Homogenization of Poroelastic Media without a Representative Elementary Volume for Seismic Applications

The substitution of a heterogeneous poroelastic medium by its homogenized viscoelastic representation is an effective technique to study its seismic response. This homogenization procedure reproduces the dispersive behaviour of the original fast P- and S-waves. This dispersive nature results from energy dissipation that occurs when a wave induces pressure gradients between the heterogenous parts of a poroelastic medium that are equilibrated by fluid exchange. The underlying homogenization approach is to apply oscillatory tests on a representative elementary volume (REV) of the poroelastic medium to find the equivalent moduli. The REV is a sample that is typical of the entire medium under consideration and that ensures results independent of boundary conditions. This is, the REV should be larger than the heterogeneities but much smaller than the medium size. Additionally, in poroelastodynamics, the size of the heterogeneities in the REV is dictated by the scale at which the wave-induced fluid exchange takes place. We focus on the mesoscale. At this scale, fluid exchange occurs between heterogeneities that are larger than the grain size but smaller than the wavelength. However, there are poroelastic media of interest that present heterogeneities of comparable size to that of the domain. Here, the REV concept does no longer apply since the poroelastic sample under examination is affected by the boundaries of the domain. For such scenarios, we propose a novel homogenization method that incorporates the boundary effects produced by the surrounding medium. In this method, we take a sample that consist of the affected poroelastic heterogeneity together with part of the embedding medium. Then, we perform the classical oscillatory tests over this ensemble. Finally, to obtain the homogenized moduli of the poroelastic medium, we perform the averaging of strain and stress only over this domain of interest. As examples, we present a poroelastic system of a single sand layer saturated with gas at the top and water at the bottom that is embedded in impermeable background. We also study a water-saturated poroelastic set consisting of a permeable fracture surrounded by a less permeable damage zone that is also embedded in impermeable background. We idealize these cases as 2D media, assuming that the poroelastic system is infinite along the layering plane but bounded perpendicular to it by impermeable half-spaces. The samples subjected to oscillatory tests consist of a piece of the semi-infinite poroelastic domain together with the corresponding bounding half-spaces. To test the viability and accuracy of the method, we compare reflectivities at the top interface of the half-space and homogenized medium against those obtained at the top interface of the half-space and the original poroelastic system. Results show that errors are of the order of 1 %. The proposed method can be readily extended to 3D and more complex models.