Pattern formation during fracture dissolution

A number of experimental and numerical studies of dissolution in fractured and porous rocks have established that the evolving topography of the pore space depends strongly on the fluid flow and mineral dissolution rates. Remarkably, there exists a wide parameter range in which positive feedback between fluid transport and mineral dissolution leads to the spontaneous formation of pronounced channels, frequently referred to as "wormholes". As dissolution proceeds the growing channels interact, competing for the available flow, and eventually the growth of the shorter ones cease. This leads to self-similar patterns of growth, with the flow becoming concentrated in fewer active channels as the dissolution proceeds. Under some conditions it is reasonable to assume that the growth of the channels is proportional to the reactant flux near the tip of each channel. Since the pressure field in the rock matrix obeys the Laplace equation, the competition between the channels may then be described in terms of a Laplacian growth model and solved using conformal mapping techniques. The figures compare the flow field around two different length channels. The upper figure was obtained from a Reynold's approximation for a parabolic channel, while the lower figure is the conformal solution, assuming a constant pressure along the channel. The Reynold's solution shows the expected enhancement of flow through the longer channel, which leads to more rapid dissolution. The conformal solution, assuming a constant pressure along the channel is remarkably similar. Combining conformal mapping with Darcy-scale simulations, we study the interaction between a large number of growing channels and the resulting dissolution patterns. Flow field around two channels, determined from a Reynold's approximation. The flow direction is along the channels.

Flow field around two channels, from a conformal mapping. The flow direction is along the channels