Probabilistic characterization of transport in heterogeneous media

The mechanics of transport and flow in a random porous medium are addressed paper. The hydraulic properties of the porous medium are modeled as spatial random processes. The random aspect of the problem is treated by introducing a new dimension along which spectral approximations are implemented. Thus, the hydraulic processes are discretized using the spectral Karhunen-Loeve expansion. This expansion represents the random spatial functions as deterministic modes of fluctuation with random amplitudes. These amplitudes form a basis in the manifold associated with the random processes. The concentrations over the whole domain are also random processes, with unknown probabilistic structure. These processes are represented using the Polynomial Chaos basis. This is a basis in the functional space described by all second order random variables. The deterministic coefficients in this expansion are calculated via a weighted residual procedure with respect to the random measure and the inner product specified by the expectation operator. Once the spatio-temporal variation of the concentrations has been specified in terms of the Polynomial Chaos expansion, individual realizations can be readily computed.