters for the water-saturated zone located towards the bottom part of the hydrocarbon zone below horizon H2. H1 and H2 represent the top of the Ilam and Sarvak formations, respectively. 5. Conclusions Anisotropy properties of rocks are important in the quality of the amplitude analysis and rock property inversion. We have introduced a formula for extracting Lame's parameters (λ and μ) in a VTI medium. Also we demonstrated application of the inversion method on this formula for extracting reflection coefficients of compressional and shear waves. This research showed that if anisotropy parameters are used in steps for extracting of λ and μ we can separate different reservoir zones with different lithology and fluid content. Acknowledgments The authors wish to thank the Institute of Geophysics, University of Tehran, for its support. The authors are also grateful to the anonymous reviewers of this paper for their valuable suggestions. References Aki K , Richards P G . , 1980 , Quantitative Seismology: Theory and Methods , vol. vol 1 San Francisco Freeman Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Assefa S , McCann C , Sothcott J . , 2003 Velocity of compressional and shear waves in limestones , Geophys. 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TI - Estimation of Lame's parameters from P-waves in a VTI medium
JF - Journal of Geophysics and Engineering
DO - 10.1088/1742-2132/5/1/004
DA - 2008-03-06
UR - https://www.deepdyve.com/lp/oxford-university-press/estimation-of-lame-s-parameters-from-p-waves-in-a-vti-medium-wieouHVDw3
SP - 37
VL - 5
IS - 1
DP - DeepDyve
ER -