Interaction between rarefaction wave and viscous fingering in a Langmuir adsorbed solute

The evolution of dissolved species in a porous medium is determined by its adsorption on the porous matrix through the classical advection-diffusion processes. The extent to which the adsorption affects the solute propagation in applications related to chromatography and contaminant transport is largely dependent upon the adsorption isotherm. Here, we examine the influence of a nonlinear Langmuir adsorbed solute on its propagation dynamics. Interfacial deformations can also be induced by classical viscous fingering (VF) instability that develops when a less viscous fluid displaces a more viscous one. We present numerical simulations of an initially step-up concentration profile of the solute that capture a rarefaction/diffusive wave solution due to the nonlinearity introduced through Langmuir adsorption and variety of pattern-forming behaviors of the solute dissolved in the displaced fluid. The fluid velocity is governed by Darcy's law, coupled with the advection-diffusion equation that determines the evolution of the solute concentration controlling the viscosity of the fluids. Numerical simulations are performed using the Fourier pseudospectral method to investigate and illustrate the role played by VF and Langmuir adsorption in the development of the patterns of the interface. We show that the solute transport proceeds by the formation of a rarefaction wave results in the enhanced spreading of the solute. Interestingly we obtained a nonmonotonic behavior in the onset of VF, which depends on the adsorption parameters and existence of an optimal value of such adsorption constant is obtained near $b=1$, for which the most delayed VF is observed. Hence, it can be concluded that the rarefaction wave formation stands out to be an effective tool for controlling the VF dynamics.

Figure 1

Schematic of the system.

Figure 2

(a–c) Concentration fields $c_m(x,y)$ with $R=0,k=0.2$, (a) initial concentration at $t=0$, (b) for $b=0$ (Linear adsorption), and (c) $b=5$ (Langmuir adsorption). (d) The corresponding transversely averaged concentration profiles ${\overline{c}}_m(x,t)$ for $b=0$ (blue lines), error function solutions away from the step-up curve at $t=0$ and $b=5$ (red lines), rarefaction solutions where the part of the curve with higher concentrations are near to the step-up curve at $t=0$.

Figure 3

(a) The mobile phase solute concentration $c_m(x,y)$ fields with $R=0,k=0.2$ for different values of $b$ at a fixed time $t={10}^4$. (b) The corresponding transversely averaged concentration profile ${\overline{c}}_m(x,t)$.

Figure 4

The spreading length $L_m$ as a function of $b$ at a fixed time $t={10}^4$. Inset: Evolution of the spreading length $L_m$ for different values of $b$ (from below $b=0,0.1,0.5,50$, and 10), with $R=0,k=0.2$.

Figure 5

The concentration plots for $R=1,k=0.2$ and (a) $b=0,$ (b) $b=1,$ (c) $b=50$.

Figure 6

The contour plots of the concentration at fixed $t=4000$, for $R=1,k=0.2$ for different values of $b$, showing nonmonotonicity in the fingering pattern. The contour lines correspond to $c_m=0.17,0.34,0.51,0.68$, and 0.85.

Figure 7

The transversely average concentration profile ${\overline{c}}_m(x,t)$ with $R=1,k=0.2$ for (a) $b=1,$ (b) $b=5$.

Figure 8

The temporal evolution of concentration corresponding to the inflexion point, $c_{m_i}$ for different values of $b$ with $R=0,k=0.2$.

Figure 9

(a) Temporal evolution of interfacial length, $I\left(t\right)$, with $R=1,k=0.2$ for different values of $b$ and compared with the ideal case of stable displacement. (b) Onset time of VF, $t_{\text{vf}}$, for different values of $b$ with $R=1,k=0.2$. It shows a nonmonotonicity with respect to $b$. Inset: Zoomed in $t_{\text{vf}}$ showing local maxima at $b\simeq 1$.

Figure 10

(a) The concentration contour for different values of $R$, with $b=5,k=0.2$ at a fixed time $t=2000$. The contour lines correspond to $c_m=0.25,0.5,0.75$, and 1. (b) Onset time of viscous fingering $t_{\text{vf}}$ as a function of $b$ for different values of $R$ with $k=0.2$.

Figure 11

(a) Characteristics showing formation of rarefaction wave for $k=1,b=1$. (b) The corresponding rarefaction wave solution of the concentration $c_m$ at $t=1$.