We present a new conceptual scheme of the interaction
between unsaturated and saturated zones of the MOBIDIC (MOdello Bilancio
Idrologico DIstributo e Continuo) hydrological model which is applicable to
shallow water table conditions. First, MODFLOW was coupled to MOBIDIC as the
physically based alternative to the conceptual groundwater component of the
MOBIDIC–MODFLOW. Then, assuming a hydrostatic equilibrium moisture profile
in the unsaturated zone, a dynamic specific
yield that is dependent on the water table level was added to MOBIDIC–MODFLOW, and calculation of the groundwater
recharge in MOBIDIC was revisited using a power-type equation based on
the infiltration rate, soil moisture deficit, and a calibration parameter linked
to the initial water table depth, soil type, and rainfall intensity. Using
the water table fluctuation (WTF) method for a homogeneous soil column, the
parameter of the proposed groundwater recharge equation was determined for
four soil types, i.e. sand, loamy sand, sandy loam, and loam under a pulse of
rain with different intensities. The fidelity of the introduced
modifications in MOBIDIC–MODFLOW was assessed by comparison of the simulated
water tables against those of MIKE SHE, a physically based integrated
hydrological modelling system simulating surface and groundwater flow, in
two numerical experiments: a two-dimensional case of a hypothetical
watershed in a vertical plane (constant slope) under a 1 cm d

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:

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 (

Schematic view of computational grids in a catchment and corresponding input and output fluxes over the saturated zone.

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,

Hypothetical soil moisture profile for

MIKE SHE is one of the widely used physically based integrated surface–subsurface hydrological models for a wide range of spatio-temporal applications, ranging from detailed theoretical (single soil column) to operational watershed-scale studies (Graham and Butts, 2005). It has a modular structure for computation of the hydrological processes with different levels of complexity, which is advantageous, particularly in large-scale watershed studies (Kollet et al., 2017). A detailed description of the computation of the hydrological processes in MIKE SHE can be found in Storm (1991) and DHI (2014). We present only the computation of flow in the unsaturated and saturated zones and their coupling approach.

The unsaturated flow is described using the one-dimensional Richards
equation as follows (Downer and Ogden, 2004):

Saturated flow in MIKE SHE is computed using the three-dimensional saturated
flow equation as the following:

The explicit coupling approach implemented in MIKE SHE has the advantage of
employing different times steps for each zone (seconds to minutes for UZ and
hours to days for SZ), which makes the system computationally less expensive
compared with the implicit coupling approach (DHI, 2014). However, employing
different time steps for the UZ and SZ may result in the generation of mass
balance errors in calculating the water flux between the two zones due to
(1) an incorrect value of

The iterative water table adjustment continues until the calculated

MOBIDIC (Castelli et al., 2009) is a distributed continuous hydrologic model in which the components of the hydrologic system are conceptualised as a system of interconnected reservoirs. Such a conceptual formulation of the model makes it computationally more efficient, especially in large-scale watershed modelling (Castillo et al., 2015). The detailed description of the model is available in Castelli et al. (2009) and Castillo et al. (2015).

In MOBIDIC, the unsaturated zone is described by two interconnected
reservoirs, i.e. the gravity and capillary reservoirs, to account for
corresponding gravity and capillary forces in the unsaturated soil. The water
content at field capacity (

Saturated flow in MOBIDIC is described either by a simplified linear reservoir (conceptual scheme) or the Dupuit assumption (Dupuit, 1848). In this study, MODFLOW (Harbaugh et al., 2000) was coupled to MOBIDIC as a physically based three-dimensional finite-difference-based alternative groundwater flow model. MOBIDIC and MODFLOW were coupled using the sequential coupling approach in which the groundwater recharge and water table depth act as the boundary conditions of the coupled MOBIDIC–MODFLOW model (Guzha, 2008). At each daily time step, the water table level determined from the solution of MODFLOW in the previous time step is used in MOBIDIC for the calculation of groundwater recharge (Eq. 12). The calculated groundwater recharge is then applied as the upper boundary condition in MODFLOW, and the water table level of the computation grid is updated. Note that, unlike in MIKE SHE, no fine spatio-temporal discretisation of the system is required in MOBIDIC–MODFLOW, in which a daily time step has been used.

