Modeling transport of charged species in pore networks: solution of the Nernst-Planck equations coupled with fluid flow and charge conservation equations

Abstract: A pore network modeling (PNM) framework for the simulation of transport of charged species, such as ions, in porous media is presented. It includes the Nernst-Planck (NP) equations for each charged species in the electrolytic solution in addition to a charge conservation equation which relates the species concentration to each other. Moreover, momentum and mass conservation equations are adopted and there solution allows for the calculation of the advective contribution to the transport in the NP equations. The proposed framework is developed by first deriving the numerical model equations (NMEs) corresponding to the partial differential equations (PDEs) based on several different time and space discretization schemes, which are compared to assess solutions accuracy. The derivation also considers various charge conservation scenarios, which also have pros and cons in terms of speed and accuracy. Ion transport problems in arbitrary pore networks were considered and solved using both PNM and finite element method (FEM) solvers. Comparisons showed an average deviation, in terms of ions concentration, between PNM and FEM below $5\%$ with the PNM simulations being over ${10}^{4}$ times faster than the FEM ones for a medium including about ${10}^{4}$ pores. The improved accuracy is achieved by utilizing more accurate discretization schemes for both the advective and migrative terms, adopted from the CFD literature. The NMEs were implemented within the open-source package OpenPNM based on the iterative Gummel algorithm with relaxation. This work presents a comprehensive approach to modeling charged species transport suitable for a wide range of applications from electrochemical devices to nanoparticle movement in the subsurface.