Attenuation and dispersion of compressional waves in fluid‐filled porous rocks with partial gas saturation (White model)—Part II: Results

Abstract

In this investigation, Biot’s (1962) theory for wave propagation in porous solids is applied to study the velocity and attenuation of compressional seismic waves in partially gas‐saturated porous rocks. The Physical model, proposed by White (1975), is solved rigorously by using Biot’s equations which describe the coupled solid‐fluid motion of a porous medium in a systematic way. The quantitative results presented here are based on the theory described in Dutta and Odé (1979, this issue). We removed several of White’s questioned approximations and examined their effects on the quantitative results. We studied the variation of the attenuation coefficient with frequency, gas saturation, and size of gas inclusions in an otherwise brine‐filled rock. Anomalously large absorption (as large as 8 dB/cycle) at the exploration seismic frequency band is predicted by this model for young, unconsolidated sandstones. For a given size of the gas pockets and their spacing, the attenuation coefficient (in dB/cycle) increases almost linearly with frequency f to a maximum value and then decreases approximately as 1/√f. A sizable velocity dispersion (of the order of 30 percent) is also predicted by this model. A low gas saturation (4–6 percent) is found to yield high absorption and dispersion. An analysis of all of the field variables (stresses and displacements) is presented in terms of Biot’s type I (the classical compressional) wave and type II (the diffusion) wave. It is pointed out that the dissipation of energy in this model is mainly due to the relative fluid flow from the type II wave. From our formulation, many of White’s equations can be derived as suitable approximations, and it is shown that the discontinuity in fluid pressure assumed by White at the gas‐water interface is the discontinuity in the fluid pressure contribution by the type II wave. Our quantitative results are in reasonably good agreement with White’s (1975) approximate theory. However, the phase velocities computed by White’s approximate treatment do not approach the correct zerofrequency limit (Gassmann‐Wood) when compared to the present theory. Most of these disagreements disappear if the corrections to White’s theory as suggested by Dutta and Seriff (1979, this issue) are incorporated.

Liquefaction and dynamic poroelasticity in soft sediments

“…The fluid displacement field w(z) (Plate 2) shows that the pore fluid flows toward the surface at low frequencies (thus equilibrating pressure) and at high frequencies there is no fluid displacement (since the timescale for pressure diffusion is longer than the period of the wave) and thus there is no pore pressure equilibrium. The low frequency region is often referred to as "relaxed" state of the pore fluid and the high frequency region is "unrelaxed" state [Mavko et al 1998, Dutta and Ode, 1979a, 1979b.…”

Dispersive surface waves along partially saturated porous media

“…This last condition allows a closed analytical solution and it is consistent with numerical calculations based on the gas pocket model. 5,6 This leads to the following relation:…”

“…In the case that only liquid saturates the pore space, the interaction between the liquid and the solid matrix can be understood in terms of the Biot theory. 2,3 This theory was previously extended in order to include the effects of gas saturation on the bulk elastic waves in partially saturated porous media by among others White, 4 Dutta and Ode, 5,6 Berryman et al, 7 Smeulders and Van Dongen, 8 Johnson, 9 and Carcione et al 10 A great deal of attention has been given to the influence of the gas saturation on the velocities and attenuation of seismic waves since the pioneering work of White. 4 The White model describes the air fraction as spherical gas pockets distributed in a cubic array in the liquid-saturated porous medium.…”

“…This model will be referred to as the "gas pocket model." Dutta and Ode 5,6 provided a more complete solution based on Biot's theory for the bulk modulus of a single bubble surrounded by a fluid-saturated porous spherical shell. Berryman et al 7 formulated a model based on variational principles for the bulk acoustic properties of a porous medium saturated with a mixture of two fluids.…”

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

“…White's concept was further extended to patchy saturation in layered form . 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.…”