An empirical method for estimation of anisotropic parameters in clastic rocks YONGYI LI, Paradigm Geophysical, Calgary, Alberta, Canada Clastic sediments, particularly shale, exhibit transverse isotropic properties with the symmetric axis perpendicular to bedding (VTI). These anisotropic rock properties are important in seismic imaging, prestack seismic analysis, and reservoir characterization. Laboratory tests are the main techniques used in measuring rock anisotropic properties. Vertical seismic profiling (VSP) and seismic refraction are the in-situ techniques used to determine anisotropic parameters. All of these methods provide significant insight to the anisotropic properties of the subsurface. However, limitations exist in the applications of these methods because of cost issues and restrictions on the number of measurements that can be conducted. Therefore, new techniques that are cost effective, easy to use, and able to measure continuous anisotropic profiles are desirable. This paper describes a methodology which involves the use of conventional rock properties to derive anisotropic rock properties. The commonly accepted cause of anisotropy in clastic rocks is the alignment of minerals. Hornby et al. (1994) confirmed this theory with evidence of the alignment of platy clay minerals using the SEM (scanning electron microscope) image technique. Johnston and Christensen (1995) performed a quantitative investigation of the alignment of clay minerals using the orientation index determined by X-ray diffraction patterns. They found a linear relationship between the orientation index (a measure of the degree of alignment of platy clay minerals in a given direction) and clay content. They also found that variation in seismic velocity due to anisotropy is proportional to the orientation index or clay content. These observations and findings provide the motivation to explore the physical linkage and quantitative relationship between conventional and anisotropic rock properties (Li, 2002). Because conventional rock properties are determined directly or indirectly from well logging, lab measurements, and insitu seismic measurements, anisotropic rock properties can be estimated after a quantitative relationship between conventional and anisotropic rock properties has been established. In this paper an empirical method is devised to establish this relationship and thus estimate anisotropy. The method is validated by comparing the estimated anisotropy values with those derived from VSP and seismic refraction data. In addition, anisotropy values inverted from prestack depth migration (PSDM) are compared. The potential for application of this method to anisotropic synthetic seismograms, and anisotropic parameter inversion from prestack seismic data is also explored. Physical background. Clay and quartz are two principal mineral constituents of clastic rocks. In general, two basic factors the volume fraction of clay and the degree of compaction determine the anisotropic properties of rocks. The first factor constitutes the physical basis for anisotropy, i.e., the platy clay minerals. The second factor forces clay minerals to align in a preferred direction while increasing seismic velocity due to a reduction in porosity. For example, the unconsolidated deposits at an ocean bottom and the lowvelocity weathering layer on land may display little or no anisotropy even though they can both have high clay content. During the process of consolidation, clay minerals tend to align in the direction perpendicular to overburden pres- Figure 1. The consolidation process results in the alignment of clay minerals. Black lines represent platy clay minerals, and red diamonds represent quartz sand grains. The degree of alignment is affected by the degree of compaction and the quantity of quartz grains in rocks. Figure 2. (a) Thin section of unconsolidated clean sand; (b) thin section of unconsolidated shaly sand; and (c) SEM image of a consolidated shale with clay minerals aligning in the direction perpendicular to overburden pressure. The arrow and dashed line indicate the direction of overburden pressure and bedding, respectively. 706 THE LEADING EDGE JUNE 2006
sure. Consequently, seismic velocity is higher in the direction of bedding forms (Figure 1). Support for this fact is shown in Figure 2. Figures 2a and 2b are examples of unconsolidated clean sand and shaly sand. These types of rocks have little anisotropy because there is no apparent mineral alignment. The SEM image in Figure 2c shows apparent alignment of platy clay minerals in a consolidated shale in the direction perpendicular to overburden pressure and indicates strong anisotropy. In order to quantitatively evaluate the anisotropic properties of rocks, the laboratory measurements published by Thomsen (1986), Vernik and Nur (1992), Johnston and Christensen (1995), and Vernik and Liu (1997) were examined. Figure 3 shows a crossplot of anisotropic parameters ε against γ, δ, and η (η=ε δ). It was observed that: (1) ε and γ are approximately equivalent. This is consistent with the