Multiscale Discretizations for Flow, Transport and Mechanics in Porous Media

Mary Fanett Wheeler

Center for Subsurface Modeling,
Institute for Computational Engineering and Sciences,
The University of Texas at Austin

A fundamental difficulty in understanding and predicting large-scale
fluid movements in porous media is that these movements depend upon
phenomena occuring on small scales in space and/or time.  The
differences in scale can be staggering.  Aquifers and reservoirs
extend for thousands of meters, while their transport properties can
vary across centimeters, reflecting the depositional and diagenetic
processes that formed the rocks.  In turn, transport properties depend
on the distribution, correlation and connectivity of micron sized
geometric features such as pore throats, and on molecular chemical
reactions.  Seepage and even pumped velocities can be extremely small
compared to the rates of phase changes and chemical
reactions. The  coupling of flow simulation with mechanical
deformations is also important in addressing the response of
reservoirs located in structurally weak geologic formations.

We will focus on the mortar mixed finite element method (MMFE) which
was first introduced by Arbogast, Cowsar, Wheeler, and Yotov for
single phase Darcy flow and later extended to multiphase flow by Lu,
Pesyznska, Wheeler, and Yotov for multiphase flow. The MMFE method is
quite general in that it allows for non-matching interfaces and the
coupling of different physical processes in a single simulation.  This
is achieved by decomposing the physical domain into a series of
subdomains (blocks) axnd using independently constructed numerical
grids and possibly different discretization techniques in each
block. Physically meaningful matching conditions are imposed on block
interfaces in a numerically stable and accurate way using mortar
finite element spaces.  The mortar approach can be viewed as a subgrid
or two scale approach.  Moreover, the use of mortars allows one to
couple MFE and discontinuous Galerkin approximations in adjacent
subdomains.  In this presentation we will discuss extensions to
couplings with reactive transport and elasticity.  Both theoretical a priori
and a posteriori results and computational results will be presented.