Low-dimensional modeling of hillslope subsurface flow: Relationship between rainfall, recharge, and unsaturated storage dynamics

We present a coupling between the one-dimensional Richards equation for vertical unsaturated flow and the one-dimensional hillslope-storage Boussinesq equation (HSB) for lateral saturated flow along complex hillslopes. Here the capillary fringe is included in the flow domain as an integral part of the Boussinesq aquifer. The coupling allows quantitative investigation of the role of unsaturated storage in the relationship between rainfall and recharge. The coupled model (HSB coupled) is compared to the original HSB model (HSB original) and a three-dimensional Richards equation (RE) based model (taken to be the benchmark) on a set of seven synthetic hillslopes, ranging from convergent to divergent. Using HSB original, the water tables are overestimated and the outflow rates are generally underestimated, and there is no delay between rainfall and recharge. The coupled model, however, shows a remarkably good match with the RE model in terms of outflow rates, and the delay between rainfall and recharge is captured well. We also see a clear improvement in the match to the water tables, even though the values are still overestimated for some hillslope shapes, in particular the convergent slopes. We show that for the hillslope configurations and scenarios examined in this paper it is possible to reproduce hydrographs and water table dynamics with a good degree of accuracy using a low-dimensional hydrological model

Analytical solution of the linearized hillslope-storage Boussinesq equation for exponential hillslope width functions

This technical note presents an analytical solution to the linearized hillslope-storage Boussinesq equation for subsurface flow along complex hillslopes with exponential width functions and discusses the application of analytical solutions to storage-based subsurface flow equations in catchment studies

Predictive Modeling of Envelope Flood Extents Using Geomorphic and Climatic-Hydrologic Catchment Characteristics

A topographic index (flood descriptor) that combines the scaling of bankfull depth with morphology was shown to describe the tendency of an area to be flooded. However, this approach depends on the quality and availability of flood maps and assumes that outcomes can be directly extrapolated and downscaled. This work attempts to relax these problems and answer two questions: (1) Can functional relationships be established between a flood descriptor and geomorphic and climatic-hydrologic catchment characteristics? (2) If so, can they be used for low-complexity predictive modeling of envelope flood extents? Linear stepwise and random forest regressions are developed based on classification outcomes of a flood descriptor, using high-resolution flood modeling results as training benchmarks, and on catchment characteristics. Elementary catchments of four river basins in Europe (Thames, Weser, Rhine, and Danube) serve as training data set, while those of the Rh\uf4ne river basin in Europe serve as testing data set. Two return periods are considered, the 10- and 10,000-year. Prediction of envelope flood extents and flood-prone areas show that both models achieve high hit rates with respect to testing benchmarks. Average values were found to be above 60% and 80% for the 10- and the 10,000-year return periods, respectively. In spite of a moderate to high false discovery rate, the critical success index value was also found to be moderate to high. It is shown that by relating classification outcomes to catchment characteristics, the prediction of envelope flood extents may be achieved for a given region, including ungauged basins

Low-dimensional modeling of hillslope sub-surface flow processes : developing and testing the hillslope-storage Boussinesq model

