Efficient Methods and Applications of Uncertainty Propagation in Non-Stationary Formations
Evaluating uncertainty in flow and solute transport under non-stationary conditions is a computationally demanding task. This is particularly true for cases with a two-point covariance function of log-conductivity depending on the actual positions of the points rather than their distance vector. The latter may be the case when the geological formation exhibits a trend. Non-stationarity can also be the result of uncertainty in trend parameters of the mean log-conductivity value, or it may originate from conditioning of the log-conductivity field to head and conductivity measurements. I present efficient numerical methods to evaluate the covariance of dependent quantities in such formations by applying a matrix-based first-order second-moment (FOSM) method for uncertainty propagation based on Fast Fourier Transformation (FFT) techniques. The applications cover the uncertainty of travel time observed at a control plane, and the uncertainty of discharges. The latter application is used to estimate the probability of failure in a permeable-reactive-barrier design with a funnel-and-gate setup for hydraulic control. I show how conditioning to head and conductivity measurements helps to reduce the uncertainty in engineering design thus allowing for a smaller margin of safety. Optimum locations for additional measurements can be determined by evaluating the expected decrease in uncertainty due the measurement prior to taking the actual measurement. These techniques may be used in the planning phase of remediation schemes in which costs for additional site characterization are to be compared to costs of designs with a larger margin of safety.