## 1. Introduction

[2] Our understanding of hillslope subsurface flow processes and their effect on catchment response to atmospheric forcing is incomplete and has been the subject of much research for several decades. Among the first to reveal the importance of hillslope subsurface flow with regard to catchment stormflow were *Hewlett and Hibbert* [1967] and *Dunne and Black* [1970]. They concluded that for humid climates water table dynamics in hillslopes have a large effect on channel stormflow through the formation of areas of saturation along the channel network, often called saturated source areas, which cause saturation excess overland flow. Since then many studies have investigated subsurface flow processes experimentally [e.g., *O'Loughlin*, 1981; *Woods et al.*, 1997; *Torres et al.*, 1998; *McGlynn et al.*, 2004] and through modeling. The two-part paper by *Freeze* [1972], where stormflow processes are examined using a three-dimensional Richards-based model to describe subsurface processes, is one of the first (computer) modeling studies of hillslope processes, and since then many have followed.

[3] In addition to the rather complex, three-dimensional models that are often used in these modeling studies [e.g., *Abbott et al.*, 1986; *Wigmosta et al.*, 1994], several simplified low-dimensional models have been proposed, because of the computational expense and parameterization difficulties associated with the former type of model. Efforts include those of *Beven and Kirkby* [1979], who describe the original version of TOPMODEL, *Duffy* [1996], who develops a two-state variable integral-balance (hillslope) model, *Reggiani et al.* [1998], who describe the “representative elementary watershed” (REW) model, and *Sloan* [2000], who describes a storage-discharge type model which is derived from hydraulic groundwater theory. Among the simplified models, many studies have focused on analysis of the Boussinesq equation [e.g., *Childs*, 1971; *Brutsaert*, 1994; *Szilagyi et al.*, 1998; *Chapman*, 2005]. Most of these studies were conducted on straight hillslopes, sometimes using linearized versions of the Boussinesq equation, aiming at an increased fundamental understanding of the flow and storage dynamics in hillslopes.

[4] In recent work by the authors, the Boussinesq equation was generalized to account for the three-dimensional soil mantle in which the flow processes take place [*Troch et al.*, 2003; *Paniconi et al.*, 2003; *Hilberts et al.*, 2004]. The effect of slope shape (i.e., convergent, divergent, and straight) and bedrock curvature on storage and outflow processes was examined for drainage and recharge scenarios, and compared to the three-dimensional Richards equation (RE) based model of *Paniconi and Putti* [1994]. A general conclusion of our recent work with the hillslope-storage Boussinesq (HSB) model is that the modeled outflow rates compared relatively well to results of the RE model, but capturing the water table dynamics was less successful. Results from a recent laboratory experiment [*Hilberts et al.*, 2005] indicated that this may be caused by the strong effect of capillarity (or, more precisely, the unsaturated storage) on groundwater dynamics, especially for shallow soils as typically encountered on hillslopes.

[5] The capillary fringe is an (almost) entirely saturated transition zone between the unsaturated zone and groundwater, of which the effects on groundwater flow are often ignored. However, the literature provides evidence that in the capillary fringe, fluxes can have considerable lateral components, thereby adding to lateral groundwater flow [e.g., *Luthin and Miller*, 1953; *Jayatilaka and Gillham*, 1996]. These flow processes were investigated numerically [e.g., *Luthin and Day*, 1955; *Vachaud and Vauclin*, 1975] and experimentally [e.g., *Berkowitz et al.*, 2004], and all of these studies clearly show large lateral flow components in the capillary fringe. *Vachaud and Vauclin* [1975] demonstrated that the fluxes in the capillary fringe are of the same order of magnitude as groundwater fluxes, and that they often have a large lateral component. They estimated that, for their experiment, roughly 14% of the lateral flux takes place above the water table.

