Stochastic Radon operators in porous media hydrodynamics

Abstract

A space transformation approach is established to study partial differential equations with space-dependent coefficients modelling porous media 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 allows the derivation of space transformations of random field products. A Plancherel representation highlights the fact that the space transformation 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 exact solutions obtained by a standard partial differential equation method. Then, exact solutions for the flow head potential in a heterogeneous porous medium are derived. The stochastic partial differential equation describing three-dimensional porous media hydrodynamics is reduced into a one-dimensional integro-differential equation involving the generalized space transformation of the head potential. Under certain conditions the latter can be further simplified to yield a first-order ordinary differential equation. Space transformation solutions for the head potential are compared with local solutions in the neighborhood of an expansion point which are derived by using finite-order Taylor series expansions of the hydraulic log-conductivity.