Modeling Preferential Flows in Porous Media

John Nieber, Professor
Aleksey Sheshukov, Research Associate
Andrey Egorov, Chief, Division of Mechanics, Kazan State University, Russia
Rafail Dautov, Professor, Kazan State University, Russia
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, macropore flows, and heterogeneity-driven flows. We also wish to develop tools from this research that will enable practitioners to readily apply preferential flow predictions tools.

Need or Impact

The preferential flow of water in soils and other porous materials leads to the rapid movement of harmful chemicals and microbial organisms. This research is directed to provide an improved understanding about the behavior of preferential flow and transport processes and thereby help to reduce their detrimental effects.

Project Status

We have continued to work on the modeling of unstable flows (leading to fingered flow) in unsaturated soils. During the past year, we have continued to work on the parameterization of models for unstable flow. In particular, we have worked on the parameterization of the non-equilibrium model that is referred to as the relaxation model; it is essentially a first-order kinetic model that relates the rate of change of water saturation with the difference between the equilibrium capillary pressure and the dynamic capillary pressure. The specific relaxation model we have examined is called the s-model. During the next year, we will also investigate the model called the p-model, and also we might be able to look at a mixed model that has features of the s-model and the p-model.

The instability that is observed in gravity-driven flows results from a pore-scale process, and this pore-scale effect is captured in the macroscopic governing equations such as the non-equilibrium capillary pressure – saturation model described in the previous paragraph. There is a perceived need to be able to develop some models that will describe these pore-scale processes that will then lead to the macroscopic observations. To meet this need, we have begun developing a pore-scale model that describes the equilibrium capillary pressure – saturation relation after upscaling to the macroscopic scale. This equilibrium model will be used as the basis for a non-equilibrium model.

As a result of complex earth surface processes, soils and deep vadose zones tend to have very complex features in terms of heterogeneity of the porous media properties. Many observations have shown that deep vadose zones can be described as being composed of a largely homogeneous material containing embedded inhomogeneities. An example of an embedded inhomogeneity is a coarse sand layer contained in a bed of fine sand. Because of the complexity of these systems it is difficult to derive finite difference or finite element solutions to flows in two dimensions or in three dimensions. We have proposed that an analytic element solution is suitable to handle the complex geometry and size of such domains. As a result, we have derived analytic element solutions for steady-state two-dimensional vadose zone flows containing circular and elliptical inhomogeneities. These inhomogeneities can be distributed randomly in space with the limitation that they cannot overlap. Analytic element solutions were found to be accurate up to machine precision. We are currently comparing the analytic element solutions to finite element solutions to assess the limitations of the analytic element solutions, and to weigh the advantages/disadvantages of the analytic element solutions relative to the finite element solutions.