The extent to which groundwater flow affects drainage density and erosion has long been debated but is still uncertain. Here, I present a new hybrid analytical and numerical model that simulates groundwater flow, overland flow, hillslope erosion and stream incision. The model is used to explore the relation between groundwater flow and the incision and persistence of streams for a set of parameters that represent average humid climate conditions. The results show that transmissivity and groundwater flow exert a strong control on drainage density. High transmissivity results in low drainage density and high incision rates (and vice versa), with drainage density varying roughly linearly with transmissivity. The model evolves by a process that is defined here as groundwater capture, whereby streams with a higher rate of incision draw the water table below neighbouring streams, which subsequently run dry and stop incising. This process is less efficient in models with low transmissivity due to the association between low transmissivity and high water table gradients. A comparison of different parameters shows that drainage density is most sensitive to transmissivity, followed by parameters that govern the initial slope and base level. The results agree with field data that show a negative correlation between transmissivity and drainage density. These results imply that permeability and transmissivity exert a strong control on drainage density, stream incision and landscape evolution. Thus, models of landscape evolution may need to explicitly include groundwater flow.

Drainage density is a fundamental property of the Earth's surface that controls erosion and the transport of water and sediments. Drainage density has been observed to vary with climate, vegetation, relief, and soil and rock properties

A number of analyses of river networks have noted a relation between drainage density, lithology and transmissivity

Most numerical landscape models use simplified representations of groundwater flow and do not simulate the water table or lateral groundwater flow directly

Here, I present a new coupled model of groundwater flow, overland flow and erosion. The model was inspired by the coupled groundwater and streamflow model of

The model code described here simulates steady-state groundwater flow, transient saturation overland flow, stream incision and hillslope diffusion in a 2D cross section of the subsurface. These processes are shown schematically in Fig.

Groundwater flow is approximated as steady state, with the dark blue line in Fig.

The model starts with a rectangular model domain with a constant slope in one direction. The rectangular model domain contains a single cross section that is oriented perpendicularly to the slope and that is used to solve the groundwater flow, overland flow and the hillslope diffusion equations. The initial topography in the direction of the cross section is randomly perturbed. The topography evolves over time as a result of stream incision and hillslope diffusion. The model simulates groundwater flow, overland flow and hillslope diffusion in the 2D cross section. All streams are assumed to run perpendicular to the cross section and are perfectly straight and parallel. Streamflow and stream incision are calculated by multiplying the water and sediment flux in the 2D cross section by the contributing area perpendicular to the cross section. Thus, water and sediment transport in the out-of-plane direction take place in a series of perfectly parallel streams that develop along an inclined topography. The workflow and equations for each component of the model are discussed in detail in the following sections.

Conceptual model showing the hydrological and erosion process represented in the new model code. The hydrological processes include groundwater flow, overland flow and streamflow. The erosion processes include hillslope diffusion and stream incision.

The model starts with a random initial topography, which is calculated as using a series of 400 linear segments with random placement and random perturbation of the elevation at the start and end points of the segments. For the model simulations shown in this study, the average initial elevation is 0 m and the initial relief is 0.5 m.

Precipitation events are quantified using rainfall intensity statistics for the Netherlands

The precipitation events per year are calculated starting with a frequency (

Precipitation–frequency curve for the Netherlands, following

The subdivision of precipitation between evapotranspiration, overland flow and groundwater flow in the model is calculated for individual precipitation events. For each precipitation event, groundwater recharge is assumed to equal the available storage in the unsaturated zone (i.e. all groundwater stored in the unsaturated zone is assumed to eventually percolate to the groundwater table). For each point in the model domain, the available storage in the unsaturated zone is calculated using the depth of the water table and the specific yield of the subsurface:

The time-averaged potential recharge rate

where

Saturation overland flow is calculated as the amount of precipitation that exceeds the available storage (

The model is based on the assumption that groundwater flow can be considered to be in steady state on the timescales of stream and hillslope erosion processes. This was judged to be reasonable because the groundwater flow system reacts much faster than the relatively slow rates of erosion. Given these assumptions, groundwater flow and the position of the water table can be calculated using analytical solutions of steady-state groundwater flow.

