Scattering theory of wave propagation in a two‐phase medium

Abstract

A theory is developed for the propagation of pressure waves in a two‐phase medium where one phase consists of spherical inclusions distributed randomly in the second phase. The theory is based on an integral equation of Foldy (1945) and Twersky (1970) for the average wave which includes almost all multiple scattering processes, but it ignores correlations among inclusions. In the low‐frequency limit, this equation is solved exactly for an analytical expression for the refractive index of compressional waves in terms of elastic parameters of the matrix and the inclusions. The theory is then applied to fluid‐solid suspensions and to fluid‐saturated porous rocks. In the former case, velocities measured by Kuster and Toksöz (1974b) as a function of the concentration of inclusions are compared with theoretical predictions of seven different models. The closest agreement is obtained for the present theory. This is attributed to systematic inclusion of multiple scattering effects including near‐field scattering interactions. Theoretically computed attenuation also agrees well with experiments. In the case of fluid‐saturated porous rocks, comparison between theoretically computed velocities and observed velocities in sandstones (Wyllie et al, 1962) suggests the importance of microcracks and nonspherical inclusions, factors which are not included in the present model. Attenuation caused by viscosity of inclusions is shown to arise in two different ways: losses caused by generation of shear waves in the fluid, and those caused by Darcy‐type flow of the fluid relative to the matrix. The former increases with viscosity, while the latter decreases. The sum, however, remains negligible at frequencies relevant to seismic prospecting and earthquakes.

An approach to combined rock physics and seismic modelling of fluid substitution effects

“…In Fig. 3, similar modelling results (except for Mehta 1983) are shown for both P‐ and S‐wave velocities versus porosity up to 30%, using a pore structure containing both spherical and ellipsoidal inclusions (pore model 1 of Table 2). In this case, within the given porosity range, the results of the non‐SC and SC models are closer to the DEM predictions.…”

“…The material properties of the solid and fluids are given in Table 1. The velocities obtained using the DEM and Mehta (1983) models are quite consistent, while the non‐SC and SC approaches are, compared with these, seen to underestimate and overestimate, respectively, the effect of spherical inclusions. The figures also show that the discrepancies in the velocity predictions increase as the bulk modulus of the pore fluid is lowered (i.e.…”

“…Before we discuss microscopic fluid‐distribution effects, we briefly compare the predicted velocities obtained from the DEM, non‐SC, SC and Mehta (1983) models. The model of Mehta (1983) predicts the P‐wave velocity for materials containing spherical fluid‐filled inclusions, where pore interaction effects, due to Twersky scattering (Ishimaru 1978), have been included. Figure 2 shows P‐wave velocities versus spherical porosity up to 50% for brine‐ and gas‐saturated materials.…”

“… Modelled P‐wave velocities as a function of (a) brine‐ and (b) gas‐saturated spherical porosity, using SC, non‐SC (KT), DEM and the multi‐order scattering model of Mehta (1983). …”

Dynamic composite elastic medium theory. Part II. Three-dimensional media

“…14 This same integral equation can also be converted to a differential equation which is then solved for a plane wave. 15 Another approach is to estimate the coherent field directly by averaging the effects of multiple scattering in the forward direction over the first Fresnel zone by the method of stationary phase. 16 Still another approach, which is similar to the one followed in this article, is to average the scattering field in the forward direction by using the parabolic approximation.…”