Stochastic Modeling of Immiscible Flow with Moment Equations

Tuesday, January 8, 2002 - 2:25pm - 2:50pm
Keller 3-180
Thomas Russell (University of Colorado Denver)
(joint work with Kenneth D. Jarman, Pacific Northwest National Laboratory)

We study a model of two-phase oil-water flow in a heterogeneous reservoir, and present a direct method of obtaining statistical moments. The method is developed as an approach either to scale-up, or to uncertainty propagation, for a general class of nonlinear hyperbolic equations. Second-order moment differential equations are derived using a perturbation expansion in the standard deviation of an underlying random process, which in this application is log permeability. The perturbation approach is taken because test results do not support the use of a multivariate Gaussian assumption to close the system. Moments may depend on location; the common assumption of statistical homogeneity is not necessary.

Classification of the resulting coupled system of nonlinear equations will be discussed. In one space dimension, the system is hyperbolic, and the analytical solution exhibits a bimodal character. The theory does not extend to 2D, but qualitative numerical results are similar. These will be compared to the results of Monte Carlo simulations, which are smoother and shock-free. Moment equations can yield approximate statistical information considerably more efficiently.