A network model for permeability in partially molten rocks

Estimation of permeability in partially molten rocks requires knowledge of the melt phase distribution at the grain-scale. The melt distribution in an isotropic two-phase (solid+melt) system under equilibrium conditions is well defined. In such a system, all of the melt channels are identical and they are either interconnected or isolated depending upon the dihedral angle and the melt fraction. A simple power-law relationship between permeability, grain size and melt fraction has been derived for such a system. However, several factors, such as non-hydrostatic stress, anisotropic interfacial energy, or the existence of a third phase, will alter this relationship. We developed a three-dimensional network model to calculate permeability as a function of melt fraction for a system with a distribution of dihedral angles. In our model, each channel is treated as a prism with a length of the grain edges. The cross-sectional area of each prism is determined by a given dihedral angle and a melt fraction. By incorporating different dihedral angles into a network model, we are able to model permeability of partially molten rocks, taking the grain-scale heterogeneity of melt distribution into account. Our results show that the permeability of a heterogeneous system can be significantly smaller than the calculated permeability of an isotropic system with the same median dihedral angle.