General solution to random advective-dispersive equation in porous media

This series of articles present general applications of functional-analytic theory to the solution of the partial differential equation describing solid transport in aquifers, when either the evolution of the system, the sources, the parameters and/or the boundary conditions are prescribed as stochastic processes in time or in space. This procedure does not require the restricting assumptions placed upon the current particular solutions on which today's stochastic transport theory is based, such as small randomness assumptions (perturbation techniques), Montecarlo simulations, restriction to small spatial stochasticity in the hydraulic conductivity, use of spectral analysis techniques, restriction to asymptotic steady state conditions, and restriction to variance of the concentration as the only model output among others. Functional analysis provides a rigorous tool in which the concentration stochastic properties can be predicted in a natural way based upon the known stochastic properties of the sources, the parameters and/or the boundary conditions. Thus the theory satisfies a more general modeling need by providing, if desired, a systematic global information on the sample functions, the mean, the variance, correlation functions or higher-order moments based on similar information of any size, anywhere, of the input functions. Part I of this series of articles presents the main relevant results of functional-analytic theory and individual cases of applications to the solution of distributed sources problems, with time as well as spatial stochasticity, and the solution subject to stochastic boundary conditions. It was found that the stochastically-forced equation may be a promising model for a variety of random source problems. When the differential equation is perturbed by a time and space stochastic process, the output is also a time and space stochastic process, in contrast with most of the existing solutions which ignore the temporal component. Stochastic boundary conditions seems to quickly dissipate as time increases.