An iterative Monte Carlo technique for estimating conditional means and variances of transmissivity and hydraulic head fields.

Abstract

Aquifer heterogeneity plays an important role in the remediation of polluted aquifers. Kriging (or cokriging) is one of the popular methods used to interpolate and extrapolate measured transmissivity data. However, depicting aquifer heterogeneity at a high resolution is a difficult task. Traditional numerical inverse approaches often suffer from numerical stability problems. Also, the classical geostatistical inverse methods, e.g. cokriging, rely heavily on the linearized form of the flow equation without incorporating the principles of continuity and boundary effect. Classical cokriging is a linear predictor approach which uses approximate covariance and cross covariance functions. Therefore, it often produces inconsistent transmissivity and head fields which results in unacceptable velocity distributions, especially for highly heterogeneous aquifers. To circumvent these difficulties, an iterative stochastic approach to estimate transmissivity and head distributions in two-dimensional, steady-state, nonuniform flow in heterogeneous confined aquifers is developed. This approach is similar to the classical cokriging technique, using a linear estimator that depends on the approximate covariance functions. The linear estimator is improved successively by solving the governing flow equation and by updating the covariances and cross-covariance functions in an iterative manner. As a result, the nonlinear relationship between transmissivity and head is incorporated. At each iteration, the objective is to minimize the estimation mean square error "MSE" of the estimate of the natural logarithm of transmissivity, Ln (T). The iterations continue until the difference in the simulated variance of the Ln (T) field between two successive iterations is less than a specified tolerance level. The results show that the estimated transmissivity and hydraulic head fields have smaller MSE than those by the classical cokriging even in aquifers with high transmissivity variances. Also, our iterative method is superior to classical cokriging method for producing mass conservative velocity fields and less biased estimation error in both Ln (T) and head fields. Similarly, a comparison study between non-iterative and iterative co-conditional Monte Carlo simulation "MCS" is conducted to show the effect of iterations on the simulation. The Conditional mean fields based on our iterative MCS are better than those non-iterative MCS, which are close to the cokriging ones, in terms of less MSE and less bias.