AVAZ inversion for fracture orientation and intensity: a physical modeling study Faranak Mahmoudian*, Gary F. Margrave, and Joe Wong, University of Calgary. CREWES fmahmoud@ucalgary.ca Summary We present a pre-stack amplitude inversion of P-wave data to determine fracture orientation and intensity. We test the method on multi-azimuth multi-offset physical model reflection data acquired over a simulated fractured medium. This medium is composed of phenolic material with controlled symmetry planes, and its elastic properties have already been determined using traveltime analysis. This experimental model represents a HTI layer. We apply amplitude inversion of small incident angle data to extract the fracture orientation (direction of isotropic plane of the medium), and are able estimate the fracture orientation quite accurately. Knowing the fracture orientation, we modified the available linear PP reflection coefficient equation to invert for the anisotropy parameters (ε (V), δ (V), γ). The results for all three anisotropy parameters from AVAZ inversion compare very favorably to those obtained previously by a traveltime inversion. This result makes it possible to compute the shear-wave splitting parameter directly related to fracture intensity, which historically had been determined from shear-wave data, from a quantitative analysis of the PP reflected data. Introduction The ultimate goal of using AVO in fracture-detection studies is to invert the pre-stack amplitude data for fracture orientation (strike) and the magnitude of fracture intensity. Fracture orientation is defined as the dominant direction of fracture faces. Fracture intensity is the number of fractures per unit volume (Nelson, 1). Assuming fractures can be described by a horizontal transverse isotropic (HTI) medium, we present the AVAZ inversion for fracture orientation, based on the method of Jenner (), and tested on physically modeled data acquired over a simulated fractured medium. We initially characterized this simulated fractured layer as a HTI layer with known symmetry axis; using traveltime analysis on transmission data we estimated all its elastic stiffness coefficients (Mahmoudian, 1a). The AVAZ inversion result for the fracture orientation is quite accurate. For fracture intensity, we propose a pre-stack amplitude inversion of large-offset P-wave data, based on Rüger s equation (1), to invert for (ε (V), δ (V), γ), where γ is directly related to fracture intensity. Theory A plane-wave approximation for the PP reflection coefficient at a boundary between two HTI media with the same orientation of the symmetry axis (along φ azimuth) is given by Rüger (1) as 1 Z 1 G ( V ) R(, ) cos ( ) sin Z G (1) 1 ( V) 4 ( V) cos ( ) sin ( )cos ( ) sin tan. where θ is the incident angle, φ is the source-receiver azimuth, α is the vertical P-wave velocity (fast P velocity), Z is the P-wave impedance, β is the vertical S-wave velocity (fast S velocity), G is the shear modulus, denotes the difference in the elastic properties across the boundary. The average values of elastic properties in the two layers are denoted by the terms with overscores. (ε (V), δ (V), γ) are GeoConvention 13: Integration 1
the Thomsen-style anisotropy parameters for HTI media, defined using stiffness coefficients by ( V ) c11 c33 c44 c55 ( V ) ( c13 c55 ) ( c33 c55 ),, and. c33 c55 c33 ( c33 c55 ) We reformulate equation (1) as a function of P- and S-wave velocities, using Z / Z / / and G/ G / /, as 1 4 1 4 R(, ) sin + 1 sin cos Fracture intensity estimation 1 4 ( V ) 4 cos ( )sin tan cos ( )sin 1 cos ( )sin 1 sin ( )tan ( V ). Assuming the fracture orientation is known (then φ is the perpendicular direction to it) equation () can ( ) ( ) be written as a function of six parameters ( /, /, /, V V,, ) : ( ) ( ) (, ) + V V R A B C D + E F, (3) where the coefficients A, B, C, D, E, and F are functions of the incident angle, azimuth, and velocity, and defined as in equation (). We use PP ray tracing of a smooth background velocity, at each azimuth for a given offset and depth to determine the incident angle, then to calculate the coefficients. Assuming that the pre-stack PP reflection amplitudes approximate reflection