A CONCEPTUAL FRACTAL MODEL FOR DESCRIBING TIME-DEPENDENT DISPERSIVITY ¹

Abstract

Field studies show that the variance of travel distance often increases nonlinearly with time elapsed after release of solute tracers. The nonlinear relationship between variance of travel distance and time is attributed to the heterogeneity of the porous media. To describe the transport in such a heterogeneous system, a time-dependent dispersivity is necessary. In this paper, the basis for the formulation of the Wheatcraft-Tyler model is discussed and its inadequacy and inconsistencies are presented. As a consequence of the above, we developed a new model, which is similar in appearance to that of Wheatcraft and Tyler. The proposed model indicates that the variance of travel distance for a tracer may grow with time raised to the power of D, where D is close to the fractal dimension of the fractal stream tube and ranges from 1.0 to 2.0. When D equals 2.0, the dispersivity will increase linearly with time, which is consistent with the Mercado model. The applicability of our model was evaluated with three field scale transport experiments: the Cape Cod, the Borden, and the Columbus sites. The obtained D values for all three sites were significantly greater than 1.0. Comparison between our model and the fractional-order advection-dispersion equation reveals that the D value in our model is related to the fractional order.