First, the out-of-plane component of groundwater flow (i.e. groundwater flow parallel to the direction of the nearest stream) is calculated using Darcy's equation and assuming that the out-of-plane hydraulic gradient is equal to the (out-of-plane) slope of the nearest stream:

Calculated out-of-plane groundwater flow for a range of values of transmissivity and stream slope, using the base-case values of the contributing area (5

The remaining in-plane groundwater flow (i.e. towards the nearest stream) is calculated using the value of recharge calculated in Eq. (

In-plane groundwater flow is calculated using the Dupuit–Forchheimer equation, which describes depth-integrated steady-state groundwater flow between two groundwater discharge points

For points at the edge of the model domain that are only bound by a discharge point on one side, the equation reduces to

First, one seepage node is picked at the lowest elevation in the model domain, and the water table is calculated using Eqs. (

Subsequently, a new seepage node is added at the node with the lowest elevation in the part of the model domain where the modelled water table exceeds the land surface.

The water table is recalculated using this new additional seepage node.

The last two steps are repeated until the modelled water table is below or at the land surface (i.e.

An example of the calculated water table and seepage locations following the procedure is shown in Fig.

Example of initial topography and calculated water table and groundwater seepage locations. The water table and seepage locations were calculated by an iterative solution of Eqs. (

The water flow in each stream consists of two components: (1) steady baseflow supplied by groundwater discharge and (2) transient flow that consists of overland flow. The calculation of both components is described in the following two sections.

The baseflow in each stream node is calculated in two steps. First, streams nodes are found by finding the node with the lowest elevation for each series of neighbouring seepage nodes in the model domain. Note that the term seepage nodes is used here to denote nodes where groundwater discharge occurs. The 2D (in-plane) value of groundwater flow toward each stream node (

Example of calculated baseflow to streams. The coloured triangles denote the magnitude of the calculated baseflow.

The volume of water that is contributing to overland flow is calculated per precipitation event as follows:

Example of calculated precipitation excess and saturation overland flow in streams. Panel

The water level in streams as a result of baseflow and overland flow is calculated using the Gauckler–Manning equation for stream discharge. The Gauckler–Manning equation for stream discharge is as follows

The discharge generated by overland flow operates on short timescales of hours to days. Modelling this process directly would require short time steps that would make the model prohibitive computational expensive. Instead, this work derives new equations for the total discharge in a stream following a single precipitation event. To keep the solution mathematically tractable, the assumption is made that each precipitation event generates a volume of overland flow that is added instantaneously to the stream channel. The volume is subsequently discharged over time.

The continuity equation for discharge of a stream channel is given by

Integration of this equation with boundary condition

The initial height of the water level in the channel (

Adding Eq. (

This equation was validated by comparison with a numerical solution for the water level over time (Fig.

Validation of the equation for the transient discharge of an instantaneously added volume of water in a stream channel (Eq.

The sediment flux in a stream channel at carrying capacity is given by the following equation

The incision of streams results in an adjustment of the stream slope (

After each time step, a new value of stream slope is calculated using the new elevation of the base of the stream and the elevation of the downstream edge of the model domain:

The sediment flux in the steam channel carried by baseflow is calculated using Eq. (

Erosion by streams caused by the discharge of overland flow is calculated by combining the equation for stream discharge due to overland flow over time (Eq.

Integrating this equation from

The eroded volume results in incision of the stream. Stream incision by overland flow is calculated by distributing the eroded volume (

Erosion of the parts of the model domain outside of the streams follows the hillslope diffusion equation

The solution of the equations for groundwater flow, overland flow, streamflow and erosion follow an iterative scheme that is detailed in Fig.

Flow chart for the iterative solution of the groundwater flow, overland flow and erosion equations.

The base-case parameter values follow

The value of the sediment transport coefficient reported by

The compilation of total sediment discharge from flume experiments and field observations by

Comparison of measured sediment discharge and the water discharge term

Base-case parameter values.

To explore the role of groundwater flow in erosion, a series of model experiments were conducted with different values for transmissivity and specific yield. To compare the effects of changes in groundwater flow with climate and erosion parameters, an additional set of experiments was conducted with different values for total precipitation (

The range of variation in the hillslope diffusion coefficient was based on

Parameter value range in model sensitivity analyses.

