Simultaneous melting and compaction in deformable two‐phase media

SUMMARY Melt generation and extraction are typically modelled using the two‐phase equations developed by McKenzie or Scott and Stevenson. Various approximations are often made to simplify the problem which may lead to some unphysical results (e.g. thermodynamically inconsistent conditions of melting and unrealistic porosity profiles). We discuss a generalized version of the set of equations introduced by Bercovici et al. that allows for mass transfer between the two phases in a single component system and consider a self‐consistent set of equations. In our description the two phases, solid and melt, are submitted to individual pressure fields whose difference is related to the surface tension at the interfaces, changes in porosity and the melting rate. A kinetic relation for the melting rate arises from the second law of thermodynamics. The condition of chemical equilibrium corresponds to the usual univariant equality of the chemical potentials of each phase when the matrix and melt are motionless. In the most general form, phase equilibrium is influenced by both the Gibbs–Thomson effect that arises naturally from thermodynamic considerations on surface tension and by the viscous deformation of the phases. We apply these new equations to a steady state problem of pressure release melting in a univariant system. We treat melting and compaction simultaneously and observe several new effects including multiple domains near the onset of melting that correspond to various force balances. A consequence of matrix compaction and melt expulsion (or matrix dilation and melt accumulation) is a pressure difference between melt and solid that facilitates (inhibits) melting. For parameters corresponding to mid‐oceanic ridge magmatism, compaction permits melting to start as much as ∼2 km below the standard solidus. Numerical solutions are necessary to determine the magnitude of the melt zone shift. Numerical results support the boundary layer solutions obtained analytically and suggest that in most of the melting zone the movement of melt and matrix should be close to the Darcy equilibrium where the buoyancy of melt is balanced by the viscous drag between the phases. The Darcy equilibrium follows an initial stage where the matrix viscous stresses balance Darcy drag. In all situations the steady state porosity profile remains a monotonic function of depth. The existence of a compaction layer following a melting zone where the porosity is maximum as described in various earlier publications has never been found.