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 twoādimensional, horizontal structure. A constant head gradient is applied on the formation boundary such that the flow is uniform in the mean. A plume of inert 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 logāconductivity covariance CY of finite integral scale I, at first order in the variance ĻY2 and for a travel distance L = Ut ā« I, dL ā ĻY2UI and transport is coined as Fickian. Ergodicity of the moments is ensured if l ā« 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 ā¼ arĪ² (ā1 < Ī² < 0) for large lag r. Under ergodic conditions, it was found that asymptotically dL ā¼ aUL1+Ī², i.e., nonāFickian 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 ā« 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 largeāscale 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 ā¼ aUl1+Ī² (Figure 3). We also discuss the case in which Y is of stationary increments and is characterized by its variogram Ī³y. Then U and dL can be defined only if Ī³Y is truncated (equivalently, an āinfrared cutoffā is carried out in the spectrum of Y). However, for a bounded U it is shown that DL depends only on Ī³Y. Furthermore, for Ī³Y = arĪ², DL ā¼ aUl2LĪ²ā1; i.e., dispersion is Fickian for 0 < Ī² < 1, whereas for 1 < Ī² < 2, transport is nonāFickian. Since Ī² < 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.