Influence of residual saturation on the spreading of a hump of liquid in a gas-filled stratum

Abstract

A study is made of the spreading of a hump of ground water in a stratum with allowance for partial retention of water in a previously occupied volume. The analogous problem under the condition of complete replacement of the water by gas has been considered earlier [1, 2]. This problem reduces to the Boussinesq equation, and if the hump of liquid is concentrated at the initial time in an infinitesimally small neighborhood of the symmetry axis the problem is self-similar and a solution of instantaneous-source type is obtained. This solution satisfies a relation which expresses conservation of the total mass of the liquid in the complete volume of the porous medium. The problem investigated in the present paper reduces to solution of the Boussinesq equation with a coefficient that has a discontinuity at the point where ∂h/∂t = 0 (h is the height of the hump), while the mass conservation condition for the liquid takes a nonintegrable form, since some of the liquid remains outside the hump. It is shown that the Boussinesq equation with discontinuous coefficient has an asymptotic self-similar solution of the second kind of the form h = At^−αf(r/Bt^β), the coefficients α and β being found during the process of solution and not on the basis of dimensional considerations. A numerical solution is obtained to the nonself-similarity problem, this approaching asymptotically to the self-similar at large values of the time.