A mathematical analog of immiscible multiphase well flow, considering three compressible fluids—two liquids and one gas—is solved with fully implicit finite differences. A Newton iteration scheme is utilized to solve the system of nonlinear difference equations. The method is applied to free surface gravity well flow, including the effect of partial penetration. The importance of capillarity, of air dissolved in water, of water compressibility, as well as the effect of the multiphase flow approach upon the shape of the free surface, are analyzed.
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Brutsaert et al. (1971) studied this question.