A hybrid-3D hillslope hydrological model for use in Earth system models
Abstract
Hillslope-scale rainfall-runoff processes leading to a fast catchment response are not explicitly included in land surface models (LSMs) for use in earth system models (ESMs) due to computational constraints. This study presents a hybrid-3D hillslope hydrological model (h3D) that couples a 1-D vertical soil column model with a lateral pseudo-2D saturated zone and overland flow model for use in ESMs. By representing vertical and lateral responses separately at different spatial resolutions, h3D is computationally efficient. The h3D model was first tested for three different hillslope planforms (uniform, convergent and divergent). We then compared h3D (with single and multiple soil columns) with a complex physically based 3-D model and a simple 1-D soil moisture model coupled with an unconfined aquifer (as typically used in LSMs). It is found that simulations obtained by the simple 1-D model vary considerably from the complex 3-D model and are not able to represent hillslope-scale variations in the lateral flow response. In contrast, the single soil column h3D model shows a much better performance and saves computational time by 2-3 orders of magnitude compared with the complex 3-D model. When multiple vertical soil columns are implemented, the resulting hydrological responses (soil moisture, water table depth, and base flow along the hillslope) from h3D are nearly identical to those predicted by the complex 3-D model, but still saves computational time. As such, the computational efficiency of the h3D model provides a valuable and promising approach to incorporating hillslope-scale hydrological processes into continental and global-scale ESMs.
Key Points:
- This study presents a hybrid-3D model for the hillslope hydrological response
- Hydrological simulations are similar to those predicted by a 3-D Richards model
- The hybrid-3D model is computationally 2-3 orders of magnitude more efficient
1 Introduction
The terrestrial hydrological system, including surface and subsurface water, is an essential component of the Earth's climate system, interacting with the terrestrial ecosystem and atmospheric boundary layer over a wide range of scales. Over the past few decades, researchers have developed one-dimensional (1-D) models to resolve the vertical flow of water through the soil column for use in Earth system models (ESMs) with a grid size of
km while accounting for subgrid variations of infiltration capacity and topography through simple parameterization schemes [Entekhabi and Eagleson, 1989; Famiglietti and Wood, 1994; Koster et al., 2000; Niu et al., 2005]. These 1-D models with coarse grid resolutions, do not explicitly represent small-scale (e.g., meters) hydrologic variations and redistribution of water over complex terrain and therefore neglect their nonlinear interactions with spatially varying terrestrial ecosystems and biogeochemical processes.
At the same time, hydrologists have developed detailed three dimensional (3-D) models (
-102 m grid size), representing small-scale hydrological (e.g., elevation, soil and vegetation) variations, across the landscape [Paniconi and Wood, 1993; Kollet and Maxwell, 2006; Rigon et al., 2006; Camporese et al., 2010] and coupled these models with land surface energy and water exchange schemes [e.g., Kollet and Maxwell, 2008; Fan et al., 2007; Shen and Phanikumar, 2010; Niu et al., 2014b]. These studies have shown the importance of vertical and lateral redistribution of surface and subsurface water on soil moisture, the surface energy balance and ecosystem dynamics [Miguez-Macho et al., 2007; Maxwell and Kollet, 2008; Niu et al., 2014b, 2014c]. However, the vast computational costs have impeded their direct use in ESMs and limited their implementation to mainly catchment scale domains [e.g., Kollet and Maxwell, 2008; Kollet et al., 2010; Niu et al., 2014b, 2014c], although recently Maxwell et al. [2015] implemented this approach over most of the continental US. It should be noted as well, that when such models are applied to larger river basins, to reduce computational costs the grid size generally increases (1–5 km grid size). As such these models are still not able to account for subgrid hillslope processes that lead to fast hydrological responses to precipitation [e.g., Miguez-Macho and Fan, 2012], and thus need to use effective subgrid parameterization schemes [e.g., Riley and Shen, 2014].
There is increasing interest in implementing 3-D hydrologic models into land surface models (LSMs) for use in ESMs to resolve small-scale (e.g., meter scale) variations of biogeochemical and ecological processes that are intimately linked to spatially varying soil moisture conditions [Niu and Zeng, 2012]. For instance, decomposition of soil organic carbon through microbial activities is extremely sensitive to soil moisture variations [Zhang et al., 2014] and thus spatial variations in soil moisture along hillslopes should be accounted for in ESMs.
During the next decade, continental and global scale LSMs are expected to achieve grid size resolutions of
km for water resource studies [Wood et al., 2011]. Even at these so called hyperresolutions, parameterization of subgrid hydrological processes at meter scale is still needed to resolve the water-sensitive biogeochemical processes. The implementation of such detailed schemes into global scale LSMs was therefore defined by Wood et al. [2011] as the next grand challenge for the hydrological modeling community.
Such an implementation should be based on sound hydrological theory, simulate the hydrological response (i.e., rainfall-runoff processes and the interaction with atmosphere) with comparable accuracy as complex physically based hydrological models, and be computationally efficient for continental or global domains at higher resolution than existing implementations (grid size
km). We are developing such a model at a grid size of
km by identifying and representing the dominant subgrid hydrological components. As part of this effort, the current study focuses on the development of an effective representation of the hillslope response. In general, the combination of relatively shallow and permeable soils under the influence of topography leads to a fast hydrological response to precipitation on hillslopes [e.g., Betson, 1964; Beven and Kirkby, 1979; McDonnell, 1990]. Hillslopes therefore play an important role in the lateral transport of water and solutes toward the riparian zone and river network [Dunne and Black, 1970; Mesa and Mifflin, 1986; Robinson et al., 1995; Woods et al., 1997; Troch et al., 2013].
Over the past few decades a number of models have been developed that simulate the hydrological behavior of hillslopes. Initially, this work mainly focused on predicting the saturated zone response [e.g., Childs, 1971; Beven, 1982; Brutsaert, 1994; Fan and Bras, 1998; Troch et al., 2003; Hilberts et al., 2004]. However, recently this work was extended to account for the interactions between the saturated and unsaturated zones [Paniconi et al., 2003; Weiler and McDonnell, 2004; Hilberts et al., 2007; Tromp-van Meerveld and Weiler, 2008; Carrillo et al., 2011]. Since the focus of this work lies in developing an efficient hillslope hydrological model for continental and global scale applications, computational constraints make it currently impossible to explicitly account for small-scale 3-D heterogeneity (i.e., plan form and soil properties) [e.g., Paniconi and Wood, 1993; Paniconi et al., 2003; Weiler and McDonnell, 2004; Broda et al., 2012]. Therefore, this study presents a hybrid 3-D model, which couples a vertical 1-D soil column model with a lateral pseudo-2D saturated zone [Troch et al., 2003] and overland flow model, while specifically accounting for the 2-D plan shape of the hillslope. The pseudo-2D model represents the 2-D variations by a single effective 1-D equation. This approach differentiates from previous work [e.g., Hilberts et al., 2007; Carrillo et al., 2011] in that it uses multiple representative vertical columns and accounts for overland flow using a process based representation. The main motivation behind this is to develop a procedure that provides a valuable and promising approach to incorporate hillslope-scale high-resolution hydrological processes into global-scale LSMs.
This study is organized as follows: Section 2 provides a detailed overview of the interactions between the different components (unsaturated zone, saturated zone, and overland flow system) of the hybrid-3D hydrological model developed here. Section 3 describes the methods pertaining to evaluating the quality and performance of the hybrid-3D hillslope model. Specifically, the response behavior of the newly developed hybrid-3D hillslope model is tested for a number of planforms, soil types and numerical discretizations. Next, these results are compared to those obtained with the 3-D physically based Richards equation model CATHY [Paniconi and Putti, 1994] and also to results from a simple representation where a single vertical column is coupled with a nonlinear reservoir representing the saturated zone response currently used in many LSMs. Section 4 presents the results of these tests and comparisons. These results are further discussed in section 5, and section 6 ends with a summary and conclusions. The Supporting information added to this document provides a numerical description of the hybrid-3D model as well as additional details about the results obtained using the hybrid-3D model.
2 A Hybrid-3D Representation of the Hillslope Response
2.1 Representing Hillslope Hydrological Processes in ESMs
As explained in the introduction hillslopes have an important impact on the hydrological response of catchments. We feel that it is therefore important to explicitly represent hillslope processes in continental and global scale ESMs. For hilly or mountainous environments, Figure 1a presents our perceptual model of the hydrological functioning of catchments within a grid cell, where the first order hillslopes interact with the river network and the higher order hillslopes interact with the riparian zone. We conceptualize this system through the identification of separate hillslopes that are representative for first order or higher order hillslopes within a grid cell (Figure 1b). Within a given grid cell two-way interaction occurs between the riparian zone and the river network.