In addition to the model sensitivity analysis, a second set of model experiments was conducted to explore the persistence of drainage density over longer timescales. These model experiments used the modelled incised topography of the base-case model run after 10 000 years as a starting point and subsequently added another 50 000 years to the simulation, although with different parameter values that follow the sensitivity analysis reported in the previous section and Table

Figure

The decrease in number of streams is caused by a process that is defined here as groundwater capture, by which faster eroding streams draw the water table below neighbouring streams and reduce the baseflow and saturation overland flow of these streams until they become dry. The process of groundwater catchment capture is illustrated in more detail in Fig.

The evolution of the stream network over time is summarized in Fig.

The initially high rate of groundwater capture is slowed down over time by the negative feedback imposed by the base level at the downstream (out-of-plane) end of the model domain. The stream slope is lower for streams that have incised deeper, which limits their incision power. The negative feedback on incision results in the establishment of a steady-state drainage network after 2500 years. For the base-case model run shown in Fig.

The incision and the reduction of active streams by groundwater capture also means that the area susceptible to saturation overland flow decreases over time, as saturation overland flow takes place near active stream channels where the water table is located close to the surface. Apart from a short phase at the start of the model run, streamflow and stream erosion are predominantly generated by groundwater flow, which is referred to as baseflow once it enters the stream channel (Fig.

Modelled change in the land surface and water table over 10 000 years for the base-case model run. The results show the evolution from a random topography with a high number of streams to an incised topography with 12 active streams. The past positions of the land surface and water table show the decrease in the number of streams over time.

Illustration of groundwater capture in the base-case model run. Panels

Changes in drainage density

Groundwater capture is dependent on the transmissivity of the subsurface. This is illustrated by three model runs with different values of transmissivity (Fig.

Modelled change in the land surface and water table over time for three model experiments with low (

Comparison of drainage density and incision to a number of parameters that govern erosion and streamflow show that drainage density is the most sensitive to transmissivity and initial slope (Fig.

The strong negative correlation of drainage density and initial downstream slope is illustrated in Fig.

Precipitation shows a complex relation with drainage density (Fig.

Specific yield only has a subtle effect on stream incision. Lower values of specific yield increase the volume of saturation overland flow and make it more difficult for streams to run completely dry, even if they are disconnected from the water table. However, given the subordinate importance of overland flow in generating streamflow and erosion in these model experiments, this effect is very modest and does not change the modelled drainage density or incision rate after a model runtime of 10 000 years (Fig.

The sediment transport coefficient exerts a strong control on the rate of incision and controls how fast the drainage network reaches a steady state. High sediment transport coefficients result in high rates of incision and a relatively fast adjustment of the stream network. However, the rate of incision is ultimately limited by the base level at the downstream end of the model domain, as explained previously; as a result, the sediment transport coefficient only has a minor effect on the modelled drainage density (Fig.

The effect of changes in the hillslope diffusion rate on the development of streams is illustrated in Fig.

Sensitivity of drainage density and stream incision to hydrological and erosion parameters. The incision rate denotes the incision rate of the stream with the lowest elevation at the end of each model run. Note that the range of parameter values reflects their variability in humid and subhumid settings, as explained in the text.

Sensitivity of drainage density and stream incision to initial slope, showing the effects of the lowest

Sensitivity of drainage density and stream incision to hillslope diffusion rate, showing the effects of the lowest

The model runs shown up until this point all started with flat topography with a random perturbation of

Figures

The asymmetry in the response of the drainage network to parameter changes is also shown in the model sensitivity analyses presented in Fig.

Modelled adjustment of a stream incision for a decrease

Sensitivity of drainage density to climate, hydrology and transmissivity parameters for different initial topographies and runtimes. Panel

The model code presented here was intentionally kept as simple as possible to keep the solutions mathematically tractable and the computational effort manageable. The main limitations are that the model is 2D and only represents a system with perfectly parallel and straight streams; groundwater flow is in a steady state and the contribution of transient groundwater discharge or subsurface storm flow to streamflow is neglected; and the treatment of erosion by overland flow is highly simplified, with an instant addition of all overland flow to the nearest active stream channel as well as a highly simplified distribution of the eroded volume over the length of the channel.

