The significance of heterogeneity of evolving scales to transport in porous formations

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 inert solute is injected at t = 0 in a volume V 0 . Under ergodic conditions the plume centroid moves with the constant, mean flow velocity U , and a longitudinal macrodispersion coefficient d L 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 C Y of finite integral scale I , at first order in the variance σ Y 2 and for a travel distance L = Ut ≫ I , d L → σ Y 2 UI 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 C Y is stationary but of unbounded integral scale with C Y ∼ ar β (−1 < β < 0) for large lag r . Under ergodic conditions, it was found that asymptotically d L ∼ aUL 1+β , 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 C Y 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 D L 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 D L ∼ aUl 1+β (Figure 3). We also discuss the case in which Y is of stationary increments and is characterized by its variogram γ y . Then U and d L can be defined only if γ Y is truncated (equivalently, an “infrared cutoff” is carried out in the spectrum of Y ). However, for a bounded U it is shown that D L depends only on γ Y . Furthermore, for γ Y = ar β , D L ∼ aUl 2 L β−1 ; i.e., dispersion is Fickian for 0 < β < 1, whereas for 1 < β < 2, transport is non‐Fickian. Since β < 2, D L 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 L 1.5 .