(a) A perceptual representation of the small-scale hydrological variability within a given high-resolution grid cell (
km) for a hilly/mountainous environment. (b) A conceptual view of the dominant hydrological components is presented, specifically accounting for the main hydrological and geomorphological characteristics.
Even though this conceptual view highly simplifies the small-scale (e.g., meters) heterogeneity in a grid cell, we believe that this representation identifies the dominant hydrological components (hillslope, riparian zone and river network) while maintaining computational efficiency. As stated, the current work focuses specifically on the effective implementation of the hillslope hydrological response. As such, this work will not deal with the interaction between hillslopes and the riparian zone and river network as this will be the focus of future contributions.
A refined schematic overview of the hybrid-3D approach for either a first or higher order hillslope is shown in Figure 2. Each hillslope is assumed to have a uniform vertical soil depth Dhs [L], a horizontal length l [L], width function w [L] indicating the width of the hillslope at the given distance from the seepage face, and a total surface area Ahs [L2].

Theoretical overview of the numerical implementation of the hillslope response as defined for the hybrid-3D model. (right) The vertical dimension of the hillslope as represented by a single soil column of depth D lying on top of an impermeable layer (solved by the 1-D Richards equation). (left) A spatial view of the hillslope. The effective width function w is assumed to vary in the direction of the hillslope. Lateral flow downslope is assumed to take place either through the saturated zone (Qlat) (solved by the hsB equation) or by overland flow in case of infiltration (
) or saturation (
) excess conditions (solved by the diffusive wave equations).
Instead of solving the full physically based 3-D equations, the hybrid-3D approach couples a vertical soil column with the lateral flow domain (i.e., saturated subsurface and overland flow). Within the soil column, which encompasses both the unsaturated and saturated zone, only flow in the vertical direction is simulated. A separate flow model simulates lateral flow through the saturated zone, where flow is assumed to be parallel to the impervious bedrock (the Dupuit-Forchheimer assumption). Furthermore, lateral flow can also occur as overland flow (i.e., infiltration or saturation excess). The following sections describe in detail all model components and their interactions.
2.2 Vertical Flow Through the Soil Column

where θ is the water content [L3 L−3],
the soil water pressure head [L], Kv the vertical hydraulic conductivity [L T−1], z the vertical direction [L] (positive upward) and G [L3 T−1 L−3] the lateral flow of water. As stated above, lateral flow through the soil matrix is assumed only to take place within the saturated zone (see section 2.3 for details). As such, this term couples the soil column with the lateral flow domain.





