A comparison of two physics-based numerical models for simulating surface water–groundwater interactions

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Abstract

Problems in hydrology and water management that involve both surface water and groundwater are best addressed with simulation models that can represent the interactions between these two flow regimes. In the current generation of coupled models, a variety of approaches is used to resolve surface–subsurface interactions and other key processes such as surface flow propagation. In this study we compare two physics-based numerical models that use a 3D Richards equation representation of subsurface flow. In one model, surface flow is represented by a fully 2D kinematic approximation to the Saint–Venant equations with a sheet flow conceptualization. In the second model, surface routing is performed via a quasi-2D diffusive formulation and surface runoff follows a rill flow conceptualization. The coupling between the land surface and the subsurface is handled via an explicit exchange term resolved by continuity principles in the first model (a fully-coupled approach) and by special treatment of atmospheric boundary conditions in the second (a sequential approach). Despite the significant differences in formulation between the two models, we found them to be in good agreement for the simulation experiments conducted. In these numerical tests, on a sloping plane and a tilted V-catchment, we examined saturation excess and infiltration excess runoff production under homogeneous and heterogeneous conditions, the dynamics of the return flow process, the differences in hydrologic response under rill flow and sheet flow parameterizations, and the effects of factors such as grid discretization, time step size, and slope angle. Low sensitivity to vertical discretization and time step size was found for the two models under saturation excess and homogeneous conditions. Larger sensitivity and differences in response were observed under infiltration excess and heterogeneous conditions, due to the different coupling approaches and spatial discretization schemes used in the two models. For these cases, the sensitivity to vertical and temporal resolution was greatest for processes such as reinfiltration and ponding, although the differences between the hydrographs of the two models decreased as mesh and step size were progressively refined. In return flow behavior, the models are in general agreement, with the largest discrepancies, during the recession phase, attributable to the different parameterizations of diffusion in the surface water propagation schemes. Our results also show that under equivalent parameterizations, the rill and sheet flow conceptualizations used in the two models produce very similar responses in terms of hydrograph shape and flow depth distribution.

Introduction

Surface and subsurface waters are not isolated components of the hydrologic cycle, but instead interact in response to topographic, soil, geologic, and climatic factors [8]. The study of these interactions has been addressed at both small (field and hillslope) (e.g. [1], [39]) and large (watershed to global) scales (e.g. [23], [37]). A number of hydrological models that incorporate some representation of groundwater–surface water interactions have been developed over the past decades, including physically-based, distributed-parameter models. This latter class of models, more rigorous but also more computationally intensive than empirical or semi-empirical approaches, uses the shallow water equations to describe surface flow, i.e., one- or two-dimensional approximations of the Saint–Venant equations for overland and/or channel flow, coupled with a subsurface component that solves the three-dimensional equation for variably saturated flow, i.e., Richards’ equation (e.g. [41], [28], [33], [18]). A comprehensive description of the types of process representation in distributed models and their inherent assumptions and limitations, together with a discussion of comparison and assessment issues, is provided in Kampf and Burges [19], Clarke [5], Furman [10], Ebel et al. [9], and Maxwell [24].

For physically-based coupled models, which are the focus of this study, various schemes have been proposed for solving the system of surface and subsurface equations and for resolving the interactions across the land surface. The solution approaches can be broadly classified as full coupling, sequential coupling, and loose coupling, whereas the formulations for the exchange fluxes are based on continuity principles, diffusion paradigms, boundary conditions switching, or other schemes. In full coupling (e.g. [41], [33], [20]), the governing equations are solved simultaneously; in sequential coupling (e.g. [12], [28], [3]), they are solved separately, with an explicit discretization used for at least one of the equations or with an iterative cycle superposed on the overall system; in loose coupling (e.g. [36], [6]), the equations are again solved separately, with the output from one regime (e.g., surface flow) simply passed as input to the other, without iteration or other conditions imposed.

