Frequency-dependent effective hydraulic conductivity of strongly heterogeneous media

The determination of the transport properties of heterogeneous porous rocks, such as an effective hydraulic conductivity, arises in a range of geoscience problems, from groundwater flow analysis to hydrocarbon reservoir modeling. In the presence of formation-scale heterogeneities, nonstationary flows, induced by pumping tests or propagating elastic waves, entail localized pressure diffusion processes with a characteristic frequency depending on the pressure diffusivity and size of the heterogeneity. Then, on a macroscale, a homogeneous equivalent medium exists, which has a frequency-dependent effective conductivity. The frequency dependence of the conductivity can be analyzed with Biot's equations of poroelasticity. In the quasistatic frequency regime of this framework, the slow compressional wave is a proxy for pressure diffusion processes. This slow compressional wave is associated with the out-of-phase motion of the fluid and solid phase, thereby creating a relative fluid-solid displacement vector field. Decoupling of the poroelasticity equations gives a diffusion equation for the fluid-solid displacement field valid in a poroelastic medium with spatial fluctuations in hydraulic conductivity. Then, an effective conductivity is found by a Green's function approach followed by a strong-contrast perturbation theory suggested earlier in the context of random dielectrics. This theory leads to closed-form expressions for the frequency-dependent effective conductivity as a function of the one- and two-point probability functions of the conductivity fluctuations. In one dimension, these expressions are consistent with exact solutions in both low- and high-frequency limits for arbitrary conductivity contrast. In 3D, the low-frequency limit depends on the details of the microstructure. However, the derived approximation for the effective conductivity is consistent with the Hashin-Shtrikman bounds.