THE SIGNIFICANCE OF HETEROGENEITY OF EVOLVING SCALES TO TRANSPORT IN POROUS FORMATIONS

Flow takes place in a heterogeneous formation of spatially variable co nductivity, 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 gra dient is applied on the formation boundary such that the flow is unifo rm in the mean. A plume of inert solute is injected at t = 0 ina volum e V-0. Under ergodic conditions the plume centroid moves with the cons tant, mean flow velocity U, and a longitudinal macrodispersion coeffic ient d(L) may be defined as half of the time rate of change of the plu me second spatial moment with respect to the centroid. For a log-condu ctivity covariance C-Y of finite integral scale I, at first order in t he variance sigma(Y)(2) and for a travel distance L = Ut much greater than I, D-L --> sigma(Y)(2)UI and transport is coined as Fickian. Ergo dicity of the moments is ensured if l much greater than I, where l is the initial plume scale. Some field observations have suggested that h eterogeneity may be of evolving scales and that the macrodispersion co efficient may grow with L without reaching a constant limit (anomalous diffusion). To model such a behavior, previous studies have assumed t hat C, is stationary but of unbounded integral scale with C-Y similar to ar(beta) (-1 < beta < 0) for large lag r. Under ergodic conditions, it was found that asymptotically d(L) similar to aUL(1+beta), i.e.,, non-Fickian; behavior and anomalous dispersion. The present study clai ms that an ergodic behavior is not possible for a given finite plume o f initial size l, since the basic requirement that l much greater than 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 effe ctive 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 experime nt, We show that in contrast with d(L), the behavior of D-L is control led by 1 and it has the Fickian limit D-L similar to aUI(1+beta) (Figu re 3). We also discuss the case in which Y is of stationary increments and is characterized by its variogram gamma(Y). Then U and d(L) Can b e defined only if gamma(Y) is truncated (equivalently, an ''infrared c utoff' is carried out in the spectrum of Y). However, for a bounded U it is shown that D-L depends only on gamma(Y). Furthermore, for gamma( Y) = ar(beta), D-L similar to aUl(2)L(beta-1); i.e., dispersion is Fic kian for 0 < beta < 1, whereas for 1 < beta < 2, transport is non-Fick ian. Since beta < 2, D-L cannot grow faster than L = Ut. This is in co ntrast with a recently proposed model (Neuman, 1990) in which the disp ersion coefficient is independent of the plume size and it grows appro ximately like L(1.5).