Unlike MIKE SHE, coupling UZ–SZ in MOBIDIC–MODFLOW is not based on an
iterative water table correction procedure. Based on the calculated water
table level in the previous time step, the recharge to groundwater can be
positive (recharge to groundwater) or negative (extraction from groundwater; Castillo, 2014). The latter occurs when the saturated storage is bigger
than moisture storage in the gravity reservoir (

In Table 1, the differences between MIKE SHE and MOBIDIC–MODFLOW with regards to the conceptualisation of the unsaturated flow, saturated flow, UZ–SZ coupling process, and limitations associated with the application in very shallow water tables are summarised. Unlike their disparity in describing the unsaturated zone, the two models share a similar formulation and numerical solution technique for the saturated flow module. The iterative water table adjustments approach in MIKE SHE takes into account the variations in the specific yield as the water table rises and falls. In MOBIDIC–MODFLOW, the specific yield is instead defined as a soil hydraulic parameter and remains unchanged during the course of simulation. This, however, leads to an underestimation of the water table rise as a result of a very small value of the specific yield as the water table approaches the ground surface (Abdul and Gillham, 1989).

Comparison of the subsurface flow processes in MOBIDIC–MODFLOW and MIKE SHE.

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

Schematic representation of actual and reference water table rise using the WTF method for a soil column subjected to a single pulse of rainfall. In this case the actual water table rise is larger than the reference value, which occurs when the water table is very close to the soil surface.

Step-by-step procedure of the WTF method used in MIKE SHE.

Hydraulic properties of the soil types used in this study, based on Rawls et al. (1982), and simulated range of initial water table depths.

Simulation results of the water table rises are shown in Fig. 5. Each dotted
curve in the plots is associated with a specific initial depth to the water table
(ranging from 0.3 to 1.5 m) and rainfall rate. The plots in Fig. 5 are
divided into two zones, i.e.

Variations of the ratio of recharge to infiltration with rainfall and
initial depth to water table in different soil types simulated by MIKE SHE.
The red dots are

For a given initial water table level, moving from low to high rainfall
intensities results in increasing the ratio of recharge to infiltration,
shifting from a situation where

Note that for all soil types analysed, the

As discussed in Sect. 3, MOBIDIC's unsaturated soil depth (

It is assumed that the unsaturated soil layer thickness,

The specific yield in calculation of the groundwater head by MODFLOW is
determined based on a soil water retention model (e.g. Brooks and Corey,
1964) and the hydrostatic equilibrium assumption in the unsaturated zone (suction
profile in unsaturated zone changes from steady state to another over the
changes in water table; Hilberts et al., 2005; Duke, 1972):

The groundwater recharge

Conceptualisation of the interaction between UZ and SZ in MOBIDIC–MODFLOW in rising and falling water table conditions.

To interpret the proposed groundwater recharge equation (Eqs. 19 and 20), let
us first analyse its limit values. As

Variations in

Figure 7 displays the values of parameter

These

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.

So far, our analysis focussed on a single soil column in which the coefficient
of the proposed groundwater recharge,

The simulated water tables of the two models are shown in Fig. 8. It can be seen that the predicted water tables of the two models are very close. The slight mismatch between the predicted water table heads of the two models can be attributed to the fact that in MOBIDIC–MODFLOW, the response of groundwater to the precipitated water is immediate, that is, within the same time step (1 day). This is not the case in MIKE SHE, where the simulation time step is much shorter at 1 s. The saturated length (the length where water table is at the soil surface) predicted by the two models is closely matched, which shows that the simplifications in the UZ–SZ interaction of MOBIDIC–MODFLOW can mimic the complex dynamical interaction between the two zones. Note that the generated saturation excess runoff by two models was removed from the soil surface, as the flow routing module was not included in the simulations.

Simulated water table level after 4 (red), 8 (blue), 12 (green), 16 (magenta), and 20 days (sky blue) with MIKE SHE (solid lines) and MOBIDIC–MODFLOW (dashed lines).

In order to further evaluate the suitability of the proposed conceptualisation of UZ–SZ in MOBIDIC–MODFLOW, the water table fluctuations of the two models at the Borden catchment for the month of May 2015 were compared. The Borden catchment is located approximately 70 km northwest of Toronto, Ontario (Jones et al., 2006). The site is about 20 m wide and 80 m long, and it was subjected to several experimental (Abdul and Gillham, 1989) and numerical rainfall–runoff studies (Jones et al., 2006; Kollet et al., 2017; VanderKwaak, 1999). It is assumed that the catchment consists of a single homogeneous sandy soil identical to the one used in the previous simulations (see Table 2 for hydraulic properties) without any vegetation cover. The motivation for simplifying the watershed physiographic characteristics here was to evaluate the MOBIDIC–MODFLOW in a “real” watershed while emphasising differences in UZ–SZ dynamics simulated by the conceptual approach as compared with a physically based numerical model.

