A multiscale model of partial melts: 2. Numerical results

In the companion paper, equations for partially molten media were derived using two scale homogenization theory. This approach begins with a grain-scale description and then coarsens it through multiple scale expansions into a macroscopic model. One advantage of homogenization is that effective material properties, such as permeability and the shear and bulk viscosity of the two-phase medium, are characterized by cell problems, boundary value problems posed on a representative microstructural cell. The solutions of these problems can be averaged to obtain macroscopic parameters that are consistent with a given microstructure. This is particularly important for estimating the "compaction length" which depends on the product of permeability and bulk viscosity and is the intrinsic length scale for viscously deformable two-phase flow. In this paper, we numerically solve ensembles of cell problems for several geometries. We begin with simple intersecting tubes, as this is a one parameter family of problems with well-known results for permeability. Using the data, we estimate relationships between the porosity and all of the effective parameters by curve fitting. For the model of intersecting tubes, permeability scales as phi(n), n similar to 2, as expected, and the bulk viscosity scales as phi(-m), m similar to 1, which has been speculated but never shown directly for deformable porous media. The second set of cell problems adds spherical inclusions where the tubes intersect. For these geometries, the permeability is controlled the pore throats and not by the total porosity, as expected. However, the bulk viscosity remains inversely proportional to the porosity, and we conjecture that this quantity is insensitive to the specific microstructure. The computational machinery developed can be applied to more general geometries, such as texturally equilibrated pore shapes. However, we suspect that the qualitative behavior of our simplified models persists in these more realistic structures. In particular, our hybrid numerical-analytical model predicts that for purely mechanical coupling at the microscale, all homogenized models will have a compaction length that vanishes as porosity goes to zero. This has implications for numerical simulations, and it suggests that these models might not resist complete compaction.