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A stochastic analysis of two‐dimensional steady state groundwater flow in a bounded domain is carried out by using Monte Carlo techniques. The flow domain is divided into a set of square blocks. A nearest‐neighbor stochastic process model is used to generate a multilateral spatial dependence between hydraulic conductivity values in the block system. Both statistically isotropic and statistically anisotropic autocorrelation functions are considered. This model leads to a realistic representation of the spatial variations in hydraulic conductivity in a discrete block medium. Results of the simulations provide estimates of the output distributions in hydraulic head. The probability distribution for hydraulic head must be interpreted in terms of the spatial variation of the expected head gradients, the standard deviation in the hydraulic conductivity distribution, the ratio of the integral scales of the autocorrelation function for conductivity to the distance between boundaries on the flow domain, and the arrangement of statistically homogeneous units within the flow domain. The standard deviation in hydraulic head increases with an increase in either the standard deviation in hydraulic conductivity or the strength of the correlation between neighboring conductivity values. The standard deviations in hydraulic head are approximately halved when a uniform, bounded, two‐dimensional flow field is reduced to one‐dimensional form. The uncertainties in the predicted hydraulic head values are strongly influenced by the presence of a spatial trend in the mean hydraulic conductivity. In evaluating the concept of an effective conductivity for a heterogeneous medium, both the nature of the spatial heterogeneities in hydraulic conductivity and the flow system operating within the flow domain must be considered.
Smith et al. (Sat,) studied this question.