Multipoint Distribution of Saturation for Stochastic Nonlinear Two-Phase Transport

Abstract

We analyze the problem of uncertainty propagation for nonlinear two-phase transport in heterogeneous porous media. Specifically, we study the evolution of the saturation field associated with nonlinear immiscible two-phase transport (i.e., Buckley--Leverett problem) in the presence of a stochastic velocity field. The uncertainty in the velocity field is due to the limited information that is usually available about the heterogeneous porosity and permeability fields of the particular subsurface formation of interest. The uncertainty in the total-velocity field leads to uncertainty in the saturation of the injected fluid phase, both in space and time. Given information about the spatial statistics of the correlated heterogeneity, we derive the multipoint cumulative distribution functions (CDF) of saturation. The methodology takes full account of the nonlinear hyperbolic nature of the species conservation law. To obtain the multipoint CDF, we first derive the partial differential equation (PDE) of the “raw” ...