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Is transmissivity a meaningful property of natural formations? Conceptual issues and model development
Author(s) -
Dagan G.,
Lessoff S. C.,
Fiori A.
Publication year - 2009
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 - 10.1029/2008wr007410
Subject(s) - hydraulic conductivity , plane (geometry) , constant (computer programming) , flow (mathematics) , mathematics , function (biology) , horizontal plane , square (algebra) , block (permutation group theory) , geometry , vertical plane , mathematical analysis , geology , soil science , structural engineering , evolutionary biology , biology , engineering , soil water , programming language , computer science
At regional scale, it is common to model groundwater flow as 2‐D in the x , y , horizontal plane, by integrating the full 3‐D equations over the vertical. Furthermore, adopting the Dupuit assumption results in the local transmissivity T as a formation property, equal to the vertically integrated hydraulic conductivity K . In practice, the related block transmissivity T b , defined for a volume of area ω (square of side L ) in the horizontal plane and height D , is the property of interest. However, most aquifers are of a heterogeneous 3‐D structure, and Y = ln K is commonly modeled as a normal and stationary random function which is characterized by the variance σ Y 2 , the horizontal I , and vertical I v integral scales. The Dupuit assumption is generally not obeyed for formations of 3‐D spatially variable Y , and transmissivity is no more a meaningful property, independent of flow conditions. Useful generalizations of local and block transmissivity are possible for steady mean uniform flow in the horizontal direction and formations of constant thickness. In that case T and T b become random stationary variables characterized by their mean, variance, and integral scales. These moments are determined for the first time in an analytical form or by a few quadratures, by adopting a first‐order approximation in σ Y 2 , and they depend on the ratio D / I v , e = I v / I and L / I . The block conductivity expected values are compared with the numerical solutions of Dykaar and Kitanidis (1993), and the agreement is very good for σ Y 2 ≤ 1. The main conclusion of the study is that for this simple flow configuration and for common parameter values, T b is practically deterministic and equal to K eff ( e ) D , where K eff is the effective conductivity in uniform flow in an unbounded formation. At regional scale, T b may be regarded as a local property which changes slowly in the horizontal plane. Analysis of numerous field data shows that this variation is also random and characterized by integral scales I reg , of the order of kilometers. The separation of scales makes possible to regard the local T b , as determined along the lines of the present study in a support volume of extent of a few D , as a point value at the regional scale. Practical implications and topics for future investigations are outlined.