Stochastic Radon operators in porous media hydrodynamics
Authors:
George Christakos and Dionissios T. Hristopulos
Journal:
Quart. Appl. Math. 55 (1997), 89-112
MSC:
Primary 76S05; Secondary 60G60, 76M35, 86A05
DOI:
https://doi.org/10.1090/qam/1433754
MathSciNet review:
MR1433754
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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.
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G. Christakos, The space transformations and their applications in systems modelling and simulation, Proc. 12th Intern. Confer, on Modelling and Simulation (AMSE) 1 (3), Athens, Greece, 1984, pp. 49–68
G. Christakos, Random Field Models in Earth Sciences, Academic Press, San Diego, CA, 1992
G. Christakos and D. T. Hristopulos, Stochastic space transformation techniques in subsurface hydrology-Part 2: Generalized spectral decompositions and Plancherel representations, Stochastic Hydrology and Hydraulics 8, no. 2, 117–138 (1994)
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M. Loeve, Probability Theory, Van Nostrand, Princeton, 1953
A. Scheidegger, Physics of Flow Through Porous Media, University of Toronto Press, Toronto, Canada, 1960
A. K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, Englewood Cliffs, NJ, 1989
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Article copyright:
© Copyright 1997
American Mathematical Society
