Gassmann equations predict effective elastic properties of an isotropic homogeneous bulk rock frame filled with a fluid. This theory has been generalized for an anisotropic porous frame by Brown and Korringa's equations. Here, we develop a new model for effective elastic properties of porous rocks -a generalization of Brown and Korringa's and Gassmann equations for a solid infill of the pore space. We derive the elastic tensor of a solid-saturated porous rock considering small deformations of the rock skeleton and the pore infill material upon loading them with the confining and porespace stresses. In the case of isotropic material, the solution reduces to two generalized Gassmann equations for the bulk and shear moduli. The applicability of the new model is tested by independent numerical simulations performed on the microscale by finite-difference and finite-element methods. The results show very good agreement between the new theory and the numerical simulations. The generalized Gassmann model introduces a new heuristic parameter, characterizing the elastic properties of average deformation of the pore-filling solid material. In many cases, these elastic moduli can be substituted by the elastic parameters of the infill grain material. They can also represent a proper viscoelastic model of the pore-filling material. Knowledge of the effective elastic properties for such a situation is required, for example, when predicting seismic velocities in some heavy oil reservoirs, where a highly viscous material fills the pores. The classical Gassmann fluid substitution is inapplicable for a configuration in which the fluid behaves as a quasi-solid.
Saturation of porous rocks with a mixture of two fluids (known as partial saturation) has a substantial effect on the seismic waves propagating through these rocks. In particular, partial saturation causes significant attenuation and dispersion of the propagating waves, due to wave-induced fluid flow. Such flow arises when a passing wave induces different fluid pressures in regions of rock saturated by different fluids. As partial fluid saturation can occur on different length scales, attenuation due to wave induced fluid flow is ubiquitous. In particular, mesoscopic fluid flow due to heterogeneities occurring on a scale greater than porescale, but less than wavelength scale, is responsible for significant attenuation in the frequency range from 10 to 1000 Hz.
Understanding the effect of stress and pore pressure on seismic velocities is important for overpressure prediction and for 4D reflection seismic interpretation. A porosity-deformation approach (originally called the piezosensitivity theory) and its anisotropic extension describe elastic moduli of rocks as nonlinear functions of the effective stress. This theory assumes a presence of stiff and compliant parts of the pore space. The stress-dependent geometry of the compliant pore space predominantly controls stress-induced changes in elastic moduli. We show how to apply this theory to a shale that is transversely isotropic (TI) under unloaded conditions. The porosity-deformation approach shows that components of the compliance tensor depend on exponential functions of the principal components of the effective stress tensor. In the case of a hydrostatic loading of a TI rock, only the diagonal elements of this tensor, expressed in contracted notation, are significantly stress dependent. Two equal shear components of the compliance will depend on a combination of two stress exponentials. Exponents of the stress exponentials are controlled by components of the stress-sensitivity tensor. This tensor is an important physical characteristic directly related to the elastic nonlinearity of the porous rock. We simplify the porosity-deformation theory for TI rocks and provide corresponding explicit equations. We apply this theory to ultrasonic measurements on saturated shale samples from the North Sea. We show that the theory explains the compliance tensor, anellipticity, and three anisotropic parameters under a broad range of loads.
SUMMARY Spatial heterogeneity of hydrocarbon reservoirs causes significant attenuation and dispersion of seismic waves due to wave‐induced flow of the pore fluid between more compliant and less compliant areas. This paper investigates the interaction between a plane elastic wave in a poroelastic medium with a spherical inhomogeneity of another porous material. The behaviour of both the inclusion and the background medium is described by the low‐frequency variant of Biot's equations of poroelasticity with the standard boundary conditions at the inclusion surface, and for the inclusion size much smaller than the wavelength of the fast compressional wave. The scattering problem is formulated as a series expansion of displacements expressed in the spherical harmonics. The resulting scattered wavefield consists of the scattered normal compressional and shear waves and Biot's slow wave, which attenuates rapidly with distance from the inclusion and represents the main difference from the elastic case. This study concentrates on the attenuation effects caused by the mode conversion into Biot's slow wave. The solution obtained for Biot's slow wave is well described by the two terms of order n= 0 and n= 2 of the scattering series. The scattering amplitude for the term of order n= 0 is given by a simple expression. The full expression for the term of order n= 2 is very complicated, but can be simplified assuming that the amplitude of the scattered fast (normal) compressional and shear waves are well approximated by the solution of the equivalent elastic problem. This assumption yields a simple approximation for the amplitude of the scattered slow wave, which is accurate for a wide range of material properties and is sufficient for the analysis of the scattering amplitude as a function of frequency. In the low‐frequency limit the scattering amplitude of the slow wave scales with ω3/2, and reduces to the asymptotic long‐wavelength solution of Berryman (1985), which is valid for inclusions much smaller than the wavelength of Biot' slow wave. For inclusions larger than the wavelength of Biot's slow wave, the scattering amplitude is proportional to ω1/2, which is consistent with the results of Gurevich et al. (1998), which were derived by the Born approximation and therefore were limited to a weak contrast between the inclusion and the background medium. Our general solution, however, does not require these assumptions on frequency and material properties. The obtained results can be used in the analysis of the effective properties, attenuation and dispersion of elastic waves in randomly inhomogeneous porous materials.
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