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low and transport phenomena in porous media are of importance in a wide range of natural and industrial processes. The flow of water containing nutrients or pollutants in soils and aquifiers is of environmental concern. Of commercial relevance is the multiphase flow of oil and water in porous and fractured rock, as well as flow and dispersion in packed beds used for catalytic and separation processes.
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| Figure 1:
Mechanical dispersion is caused by random spatial variations of the velocity
field. Particles in close spatial proximity at t = 0 (o) are dispersed
at later times (*) as they flow along diverging stream lines.
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In order to model these processes, it is necessary to understand the pore scale mechanisms of flow and transport. This requires an understanding of how the structure of the porous media affects the distribution, flow, and displacement of one or more fluid phases as well as the dispersion (i.e., mixing) of one fluid in another. Recent advances in understanding these phenomena are due, in large part, to the refinement of experimental techniques such as pulsed field gradient spin-echo Nuclear Magenetic Resonance (NMR) and the development of new computational methods and technologies.
When two miscible fluids are brought into contact, a transition zone develops
across the interface, and the two fluids slowly diffuse into one another. However,
if the two fluids are also flowing, there will be some additional mixing. This
mixing is generated by a non-uniform velocity field, which may be caused by
the morphology of the medium, the flow conditions, or chemical and physical
interactions with the solid surfaces of the medium; this effect is called hydrodynamic
dispersion. The relative importance of diffusion and hydrodynamic dispersion
in spreading the transition zone is described by the Péclet number
Pe = vd/Dm, where v is the mean pore
velocity, d is the average grain size of the media, and
Dm is the relevant molecular diffusion constant. The larger the Péclet number, the greater the influence of hydrodynamic dispersion. In most random porous media, the dominant contribution to dispersion at large Péclet numbers comes from random spatial variations of the velocity field as shown in figure 1.
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| Figure 2: Color coded flow structure in a simple cubic bead pack. High-velocity flow regions are colored yellow and white. The red regions are recirculation zones.
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One approach to the analysis of pore scale flow and transport is to solve the Navier-Stokes equation and mass-transport equations in the pore space. An alternative approach has been developed by researchers at the University of Minnesota under the direction of Supercomputing Institute Fellow Regents¹ Professor H. Ted Davis of the Chemical Engineering and Materials Science Department and IT Dean, Robert Maier of the Army High Performance Computing Research Center, and Professor Daniel Kroll of the Department of Medicinal Chemistry. These researchers are working in collaboration with United States Army Corps of Engineers Waterways Experiment Station researchers Robert S. Bernard, Stacy E. Howington, and John F. Peters.
This approach simulates flow using the lattice-Boltzmann (LB) method. A discretized version of the Boltzmann kinetic equation is solved for the single particle distribution function of the fluid. A random-walk particle tracking method is then used to simulate tracer dispersion using the LB flow fields.
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| Figure 3:
Comparison of simulation results (*) for the longitudinal dispersion coefficient
with Nuclear Magnetic Resonance data (*).
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Computer codes for the LB method are relatively simple, and the method lends itself quite naturally to parallel processing. Using this approach, high-resolution studies of flow and transport in random bead packs have been performed. The velocity distribution in a cubic packing of monodisperse spheres is shown in figure 2, where it can be seen that no-slip boundary conditions at the bead surfaces results in most flow occuring in high-velocity channels. A comparison of simulation results for dispersion in random porous media with recent NMR spectroscopy studies show agreement on transient, as well as asymptotic, dispersion rates. In particular, the results support NMR findings that longitudinal dispersion rates are significantly lower than reported in earlier experimental literature and that asymptotic rates are achieved earlier than previously reported. Results for the longitudinal dispersion constant, DL, as a function of the Péclet number are compared with recent NMR data in figure 3.
Related work involving the transport of solute in packed bed liquid chromatography and electrochromatography is being performed in collaboration with Dr. Mark Schure of Rohm and Haas Company, who is currently an Industrial Fellow at the University of Minnesota's Center for Interfacial Engineering. Dr. Thomas Ihle, a Supercomputing Institute Research Scholar, is developing LB methods for treating multiphase immiscible flow and transport.
Future work will involve studies of multiphase flow. A number of issues need to be addressed, but one of the most important involves nonaqueous-aqueous mass transfer. In addition to its obvious importance in advanced oil recovery processes, it is also of great interest for environmental studies since nonaqueous liquids are a frequent source of contamination.
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