Reactive transport processes in natural environments often involve many ionic species. The diffusivities of ionic species vary. Since assigning different diffusivities in the advection–diffusion equation leads to charge imbalance, a single diffusivity is usually used for all species. In this work, we apply the Nernst–Planck equation, which resolves unequal diffusivities of the species in an electroneutral manner, to model reactive transport. To demonstrate the advantages of the Nernst–Planck model, we compare the simulation results of transport under reaction-driven flow conditions using the Nernst–Planck model with those of the commonly used single-diffusivity model. All simulations are also compared to well-defined experiments on the scale of centimeters. Our results show that the Nernst–Planck model is valid and particularly relevant for modeling reactive transport processes with an intricate interplay among diffusion, reaction, electromigration, and density-driven convection.

Reactive transport processes are fundamental in large-scale subsurface applications such as geological hydrogen storage

Modeling electrochemical processes using the Nernst–Planck equation is usually coupled with the Poisson equation for electrical potential, also known as the Poisson–Nernst–Planck (PNP) model. The PNP model has been applied in various electrochemical applications

The above benchmark and validation studies focused on reactions and the mixing of fluids, which are subject to flow introduced by fluid pressure or electric fields. In this work, we propose another set of benchmark problems for numerical models, based on chemically driven convection experiments

In addition, we present simulations of convective dissolution of

In the following sections, we first establish the fundamental equations of the conservation laws and the accompanying numerical methods. We then provide details concerning the two reaction-driven flow experiments and a

In this section, we delineate the fundamental equations describing the reactive transport of aqueous species.

We consider a single-phase fluid composed of

We use standard Fickian diffusion to model the diffusive flux,

The type of experiments of interest for this work is performed in a Hele-Shaw cell, for which fluid flow can be described by Darcy's law,

When the components in the fluids are electrically charged, for example, the ions

The PNP model aims at resolving both the electric potential and the molar concentrations subject to the boundary conditions of the electric potential. The mathematical properties of the PNP model, with no external flow

In this work, we enforce the null current condition

Combining Eqs. (

Another aspect of mass balance amongst the fluid components, which we now denote as the chemical species, is the reaction that changes the amount of the chemical species while conserving the mass balance of chemical elements. The reaction source or sink term in the mass balance of each fluid component,

We use Reaktoro to solve the chemical equilibrium problem. The numerical methods and implementations of the Gibbs energy minimization problem are detailed in

This section elaborates on the numerical schemes we use for solving the fundamental equations of the reactive transport processes.

We utilize the mixed finite-element formulation to solve for fluid velocity and fluid pressure under the constraints of mass balance, Eq. (

In the previous section, we selected an LBB-compatible velocity and pressure pair. The basis function of pressure is a piecewise constant, which can be described as the average pressure of a certain cell volume in a given mesh. Therefore, it is straightforward to define the molar concentrations as piecewise constants. We use the finite-volume method to discretize the transport of the fluid components, Eq. (

With regard to the time-stepping schemes, we use an explicit scheme for the upwind advection term and the Crank–Nicolson scheme for the diffusion and Nernst–Planck terms. The time step size

Of many coupling approaches in reactive transport modeling

Due to the simplicity and effectiveness in applying the SNIA coupling between the transport solvers and the chemical equilibrium codes, this coupling method is widely adopted in the following software: OpenGeoSys

Flowchart of the simulation procedures that couple flow, transport, and chemical equilibrium using the sequential non-iterative approach (SNIA).

