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Stochastic analysis of spatial variability in subsurface flows: 2. Evaluation and application
Author(s) -
Gutjahr Allan L.,
Gelhar Lynn W.,
Bakr Adel A.,
MacMillan John R.
Publication year - 1978
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/wr014i005p00953
Subject(s) - hydraulic conductivity , harmonic mean , logarithm , geometric mean , mathematics , standard deviation , flow (mathematics) , homogeneous , mathematical analysis , porous medium , mean flow , conductivity , statistics , geometry , mechanics , porosity , physics , soil science , geotechnical engineering , geology , turbulence , combinatorics , quantum mechanics , soil water
The stochastic differential equation describing one‐dimensional flow in a statistically homogeneous porous medium is solved exactly, and the results are compared with an approximate solution considering small perturbations in the logarithm of the hydraulic conductivity. The results show that the logarithmic approximation is valid when the standard deviation of the natural logarithm of the hydraulic conductivity σ f is less than 1; the errors increase rapidly for σ f > 1. The effective hydraulic conductivity of statistically homogeneous media with one‐, two‐, and three‐dimensional perturbations is determined to the first order in σ f 2 . The effective conductivity is found to be the harmonic mean for one‐dimensional flow, the geometric mean for two‐dimensional flow, and (1 + σ f 2 /6) times the geometric mean for three‐dimensional flow. The application of stochastic analysis is illustrated through two elementary network design problems that demonstrate the importance of the correlation length of the hydraulic conductivity and the role of measurement error.