The Effect of Solute Immobilization on a Stability of a Diffusion Front in Porous Media Under Gravity Field

Abstract

In the present paper, we consider the effect of solute immobilization on a stability of planar diffusion front in gravity field. The case when heavy admixture diffuses from upper boundary into a bulk of semi-infinite domain of porous medium is investigated. In this situation, the base state formed by the diffusion corresponds to the layer of heavier liquid located above the lighter one. This unsteady base state is potentially unstable to the Rayleigh–Taylor fingering instability of front. The instability conditions are examined using the quasi-static method within the fMIM diffusion model. It is found that the immobilization results in the increase in the critical time for instability significantly, whereas the effect of immobilization on the critical wave number is weak. The stability maps in the parameter space are obtained.

Appendices

Appendix 1: The Evolution Method

In order to justify the applicability of the quasi-static approach, we solve the problem by the evolution method. In this case, unsteady linearized problem (11) for small perturbations of base state is:

$$\begin{aligned} \begin{aligned}&\frac{\partial }{\partial t}\left( c+\lambda I_{0,+}^{1-\alpha }c\right) = -k\psi \frac{\partial C}{\partial z} +\frac{\partial ^2 c}{\partial x^2}+\frac{\partial ^2 c}{\partial z^2},\\&\frac{\partial ^2 \psi }{\partial x^2}+\frac{\partial ^2 \psi }{\partial z^2}-kc=0, \quad \left. c,\psi \right| _{z=0,\infty }=0. \end{aligned} \end{aligned}$$

(15)

and unsteady problem (9) for the base state concentration is

$$\begin{aligned}&\frac{\partial }{\partial t}\left( C+\lambda I_{0,+}^{1-\alpha }C\right) = \frac{\partial ^2 C}{\partial z^2}, \quad \left. C \right| _{z=0}=C_0, \quad \left. C \right| _{z=L}=0. \end{aligned}$$

(16)

We solved these two problems in a coupled manner.

The initial conditions for the base state were taken as $\left. C \right| _{z=0,t=0}=1$ and $\left. C \right| _{z>0,t=0}=0$ and for perturbations as $\left. c \right| _{z=L/2,t=0}=1$, $\left. c \right| _{z\ne L/2,t=0}=0$, $\left. \psi \right| _{t=0}=0$. We also used the perturbations located in the other points ($\left. c \right| _{z=z^*,t=0}=1$) and obtained the same results. The calculations have shown that during the transient stage, any used initial perturbation fades up to small value and remains small until sharp growth at the critical time moment $t=\tau$. The initial perturbation of any type can be constructed as the superposition of perturbations located in different points, because of that we assumed that the behavior of any perturbation will be the same.

Direct numerical simulation of the perturbations evolution was performed for several fixed values of k and the time moment when the perturbation c growth starts was detected. In this way, the neutral curve $\tau (k)$ was obtained. The results of such calculations are presented in Fig. 6.

Fig. 6

Neutral curve in the plane $\left( \tau ,\,k \right)$ obtained for the fMIM diffusion model for $\lambda =0.1$ and $\alpha =0.75$. The solid line shows the results obtained within the quasi-static approach and the points show the results of direct numerical simulation (DNS)

The direct numerical calculations were performed using the implicit finite differences scheme of second-order accuracy in time and space with Diethelm’s (Diethelm et al. 2005) approximation for the fractional integral. The values of time and space steps were $\triangle t =10^{-4}$ and $h=10^{-2}$. The difference between the values of the critical time (obtained by the quasi-static method and DNS) is less than two percents near the minimum value. Thus, we can conclude that the quasi-static method can be applied for the correct determination of critical time.

Appendix 2: Deep-Pool Approximation

This paper is devoted to the investigation of planar diffusion front stability in the semi-infinite porous domain. The modeling of infinite domains is not possible, and all calculations were performed for the layer of large enough but finite depth. This appendix is devoted to the determination of “deep layer” criteria. It was accepted that the layer is deep if the increasing depth does not affect the neutral curve $\tau (k)$. The fastest diffusion corresponds to the case $\lambda =0$ (see Fig. 6). In order to obtain the quantitative estimation, the dependence of relative error $\varepsilon$ on layer depth L was plotted (see Fig. 7).

Fig. 7

Relative error versus layer depth

The relative error was determined by the formula

$$\begin{aligned} \varepsilon \left( L\right) = \left| \frac{\tau \left( L\right) -\tau _0}{\tau _0} \right| , \end{aligned}$$

(17)

where $\tau _0=\left. \tau \left( k_{\min }=0.055\right) \right| _{L=\infty }=55.8$ is the critical time for the classical diffusion model (Rees et al. 2008) and $\tau \left( L\right) =\left. \tau \left( k_{\min }=0.055\right) \right| _{L}$ is the critical time for the layer of depth L. All calculations were performed for the case of fastest diffusion $\lambda =0$. One can see from Fig. 7 that $\varepsilon \left( L=100\right) \ll 1\,\%$. It is a satisfactory accuracy, and the value $L=100$ is chosen for the calculations.