Postbreakthrough behavior in flow through porous media

We numerically simulate the traveling time of a tracer in convective flow between two points (injection and extraction) separated by a distance r in a model of porous media, [Formula presented] percolation. We calculate and analyze the traveling time probability density function for two values of the fraction of connecting bonds p: the homogeneous case [Formula presented] and the inhomogeneous critical threshold case [Formula presented] We analyze both constant current and constant pressure conditions at [Formula presented] The homogeneous [Formula presented] case serves as a comparison base for the more complicated [Formula presented] situation. We find several regions in the probability density of the traveling times for the homogeneous case [Formula presented] and also for the critical case [Formula presented] for both constant pressure and constant current conditions. For constant pressure, the first region [Formula presented] corresponds to the short times before the flow breakthrough occurs, when the probability distribution is strictly zero. The second region [Formula presented] corresponds to numerous fast flow lines reaching the extraction point, with the probability distribution reaching its maximum. The third region [Formula presented] corresponds to intermediate times and is characterized by a power-law decay. The fourth region [Formula presented] corresponds to very long traveling times, and is characterized by a different power-law decaying tail. The power-law characterizing region [Formula presented] is related to the multifractal properties of flow in percolation, and an expression for its dependence on the system size L is presented. The constant current behavior is different from the constant pressure behavior, and can be related analytically to the constant pressure case. We present theoretical arguments for the values of the exponents characterizing each region and crossover times. Our results are summarized in two scaling assumptions for the traveling time probability density; one for constant pressure and one for constant current. We also present the production curve associated with the probability of traveling times, which is of interest to oil recovery.