Characterising groundwater–surface water interactions in idealised ephemeral stream systems

Transmission losses from the beds of ephemeral streams are thought to be a widespread mechanism of groundwater recharge in arid and semi‐arid regions and support a range of dryland hydro‐ecology. Dryland areas cover ~40% of the Earth's land surface and groundwater resources are often the main source of freshwater. It is commonly assumed that where an unsaturated zone exists beneath a stream, the interaction between surface water and groundwater is unidirectional and that groundwater does not exert a significant feedback on transmission losses. To test this assumption, we conducted a series of numerical model experiments using idealised two‐dimensional channel‐transects to assess the sensitivity and degree of interaction between surface and groundwater for typical dryland ephemeral stream geometries, hydraulic properties and flow regimes. We broaden the use of the term ‘stream–aquifer interactions’ to refer not just to fluxes and water exchange but also to include the ways in which the stream and aquifer have a hydraulic effect on one another. Our results indicate that deep water tables, less frequent streamflow events and/or highly permeable sediments tend to result in limited bi‐directional hydraulic interaction between the stream and the underlying groundwater which, in turn, results in high amounts of infiltration. With shallower initial depth to the water table, higher streamflow frequency and/or lower bed permeability, greater ‘negative’ hydraulic feedback from the groundwater occurs which in turn results in lower amounts of infiltration. Streambed losses eventually reach a constant rate as initial water table depths increase, but only at depths of 10s of metres in some of the cases studied. Our results highlight that bi‐directional stream–aquifer hydraulic interactions in ephemeral streams may be more widespread than is commonly assumed. We conclude that groundwater and surface water should be considered as connected systems for water resource management unless there is clear evidence to the contrary.


| INTRODUCTION
Loss of water through the streambeds of ephemeral streams is thought to be a key pathway of aquifer recharge in arid and semi-arid dryland regions (Costa, Bronstert, &  Despite this nomenclature becoming widespread in the literature, we note that, even during a so-called disconnected state, flow of water still occurs between the stream and the aquifer-there is no hydraulic disconnection between SW-GW in real terms. Rather, the term 'disconnected' simply refers to the fact that additional lowering of the water table cannot induce a greater loss from the stream for that particular set of conditions. The 'stream-aquifer' research community often use the term 'interaction' synonymously with 'exchange' of fluxes (Brunke & Gonser, 1997;Brunner, Cook, & Simmons, 2011;Winter, 1995). However, here we are using the term 'interaction' in a broader sense to encompass the ways in which the stream and aquifer have a hydraulic effect on one another. Thus, we consider that the hydraulic interaction between surface and groundwater in the 'disconnected' state is still uni-directional, whereas in the 'connected' state there can be feedback from the groundwater to the surface water and thus the hydraulic interaction can be said to be bi-directional.
We also note that the state of the system may change through time (see, e.g., Rau et al., 2017), a further reason that categorising SW-GW interactions as connected or disconnected may be misleading.
An important characteristic of SW-GW interactions that has been also shown in previous studies is the development of an inverted water  Fox & Durnford, 2003). Under changes in stream stage, for thin streambeds, the extension of the IWT may increase or decrease immediately below the streambed, whereas for thicker streambeds the development of the IWT will gradually increase its size for any change in stream stage (Xian et al., 2017). In ephemeral streams we anticipate that the development of the IWT should be controlled by factors such as the degree of saturation, initial water table depth, the magnitude, timing and sequencing of streamflow events and hydraulic properties, including anisotropy, of the streambed sediments. However, these factors have not yet been evaluated in the literature, despite recent advances in understanding the nature of groundwater mounding beneath ephemeral streams (Cuthbert et al., 2016).
In addition to a lack of fundamental research on the general understanding of SW-GW interactions in ephemeral streams at small scales, its importance at larger scales has also been neglected ( Here, we first propose a general conceptual model for characterising the main factors that control SW-GW interactions in ephemeral streams and their role in affecting the water balance of arid and semiarid regions. These concepts are then tested using a series of numerical model simulations, enabling the quantitative evaluation of different scenarios of stream-aquifer interactions.

