STOCHASTIC RADON OPERATORS IN POROUS-MEDIA HYDRODYNAMICS

A space transformation approach is established to study partial differ ential equations with space-dependent coefficients modelling porous me dia hydrodynamics. The approach reduces the original multi-dimensional problem to the one-dimensional space and is developed on the basis of Radon and Hilbert operators and generalized functions. In particular, the approach involves a generalized spectral decomposition that allow s the derivation of space transformations of random field products, A Plancherel representation highlights the fact that the space transform ation of the product of random fields inherently contains integration over a ''dummy'' hyperplane. Space transformation is first examined by means of a test problem, where the results are compared with the exac t solutions obtained by a standard partial differential equation metho d. Then, exact solutions for the flow head potential in a heterogeneou s porous medium are derived. The stochastic partial differential equat ion describing three-dimensional porous media hydrodynamics is reduced into a one-dimensional integro-differential equation involving the ge neralized space transformation of the head potential, Under certain co nditions the latter can be further simplified to yield a first-order o rdinary differential equation. Space transformation solutions for the head potential are compared with local solutions in the neighborhood o f an expansion point which are derived by using finite-order Taylor se ries expansions of the hydraulic log-conductivity.