ANALYSIS OF MACRODISPERSION THROUGH VOLUME-AVERAGING - MOMENT EQUATIONS

Macrodispersion is spreading of a substance induced by spatial variations in local advective velocity at field scales. Consider the case that the steady-state seepage velocity and the local dispersion coefficients in a heterogeneous formation may be modeled as periodic in all directions in an unbounded domain. The equations satisfied by the first two spatial moments of the concentration are derived for the case of a conservative non-reacting solute. It is shown that the moments can be calculated from the solution of well-defined deterministic boundary value problems. Then, it is described how the rate of increase of the first two moments can be calculated at large times using a Taylor-Aris analysis as generalized by Brenner. It is demonstrated that the second-order tensor of macrodispersion (or effective dispersion) can be computed through the solution of steady-state boundary-value problems followed by the determination of volume averages. The analysis is based solely on volume averaging and is not limited by the assumption that the fluctuations are small. The large-time results are valid when the system is in a form of equilibrium in which a tagged particle samples all locations in an appropriately defined "phase space" with equal probability.