coefficients, incorporating n reflection amplitudes up to large incident angles (e.g., far-offset up to 45⁰) from m different azimuths as the input data, we use equation (3) to form a linear system of equations to invert for the six parameters solved using a damped least-squares inversion. Fracture orientation estimation Considering only small incident angle data (e.g., less than 35⁰), for which the sin tan term can be neglected, equation (1) can then be written as: R(, ) I G1 G cos ( ) sin, (4) ( V ) where I (.5) Z / Z, G1.5( / ( / ) G / G), and G.5( ( / ) ). Equation (4) has the form of AVO intercept and gradient. The gradient term Q G 1 G cos ( ) () is nonlinear with respect to three unknowns ( G1, G, ).The goal here is to invert the PP amplitude data for these three unknowns. The following describes how to bypass this non-linearity and to apply a linear inversion. Using the identity sin ( ) cos ( ) 1, the gradient becomes 1 1 Q = ( G G )cos ( ) G sin ( ). If the AVO gradient does not change sign azimuthally, the gradient versus azimuth vector delineates a curve that closely resembles an ellipse (Rüger, 1) with the semi-axes aligned with the symmetry plane directions of the fracture system (Figure 1). For a coordinate system aligned with the fracture system, every point of the gradient ellipse is ( y1, y) r(cos( ),sin( )), where r is the vector magnitude. In this coordinate system the gradient term can be written as: Q r(( G G ) y G y ). (5) 1 1 1 GeoConvention 13: Integration
Figure 1: Reference coordinate systems. (x 1,x ) is the acquisition coordinate system, and (y 1,y ) is the coordinate system aligned with the fracture system. φ is the source-receiver azimuth, and φ is the fracture orientation azimuth. On the other hand by expanding trigonometric terms, the gradient term can be written as Q r( W11 cos W 1 cos sin W sin ), where W ij are functions of ( G1, G, ). In the acquisition coordinate system (Figure 1), every point on the gradient ellipse is ( x1, x) r(cos,sin ), so the last equation becomes Q r( W x W x x W x ), a quadratic form and can be written in a matrix form, 11 1 1 1 W11 W1 x1 T [ x1 x] X WX W1 W x. If we wish to eliminate the mixed term xx 1 using the eigenvalues of matrix W ( 1, ), it reads Q r( 1Y 1 Y ), (6) where ( Y1, Y) are the eigenvectors of matrix W. Comparing coefficients from equations (5) and (6), we obtain: G1 G 1 and G1, so G1 and G can be obtained from eigenvalues. From the eigenvalue problem, we also know that a orthogonal rotation matrix, relates the two coordinate systems as [ y y ] R [ x x ], where the rotation angle obeys (e.g., Lax, 1997) 1 1 This results in two values for where W 1 tan. W W11 () (1) /, as (1),() 1 W11 W ( W11 W ) 4W1 W1 tan. (8) Equation (8) is used by Jenner (), and is equivalent to that used by Grechka et al. (1999) in solving for fracture orientation from the azimuthal variation of NMO velocity. Incorporating the amplitudes from different azimuths and small incident angles (e.g., up to 35⁰) as in the input data, we apply a linear inversion to solve for ( W11, W1, W ). Then, knowing W ij we use equation (8) to estimate the fracture orientation. Physical model reflection data We test the proposed AVAZ inversions for fracture orientation and intensity on physical model reflection data acquired over a four-layered model (Figure ). Our model consists of isotropic water and plexiglas and an HTI layer composed of phenolic material. The construction of the experimental phenolic layer and its initial characterization is presented in Mahmoudian et al. (1a). The experimental phenolic layer approximates a weakly anisotropic HTI layer, or equivalently a vertically fractured transversely isotropic layer with known fracture orientation. The five independent parameters required to describe ( V) ( V) our simulated fractured medium (,,,, ) are listed in Table 1. Table 1: Elastic properties of the simulated fractured layer. α (m/s) β (m/s) ε (V) δ (V) γ 35 17 -.145 -.185.117 R, (7) GeoConvention 13: Integration 3
