Dynamics of wetting fronts in porous media

We propose a phenomenological approach for describing the dynamics of wetting front propagation in porous media. Unlike traditional models, the proposed approach is based on the dynamic nature of the relation between capillary pressure and medium saturation. We choose a modified phase-field model of solidification as a particular case of such a dynamic relation. We show that in the traveling wave regime the results obtained from our approach reproduce those derived from the standard model of flow in porous media. In a more general case, the proposed approach reveals the dependence of front dynamics upon the flow regime. @S1063-651X~98!51411-0# PACS number~s!: 47.55.Mh, 47.55.Kf, 68.45.Gd, 87.22.As The dynamics of fluids in porous media has been a subject of numerous theoretical and experimental studies because of its importance for engineering and environmental applications @1‐3#. Among the most challenging problems in this area is modeling fluid flow through a partially saturated medium, in particular the propagation of wetting ~or drying! fronts. This has been addressed on both ‘‘microscopic’’ ~or pore-scale! and ‘‘macroscopic’’ ~on scales larger than poresize! levels. The fluid dynamics in the pore networks has been studied by numerous authors both theoretically @4# and experimentally @5,6#. These studies on the microscopic scale provide valuable insight into the underlying mechanisms of fluid transport in porous media. A microscopic description requires the detailed information of the pore structure and pore-size distribution. When this information is not available, as often happens for large domains, one has to rely upon a macroscopic description. This description is useful for determining such integral characteristics as the width and propagation velocity of moving wetting fronts, the influence of its curvature on the dynamics, etc. The subject of the present work is the dynamics of wetting fronts in porous media when a liquid phase ~water! displaces air. A straightforward description of this process consists of treating wetting fronts as sharp interfaces, which separates completely wet and dry regions @5,7#. However, in many realistic cases the structure of the transitional zone ~wherein water saturation varies gradually! cannot be neglected. One can either describe the dynamics of the liquid and air phases separately, or noting that air pressure is close to atmospheric, consider only the dynamics of the liquid phase. These two approaches generally produce similar results ~Ref. @8# ,p . 213 !. The latter approach seems to be more attractive due to its simplicity. Within the framework of the chosen approach water flow is described by Darcy’s law which, similarly to Ohm’s law for electric current, stipulates the proportionality between the flux and gradient of a potential ~see, e.g., @7#!. For flow through porous media a water pressure, averaged over many pore sizes, plays the role of the potential. Richards @9# has empirically generalized Darcy’s law onto flow in partially saturated porous media ~PSPM! by letting the proportionality coefficient depend on saturation u of the medium ~0,u,1!. Coupled with the mass conservation law, the generalized Darcy’s law constitutes the existing macroscopic description of the fluid dynamics in PSPM. To complete this description, one needs to specify a relation between u and capillary pressure ~normalized by the product of fluid density r and gravitational acceleration g) c. Traditionally this relation is assumed to be algebraic @1,10#. However, this contradicts the numerous experimental evidences revealing the dependence of the u-c relation upon the conditions of the experiment. In particular, this relation exhibits hysteresis for wetting and drying of a medium @1,10,11#. In the present Rapid Communication we propose a phenomenological approach to describe the propagation of wetting fronts in porous media, which is based on a dynamic u-c relation. Our description consists of two dynamic equations for u and c coupled by nonlinear sources. This implies that the traditional algebraic u-c correspondence is replaced by a nonlocal ~integro-differential! relationship. The proposed model is an application of the well-known ‘‘phasefield’’ approach used to describe solidification, electrodeposition, and other physical problems @12‐17#. We show that, under certain conditions, dynamics of the wetting front in our approach reproduce that in the traditional approach. The same is true for the pressure profiles associated with the wetting fronts obtained from both models. We demonstrate that, under different conditions, this equivalence breaks down, with our model revealing a dynamic nature of the u-c relation. The generalized Darcy’s law for flux q through PSPM has a form q52K(u)i(c2x3), where K is the saturationdependent conductivity of the medium and x 3 is the vertical coordinate ~positive downward! that stands for the gravitational component of pressure ~normalized by rg). Here c is measured relative to the ambient atmospheric pressure, so that c,0 for the partially saturated and c>0 for the fully