The general mathematical constraints for thermomechanical equilibrium of partial melts under hydrostatic or nearly hydrostatic stress conditions are well known. However, melt phase geometries under equilibrium conditions have previously been understood only qualitatively. Because of its importance in determining bulk physical and chemical properties of partially molten regions of the crust and the mantle, we have computed the geometries of the liquid phase for a range of melt fractions and dihedral angles. We describe here a theoretical interfacial topology model and its associated calculational algorithms which yield numerically precise equilibrium solid‐liquid interfacial surfaces where the only constraints imposed on the system are constant mean curvature and constant wetting angle in accordance with equilibrium. The calculated geometries are used to determine the permeability of partial melts to fluid flow, interfacial areas, surface energy, minimum channel cross section, surface mean curvature, and maximum trappable melt fraction for melt fractions up to 5% and wetting angles between 20° and 80°. For wetting angles less than 60° the melt is distributed in an interconnected network of channels along the grain edges, and therefore melt mobilization is possible as soon as melt forms. For wetting angles greater than 60° the melt phase will be distributed in isolated pockets at the grain corners if the melt fraction is less than a critical value, whereas an interconnected network of melt channels will be established if the critical melt fraction is exceeded. Wetting angle exerts a first‐order control on the permeability of partially molten systems through its effect on the connectivity of the melt phase. Once a connected melt phase has formed, the permeability is only mildly sensitive to wetting angle.

Electrical conductivities of molten Hawaiian tholeiite and Crater Lake andesite were measured between 1200°C and 1400°C at atmospheric pressure and at pressures up to 17and 25 kbar, respectively. Isobaric plots of log σ versus 1/T (σ is electrical conductivity) are linear, with the exception of the zero pressure tholeiite melt data. Conductivities decrease with increasing pressure in both melts, with the andesitic melt exhibiting a greater pressure dependence. Between 5 and 10 kbar, abrupt decreases in the slopes of isothermal log σ versus P plots (i.e., decreases in activation volume) are observed for both rock melts. This discontinuity probably reflects changes in melt structure, as opposed to changes in conduction mechanism. In each pressure range, the data for each rock melt can be described reasonably well by an equation of the form σ = σ′0 exp[−(Ea + PΔV′σ)/kT], where σ′0 is a preexponential constant, Ea is the activation energy, and ΔV′σ is the activation volume. A qualitative model involving depolymerization of the melt with increased pressure leading to increased efficiency of packing can explain the observed discontinuity in activation volume as well as the observed pressure dependences of other melt physical properties such as viscosity and density. Conductivity versus melt fraction curves for partially molten peridotite are reevaluated using high pressure tholeiitic melt conductivities and crystalline conductivity values recently determined by other workers. Minimum melt fraction estimates of 5–10% are required to explain upper mantle regions of anomalously high electrical conductivity in terms of a partial melting hypothesis.

We describe a method to analyze the melt distribution in experimentally produced ultramafic partial melts. It is shown that the melt inclusions can be approximated by ellipses in two dimensions and by penny‐shaped ellipsoids in three dimensions. The aspect ratios of these ellipses (the ratio of the minor to the major axis) can in turn be used to calculate bulk physical properties of partial melts. We apply this method to two olivine‐basalt samples with 3.2% and 0.75% melt fraction. In the samples analyzed approximately 75% of the melt is contained in inclusions with much smaller aspect ratios than triple junction tubules. The reduction of P‐wave velocities calculated for this melt distribution is twice as large as for melt distributed solely in triple junction tubules.

Relationships between bulk effective electrical conductivity, melt fraction, and liquid path connectivity are derived for a partially melted material. Hashin‐Shtrikman bounds are determined for the conductivity on the basis of entropy production and compared with results obtained for exact geometrical models for the limiting cases of isolated melt pockets and complete grain boundary wetting. Models used are (1) a spherical particle assemblage of an infinite number of different‐sized composite spheres, each containing an inner core of one phase, surrounded by an outer shell of a second phase, and (2) a three‐dimensional periodic array of identical‐sized cubes of one phase, surrounded by a second phase that extends continuously throughout the body. Both models yield the same expressions as those of the upper Hashin‐Shtrikman bound for the bulk conductivity dependence on melt fraction for small melt fractions with complete liquid bridging. A small dependence on grain size and shape is inferred. Partial melting in the form of isolated melt pockets is found to be ineffective in raising bulk conductivity significantly over that of the solid phase material. Quasi‐continuous grain boundary wetting is considered in terms of analog modeling. Melt fraction and liquid path connectivity are found to be main determining factors for electrical conductivity. It is concluded that grain boundary and/or edge wetting is necessary for partial melting to raise bulk conduction in the mantle appreciably over that of its solid phase material. Such wetting is therefore inferred to be a highly probable mechanism responsible for high conductivity anomalies observed in deep geomagnetic soundings. Implications for geothermometry are considered.

For mantle regions of anomalously high electrical conductivity (greater than 0.1 S/m) the bulk conductivity is modeled by effective medium theory as a basalt melt fraction within a mainly olivine matrix. In order for the highly conducting melt to affect the bulk conductivity it must form interconnections, so that the very existence of mantle conductivity anomalies constitutes evidence for such interconnections. The inclusion of petrological data on the partial melting of peridotite strongly constrains the range of temperatures and melt fractions that can be used to yield an observed electrical conductivity. Thus from anomalous conductivities which are observed under rift zones, volcanic belts, geothermal areas, and beneath the oceans, it is possible to estimate both the temperature and the degree of partial melting. While other mechanisms for mantle conductivity enhancement may exist, e.g., contributions from contaminated grain boundaries or high volatile contents, these explanations associate a chemical differentiation in the mantle with thermal manifestations and in most cases create conditions that favor melting.