relation γ = 0.956ε - 0.01049 with R 2 = 0.7463 obtained by Wang (2001), where R is the correlation coefficient; (2) best fitting to δ yields a relation of δ=0.32; and (3) in general, anisotropic parameters δ and η increase with ε but have multiple values with respect to a given ε. In Figure 3, it is evident that the values of ε and γ range from 0.0 to approximately 0.7. This indicates that both weak and strong anisotropy exist in the subsurface. Anisotropic parameter estimation. The facts described in the previous section indicate a direction for the development of an empirical method to estimate anisotropic parameters. Figure 4 shows a crossplot of V P (0) and V S (0) against anisotropic parameters ε and γ using the same data as in Figure 3, where V P (0) and V S (0) are the velocities measured in the direction of the symmetric axis of VTI media or in the direction perpendicular to bedding. There are three key points in Figure 4a or Figure 4b. They are called critical porosity sand point, zero porosity sand point (or quartz point), and zero porosity clay point (or clay mineral point). The critical porosity sand point has zero effective shear modulus and zero shear velocity which indicates that the rock is in the suspension domain. The effective compressional velocity of clastic rocks at this point may be approximated by the velocity of brine. The properties of quartz are used as an approximation of sand with zero porosity. Notice that both the critical porosity sand point and zero porosity sand point are associated with zero anisotropy. Finally, the clay mineral point is determined using the mean of the data points with the largest anisotropic values. The lines linking the key points have the following physical meaning. First, the base lines represent isotropic clean sands. Second, the line linking the critical porosity sand point and the clay mineral point represents a consolidation process of the rocks with 100% clay. Along this line anisotropic parameters have the largest increase. Third, the line linking the quartz point and the clay mineral point represents the process of clay content increasing in tight rocks. Along this line anisotropic parameters experiences a transition from isotropic rocks to the rocks with maximum anisotropy. One of the relationships in Figure 4 is that for a given clay volume the anisotropy parameters increase linearly with V P (0) or V S (0), and the greater the clay volume, the greater the increase. Using Figure 4, one can graphically calculate anisotropic parameters when conventional rock properties, clay volume, and velocity, are known. In Figure 4, the clay mineral point has values of 3.4 km/s and 1.8 km/s values for P- and S-wave velocities, respectively. These values are consistent with the clay velocities measured by Tosaya and Nur (1982), Castagna et al. (1985), and Han et al. (1986) (Table 1). The anisotropic parameters of ε = Figure 3. Crossplot of anisotropic parameter ε against anisotropic parameters γ, δ, and η. Figure 4. The relationships between anisotropic parameters ε and γ, the P- and S-wave velocities perpendicular to bedding (V P (0) or V S (0)), and clay volume V clay. JUNE 2006 THE LEADING EDGE 707
Figure 5. Conventional well logs, estimated anisotropic well logs, and VSP measurements in the North Sea. The triangles are receivers in the wellbore. The thick solid blue line represents the VSP measurement. 0.6 and γ = 0.67 derived are especially useful because they are rarely determined in laboratory or in in-situ conditions. The following steps were used to derive the equation for calculating ε. First, the slope was determined as m = (y 1 y o )/(x 1 x o ) for an arbitrary line as indicated in Figure 4a. It can be seen that y 1 y o = 0.6V clay and x 1 x o = V Pquartz V Pwater (V Pquartz V Pclay )V clay, where 0.6 is the P-wave anisotropic parameter for clay. ε is then calculated using ε = m(v P V Pwater ) that is Using the same approach the equation for calculating γ was derived: (1) where V clay = volume of clay, V P (or V P (0)) = P-wave velocity perpendicular to bedding, V Pwater = an approximation of P-wave velocity at critical porosity, V Pquartz = P-wave velocity of quartz, V S (or V S (0)) = S-wave velocity perpendicular to bedding, and V Squartz = S-wave velocity of quartz. The constants used in equations 1 and 2 are: V Pwater = 1.5 km/s, V Pquartz = 6.05 km/s, and V Squartz = 4.09 km/s. When velocity and clay volume logs are used, the estimated anisotropic parameters derived by using equations 1 and 2 are in a form of well logs. They are called anisotropic well logs. Attempts to derive an equation using a similar approach for anisotropic parameter δ were unsuccessful. Thus, the following equation is recommended: This was derived through best fitting of δ against ε in Figure (2) (3) 3. Validation. VSP measurement. To validate the method developed in this study, a VSP measurement by Armstrong et al. (1995) in the North Sea was used. The measurement targeted a shale layer overlying a reservoir sand layer. The study area has a regional dip of less than 5 so it was safe to assume a VTI medium. An available full suite of well logs including gamma ray and dipole sonics enabled the estimation of the anisotropic well logs (Figure 5). In the shale interval, the estimated average value of ε was 0.135 which is close to 0.14 measured by the VSP. The estimated γ and δ were 0.153 and 0.0432, respectively. Prestack depth migration. Anisotropic prestack depth migration (PSDM) is becoming a common practice in seismic imaging. In comparison with isotropic PSDM, it is more accurate in correcting lateral image displacements, positioning events in depth, and focusing diffracted energy. In 708 THE LEADING EDGE JUNE 2006
