### TITLE: Probabilistic Analysis of Flow in Random Porous Media by
Stochastic Boundary Elements

### AUTHOR: Roy-R-Valery. Grilli-S

### SOURCE: Engineering Analysis with
Boundary Elements (in review)

### PUBLICATION: JA (Journal Article).

### 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.