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.

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