| A CONCEPTUAL MODEL OF EPHEMERAL STREAM-AQUIFER INTERACTIONS
Despite the paucity of research on ephemeral stream-aquifer interactions, existing hydrological theory can inform the likely range of controls on these interactions. We propose that the following factors will be most important in controlling the degree of bi-directional hydraulic interactions: water table depth, stream stage, hydrograph shape, time between events, channel shape, channel boundary permeability and water retention characteristics of the subsurface materials. All these factors may vary individually or in combination in real systems. For example, channel shape will impact the infiltrated volume by increasing or decreasing the wetted perimeter and, consequently, the rate of infiltration through the streambed. Streambed permeability will increase infiltration rates for high values of permeability and reducing it for low values. Flow duration and frequency affect the amount of water available ultimately available for infiltration, and the amount of water than can infiltrate will depend on the degree of saturation and the water table depth. To illustrate the general way in which interactions may occur, we can characterise two end-member responses for 'deep' and 'shallow' water table systems that depend on the variations between these parameters as shown in Figure 1.
In the case of a deep water table, the frequency of events will affect the degree of saturation based on the prevailing time of drainage between events and consequently, the rate at which the channel bed can infiltrate newly arrived water. The process of water flowing through a thick variably saturated zone is depicted in Figure 1a in a two-dimensional cross section. When the stream stage starts to rise the IWT starts to develop, at a growth rate and size that are controlled by the antecedent saturation and the hydraulic conductivity of the sediments. Under lower antecedent saturation, which occurs under long time periods between flood events, more water will infiltrate below the streambed due to higher hydraulic gradients. The rate of movement of the IWT will depend on the degree of saturation, and for lower values of saturation the IWT will move more slowly downwards. The movement of the IWT will also be affected by anisotropic characteristics of the sediments, which may favour increased horizontal spreading of water. At the end of the event, the IWT becomes separated from the streambed as it descends due to gravitational drainage. At the same time, it decreases in size (areas decrease from t1 to t5 in Figure 1a) due to the losses associated with the spreading of water due to capillary forces. No influence of the water table depth is expected during the advance of the IWT for this case of a deep water table, and the rate of IWT movement is only a function of the saturation state of the sediment surrounding and below the channel.
For the case of a shallow groundwater system, the frequency of streamflow events combined with the antecedent water table depth will influence the infiltration rate. This process is shown in Figure 1b.
Under this scenario, as the IWT develops within the thin variably saturated zone it rapidly interacts with the shallow water table creating a continuous zone of saturation beneath the stream; the hydraulic gradient is thus reduced and consequently the infiltration rate declines.
For both shallow and deep water tables, the change in saturation within the material surrounding and below the channel under different pressure heads will depend on their hydraulic and water retention properties (hydraulic conductivity and soil moisture retention curve).
F I G U R E 1 Conceptual process model of interactions between ephemeral streams and an underlying homogeneous aquifer for (a) deep and (b) shallow water tables. Dashed lines represent the evolution of the inverted water table (IWT) and the water table mound at time t i during and after a streamflow event. The hypothetical shape and size of the IWT depend on the magnitude, shape and duration of the streamflow hydrograph and the antecedent conditions of saturation (inherited from the previous dry period), as well as hydraulic and soil moisture retention properties of the sediments In this article, we seek to test and generalise this conceptual understanding via idealised numerical modelling, which enables quantification and insight into a poorly understood process that is very common in drylands.

