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Groundwater Pollution: Theory, Methodology, Modelling and by Jean J. Fried

By Jean J. Fried

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We now see that to obtain the equivalence between the two systems is not a simple problem and probably still is an interesting field of research. , which is the approximation of an infinitesimal element of macroscopic medium in the sense we just defined) are twice continuously differentiable except at some points, curves and surfaces of the porous medium, which is not consistent with the assumption that these variables verify second-order partial differential equations everywhere. Thus our purpose is to give some hints at possible developments of this field of research, showing that if the boundary is taken into account, an equation is derived, which is not the classical dispersion equation.

2), and retaining his phenomenological interpretation of the last flux in terms of dispersion coefficients, we obtain the classical dispersion equation with supplementary terms due to uo and uv: x,* 0,6(s) i s a distribution with support on S and xr* a / a v [ o o 6 ( s ) ]is the resultant of a set of doublets of moment densities -oo on the normal to S at each point of S. They represent the influence of the boundary. It seems thus that even if the boundary S between void space and solid space is physicochemically inert, it has an effect as a set of discontinuity for some physical quantities: on this set, microscopic equations do not hold and any regularizing process should take this into account.

V. But then concentration, which is a measure of the void space, is no longer differentiable everywhere, because it is discontinuous at the boundary between solid and void: indeed concentration actually measured during experiments is twice continuously differentiable at every point of the void space; it is extended by a constant in the solid and is discontinuous at the boundary (Fig. 6). I /Solid 1 "Oid Fig. 6. Concentration discontinuities in a porous medium. This set of contradictions shows what appears to be a major difficulty in the theory of dispersion: how should the influence of the boundary between solid space and void space be taken into account?

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