Method of model reduction and multifidelity models for solute transport in random layered porous media

This work presents a method of model reduction that leads to models with three solutions of increasing fidelity (multifidelity models) for solute transport in a bounded layered porous media with random permeability. The model generalizes the Taylor-Aris dispersion theory to stochastic transport in random layered porous media with a known velocity covariance function. In the reduced model, we represent (random) concentration in terms of its cross-sectional average and a variation function. We derive a one-dimensional stochastic advection-dispersion-type equation for the average concentration and a stochastic Poisson equation for the variation function, as well as expressions for the effective velocity and dispersion coefficient. In contrast to the linear scaling with the correlation length and the mean velocity from macrodispersion theory, our model predicts a nonlinear and a quadratic dependence of the effective dispersion on the correlation length and the mean velocity, respectively. We observe that velocity fluctuations enhance dispersion in a nonmonotonic fashion (a stochastic spike phenomenon): The dispersion initially increases with correlation length λ, reaches a maximum, and decreases to zero at infinity (correlation). Maximum enhancement in dispersion can be obtained at a correlation length about 0.25 the size of the porous media perpendicular to flow. This information can be useful for engineering such random layered porous media. Numerical simulations are implemented to compare solutions with varying fidelity.