Valery R. Roy and Stephan T. Grilli

Associate Professor Associate
Professor

Department of Mechanical Eng.
Department of Ocean Eng.

University of Delaware
University of Rhode Island

Newark, DE 19716, USA
Narragansett,
RI 02882, USA

**Abstract :
**The mathematical and numerical modeling of groundwater flows
in random porous media is studied assuming that the formation's hydraulic
log-transmissivity is a statistically homogeneous, Gaussian, random field
with given mean and covariance function. In the model, log-transmissivity
may be conditioned to take exact field values measured at a few locations.
Our method first assumes that the log-transmissivity may be expanded in
a Fourier-type series with random coefficients, known as the Karhunen-Loeve
(KL) expansion. This expansion has optimal properties and is valid for
both homogeneous and non-homogeneous fields. By combining the KL expansion
with a small parameter perturbation expansion, we transform the original
stochastic boundary value problem into a hierarchy of deterministic problems.
To the first order of perturbation, the hydraulic head is expanded on the
same set of random variables as in the KL representation of log-transmissivity.
To solve for the corresponding coefficients of this expansion, we adopt
a boundary integral formulation whose numerical solution is carried out
by using boundary elements and the dual reciprocity (DR-BEM). To illustrate
and validate our scheme, we solve three test problems and compare the numerical
solutions against Monte Carlo simulations based on a finite difference
formulation of the original flow problem. In all three cases we obtain
good quantitative agreement and the present approach is shown to provide
both a more efficient and accurate way of solving the problem.