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

Abstract

Two-dimensional steady-state flow towards a well that fully penetrates a randomly heterogeneous aquifer with a deterministically prescribed constant head boundary is considered. 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. A lower and upper cut-off of the hierarchy, respectively related to the length scale of the domain and data support (sample volume), is introduced. This allows the direct incorporation of 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). An analytical solution for the hydraulic head mean and variance is developed based on recursive approximations of exact nonlocal moment equations. The solution allows assessment of the 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.