The digital elevation model (DEM) of the Borden catchment has a spatial
resolution of 0.5 m, is available at

The difference in the simulated water table level of the two models is shown in Fig. 10 for the selected simulation days after the rainy events. The predicted groundwater head of the models at the outlet of the catchment is also compared in Fig. 11. It can be seen that the two models generally compare well.

The difference in simulated water tables by MIKE SHE and
MOBIDIC–MODFLOW (

However, MOBIDIC–MODFLOW slightly predicts a higher water table rise as compared with MIKE SHE following rainfall events (e.g. events on days 11, 18, and 25). At day 6, MOBIDIC–MODFLOW simulates higher water table levels compared with MIKE SHE because of the higher groundwater recharge received by the grid cells in zone 2 and subsequent lateral saturated flow from zone 2 to zone 1. Note that in the original UZ–SZ scheme of MOBIDIC, if the groundwater level is below the modelled soil layer (as in zone 2), the groundwater recharge is a linear function of the moisture content of the gravity reservoir (see Eq. 12). However, in MIKE SHE, the groundwater recharge is calculated from the solution of the Richards equation, considering the water table level to be the lower boundary condition. Therefore, the magnitude of groundwater recharge simulated by MIKE SHE for zone 2 is smaller than MOBIDIC–MODFLOW. The spatial pattern of water table depth at day 14 and following events changes as the direction of saturated lateral flow between the grids changes.

Simulated water table depth by MIKE SHE (solid lines) and MOBIDIC–MODFLOW (dashed lines) at the outlet grid of Borden catchment.

Hence, the saturated flow exchange between the zones with different characteristics of the interaction between the UZ and SZ (zone 1 with inverse relation between the unsaturated and saturated storage, and zone 2 with a direct relationship between them as classified in the Introduction) can affect the overall behaviour of the water table fluctuations at the catchment scale. In Fig. 11, the water table changes (sharp in MIKE SHE and smooth in MOBIDIC–MODFLOW) are related to the calculation of groundwater recharge in two models. Equation (19) (revised groundwater recharge) shows that in the revised version of MOBIDIC–MODFLOW, the groundwater recharge is zero during days without infiltration. This means that the response of the water table to the fallen precipitation is rapid, and the unsaturated moisture profile quickly reaches its equilibrium state as discussed in Sect. 5. Therefore, on rainy days, the MOBIDIC–MODFLOW water table quickly rises as opposed to that of MIKE SHE, in which the groundwater recharge is determined from the solution of the Richards equation and the soil profile may take a longer time before reaching its equilibrium state. Note that the slight increase in the water table level of MOBIDIC–MODFLOW between days 14 and 18 (no rainfall period) is due to the incoming lateral saturated flow from adjacent cells. That is why the predicted water able in MOBIDIC–MODFLOW is still slightly rising between 2 rainy days (from day 12 to day 17 and day 19 to day 24), as shown in Fig. 11. Consequently, the simulated water tables of the two models gradually converge in the days following a rainfall event (for example between the days 14 to 17 and 19 to 24) as the transient behaviour simulated by MIKE SHE dissipates following a long redistribution period.

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

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 (

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

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

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 (

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.

The presented analysis showed a complex relation between the water table rise and the rainfall rate, soil type, and depth to water tables. The simulation over different rainfall rates and water table levels showed that fine-textured soils with a large capillary fringe (consequently small unsaturated soil moisture deficit) could have the water table at the soil surface even with a low rainfall intensity. This is important for the investigation of saturation excess runoff in lowland near river zones of the catchments. Therefore, it is important for the model to distinguish such differences in the expected water table responses of different soil types for its suitability at the catchment scale.

The comparison of the simulated water tables of the two models at Borden
case (mean absolute error equals 1.3 cm) along with their execution times
(30 h with MIKE SHE against 10 min with MOBIDIC–MODFLOW) clearly
demonstrates the applicability of a simplified approach implemented in
MOBIDIC–MODFLOW in investigation of the groundwater–surface-water
interaction. Further improvements of the approach with inclusion of other
hydrological processes such as the effect of evapotranspiration on
parameterization of the (

The digital elevation model (DEM) of the Borden catchment can be accessed
from

MB, RL, and MN contributed to the development of the methodology. The simulations with MIKE SHE and modifications in MOBIDIC were made by MB. Preparation and revision of the manuscript were done by MB under the supervision of RL and MN.

The authors declare that they have no conflict of interest.

This research was partly funded by a Discovery Grant from the National Science and Engineering Research Council of Canada and by the Groupe de recherché sur l'eau de l'Université de Sherbrooke (GREAUS).

This paper was edited by Thom Bogaard and reviewed by two anonymous referees.