Percolation and magma segregation

Percolation models differ from the zone-refining model essentially by the absence of mixing in the liquid, giving the liquid position-dependent properties. A simplified account of these models was described in Chapter 8. We will now provide a reasonably comprehensive account which may prove useful to the demanding reader, and then examine some properties of the chromatographic effect in a simple configuration. Let p be the open volume porosity of the medium, pso, and pliq the density of the solid matrix and melt, respectively, vliq the liquid velocity relative to the matrix, and Cso, and CHq the concentration of element i in the matrix and melt, respectively. Let us rewrite equation (8.3.14) as 

Significant simplification can be achieved through the rather innocuous assumption of constant pwi. Given the differential form of equation (9.3.22) 

It should be kept in mind that the reference frame is attached to the solid matrix. In order to solve this equation, solid-liquid fractionation for element i, the flow field 

A simple solution is arrived at for local equilibrium obeying Henry s law and constant porosity. It resembles the solution which was worked out in the advection section of Chapter 8 except that now it is expressed in terms of concentrations in the liquid 

An isopleth moves with an apparent rate v such that 

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