Unsaturated hydraulic conductivity modeling for porous media with two fractal regimes Allen G. Hunt a,b , Behzad Ghanbarian b, ⁎, Kenneth C. Saville b a Dept. of Physics, Wright State University, Dayton, OH 45435, USA b Dept. of Earth and Environmental Sci., Wright State University, Dayton, OH 45435, USA abstract article info Article history: Received 21 July 2012 Received in revised form 7 May 2013 Accepted 27 May 2013 Available online 21 June 2013 Keywords: Fractal regime Percolation theory Pore–solid fractal model Soil water retention curve Unsaturated hydraulic conductivity A reliable means to predict the saturation-dependence of the hydraulic conductivity would have important applications and implications across soil science. In our efforts to improve predictive capabilities we apply a bimodal pore size distribution to generate simultaneously the soil water retention curve (SWRC) and the unsaturated hydraulic conductivity K in porous media. Our specific pore size model incorporates two fractal regimes, which we treat within the pore–solid fractal approach. The calculation of the hydraulic conductivity employs critical path analysis from percolation theory, which has already been shown to perform the best overall among models commonly employed. To evaluate the developed piecewise functions, 8 soil samples with different textures, e.g., loam, silt loam, sandy loam and clay are selected. All soils show almost the same cross-over point on both water retention and hydraulic conductivity curves on semi-log plots. We find that the piecewise water retention and unsaturated hydraulic conductivity models fit well the measured data. How- ever, the hydraulic conductivity curves predicted from the water retention data agree relatively well with the measured one just for the first regime and tend to underestimate K in the second. We also compare our results with those obtained from unimodal pore-size distribution reported by Ghanbarian-Alavijeh and Hunt (2012). Comparing the measured data with the unimodal and bimodal models indicates that the bimodal distribution provide somewhat more realistic predictions than the unimodal one. If prediction is sacrificed and we simply try to model K using our results, we find that we can generate a very accurate phenomenological description of K with only a slight change in the values of the fractal dimensionality. Reasons for this discrepancy are discussed. © 2013 Elsevier B.V. All rights reserved. 1. Introduction A long-cherished goal of soil physics is the ability to predict the volu- metric water content (θ) dependence of the hydraulic conductivity, K(θ), from knowledge of the water retention curve, θ(h) (where h is tension head). We have recently addressed this problem (Ghanbarian-Alavijeh and Hunt, 2012), showing that application of critical path analysis to a rather simple (monomodal) model, the pore–solid fractal model, gener- ated relatively good predictions, especially compared with other models, e.g., van Genuchten–Mualem (Mualem, 1976; van Genuchten, 1980) in common use. However, we noted some cases where such a simple model of the medium was not realistic enough to capture the structure of the SWRC, which makes the further comparison with experimental values of K(θ) questionable. In some of these cases the SWRC presented a distinct change in slope at an intermediate water content, indicating that the appropriate model of the medium must at least contain two dis- tribution modes, i.e., be a bimodal distribution. In fact, however, this is not a surprising conclusion as it has already been the subject of discussion elsewhere (Hunt and Gee, 2002). The disordered structure of soils can be quantified using statistically self-similar fractal models. However, a fractal model is never more than an approximation to the true structure of soil (Crawford et al., 1995). Typical fractal models presented in the literature consider one fractal di- mension which scales the hierarchical property of a medium within the range of lower and upper limits (e.g., smallest and largest pore or parti- cle radii). However, porous media may have more than one fractal regime. In the literature, soils which show two or more fractal regimes have already been reported (Bittelli et al., 1999; Hunt and Gee, 2002; Pachepsky et al., 1995; Rieu and Sposito, 1991b; Wu et al., 1993). Thus, piecewise fractal approaches have been developed to model particle-size distributions (Millán et al., 2003) and soil water retention curves (Hunt and Gee, 2002; Millán and González-Posada, 2005; Ojeda et al., 2006; Russell, 2010) of soils with two fractal regimes. In addition to bimodal power law distributions, bi- and multi-modal approaches have been also applied to log-normal distributions (Kutilek, 2004; Kutilek and Jendele, 2008; Kutilek et al., 2009; Romano et al., 2011) and sigmoidal functions (Coppola, 2000; Durner, 1994; Othmer et al., 1991; Priesack and Durner, 2006; Spohrer et al., 2006; Zhang Geoderma 207–208 (2013) 268–278 ⁎ Corresponding author. Tel.: +1 937 775 3116; fax; +1 937 775 4997. E-mail address: ghanbarian-alavijeh.2@wright.edu (B. Ghanbarian). 0016-7061/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.geoderma.2013.05.023 Contents lists available at SciVerse ScienceDirect Geoderma journal homepage: www.elsevier.com/locate/geoderma