A Network Model of Channel Competition in Geochemical Flows

Dissolution in porous media introduces a positive feedback between fluid transport and chemical reactions at the mineral surfaces leading to the formation of pronounced wormhole-like channels [1]. As the dissolution proceeds, the channels interact and compete for the flow; eventually the growth of the shorter ones ceases. Thus the number of channels decreases with time while the characteristic distance between them increases, which leads to a scale-invariant power-law distribution of channel lengths. We present several resistor network models of the evolution of dissolving channels and explore them using computer simulation and theoretical methods such as a theory of branched growth [2]. First, the simplest case of two competing channels is considered, and the conditions under which one of them captures the flow from the other are explored. Next, the case of N competing channels is analyzed and the numerical results for N ~ 103 are presented. The results are compared with pore-scale simulations of fracture dissolution using the microscopic, three-dimensional numerical model developed in [3]. It is seen that despite their simplicity, the resistor models retain the essential features of the nonlinear interaction between the channels. [1] P. Ortoleva, J. Chadam, E. Merino, and A. Sen, Geochemical Self Organisation II: The Reactive-Infiltration Instability, Am. J. Sci., 287, 1008, 1987 [2] T.C. Halsey, M. Leibig, Theory of Branched Growth, Phys. Rev. A, 46, 7793, 1992 [3] P. Szymczak, A. J. C. Ladd, Microscopic Simulations of Fracture Dissolution, Geophys. Res. Lett., 31, L23606, doi:10.1029/2004GL021297, 2004