  The generalized polynomial chaos method is applied to the Buckley-Leverett
equation. We consider a spatially homogeneous domain modeled as a random field.
The problem is projected onto stochastic basis functions which yields an
extended system of partial differential equations. Analysis and numerical
methods leading to reduced computational cost are presented for the extended
system of equations.
  The accurate representation of the evolution of a discontinuous stochastic
solution over time requires a large number of stochastic basis functions.
Adaptivity of the stochastic basis to reduce computational cost is challenging
in the stochastic Galerkin setting since the change of basis affects the system
matrix itself. To achieve adaptivity without adding overhead by rewriting the
entire system of equations for every grid cell, we devise a basis reduction
method that distinguishes between locally significant and insignificant modes
without changing the actual system matrices.
  Results are presented for problems in one and two spatial dimensions, with
varying number of stochastic dimensions. We show how to obtain stochastic
velocity fields from realistic permeability fields and demonstrate the
performance of the stochastic Galerkin method with local basis reduction. The
system of conservation laws is discretized with a finite volume method and we
demonstrate numerical convergence to the reference solution obtained through
Monte Carlo sampling.
