Todd Arbogast

Derivation of the double porosity model of single phase flow via homogenization theory

A general form of the double porosity model of single phase flow in a naturally fractured reservoir is derived from homogenization theory. The microscopic model consists of the usual equations describing Darcy flow in a reservoir, except that the porosity and permeability coefficients are highly discontinuous. Over the matrix domain, the coefficients are scaled by a parameter

Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences

\epsilon

A characteristics-mixed finite element method for advection-dominated transport problems

representing the size of the matrix blocks. This scaling preserves the physics of the flow in the matrix as

Implementation of a Locally Conservative Numerical Subgrid Upscaling Scheme for Two-Phase Darcy Flow

\epsilon

Mixed Finite Element Methods on Nonmatching Multiblock Grids

tends to zero. An effective macroscopic limit model is obtained that includes the usual Darcy equations in the matrix blocks and a similar equation for the fracture system that contains a term representing a source of fluid from the matrix. The convergence is shown by extracting weak limits in appropriate Hilbert spaces. A dilation operator is utilized to see the otherwise vanishing physics in the matrix blocks as

A MULTISCALE MORTAR MIXED FINITE ELEMENT METHOD

\epsilon

A Nonlinear Mixed Finite Eelement Method for a Degenerate Parabolic Equation Arising in Flow in Porous Media

tends to zero.

On the implementation of mixed methods as nonconforming methods for second-order elliptic problems

We present an expanded mixed finite element approximation of second-order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coefficient times the negative gradient). The resulting linear system is a saddle point problem. In the case of the lowest order Raviart--Thomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cell-centered finite difference method requiring the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a 9-point stencil in two dimensions and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order approximations in the

Numerical Subgrid Upscaling of Two-Phase Flow in Porous Media

L^2

The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow

- and