An ELLAM approximation for advective-dispersive transport with nonlinear equilibrium and nonequilibrium sorption

We consider an Eulerian-Lagrangian localized adjoint method (ELLAM) applied to nonlinear model equations governing solute transport and sorption in porous media. Solute transport in the aqueous phase is modeled by standard advection and hydrodynamic dispersion, while two types of solid phase are distinguished --- a fraction which achieves equilibrium with the aqueous phase quickly, and another which does not. The rapidly sorbing fraction is modeled using a local equilibrium assumption, while a first-order rate expression is used for the slowly sorbing fraction. The presence of both equilibrium and non-equilibrium sorption can be challenging for Eulerian-Lagrangian methods, since information may propagate along different characteristic directions in the space-time domain. Here, we present an implementation of a finite element ELLAM (FE-ELLAM) discretization in both fully coupled and operator-split frameworks for the reactive transport model. We then evaluate our method for several test problems spanning a range of auxiliary and physical conditions and compare its performance to more standard approaches.