Simulation of mineral dissolution at the pore scale with evolving fluid-solid interfaces: review of approaches and benchmark problem set

Abstract

This manuscript presents a benchmark problem for the simulation of single-phase flow, reactive transport, and solid geometry evolution at the pore scale. The problem is organized in three parts that focus on specific aspects: flow and reactive transport (part I), dissolution-driven geometry evolution in two dimensions (part II), and an experimental validation of three-dimensional dissolution-driven geometry evolution (part III). Five codes are used to obtain the solution to this benchmark problem, including Chombo-Crunch, OpenFOAM-DBS, a lattice Boltzman code, Vortex, and dissolFoam. These codes cover a good portion of the wide range of approaches typically employed for solving pore-scale problems in the literature, including discretization methods, characterization of the fluid-solid interfaces, and methods to move these interfaces as a result of fluid-solid reactions. A short review of these approaches is given in relation to selected published studies. Results from the simulations performed by the five codes show remarkable agreement both quantitatively—based on upscaled parameters such as surface area, solid volume, and effective reaction rate—and qualitatively—based on comparisons of shape evolution. This outcome is especially notable given the disparity of approaches used by the codes. Therefore, these results establish a strong benchmark for the validation and testing of pore-scale codes developed for the simulation of flow and reactive transport with evolving geometries. They also underscore the significant advances seen in the last decade in tools and approaches for simulating this type of problem.

Appendices

Appendix A: Additional/alternative equations

Flow at the pore scale may be described by the incompressible Navier-Stokes (33) and (34):

$$ \nabla\cdot \textbf{u} = 0, $$

(33)

$$ \frac{\partial \textbf{u}}{\partial t}+\left( \textbf{u}\cdot\nabla \right)\textbf{u}+\frac{1}{\rho}\nabla p=\nu\nabla^{2}\textbf{u}, $$

(34)

as well as the Stokes (1) and (2). In these benchmarks, the Reynolds number is sufficiently small that fluid inertia can be neglected; thus, these two approaches are equivalent.

In the dissolution benchmarks (parts II and III), codes may take advantage of the large time scale separation between boundary motion and transport processes to solve the steady-state transport equation directly,

$$ \boldmath{\nabla} \cdot (\mathbf{u} c) = D \nabla^{2}c. $$

(35)

Time-dependent solutions of transport and reaction (part I) are more tightly coupled than dissolution (parts II and III), because t_A and t_R are often of the same order, especially for relatively fast reacting minerals such as carbonates. Both global implicit and operator splitting approaches have been used for time-dependent transport, with the time stepping in the operator splitting constrained by the Courant-Friedrichs-Levy (CFL) criterion

$$ {\Delta} t<\frac{\Delta x}{\max{(u)}}. $$

(36)

Appendix B: Analysis and comparison of results

B.1. Upscaled parameters

Simulation results are compared in terms of the evolution with time of upscaled parameters. These upscaled parameters include the volume (V ) and surface area of all reacting reacting mineral surfaces (A) and the average reaction rate (R). The average rate is calculated as follows:

$$ R=\frac {Q (c_{out}-c_{in})} {\xi A}, $$

(37)

where ξ is the stoichiometric coefficient, c_in is the (uniform) concentration at the inlet, given by the boundary conditions, and c_out is the flux-weighted-average outlet concentration,

$$ c_{out}= \frac{{\int}_{\delta S} c {\textbf{u}} \cdot d \boldsymbol{s}} {Q}. $$

(38)

The volumetric flux at the outlet Q is found by integrating over the outlet area

$$ Q={\int}_{\delta S} \mathbf{u} \cdot ds. $$

(39)

In addition to these upscaled parameters, simulation results are compared on the basis of the geometry of the grain at different time points and the concentration contours are prescribed times.

B.2. Grid convergence

As methods for simulating of moving boundary problems vary greatly, we want to investigate the impact of grid resolution on results for each method separately. For this purpose, the simulations were run at different resolutions (Figs. 14, 15, and 9) in the main text. Results for the grain volume and surface area were analyzed to ensure grid convergence of the methods, and choose a resolution for which results will be assumed to have converged within a reasonable accuracy.

