STOCHASTIC PROPERTIES OF THE SCALAR BUCKLEY-LEVERETT EQUATION

The Riemann problem for the one-dimensional, scalar Buckley-Leverett equation u(x, t)t + f(u(x, t))x = 0, u(x, 0) = u(l) if x less-than-or-equal-to 0, u(x, 0) = u(r) if x > 0 with a stochastic flux function f or stochastic initial data u(l), u(r) is analyzed. More precisely, this conservation law when f is measured at certain points u1,...,u(N) with a piecewise linear interpolation between the points u(i) is considered, and the expectation value of the solution u under certain assumptions on f is computed explicitly. The case in which the relative permeabilities are given by a power law, viz. k(u) = u(a), where a is considered f is a known continuous function is studied. The Buckley-Leverett equation is commonly used to model, e.g., two-phase flow in a porous medium.