Flow takes place in a heterogeneous formation of spatially variable conductivity, which is modeled as a stationary space random function. To model the variability at the regional scale, the formation is viewed as one of a two-dimensional, horizontal structure. A constant head gradient is applied on the formation boundary such that the flow is uniform in the mean. A plume of insert solute is injected at t=0 in a volume V0. Under ergodic conditions the plume centroid moves with the constant, mean flow velocity U, and a longitudinal macrodispersion coefficient dL may be defined as half of the time rate of change of the plume second spatial moment with respect to the centroid. For a log-conductivity covariance CY of finite integral scale I, at first order in the variance &sgr;2Y and for a travel distance L=Ut≫I, dL→&sgr;2YUI and transport is coined as Fickian. Ergodicity of the moments is ensured if l≫I, where l is the initial plume scale. Some field observations have suggested that heterogeneity may be of evolving scales and that the macrodispersion coefficient may grow with L without reaching a constant limit (anomalous diffusion). To model such a behavior, previous studies have assumed that CY is stationary but of unbounded integral scale with CY~ar&bgr; (-1<&bgr;<0) for large lag r. Under ergodic conditions, it was found that asymptotically dL~aUL1+&bgr;, i.e., non-Fickian behavior and anomalous dispersion.

The present study claims that an ergodic behavior is not possible for a given finite plume of initial size l, since the basic requirement that l≫I cannot be satisfied for CY of unbounded scale. For instance, the centroid does not move any more with U but is random (Figure 1), owing to the large-scale heterogeneity. In such a situation the actual effective dispersion coefficient DL is defined as half the rate of change of the mean second spatial moment with respect to the plume centroid in each realization. This is the accessible entity in a given experiment. We show that in contrast with dL, the behavior of DL is controlled by l and it has the Fickian limit DL~aUI1+&bgr; (Figure 3). We also discuss the case in which Y is of stationary increments and is characterized by its variogram &ggr;Y. Then U and dL can be defined only if &ggr;Y is truncated (equivalently, an ''infrared cutoff'' is carried out in the spectrum of Y). However, for a bounded U it is shown that DL depends only on &ggr;Y. Furthermore, for &ggr;Y=ar&bgr;, DL~aUl2L&bgr;-1; i.e., dispersion is Fickian for 0<&bgr;<1, where for 1<&bgr;<2, transport is non-Fickian. Since &bgr;<2, DL cannot grow faster than L=Ut. This is in contrast with a recently proposed model (Neuman, 1990) in which the dispersion coefficient is independent of the plume size and it grows approximately like L1.5. ¿ American Geophysical Union 1994