Effects of evolving scales of heterogeneity on hydraulic head predictions under convergent flow conditions

We consider two-dimensional steady state flow towards a well that fully penetrates a randomly heterogeneous aquifer, with deterministically prescribed constant head boundary. Flow occurs over an infinite hierarchy of mutually uncorrelated, statistically homogeneous and isotropic random fields (modes) of natural log-transmissivity, Y, each of which is associated with a Gaussian variogram. We consider a lower and upper cut-off of the hierarchy, respectively related to the length scale of the domain and data support (sample volume). This allows directly incorporating the scale dependence of the integral scale of Y into groundwater flow (ensemble) moments and leads to a geostatistical description of the system in terms of a (stationary) Truncated Power Variogram (TPV). We then develop an analytical solution for hydraulic head mean and variance based on recursive approximations of exact nonlocal moment equations. Our solution allows to assess functional dependences of the distribution of the leading (statistical) moments of hydraulic head on parameters of the variogram associated with the hierarchy of log-transmissivity modes. The latter can be determined, for instance, along the lines of Neuman et al. (2008).