A multiscale model of partial melts: 1. Effective equations

Developing accurate and tractable mathematical models for partially molten systems is critical for understanding the dynamics of magmatic plate boundaries as well as the geochemical evolution of the planet. Because these systems include interacting fluid and solid phases, developing such models can be challenging. The composite material of melt and solid may have emergent properties, such as permeability and compressibility that are absent in each phase alone. Previous work by several authors have used multiphase flow theory to derive macroscopic equations based on conservation principles and assumptions about interphase forces and interactions. Here we present a complementary approach using homogenization, a multiple scale theory. Our point of departure is a model of the microstructure, assumed to possess an arbitrary, but periodic, microscopic geometry of interpenetrating melt and matrix. At this scale, incompressible Stokes flow is assumed to govern both phases, with appropriate interface conditions. Homogenization systematically leads to macroscopic equations for the melt and matrix velocities, as well as the bulk parameters, permeability and bulk viscosity, without requiring ad hoc closures for interphase forces. We show that homogenization can lead to a range of macroscopic models depending on the relative contrast in melt and solid properties such as viscosity or velocity. In particular, we identify a regime that is in good agreement with previous formulations, without including their attendant assumptions. Thus, this work serves as independent verification of these models. In addition, homogenization provides a consistent machinery for computing consistent macroscopic constitutive relations such as permeability and bulk viscosity that are consistent with a given microstructure. These relations are explored numerically in the companion paper.