Figure 6. Conventional well logs and estimated anisotropic well log at the well near Jumpingpound Creek, the foothills area of Alberta, Canada. try or scanning technique. This technique results in a single set of anisotropic parameters (ε and δ) when best results for imaging and depth correlation with well control at the target horizons are achieved. These anisotropic parameters are called PSDM effective anisotropic parameters. Apparently, these anisotropic parameters do not reflect the variation of geology. The method developed in this study has the advantage of estimation of continuous and geologically consistent anisotropic models. Figure 6 shows an estimated P-wave anisotropic profile near the Jumpingpound Creek in the Foothills area of Alberta, Canada. First, we can see that there are five major layers with distinct anisotropic properties to a depth of 4000 m. Secondly, the estimated ε values range from 0.0 to 0.36 with an average value of 0.1726. Notice that the ε and δ in the carbonate units are approximately equal to zero. This indicates that this method is valid to carbonate rocks when fractures are not present. In addition, the variation of anisotropic properties with depth as shown in Figure 6 suggests that geologically consistent anisotropic models should improve imaging and depth correlation. The estimated results were compared with the inverted PSDM effective anisotropic parameters in the Foothills area of Alberta derived by Yan et al. (2004). Our estimated average values of ε = 0.1726 and δ = 0.055 are consistent with ε = 0.18 and δ = 0.04 that were inverted from PSDM. Recent applications in 2D PSDM projects also showed that the use of the geologically consistent anisotropic models generated by the method developed in this study resulted in high accuracy of depth correlation and better imaging. To further validate our method, the estimates of anisotropy from well logs were compared to measurements using refraction seismic by Leslie and Lawton (1999) for the shallow Wapiabi Formation in the Jumpingpound Creek area of Alberta. In Figure 6, the estimated ε and δ values of 0.1465 and 0.04688 for this formation are in agreement with ε= 0.14±0.05 and δ=0.00±0.08 measured by the refraction seismic method. Prestack seismic analysis. Anisotropic contrasts are defined as ε = ε 2 ε 1, γ = γ 2 γ 1, and δ = δ 2 δ 1, and can be expressed in the form of reflectivity as R ε, R γ, and R δ. These anisotropic contrasts have a magnitude similar to P- or S-wave reflectivities. Theoretically, they can be inverted from prestack seismic data. However, this is complicated by the fact that both density and anisotropic effects on amplitude are most pronounced at far source-receiver offsets. The anisotropic profiles estimated using the method developed in this paper enable the study of anisotropic contrasts, anisotropic contributions in a CMP gather, and anisotropic synthetic seismograms. To illustrate the importance of anisotropic properties in reservoir characterization, Figure 7 shows a synthetic CMP gather from an oil sand play in the Western Canadian Basin. The CMP gather was generated using the isotropic Zoeppritz equations and ray tracing. In Figure 7, P- and S-wave reflectivities were inverted from the CMP gather. For the purpose of comparison, a zero-offset P-wave anisotropic synthetic was generated. It is shown that the reservoir at 0.475 s is better defined by R ε than by P-wave reflectivity with a trough followed by a peak. In addition, a strong peak of R ε at 0.445 s corresponds to the interface between the shaly sand above and shale below, where a low impedance contrast presents. In Figure 7, an opposite case can be observed that is R ε has a weak response to the tight streak at 0.43 s but R P and R S have a strong response. These observations indicate that anisotropic properties and their contrasts do provide additional information on subsurface lithology. The ultimate question is whether anisotropic parameters for transversely isotropic media can be inverted from prestack seismic data. To answer this question, one should start with the AVO equation for VTI media (Thomsen, 1993): where Using the well logs of V P, V S, density, and the estimated ε and δ anisotropic as input, anisotropic ray tracing can be performed and synthetic CMP gathers can be generated. In order to solve the anisotropic parameters, a three-term AVO equation was developed based on simplification of a equation that is the combination of Fatti et al. s isotropic threeterm AVO equation (1994) and equation 5. In the simplification, we assumed density-velocity follows (4) (5) JUNE 2006 THE LEADING EDGE 709
