The problem of the vertical one-dimensional transient gravity drainage from soils is investigated theoretically. During drainage, a non-steady state exists which is described by the three equations of motion, continuity and state. The resulting equivalent problem in mathematical terms is a non-linear parabolic partial differential equation in which the dependent variable is the pressure of the water or the volumetric water content of the soil and the independent variable are the time and the position along the vertical direction. Because of the strong non-linearity in its terms, the equation cannot be solved by analytical methods and therefore numerical integration is employed. The differentials of the governing differential equation are approximated by finite differences at discrete points in the solution domain and integrated numerically according to corresponding initial and boundary conditions. Both explicit and implicit difference schemes may be used. For a fast solution the implicit scheme should be used since larger time and integration network distance increments ean be employed. The use of the implicit scheme generates a system ofsimultaneous non-linear equations that has to be solved for each time increment. The solution of the system is obtained by matrix inversion and particularly with back substitution technique. The results of the theoretical solution are shown together with experimental data obtained from tests on a vertical column, packed uniformly with very fine sand. Comparison shows that complete agreement exists between values predicted from theory and values predicted from thcory and values obtained experimentally.
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