Equation 1 is solved by discretizing the vertical soil column into a number of vertical layers. For each node at the middle of a layer, a numerically implicit iterative scheme was used to calculate the soil water pressure head (
) and water content (θ) [e.g., Celia et al., 1990; van Dam and Feddes, 2000], influenced by the fluxes across the boundary of each layer (see supporting information for details). At the upper boundary, the input flux accounts for both the effective precipitation flux (P-Ea) [L T−1]) and the possibility of recharge from a ponding layer with depth dpond [L]. Ponding occurs in case the water is not able to infiltrate during previous time steps and the lateral overland flow rate by infiltration excess mechanism
in the downslope direction of the hillslope (see Figure 2 and section 2.4) is unable to completely drain this layer. The ponding layer is therefore able to infiltrate during subsequent time steps (see supporting information for details). At the bottom boundary a zero flux condition (qbot = 0) is applied as the soil column is assumed to lie on top of an impervious layer (see supporting information for details).
2.3 Lateral Flow Through the Saturated Zone



The hsB equation enables simulations of the 2-D lateral response by a 1-D equation and has been shown to effectively represent the groundwater response of a hillslope [Paniconi et al., 2003; Hilberts et al., 2004]. As such, a main assumption behind the hsB equation is that lateral flow only takes place in the downslope direction. The current implementation as given by equation 6 is modified from the original version defined by Troch et al. [2003] who assumed the lateral conductivity to be constant.






In the current work, the drainable porosity f is assumed to be constant even though we are aware that in reality, this parameter varies. A number of authors have derived expressions for the drainable porosity f as a function of the groundwater level by assuming a quasi steady-state relation for the vertical pressure distribution [e.g., Bierkens, 1998; Hilberts et al., 2005; Acharya et al., 2012] using a modified version of the Van Genuchten soil hydraulic functions [van Genuchten, 1980]. A similar assumption has been used elsewhere to obtain an estimate of the groundwater table in LSMs [e.g., Abramopoulos et al., 1988; Chen and Kumar, 2001]. However, when the Campbell [1974] relations are applied to estimate drainable porosity as a function of saturated zone depth, the drainable porosity estimates become too small (f < 0.15) when the water table is deep (> 1 m from the surface, see supporting information). Paniconi et al. [2003], who used a 3-D Richards equation model, showed that the porosity is similar in magnitude to the drainable porosity, while Hilberts et al. [2005] showed that the drainable porosity only becomes smaller than the porosity when the water table is close to the surface. The use of a constant value of the drainable porosity is therefore expected to lead to better results as compared to using a time-varying value based on the Campbell [1974] relations. Therefore, it was decided to set drainable porosity to the porosity
. One extra benefit of using a constant value is that it improves computation time.
Equation 6 is solved by discretizing the hillslope into a number of lateral sections with average width w. For each node in the middle of a layer the saturated zone level h is estimated using a numerically implicit iterative scheme. At the upper lateral boundary, a zero flux condition (Qsat = 0) is assumed, while at the lower lateral boundary at the seepage face a kinematic boundary condition (
) was implemented (see supporting information for details).
2.4 Overland Flow Due to Infiltration or Saturation Excess Mechanisms


These equations assume that overland flow takes place in the form of sheet flow, with a constant overland flow level dtp [L] and width w [L] at a given distance from the lateral outflow point at the bottom of the hillslope. As such, similar to the hsB equation in section 2.4, a pseudo-2D equation is used to represent the overland flow response.
In equation 9, the subscript tp indicates either infiltration excess (ie) [Horton, 1933] or saturation excess (se) conditions [Dunne and Black, 1970]. It should be noted that x in equation 9 is defined in the same direction as in equation 5 (see also Figure 2), as the depth of the soil column is assumed constant along the hillslope. Saturated excess conditions mainly occur near the bottom of the hillslope once the saturated level h (equation 5) exceeds the soil depth
. For each condition the recharge rate
[L T−1] is defined as
and Rse for infiltration and saturation excess conditions, respectively. Here the
term was added to account for the fact that lateral flow is not strictly perpendicular to the vertical flow direction. To properly account for the origin of the water, equation 9 is solved separately for infiltration and saturation excess conditions, as will be shown in the next section. However, the joint overland flow depth
[L] is used to solve equation 10.


Equations 9 and 10 were implemented using a similar discretization and implicit solution as the hsB equation. The upper lateral boundary assumes a zero flux conditions (Qof = 0) while for the lower lateral boundary the kinematic boundary condition (
) was used (see supporting information for details).
2.5 Coupling of the Three Components
In the previous sections the different aspects of the hybrid-3D hillslope hydrological model were presented. The current section shows how each component dynamically interacts, as shown in Figure 3a.

Schematic overview of the different fluxes interacting between the vertical and lateral components of the hybrid-3D hillslope model. (a) A conceptual overview of a single vertical soil column coupled with the lateral response model (1c-h3D). This representation leads to a uniform recharge rate Rsat for each lateral flow node of the hsB equation (equation (6)), while the hillslope total lateral saturated zone (Qlat) and overland flow (Qof) fluxes are coupled with the vertical column (equation (1)). In case the hsB solution results in saturated conditions along the hillslope, the positive recharge rate by saturated excess (Rse) couples the hsB equation with the overland flow equations (equations (9) and (10)). (b) The number of vertical columns is extended to two (2c-h3D), leading to variations in Rsat, Qlat and Qof in the upper (up) and lower (down) part of the hillslope. (c) The number of vertical soil columns is further extended to twenty (20c-h3D). By increasing the number of vertical columns, variations in Rsat, Qlat and Qof along the hillslope can be taken into account.



