Effective fluid and grain bulk moduli for heterogeneous thinly layered poroelastic media

Abstract

Consider a package of parallel porous layers in which the porosity, the bulk modulus of the pore fluid, and the bulk modulus of the mineral phase are constant within each layer but may vary along the package. We wished to apply Gassmann’s fluid substitution to the entire package now treated as an effective medium (long-wavelength case). The question is: What are the effective properties, namely, the mineral’s and fluid’s bulk moduli, to be used in this operation? The answer depends on the degree of hydraulic communication between the layers. We have examined two limiting cases: (1) all layers in the package are in perfect communication and (2) all layers are hydraulically isolated from each other. The two mathematical methods relevant to these cases are (1) the poroelastic Backus average for the former and (2) the elastic Backus average, respectively. By conducting a sufficient number of numerical experiments, we have found that, in both cases, the effective mineral bulk modulus for the package is very close to the value of the Hill’s average of the bulk moduli of individual layers weighted by their solid fraction. In the case of perfect hydraulic communication, the effective fluid bulk modulus is very close to the harmonic average of the individual moduli weighted by individual porosity. For the case of hydraulically isolated layers, the effective fluid bulk modulus falls between the porosity-weighted harmonic and arithmetic averages. We have evaluated approximate close-form solutions for both cases and found that these approximations cause less than 6% average relative error in the computed stiffness components. Bearing in mind that the full communication and complete isolation scenarios are relevant to very low- and very high-frequency wave propagation, respectively, our results can be interpreted as the frequency dispersion of the effective fluid bulk modulus.