Correction of the Buckingham–Darcy Law for flow of high strength salts in variably saturated porous media Erick R. Burns a, * , Maria I. Dragila b , John S. Selker c , Ronald B. Guenther d , Jean-Yves Parlange e , Noam Weisbrod f a Department of Geology and Geophysics, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N 1N4 b Department of Crop and Soil Science, Oregon State University, 3017 Agriculture and Life Science Building, Corvallis, OR 97331-3002, United States c Department of Bioengineering, Oregon State University, 116 Gilmore Hall, Corvallis, OR 97331, United States d Department of Mathematics, Oregon State University, Kidder Hall 368, Corvallis, OR 97331-4605, United States e Department of Biological and Environmental Engineering, Cornell University, 228 Riley-Robb Hall, Ithaca, NY 14853-5701, United States f Department of Environmental Hydrology & Microbiology, Zuckerberg Institute for Water Research, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boker Campus 84990, Israel Received 20 September 2005; received in revised form 3 April 2006; accepted 17 April 2006 Available online 13 June 2006 Abstract The Buckingham–Darcy Law is used to describe fluid flow in unsaturated porous media at low Reynolds number. In order to provide a priori corrections to this law, a process thermodynamic approach is utilized to ascertain the functional dependence. Using this knowl- edge, corrections to the hydraulic conductivity coefficient are proposed and compared with available data. The proposed corrections sub- stantially predict the observed behavior of flow of high concentration (saturated) sodium chloride solutions in porous media. During the derivation, physical principles consistent with the thermodynamics of the system were utilized. A review of these principles and their results provides an alternative form of the generalized Gibbs–Duhem Relation for continuous processes, indicating that the identical equivalence to zero is unlikely to occur for dissipative processes. Further, the postulated Gibbs and Gibbs–Duhem Relations indicate that special differential operators need to be used for continuous processes rather than the usual use of a generic differential. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Process thermodynamics; Darcy; Buckingham; Salt; Saline; Unsaturated 1. Introduction The goal of this paper is to place the Buckingham– Darcy Law into the proper context with respect to process thermodynamics. Then, using this context, the proper cor- rections to the Buckingham–Darcy Law coefficient are pos- tulated for high salt strength aqueous solutions in unsaturated porous media. Finally, the postulated correc- tions are tested against data collected by Scotter [22]. Darcy’s Law is an empirical linear relationship that has been shown to describe the flow (at sufficiently low Rey- nolds number) of liquid in saturated porous media. Buck- ingham subsequently extended Darcy’s Law to the case of unsaturated flows in porous media [4]. Since this time, other important gradients have been examined for their effects on liquid flow. These include gradients in salt con- centration (cf., [15,16]) and temperature (cf., [17,21,24]). Approaches to handle salt gradients include postulation of an osmotic pressure for incorporation into a Darcy-like law (cf., [16]) and the use of Onsager Relations in process thermodynamic formulations of the flow problem and sub- sequent experimental determination of osmotic efficiency (cf., [15]). The following uses a process thermodynamic approach, but rather than assuming that formulation of 0309-1708/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2006.04.010 * Corresponding author. E-mail addresses: erick@lifetime.oregonstate.edu (E.R. Burns), Maria. Dragila@oregonstate.edu (M.I. Dragila), selkerj@engr.orst.edu (J.S. Selker), guenth@math.orst.edu (R.B. Guenther), jp58@cornell.edu (J.-Y. Parlange), weisbrod@bgu.ac.il (N. Weisbrod). www.elsevier.com/locate/advwatres Advances in Water Resources 30 (2007) 469–482