Equations 12 and 13 explicitly assume that the lateral saturated zone flux G within the vertical soil column originates either from saturated subsurface flow (section 2.3) or from saturated excess overland flow (section 2.4) (see also the vertical response of Figure 2). In equation 13 the recharge rate Rsat can become either positive or negative.





For such locally saturated conditions it is assumed that the excess water does not immediately leave the hillslope. Only once the saturation excess flux reaches the lateral outflow point at the bottom of the hillslope (equation 16) it is accounted for in equations 12 and 13.



For a complete description of the numerical implementation of each of the different components and their interactions we refer the reader to the supporting information.
3 Experimental Setup and Model Comparison
3.1 Hillslope Description and Experimental Setup
Three different theoretical planforms (uniform, convergent and divergent) are used to test the hybrid-3D model (see Figure 4). These shapes are generalized forms for those observed in nature. Each hillslope has a horizontal length L = 100 m, slope α = 10%, soil depth D = 3.4 m and horizontal surface area Ahs = 5000 m2. The width w at the bottom (top) for the uniform, convergent and divergent hillslope is set to 50 (50), 10 (90) and 90 (10) m, respectively. By calculating the flow distance to the seepage face of the hillslope (see colors in Figure 4), the width function w can be estimated [Bogaart and Troch, 2006; Liu et al., 2012]. Width functions for uniform, convergent, and divergent hillslope are shown in the bottom panels of Figure 4).

(top) The plan shape of three different hillslope planforms used in current study. Different colors define the flow distance to the seepage face of the hillslope for a grid resolution of dx=2 m and dy=5 m. (bottom) Based on the flow distance the hillslope width function was derived is indicated by the black line.
The response of the hybrid-3D hillslope model is tested for a combined recharge-drainage experiment for a simulation period of 100 days. During the first 50 days the precipitation rate is set to 30 mm d−1, followed by a period without precipitation. This relatively large precipitation intensity enables testing of the performance of the model under extreme conditions.
Simulations are performed for both a sandy loam and clay loam soil for which the corresponding Campbell [1974] parameters and typical values of β and Dmp are given in Table 1. At the start of each experiment a hydrostatic pressure distribution was assumed for the soil column without the presence of a water table.

Soil Type | b |
![]() |
θs | Ks (cm h−1) | Dmp (cm) | β (cm) |
---|---|---|---|---|---|---|
Sandy loam | 4.90 | −21.8 | 0.435 | 12.480 | 100 | 0.008 |
Clay loam | 8.52 | −63.0 | 0.476 | 0.882 | 100 | 0.008 |
3.2 Numerical Setup and Varying Numbers of Vertical Columns
In one set of simulations, a vertical single soil column is used to simulate the vertical response of the hillslope (1c-h3D) as shown in the left panel of Figure 3. Two different vertical grid resolutions are used to discretize the soil column. The majority of the simulations use an equally spaced high resolution soil column with
cm (henceforth referred to as the high-fixed vertical resolution), which results in a total of 340 vertical nodes. This detailed representation reduces the occurrence of numerical artifacts [e.g., van Dam and Feddes, 2000] and can be assumed to provide the most accurate results. In addition, simulations are also performed using a much coarser variable vertical resolution of only 11 nodes at depths
−0.71, −2.79, −6.23, −11.89, −21.22, −36.61, −61.98, −103.80, −172.76, −277.31 and −339.50 cm from the surface (henceforth referred to as the coarse-variable vertical resolution). This resolution corresponds to the vertical column node setup as currently used within the Community Land Model (CLM) [Oleson et al., 2013] and improves runtime performance. It was chosen to add a small layer (
cm) at the bottom of the soil column (at −339.50 cm) to improve the identification of the saturated zone level hsat (equations 12 and 13) in the vertical soil column.
The lateral node spacing for the hsB and the overland flow portion of the hybrid-3D hillslope model for each of the three hillslope planforms is set to
m resulting in 20 lateral nodes for the uniform and divergent hillslopes and 21 for the convergent hillslope. For the convergent hillslope, the flow distance to the seepage face for the top left and right corners exceeds 100 m (see Figure 4), and results in the use of an extra lateral node.
The iteration criterion for the implicit numerical solution of the 1-D Richards equation (equation 1) and hsB equation (equation 6) is set to
and
m, respectively. These values result in stable simulations. Even though simulations are saved at the hourly resolution, the model makes use of an internal flexible time step, initially set to
h. If either iteration criterion is not satisfied, the time resolution is doubled up to a minimum value of
h. For the simulated results presented in this study, this minimum value was never encountered.
So far this work has focused on the coupling of a single vertical soil column with the lateral saturated zone and overland flow model (1c-h3D) as shown in Figure 3a. In addition to a one column implementation, the hybrid-3D model is also implemented using two (2c-h3D), four (4c-h3D), ten (10c-h3D) and twenty (20c-h3D) vertical columns along the hillslope. Figures 3b and 3c present the schematic overview of the model setup coupling two and twenty vertical soil columns with the lateral zone, respectively. Increasing the number of soil columns is expected to lead to improved simulations of the vertical soil moisture distribution and recharge rate to the saturated zone Rsat along the hillslope. Using more soil columns also allows for the simulation of lateral variations in Qlat and Qof (both saturation and infiltration excess) along the hillslope. Note that when 20 vertical columns are used (21 for the convergent hillslope), each individual column corresponds to a single lateral node (see Figure 3c) used to solve the hsB and overland flow equations. Furthermore, the maximum number of vertical soil columns that can be simulated by the hybrid-3D model corresponds to the number of lateral nodes used to solve the hsB and overland flow equations.
3.3 Model Comparison








