A Free Boundary Problem Describing the Saturated-unsaturated Flow in a Porous Medium. Part Ii. Existence of the Free Boundary in the 3d Case

During a rainfall water infiltration into an unsaturated soil, zones of saturation may be developed anywhere within the flow domain. Consequently, a natural question arises: under which conditions depending on the rate at which rain water is supplied, the initial moisture distribution in the soil, the presence of underground sources and the boundary permeability, the flow domain may be separated into two parts, one saturated and the other unsaturated? Related to that, when saturation may be observed first at the ground surface? If this happens, then beginning with the time at which the soil surface reaches saturation, the so-called saturation time, a waterfront starts to move downwards and this represents the unknown interface between the saturated and unsaturated flow regimes. Situations under question have been experimentally put in evidence and various studies focusing especially on the determination of an approximate analytical solution have been done in the 1D case, for a special hydraulic model introduced in [6]. Besides it, we may cite, for example, [5, 7]. In a recent paper dealing with the study of a rainfall infiltration problem (see [4]) the main feature of the model focuses on a switching boundary condition on the ground surface, that is, changing the type at the moment when this one reaches saturation. In this paper, we present a functional approach to a rainfall infiltration, find sufficient conditions under which the saturation may be first generated at the soil surface and prove in fact the existence of the free boundary in the 3D case. We mention that the hydraulic model for which we develop the theory obeys the particularities of the most used hydraulic models in soil sciences, covering a wide range of soil types, that is, those of Broadbridge-White (see [6]) and van Genuchten (see [10]). The paper extends in fact a previous study on this subject, that is [9] which was conceived in four main parts.