Statistical Theory of Groundwater Flow and Transport: Pore to Laboratory, Laboratory to Formation, and Formation to Regional Scale

The statistical approach has been applied increasingly to groundwater flow problems in the last decade, as is illustrated in Figure 3 by the cumulative number of articles published in this field in Water Resources Reseach. This development has been motivated by the recognition of the fact that porous formations are heterogeneous, i.e., with properties which vary in an irregular manner in space. Flow domains are characterized by the length scale L of their spatial extent and three such scales of a fundamental nature are introduced: the laboratory, the local, and the regional scale. Heterogeneity is characterized by the spatial correlation scale I of the property of interest, the three scales corresponding to the above ones being the pore scale, the log hydraulic conductivity, and the log transmissivity integral scales. The medium properties and related flow variables are regarded as random space functions which satisfy two basic requirements: they enjoy some type of stationarity and I ≪ L. An additional scale D is the measurement or computational scale, characterizing the size of the measurement device of a flow variable or the element over which the variable is averaged for computational purposes. In both cases, the interest resides in the space average of the flow variable over a volume or area of length scale D. The primary aim of the theory of flow and transport through porous media is to determine the statistical moments of the space-averaged variables, given the statistical structure of the spatially variable property. The main objective of the study is to show that flow and transport problems at the three fundamental scales can be treated by a unified statistical approach, along this line. The specific aspects of each scale are examined separately, and areas of interest for future research are indicated. In the concluding remarks it is submitted that the statistical approach to groundwater flow has become a comprehensive theory, beyond the stage of an ad hoc modeling technique.