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

Abstract

[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 e T 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 e Y = ln e T accurate to the first order in s 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 hỸ i is approximately but not exactly equal to ln T G (the geometric mean) and that the variance ratio s e Y 2 /s Y 2 , which depends on r w /I and the shape of the correlation r, 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 ffi 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.Citation: Dagan, G., and S. C. Lessoff (2007), Transmissivity upscaling in numerical aquifer models of steady well flow: Unconditional statistics, Water Resour. Res., 43, W05431,

Estimating parameters of aquifer heterogeneity using pumping tests – implications for field applications

“…Moreover, this function has to depend on the variance and correlation scale. Most of the studies describe well flow in two-dimensional heterogeneous porous media, thereby representing large scale pumping tests [9][10][11][12][13][14][15][16][17][18]. For small scale pumping tests, vertical flow in the vicinity of the pumping well is a http://dx.doi.org/10.1016/j.advwatres.2015.05.021 0309-1708/© 2015 Elsevier Ltd. All rights reserved.…”

Beyond Thiem: A new method for interpreting large scale pumping tests in heterogeneous aquifers

“…To this end, effective and equivalent quantities differ not only by how they are derived (ensemble averaging or spatial averaging) but also in what they are used for since they address different features of radially convergent flows as we will show in this study. An excellent review on existing literature is presented by Sánchez‐Vila and coworkers [ Sánchez‐Vila et al , 2006] as well as by Dagan and Lessoff [ Dagan and Lessoff , 2007].…”

“…On large scales, the aquifer thickness can be considered as much smaller than the horizontal range of the influence of the pumping test and such large aquifers can be treated as effectively two dimensional systems. Dagan and Lessoff [ Dagan and Lessoff , 2007, 30] conclude that K efu times the aquifer thickness determines the large scale transmissivity to be used in large scale flow simulations. On this large scale, transmissivity is a point value and due to large scale heterogeneities again heterogeneously distributed in space [ Indelman and Dagan , 2006].…”

Technical note: Analytical drawdown solution for steady-state pumping tests in two-dimensional isotropic heterogeneous aquifers

“…Effective or equivalent descriptions of transmissivity in pumping tests have been investigated, e.g. by Desbarats (1992), Sánchez-Vila et al (1999), Neuman et al (2007), Dagan and Lessoff (2007), Schneider and Attinger (2008), and many more. For a detailed review see Sánchez-Vila et al (2006).…”

“…T eq (r) = − Q w 2π(h(r)−h(r w )) ln r r w was derived from Thiem's solution (Eq. 1) (Matheron, 1967;Indelman et al, 1996;Dagan and Lessoff, 2007). In this sense, the equivalent transmissivity T eq is defined as the value for …”