CATHY was implemented for each of the three planforms using the same parameter values and vertical and lateral discretization as the hybrid-3D model. Also a lateral zero-flux condition was set for all boundary conditions to match the hybrid-3D simulation, except the for seepage face boundary where CATHY uses a constant zero head pressure (
) condition. This condition differs from the kinematic wave (
) approximation implemented for the hybrid-3D model. The iteration criterion for CATHY was set to
m.
4 Results
The following section shows how the hybrid-3D model performs in the idealized hillslope simulations. Section 4.1 presents the soil moisture and base flow dynamics for the recharge-drainage experiment using the high-fixed vertical column discretization (
cm). In section 4.2, the impact of using the coarse-variable vertical column discretization is shown. The runtime performance of the hybrid-3D model, in comparison with the two other models, is presented in section 4.3. Section 4.4 presents some of the issues related to not accounting for recharge rate to the saturated zone (Rsat) variations along the hillslope.
4.1 Temporal Soil Moisture and Base flow Dynamics
Figure 5 presents the temporal evolution of the vertical soil moisture distribution for the uniform hillslope for two versions of the hybrid-3D model (1c-h3D and 20c-h3D), the 1c-CLM model and CATHY using the fixed (
cm) vertical soil column discretization. There are considerable similarities between the two hybrid-3D implementations and CATHY. Although the 1c-h3D model predicts a thicker saturated zone at the end of the recharge period as compared to CATHY (white numbers). The 20c-h3D model also overpredicts the thickness of the saturated zone compared to the CATHY simulations, but its value lies much closer to CATHY. Much wetter conditions are simulated by the 1c-CLM simulations for the sandy loam soil compared to the other models. The difference between the 1c-CLM and other simulations are smaller, but still apparent, for the clay loam soil simulations.

Temporal soil moisture profile for the uniform hillslope (see Figure 3) for two different soil types. The white number at the bottom of each plot describes the depth of the saturated zone (cm) at the end of the recharge period. Note that for the 20c-h3D and CATHY model the average value of the lateral variations in the vertical soil column are presented.
To gain more information about the temporal soil moisture distribution Figure 6 shows the difference between CATHY simulations and the 1c-CLM, 1c-h3D, and 20c-h3D simulations for uniform, convergent and divergent hillslopes with a sandy loam soil. For the 1c-h3D model the main differences with respect to CATHY are limited to the interface between the saturated-unsaturated zones: the 1c-h3D model simulates a thicker saturated zone (as also shown in Figure 5) and lower soil moisture values in the overlying unsaturated zone. The difference between the soil moisture distribution predicted by the 20c-h3D and CATHY are very small (< 0.03, see Figure 5). At the end of the drainage period this model simulates slightly wetter (> 0.05) conditions for the region just above the impervious layer. At the same time, the 1c-CLM model predicts much wetter soil moisture conditions than CATHY with differences close to 0.15 (blue area in Figure 5). These values correspond to
of the available pore space.

Temporal soil moisture profile difference for the different hybrid-3D models and the CATHY model for the (top) uniform, (middle) convergent, and (bottom) divergent hillslopes for a sandy loam soil.
Figure 7 presents the resulting base flow response at the lateral outflow point. Overland flow is only simulated by the 20c-h3D model and CATHY for the convergent hillslope with clay loam soil. For all other cases, only base flow is simulated. The results between the 1c-h3D model and CATHY are similar during the drainage period and for the initial and equilibrium phase of the recharge period. However, for the period in between, base flow is initially underestimated and then overestimated. Peak base flow values during this overestimation phase are higher than the equilibrium base flow rate and is largest for the convergent hillslope (see also Table 2). Section 4.4 provides a detailed explanation of the mechanism causing this behavior.

Temporal base flow response for the (top) uniform, (middle) convergent, and (bottom) divergent hillslopes for a (left) sandy loam and (right) clay loam soil for the four different models using a high-fixed (1 cm) vertical resolution. Also presented in these panels is the precipitation rate P.

