Reactive-infiltration instability in radial geometry: From dissolution fingers to star patterns

We consider the process of chemical erosion of a porous medium infiltrated by a reactive fluid in a thin-front limit, in which the width of the reactive front is negligible with respect to the diffusive length. We show that in the radial geometry the advancing front becomes unstable only if the flow rate in the system is sufficiently high. The existence of such a stable region in parameter space is in contrast to the Saffman-Taylor instability in radial geometry, where for a given flow rate the front always eventually becomes unstable, after reaching a certain critical radius. We also examine the similarities between the reactive-infiltration instability and the similar instability in the heat transfer, which is driving the formation of star-like patterns on frozen lakes.

Figure 1

Geometry of the system. (a) Rectangular geometry: a reactive fluid is injected from the left side and dissolves the porous matrix through chemical reactions. As the dissolution progresses, the reaction front $\zeta \left(y\right)$, shown by the solid line, advances into the matrix, separating the fully dissolved, upstream domain ($\mathrm{\Omega }$) of porosity ${\phi }_1$ from the undissolved medium of initial porosity ${\phi }_0$. (b) Radial geometry: the reactive fluid is injected at the center of the system and dissolves the surrounding matrix. The reaction front, $R\left(\theta \right)$ advances outwards.

Figure 2

Concentration and porosity profiles. The concentration profile decays with different length scales, $l_u$ and $l_d$, in the upstream $[x<\zeta (t\left)\right]$ and downstream $[x>\zeta (t\left)\right]$ regions, with $\zeta \left(t\right)$ marking the position of a reaction front. Right panel illustrates the thin-front limit, in which $l_u\gg l_d$.

Figure 3

Dispersion relation in radial geometry for $\text{Pe}=100$, $\mathrm{\Gamma }=0$ (solid line); $\text{Pe}=100$, $\mathrm{\Gamma }=0.2$ (dashed); $\text{Pe}=14$, $\mathrm{\Gamma }=0.2$ (dotted).

Figure 4

The threshold Péclet number ${\text{Pe}}_{\text{thr}}$ as a function of the permeability ratio, $\mathrm{\Gamma }$.

Figure 5

Starlike patterns on a frozen lake. The photos are courtesy of Martin Mecnarowski (www.photomecan.eu).