ANALYSIS OF MACRODISPERSION THROUGH VOLUME-AVERAGING - MOMENT EQUATIONS

Macrodispersion is spreading of a substance induced by spatial variati ons in local advective velocity at field scales. Consider the case tha t the steady-state seepage velocity and the local dispersion coefficie nts in a heterogeneous formation may be modeled as periodic in all dir ections in an unbounded domain. The equations satisfied by the first t wo 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 va lue problems. Then, it is described how the rate of increase of the fi rst two moments can be calculated at large times using a Taylor-Aris a nalysis as generalized by Brenner. It is demonstrated that the second- order tensor of macrodispersion (or effective dispersion) can be compu ted through the solution of steady-state boundary-value problems follo wed by the determination of volume averages. The analysis is based sol ely on volume averaging and is not limited by the assumption that the fluctuations are small. The large-time results are valid when the syst em is in a form of equilibrium in which a tagged particle samples all locations in an appropriately defined "phase space" with equal probabi lity.