Numerical solution of regional scale aquifer flow requires discretizing the transmissivity T. For spatially varying T and by the stochastic approach, T is modeled as a random field. Typically, both the numerical element scale R and the log transmissivity integral scale I are of order of hundreds to thousands meters. Consequently, the upscaled block transmissivity is random and its statistical moments depend on those of the fine scale T, on flow conditions, and on R/I. Modeling Y = ln T as a two‐dimensional normally distributed stationary random field, we have derived [Dagan and Lessoff, Water Resources Research, 43, W05431, doi:10.1029/2006WR005235 2007] the unconditional statistics of the upscaled = ln , accurate to the first‐order in σY2, the log transmissivity variance. Both cases of mean uniform flow [solved previously by Indelman and Dagan, Transport in Porous Media, 12, 161‐183, 1993] and of strongly nonuniform flow in a circular block of radius R centered at a well of radius rw, were considered. Such upscaling applies either to elements in regions of natural gradient flow or those surrounding a projected well, respectively, which are sufficiently far from points of measurement of T. The present article extends the analysis to account for the presence of a measurement removed in the center of the element (e.g., result of a pumping test). The fine scale conditional log transmissivity Yc is modeled as multi‐Gaussian, of nonstationary mean and of two point covariance which depend on the measured value and distance from the measurement location. Upscaling in mean uniform flow is related, for instance, to the case of a pumping well which is not operative, but served to determine T while natural gradient flow conditions prevail in the area. It was found that upscaling has a significant effect upon the mean upscaled log transmissivity that changes from the measured value for R/I ≪ 1 to the effective one 〈Y〉 = ln TG for R/I > 6. Similarly, the conditional log transmissivity variance is greatly reduced compared to the unconditional case. It tends to zero for R/I → 0 or R/I → ∞, and it reaches a maximal value of around 0.15 σY2 for R/I ≃ 1. The second case, of upscaling in well flow, applies to an element with an operative well at its center, which was used to determine the transmissivity Tw. Unlike mean uniform flow, it is found that upscaling has a very small effect for the realistic values of rw/I = 0(10−3) and R/I < 5. In this case the upscaled transmissivity is practically deterministic and equal to the given Tw. A summary of the two articles, (Dagan and Lessoff, 2007) and the present one concludes the paper.
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