Over the last decades, a number of integrated surface–subsurface hydrologic models have been developed. The main objective of such models is to conceptualise the hydrologic cycle in an integrated way, particularly by coupling the surface and subsurface (unsaturated and saturated zones) hydrological processes. Such integration is particularly important in humid regions where the water table is close to the surface and runoff generation is dominated by variable source area mechanisms (Dunne and Black, 1970; McDonnell and Taylor, 1987). In this runoff generation mechanism, infiltrated water enters the water table, which rises until it reaches the surface, often in valley bottoms, creating areas where any additional precipitation results in saturation excess runoff (McDonnell and Taylor, 1987). Investigation of such a runoff mechanism at the catchment scale can be addressed using physically based models in which the unsaturated zone (UZ) and the saturated zone (SZ) are either explicitly or implicitly coupled. In an explicit coupling approach, a one-dimensional Richards equation for the unsaturated zone is coupled to a three-dimensional saturated flow. In this approach, it is assumed that flow in the unsaturated zone is only vertical and that the groundwater recharge is computed using an iterative water table correction process. MIKE SHE (Refsgaard and Storm, 1995) is an example of one such coupling approach. In the implicit coupling of the unsaturated–saturated zones, the whole subsurface flow process is described using a three-dimensional variably saturated flow equation without an explicit distinction in the interaction between unsaturated and saturated zones (Camporese et al., 2010; Kollet and Maxwell, 2006). This approach more truly reflects the physical processes governing flow but is computationally more expensive when compared with explicit approaches, which themselves require considerable computer resources to solve the unsaturated flow equation at the watershed scale. There is a third group of integrated surface–subsurface hydrologic models, namely externally coupled hydrologic models, where already-existing hydrologic and groundwater models are coupled, such as SWAT–MODFLOW (Chung et al., 2010), TOPNET–MODFLOW (Chenjerayi Guzha and Hardy, 2010), and GSFLOW (Markstrom et al., 2008). MOBIDIC–MODFLOW, the integrated surface–subsurface hydrologic model developed in this study, belongs to that last category of models. Unlike in physically based integrated hydrologic models, the description of the flow in the unsaturated zone in externally coupled models is not based on the Richards equation, and their simplified unsaturated–saturated coupling scheme can restrict their applicability in modelling of shallow water table fluctuations (WTFs).

Seibert et al. (2003) distinguishes three types of interactions between unsaturated and saturated zones based on the water table levels:

Type 1. The water table is relatively deep, and there is only one-directional interaction between UZ and SZ. This means that the soil moisture state in the unsaturated soil is independent of the groundwater level and that the role of groundwater in the runoff generation process is not considered.

With regard to these limitations, the objective of this study is to propose a series of modifications to the original conceptualisation of the unsaturated and saturated flow process of MOBIDIC in order to extend its applicability for modelling of shallow water table fluctuations while retaining its computational efficiency. To this aim, the conceptual saturated flow scheme of MOBIDIC was replaced with MODFLOW as a physically based three-dimensional groundwater model using the sequential coupling approach (Chenjerayi Guzha and Hardy, 2010). Then, a novel methodology for revisiting the calculation of the groundwater recharge in MOBIDIC, the specific yield in MODFLOW, and the interaction between the unsaturated and saturated zones in MOBIDIC–MODFLOW was developed. The fully coupled surface–subsurface model MIKE SHE is used as a reference for comparison; hence the methodology is based on numerical benchmarking on hypothetical and realistic catchments. Using the WTF method (Healy and Cook, 2002) in MIKE SHE, the rises of a shallow water table were simulated under different sets of rainfall intensity, soil property, and depth to the water table. The simulated responses were then used to reformulate the groundwater recharge of MOBIDIC based on the assumption of a quasi-steady pressure profile in the unsaturated zone as the water table fluctuates. The accuracy of the proposed modifications was first evaluated in a two-dimensional case (constant slope), where the simulated water table rises of the two models under a uniform rainfall rate were compared. In a second experiment, the approach was tested at the catchment scale and under unsteady rainfall conditions. Comparison of the simulated water table responses of the MOBIDIC–MODFLOW against those of MIKE SHE allowed us to evaluate how the unsaturated–saturated interaction scheme of the externally coupled models can be adapted for applications in shallow water table regions.

The water table fluctuation (WTF) method is a simplified approach for the determination of groundwater recharge of an unconfined aquifer based on groundwater level fluctuations. This method is based on the assumption that the rise in groundwater levels is due to the groundwater recharge (Healy and Cook, 2002). Considering the groundwater budget for a representative element (Fig. 1), any change in the water table level (groundwater storage) would be due to a combination of recharge to groundwater (R), inflow from the upstream cell (Qu), outflow to the downstream cell (Qd), and evapotranspiration from groundwater (ETGW) as follows: 1ΔSGW=R+Qu-Qd-ETGW, where ΔSGW (L T-1) is the change in groundwater storage. Evapotranspiration from groundwater is taken as the direct root water uptake from the saturated zone. Unlike in deep water table conditions, groundwater evapotranspiration in shallow water table regions can be much greater than evapotranspiration from the unsaturated zone (Shah et al., 2007).