In spite of these limitations, the conclusions of the importance of groundwater flow for drainage density and stream incision are arguably relatively robust. The model results show that groundwater capture is a somewhat inevitable consequence of coupling groundwater flow, streamflow and erosion equations. Regardless of the exact equations and numerical implementation used, the water table will always crop out at perennial streams, and the incision of these streams will draw the water table down. Thus, smaller streams losing their groundwater discharge and eventually falling dry is a logical consequence of differences in incision rate between streams. These conclusions are also supported by previous model studies that found that groundwater flow had a strong effect on erosion

However, one thing that the model does not represent well is the initiation of stream channels. Due to the 2D nature of the model, the initial topography is represented only in the modelled cross section, and all streams are effectively parallel. This means that, initially, small streams develop in most small depressions. The presence of many small streams divides the water to be discharged over many streams, which each have a very low incision power but are nonetheless considered active streams in the model. In reality, many of these streams would not initiate, as the initial topography would not consist of a set of small depressions that are lined up and connected along an inclined slope. Instead, most of these depressions would be isolated. Therefore, the number of active streams at low runtimes should be much smaller than the high number in the model runs reported here (see e.g. Fig.

To my knowledge, the process of groundwater capture that is illustrated here has not been proposed before. Studies by

The process of groundwater capture is expected to be important wherever streamflow is dominated by either groundwater discharge or saturation overland flow. In systems where infiltration-excess flow is dominant, which include most semi-arid and arid regions

Note that the difference between the groundwater capture process and the more well-studied process of stream capture

The model study presented here used climate, hydrology and erosion parameters that were based on the southern Netherlands

The negative correlation between transmissivity and drainage density that the model results shows (Fig.

Figure

Although the field data show the same trend as the model results, the predicted drainage densities for low transmissivities (

Comparison of the modelled relation between transmissivity and drainage density with field observations

The strong correlation between drainage density, transmissivity and groundwater flow shown in both the model results and field data means that the inclusion of groundwater flow and a representation of the water table could strongly improve models of stream network evolution, landscape evolution, and water and sediment fluxes over geological (

In addition to being potentially beneficial for models of landscape evolution, a tighter integration of landscape evolution and groundwater models could also improve knowledge of groundwater and surface water systems. The groundwater table is often assumed to follow the topography to some degree, and groundwater divides are often assumed to coincide with surface water divides

A new model is presented that couples equations that govern groundwater flow, overland flow, streamflow and erosion and that represents humid regions where infiltration-excess overland flow is negligible. The coupling of these equations reveals a strong dependence of drainage density on groundwater flow and transmissivity. The dependence of drainage density on groundwater flow is controlled by a process identified here as groundwater capture, whereby faster incising streams draw the water table below neighbouring streams, which become dry and lose the ability to incise any further. This process is more efficient when the water table is relatively flat, which can be due to either low groundwater recharge or high transmissivity, with transmissivity often being the dominant control due to its high variability. Sensitivity analysis shows that the importance of transmissivity for drainage density is roughly equal to the base level and initial slope and exceeds other hydrological and erosion parameters such as precipitation and the hillslope diffusion coefficient. These results agree with published field data from the Thames Basin, UK, and the Cascades, USA, that show a negative correlation between transmissivity and drainage density. The results also imply that groundwater and transmissivity may be a dominant control on drainage density in humid regions. Landscape evolution models may need to include groundwater flow and the water table. At the same time a closer integration of landscape and groundwater data and models may help improve knowledge of the evolution of the Earth's surface.

The model code presented in this paper is named the “groundwater flow, overland flow and erosion model” (GOEMod) and has been published on Zenodo (

All data used in this study are available via the cited sources. The sediment discharge data shown in Fig. 9 are available in the Supplement of Lammers and Bledsoe (2018). The drainage density and transmissivity data shown in Fig. 19 are available from Table II in Bloomfield et al. (2011) and numbers reported by Luo and Stepinski (2008), Luo et al. (2010) and de Vries (1994).

The contact author has declared that there are no competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

I would like to thank Jacobus de Vries for sparking my interest in the relation between groundwater systems and landscape evolution.

This paper was edited by Wolfgang Schwanghart and reviewed by Marijn van der Meij and Stefan Hergarten.