# Browsing by Subject "Porous materials--Mathematical models"

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Item Analysis of a Darcy-Stokes system modeling flow through vuggy porous media(2004) Lehr, Heather Lyn; Arbogast, Todd James, 1957-Show more Our goal is to accurately model flow through subsurface systems composed of vuggy porous media. A vug is a small cavity in a porous medium which is large relative to the intergranular pore size. A vuggy porous medium is a porous medium with vugs scattered throughout it. While the vugs are often small, they can have a tremendous effect on the flow of fluid through the medium. We first introduce our microscale mathematical model for flow of an incompressible, viscous fluid in vuggy porous media. Our next step is to obtain a homogenized macroscale model. In order to do so, we assume periodicity of the medium. We obtain necessary existence and uniqueness results, error estimates, and slight generalizations of two-scale convergence results for bi-modal media. First using formal homogenization and then the rigorous two-scale convergence method, we show that our microscale model homogenizes to give a much simpler modified Darcy’s law macroscale model. In this homogenized model, the permeability tensor is modified to capture the effects of the vugs on the flow through the medium. In order to compute the homogenized permeability tensor, we essentially compute our microscale system on a (much smaller) representative cell. Toward this end, we introduce two numerical methods for the microscale model. We combine a discontinuous Galerkin method with a low order RaviartThomas element and obtain suboptimal convergence rates for the first method. The second method differs only slightly from the first, but yields optimal convergence rates. Unfortunately, it is less efficient in practical implementations.Show more Item Simulating fluid flow in vuggy porous media(2005) Brunson, Dana Sue; Arbogast, Todd James, 1957-Show more We develop and analyze a mixed finite element method for the solution of an elliptic system modeling a porous medium with large cavities, called vugs. It consists of a second order elliptic (i.e., Darcy) equation on part of the domain coupled to a Stokes equation on the rest of the domain, and a slip boundary condition (due to Beavers, Joseph, and Saffman) on the interface between them. The tangential velocity is not continuous on the interface. We consider a vuggy porous medium with many small cavities throughout its extent, so the interface is not isolated. We use a certain conforming Stokes element on rectangles, slightly modified near the interface to account for the tangential discontinuity. This gives a mixed finite element method for the entire Darcy-Stokes system with a regular sparsity pattern that is easy to implement, no matter how complex the vug geometry may be. We prove optimal global first order L 2 convergence of the velocity and pressure, as well as of the velocity gradient, in the Stokes domain. Numerical results verify these rates of convergence, and even suggest somewhat better convergence in v certain situations. Finally, we present a lower dimensional space that uses Raviart-Thomas elements in the Darcy domain and uses our new modified elements near the interface in transition to the Stokes elements. We present two computational studies to illustrate and verify an homogenized macro-model of flow in a vuggy medium. And finally, we compare the effect of the Beavers-Joseph slip condition to using a no slip condition on the interface in a few simple examples.Show more Item A three dimensional finite element method and multigrid solver for a Darcy-Stokes system and applications to vuggy porous media(2007-05) San Martin Gomez, Mario, 1968-; Arbogast, Todd James, 1957-Show more A vuggy porous medium is one with many small cavities called vugs, which are interconnected in complex ways forming channels that can support high flow rates. Flow in such a medium can be modeled by combining Darcy flow in the rock matrix with Stokes flow in the vugs. We develop a finite element for the numerical solution of this problem in three dimensions, which converges at the optimal rate. We design a multigrid method to solve a saddle point linear system that comes from this discretization. The intertwining of the Darcy and Stokes subdomains in a natural vuggy medium makes the resulting matrix highly oscillating, or ill-conditioned. The velocity field we are trying to compute is also very irregular and its tangential component might be discontinuous at the Darcy-Stokes interface. This imposes a difficulty in defining intergrid transfer v operators. Our definition is based on mass conservation and the analysis of the orders of magnitude of the solution. A new smoother is developed that works well for this ill-conditioned problem. We prove that coarse grid equations at all levels are well posed saddle point systems. Our algorithm has a measured convergence factor independent of the size of the system. We then use our solver to study transport and flow properties of vuggy media by simulations. We analyze the results of our transport simulations and compare them to experimental results. We study the influence of vug geometry on the macroscopic flow properties of a three dimensional vuggy porous medium.Show more