Upscaling in porous media

Abstract

Effective-medium approaches are widely used to model initially heterogeneous systems: it saves computational time. In poroelasticity (two-phase media), it is advantageous to use one-phase effective medium if possible: it simplifies computations even more. In this paper we discuss situations where phase reduction introduces significant inaccuracy based on the example with homogenization of periodically layered porous media. The layers represent mesoscopic inhomogeneities (larger than the pore and grain sizes but smaller than the wavelength). Each layer is homogeneous and behaves according to Biot’s equations of poroelasticity. The effective model is characterized by the frequency-dependent P-wave modulus, and is validated with the exact analytical solution (Floquet’s theory). The reduced-phase model is in agreement with the exact solution for stiff-frame materials (such as rocks) but introduces inaccuracy for weaker sandy sediments. The cause of the inaccuracy might be the no-flow boundary conditions at the edges of the representative element that do not allow flow at the macroscopic scale. Results show that the discrepancy mainly depends on the values of permeability, frame bulk and shear moduli, as well as saturation for patchy-saturated sands. The Analysis is based on comparison of phase velocity, attenuation, transient point-source response and reflection from a fluid half-space for the effective and exact solutions. The results of the study are envisaged to be of importance for application of the effective models to highly permeable sandstones and sandy sediments.