CENTER FOR COMPUTATIONAL MATHEMATICS COLLOQUIUM

                  UNIVERSITY OF COLORADO AT DENVER



TITLE:   An Analysis of the Stochastic Approaches to the 
         Problems of Flow and  Transport in Porous Media 
 

SPEAKER: David W. Dean, Department of Mathematics, UCD

DATE:    Tuesday, May 14, 1996  (PLEASE NOTE UNUSUAL DAY AND TIME)

PLACE:   Math Conference Room - Suite 540
         UCD Building, 1250 14th St., Denver

TIME:    1:00 - 2:00 pm 



ABSTRACT


One need in the current theory of subsurface transport in porous media
is an improved understanding of the basic transport physics in highly
heterogeneous subsurface environments using models that are valid at
multiple scales. The thesis begins by developing a theoretical background
for the spectral representation of stochastic processes. These representations
are then used to illustrate the more common aspects of the theoretical
descriptions of dispersion. The analysis shows how the dispersion tensor
in the homogeneous case must be modified in order to include mildly
heterogeneous permeability fields and provides a transformation law for
the conversion of the spectrum of velocity perturbations to the spectrum
of log hydraulic conductivities. This theoretical connection is important
because in Chapter II a Lagrangian approach is used to develop a
description of dispersion in terms of the covariance of the hydraulic
conductivities using a particle tracking algorithm. Chapter III
describes the numerical methods used to implement the algorithm.
Chapter IV treats the transport equation using stochastic calculus, specifically
Ito's lemma, from which weak formulations of the mean and covariance
equations can be derived. Chapter V considers the application of the theory
of stochastic evolution equations to the problem of transport.
By allowing both the dispersion and velocity to have random components,
the evolution equation can be split into deterministic and 
stochastic parts. Using semigroup methods, the solution is given in terms
of a Neumann expansion. Finally, Chapter VI uses the operator
splitting method of Section V to illustrate a stochastic finite element
method for solving the transport equation that uses the
Karhunen-Loeve expansion, the Galerkin method and the Homogeneous
Chaos spaces of Wiener.