Avalanches dynamics in reaction fronts in disordered flows

We report on numerical studies of avalanches of an autocatalytic reaction front in a porous medium. The front propagation is controlled by an adverse flow resulting in upstream, static, or downstream regimes. In an earlier study focusing on front shape, we identified three different universality classes associated with this system by following the front dynamics experimentally and numerically. Here, using numerical simulations in the vicinity of the second-order transition, we identify an avalanche dynamics characterized by power-law distributions of avalanche sizes, durations, and lateral extensions. The related exponents agree well with the quenched-Kardar-Parisi-Zhang theory, which describes the front dynamics. However, the geometry of the propagating front differs slightly from that of the theoretical one. We show that this discrepancy can be understood in terms of the nonquasistatic correction induced by the finite front velocity.

Figure 1

(a) Front dynamic for $\epsilon =-1.13$. In black: front positions at different constant interval of time ($\delta t=100\lambda /V_{\chi }$). In gray scale: logarithm of the normalized velocity field, ${log}_{10}(U_x/V_{\chi })$. The mean flow is oriented from left to right, and the chemical reaction propagates from right to left. The initial front location is flat and located at the rightmost end of the domain. (b) Spatiotemporal map of the logarithm of the normalized front velocity, $log\left[\right|v(x,t)/V_{\chi }\left|\right]$. The mean front velocity is $V_f/V_{\chi }=0.057$ (c) associated spatiotemporal identified clusters.

Figure 2

Distribution of avalanche lengths (a), sizes (b), and durations (c) for different distances to the critical point ${\epsilon }_c=-1.17$. Insets: evolution of the corresponding cutoff as a function of the distance to the critical point $\epsilon -{\epsilon }_c$.

Figure 3

Data collapse for lengths, sizes, and durations using the exponents of Table 1 and the scaling ansatz.

Figure 4

Plots for (a) roughness and (b) area vs durations. Lines scale as (a) ${\langle W\rangle }_L\propto L^{\zeta }$ with $\zeta \approx 0.74$ and (b) ${\langle S\rangle }_T\propto T^{\gamma }$ with $\gamma \approx 1.79$.

Figure 5

Example of the transfer-matrix algorithm applied to the velocity field at the percolation threshold. Left: in black are the points connected to the top border by a percolating path. Right: binarized velocity field (black, $\left|U\right|>V_{\chi }$; white, $\left|U\right|<V_{\chi }$). In both figures, the red path corresponds to the first barrier from left to right.

Figure 6

Left: two-point correlation function $W\left(l\right)=\sqrt{\langle {[h\left(y\right)-h(y+l)]}^2\rangle }$ of the directed percolation path for two types of disorder. Blue triangles: uncorrelated white noise. Green circles: velocity field from the log-normal permeability distribution. The two lines correspond to the fitted scaling $W\left(l\right)\propto l^{\zeta }$. Right: distribution roughness exponent measured for the log-normal permeability. The mean of the distribution is $\langle \zeta \rangle =0.63$ and the standard deviation is $\mathrm{std}\left(\zeta \right)=0.04$.