Whereas the accuracy, robustness, and other performance features of surface and subsurface numerical models have been extensively documented (e.g. [35], [42] for Saint–Venant approximations; [16], [29] for Richards’ equation), there have been very few assessments of coupled models based on these equations. The purpose of this study is to provide such an assessment via a comparative analysis of two process-based groundwater–surface water models. One model, ParFlow [20], [21], uses a full coupling approach and continuity of pressures and fluxes across the land surface to resolve the surface–subsurface interactions; the other model, CATHY [2], [3], is based on sequential coupling with boundary condition switching to partition atmospheric fluxes into infiltration (or exfiltration) and a change in surface water storage. A comparison of these two very different models provides a first opportunity to critically examine some key features of coupled hydrological models. In addition to different schemes for coupling and exchange flux resolution, the two models use different conceptualizations of surface routing: sheet flow representation and a kinematic wave equation in ParFlow; rill flow representation and a diffusion wave equation in CATHY. Although not directly inherent to coupling issues, these additional differences are also worthy of assessment, given the high interest in applying coupled hydrological models at catchment and river basin scales where terrain features (slope, roughness, etc.), and consequently surface flow conditions, can vary greatly. Other differences between the models (e.g., ParFlow uses a finite difference/finite volume discretization whereas CATHY uses finite elements for the subsurface and finite differences for the surface) will also have an effect on the intercomparison tests and will be duly considered.

The intercomparison study is carried out through a series of simple test cases subjected to step functions of rainfall followed by a recession or evaporation period. The test cases involve a sloping plane [11] and a tilted V-catchment [33]. The simulations are designed to clearly expose model differences and similarities under complex and realistic physical conditions. The first tests focus on the different treatments of the exchange fluxes between the subsurface and surface domains and their sensitivity to factors such vertical mesh resolution, time step size, and slope angle. A second set of tests is intended to evaluate the impact of the different conceptualizations for propagation of surface runoff in terms of water depth distribution at the ground surface and timing and shape of the hydrograph.

Section snippets

Description of the models

The governing equations for the ParFlow model [20], [25] are the three-dimensional (3D) Richards equation for subsurface flow in variably saturated soils and the kinematic wave approximation of the Saint–Venant equation for overland and channel flow:SsSwψt+ϕSwt=-·q+qs,q=-KsKr(ψ-z),ψst=·(ψsν)+qr(x),Sf,i=So,i,where Ss is the specific storage coefficient [1/L], Sw = Sw(ψ) is the relative saturation [–], ψ is the subsurface pressure head [L], t is time [T], ϕ is the porosity [–], ∇ is the

Test case descriptions

In the sloping plane simulations, surface–subsurface interactions were investigated for the infiltration excess (Horton) and saturation excess (Dunne) runoff generation mechanisms under homogeneous and heterogeneous conditions, and for the return flow process under homogeneous conditions. Rill flow vs. sheet flow routing for overland flow was also examined for this test case (channel flow was not considered for this test). The test catchment (Fig. 1) is 400 m long by 320 m wide. A surface grid of

Runoff generation mechanisms, homogeneous conditions

Surface–subsurface water exchanges under Dunne and Horton saturation processes were first investigated for the homogeneous case. In the saturation excess test two initial water table configurations, at 0.5 m and 1.0 m from the ground surface, were simulated with a saturated hydraulic conductivity Ks of 6.94 × 10−4 m/min. In the infiltration excess test the initial water table depth was fixed at 1.0 m and two Ks values were used, 6.94 × 10−5 and 6.94 × 10−6 m/min. The influence of vertical mesh size

Conclusions

Two physically-based, spatially-distributed models of conjunctive surface and subsurface flow, ParFlow and CATHY, have been compared. The analysis has been focused on examining the coupling approaches implemented in the two models and the different conceptualizations used to describe the propagation of surface runoff. The theoretical and numerical bases for ParFlow and CATHY were briefly presented, highlighting key features and differences between the models, and two test problems were

Acknowledgments

We acknowledge the financial support of the Ouranos Consortium and the Natural Sciences and Engineering Research Council of Canada (Grant CRDPJ-319968-04) and of the CARIPARO foundation, Italy (project “Transport phenomena in hydrological catchments: hydrological and geophysical experiments and modelling”). We also wish to thank the three anonymous reviewers for their helpful comments.

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