An Examination of Scale‐Dependent Dispersion Coefficients

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Abstract
Many hydrologists have observed that dispersion coefficients, when measured in the field, turn out to be scale‐dependent. That is, the greater the travel distance in a tracer test used to measure dispersivity, the larger the dispersivity value that is calculated. Recently, Güven et al.(1983) presented a study which contains a basis for understanding the phenomenon of scale‐dependent dispersion within a deterministic framework. The results of that study are used as a basis for defining a scale‐dependent macro‐ dispersion coefficient for unidirectional flow in a stratified aquifer. Theoretical expressions are then obtained for the macrodispersivity, its various components, and its small time and large time limits. Using the data of Pickens and Grisak (1981a, b), numerical values are calculated for the macrodispersivity, and estimates are made of the travel distance required in order to reach Fickian conditions. Asymptotic large time macrodispersivity values of 49.5 m to 990 m and travel distances of 5.4 km to 109 km result for the particular aquifer studied. Based on these results it appears that Fickian conditions will seldom apply in practice to porous media flow. This study and previous studies show that the primary physical mechanism that causes spreading of a solute near the source is different advection rates at different elevations in the aquifer. Present results and comparisons with field data indicate that this phenomenon is not represented well by a scale‐ dependent dispersion coefficient. In modeling dispersion phenomena, it appears that more emphasis should be placed on field study and the accurate determination of hydraulic conductivity variations and other nonhomogeneities, and less on incorporating somewhat arbitrary dispersion coefficients into complex mathematical models.

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