Dr. Lea Jenkins
Dept. of Mathematical Sciences
Dr. Christopher Kees
Coastal and Hydraulics Laboratory
US Army Engineer Research and Development Center
The modeling equations for saturated flow through porous media have been studied for over a century. Henri Darcy developed the fundamental equations for saturated flow through soils in the mid 1800’s, in which he derived a proportionality relationship between the velocity of the fluid through the medium and the pressure gradient. The proportionality constant was dependent on the type of soil; it is higher for sandy soils, and lower for clay soils.
Historically, the study of porous media flow has been connected to problems in the subsurface. Members of the groundwater and petroleum engineering communities have been particularly involved in the development of model equations and simulation tools, as tracking of a variety of fluids, including oil, water, and contaminants, has always been of interest. More recently, however, the study of porous media flows has become connected to industrial problems. Filtration processes are ubiquitous; they appear in cars, industrial processing, our bodies, pharmaceutical manufacturing, and occupational safety equipment, just to name a few. The filter is a porous media, and the flow through the filter is governed by the media, the fluid type, and the requirements associated with the underlying process.
A more accurate representation of the flow through a porous media would be realized if one used the Navier-Stokes equations to model the flow through the porous network. This is a difficult undertaking, as one must know what happens inside the porous medium on the microscale. This requires information about the pore sizes throughout the medium, the connectivity of pore channels, and the tortuosity of these channels. This level of detail can be impossible to acquire for a relatively small porous medium, much less a larger area of study. In addition, the microscopic description of the medium can change over time, as happens when filters are used to capture debris particles. The size of the system that would need to be solved, requiring 3 velocity unknowns and a pressure unknown, also adds to the complexity of the problem.
Darcy’s equation is based on an averaged description of the flow through a given porous medium, and is only valid for relatively slow velocities. An example where Darcy’s flow would likely not be valid is in the pharmaceutical industry, where membrane filters are used to separate proteins during the drug purification process. The industry requires high-volume throughput, which means that the unprocessed fluids are pushed through the filtration membrane at a high velocity. We are interested in developing simulation tools that can guide the design of efficient membranes, but development of an accurate simulation tool requires accurate modeling equations.
In this workshop, we will study effective models for convective flows in porous media, which will allow us to handle the high velocity throughput associated with protein separations. We will consider the appropriateness of existing model equations for convective flow, and we will develop and use simulation tools to compare the results of a variety of flow equations with existing data. We will make recommendations for generalized models of porous media flows, which will allow one to solve relatively large (in scale) porous media problems at the macroscale.
Donald A. Nield and Adrian Bejan, Convection in porous media.
M. Peszynska, A. Trykozko, andK. Kennedy, Sensitivity to anisotropy in non-Darcy flow model from porescale through mesoscale, Proceedings of CMWR 2010.
Luc Tartar, The General Theory of Homogenization: A Personalized Introduction.
A. Bourgeat, E. Marusic-Paloka, and A. Mikelic, Weak nonlinear corrections for Darcy’s law, Mathematical Models and Methods in Applied Sciences, 8(6), 1996, pp. 1-13.
Oleg Iliev, Andro Mikelic, and Peter Popov, On Upscaling Certain Flows in Deformable Porous Media, Simulation , 7(1), 2008, pp. 93-123.
L. C. Evans, Chapter 4.5.4, Homogenenization Partial Differential Equations.
A. Bensoussan, J. Lions, and G. Papanicolaou, “Asymptotic analysis for periodic structures“, Stud. Math. Appl. 5, North-Holland, Amsterdam, 1978
P. Forchheimer, “Wasserbewegung durch boden“, Z. Ver. Deutsch. Ing., 45, 1901, pp. 1782-1788
H. Darcy, “Les fontaines publique de la ville de Dijon,” Librairie des Corps Imperiaux des Ponts et Chaussees et des Mines, Paris, 1856.