A numerical study of fronts in random media using a reactive solute transport model

We simulate the random front solutions of a nonlinear solute transport equation with spatial random coefficients modeling inhomogeneous sorption sites in porous media. The nonlinear sorption function is chosen to be Langmuir type, and the random coefficients are two independent stationary processes with fast decay of correlations. The model equation is in conservation form, and the random fronts are similar to random viscous shocks. We find that the average front speed is given by an ensemble averaged explicit Rankine–Hugoniot relation, and the front position fluctuates about its mean. Our numerical calculations show that the standard deviation is of the order O( $sqrt t $ ) for large time, and the front fluctuation scaled by $sqrt t $ converges to a Gaussian random variable wih mean zero. We come up with a formal theory of front fluctuation, yielding an explicit expression of the root t normalized front standard deviation in terms of the random media statistics. The theory agrees remarkably with the numerically discovered empirical formula.