| Occurrence of gravity-driven fingers, due to wetting front (WF) instability, have long been observed in initially dry, homogeneous, and highly nonlinear materials, when the applied surface flux is less than the medium's saturated hydraulic conductivity. A distinguishing characteristic of such fingers is the existence of nearly saturated tips that drain at a certain distance behind and ultimately result in nonmonotonic pressure and saturation fields along the vertical extent of the fingers. This dissertation concentrates on the continuum-scale numerical modeling of gravity-driven fingers, particularly, capturing the underlying nonmonotonicity in pressure and/or saturation fields. To conduct the modeling of fingers, we first consider the ability of the Richards equation (RE) with standard monotonic constitutive relations and hysteretic equations of state. A number of recent papers purport on successfully modeling gravity-driven fingers using the RE numerical solution with a downwind averaging method. However, we find these fingers to be numerical artifacts, generated by the combined effects of a truncation error induced oscillation at the WF and false hysteretic reversals. As we remove the oscillations, either through grid refinement or the use of inherently monotone schemes, the finger-like behavior fades and the solution yields a monotonic response. When we consider the inclusions of the air-phase and/or small scale heterogeneity in the porous media, oscillation-free numerical solutions with the continuum scale models still yield monotonic responses. Thus, we conclude the RE or its two-phase variants, along with standard monotonic hydraulic properties do not contain the critical physics required to model gravity-driven fingers and are inadequate under the conditions where these fingers occur.; To extend the existing standard theory, we assume the standard forms of the constitutive relations are still valid and hypothesize a mathematical requirement for the continuum-scale simulation of gravity-driven fingers. The mathematical representation for this requirement we refer to as the hold-back-pile-up effect.; Next, we consider the hypo-diffusive form of the RE (HDRE) and illustrate its ability to simulate gravity-driven fingers. (Abstract shortened by UMI.)... |