Figure : The four-layered earth model used to acquire reflection data. The acquisition ⁰ azimuth is parallel to the fracture symmetry axis. Figure 3: (left) 9⁰ azimuth data with a long gate automatic gain control applied. Our target is event "B", the PP reflection from the top of the fractured layer. (right) 9⁰ azimuth data, zoomed on the target reflector. We used reflection amplitudes from nine long offset common midpoint (CMP) gathers acquired along azimuths between ⁰ and 9⁰. Figure 3(left) shows the CMP seismic line acquired along the 9⁰ azimuth (fracture plane). The amplitudes reflected from the top of the simulated fractured layer are inputs to AVAZ inversion. We manually picked the amplitudes and applied deterministic corrections to make them represent reflection coefficients required by an amplitude inversion. The corrections for geometrical spreading, emergence angle, transmission loss, and source-receiver transducer directivity (specific to physical model transducers) have been applied. A detailed description for the data acquisition and corrections is given in Mahmoudian et al. (1b). The corrected reflected amplitudes from the top of the fractured layer for nine azimuths between ⁰ and 9⁰ are shown in Figure 4(left). The large oscillations in the amplitude data are due to the interference with the top reflector reverberations which has a different dip than the target event, Figure 3(right) shows the wave interference effect on the target amplitudes. The presence of large oscillations in the data causes the AVAZ inversion to be unstable depending on the maximum incident angle used. To avoid unstable inversion, we smooth the amplitude data prior to inversion. Smoothing is applied by using the best fit polynomial of degree n = 1 to the amplitude data. The smoothed amplitude data are shown in Figure 4(right). Estimated orientation and anisotropy parameters of the simulated fractured layer We use small incident angle data (maximum incident angle of 35⁰) in the AVAZ inversion. Since the symmetry of the simulated fractured medium is known and the physical model data acquisition coordinate was aligned with the fracture system, we rotate our acquisition coordinate system to arbitrary directions and used the proposed AVAZ inversion to estimate the fracture orientation. Table shows the predicted fracture orientations. Table : Estimated fracture orientation from the AVAZ inversion. True φ ⁰ ⁰ 4⁰ 5⁰ 6⁰ 8⁰ 9⁰ -1.5⁰ 18.5⁰ 38.5⁰ 48.5⁰ 58.5⁰ 78.5⁰ 88.5⁰ Estimated φ 88.5⁰ 18.5⁰ 18.5⁰ 138.5⁰ 148.5⁰ 168.5⁰ 178.5⁰ GeoConvention 13: Integration 4
Amplitude Amplitude.3.5. Az 9 Az 76 Az 63 Az 53 Az 45 Az 7 Az 37 Az 14 Az.3.5..15.15 1 3 4 5 Incident angle (degrees) 1 3 4 5 Incident angle (degrees) Figure 4: (left) Fracture top corrected reflection amplitudes from all nine azimuths and incident angles before the critical angle. (right) Smooth amplitude data input to the AVAZ inversions. The method is successful in predicting the fracture orientation; however, there is an ambiguity in the estimation as the method predicts both the fracture orientation and the direction normal (symmetry axis) to it. Therefore, a priori knowledge of the fracture orientation is required, perhaps from NMO analysis. To estimate the anisotropy parameters, knowing the fracture orientation for our simulated fractured medium, we tested the proposed six-parameter AVAZ inversion on the physical model data. In the first implementation, we estimated all six parameters simultaneously. Figure 5 shows the six-parameter AVAZ inversion for different maximum incorporated incident angles. The six-parameter AVAZ inversion using small incident angles does not produce good estimates for any of the six parameters. Incorporating large incident angles (e.g., 4⁰) within 1⁰ of the critical angle can result in good estimates for the first three isotropic terms, showing of the larger influence of the isotopic terms on the reflection coefficients, while the estimates of anisotropy parameters still have large errors. Incorporating larger incident angles, closer to the critical angle, results in better estimates for the anisotropy parameters as the azimuthal anisotropy is more pronounced at larger incident angle data. However, it produces large errors for the estimates of the three isotropic terms. The ε (V) and γ parameters are more accurately estimated, but the δ (V) parameter which governs the near-vertical wave propagation loses its accuracy. The overall error for all six parameters is larger when incorporating incident angles close to the critical angle, since the linear Rüger s equation is not valid in this region. 