Consistent parameter constraints for soil hydraulic functions Andre Peters a,⇑ , Wolfgang Durner b , Gerd Wessolek a a Institut für Ökologie, Technische Universität Berlin, Germany b Institut für Geoökologie, Technische Universität Braunschweig, Germany article info Article history: Received 11 March 2011 Received in revised form 19 July 2011 Accepted 19 July 2011 Available online 4 August 2011 Keywords: Soil hydraulic properties Parameter estimation Water retention Hydraulic conductivity abstract Parameters of functions to describe soil hydraulic properties are derived from measurements by means of parameter estimation. Of crucial importance here is the choice of correct constraints in the parameter space. Often, the parameters are mere shape parameters without physical meaning, giving flexibility to the model. A fundamental requirement is that the hydraulic functions are monotonic: the retention func- tion and the conductivity function can only decrease as the capillary suction increases. A stricter physical requirement for the conductivity function is that its decrease with respect to saturation is at least linear. This linear decrease would occur if all pores of a capillary bundle had an equal radius. In the first part of this contribution, we derive constraints for the so-called tortuosity parameter of the Mualem conductiv- ity model, which allow highest possible flexibility on one hand and guarantee physical consistency on the other hand. In combination with the retention functions of Brooks and Corey, van Genuchten, or Durner, such a constraint can be expressed as a function of the pore-size distribution parameters. In the second part, we show that a common modification of retention models, which is applied to reach zero water con- tent at finite suction, can lead to the physically unrealistic case of increasing water content with increas- ing suction. We propose a solution for this problem by slightly modifying these models and introducing a correct parameter constraint. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Water transport in unsaturated soils is usually described by the Richards equation. Solving this equation requires specified initial and boundary conditions and knowledge of the constitutive rela- tionships between the volumetric water content, h [–], capillary pressure head, h [L] and hydraulic conductivity, K [L T 1 ]. These relationships are expressed by two hydraulic functions, i.e. the water-retention function, h(h), and the hydraulic conductivity func- tion, K(h). If evaporation and/or root water uptake processes shall be modeled, these functions must be known from the wet to the very dry range, where measurements are difficult to conduct. The most widely used unimodal mathematical expressions for h(h) are the models of Brooks and Corey [1], van Genuchten [2] and the more recently published model of Kosugi [3]. A well estab- lished h(h)-model for structured soils is the bimodal model of Durner [4]. Usually the water retention functions assume a distinct residual water content, h r , and reach this value asymptotically with a water capacity (i.e., the slope @h/@h) that approaches zero. The parameter h r is either interpreted as the water held by adsorptive forces [5] or as a mere fitting parameter. In the very dry range, even the adsorptive water content finally reaches a value of 0 and thus the concept of residual water content does not hold. To account for this discrepancy, Campbell and Shiozawa [6] suggested that the water content of dry porous media decreases linearly on a semi-log plot of the retention function. Fayer and Simmons [7] combined this approach with the common h(h) functions of Brooks and Corey and van Genuchten. More recently, this approach was also used by Khlosi et al. [8] in combi- nation with the Kosugi model. The commonly used hydraulic conductivity functions are de- rived from retention functions by means of pore-bundle models [9–11], using the law of Hagen–Poiseuille of capillary flow and some assumptions about tortuosity, connectivity and spatial distri- bution of the capillaries. One of the main advantages of this approach is the small number of additional parameters that are required to describe K(h). The Mualem model [10] for instance needs only two extra parameters: the saturated conductivity, K s [L T 1 ], or any other measured conductivity at a certain pressure head to scale the relative conductivity function, and an empirical parameter, s [–], accounting for tortuosity and connectivity. Since the K(h) functions are often used for predictive purposes, where no information for K(h) is available, s is mostly fixed to 0.5 in accordance to the original work of Mualem [10]. However, the physical meaning of s is questioned [12]. If data of K(h) are avail- able, s is often treated as a free fitting parameter that is frequently negative [13]. Negative values express an apparent tortuosity reduction with decreasing water content. This is physically not fea- sible, and indicates a compensation for a conceptual deficiency of 0309-1708/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2011.07.006 ⇑ Corresponding author. E-mail address: andre.peters@tu-berlin.de (A. Peters). Advances in Water Resources 34 (2011) 1352–1365 Contents lists available at ScienceDirect Advances in Water Resources journal homepage: www.elsevier.com/locate/advwatres