On solving the unsaturated flow equation: 2. critique of parlangeʼs method
1973
KNIGHT, J. H. | PHILIP, J. R.
The physical content of Parlangeʼs method of solving the flow equation is explored. His first approximation, which was developed by Macey (1959), satisfies continuity in the integral sense, but the second and higher approximations do not. Approximations beyond the first lack any constraining link between the separate steps of ‘satisfying continuity’ and ‘satisfying Darcyʼs law’, which make up each iteration. There is, consequently, nothing in the procedure to ensure convergence.A detailed investigation establishes the nonconvergence of the method when applied to one-dimensional sorption. It is found that the first approximation is best, and that the higher approximations make oscillations of increasing magnitude about the exact solution. Two illustrative examples are given. There is no reason to expect the procedure to be any more useful in other cases.The utility of Parlangeʼs method is thus simply the utility of the first approximation: the dependence of this on the shape of the diffusivity function and on the flux-concentration relation is discussed.Contrary to Parlangeʼs claim, the method cannot be applied to two-and three-dimensional systems other than radially symmetrical ones.
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