Key words: hillslope hydrology, low-dimensional modeling, Boussinesq equation, Richards equation, water table dynamics.In this thesis the focus is on investigating the hillslope hydrological behavior, as a crucial part in understanding the catchment hydrological response. To overcome difficulties associated with most of the existing models, i.e., high computational costs and difficulties in model parameterization and calibration, the focus is on low-dimensional physically based models. The central question is whether it is possible to formulate low-dimensional physically based models such that the essential physical behavior of the natural system is preserved.Several models are compared to each other and to measurements from a laboratory and from a field site. The comparisons are carried out for a diversity of bedrock slopes, hillslope shapes, and profile curvatures. The low-dimensional models that are formulated are essentially based on the hillslope-storage Boussinesq (HSB) model. Several modifications to the model are conducted in order to investigate the model response under different conditions: a) a generalized HSB formulation is derived that allows model evaluation on curved bedrock profiles, b) a new HSB model is derived in which the effects of capillarity are partly accounted for based on the assumption of hydraulic equilibrium in the unsaturated zone, and c) a coupled saturated-unsaturated HSB model is derived in which the unsaturated zone dynamics are described with a one-dimensional Richards model. Evaluation of the developed low-dimensional models generally shows that: a) the HSB model saturated zone dynamics alone can describe the outflow rates and water table dynamics accurately when the influence of capillarity is small; b) for very shallow soils or water tables in the proximity of the soil surface, the effects of capillarity are more clearly noticeable and it is possible to improve the simulated water table dynamics during drainage experiments significantly by regarding the parameter drainable porosity as a state-dependent parameter of which the value depends on the unsaturated depth; c) for drainage and recharge scenario's, a low-dimensional coupled saturated-unsaturated model that is developed to dynamically account for the unsaturated zone processes is very well capable of describing outflow rates and water table dynamics; d) the comparison of the results of the coupled model run in an uncalibrated mode to field data yields a reasonable match in terms of hydrographs and water table dynamics. However, the water table response is highly sensitive due to limited free pore space caused by capillarity effects

Storage-dependent drainable porosity for complex hillslopes

In hydraulic groundwater theory the parameter drainable porosity f (a storage coefficient that accounts for the effect of the unsaturated zone on water table dynamics) is usually treated as a constant. For shallow unconfined aquifers the value of this parameter, however, depends on the depth to the water table and the water retention characteristics of the soil. In this study an analytical expression for f as a function of water table depth is derived under the assumption of quasi-steady state hydraulic equilibrium, in this way accounting, in part, for the effects of the unsaturated zone on groundwater dynamics. The derived expression is implemented in the nonlinear hillslope-storage Boussinesq (HSB) model (Troch et al., 2003) to simulate the drainage response of complex hillslopes. The model's behavior is analyzed by comparison to (1) the HSB model with a constant value for f and (2) measurements of water tables and outflow hydrographs on a 6.0 × 2.5 × 0.5 m laboratory hillslope experiment. The comparison is conducted for a pure drainage case on two different hillslope shapes (linearly convergent and divergent) and for three different slope inclinations (5%, 10%, and 15%). Comparison 1 is run in an uncalibrated and a fully calibrated mode, and it enables us to evaluate the effect of a dynamic, state-dependent value for f on model output. Comparison 2 allows us to test the HSB model on several hillslope configurations and to analyze whether the concept of a storage-dependent f enhances the model performance. The comparison of the HSB models to the measurements from the laboratory hillslopes shows that it is possible to capture the general features of the outflow hydrograph during a drainage experiment using either one of the HSB models. Overall, the original (constant f) HSB model, with one fitting parameter more than the revised HSB model, shows a slightly better fit on the hydrographs when compared to the revised (variable f) HSB model. However, the peak outflow values (the first few minutes after initiation of the experiments) are better captured by the revised HSB model. The revised HSB model's performance in simulating water table movements is much more accurate than that of the original HSB model. The improved match of the revised HSB model to piezometric measurements is worth stressing because the ability to model water tables is a key attribute of the model, making it possible to investigate phenomena such as saturation excess runoff. Also noteworthy is the good match between the revised HSB model and the outflow measurements, without any calibration, for the divergent slopes. The changing values of the calibrated drainable porosity parameter for the original HSB model as different configurations are simulated (slope angle, plan shape, initial conditions), together with the ability of the revised HSB model to more accurately simulate water table dynamics, clearly demonstrates the importance of regarding drainable porosity as a dynamic, storage-dependent paramete

Analytical solution of the linearized hillslope-storage Boussinesq equation for exponential hillslope width functions

