Theory of frequency dependent acoustics in patchy-saturated porous media

Abstract

The theory of the dynamic bulk modulus, K (), of a porous rock, whose saturation occurs in patches of 100% saturation each of two different fluids, is developed within the context of the quasi-static Biot theory. The theory describes the crossover from the Biot-Gassmann-Woods result at low frequencies to the Biot-Gassmann-Hill result at high. Exact results for the approach to the low and the high frequency limits are derived. A simple closed-form analytic model based on these exact results, as well as on the properties of K () extended to the complex-plane, is presented. Comparison against the exact solution in simple geometries for the case of a gas and water saturated rock demonstrates that the analytic theory is extremely accurate over the entire frequency range. Aside from the usual parameters of the Biot theory, the model has two geometrical parameters, one of which is the specific surface area, S/V, of the patches. In the special case that one of the fluids is a gas, the second parameter is a different, but also simple, measure of the patch size of the stiff fluid. The theory, in conjunction with relevant experiments, would allow one to deduce information about the sizes and shapes of the patches or, conversely, to make an accurate sonic-to-seismic conversion if the size and saturation values are approximately known.

Using a Poroelastic Theory to Reconstruct Subsurface Properties: Numerical Investigation

“…The attenuation mechanism that prevails at low frequency comes from the mesoscopic scale (Pride et al, 2004), and it is due to fluid flow that occurs at boundaries between any medium heterogeneities whose sizes are between the grain sizes and the seismic wavelengths. This is particularly true for layered media (Gurevich et al, 1997;Pride et al, 2002) or when the medium contains 1) inclusions of different materials such as composite medium or double porosity medium (Berryman & Wang, 2000;Pride et al, 2004;Santos et al, 2006), or 2) different fluids (Santos et al, 1990) or patches of different saturation (Johnson, 2001). …”

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory

“…Another necessary condition must be satisfied to reduce equation (20) to equation (1a). By assuming pore fluid have the same static deformation as solid skeleton under hydrostatic loading, which means solid and fluid share the same constitutive relation, both in "small" dynamic and "large" static part, and then substituting    in equation (20),…”

“…Based on the same linear assumption, some new expansions on Biot's theory have included local fluid flow, dynamic permeability and multi-scale heterogeneity in last two decades [17][18][19][20][21][22][23]. Around linear poroelasticity, the research interests of recent years are focused on the frequency-dependent P-and S-waves' velocity and attenuation features which are influenced by patchy saturation, pore distribution and rock microstructure.…”

Computation of Wave Attenuation and Dispersion, by Using Quasi-Static Finite Difference Modeling Method in Frequency Domain

“…In subsequent studies, Dutta and Ode [Dutta et al 1979] further investigated gas-water patchy saturation in terms of Biot's poroelastic theory [M. A. Biot 1956]. Johnson [Johnson 2001] analyzed consequences of arbitrary shape patchy saturation on wave characteristics and figured out the low and high-frequency limits for p-wave velocity. Additionally, with the growing importance of mesoscopic scale theory, upcoming researchers analyzed wave attenuation and velocity dispersion due to the presence of three-dimensional heterogeneities at mesoscopic scale [Müller et al 2005, Toms et al 2007.…”