Scaling properties of the dissolving channels in a fractured rock

During dissolution in porous or fractured rock, a positive feedback between fluid transport and mineral erosion leads to the spontaneous formation of wormhole-like channels. An important ingredient in the evolution of such a system is the competition between the growing channels which leads to the appearance of a scale-invariant distribution of channel lengths. A simplified model of the dissolving medium is studied, in which the channels are considered to be one-dimensional needles, growing only at the tip [1]. It is shown that the needle model posseses analogous scaling properties to the full system [2]. In particular, a limit of small permeability ratio, r=k_0/k, between the undissolved and dissolved medium is studied. This limit is shown to be singular: the scaling properties for r=0 are fundamentaly different form those obtained for small but finite r. [1] P. Szymczak and A.J.C. Ladd. A Network model of channel competition in fracture dissolution Geophys. Rev. Lett., 33:L05401, 2006. [2] P. Szymczak and A.J.C. Ladd. Microscopic simulations of fracture dissolution. Geophys. Res. Lett., 31:L23606, 2004.