Melt Extraction in Two-Phase Continuum Theory

Melt generation and extraction are often modeled using the two-phase equations developed by McKenzie (1984). Usually the generation of melt and its subsequent propagation are treated independently which may lead to some unphysical results. Here, we discuss a generalized version of the set of equations introduced by Bercovici et al. (2001) that allows for mass transfer between the two phases. In our description the two phases are submitted to different pressure fields whose difference is related to the surface tension at the interfaces and to the changes in porosity. 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, the Gibbs-Thomson effect comes out naturally from thermodynamic equilibrium considerations. We apply these new equations to a steady state problem of pressure release melting under oceanic spreading centers. We treat melting and compaction simultaneously and we observe several new effects. A consequence of matrix compaction is a pressure difference between melt and solid which favors melting. Melting thus starts deeper than what would be predicted from the average pressure. Numerical results show that for Earth like parameters melting could start at most ∼10 km below the standard solidus. Simple numerical analysis suggests that the movement of melt and matrix should be close to the Darcy equilibrium where the buoyancy of melt is equilibrated by the mechanical interaction between the phases. This near equilibrium state implies an upper limit on porosity of the two-phase medium. The limit value is found to be ∼10 % for a matrix upwelling at 10 cm yr^-1; the dependence on different parameters will be subject to discussion. Our set of equations can also mimic multivariant phase transformation. We will show how the predicted porosity and degree of melting aryaccording to the mechanical conditions that prevail (Darcy equilibrium or viscous equilibrium).