TY - JOUR

T1 - Transmissivity upscaling in numerical aquifer models of steady well flow

T2 - Unconditional statistics

AU - Dagan, G.

AU - Lessoff, S. C.

PY - 2007/5

Y1 - 2007/5

N2 - 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 T̃ 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 Ỹ = ln T̃ accurate to the first order in σY2 for strongly nonuniform flow in a circular block of radius R centered on a well of radius rw in an unbounded domain. After a proper definition, it was found that the ensemble mean 〈Ỹ〉 is approximately but not exactly equal to ln TG (the geometric mean) and that the variance ratio σỸ 2/σY2 which depends on rw/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 Teq 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 Teq ≅ TH the harmonic mean and grows slowly with increasing R/I, whereas for mean uniform flow Teq = TG. Application of the procedure is illustrated for an example of aquifer with selected values of parameters.

AB - 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 T̃ 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 Ỹ = ln T̃ accurate to the first order in σY2 for strongly nonuniform flow in a circular block of radius R centered on a well of radius rw in an unbounded domain. After a proper definition, it was found that the ensemble mean 〈Ỹ〉 is approximately but not exactly equal to ln TG (the geometric mean) and that the variance ratio σỸ 2/σY2 which depends on rw/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 Teq 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 Teq ≅ TH the harmonic mean and grows slowly with increasing R/I, whereas for mean uniform flow Teq = TG. Application of the procedure is illustrated for an example of aquifer with selected values of parameters.

UR - http://www.scopus.com/inward/record.url?scp=34347259547&partnerID=8YFLogxK

U2 - 10.1029/2006WR005235

DO - 10.1029/2006WR005235

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AN - SCOPUS:34347259547

VL - 43

JO - Water Resources Research

JF - Water Resources Research

SN - 0043-1397

IS - 5

M1 - W05431

ER -