CDF Solutions of Buckley-Leverett Equation with Uncertain Parameters

The Buckley–Leverett (nonlinear advection) equation is often used to describe twophase flow in porous media. We develop a new probabilistic method to quantify parametric uncertainty in the Buckley–Leverett model. Our approach is based on the concept of a fine-grained cumulative density function (CDF) and provides a full statistical description of the system states. Hence, it enables one to obtain not only average system response but also the probability of rare events, which is critical for risk assessment. We obtain a closed-form, semianalytical solution for the CDF of the state variable (fluid saturation) and test it against the results from Monte Carlo simulations.