2002 Annual Report
Research
Modeling Preferential Flows in Porous Media
John Nieber, Professor
Aleksey Sheshukov, Visiting Assistant Professor
Andrey Egorov, Chief, Division of Mechanics, Kazan State University, Russia
Rafail Dautov, Professor, Kazan State University, Russia
Tammo Steenhuis, Professor, Cornell University
Yves Parlange, Professor, Cornell University
Coen Ritsema, SC-DLO, Wageningen University, The Netherlands
Reinder Feddes, Department of Water Resources, Wageningen University,
The Netherlands
Ge van den Eertwegh, Hooheemraadschap van Rijnland, The Netherlands
Funding Source
Minnesota Agricultural Experiment Station
Objective
To investigate new forms of equations governing the processes of preferential
flows in porous media, including fingering and macropore flows.
Project Description
Many government agencies have a major focus on protection of the environment
and remediation of past environmental damage. One area of interest is
fluid flow and contaminant transport in the unsaturated zone and in groundwater
aquifers. Current efforts are focused on examining extended flux laws
and assessing how the use of these extended laws might affect the character
of flows.
Results
Studies of the stability of the Richards equation (RE) have been published
by several authors in the literature, with different conclusion among
them. In general, most of the studies have indicated that the RE can be
unstable for certain conditions. During this past year, linear stability
analyses and nonlinear stability analyses of the RE were performed. The
initial nonlinear stability analysis was for the relatively simpler case
of homogeneous porous media. The conclusions were similar to those reported
earlier in the mathematical literature, that the RE type of equation is
unconditionally stable. This nonlinear stability analysis was then extended
to the case of heterogeneous porous media to come up with a very strong
statement about the stability of the RE. The results of this extended
nonlinear stability analysis confirmed the results of the first nonlinear
stability analysis, that is, the RE is unconditionally stable.
To determine what forms of equations might lead to conditionally stable
conditions during gravity-driven flow, linear stability analysis was performed
on the RE, and two alternative mass balance equations, a Sharp Front Richards
Equation (SFRE) and a Relaxation Richards Equation (RRE). The stability
analyses were performed through the perturbed traveling wave solution
for each of these mass balance equations. While these linear stability
analyses do not lead to as strong a conclusion as the nonlinear stability
analysis, the linear stability analysis was used for these equations because
at present the SFRE and the RRE are not tractable to the nonlinear stability
analysis.
The results of the linear stability analysis showed that the RE is unconditionally
stable, the SFRE is unconditionally unstable, and the RRE is conditionally
stable. Because the geometry of the flow field in experiments of gravity-driven
unstable flow indicate that the instabilities are controlled by a conditional
stability, the RRE model is considered to be the only equation capable
of simulating gravity-driven unstable flows, at least among those studied
thus far. The results showed that the dynamic CPS is necessary to produce
instabilities in the flow, and that hysteresis in the equilibrium CPS
is necessary for the persistence of fingers that form from the instability.
A numerical solution to the RRE equation for two-dimensional flow was
developed and applied to investigate the effect of boundary conditions
on the formation of fingers in unstable flows. This solution accounts
for hysteresis in the equilibrium CPS. The model has also been applied
successfully to the simulation of flow instability for water repellent
soils.