15 Max incident angle = 37 15 Max incident angle = 41 15 Max incident angle = 45 1 1 1 5 5 5-5 / / / (v) (v) -5 / / / (v) (v) -5 / / / (v) (v) Figure 5: Six-parameter AVAZ inversion for three maximum incorporated incident angles. The estimates are compared to the values previously estimated from traveltime inversion. In an effort to obtain better estimates for the anisotropic terms, we used a second implementation in which we applied some constraints to the three isotropic terms. We put constraints on these first three variables using their estimated values from an isotropic AVA inversion of 9⁰ (isotropic plane) azimuth data. With such constraints, we examined the three-parameter inversion results for various maximum incorporated incident angles. Similar to six-parameter inversion the best results come from a large maximum incident angle but not close to critical angle. Figure 6 shows the constrained AVAZ inversion GeoConvention 13: Integration 5
for three different maximum incident angles. As a result of these constraints, for the right choice of the maximum incident angle, the inversion results for the three anisotropy parameters agree very favorably with those obtained previously by traveltime inversion. The estimate for anisotropy parameters from constrained AVAZ inversion is within 1%. These estimates are better than those from the simultaneous six-parameter inversion. Max incident angle = 35 Max incident angle = 39 Max incident angle = 43 - - - (v) (v) (v) (v) (v) (v) Figure 6: Constrained AVAZ inversion for anisotropy parameters. Conclusions We presented pre-stack amplitude inversion procedures to extract the anisotropy parameters (ε (V), δ (V), γ), and fracture orientation from the azimuthal variations in the PP reflection amplitudes. Since the shear-wave splitting factor, γ, is directly related to fracture intensity. We showed that it is possible to relate the difference in P-wave azimuthal AVO variations directly to the fracture intensity of our simulated fracture layer. Accurate linear inversion for the anisotropy parameters requires the use of large-offset data. However, incorporating very large offset data close to the critical angle should be avoided as the linear Rüger s equation is a plane wave solution and not valid close to the critical angle. The AVAZ inversion determines the fracture orientation with an inherent ambiguity, as it predicts both the directions of the isotropic plane and symmetry axis of an HTI medium. To resolve the ambiguity some other information is required, such as azimuthal NMO or shear-wave splitting. These effects are qualitatively different from azimuthal AVO and can be combined effectively to invert for fracture orientation. Our inversion is based on the approximate reflection coefficients by Rüger (1), the inversion estimates demonstrate that the Rüger s equation is suitable for quantitative amplitude analysis of anisotropic targets, and can be employed for numerical inversion algorithms. Acknowledgements We thank the sponsors of CREWES for their financial support. Dr. P.F. Daley, Dr. Kris Innanen, Dave Henley and Marcus Wilson are acknowledged for discussions. References Grechka, V., and Tsvankin, I., 1998, 3-D description of normal moveout in anisotropic inhomogeneous media, 63, 179-19. Jenner, E.,, Azimuthal AVO: methodology and data example: The Leading Edge, 1, No. 8, 78 786. Lax, P. D., 1997, Linear algebra and its applications. Mahmoudian F., Margrave G. F., Daley P. F., and Wong j., 1a, Anisotropy estimation for a simulated fractured medium using traveltime inversion: A physical modeling study, SEG expanded abstracts. Mahmoudian F., Wong J, Margrave G. F., 1b, Azimuthal AVO over a simulated fractured medium: A physical modeling experiment, SEG expanded abstracts. Nelson, R. A., 1, Geologic analysis of naturally fractured reservoirs, Second edition: Elsevier. Rüger, A., 1, Reflection coefficients and azimuthal AVO analysis in anisotropic media: Geophysical Monograph Series. GeoConvention 13: Integration 6