Assuming that the water table rise is solely due to the recharge of groundwater requires the sum of other fluxes in Eq. (1) to be zero. This means that the determination of the groundwater recharge using WTF is best applicable over short periods (hours to days) after the onset of rainfall (before any significant redistribution of groundwater recharge to the other fluxes; Healy and Cook, 2002). Therefore, 2R=SyΔhΔt, where Sy (–) is the specific yield and ΔhΔt is the change in the water table over Δt. Application of the WTF method requires an estimation of the specific yield which is defined as the volume of drained water per unit of the drop in the water table and aquifer area (Nachabe, 2002): 3Sy=VwAΔh, where Vw (L3) is the volume of drained water, A (L2) is the area of the aquifer, and Δh (L) is the change in the water table level. The specific yield is also defined as the difference between water contents at saturation θsat and at field capacity θfld (the moisture level below which water cannot be drained by gravity; Nachabe, 2002). Such a constant value for the specific yield, however, holds only for deep water tables where changes of the soil moisture profile in the unsaturated zone due to the water table drop are relatively small and the volume of drained water can be approximated as (θsat-θfld)×Δh (Fig. 2a). In shallow water tables, however, the specific yield is small (in Fig. 2b, the shaded area is smaller than (θsat-θfld)×Δh and approaches zero when the capillary fringe zone extends up to the soil surface; Nachabe, 2002).

We use the WTF method for a soil column to understand how the rise in groundwater is affected by rainfall intensity, soil type, and depth to the water table and how much of the infiltrated rain will percolate to the groundwater. The step-by-step procedure is shown in Fig. 4. Using a soil column with closed boundaries on the sides and bottom, the rise in groundwater level will be only due to the groundwater recharge (see Eqs. 1 and 2). In shallow water table regions, as the water table rises, the specific yield nonlinearly decreases, and, therefore, the actual rise in the water table might be greater than what would be expected by assuming that Sy=θsat-θfld. This is defined as “reference water table rise” (see Fig. 3) and is calculated as 17Δh=Iθs-θfld, where I (L T-1) is the infiltrated water and Δh (L) is the reference water table rise, which is the expected rise in the water table if the specific yield remains at its ultimate value (θsat-θfld) and the infiltrated water completely reaches the water table. However, depending on the initial water table level and rainfall intensity, it is possible for the water table rise to be smaller than Δh (e.g. for relatively deeper water table and/or low rainfall rate) or that the groundwater recharge takes a value greater than the infiltration rate. The actual water table rise was determined through the coupled unsaturated–saturated flow simulation in MIKE SHE, and knowledge of the ultimate specific yield allowed the calculation of groundwater recharge using Eq. (17). Therefore, by defining (ReI), we can evaluate how the rise of the water table is related to the precipitation rate (assuming no infiltration excess runoff), soil hydraulic property, and depth to the water table. ReI>1, ReI=1, and ReI<1 mean that the actual rise of the water table is larger than, equal to, or smaller than the reference rise, respectively. Once again, in MIKE SHE, Sy is a constant input of the saturated flow module, and changes regarding its value (decrease in its value when depth to the water table decreases) are captured by the iterative unsaturated–saturated coupling procedure explained in Sect. 3.1.3. The procedure was repeated for four soil types (see Table 2 for the hydraulic properties) and different rainfall intensities (up to the minimum of saturated hydraulic conductivity of the four soils to prevent infiltration excess runoff generation on the soil surface) to further investigate the significance of the rise in the water table of soil types with different characteristics in response to a given pulse of rain. In shallow water tables, loamy soils with a larger capillary fringe range have substantially smaller specific yield values which result in a higher rise of the water table compared with the sandy soils (Abdul and Gillham, 1989). Therefore, its actual rise in the water table is expected to be larger than the reference rise. However, as loamy soils have smaller hydraulic conductivities compared with sand, their water table response will be delayed, especially for deep water tables. To avoid this delayed response, we focussed our analysis on cases where the initial depths to the water table are at a maximum of 1.5 m. Computational time steps were 1 s and 1 min for UZ and SZ, respectively, to avoid numerical instabilities in the simulated water tables by MIKE SHE.

