A mineral precipitation model based on the volume of fluid method

Abstract

A novel volume of fluid method is presented for mineral precipitation coupled with fluid flow and reactive transport. The approach describes the fluid-solid interface as a smooth transitional region, which is designed to provide the same precipitation rate and viscous drag force as a sharp interface. Specifically, the governing equation of mineral precipitation is discretized by an upwind scheme, and a rigorous effective viscosity model is derived around the interface. The model is validated against analytical solutions for mineral precipitation in channel and ring-shaped structures. It also compares well with interface tracking simulations of advection-diffusion-reaction problems. The methodology is finally employed to model mineral precipitation in fracture networks, which is challenging due to the low porosity and complex geometry. Compared to other approaches, the proposed model has a concise algorithm and contains no free parameters. In the modeling, only the pore space requires meshing, which improves the computational efficiency especially for low-porosity media.

Appendix A Effective viscosity model

In this appendix we will prove that the effective viscosity model, described by Eq. (2), can guarantee that the smooth interface exerts the same viscous drag force on the fluid as a sharp interface. We start by constructing a local coordinate system, where the x axis is normal to the interface. The transitional region of the smooth interface is from $x_1$ to $x_2$, with $x_1$ and $x_2$ in the liquid and solid phases, respectively:

$$\begin{aligned}&\phi (x_1) = 1, \;\; \phi (x_2) = 0, \end{aligned}$$

(1a)

$$\begin{aligned}&u(x_1) = u_1, \;\; u(x_2) = 0, \end{aligned}$$

(1b)

where $u = \Vert \textbf{u}\Vert$ is the velocity magnitude. Since the viscous force is dominant near the wall boundary, we can neglect the inertial and pressure terms in Eq. () and only keep the viscous term:

$$\begin{aligned}&\nabla \cdot ( \mu _{eff} \nabla \textbf{u}) = 0. \end{aligned}$$

(2)

We assume the derivative in the normal direction (x direction) is much larger than in the tangential direction and there is no penetration ($\textbf{u}\cdot \textbf{x}= 0)$:

$$\begin{aligned}&\frac{d}{dx} (\mu _{eff} \frac{d u}{dx}) = 0, \end{aligned}$$

(3)

$$\begin{aligned}&f = -\mu _{eff} \frac{d u}{dx} = const, \end{aligned}$$

(4)

where f is the viscous drag force exerted on the fluid per unit interface area.

Next we consider a sharp interface. To ensure the fluid occupies the same volume as in the smooth-interface case and the sharp-interface case, the sharp interface location ($x_0$) should be:

$$\begin{aligned}&x_0 = x_1 + \int _{x_1}^{x_2} \phi (x) dx. \end{aligned}$$

(5)

The governing equation and the boundary conditions are:

$$\begin{aligned}&\nabla \cdot ( \mu _{f} \nabla \textbf{u}^*) = 0, \end{aligned}$$

(6)

$$\begin{aligned}&u^*(x_1) = u_1, \;\; u^*(x_0) = 0, \end{aligned}$$

(7a)

where $\textbf{u}^*$ is the velocity in the sharp-interface case. With the same assumptions, the viscous drag force from the sharp interface, per unit area, is:

$$\begin{aligned}&f = \mu _{f} \frac{u_1}{x_0 - x_1}. \end{aligned}$$

(7b)

The smooth interface and the sharp interface should provide the same viscous drag force, to ensure the momentum balance is accurately described. Thus, we let f in Eq. A4 and Eq. A8 be the same and combine them with Eq. A5:

$$\begin{aligned}&- \frac{d u}{dx} = \frac{\mu _f}{\mu _{eff}} \frac{u_1}{\int _{x_1}^{x_2} \phi (x) dx}. \end{aligned}$$

(7c)

Integrate both sides from $x_1$ to $x_2$:

$$\begin{aligned}&\int _{x_1}^{x_2} - \frac{d u}{dx} dx = \int _{x_1}^{x_2} \frac{\mu _{f}u_1}{\mu _{eff} \int _{x_1}^{x_2} \phi (x^*) dx^*} dx. \end{aligned}$$

(8a)

Substitute the boundary conditions in Eq. A1b:

$$\begin{aligned}&u_1 = \int _{x_1}^{x_2} \frac{\mu _{f}u_1}{\mu _{eff} \int _{x_1}^{x_2} \phi (x^*) dx^*} dx, \end{aligned}$$

(8b)

$$\begin{aligned}&\int _{x_1}^{x_2} \phi (x) dx = \int _{x_1}^{x_2} \frac{\mu _{f}}{\mu _{eff} } dx. \end{aligned}$$

(8c)

To ensure Eq. A12 is valid for arbitrary $\phi (x)$, we have:

$$\begin{aligned}&\mu _{eff} = \frac{\mu _{f}}{\phi (x)}. \end{aligned}$$

(8d)