Influence of the first-order exchange coefficient on simulation of coupled surface–subsurface flow

Summary

Presently, there is little guidance for model users on the selection of the first-order exchange coefficient (FOEC; or “conductance”) commonly used in simulating surface–subsurface interactions (e.g. infiltration). In this study, relationships between the FOEC and surface–subsurface exchange flux, surface-subsurface head difference and time to initiate overland flow are systematically explored using 1D soil column simulations with the fully integrated code HydroGeoSphere. Numerical experiments adopt five different hydrological scenarios and nine different soil profiles. Results converge on the more accurate, but sometimes more computationally intensive, continuity of pressure (COP) coupling approach as the coupling length (l_e) parameter within the FOEC is decreased (i.e. FOEC increased). Threshold l_e values that produce results converged on the COP approach vary considerably with hydrological scenario, soil type and total obstruction height (H_s; accounting for sub-grid depression storage), with most threshold l_e values ⩽10⁻² m. Lower l_e values are required for infiltration under Hortonian conditions, under non-Hortonian conditions in lower permeability soils, and to capture timing of initiation of overland flow. The condition l_e > H_s precludes top-down saturation under Hortonian conditions. Steady-state exchange flux and time to initiate overland flow are within 0.05% and 24%, respectively, of COP results when l_e = H_s = 1 mm. 3D simulation of a hypothetical catchment demonstrates that the general FOEC sensitivities obtained through 1D simulation are transferrable to the 3D case. This study shows that a value of l_e = H_s provides an appropriate initial value for modelling applications. We suggest a FOEC parameter sensitivity assessment on a case-by-case basis to ensure adequately converged results and to avoid unrealistic model behaviour.

Introduction

The development of fully integrated hydrological codes which enable the catchment-scale simulation of water movement both within and between the surface and subsurface has helped fulfil Freeze and Harlan’s (1969) blueprint for physically based, hydrological response modelling (Loague et al., 2006). Fully integrated codes are defined here as those which solve the surface and subsurface domains simultaneously in a single matrix of equations (Morita and Yen, 2002, Furman, 2008). Integrating surface and subsurface processes can aid studies of catchment behaviour and water management, where interdependency of surface and subsurface domains is an important aspect of the spatial and temporal variability in catchment hydrological functioning (Loague and VanderKwaak, 2004).

Popular fully integrated codes include HydroGeoSphere (HGS; Therrien et al., 2009), Integrated Hydrology Model (InHM; VanderKwaak, 1999, VanderKwaak and Loague, 2001), MODHMS (Panday and Huyakorn, 2004; HydroGeoLogic Inc., 2006) and ParFlow (Kollet and Maxwell, 2006). These types of codes have been successfully applied to a wide range of hydrological problems that include simulation of overland flow (e.g. VanderKwaak and Loague, 2001, Kollet and Maxwell, 2006, Mirus et al., 2009), surface water and groundwater interaction (Werner et al., 2006, Smerdon et al., 2007, Cardenas, 2008, Brookfield et al., 2009), diffuse recharge (Lemieux et al., 2008, Smerdon et al., 2008), and atmosphere–surface–subsurface interactions (Maxwell and Kollet, 2008a, Kollet et al., 2010).

Determining the most effective method for coupling the surface and the subsurface is one of the most significant conceptual challenges in fully integrated catchment simulation (Ebel et al., 2009). The surface–subsurface coupling approach can influence catchment rainfall–runoff behaviour by potentially imposing controls on infiltration, recharge, overland flow, groundwater exfiltration, and exchanges between surface water bodies (lakes, rivers, wetlands, etc.) and the subsurface (e.g. Ebel et al., 2009, Gaukroger and Werner, 2011). A number of methods for surface–subsurface coupling have been identified (Morita and Yen, 2000, Furman, 2008); however, the most common coupling methods in fully integrated codes are the continuity of pressure (COP) and first-order exchange coefficient (FOEC) approaches (Ebel et al., 2009).

