Effective media: A forward modeling view

Abstract

The existing effective media theories, such as Backus averaging, can be only used in media that possess certain characteristics regarding to concentration, shape, or geometry of their heterogeneities. These limitations originate from necessity of having analytical description of complex stress and strain fields that normally arise in microheterogeneous solids. The need for explicit solutions can be eliminated by computing the stresses and strains numerically. As a result, effective media can be in principle constructed for solids of arbitrary complexity. This simple idea is tested on two 2D models, where conventional analytical effective media theories are likely to break down. The first model is an isotropic layered solid with cracks that intersect the layer interfaces, the second is a layered medium containing random inclusions. In both cases, some differences are observed between the effective stiffness coefficients obtained numerically and those derived using the existing effective media theories. For instance, it is demonstrated that Backus averaging (improperly) applied to horizontal isotropic layers with random inclusions leads to biases in the calculated vertical velocities. Overall, the proposed technique enables us to establish the limits of applicability of conventional effective media theories. It also makes it possible to obtain quantitative estimates of the errors incurred because of violating certain assumptions of a given theory.