Depth‐averaging effects on hydraulic head for media with stochastic hydraulic conductivity

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Abstract
Hydraulic conductivity of a porous medium frequently is considered to be a single realization of a three‐dimensional spatial stochastic process. The most common observations of flow in porous media are hydraulic‐head measurements obtained from wells which are screened over extensive sections of the medium. These measurements represent, approximately, a one‐dimensional spatial average of the actual three‐dimensional head distribution, the actual head distribution being a stochastic process resulting from flow through a random hydraulic‐conductivity field. This paper examines, via ensemble averages, the effect of such spatial averages of groundwater flow on the spatial autocovariance function for a simple, yet viable, stochastic model of a bounded medium. The model is taken to be three‐dimensional flow in a medium that is bounded above and below and in which the hydraulic conductivity is a second‐order stationary stochastic process. Spatial averaging of the hydraulic heads is considered to take place over the entire thickness of the medium. Ensemble variances and autocorrelations for depth‐averaged heads are computed for the resulting two‐dimensional flow system and compared with those from a fully three‐dimensional flow system.