| Governing equations and numerical methods
Flow under unsaturated conditions can be described by the following equation (Richards, 1931): where H p is the pressure head [L], K is the saturated hydraulic conduc- is the effective saturation estimated by: where θ s and θ r represent the saturated and residual liquid volume fraction, respectively.
θ is described by using the van Genuchten soil moisture retention equation (van Genuchten, 1980): Relative permeability k r is also estimated by the van Genuchten method in the following way: F I G U R E 2 (a) Shape of the flow event is implemented as a specified head boundary condition at the stream base and sides. Before and after the flow event in the channel, the boundary condition switches to become 'no flow'; (b) cross section of the idealised transect considered in the numerical model, including a list of the boundary conditions and parameters of the base case model The specific moisture capacity C m [L −1 ] is defined by the following equation: We chose to use COMSOL V5.1 Multiphysics for the numerical model in order to have the necessary flexibility in the applied equations and boundary conditions.

| Boundary conditions
where: H r = z + y represents the hydraulic head in the channel, and

| Initial conditions
Choice of initial conditions in ephemeral streams is non-trivial due to complex antecedent moisture conditions implicit in such systems. Two options for initial conditions often used in unsaturated zone models are either a hydrostatic initial state of the water table and unsaturated zone or a periodic steady state for a specific dry period length. However, both of these can be unrealistic considering the typically highly variable frequency of flow events in ephemeral systems. Thus, we implemented a compromise between these end members as follows.
First, a steady-state condition for a small stream stage corresponding to 0.5 cm was specified in order to raise the moisture state of the unsaturated zone above the unrealistically dry conditions that hydrostatic conditions would imply. Second, using this initial steady-state condition the stream stage was then set to zero in order to let the sediment drain and to allow the dissipation of groundwater mound for a period of approximately a year (360 days) of no flow. Third, at the end of this no flow period, a pair of identical flow events was modelled using the various types of flow event described below, separated in time by a dry period whose duration was also varied as described below. The second event of the pair was then analysed and included in the results presented in the following sections.

| Base case scenario and sensitivity analysis
A base case model was defined with a K of 1.45 m day −1 , which corresponds to sandy loam sediments (Carsel & Parrish, 1988). This is

| Streamflow duration and water table depth
The shape of the event hydrograph was varied by changing the total duration of the event from 7 to 16 days. The rising and falling limb of the hydrograph were kept the same as the base case scenario (i.e., 1 and 5 days for the rising and falling limb, respectively) but the duration of the peak of the event was varied with values of 1 (base case), 5 and 10 days.

| Length of dry period between streamflow events
The influence of the dry conditions is evaluated by two streamflow . Therefore, we have used a range of 10-360 days for the duration of the dry period in order to include seasonal variations ( Table 1).
The analysed event corresponded to an event peak of 5-day duration event for the simulations.

| Soil hydraulic and water retention properties
The characteristics considered in the sensitivity analysis were: (a) hydraulic conductivity, (b) water retention curve and (c) storage capacity. These were evaluated separately and are summarised in

| Transmissivity
The influence of aquifer transmissivity was evaluated by increasing and reducing the height of the model domain by 10 m while keeping the K value constant.

| Channel cross-section shape and channel width
Channel cross-section was evaluated by changing the channel width in relation to the base case scenario. For a channel width larger than the base case scenario, the model domain was also increased in order to reduce the influence of lateral boundary conditions. Since it is intuitive that the increase in channel width increases the total infiltration, the infiltration per unit length flowing through the streambed was used for comparative analysis. Channel cross section shape was also considered by simulating and comparing results for rectangular, triangular and trapezoidal shapes. For the latter two cross sections, a slope of 1:1 was specified for the channel banks.

| Combinations of parameters used in sensitivity simulations
All variations of the above parameter variations where carried out in combination with variations in initial water table depth values of: 1, 3, 5, 10, 15, 20 m below the streambed (Table 1). In addition, the length of dry period and event peak duration were also varied in combination. (Table 1).

