Pore-scale dilution of conservative solutes: An example

We simulate flow and transport of a conservative nonsorbing tracer in an idealized periodic pore channel using finite element techniques. The concentration is computed; then the slowly varying concentration mean, variance, coefficient of variation, and reactor ratio are calculated through averaging over every cell. The coefficient of variation and reactor ratio are related and quantify the degree of dilution. Then a novel methodology is developed for the evaluation of macroscopic parameters (homogenization), including the variance decay coefficient, which measures the rate with which small-scale concentration fluctuations tend to diminish, and the large-time coefficient of proportionality between the concentration variance and the square of the mean concentration variance. The methodology is based on the solution of a steady advection-dispersion problem in a single cell (which acts as a representative elementary volume); the computed result is then integrated in order to compute the macroscopic parameters. These parameters are compared with the parameters computed through direct simulation on a cell-by-cell basis, and they are found to be in reasonably good agreement. When the macroscopic parameters are used in the macroscopic equations, they produce estimates of the concentration mean and variance that are in agreement with the results of direct simulation.