Is the Dupuit assumption suitable for predicting the groundwater seepage area in hillslopes?

Abstract

Many physically based hydrological/hydrogeological models used for predicting groundwater seepage areas, including topography-based index models such as TOPMODEL, rely on the Dupuit assumption. To ensure the sound use of these simplified models, knowledge of the conditions under which they provide a reasonable approximation is critical. In this study, a Dupuit solution for the seepage length in hillslope cross sections is tested against a full-depth solution of saturated groundwater flow. In homogeneous hillslopes with horizontal impervious base and constant-slope topography, the comparison reveals that the validity of the Dupuit solution depends not only on the ratio of depth to hillslope length d/L (as might be expected), but also on the ratio of hydraulic conductivity to recharge K/R and on the topographic slope s. The validity of the Dupuit solution is shown to be in fact a unique function of another ratio, the ratio of depth to seepage length d/L S . For d/L S < 0.2, the relative difference between the two solutions is quite small (<14% for the wide range of parameter values tested), whereas for d/L S > 0.2, it increases dramatically. In practice, this criterion can be used to test the validity of Dupuit solutions. When d/L S increases beyond that cutoff, the ratio of seepage length to hillslope length L S /L given by the full-depth solution tends toward a nonzero asymptotic value. This asymptotic value is shown to be controlled by (and in many cases equal to) the parameter R/(sK). Generalization of the findings to cases featuring heterogeneity, nonhorizontal impervious base and variable-slope topography is discussed.

Improvement of the hillslope‐storage Boussinesq model by considering lateral flow in the unsaturated zone

“…By incorporating a width function along the length of a hillslope into the traditional Boussinesq equation, Troch et al [2003] developed the Hillslope-storage Boussinesq (HSB) model to more efficiently simulate the subsurface flow in a sloping, unconfined aquifer. The efficiency of the HSB model primarily lies in its one-dimensional (1-D) form and its simplicity of being analytically solved for different problems [Serrano, 1995;Troch et al, 2002;Rocha et al, 2007] based on the Dupuit assumption [Bresciani et al, 2014]. In particular, it allows for a better understanding of the hillslope storage drainage process by resetting the source/sink term within the model [Brutsaert, 1994;Verhoest and Troch, 2000;Troch et al, 2004;Hogarth et al, 2014].…”

Hydrogeological controls of water table‐land surface interactions

“…First, the third dimension (i.e., second horizontal dimension) is neglected. However, where this occurs, interpreting the homogeneous hydraulic conductivity of equation (1) as an equivalent parameter may be a reasonable approach [Bresciani et al, 2014]. The opposite can be expected if the flow is mostly convergent.…”

“…Vertical stratification can yield important variations of the flux with depth. However, where this occurs, interpreting the homogeneous hydraulic conductivity of equation (1) as an equivalent parameter may be a reasonable approach [Bresciani et al, 2014]. Apparently, random variations in hydraulic conductivity may have negligible effects on the water table elevation if the correlation length is small relative to the system dimensions.…”

“…These relationships are governed by highly nonlinear dynamics precluding the existence of simple, exact analytical solutions. These dynamics have been quite well described in hillslopes under the assumption that these form closed hydrologic systems [Beven and Kirkby, 1979;O'Loughlin, 1981;Forster and Smith, 1988;Salvucci and Entekhabi, 1995;Bresciani et al, 2014]. However, it is well known that groundwater divides do not necessarily coincide with topographic divides [Winter, 1999] and that significant intercatchment transfers of both water and solutes can occur at various scales [Cardenas, 2007;Schaller and Fan, 2009].…”

A Velocity-Oriented Approach for Modflow

“…The conditions under which the Dupuit approximation can be justified have been investigated by many authors, see, for instance, Bresciani et al (2014). Using the perturbation method (Nayfeh 1973;Van Dyke 1975), Dupuit approximation (6) can be derived as the approximation of order zero; see Zijl and Nawalany (1993).…”