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.

中文翻译：

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模拟孔隙网络中带电物质的传输：Nernst-Planck 方程与流体流动和电荷守恒方程耦合的解

提出了用于模拟多孔介质中带电物质（例如离子）传输的孔隙网络建模 (PNM) 框架。除了将物种浓度相互关联的电荷守恒方程之外，它还包括电解溶液中每个带电物种的 Nernst-Planck (NP) 方程。此外，采用动量和质量守恒方程，并且该解允许计算对流对 NP 方程中输运的贡献。所提出的框架是通过首先导出对应于基于几种不同时间和空间离散化方案的偏微分方程 (PDE) 的数值模型方程 (NME) 来开发的，并进行比较以评估解决方案的准确性。推导还考虑了各种电荷守恒场景，这在速度和准确性方面也有利有弊。使用 PNM 和有限元方法 (FEM) 求解器考虑并解决了任意孔隙网络中的离子传输问题。比较显示 PNM 和 FEM 之间在离子浓度方面的平均偏差低于 $5\%$，对于包含约 ${10 的介质，PNM 模拟比 FEM 模拟快 ${10}^{4}$ 倍}^{4}$ 毛孔。通过对从 CFD 文献中采用的平流项和迁移项使用更准确的离散化方案来提高精度。NME 在开源包 OpenPNM 中实现，基于带松弛的迭代 Gummel 算法。