Non-Linear Growth of the Dissolution Front

A planar dissolution front is unstable to small perturbations; regions where the front is slightly extended obtain more flow, which serves to amplify the initial perturbation. The upper image shows a dissolution front advancing into a spatially homogeneous porous medium, at a point in time where the initial sinusoidal perturbations start to develop into distinct channels. A selection mechanism then begins to operate, leading to more rapid growth of long channels at the expense of the shorter ones, as can be seen in the second panel where there are fewer but deeper channels.

The color contours indicate the depth of erosion of the porous matrix; blue represents minimal erosion and red represent maximal erosion. The flow is from top to bottom and the erosion patterns are shown at early (top) and late (bottom) times. The initial porous matrix is statistically homogeneous, with only small scale variations in porosity. These random fluctuations are amplified by the non-linear feedback between fluid flow, solute transport and reaction kinetics at the fluid-rock interface. The inset figures illustrate the basic physics behind the channel competition. The cartoon on the left indicates a long and a short channel with the flow draining into the longer channel (B) and then being expelled near the outlet. The reason for this is made clear by the sketch of the pressure in the two channels.

The mechanism for the flow focusing can be understood in terms of the model [1] shown in the inset figure. The longer channel (B) has the higher flow rate, so that the pressure gradient near the inlet is larger in the long channel (B) than in the short one (A), leading to a lower pressure in the longer channel. Thus under-saturated fluid, near the inlet, is drawn towards the long channels, further extending them at the expense of the short ones. Further into the fracture the situation is reversed. Near the outlet the pressure is higher in the long channel, as shown in the right-hand inset, so that fluid tends to flow from the channel into the surrounding porous matrix, leading to the well known "tip splitting" seen in the lower panel. This inhomogeneous growth is transport controlled, so the channel competition is qualitatively independent of the detailed reaction kinetics.

References

  1. P. Szymczak and A. J. C. Ladd, Geophys. Rev. Lett., 33:L05401, 2006.



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