[6] The effect of capillarity on water table dynamics, which in hydraulic groundwater models is usually accounted for through parameters such as specific yield, effective porosity, or drainable porosity, has been noted by *Hooghoudt* [1947], who referred to it as the “Wieringermeer effect,” and later by *Gillham* [1984], *Abdul and Gillham* [1989], *Parlange et al.* [1990], *Kim and Bierkens* [1995], and *Nielsen and Perrochet* [2000]. However, only few have tried to account for the effect. In a benchmark paper by *Parlange and Brutsaert* [1987] the capillarity effect on groundwater systems is modeled, assuming a deep profile for which θ = θ_{r} holds at the land surface. Assuming instantaneous equilibrium in the unsaturated zone, an analytical expression to account for capillarity effects is derived that can be added to the Boussinesq equation. *Barry et al.* [1996] extended the equations derived by *Parlange and Brutsaert* [1987] to include higher-order capillarity effects, and they are used to investigate the inland propagation of oscillations in water tables for a coastal aquifer. *Nachabe* [2002] derives an analytical expression to account for dynamic capillarity effects, which also includes delayed recharge due to rapidly dropping water tables. *Hilberts et al.* [2005] derived an analytical expression to account for capillarity effects under equilibrium in shallow groundwater systems, and its influence on hillslope dynamics is investigated. All of these studies are mainly applicable in situations where recharge is negligibly small. To extend the investigation of the effect of unsaturated zone storage on saturated flow to recharge scenarios, a coupling of the saturated zone model to a dynamic unsaturated zone model is needed, and the impact of the capillary fringe on water table dynamics needs to be incorporated.

[7] Hydrological studies at the hillslope and catchment scale, as well as land surface modeling, have put much emphasis on the processes that occur in the soil layer close to the soil surface. This is because the interactions of the unsaturated zone with the atmospheric boundary layer are known to have an important effect on surface fluxes and therefore also on climate [*Koster et al.*, 2003]. It is well known that subsurface flow processes are currently not well simulated in land surface models [*Liang et al.*, 2003]. A more thorough understanding of the interactions between (shallow) groundwater and soil moisture in the unsaturated zone is needed if we are to improve model results significantly [*Koster et al.*, 2000].

[8] Several authors have described a coupling of separate models for unsaturated and saturated flow under a diversity of assumptions. *Pikul et al.* [1974] coupled a one-dimensional Boussinesq model to a one-dimensional RE model for the unsaturated zone. The coupled system was solved as a boundary value problem, and the drainable porosity for the saturated zone model was taken to be a constant (namely, θ_{s} − θ_{m}), where θ_{s} (dimensionless) is saturated soil moisture content and θ_{m} (dimensionless) is “the minimum soil moisture content below the depth from which moisture may be removed directly by evapotranspiration.” As no functional form is given for θ_{m}, its value is somewhat arbitrary [*Vachaud and Vauclin*, 1975]. A very similar approach was used by *Kim et al.* [1999], but they also did not give a relationship describing the drainable porosity. *Smith and Hebbert* [1983] coupled a Boussinesq model to a kinematic wave model for the unsaturated zone. To calculate the recharge from the unsaturated zone to the saturated zone, they assumed that the soil moisture pulses in the unsaturated zone have attenuated when they reach the groundwater table. The coupled system was solved as a system of ordinary differential equations. A similar approach was taken by *Beven* [1982]; however, in his work, two kinematic wave models were coupled. *Liang et al.* [2003] described a coupling between a one-dimensional RE model for the unsaturated zone and a generalized bucket model for lateral subsurface flow, which was linked to the VIC model [*Wood et al.*, 1992]. The coupled system was also solved as a boundary value problem. However, in none of the mentioned papers that deal with coupling of models, the actual functional interactions between the saturated zone and the unsaturated zone are investigated.

[9] In this work the one-dimensional Richards equation is coupled to the HSB equation. In the coupled model, the capillary fringe is treated as an integral part of a Boussinesq aquifer, i.e., lateral groundwater transport takes place over the entire saturated depth (and not only below the atmospheric pressure plane). By introducing the unsaturated zone matric pressure head as a system state and reformulating the derived equations in state-space notation, we solve the coupled system simultaneously as a set of ordinary differential equations, and obtain a functional state-dependent expression for the drainable porosity. With a Richards equation representation for the unsaturated zone and a functional form for the drainable porosity, this coupled model allows us to investigate more accurately the interactions between the saturated and unsaturated zone and the relationship between rainfall intensity, unsaturated storage (and drainable porosity), and recharge. We assume that a single (space-averaged) soil moisture profile can sufficiently describe the unsaturated zone processes, which is an assumption that is done to retain the coupled model's low dimensionality. A similar assumption underlies the work of *Boussinesq* [1877], where recharge was assumed uniform over a hillslope. The coupled HSB model's behavior is compared to the original HSB model of *Troch et al.* [2003] and the three-dimensional RE model of *Paniconi and Putti* [1994] (which is taken to be a benchmark model) for a set of seven synthetic hillslopes.