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 ) = uₗ if x 0,u( x,0 ) = uᵣ if $x > 0$ with a stochastic flux function f or stochastic initial data u₁ ,uᵣ is analyzed. More precisely, this conservation law when f is measured at certain points u₁ , ⋯ ,uN with a piecewise linear interpolation between the points uᵢ 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 ) = uᵃ, where a is considered to be stochastic, is also analyzed. Thirdly, the case where the initial data are random, but the flux function 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.