The COP approach assumes that the surface and uppermost subsurface pressure heads are the same, enforcing a direct connection between the domains (Fig. 1a). The surface water and porous media flow equations are solved simultaneously at a single surface–subsurface interface node. There are no additional parameters needed to define the surface–subsurface exchange processes beyond those needed for independent surface and subsurface flow simulation. This approach is arguably the most physically based manner for coupling the domains (Kollet and Maxwell, 2006); however, rapid changes in surface pressure can lead to numerical instabilities at the subsurface boundary, which can be overcome using small time steps, but may lead to very long simulation times (Beven, 1985, Ebel et al., 2009, Huang and Yeh, 2009).

A commonly used alternative to the COP approach is the FOEC or ‘conductance’ approach (Ebel et al., 2009) (Fig. 1b). Here, the surface–subsurface exchange flux is proportional to the head difference between separate surface and subsurface nodes, and the FOEC. High values of FOEC promote surface–subsurface exchange, and Huang and Yeh (2009) demonstrated that the FOEC approach can approximate the COP approach. They used an iteratively coupled scheme to assess FOEC relationships using a hypothetical 3D watershed. Ebel et al. (2009) explored relationships between the FOEC parameterisation and hydrological processes occurring within a simulated instrumented watershed (the R5 catchment). They concluded that the FOEC approach can be applied both to balance simulation run times and to minimise the head difference across the surface–subsurface interface under saturated conditions. They found that a critical threshold of FOEC parameters existed, below which consistent results were obtained in terms of both the integrated and distributed catchment responses. These were assessed using the discharge hydrograph and surface–subsurface head differences, respectively. Ebel et al. (2009) based their analysis at the catchment scale, with a heterogeneous saturated hydraulic conductivity field and variable topography and rainfall. It is our intention to consider a smaller scale than that adopted by Ebel et al. (2009) and Huang and Yeh (2009) in order to isolate the effects of the FOEC on specific hydrological scenarios, rather than on the whole catchment response.

Delfs et al. (2009) used 1D simulations to explore relationships between FOEC parameterisation and the prediction of hydrological processes associated with Hortonian overland flow (i.e. driven by infiltration excess or top-down saturation; Horton, 1933). Their results also show that infiltration becomes relatively insensitive to FOEC, as FOEC increases. The Delfs et al. (2009) analysis was constrained to sensitivities associated with Hortonian conditions, and further testing is needed to extend the analysis to other surface–subsurface interactions. Additionally, neither Delfs et al. (2009) nor Ebel et al. (2009) compared the FOEC approach directly to the COP approach.

While Ebel et al. (2009) found a critical value for the FOEC parameter to accurately simulate their catchment-scale model, they point out that this critical value may vary depending on the specific runoff generation mechanism, soil hydraulic properties, mesh discretization, surface flow properties and topography. The primary aim of the current paper is to extend the work of Ebel et al., 2009, Delfs et al., 2009 and Huang and Yeh (2009) by analysing the sensitivity of overland flow generation mechanisms to FOEC parameters for a range of simple physical conditions and basic hydrological scenarios, using homogeneous 1D soil column simulations. Hypothetical scenarios are used to isolate the effect of FOEC parameterisation on specific surface–subsurface interactions, where aspects of the expected model behaviour are known a priori with simple theory and calculations.

The influence of FOEC parameters on rainfall partitioning into overland flow and infiltration, plus situations producing exfiltration is systematically explored for different hydrological scenarios. These include Hortonian overland flow, Dunne overland flow (i.e. driven by saturation excess or bottom-up saturation; Dunne, 1978) and exfiltration. FOEC parameter values that produce a suitably accurate solution compared to the COP approach are determined for each of the hydrological scenarios and nine different soil columns. The findings from the 1D analysis are then compared to a hypothetical 3D catchment example, which was originally devised by Panday and Huyakorn (2004). In doing this, we explore the transferability of 1D interpretations of specific hydrological scenarios to a 3D simulation with a combination of hydrological processes, in which FOEC is known to influence the predictions of catchment hydrology (e.g. Gaukroger and Werner, 2011).