Effective hydraulic properties of fractured reservoirs and composite porous media

An integral equation method (Green's function technique) is used here to derive approximations for the effective (absolute) permeability tensors of fractured reservoirs and composite porous media. First, I derive an exact formal solution for the effective permeability tensor of a general (statistically homogeneous) random medium, in terms of the so-called T-matrix for the medium, which satisfies an integral equation of the Lippmann-Schwinger type (originally associated with quantum scattering theory). Then, I solve the Lippmann-Schwinger equation approximately under the assumption that all heterogeneities can be represented by ellipsoidal inclusions. It is assumed that the inclusions are distributed randomly in space in accordance with two-point correlation functions of ellipsoidal symmetry, and the interactions between more than two inclusions are ignored. On the basis of this T-matrix approach, I have derived novel expressions for the effective permeability tensors of a wide class of random media (including fractured reservoirs and sand-shale formations) of interest to the petroleum industry. The present method is extremely cheap computationally, and it complements the numerical methods of permeability upscaling (in the limit of complete scale separation) that are commonly used by industry. The effective medium approximations derived here may help us to understand the link between (frequency-dependent) seismic anisotropy and permeability, and is particularly relevant for the integration of seismic and production data from mesoscopically, heterogeneous reservoirs.