The solution character of capillary retention in porous media at steady state

The saturation solution to the capillary retention phenomenon is critical to understand how much and what type of phase is retained by a heterogeneous medium. Hence, this paper studies the solution qualitatively using methods from the qualitative theory of differential equations. In particular, when the capillary function, defined as the square root of the ratio of permeability to porosity, is differentiable and monotonic is examined. The analysis shows that the saturation solution is bell shaped with a local minimum or maximum point depending on the Leverett curve and the derivative of the capillary function. The case with a discontinuous capillary function is identified here as the limiting case when the regularisation parameter approaches zero. Analytical solution of the linearised differential equation enables the derivation of a parameter that characterises the local flow regime of the solution. The parameter is a function of the capillary number, the viscous limit fractional flow and the diffusivity functions. The results of the analysis are validated and demonstrated numerically.