Dynamic effective properties of heterogeneous geological formations with spherical inclusions under periodic time variations

In unsteady groundwater flow (or similar processes of heat/electrical conduction), the heterogeneous medium structure is characterized by two random properties, the conductivity K and the specific storativity S . The average head field ⟨ H  ⟩and the associated effective properties K ef , S ef are determined for a layer with a periodic head drop between boundaries, such that H is periodic in time, and a medium made up of a matrix with a dilute concentration of spherical inclusions. In the common quasi‐steady approximation, K ef is equal to the classical steady solution while S ef  =  S A , the arithmetic mean. We derive expressions for the frequency dependent K ef , S ef , which are generally complex, i.e., dynamic. The main result is the delineation of the ranges of the parameters: dimensionless frequency ( ω ) and contrasts of conductivity ( κ ) and storativity ( s ) between the matrix and the inclusions, for which dynamic effects are significant.