Fig. 14

Grid convergence tests results for the time evolution of the grain volume (part II) from a OpenFOAM-DBS, b lattice Boltzmann, c vortex, d Chombo-Crunch, and e dissolFoam simulations

Fig. 15

Grid convergence tests results for the time evolution of the grain surface area (Part II) from a OpenFOAM-DBS, b lattice Boltzmann, c vortex, d Chombo-Crunch, and e dissolFoam simulations

Appendix C: Notes on unit conversion for concentrations and rates

The conversions from the parameters reported by [96] to the units used in Part III are presented. Mass fraction is converted to molar concentration using

$$ c = \frac{\rho f} {M}, $$

(40)

where c is the molar concentration of protons (molcm^− 3), M is the molar weight of acid (gmol^− 1), ρ is the fluid density (gcm^− 3), and f is the mass fraction of acid. The inlet concentration (0.05%) is converted to mol cm^− 3 as follows:

$$ c_{in} = \frac{0.92 \text{g} \text{cm}^{-3} \times 0.0005} {36.5 \text{g} \text{mol}^{-1}} = 1.26 \cdot 10^{-5} \text{mol} \text{cm}^{-3}. $$

(41)

In the formulation used in this manuscript (Section 2), the first-order reaction is expressed as a function of the activity coefficient and the molar concentration of H⁺ (26). Assuming that $\gamma _{\text {H}^{+}}=1000 \text {cm}^{3} \text {mol}^{-1}$, the proton concentration $c_{\text {H}^{+}}$ must be in mol cm^− 3 so that the product $k_{\text {H}^{+}} \gamma _{\text {H}^{+}}$ has units of cm s^− 1. The conversion from the rate constant used in [96] ($k_{\text {H}^{+}} \gamma _{\text {H}^{+}} = 0.5 \text {cm\ s} ^{-1}$) is

$$ k_{\text{H}^{+}} = \frac{0.5 \text{cm} \text{s} ^{-1}}{1000 \text{cm}^{3} \text{mol}^{-1}} = 5 \times 10^{-4} \text{mol} \text{cm}^{-2} \text{s} ^{-1}. $$

(42)

However, in [96], this rate is applicable to the rate of HCl consumption when reacting with calcite according to the following stoichiometry

$$ \text{CaCO}3_{(\text{s})} + \text{2HCl} -> \text{CaCl}_{2} + \text{H}_{2}\text{CO}_{3}. $$

(43)

To maintain consistency with the rate expressed for calcite, one must multiply by the stoichiometric coefficient of HCl in Eq. 43,

$$ k_{\text{H}^{+}} = 10^{-3} \text{mol} \text{cm}^{-2} \text{s} ^{-1}. $$

(44)

Appendix D: Additional information on numerical choices and parameters

D.1. Lattice Boltzmann dimensionalization

Dimensionalization of the LB computations is a process that needs special care. Lattice Boltzmann unit conversion to physical units can be done after matching the characteristic non-dimensional Reynolds, Péclet, and Damköhler numbers. For a 256 × 128 discretization grid L_y = 128 (in lattice units), each lattice space unit in parts I and II corresponds to w/128 = 3.91 × 10^− 4cm. For the current setup, viscosity is defined as ν = τ_fρT. The relaxation parameter for the fluid phase, τ_f, is set to τ_f = 0.5 in lattice units. By equating Re=Re_LB= 0.6, using the aforementioned viscosity, the inlet velocity can be calculated as u_inLB = 0.00078125 (in lattice units), which corresponds to $\textbf {u}_{in}=0.12 \text {cm\ s} ^{-1}$. Once the lattice velocity is set, the duration of the time step δt can be calculated by equating the inlet velocities: δt = 2.54 × 10^− 6 s. Note that the time step is dictated by the slow advective flow, and by choosing to keep the same time step for all processes. This leads to a fully coupled advection-diffusion-reaction scheme applicable to all flow and chemical conditions. Separation of time scales is possible by solving for steady-state flow, then steady-state reactive transport, and finally the solid geometry update. Such an approach would be sufficient for these benchmarks and would greatly reduce the number of time steps to reach the solution.