| Conceptualising a single flow event in time and space
Based on the results of the numerical simulations, the hydraulic processes governing the loss of water from an ephemeral stream transect can be described as follows for a 3-day streamflow event with a 1-day peak (Figure 3, stage hydrograph shown in Figure 4).    These results are intuitive because for events occurring after short dry periods, we would expect the rate of decay of the degree of saturation to be higher after the event has ceased caused by the downward movement of the IWT (Figure 4c). For longer dry periods following an event, the rate of change in the degree of saturation F I G U R E 4 Temporal variation modelled at 1 m below the centre of stream (left side of model half-space) for the same scenarios of Figure 3: (a) infiltration rate, with stream stage shown for comparison on right-hand axis, (b) pressure head and (c) degree of saturation for deeper (20 m) and shallower (5 m) water tables (WT). Soil parameters correspond to the base case scenario ( Figure 2) slows considerably, and becomes nearly constant (Figure 4c). This reduced variation of saturation states for long dry periods between events means that the infiltrated volume does not vary much when an event occurs, reaching an almost constant value depending on the depth of the water table (Figure 5a).
For shallow water tables of <3 m in our simulations, the range of variation of total infiltrated volume due to the length of dry period is also restricted, but in this case due to the rapid connection of the IWT with the water table and the influence of the capillary fringe ( Figure 5a). The extension of the capillary fringe represents a region in which the degree of saturation reaches a constant value. Therefore, the initial conditions for a shallow water table will be similar for any dry period length, which in consequence will result in a similar volume of water losses for events, irrespective of dry period duration between events.

| The influence of streamflow duration
A summary of the simulation results used to test the influence of streamflow duration on total streambed infiltration volumes are shown in Figure 6. As expected, infiltrated volumes increase with the duration of the event. Variation in flow event duration shows that the maximum value of infiltrated volume is asymptotically reached later for longer streamflow durations and for deeper water table depths.
For example, the increase in infiltrated volume reaches a steady value at water table depths of around 10, 15 and >20 m for 1-, 5-and 10-day-long flow events, respectively.
For shallow water tables, the increase of infiltration losses is limited due to the rise and lateral expansion of the groundwater mound below the stream, which quickly reduces the hydraulic gradient and regulates the infiltration rate (Figure 3i). For deep water tables, there is more pore-space available to enable continued lowering of the IWT which enables higher infiltration and, consequently, a larger increase in total infiltration volume ( Figure 6). As the streamflow duration increases, the maximum depth at which this F I G U R E 5 Variation of the total infiltrated volume into the streambed during one event against varying: (a) dry period durations between events (specified as the number of days with zero streamflow)-with different data series representing a different water table depth, dashed line represents the dry period at which the rate of variation of infiltrated water becomes log -linear, and (b) water table depths, with the variation due to different duration of dry periods indicated by the shaded area, dashed line represent the approximate apparent water table depth threshold F I G U R E 6 Variation of streambed infiltration volume during a single event as a function of the duration of the streamflow event, the length of dry period between events (shaded range in the style of Figure 5), and water table depth feedback from the water table occurs is therefore also greater. Thus, the limit to the depth of eventual SW-GW bi-directional interactions may be 10s of metres in the scenarios simulated, but in principle even greater for other combinations of high permeability sediment and long flow durations.

| The influence of sediment properties
Overall our simulations showed that the total streambed infiltration volume per event increases as the sediment hydraulic conductivity increases (Figure 7a), the 'coarseness' of moisture retention curve increases (Figure 7b) or the amount of total pore space available increases (Figure 7c).
Total infiltration is particularly sensitive to changes in hydraulic conductivity because infiltration rate is proportional to the hydraulic conductivity and the hydraulic gradient. For a specific stream stage, the hydraulic head and consequently the hydraulic gradient remain similar. The opposite occurs for low values of hydraulic conductivity.
However, when the shape of the moisture retention curve is changed, rates of infiltration also change due to changes in hydraulic gra-

