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.