Counter-current imbibition and non-linear diffusion in fractured porous media
Résumé
The displacement of non-aqueous phase liquid (NAPL) in permeable porous rock by water saturating the surrounding fractures is studied. In many situations of practical interest, capillarity is the dominant driving force. Using the global pressure concept, it is shown that the water saturation is driven by a generic non-linear diffusion equation governing the matrix block counter-current spontaneous imbibition. In addition, in most cases, the saturation-dependent diffusion coefficient vanishes at the saturation end points, that renders the driving equation highly singular. In this paper, two exact asymptotic solutions valid for short and long times are presented, under the assumption that the conductivity vanishes as a power-law of both phase saturations at the extreme values of the fluid saturation. Focusing on the late-time domain, the asymptotic solution is derived using an Ansatz that is written under the form of a power-law time decay of the NAPL saturation. In the spatial domain, this solution is an eigenvector of the non-linear diffusion operator driving the saturation, for a problem with Dirichlet boundary conditions. If the diffusion coefficient varies as a power law of the NAPL saturation, the spatial variations of the solution are given analytically for a one-dimensional porous medium corresponding to parallel fracture planes. The analytical solution is in very good agreement with results of numerical simulations involving various realistic sets of input transport parameters. Generalization to the case of two- or three-dimensional matrix blocks of arbitrary shape is proposed using a similar Ansatz, solution of a non-linear eigenvalue problem. A fast converging algorithm based on a fixed-point sequence starting from a suitable first guess was developed. Comparisons with full-time simulations for several typical block geometries show an excellent agreement. These results permit to set-up an analytical formulation generalizing linear single-phase representation of matrix-to-fracture exchange term. It accounts for the non-linearity of the local flow equations using the power-law dependence of the conductivity for low NAPL saturation. The corresponding exponent can be predicted from the input conductivity parameters. Similar findings are also presented and validated numerically for two- or three-dimensional matrix blocks. That original approach paves the way to research leading to a more faithful description of matrix-to-fracture exchanges when considering a realistic fractured medium composed of a population of matrix blocks of various size and shapes.