Strong pinning of propagation fronts in adverse flow

Reaction fronts evolving in a porous medium exhibit a rich dynamical behavior. In the presence of an adverse flow, experiments show that the front slows down and eventually gets pinned, displaying a particular sawtooth shape. Extensive numerical simulations of the hydrodynamic equations confirm the experimental observations. Here we propose a stylized model, predicting two possible outcomes of the experiments for large adverse flow: either the front develops a sawtooth shape or it acquires a complicated structure with islands and overhangs. A simple criterion allows one to distinguish between the two scenarios and its validity is reproduced by direct hydrodynamical simulations. Our model gives a better understanding of the transition and is relevant in a variety of domains, when the pinning regime is strong and only relies on a small number of sites.

Figure 1

Average speed of the front $V_f$ as a function of the injection speed $\overline{U}$ (the convention chosen is $\overline{U}>0$ for a flow from top to bottom of the cell). Depending on the sign of $V_f$, the different regimes $\text{Up}$, $S$, and $D$ are defined. For $\overline{U}=U_{SD}$, we observe a transition between a static front ($V_f=0$) and a downstream front ($V_f>0$). Bottom part: Hydrodynamical simulations of the front, for $\overline{U}\sim U_{SD}$ and for different permeability distributions. Two scenarios are observed: a regular sawtooth shape (right) or a complicated shape with overhangs (left).

Figure 2

Sketch of the stylized model: in the thin eikonal limit ${\ell }_d\gg {\ell }_{\chi }$, the system can be described as a propagating front. Close to $U_{SD}$, the density of stagnant regions becomes small and the interface adopts a sawtooth structure.

Figure 3

Probability distribution of the velocity threshold $U_{SD}$. The histogram corresponds to the hydrodynamical simulations of $N=2000$ samples with a log-normal permeability, setting $L=512$, $V_{\chi }=0.0016$, $\overline{U}=0.0036$, and ${\ell }_d=5.0$. The dashed line corresponds to the prediction of the stylized model [Eq. (9)] for a log normal $\varphi \left(v\right)$ with a scale parameter $\sigma =0.315$ (see main text). Inset: Sketch of the algorithmic recursive procedure.

Figure 4

Distribution of $l_{\Delta }$ for the stylized model with parameters $\lambda =0.5$ and $\theta =\pi /2$. The system size is $L=100$ and the simulation is performed over $N=3000$ samples. The dashed line corresponds to the asymptotic prediction of Eq. (15). Inset: Interface pinned between two adjacent stagnant patches of coordinates ${\mathbf{x}}_{\mathbf{1}}$ and ${\mathbf{x}}_{\mathbf{2}}$.