We select three experiments, performed (by others) in a Hele-Shaw cell, to evaluate the effectiveness of the Nernst–Planck model in reactive transport situations. The diffusivities of the aqueous species are listed in Table

Diffusivities of aqueous species in 25

The diffusivities of aqueous species other than

Dynamic viscosity and density of aqueous solutions and water at 25

Linearly interpolated using

The chemical compositions of the aqueous solutions. The chemical compositions are in units of mol L

For the second modeling case, we choose a similar experiment with a strikingly different instability process. In their chemically driven convection experiments,

Numerical studies of

Using lattice Boltzmann methods with consideration of electrostatic forces,

In the third case of our numerical study, we model the experiment of

The selected experimental setups. On the left, the chemically driven convection experiments with 1 M

This section compares the simulations and the experimental results of the chemically driven convection problems and the convective dissolution of

We present the results of the chemically driven convection between 1 M

Figure

Comparing the simulations using the Nernst–Planck model and those using the single-diffusivity model with the experiments of chemically driven convection in

Comparing the simulation results using the Nernst–Planck model with the experimental results of chemically driven convection in

Figure

In Fig.

Comparing the simulations using the Nernst–Planck model and those using the single-diffusivity model with the experiments of chemically driven convection in

Comparing the simulations using the Nernst–Planck model and those using the single-diffusivity model in modeling chemically driven convection of

Figure

Figure

To further quantify the simulation results, Fig.

Comparing the simulations using the Nernst–Planck model and those using the single-diffusivity model with the experiments of convective dissolution of

Comparing the simulations using the Nernst–Planck model and those using the single-diffusivity model, while numerically modeling the convective dissolution of

Comparing the simulations using the Nernst–Planck model and those using the single-diffusivity model, while numerically modeling convective dissolution of

In this section, we discuss whether the Nernst–Planck model is valid, and its differences are compared to the commonly used single-diffusivity model.

Comparing to two chemically driven convection experiments conducted by

In the 1.5 M

In the case of

Although the comparison between the experiment and the simulations of the two models in Fig.

To summarize, we use the experiments conducted by

Using the simulations of chemically driven convection in 1 M

Comparing the simulations using the Nernst–Planck model and those using the single-diffusivity model in modeling chemically driven convection in 1 M

Comparing the simulations using the Nernst–Planck model and those using the single-diffusivity model in modeling convective dissolution of

Simulation results using the Nernst–Planck model, showing the molar concentration of

Moreover, the only reaction that occurs in this experiment is the neutralization reaction

To further demonstrate the electromigration effects of the inert species, Fig.

Focusing on the density plots in Fig.

By comparing the single-diffusivity model and the Nernst–Planck model with reaction-driven flow experiments, we have demonstrated the importance and necessity under certain conditions of modeling the electromigration effects between charged species in aqueous environments using the Nernst–Planck model. Our results of simulating the reaction-driven flow experiments and convective dissolution of

By comparing our numerical modeling results to previously published flow experiments, we have shown that the Nernst–Planck model is valid for modeling reactive transport processes. The processes in the experiments considered here are characterized by an intricate interplay between diffusion, reaction, electromigration, and density-driven convection.

Compared to the often-used single-diffusivity model, the Nernst–Planck model enables the numerical modeling of the electromigration of ionic species introduced by differing species diffusivities, resulting in more numerous and improved physical insights.

These reaction-driven flow experiments from literature can be further utilized for benchmarking reactive transport codes.

The RetroPy code is available on GitHub (

We provide the simulation results of Figs.

PWH: Conceptualization, Methodology, Software, Formal analysis, Writing – Original Draft, Writing – Review & Editing, Visualization. BF: Writing – Review & Editing. CZQ: Writing – Review & Editing. MOS: Supervision, Writing – Review & Editing. AE: Conceptualization, Writing – Review & Editing, Project administration, Funding acquisition.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank the Werner Siemens-Stiftung (Werner Siemens Foundation) for its support of the Geothermal Energy and Geofluids (

We would like to thank our colleagues, Xiang-Zhao Kong and Isamu Naets, for all the helpful discussions. Furthermore, this work could not have been done without the visualization tool Matplotlib

This research has been supported by the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung project entitled “Analysing spatial scaling effects in mineral reaction rates in porous media with a hybrid numerical model” (grant no. 175673).

This paper was edited by Sylwester Arabas and reviewed by Lucjan Sapa and one anonymous referee.