This technical note presents an analytical solution to the linearized hillslope-storage Boussinesq equation for subsurface flow along complex hillslopes with exponential width functions and discusses the application of analytical solutions to storage-based subsurface flow equations in catchment studies

Hillslope-storage Boussinesq model for subsurface flow and variable source areas along complex hillslopes: 2. Intercomparison with a three-dimensional Richards equation model

The Boussinesq equation for subsurface flow in an idealized sloping aquifer of unit width has recently been extended to hillslopes of arbitrary geometry by incorporating the hillslope width function w(x) into the governing equation, where x is the flow distance along the length of the hillslope [ Troch et al., 2003 ]. Introduction of a source/sink term N allows simulation of storm-interstorm sequences in addition to drainage processes, while a function S c (x) representing the maximum subsurface water storage can be used to account for surface saturation response in variable source areas activated by the saturation excess mechanism of runoff generation. The model can thus simulate subsurface flow and storage dynamics for nonidealized (more realistic) hillslope configurations. In this paper we assess the behavior of this relatively simple, one-dimensional model in a series of intercomparison tests with a fully three-dimensional Richards equation model. Special attention is given to the discretization and setup of the boundary and initial conditions for seven representative hillslopes of uniform, convergent, and divergent plan shape. Drainage and recharge experiments are conducted on these hillslopes for both gentle (5%) and steep (30%) bedrock slope angles. The treatment and influence of the drainable porosity parameter are also considered, and for the uniform (idealized) hillslope case the impact of the unsaturated zone is examined by running simulations for different capillary fringe heights. In general terms, the intercomparison results show that the hillslope-storage Boussinesq model is able to capture the broad shapes of the storage and outflow profiles for all of the hillslope configurations. In specific terms, agreement with the Richards equation results varies according to the scenario being simulated. The best matches in outflow hydrographs were obtained for the drainage experiments, suggesting a greater influence of the unsaturated zone under recharge conditions due to transmission of water throughout the hillslope. In the spatiotemporal water table response a better match was observed for convergent than divergent hillslopes, and the bedrock slope angle was not found to greatly influence the quality of the agreement between the two models. On the basis of the intercomparison experiments we make some suggestions for further development and testing of the hillslope-storage mode

The hillslope-storage Boussinesq model for non-constant bedrock slope

In this study the recently introduced hill slope-storage Boussinesq (hsB) model is cast in a generalized formulation enabling the model to handle non-constant bedrock slopes (i.e. bedrock profile curvature). This generalization extends the analysis of hydrological behavior to hillslopes of arbitrary geometrical shape, including hillslopes having curved profile shapes. The generalized hsB model performance for a free drainage scenario is evaluated by comparison to a full three-dimensional Richards equation (RE) based model. The rnodel results are presented in the form of dimensionless storage profiles and dimensionless outflow hydrographs. In addition, comparison of both models to a storage based kinematic wave (KW) model enables us to assess the relative importance of diffusion processes for different hillslope shapes, and to analyze the influence of profile curvature on storage and flow patterns specifically. The comparison setup consists of a set of nine gentle (5% bedrock slope) and nine steep (30% bedrock slope) hillslopes of varying plan shape and profile curvature. Interpretation of the results shows that for highly conductive soils the simulated storage profiles and outflow hydrographs of the generalized hsB model and RE model match remarkably for 5% bedrock slope and for all plan and profile curvatures. The match is slightly poorer on average for 30% bedrock slope, in particular, on divergently shaped hillslopes. In the assessment of the influence of hydraulic diffusion, we find good agreement in Simulation results for the KW model compared to results from the generalized hsB model and the RE model for steep divergent and uniform hillslopes, due to a relatively low ratio between water table gradient and bedrock slope compared to convergent or gentle hillslopes. Overall, we demonstrate that, in addition to bedrock slope, hillslope shape as represented by plan and profile curvature is an important control on subsurface flow response. (C) 2004 Elsevier B.V. All rights reserved