Instability of the Dissolution Front in a Smooth Fracture

In the standard model for fracture dissolution, the aperture is assumed to be constant in the direction perpendicular to the flow [1]. However, such models do not predict erosion deep into the fracture without a strong concentration dependence of the reaction rate. This kinetic "trigger" mechanism for Karst formation is well verified in calcite [1]; near saturation it is known that the dissolution rate drops by several orders of magnitude. However in other minerals, for instance Gypsum, the rate coefficient is essentially independent of concentration yet caverns develop in these formations as well.

However, we have recently found that a planar dissolution front in a smooth fracture is unstable to infinitessimally small perturbations [2]. The most striking result of the linear stability analysis is that there is a maximal growth rate at a specific wavelength. We have confirmed this prediction by numerical simulations; an example of a growing instability is illustrated in the figure below.

Concentration profiles in a dissolving fracture. Contour plots of the normalized undersaturation are plotted at successive times (from left to right). The flow is from left to right. The fracture is 20m by 10m, with a mean aperture of 0.2mm; the statistical variation in the aperture (roughness) is 1 part in 10000. A movie of the evolution in concentration is shown below or can be downloaded here.

As a result of the instability, an individual fracture develops a strongly heterogeneous permeability during dissolution, with an inherent length scale that depends on the kinetics and flow rate, but not the initial topography. Furthermore, there is no lower limit to the reaction rate for unstable dissolution. Only the wavelength and penetration depth are affected, scaling with the inverse of the Damköhler number.

Concentration profiles in a dissolving fracture. Contour plots of the normalized undersaturation are plotted for different Da: left to right Da = 0.000125 , 0.00025, 0.0005, and 0.001.

References

  1. W. Dreybrodt, Water Resources Res., 32:2923, 1996.
  2. P. Szymczak and A. J. C. Ladd, EPSL, 201:424-432, 2011.



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