Model | Uniform | Convergent | Divergent | ||||||
---|---|---|---|---|---|---|---|---|---|
1c-CLM | 1c-h3D | 20c-h3D | 1c-CLM | 1c-h3D | 20c-h3D | 1c-CLM | 1c-h3D | 20c-h3D | |
![]() |
0.00 | 9.30 | 0.14 | 0.00 | 20.77 | 0.13 | 0.00 | 4.25 | 0.13 |
TBE | 236.61 | 89.16 | 37.25 | 228.47 | 137.65 | 36.30 | 246.45 | 65.44 | 46.39 |
RMSE | 0.21 | 0.06 | 0.02 | 0.20 | 0.11 | 0.02 | 0.23 | 0.03 | 0.03 |
NS | 0.849 | 0.989 | 0.998 | 0.864 | 0.956 | 0.998 | 0.825 | 0.996 | 0.998 |
As with the vertical soil moisture distribution the 20c-h3D model and CATHY show an almost identical base flow response (small total bias error and Nash-Sutcliffe coefficient close to one, see also Table 2). It is only during the latter stage of the drainage period that the 20c-h3D model predicts smaller base flow values compared to CATHY, which corresponds to the slightly wetter soil moisture conditions just above the impervious layer that are shown in Figure 6 (right column). For the clay loam soil a slight delay in the base flow response for the 20c-h3D model is observed during the initial recharge phase. This difference results from the fact that the hybrid-3D model only simulates lateral flow within the saturated zone (where
). For the clay loam soil this assumption might have been too strict, as CATHY shows that lateral flow already occurs at close to saturated conditions.
The base flow response of the 1c-CLM simulations for the sandy loam soil increases much slower during the recharge phase and decreases much faster during the drainage phase in comparison to the other models (see Figure 7). This results in higher soil moisture values shown in Figures 5 and 6. Conversely, the clay loam simulations of the base flow response compares reasonably well with CATHY (though base flow still declines too quickly during the drainage phase). As explained in section 3.3, the parameters of the nonlinear reservoir groundwater model are constant and irrespective of soil type and geomorphological shape. It appears that these parameters are more representative of clay loam soils than sandy loam soils and perhaps this explains why these 1c-CLM simulations with clay loam soils have similar soil moisture and base flow response compared to the other models.
For the current simulations overland flow from saturated excess is only simulated by the 20c-h3D and CATHY for the convergent hillslope with clay loam soil. As a result, the simulated base flow is smaller than the precipitation rate, since part of the outflow occurs as overland flow. The slightly larger base flow rate for the 20c-h3D as compared to CATHY might be related to assumption of a constant value for the drainable porosity f (see section 2.3). Hilberts et al. [2007] showed that when the water level h reaches the surface, the drainable porosity becomes smaller and this will cause overland flow to be simulated earlier resulting in smaller base flow values. However, in the current hybrid-3D implementation temporal changes in the drainable porosity were not accounted for. For this hillslope setup the one column models, 1c-CLM and 1c-h3D, did not simulate saturated conditions at the outflow point of the hillslope. In principle the 1c-CLM model is not able to take such lateral variations into account, and a single lateral recharge rate along the hillslope in the 1c-h3D simulations does not result in local saturated conditions (see also section 4.4)
Table 2 shows the statistical correspondence of the base flow response for the three models with respect to CATHY. As expected, the best results are obtained for the 20c-h3D model followed by the 1c-h3D model (similar to Figure 7), although the latter leads to an overestimation of the maximum base flow rate.
4.2 Impact of Vertical Resolution
The previous section described the results using the high-fixed vertical resolution (
=1 cm). A variable vertical resolution, similar to the current implementation of the CLM [Oleson et al., 2013] land surface model (see section 3.2), was also tested. The impact of using the coarse-variable vertical resolution on the simulated base flow for the sandy loam soil is shown in Figure 8. At the lower vertical resolution, the subsurface runoff response has a number of inaccuracies as shown by the wiggles for all four models used. However, the overall behavior is similar to the response obtained using the high-fixed resolution shown in Figure 7.

Temporal base flow response for the three different planforms for the sandy loam soil for the four different models using a coarse-variable vertical node resolution.
For the hybrid-3D implementation, a number of subtle changes (bends) in the base flow response can be observed. These are related to difficulties in identifying the exact location of the saturated zone level hsat using the simulated pressure head value
for the nodes just above this layer at the variable vertical resolution. Changes in the saturated zone level are used to estimate the recharge rate Rsat (see equation 13). However, these changes are influenced by the location of the saturated zone with respect to the nearest vertical node. For the high-fixed vertical resolution simulations this effect is very small, however, at a coarser vertical resolution the impact of these changes is more apparent. Figure 8 also shows that for the variable vertical resolution, CATHY simulates an immediate base flow response, which is also incorrect and a result of the coarse discretization.
In Figure 9 the temporal evolution of the total vertical column soil moisture difference with respect to CATHY is presented using both the fixed and variable vertical resolutions. Overall, the two discretizations yield similar results. For the sandy loam soil, the differences between the two hybrid-3D models (1c-h3D and 20c-h3D simulations) and CATHY are small (< 5 cm) during both the recharge and drainage period. For this soil type the 1c-CLM model again simulates much wetter conditions. For the clay loam soil, it can be observed that during the recharge period all three models agree well with CATHY (difference close to zero). However, during the drainage phase, slightly larger base flow rates are simulated by CATHY (see also Figures 7d and 7f), leading to an increase in total soil moisture differences. As explained above, base flow is generated only within the saturated part of the vertical column in the hybrid-3D implementations and the 1c-CLM model. For the clay loam soil this assumption was too strict.

