Transmissivity upscaling in numerical aquifer models of steady well flow: Unconditional statistics

[1] Numerical solution of regional-scale aquifer flow requires discretizing the transmissivity T. Typically, the numerical element scale R and the transmissivity integral scale I are both of the order of hundreds of meters. Then the upscaled block transmissivity (T) over tilde is random, and its statistical moments depend on those of T, on flow conditions, and on R/I. Modeling Y = ln T as a two-dimensional normally distributed stationary random field, we derive unconditional statistics of the upscaled (Y) over tilde = ln (T) over tilde accurate to the first order in sigma(2)(Y) for strongly nonuniform flow in a circular block of radius R centered on a well of radius r(w) in an unbounded domain. After a proper definition, it was found that the ensemble mean [(Y) over tilde] is approximately but not exactly equal to ln T-G ( the geometric mean) and that the variance ratio sigma(2)(Y)/sigma(2)(Y), which depends on r(w)/I and the shape of the correlation rho, drops slowly from unity for R/I = 0 to approximately 0.6 for R/I = 10. Hence the variance of the upscaled transmissivity in radial flow is larger than that determined previously for uniform flow. Additionally, defining the equivalent T-eq as a deterministic value for which the solution of the flow problem renders directly the mean upscaled head drop and specific discharge, we find that in radial flow T-eq congruent to T-H the harmonic mean and grows slowly with increasing R/I, whereas for mean uniform flow T-eq = T-G. Application of the procedure is illustrated for an example of aquifer with selected values of parameters.