Fragmentation and coalescence in simulations of migration in a one-dimensional random medium

A simple one-dimensional model of fluid migration through a disordered medium is presented. The model is based on invasion percolation and is motivated by two-phase flow experiments in porous media. A uniform pressure gradient g drives fluid clusters through a random medium. The clusters may both coalesce and fragment during migration. The leading fragment advances stepwise. The pressure gradient g is increased continuously. The evolution of the system is characterized by stagnation periods. Simulation results are described and analyzed using probability theory. The fragment length distribution is characterized by a crossover length s∗ (g) ∼ g−12 and the length of the leading fragment scales as sp(g) ∼ g−1. The mean fragment length is found to scale with the initial cluster length s0and g as 〈s〉 = s012f(gs034).