Temporal total column soil moisture difference between the different hybrid 3-D models, the 1c-CLM model and CATHY using both a high-fixed and coarse-variable vertical resolution for three different planforms for the sandy loam and clay loam soil with depth D=3.4 m.
4.3 Runtime Comparison
In Table 3, the runtime performance of the hybrid-3D approach using different numbers of vertical columns, the 1c-CLM model and CATHY are presented. As expected, increasing the number of vertical columns leads to an increase in the total computation time. However, even the most detailed hybrid-3D implementation (20c-h3D) is still more efficient as compared to CATHY.
Model | Uniform | Convergent | Divergent | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
SL | CL | SL | CL | SL | CL | |||||||
F | V | F | V | F | V | F | V | F | V | F | V | |
1c-CLM | 15 | 1.2 | 15 | 1.4 | ||||||||
1c-h3D | 13 | 2.0 | 16 | 2.7 | 13 | 2.1 | 18 | 3.9 | 13 | 2.0 | 16 | 2.4 |
2c-hsB | 24 | 2.7 | 30 | 3.5 | 24 | 2.7 | 29 | 5.0 | 24 | 2.7 | 27 | 3.0 |
4c-hsB | 47 | 3.9 | 56 | 4.6 | 48 | 4.0 | 55 | 5.9 | 46 | 3.9 | 57 | 5.0 |
10c-hsB | 115 | 7.7 | 134 | 9.3 | 117 | 7.8 | 132 | 11 | 113 | 7.7 | 141 | 9.6 |
20c-h3D | 230 | 14 | 270 | 17.2 | 243 | 15 | 281 | 19 | 250 | 18 | 274 | 17.3 |
CATHY | 6315 | 165 | 9731 | 216 | 2399 | 57 | 3112 | 52 | 1984 | 52 | 2566 | 68 |
- a Results are presented using both for a high-fixed 1 cm (F) and coarse-variable (V) vertical resolution. These results were obtained using a single core implementation without accounting for model initialization.
For the high-fixed vertical resolution of
cm the runtime for the single vertical column (1c-h3D) implementation was, on average, about 250 times faster than CATHY. This clearly reveals the computational efficiency of the hybrid-3D approach. At this resolution, the runtime of both 1c-CLM and 1c-h3D are comparable. However, as shown in section 4.1, the hybrid-3D approach leads to improved soil moisture and base flow simulations, especially for the sandy loam soil (see Figures 5 and 7).
The overall computation time is considerably reduced for coarse-variable vertical resolution simulations (see Table 3). For these simulations the 1c-h3D model is about 55 times faster than CATHY. For the coarse-variable vertical discretization, the 1c-CLM model is slightly faster than the 1c-h3D. This is because number of lateral nodes (20) used to solve equation 6 is relatively small as compared to the number of vertical nodes (340) used to solve equation 1 using the high-fixed vertical resolution. However, for the coarse-variable vertical resolution simulation the additional number of lateral nodes for the 1c-h3D model is larger than the number of vertical nodes (20 lateral versus 11 vertical nodes). For all simulations the 1c-CLM model uses only a single lateral node. Therefore, the overall impact of solving the lateral flow equations in the 1c-h3D model is much larger for the variable vertical resolution simulations and leads to a slight increase overall computation time in comparison with the 1c-CLM model.
4.4 Spatial Variations of Recharge Rates
Figure 7 shows that there was an overestimation of the base flow rate before reaching equilibrium conditions in the 1c-h3D simulations, while this behavior was not observed using the 20c-h3D model. More details about these differences are provided in the current section.
Figure 10 presents the simulated base flow response using 1, 2, 4, 10 and 20 vertical soil columns in the hybrid-3D hillslope model for the first 35 days of the recharge period. It shows that an overestimation mainly occurs for the single column implementation (1c-h3D) although maximum base flow rates also exceeds equilibrium value for the two column case (2c-hsB). When there are four or more vertical columns, the simulated base flow response are nearly identical, and the peak values are not overestimated.

Impact of the number of vertical columns on the base flow response during the recharge period for the three different planforms for the sandy loam soil.
The main reason for the overestimation of the base flow rate is that the one and two column implementations do not properly account for lateral variations of the recharge rate toward the saturated zone Rsat. Figure 11 illustrates this effect by showing the spatial variation in Rsat for the 2, 4, 10 and 20 column implementations for the first 20 days of the recharge experiment. Each panel also shows the simulated recharge rate using a single vertical column (dashed black line). Once the bottom of the soil column becomes saturated (when the wetting front reaches the bottom of the soil column), lateral variations in Rsat become large, with the largest recharge rates Rsat in the vertical column situated closest to the lateral outflow point of the hillslope. This column receives the largest lateral base flow flux from the saturated zone system further up the hillslope causing the saturated zone in this area to grow. Due to these lateral variations, the mean recharge rate for the individual columns differs considerably from the single column implementation (see dashed red line in Figure 11). Note that once the wetting front reaches the saturated zone, the actual recharge flux Rsat is much larger than the precipitation rate (blue dashed line). These large values arise because both precipitation and unsaturated zone water become part of the saturated zone. Therefore, a given amount of precipitation leads to a large addition to the saturated zone [Abdul and Gillham, 1984, 1989].

Lateral variations of Rsat for multiple vertical column implementations. Different solid gray lines represent different vertical soil columns of which their lateral distance with respect to the seepage face of the hillslope is presented in the inset of each plot. The red dashed line presents the average recharge value for all vertical columns, while the black dashed line shows the recharge rate obtained using a single vertical column.
By using a single recharge rate Rsat the 1c-h3D model underestimates the height of the saturated zone at the seepage face, as Rsat is larger at the bottom of the hillslope in simulations with more columns. At the top of the hillslope, the opposite situation occurs as the recharge rate is overestimated. Figure 12 illustrates the impact of this by showing the temporal evolution of the water table along the hillslope for the five different hybrid-3D implementations in comparison with CATHY. For the period when Rsat exceeds the precipitation rate (first 20 days) the one (1c-h3D) and two (2c-h3D) column implementations have a shallower saturated zone close to the seepage face, while at the top of the hillslope the height of the saturated zone is overestimated. This causes a lower initial base flow rate. However, once the excess water at the top of the hillslope reaches the seepage face, there is an overestimation of base flow (see also Figure 7). Note that the slightly thicker saturated zone close to the seepage face of the hillslope in the hybrid-3D simulations as compared to the CATHY simulations are the result of the two different boundary conditions used to solve the hsB equation and CATHY. More specifically, the hybrid-3D model uses a kinematic boundary conditions at the seepage face to solve the hsB equation while CATHY assumes a zero pressure head condition (see sections 2.3 and 3.3). Lastly, errors in the accounting of Rsat as functions of distance from outlet will also impact the timing of flow responses in addition to amount of flow.