As discussed in Sect. 3, MOBIDIC's unsaturated soil depth (d in Eqs. 7 and 8) remains constant during the course of a simulation (both in rainy and subsequent draining periods). This is due to its conceptualisation of UZ and SZ, as their interaction is not aimed to address the reverse relationship between them in very shallow water table cases. The proposed changes in this paper aim to make the model applicable to such cases. The changes are as follows:

It is assumed that the unsaturated soil layer thickness, d, is no longer a constant input of the model and changes with water table fluctuations. Hence, the total moisture capacity of the reservoirs (Wgmax and Wcmax) is determined similarly to Eqs. (7) and (8), but with replacing d by the depth to the water table (zw). Therefore, the water table rise or fall results in a decrease or increase in moisture capacities of the reservoirs. Such assumption is valid when the groundwater level is treated as a moving boundary and there is a continuous transfer of moisture between the unsaturated and saturated zones.

The appropriateness of the proposed changes in the conceptualisation of UZ–SZ interactions of MOBIDIC–MODFLOW was tested by comparing its simulated water table results against those of MIKE SHE in two test cases. The description of the test cases follows.

The quick rise of the shallow water table in response to the precipitation was also observed in the experimental work of Abdul and Gillham (1984). In their study, the water table response of a sandy soil packed in a 160 cm×120 cm×8 cm box with a 12∘ surface slope with different initial water table levels under a uniform rainfall rate was investigated. The objective of the experiment was to evaluate the effect of the capillary fringe on the rise in the water table and on streamflow generation. The results revealed that when the water table is very shallow, that is, for the downhill regions of the slope where the capillary fringe extended to the soil surface, the water table rise was much greater than expected by the constant specific yield. The uphill regions, with a deeper water table depth, however, showed a delayed response due to the presence of the moisture deficit in the unsaturated zone. The simulated responses using MIKE SHE and MOBIDIC–MODFLOW (Fig. 8) are consistent with the findings of Abdul and Gillham (1984), attesting to their capability in capturing the effect of the capillary fringe on the water table rise. Note that the coefficient ω in MOBIDIC–MODFLOW (Eq. 19) changes as the water table level rises and falls, and, therefore, each computation grid in Fig. 8 has a different value for ω at each time step.

Comparing the simulated water table levels of the two models at Borden catchment (Figs. 10 and 11) attests to the soundness of the proposed UZ–SZ interaction scheme of MOBIDIC–MODFLOW at the catchment scale. The mean absolute error of the predicted water table by MOBIDIC–MODFLOW compared to MIKE SHE for the outlet grid cell is 0.013 m. As the water table rises, the water table could be much shallower than expected by the ultimate value for the specific yield (θsat-θfld). Therefore, the inclusion of a water-table-dependent specific yield in MOBIDIC–MODFLOW with the proposed groundwater recharge (Eq. 19) allowed for modelling of such a significant water table rise. Note that the modified MOBIDIC–MODFLOW does not have any additional calibration parameters, thereby the parameter uncertainties of the model remain unchanged. Comparison of the original and the modified equations of groundwater recharge in MOBIDIC (Eqs. 12 and 19, respectively) shows that both equations have only one calibration parameter (γ in Eq. 12 and ω in Eq. 19). The proposed modifications, however, have the advantage of eliminating the specific yield from the calibration parameters of MODFLOW using the expression dependent on the water table that is given in Eq. (18). The required soil hydraulic parameters for application of Eq. (18) are derived based on the soil texture database from Rawls et al. (1982). Therefore, the modified MOBIDIC–MODFLOW has one less calibration parameter (specific yield) compared with the original structure of the MOBIDIC–MODFLOW.

In addition, the proposed modifications in the conceptualisation of the unsaturated–saturated flow interaction of MOBIDIC–MODFLOW have the capability to take into account the inverse relation between the unsaturated and saturated moisture storage using a conceptual and computationally efficient framework. Modelling of such cases has been mostly addressed using physically based numerical models in which the subsurface flow is described either using a three-dimensional variably saturated flow or a one-dimensional Richards equation in the unsaturated zone coupled to a three-dimensional saturated flow equation. However, the application of these complex models at the catchment scale can be cumbersome due to the required data and computational burden to run such models. On the other hand, the conceptual shallow water table unsaturated–saturated interaction schemes in the literature (e.g. Seibert et al., 2003) do not allow for capturing the different water table responses of different soil types to the fallen pulses of precipitation shown in Fig. 5. The coefficients of such conceptual models are often determined through the calibration process against a few water table observations, which does not represent the disparity in water table response of different soil types of the catchment. However, using responses of the physically based model MIKE SHE for the calibration of the new groundwater recharge, it has been possible to model, with a good computational efficiency, the difference in the water table response of different soil types of the catchment. Note that, although the coefficient of ω in Eq. (19) in MOBIDIC–MODFLOW was calibrated against the simulated water table responses of MIKE SHE, the model works independently. In this study, the purpose of comparing the water table responses of the two models in test cases was to verify whether MOBIDIC–MODFLOW can reproduce the catchment-scale simulated water table responses of MIKE SHE as the expected response of the catchment in the absence of water table measurements.

