Effective Parameter Functions for the Richards Equation in Layered Porous Media

Abstract

Upscaling of the Richards equation is important for large‐scale modeling of water flow in the unsaturated zone. We derive an upscaled model for Richards' equation in a layered porous medium. Homogenization theory is applied to derive the upscaled equations for slow flow processes. Two flow regimes are compared. The first flow regime is quantified by small Bond numbers, meaning forces due to pressure gradients are dominant on the small scale. The second flow regime is quantified by large Bond numbers, meaning that forces due to pressure gradients and gravity contribute equally on the small scale, while the large scale is dominated by gravity forces. The case of intermediate Bond numbers is also addressed. We compare the effective relative permeability function and the effective capillary pressure head–saturation function for both flow regimes. In the case of small Bond numbers the effective curves can be derived based on capillary equilibrium. The procedure to calculate effective curves is then quite simple. In the case of large Bond numbers the procedure is more complex. However, comparing effective curves of different test cases showed that the difference between the effective curves for different Bond numbers is not that large for moderate parameter contrasts and when the gradients of the capillary–saturation curve do not become too small. The effective parameter functions calculated with a capillary equilibrium assumption are therefore often a good estimate.

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

“…The presented results are consistent with findings of other relevant studies of unsaturated hydraulic conductivity (e.g., Sarris and Paleologos, 2003;Warrick, 2005). Based on homogenization theory, Neuweiler and Eichel (2006) validated this result analytically for the same periodic heterogeneity structure with layer thicknesses much smaller than the profile depth. An advantage of the presented upscaling approach is that it is not restricted to binary heterogeneity structures, which is considered a limitation of homogenization theory.…”

“…When the layer thicknesses are decreased, the effective curve approaches a homogeneous soil profile. This result is consistent with ''homogenization theory'' where a microscopically heterogeneous soil profile is represented as homogeneous at the macroscopic scale when the scale of heterogeneity is much smaller than the scale considered for modeling (e.g., Neuweiler and Eichel, 2006). Fig.…”

“…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;…”

Unsaturated Hydraulic Conductivity for Evaporation in Heterogeneous Soils

“…Two widely used upscaling approaches in vadose zone flows include homogenization theory and Monte Carlo type of simulations based on the stream tube approximation. In the homogenization theory, an upscaled flow equation is developed based on a separation of length scales in the medium for the limit at which the typical length scale of heterogeneities became negligible compared to the size of the medium (e.g., Sviercoski et al, 2009;Neuweiler & Eichel, 2006;Neuweiler & Cirpka, 2005;Lewandowska & Laurent, 2001). The second approach is the stream tube approach in which the heterogeneous field is conceptualized as a series of vertically homogeneous and horizontally independent stream tubes or parallel columns Govindaraju et al, 1992;Rubin & Or, 1993;Chen et al, 1994a,b;Toride & Leij, 1996a, b;Kim et al, 1997;Wildenschild & Jensen, 1999;Zhu & Mohanty, 2002b).…”

Introduction to the Special Section in Vadose Zone Journal: Parameter Identification and Uncertainty Assessment in the Unsaturated Zone

“…Neuweiler and Eichel (2006) derive an upscaled model for Richards' equation in a layered porous medium using homogenization theory for small‐scale capillary‐dominated, intermediate, and large‐scale gravity‐dominated flow regimes. They find that effective parameter functions describing relative permeability and capillary pressure head versus saturation can be estimated with reasonable accuracy based on a capillary equilibrium assumption.…”