Sensitivity of AVO reflectivity to fluid properties in porous media

Abstract

The Zoeppritz equations used in a typical reflection amplitude versus source-receiver offset (AVO) study to calculate the reflection and transmission coefficients do not directly consider the fluids filling the pore space in a porous solid medium. Although they account for the effects on the density and P-wave and S wave velocities in porous solids, these equations neglect the movements of fluids with respect to the porous framework. In doing so, the effects of the permeability and viscosity of the fluids during flow are ignored. These properties may affect the energy reflected and transmitted at a boundary; therefore, they must be accounted for to give an accurate wave propagation model. Biot theory considers the propagation of elastic waves in a porous elastic solid saturated by a viscous fluid. This theory accounts for the motion of fluids in the interconnected voids of a porous solid, assuming Darcian fluid flow. Biot theory accounts for the propagation of three waves, one rotational (shear) wave and two dilational waves (P-wave and slow wave). Reflection and transmission coefficients are calculated including Biot theory, showing potentially observable differences from the coefficients calculated using the Zoeppritz equations, for different physical situations. The sensitivity of the reflection coefficients to different physical parameters is examined. The goal is to evaluate how the reflection coefficients change as individual parameters, such as viscosity or permeability, are varied, and which parameters affect the reflection coefficients the most. If the reflection coefficient does not change as a parameter is varied, there is no sensitivity to that parameter and information about that parameter cannot be extracted from the data. The sensitivity analysis is complimented by calculating partial derivatives of the expressions for the reflection coefficients with respect to individual parameters, particularly fluid parameters. With this approach, large values of the partial derivative imply large changes in reflection coefficients with respect to a physical parameter indicate the most sensitivity to that parameter in the reflection coefficient. In Biot theory, the solid properties dominate over those of the fluids alone. The fluid properties only impact the reflection coefficients in a significant manner when there is a small contrast in the solid properties across a boundary. If the contrast in solid properties is too large, any effects caused by the fluid properties are insignificant compared to the solid effects. The three shale over sandstone models have too large of a contrast in solid properties to see fluid effects. Conversely, the six models of fluid boundaries within a reservoir sand all have little to no contrast in solid properties, so the fluid effects are evident. For gas-water interfaces, the observable changes in the P-P reflectivity are estimated to be as large as 5% for a 1% change in permeability and 15% for a 1% change in viscosity. When the above criteria for observing the fluid effects are met, the P-wave has sensitivity to viscosity, sensitivity, and porosity, with the reflection coefficients giving the most sensitivity to changes in the fluid viscosity. The apparent sensitivity to porosity is mostly a response to the density change caused by the change in porosity, rather than direct effects of the porosity. Theoretical AVO reflection coefficient curves based on Biot theory are inverted using two and three term AVO inversions based on approximations of Zoeppritz reflectivity. There is significant error in the parameters extracted by the inversion for both the two- and three-term AVO inversions. The three-term Aki and Richards (2002) inversion produces inaccurate values of the physical parameters across the boundary. Standard AVO inversion algorithms based on Zoeppritz reflectivity have problems accurately calculating parameters for a porous medium where fluids can move. An intercept and gradient interpretation algorithm based on Biot theory is desirable to accurately extract physical properties in porous media. A second formulation for reflectivity in a porous elastic solid is examined. In this study the theory developed by de la Cruz and Spanos (1985) is modified from their high viscosity limits, to fit more common lighter-oil viscosity regimes. The equations of motion and boundary conditions developed as part of the Spanos theory are adapted for this application. The reflectivity problem is simplified to an eigenvalue problem, based on a number of assumptions. De la Cruz, Hube, and Spanos (1992) published their computed values of reflection coefficients for high viscosity fluids. However, the complexity of this theory makes it impractical, in this study, to follow through to calculation of reflection coefficients in a porous elastic solid