The Buckley-Leverett Equation with Spatially Stochastic Flux Function

When the reservoir parameters are stochastic, then the flow in a reservoir is described by stochastic partial differential equations. Spatial stochastic relative permeability in one spatial dimension is modeled by the stochastic Buckley–Leverett equation s(x, t)t + f(s(x, t), x)x = 0 for x > 0 and t > 0. f is the stochastic flux function and s is the saturation. This equation is analyzed, and it is proved that the solution of this equation with Riemann initial data converges to the solution of s(x, t)t + f̄(s(x, t))x = 0, where f̄(s) is the spatial average of f(s, x) when f(s, x) varies randomly with position.