Homogenization of Richards equation in permeability fields with different connectivities

Abstract

[1] Large-scale modeling of transient flow in the unsaturated zone is important for the estimation of the water budget and solute transport in the vadose zone. Upscaled flow models need to capture the impact of small-scale heterogeneities, which are not resolved by the model, on large-scale flow. We perform upscaling of the Richards equation in heterogeneous porous media with continuous distributions of the soil hydraulic parameters using homogenization theory and stochastic averaging techniques. We restrict the analysis to flow regimes in which the capillary-equilibrium assumption holds on the small scale. In order to account for effects of capillary entry pressure we apply the Brooks-Corey model for the soil retention and relative permeability curves and consider Leverett scaling for the coupling of intrinsic permeability and entry pressure. For this model we derive and analyze the ensemble-averaged parameter functions for the macroscopic flow equations. The effects of a definite entry pressure vanish with increasing variance of the log intrinsic permeability. We compare the statistically averaged parameter functions to numerically calculated effective functions for parameter fields with different connectivity properties. These results illustrate that soils with wellconnected coarse materials differ in the relative permeability from those with wellconnected fine materials or those without particular connectedness.

Column-scale unsaturated hydraulic conductivity estimates in coarse-textured homogeneous and layered soils derived under steady-state evaporation from a water table

“…A vast body of literature exists on upscaling issues for various soil water processes including steady-state flow (Yeh and Harvey, 1990;Zhu and Mohanty, 2002;Warrick, 2005;Neuweiler and Eichel, 2006). There are different approaches with regard to scaling of soil hydraulic properties and soil water processes in heterogeneous porous media such as averaging methods (Kitanidis, 1990;Zhu et al, 2004;Zhu and Mohanty, 2006), inverse solutions (van Dam et al, 1994;Hopmans and Simunek, 1999;Zhang et al, 2004;Javaux and Vanclooster, 2006;Vrugt et al, 2008;Erdal et al, 2012), percolation theory (Hunt, 1998;Samouelian et al, 2007;Hunt and Idriss, 2009), homogenization theory (Neuweiler and Cirpka, 2005;Neuweiler and Eichel, 2006;Neuweiler and Vogel, 2007), hybrid mixture theory , 1980, similar media scaling theory (Miller and Miller, 1956;Warrick and Amoozegar-Fard, 1979;Kosugi and Hopmans, 1998;Sadeghi and Jones, 2012;Ojha et al, 2014), dissimilar media scaling methods (Sadeghi et al, 2012b,c), stochastic methods (Yeh et al, 1985;Zhang and Lu, 2002), and renormalization methods (King, 1989;Saucier, 1992;King and Neuweiler, 2002). Many of these methods and their strengths and limitations have been previously reviewed and discussed (e.g., Wen and Gomez-Hernandez, 1996;Pachepsky et al, 2003;Cushman et al, 2002;Chen, 2006;…”

Control-Relevant Upscaling

“…However, we have to be aware of the strong assumptions and approximations made. For further details and reviews of homogenization applications in porous media, we refer to Renard & Marsily (1997), Farmer (2002, Hornung (1997), Van Duijn et al (2002), Neuweiler & Cirpka (2005), and Salimi & Bruining (2009).…”

Effective equations for two-phase flow in porous media: the effect of trapping on the microscale

“…The essence of this approach, together with many examples, is explained in Hornung (1997). Homogenization techniques have been successfully applied by Neuweiler and Cirpka (2005) and Lewandowska et al (2004) to upscale flows in unsaturated porous media. This article is a continuation of van Duijn et al (2002a), where two-phase flow with trapping at the microscale is considered.…”