Figure 7. CMP gather generated by the isotropic Zoeppritz equations with ray tracing, and P-, S- and ε-reflectivity for an oil sand reservoir in the Western Canadian Basin. Figure 8. Synthetic CMP gather and anisotropic reflectivities. The CMP gather was generated using the Zoeppritz equations with anisotropic terms and anisotropic ray tracing. Anisotropic reflectivity R ε1 and R ε2 were calculated directly from the ε log, one with and other without band-pass filtering. Anisotropic reflectivity R ε3 was inverted from the CMP gather. Gardner s relation ρ = AVp g. The equation is: where I P and I S are P- and S-impedance, the constant and the incident angle dependent variable is (6) (7) (8) The three unknowns in equation 6 are P-wave reflectivity R P in the first term, a reflectivity that is a combination of S-wave and δ-reflectivity in the second term, and P-wave anisotropic reflectivity R ε in the third term. An example of P-wave anisotropic parameter inversion is shown in Figure 8, where R ε1 and R ε2 are the anisotropic reflectivities directly calculated from well logs, one with and the other without band-pass filtering, and R ε3 is the P- wave anisotropic reflectivity inverted from the CMP gather. After anisotropic reflectivity inversion, P-wave anisotropic reflectivity R ε3 was inverted to the P-wave anisotropic parameter ε with an input of low frequency background in a similar fashion to the impedance inversion method. As a final step the P-impedance parallel to bedding was calculated using I P = εi P I P (Figure 9). It is apparent that the inverted anisotropic profile clearly shows the variation of the lithology. 710 THE LEADING EDGE JUNE 2006
1985). Effects of porosity and clay content on wave velocities in sandstones by Han et al. (GEOPHYSICS, 1986). Detection of gas in sandstone reservoirs using AVO analysis: A 3D seismic case history using the geostack technique by Fatti et al. (GEOPHYSICS, 1994). Seismic anisotropy in sedimentary rocks by Wang (SEG 2001 Expanded Abstracts). Influence of seismic anisotropy on prestack depth migration by Yan et al. (TLE, 2004). TLE Figure 9. Anisotropic reflectivity (R ε3 in Figure 8) was inverted into the anisotropic parameter ε. P- impedance parallel to bedding was calculated, where gamma ray, ε, and impedances have the same color scale but with a value range of 70 140 (in API), 0.1 0.4, and 7 14 km/s g/cc, respectively. Conclusions. Estimation of the anisotropic properties of subsurface lithology is a task with many challenges. This paper has presented an empirical method to accomplish this task using conventional rock properties: velocity and clay volume. The method is straightforward, easy to use, and produces continuous anisotropic profiles. Anisotropic parameter estimations using this method have shown consistency with a VSP measurement and agree well with PSDM inversion and anisotropic measurements from seismic refraction data in the foothills of Alberta, Canada. Recent applications in 2D PSDM studies using the method developed in this study showed accurate depth correlation and better imaging. Preliminary studies in prestack seismic analysis indicate that this method may also aid in reservoir characterization. It is expected that the accuracy of this method will be improved with further use and as more laboratory measurements are included in the benchmarking of results. Acknowledgments: The author thanks Bob Somerville for his help in coding the programs for anisotropic ray tracing and AVO modeling, Doug Schmitt at the University of Alberta for providing the SEM image, and Michael West, Glen Larsen, and Paul Hewitt for their valuable comments and discussions. The author would also like to acknowledge Michael West who pioneered the practical use of this anisotropy estimation method in his PSDM projects. Corresponding author: yli@paradigmgeo.ca Suggested reading. Anisotropic well logs and their applications in seismic analysis by Li (SEG 2002 Expanded Abstracts). AVO calibration using borehole data by Armstrong et al. (First Break, 1995). Anisotropic effective-medium modeling of the elastic properties of shales by Hornby et al. (GEOPHYSICS, 1994). Seismic anisotropy of shales by Johnston and Christensen (JGR, 1995). A refraction-seismic field study to determine the anisotropic parameters of shale by Leslie and Lawton (GEOPHYSICS, 1999). Weak elastic anisotropy by Thomsen (GEOPHYSICS, 1986). Velocity anisotropy in shales: A petrophysical study by Vernik and Liu (GEOPHYSICS, 1997). Ultrasonic velocity and anisotropy of hydrocarbon source rocks by Vernik and Nur (GEOPHYSICS, 1992). Effects of diagenesis and clays on compressional velocities in rocks by Tosaya and Nur (Geophysical Research Letters, 1982). Relationships between compressional-wave and shear-wave velocities in clastic silicate rocks by Castagna et al. (GEOPHYSICS, JUNE 2006 THE LEADING EDGE 711