(top) Five plots show the difference in saturated zone height between the hybrid-3D model for different vertical column numbers and CATHY for three different planforms for the sandy loam soil. (bottom) The temporal height (cm) of the saturated zone level along the hillslope as simulated by CATHY.
5 Discussion
The results reported above show the ability of the hybrid-3D model to simulate the hydrological response of three different hillslope planforms for two different soil types in comparison with CATHY. As explained in the introduction the main motivation behind the development of the hybrid-3D approach is to create an efficient hillslope hydrological model to be incorporated into a continental and global scale LSM. Currently, these models are not able to take such small-scale features (e.g., elevation, soil and vegetation variations) into account even though these features have an important impact on the short term response to precipitation [Miguez-Macho and Fan, 2012]. During the next decade, it is anticipated that the grid cell size of these models will decrease to 1–5 km. Currently, it is computationally not feasible to solve the full 3-D Richards equation at these scales [Wood et al., 2011], although it should be noted that their scale of application increases [Maxwell et al., 2015]. The hybrid-3D simulations closely resemble the results obtained by the 3-D Richards equation model CATHY but is computationally more efficient. This motivates further development of the hybrid-3D approach presented here and to identify its impact on continental and global scale applications. These results have not been presented here, and will be the focus of future contributions. Instead, the current study presents the possibilities of the hybrid-3D model at the hillslope scale. At these small scales the presented results show that the single column simulations (1c-h3D) have similar runtime performance as using a nonlinear reservoir model (1c-CLM) to represent the groundwater response. However, the hydrological response predicted by the hybrid-3D model better matches the simulations performed by CATHY both the vertical soil moisture and Nash-Sutcliffe coefficient of the base flow response). Because CATHY has been shown to closely resemble real-world observations [Niu et al., 2014a], we believe that the hybrid-3D approach is able to realistically simulate the hydrological response of a hillslope.
Even though peak base flow values were overestimated for the single column hybrid-3D implementation (1c-h3D), the temporal behavior of predicted soil moisture is similar to that predicted by CATHY (see Figures 5 and 6). Section 4.4 explained that the overestimation of the base flow results from the fact that lateral variations of the soil moisture distribution and recharge rate to the saturated zone Rsat are not properly accounted for by the single column model (1c-h3D). However, it can also be observed from Figure 11 that during the first hours of positive recharge rate Rsat, the difference between the single column (dashed black line) and the average of using multiple columns (dashed red line) is small. Real-world precipitation events generally only last a few hours. Therefore, it is expected that the total impact of the base flow overestimation using a single column is much smaller, especially when the initial saturated zone levels are small and show little lateral variability. As the single column implementation of the hybrid-3D hillslope model is the most computationally efficient, we feel that this approach is preferred for continental and global scale applications. The impact of the hybrid-3D model using one vertical soil column will therefore be the focus of future contributions.
In this study, results were shown for two different soil types that have dramatically different permeability and conductivities. Different parameter sets or forcing will cause changes to the simulated soil moisture and lateral flow behavior. However, in general these changes are similar for the different models. We have provided some more information about this in the supporting information.
6 Conclusion
- The hybrid-3D approach is able to simulate the hydrological response of hillslopes for a recharge-drainage experiment both for the vertical soil moisture distribution as well as the base flow response. These results are in correspondence to simulations obtained by the full 3-D Richards equation model CATHY.
- The current setup of land surface models, consisting of a vertical soil column coupled with a parameterized representation of the groundwater to provide the interaction with the the river network, is not able to account for lateral saturated zone variations and the geomorphological characteristics of the landscape. The simulations performed in the current work by the 1c-CLM model lead to incorrect simulation of the soil moisture distribution and base flow response for both the sandy loam and clay loam soil in comparison to CATHY.
- The work presented here shows that the hybrid-3D model is computationally efficient in comparison with CATHY. When a single soil column is used the runtime performance is about 250 times faster using a high-fixed (
cm) vertical resolution and 55 times faster using a coarse-variable vertical resolution. These runtime results are of similar order of magnitude as those obtained using the land surface model based (1c-CLM) hydrological representation. However, the hybrid-3D model results in much better hydrological simulation as compared to those obtain using the 1c-CLM model.
- The hybrid-3D approach is able to take lateral variations in the saturated zone and overland flow layer into account. However, using a single column to represents the vertical response (1c-h3D) the model has limited capabilities to account for the lateral variations in the recharge rate along the hillslope. As a result, saturated water levels at the seepage face (top) of the hillslope are underestimated (overestimated), which leads to a slight (
) overestimation of the maximum base flow rate.
In the current study only 3-D Richards equation model results (rather than direct observations) were used for comparison. Currently, we are testing the ability of the hybrid-3D hillslope to simulate real-world observations within a controlled environment. As explained in the introduction, the main motivation behind this study is to develop an effective approach to incorporate subgrid processes like the response of hillslopes within a continental and global scale land surface model as part of an Earth system model. The actual impact of this newly developed approach within such a model environment will be the focus of study in future work.
Acknowledgments
This research has been financially supported by a Department of Energy research grant (DE-SC0006773) involving the development of a hybrid 3-D hydrological model for the NCAR Community Earth System Model (CESM). Y. Fang has been financially supported by a grant from the National Science Foundation of China (51420105014). The data for this paper are available by contacting the corresponding author. Three anonymous reviewers are thanked for constructive and helpful comments that have greatly improved the clarity of our presentation.