Solving the Nernst-Planck Equation in Heterogeneous Porous Media With Finite Volume Methods: Averaging Approaches at Interfaces

Molecular diffusion of dissolved species is a fundamental mass transport process affecting many environmental and technical processes. Whereas diffusive transport of single tracers can be described by Fick's law, a multicomponent approach based on the Nernst-Planck equation is required for charge-coupled transport of ions. The numerical solution of the Nernst-Planck equation requires special attention with regard to properties that are required at interfaces of numerical cells when using a finite difference or finite volume method. Weighted arithmetic and harmonic averages are used in most codes that can solve the Nernst-Planck equation. This way of averaging is correct for diffusion coefficients but inappropriate for solute concentrations at interfaces. This averaging approach leads to charge balance problems and thus to numerical instabilities near interfaces separating grid volumes with contrasting properties. We argue that a logarithmic-differential average should be used. Here this result is generalized, and it is demonstrated that it generally leads to improved numerical stability and accuracy of concentrations computed near material interfaces. It is particularly relevant when modeling semipermeable clay membranes or membranes used in water treatment processes.