Influence of Langmuir adsorption and viscous fingering on transport of finite size samples in porous media

We examine the transport in a homogeneous porous medium of a finite slice of a solute which adsorbs on the porous matrix following a Langmuir adsorption isotherm and can influence the dynamic viscosity of the solution. In the absence of any viscosity variation, the Langmuir adsorption induces the formation of a shock layer wave at the frontal interface and of a rarefaction wave at the rear interface of the sample. For a finite width sample, these waves interact after a given time that varies nonlinearly with the adsorption properties to give a triangle-like concentration profile in which the mixing efficiency of the solute is larger in comparison to the linear or no-adsorption cases. In the presence of a viscosity contrast such that a less viscous carrier fluid displaces the more viscous finite slice, viscous fingers are formed at the rear rarefaction interface. The fingers propagate through the finite sample to preempt the shock layer at the viscously stable front. In the reverse case, i.e., when the shock layer front features viscous fingering, the fingers are unable to intrude through the rarefaction zone and the qualitative properties of the expanding rear wave are preserved. A nonmonotonic dependence with respect to the Langmuir adsorption parameter $b$ is observed in the onset time of interaction between the nonlinear waves and viscous fingering. The coupled effect of viscous fingering at the rear interface and of Langmuir adsorption provides a powerful mechanism to enhance the mixing efficiency of the adsorbed solute.

Figure 1

Schematic of the displacement of a finite slice of miscible fluid in a two- dimensional homogeneous porous media.

Figure 2

(a) Concentration field $c_m(x,y,t)$ for $R=0,k=0.2$ with $b=0$ and $b=2$ at different times. (b) Corresponding transverse averaged concentration profiles $\overline{c_m}(x,t)$ for $R=0,k=0.2$ and $b=0$ (red), $b=2$ (blue).

Figure 3

(a) 1D concentration profiles of a Langmuir adsorbed solute for $k=0.2,b=1$ at time $t=0, t<t_{\mathrm{on}}$ showing the definition of $L_{\mathrm{rf}}\text{and}{L}_{\mathrm{sl}}$ (blue), $t=t_{\mathrm{on}}$ (time of onset of the interaction between SL and RF, black), $t>t_{\mathrm{on}}$ (long time triangle-like profile, red). (b) Interaction times $t_{\mathrm{on}}$ (SL with RF), $t_{\mathrm{on},\mathrm{sl}}$ (VF with SL), and $t_{\mathrm{on},\mathrm{rf}}$ (VF with RF) exhibit a nonmonotonic dependence on $b$. Here $k=0.2$. The vertical lines correspond to $b^{★}$ (dotted) and $b^{†}$ (dash-dotted).

Figure 4

Shock layer thickness $L_{\mathrm{sl}}$ as a function of $b$ shows a global minimum at $b^{★}=\sqrt{2(1+k)}$.

Figure 5

The temporal evolution of the rarefaction thickness $L_{\mathrm{rf}}$ for different $b$ with $k=0.2$ shows a decrease in $L_{\mathrm{rf}}$ beyond $b=20$.

Figure 6

Concentration maps for $R=1,k=0.2$: (a) $b=0$, (b) 1.55 ($=b^{★}$), (c) 10, and (d) ${10}^3$ (the saturated case) at the time written in the panel. The solid contours are $c_m=0.1$ to 0.9 with step 0.2. The dashed contour corresponds to $c_m=0.001$. The concentration maps are shown at the onset of VF (top row), at $t\approx t_{\mathrm{on},\mathrm{sl}}$ (middle row), and at $t>t_{\mathrm{on},\mathrm{sl}}$.

Figure 7

Comparison of transverse averaged concentration profiles ${\overline{c}}_m\left(x\right)$ of the solute at time $t=10000$ for (a) $k=0.2,b=1.55$, (b) $k=1,b=2$, (c) $k=0,b=0$, i.e., no-adsorption case and $R=1$ or $R=0$. (d) Transverse averaged viscosity profiles $\overline{\mu }(x,t)$ for no-adsorption ($k=0,b=0$) and Langmuir adsorption ($k=0.2,b=1.55$) with $R=1$ at time $t=10000$.

Figure 8

Concentration maps for $R=-1,k=0.2$: (a) $b=0$, (b) 1.55 ($=b^{★}$), (c) 10, and (d) ${10}^3$ (the saturated case) at the time written in the panel. The solid contours are $c_m=0.1$ to 0.9 with step 0.2. The dashed contour corresponds to $c_m=0.001$. The concentration maps are shown at the onset of VF (top row), at $t\approx t_{\mathrm{on},\mathrm{rf}}$ (middle row), and at $t>t_{\mathrm{on},\mathrm{rf}}$.

Figure 9

Transverse averaged concentration profiles of the solute ${\overline{c}}_m\left(x\right)$ (a) for different values of $b$ at $t=6000$ with $k=0.2,R=-1$; (b) for different values of $R$ at $t=8000$ with $k=0.2,b=b^{★}$.

Figure 10

Temporal evolution of the degree of mixing $\chi \left(t\right)$ for no-adsorption ($k=0$), linear adsorption ($b=0$), and Langmuir adsorption ($b=2$) cases with $R=0,1,-1$.

Figure 11

Temporal evolution of $\dot{\chi }\left(t\right)$ for $k=0.2$ and different values of $b$: $R=$ (a) 1, (b) $-1$.