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

Abstract

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 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 = ln accurate to the first order in σ Y 2 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 〈 〉 is approximately but not exactly equal to ln T G (the geometric mean) and that the variance ratio 2 / σ Y 2 , which depends on r w / I and the shape of the correlation ρ , 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 ≅ 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.