Explicit Monte Carlo Simulation head moment estimates for stochastic confined groundwater flow

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Abstract
A methodology is presented for expressing the head moment estimates as explicit functions of the pumping rates for confined groundwater flow with parameter uncertainty. The technique combines a single Monte Carlo simulation (MCS) and the statistical moment estimators to express the mean head estimate as an explicit linear function of the pumping rates and the head variance estimate as an explicit quadratic function of the pumping rates. With a change in the pumping rates, instead of again applying MCS, the MCS head moment estimates can be found from the explicit equations with trivial computational effort. The results are limited to systems where the dependent variable can be expressed as a linear function of the control variable. To demonstrate the results, a simulation example is presented for steady confined groundwater flow, with the transmissivity treated as a spatially correlated, lognormally distributed random variable. The explicit MCS head moment estimates are also used in a simple management model for dewatering under parameter uncertainty. The objective is to minimize the total pumping, while meeting constraints on the probability of the head exceeding an upper bound at the control nodes. The explicit MCS head moment estimates and a Gaussian assumption for the head distribution are used in a chance constraint formulation to express the head level constraints as explicit functions of the pumping rates. Solution of the management model requires modest computational effort due to the explicit nature of the head level constraints. Also examined is the effect on the optimal solution of the management model of (1) changes in the acceptable probability of exceeding the head upper bound, (2) changes in the number of samples used for the explicit MCS head moment estimates, and (3) the Gaussian assumption for the head distribution.