Advection and percolation

In a frame fixed to the matrix, the mass conservation equation for the solid and melt in a one-dimensional melting column is (McKenzie, 1984 Richter and McKenzie, 1984 Navon and Stolper, 1987) 

Assuming complete local chemical equilibrium between the solid and melt (Cj = KjCf), Eq. (13.17) may be written as 

is the rate at which a point of constant concentration for an element moves through the column, that is 

This is the fundamental transport equation for the species i, which does not depend on any assumption other than mass conservation. 

This equation is valid for a fixed reference frame, such as, using a comparison borrowed from Bird et al. (1960), counting fish in a river from a bridge. Occasionally, the fisherman wants to track the concentration in a given parcel of matter, say that he is now counting fishes from a boat carried by the stream. Let us write the total differential of C  

If concentrations have to be known at a point moving with an arbitrary velocity, the increment ratios dx/dt, dy/dt, and dz/dt can be constrained accordingly. Commonly, the velocity will be that of the medium itself (the boat is now drifting freely) and 

In the case of pure advection (no molecular transport), the diffusion term in the general transport equation (8.2.5) is made equal to zero and time-dependent mass balance is expressed as 

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Contaminants in the soil compartment are associated with the soil, water, air, and biota phases present. Transport of the contaminant, therefore, can occur within the water and air phases by advection, diffusion, or dispersion, as previously described. In addition to these processes, chemicals dissolved in soil water are transported by wicking and percolation in the unsaturated zone.26 Chemicals can be transported in soil air by a process known as barometric pumping that is caused by sporadic changes in atmospheric pressure and soil-water displacement. Relevant physical properties of the soil matrix that are useful in modeling transport of a chemical include its hydraulic conductivity and tortuosity. The dif-fusivities of the chemicals in air and water are also used for this purpose. 

Even if a chemical does not partition into an environmental medium, it may nevertheless undergo advective transport. For example, oil spilled at sea dissolves poorly in water, and most of it floats on the water s surface. The undissolved oil is transported advectively by the water, its pathways determined by wind, wave action, and ocean currents. Groundwater, too, may transport undissolved organic chemicals advectively, e.g., trichloroethylene that was spilled on the ground at an industrial site and percolated into a groundwater aquifer beneath the site. 

Figure 24 Chondrite-normalized abundances of REEs in a wall-rock harzburgite from Lherz (dotted lines— whole-rock analyses), compared with numerical experiments of ID porous melt flow, after Bodinier et al. (1990). The harzburgite samples were collected at 25-65 cm from an amphibole-pyroxenite dike. In contrast with the 0-25 cm wall-rock adjacent to the dike, they are devoid of amphibole but contain minute amounts of apatite (Woodland et al., 1996). The strong REE fractionation observed in these samples is explained by chromatographic fractionation due to diffusional exchange of the elements between peridotite minerals and advective interstitial melt (Navon and Stolper, 1987 Vasseur et al, 1991). The results are shown in (a) for variable t t ratio, where t is the duration of the infiltration process and t the time it takes for the melt to percolate throughout the percolation column (Navon and Stolper, 1987). This parameter is proportional to the average melt/rock ratio in the percolation column. In (b), the results are shown for constant f/fc but variable proportion of clinopyroxene at the scale of the studied peridotite slices (<5 cm). All model parameters may be found in Bodinier et al. (1990). As discussed in the text, this model was criticized by Nielson and Wilshire (1993). An improved version taking into account the gradual solidiflcation of melt down the wall-rock thermal gradient and the isotopic variations was recently proposed by Bodinier et al. (2003).

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