| The influence of geometrical characteristics of the stream channel
We found that infiltration through the streambed for both trapezoidal and rectangular channel geometries showed differences with higher values (6%, not shown) for the rectangular shape which are consistent with the shorter wetted perimeter in comparison with the trapezoidal shape that reduces the influence of lateral flow due to capillary flux during the advance of the IWT.
Since the triangular channel geometry does not have a 'streambed' as such (only a channel invert), for comparison of all three geometries tested, we compared infiltration rates just through the streams' banks.  Table 1)   is also narrower, is more easily spread laterally due to the higher hydraulic gradient, resulting in less feedback to the infiltration rate.
As the channel width increases, the interaction with the water table last longer due to the development of a bigger groundwater mound which reduces the hydraulic gradient and consequently the infiltration rate.
The variation of the infiltration rate through both the streambed and streambank shows a non-linear relation with the stream width.

| DISCUSSION AND CONCLUSION
We set out to understand the process controls on transmission losses from idealised ephemeral stream beds in dryland environments. We first developed a conceptual model of factors that control infiltration through the variably saturated zone around, and below, an ephemeral streambed, and then quantified the relative importance of these factors using a suite of numerical model simulations. Specifically, we evaluated streamflow characteristics, time duration between streamflow events, water table depth, aquifer hydraulic properties and channel geometry.
For a given streamflow event, the initial saturation conditions characterised by the duration of the antecedent dry period, the hydraulic conductivity of the sediments, and the water table depth all provide strong controls of the infiltration rates lost from the stream.
As expected from the conceptual model, deeper water tables combined with longer dry periods and higher hydraulic conductivity increase the amount of infiltrated water; the opposite occurs when these parameters decrease.
Our analyses of the variability of infiltration rates when the geometrical characteristics of the channel change, have important implications for hydrologic and land surface models, especially for large scale models where narrow channels are difficult to represent, which can result in the gross underestimation of infiltration rates. At smaller scales, the variation of infiltration rates through streambanks due to changes in the cross section will also impact the availability of water for biochemical processes occurring within the streambed.
Our simulations show that infiltration rates vary non-linearly with water table depth, although they become constant, dependent on the local conditions, when a threshold in the water table depth is reached.
For a homogeneous aquifer with hydraulic properties corresponding to a sandy loam material, the threshold for a 7-day streamflow event (with a 1-day peak) is reached for water table depths greater than approximately 10 m. This threshold, beyond which bi-directional SW-GW interactions become limited, increases for longer events and can be 10s of metres in some of the scenarios tested. For different values of hydraulic conductivity, including anisotropy, a similar behaviour will be expected with a threshold that will vary depending on the exact combination of all components involved a particular situation.
In all these cases, the initial condition beneath the stream is one of partial saturation, and yet we demonstrate that feedback from the underlying groundwater is common during the simulated ephemeral streamflow events. Hence, we conclude that the paradigm of assumes 'hydraulic disconnection' between SW and GW in dryland regions for water management purposes may be misleading since any increase or decrease in water table depth caused by natural or human activities could still affect the amount of recharge that the aquifer receives in many cases. Such 'capture' of additional recharge (Theis, 1940) is generally ignored for dryland regions (Bredehoeft, 1997(Bredehoeft, , 2002Bredehoeft, Papadopoulos, & Cooper, 1982). We suggest that there is a broad spectrum of channels present within dryland environments that function in a 'transitional', rather than a disconnected state. Since dryland groundwater supplies a significant proportion of the world's water for irrigated agriculture, and that the depletion of groundwater of such regions is a major global issue, more ongoing research into the potential feedbacks between SW and GW in these contexts in still needed.

ACKNOWLEDGMENTS
We gratefully acknowledge financial support from Cardiff University.

DATA AVAILABILITY STATEMENT
All data from the modeling used to generate the figures will be available via the Natural Environment Research Council's Environmental Information Data Centre (Terrestrial and freshwater).