Is transmissivity a meaningful property of natural formations? Conceptual issues and model development

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 Γas 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= 1nK is commonly modeled as a normal and stationary random function which is characterized by the variance σ²y, the horizontal /, 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 σ²_Y, 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 σ²_Y < 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.