An important advantage of the coupled MOBIDIC–MODFLOW presented in this paper lies in its capability to investigate phenomena such as saturation excess runoff without expensive computational burden. With regard to the similarities in the formulation and numerical solution technique of saturated flow in MIKE SHE and MOBIDIC–MODFLOW, the computational efficiency of the revised MOBIDIC–MODFLOW model is related to the conceptualisation of the unsaturated flow and its interaction with the saturated zone. The physically based models such as MIKE SHE require very fine discretisation of the unsaturated soil domain and require short time steps to avoid numerical instabilities. In addition, the iterative unsaturated–saturated coupling method in each unsaturated zone time step can remarkably increase the execution time of the system. However, the proposed approach in MOBIDIC–MODFLOW works with identical time steps in both unsaturated and saturated zones, without discretisation of the soil layer, which greatly improves the computational process. For example, the execution time for the Borden case took 10 min using a PC Core i7 with 8 GB of RAM for MOBIDIC–MODFLOW, while taking 30 h with MIKE SHE. This clearly indicates why such alternative modelling approaches are attractive for watershed-scale modelling purposes.

It should be noted that, while the coefficient ω in Eq. (19) was determined for sand, loamy sand, sandy loam, and loam, the model was tested for sandy soils where the moisture retention in unsaturated zone is small and the assumption of a quasi-steady equilibrium moisture profile in the unsaturated zone is legitimate, as discussed by Bierkens (1998). For finer textured soils, however, such an assumption may be questionable, and, therefore, the approach presented in this study must be fully tested over a variety of soil textures to further evaluate its exactitude to broader watershed conditions. In addition to testing the proposed approach over a variety of soil textures, a next step will consist of extending the applicability of the approach against observations in real catchments where other hydrological processes, such as summer–fall evapotranspiration and spring snowmelt, affect groundwater recharge in shallow aquifers.

In this paper, we have presented a series of modifications to the conceptualisation of the unsaturated and saturated flows of MOBIDIC, a conceptual hydrologic model, to extend its applications to shallow water tables where an inverse relationship between the unsaturated and saturated moisture storage exists. The modifications were as follows: (1) replacement of the model's conceptual saturated flow scheme with MODFLOW, (2) revisiting the conceptualisation of the interaction between the unsaturated and saturated zones and calculation of groundwater recharge, and (3) inclusion of a dynamic specific yield based on the quasi-steady-state pressure profile in the unsaturated zone. The key variables in modelling of shallow water table regions are (1) groundwater recharge and (2) the specific yield. Groundwater recharge is usually considered to be a function of the unsaturated moisture state and is a fraction of infiltrated moisture into the unsaturated soil layer. However, in shallow water table cases, its value can be larger than infiltration due to the capillary fringe above the water table, which causes a quick and significant water table rise (Jayatilaka and Gillham, 1996). The specific yield in the shallow water table regions is no longer a constant soil property and significantly decreases as the water table reaches the soil surface. Investigation of such cases is mostly addressed using physical based models (e.g. Cloke et al., 2006) or using fully conceptual approaches (e.g. Seibert et al., 2003). Through the coupling of MODFLOW to MOBIDIC, we have modified the unsaturated–saturated flow interaction of the model based on the assumption of the quasi-steady pressure profile in the unsaturated zone as the water table changes. In turn, the static specific yield in MODFLOW could be replaced with the dynamic specific yield that is dependent on the water table, such as derived by Duke (1972). Such modifications allowed the rapid and dynamic water table responses of shallow water table regions to be adequately captured. Also, the groundwater recharge in the model was defined with a power-type equation whose parameter (ω) was determined using the WTF method along with the physically based MIKE SHE model for a homogeneous soil column with different initial water table levels under different rainfall intensities. The appropriateness of the proposed changes were evaluated by comparing the simulated water table responses of the model against those of MIKE SHE in a two-dimensional (single slope) case under a continuous uniform rainfall rate and in a three-dimensional quasi-real catchment case (simplified Borden catchment) over a month of a measured daily rainfall hyetograph. In the light of simulation results, the following conclusions can be derived:

Inclusion of a dynamic specific yield in investigation of shallow water table responses is essential. Indeed, when the water table is close to the soil surface, the significance of rise is much greater than expected based on the ultimate value of the specific yield, e.g. θsat-θfld, as investigated by Jayatilaka and Gillham (1996). While in MIKE SHE, this is accounted for by using the iterative water table correction method, in MOBIDIC–MODFLOW, this is addressed by assuming a hydrostatic equilibrium state between UZ and SZ. To the authors' knowledge, this has not been taken into account in simplified coupling of a hydrologic and a groundwater model.