Diffusivity is defined as D = τ_gT. By equating the Péclet numbers Pe=Pe_LB= 600, the relaxation parameter τ_g, which corresponds to the diffusive time scale, is set to τ_D = 0.0005, for the species that follow the advection-diffusion equation. Finally, by equating the Damköler numbers Da_II=Da_II-LB= 178.15, the rate constant $k_{\text {H}^{+} \text {LB}}=10^{-3.2364}$. For this dimensionalization Ma<Kn<< 1, thus recovering the incompresible Navier-Stokes equations.

D.2. Discussion on interpolation kernel for Lagrangian methods

The choice of the kernel Λ used for re-meshing the particle is crucial for the accuracy of vortex and particle methods. Indeed, in order to avoid holes and accumulation of particles that would ruin the convergence, particle information F_p (including vorticity, concentration, ...) in volumes v_p located at positions ξ_p is remeshed on to a new structured mesh (with cell volumes $\tilde {v}_{q}$). This mesh defines a new set of particles $\tilde {F}_{q}$ at locations $\tilde {\xi }_{q}$ by means of the following convolution:

$$ \begin{array}{@{}rcl@{}} \tilde{F}_{q}&=& F*{\Lambda} (\tilde{x}_{q})=\int F(y){\Lambda}(\tilde{x}_{q}-y)dy\\ &=& \sum\limits_{p} F_{p}{\Lambda}(\tilde{x}_{q}-x_{p})v_{p}, \end{array} $$

(45)

since the set of particles is mathematically defined by the generalized function $\displaystyle F=\sum \limits _{p} F_{p}\delta _{x_{p}}v_{p}$, based of Dirac functions at x_p. In practice, when Λ is the “hat” (or “tent”) function, the reaction stays confined on the fluid/solid interface, but exhibits a pH over-estimation close to the stagnation points, thus over-estimating the reaction rate. When this kernel is smoother but positive in order to be sign preserving, such as the first-order cubic spline M₄, the fluid/solid boundary becomes fuzzy and requires us to force the reaction on the interface by means of the function ∥∇𝜖∥, as in [96]. When using the second-order kernel $M_{4}^{\prime }$ from [67], which is non sign preserving since the integral of $x^{2}M_{4}^{\prime }(x)$ is zero, no negative concentration appears despite the jump of acid concentration at the body but it leads underestimation of reaction rate. However, the hydrodynamic flow is computed with better accuracy using $M_{4}^{\prime }$, as expected [28]. Consequently, the short-supported function M₃, smoother than the hat function with a support smaller than M₄, has been chosen for interpolating and remeshing the chemical concentrations, while $M_{4}^{\prime }$ has been chosen for the interpolation hydrodynamic values (velocity and vorticity).

In practice for the present benchmark, for which the reaction properties (bounds and positivity) have to be strictly satisfied, the choice of the remeshing kernel is mainly driven by the following arguments:

• The hat function, is good for the estimation of reaction rate but does respect the pH bounds (pH overshoots below 2 can occur),

• • The kernel M₄ is smooth but M₄(0) = 2/3 ≠ 1; thus, it is diffusive: pH bounds are good but reaction rate is under-estimated (see formula A.4 of [18] for definition),

• $M_{4}^{\prime }$ (formula 4.5 of [20]) is algebraically mass-conservative, smooth, and second order, but its negative values induce oscillations at concentration jumps and over-estimate the reaction rate. Furthermore, it is not mathematically sign preserving, although negative concentrations were never been observed in this benchmark,

• M₃ (formula A.3 of [18]) is smoother than hat, first order and sign preserving, with short support. It is the best choice for reactive flows like the one considered in the present study; the reaction rate is well estimated (a bit higher than the hat function and closer to other curves) and does not go lower than the initial pH= 2 bound, consistent with this purely dissolution process,

• M₆ and $M_{6}^{\prime }$ supports are too large for this geometry, and cannot handle correctly the final state of the dissolution.

Consequently, the kernel $M_{4}^{\prime }$ is the best choice for hydrodynamic computations (for particle remeshing and interpolation of velocity and vorticity from and to grids), while M₃ is the best choice for interpolation and transfer of concentrations.

D.3. Darcy-Brinkman-Stokes parameter values

Parameters specific to Darcy-Brinkman-Stokes code simulations are presented in Table 6.

Table 6 Parameters for Darcy-Brinkman-Stokes equations