Application of stochastic modelling to numerical solution of groundwater flow: transmissivity upscaling

Stochastic modelling of groundwater flow and transport has undergone a tremendous development in the last 30 years. However, its use in application still lags behind the theoretical developments. Following a strategy outlined in the past (Dagan, 2002), it is suggested that stochastic concepts be applied to numerical solution of groundwater at the regional scale, which is one of the common hydrological modelling activities. The basic approach is to regard the log-transmissivity of the modelled aquifer as random and stationary, characterized by a normal probability distribution function and a two-point covariance (variance, integral scale). Then, the dependent variables to be determined by the numerical solution (head, water flux at grid points) are also random and characterized statistically, in terms of their mean and variance. These values provide measures of uncertainty of the model output as related to the transmissivity spatial variability. Among the various steps required to implement this goal, the one discussed here is that of upscaling, i.e. of attaching values of transmissivity to numerical blocks. Such blocks generally have dimensions of the order of the integral scale of log-transmissivity. The latter was found, by analysing field data, to be of the order of hundreds to thousands of metres. Upscaling procedures are developed in two modes: regarding the upscaled transmissivity as a random field, to be used in Monte Carlo simulations; or determining equivalent transmissivities, that lead directly to the expected value of the dependent variables. Upscaling is carried out for conditions of mean uniform flows, which apply to natural gradients, or to strongly non-uniform but common, well flows. For each case solutions are provided in the unconditional mode (for regions far from measurement points) or the conditional one, near points of transmissivity measurements. By using a firstorder approximation in the log-transmissivity variance, simple upscaling rules are provided.