Spatially variable transmissivity T of aquifers is modeled as random. Analysis of field data [Water Resour. Res. 21 (1985) 563] indicate that the logtransmissivity Y = ln T is normal and its covariance can be characterized by three parameters: the variance sigma(Yc)(2) and the integral scale I-gamma of correlated residuals and a nugget sigma(Yn)(2), representing variability of small support. The equation of flow is stochastic and the head H is also random. The head-logtransmissivity cross-covariance C-HY and the head variogram Gamma(H) can be used conveniently to solve the direct and inverse problems. These covariances are derived for an unbounded domain and for mean uniform flow of constant head gradient - J. Under these conditions, analytical expressions were determined in the past by first-order approximation in sigma(Yc)(2), pertinent to weak heterogeneity. The aim of the present study is to derive C-HY and Gamma(H) for highly heterogeneous aquifers of total variance sigma(Yc)(2) less than or equal to 4. This goal is achieved by adopting a multi- indicator model of the aquifer consisting of circular inclusions of radius R and of normal logtransmissivity of variance sigma(Y)(2), submerged in a matrix of effective transmissivity T-G (geometric mean). The system is characterized by sigma(Y)(2), the integral scale I-gamma = 8R/(3pi) and the volume fraction of inclusions n, which are simply related to the aquifer parameters sigma(Yc)(2), sigma(Yn)(2) and I-gamma. The flow problem is solved numerically at high accuracy by the analytic element method. The medium is modeled by 50,000 inclusions and parameters values are sigma(Y)(2) = 0.1, 1, 2, 4 and n = 0.4, 0.65, 0.9. Analytical solutions' are derived by the effective medium approximation (EMA), in which each inclusion is regarded as submerged in a medium of effective transmissivity, and by first-order approximation (FAO in sigma(Y)(2)). Comparison between the numerical and analytical solutions shows that C-YH is overestimated by FOA and is in agreement with the EMA. The head variogram is in agreement with EMA for n less than or equal to 0.65, but underestimated for n = 0.9, when it is close to the FOA. The latter effect results from cancellation of errors. An outline of application of results concludes the study. (C) 2004 Elsevier B.V. All rights reserved.
Dagan, G., Fiori, A., Jankovic, I. (2004). Transmissivity and head covariances for flow in highly heterogeneous aquifers RID A-2321-2010. JOURNAL OF HYDROLOGY, 294(1-3), 39-56 [10.1016/j.jhydrol.2003.10.022].
Transmissivity and head covariances for flow in highly heterogeneous aquifers RID A-2321-2010
FIORI, ALDO;
2004-01-01
Abstract
Spatially variable transmissivity T of aquifers is modeled as random. Analysis of field data [Water Resour. Res. 21 (1985) 563] indicate that the logtransmissivity Y = ln T is normal and its covariance can be characterized by three parameters: the variance sigma(Yc)(2) and the integral scale I-gamma of correlated residuals and a nugget sigma(Yn)(2), representing variability of small support. The equation of flow is stochastic and the head H is also random. The head-logtransmissivity cross-covariance C-HY and the head variogram Gamma(H) can be used conveniently to solve the direct and inverse problems. These covariances are derived for an unbounded domain and for mean uniform flow of constant head gradient - J. Under these conditions, analytical expressions were determined in the past by first-order approximation in sigma(Yc)(2), pertinent to weak heterogeneity. The aim of the present study is to derive C-HY and Gamma(H) for highly heterogeneous aquifers of total variance sigma(Yc)(2) less than or equal to 4. This goal is achieved by adopting a multi- indicator model of the aquifer consisting of circular inclusions of radius R and of normal logtransmissivity of variance sigma(Y)(2), submerged in a matrix of effective transmissivity T-G (geometric mean). The system is characterized by sigma(Y)(2), the integral scale I-gamma = 8R/(3pi) and the volume fraction of inclusions n, which are simply related to the aquifer parameters sigma(Yc)(2), sigma(Yn)(2) and I-gamma. The flow problem is solved numerically at high accuracy by the analytic element method. The medium is modeled by 50,000 inclusions and parameters values are sigma(Y)(2) = 0.1, 1, 2, 4 and n = 0.4, 0.65, 0.9. Analytical solutions' are derived by the effective medium approximation (EMA), in which each inclusion is regarded as submerged in a medium of effective transmissivity, and by first-order approximation (FAO in sigma(Y)(2)). Comparison between the numerical and analytical solutions shows that C-YH is overestimated by FOA and is in agreement with the EMA. The head variogram is in agreement with EMA for n less than or equal to 0.65, but underestimated for n = 0.9, when it is close to the FOA. The latter effect results from cancellation of errors. An outline of application of results concludes the study. (C) 2004 Elsevier B.V. All rights reserved.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
