Percolation and infiltration metasomatism

Migration of fluids in a porous matrix with solid-liquid fractionation results in a process much similar to the chromatographic separation of elements (DeVault, 1943 Korzhinskii, 1970, Hofmann, 1972). This mechanism has recently been revived in the context of mantle metasomatism by Navon and Stolper (1987), Bodinier et al. (1990), Vasseur et al. (1991), in the context of hydrothermal systems by Lichtner (1985) and, for stable isotopes, by Baumgartner and Rumble (1988). Only a simplified account of this model will be given here. Let p be the porosity of the medium, and pHq the density of the solid matrix and melt, respectively, and vHq the fluid velocity relative 

Exchanging the order of derivation and integration on the left-hand side integral and using the divergence theorem on the right-hand side, we get 

What is true for an arbitrary volume Q is true for the volume element dK, hence we can drop the integration signs and AV on both sides of the equation 

Dispersive forces due to the tortuous path of the fluid, matrix deformation, and diffusion are neglected. Taking the same steps as for the continuity equation, we get 

Taking the derivatives of the products, rearranging the left-hand side and expanding the divergence on the right-hand side gives 

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