A Multiscale Model of Partial Melts 1: Effective Equations

TL;DR

This paper develops a multiscale homogenization model for partial melts, deriving effective macroscopic equations from grain-scale physics, and verifies its consistency with existing models in geophysics.

Contribution

It introduces a novel application of two-scale homogenization to model partial melts in the solid Earth, deriving effective equations and constitutive relations.

Findings

Model aligns with previous geophysical work

Provides a basis for numerical analysis of permeability and viscosity

Establishes a new multiscale approach for Earth's partial melts

Abstract

In this paper a model for partial melts is constructed using two-scale homogenization theory. While this technique is well known to the mathematics and materials communities, it is relatively novel to problems in the solid Earth. This approach begins with a grain scale model of the medium, coarsening it into a macroscopic one. The emergent model is in good agreement with previous work, including D. McKenzie's, and serves as verification. This methodology also yields a series of Stokes problems whose solutions provide constitutive relations for permeability and viscosity. A numerical investigation of these relations appears in a companion paper.