Introduction

To improve the predictive capability of groundwater models, it is necessary to enhance the representations of both the aquifer characteristics, such as hydraulic conductivity and storativity, and the model-specified fluxes, such as lake and river interactions, recharge and evapotranspiration (ET). While improvements to estimations of hydraulic conductivity have been outlined in review papers, for example Wen and Gómez-Hernández (1996), this paper considers methods for improved representation of recharge and ET in groundwater models.

Historically, recharge has been represented in saturated groundwater models as a single input per cell per time step. ET has been conceptualised as a linear or piecewise linear relationship with depth (Banta 2000; Harbaugh 2005). These were reasonable approximations as field-based data, such as that from ET flux towers, were relatively sparse; therefore very little information existed with which to calibrate a model. Recently however, estimates of recharge and ET derived by remotely sensed methods such as reflectance and land-surface temperature have become more readily available (Guerschman et al. 2009; Nagler et al. 2005), and are being used as inputs to, and for calibration of groundwater flow models (Morway et al. 2013).

Parallel to this, there has been an increasing volume of research on water resources in areas with shallow water tables (water tables within the root zone, usually < 7 m deep), involving mechanisms such as surface-water/groundwater interactions (Brunner et al. 2011; Brunner et al. 2009; Doble et al. 2012; Lamontagne et al. 2014), ecohydrology and plant use of groundwater (Baird et al. 2005; Benyon et al. 2006; Crosbie et al. 2008; Goodrich et al. 2000; Holland et al. 2006; Lamontagne et al. 2005) and salinisation (Doble et al. 2006; Jolly et al. 2008). The time seems opportune, therefore, to rethink recharge and ET processes, particularly for conditions with shallow groundwater, and to determine what assumptions are physically realistic for these conditions.

This paper provides a review of the current and emerging methods used to incorporate recharge and ET as boundary conditions and as outputs from catchment-scale groundwater models. A robust conceptualisation of recharge and ET is particularly important where groundwater is shallow and these surface processes are more pronounced. The paper does not provide a review of methods for estimating recharge as there are already many quality papers that address this topic (Crosbie et al. 2010; Gee and Hillel 1998; Healy 2010; Kim and Jackson 2012; Petheram et al. 2002; Scanlon et al. 2006). Similarly, good review articles are available for ET processes, particularly in relation to remote sensing (Glenn et al. 2011; Kalma et al. 2008) and ET from groundwater in Australia (O’Grady et al. 2011).

This paper presents a conceptual understanding of recharge and ET processes including factors affecting recharge and ET functions, and the evaluation of field evidence for recharge and ET being dependent on groundwater depth. It outlines various approaches for modelling recharge and ET, discussing advantages and disadvantages and gives general considerations for the representation of recharge and ET, including the use of remote sensing data and uncertainty analysis. Some future research opportunities are also suggested.

Conceptual understanding of recharge and ET processes

Understanding methods for incorporating recharge and ET functions into a groundwater model requires a brief review of the components of the soil-moisture mass balance equation (Delleur 2006):

$$ {R}_{\mathrm{gross}}=P-\mathrm{I}\mathrm{n}\mathrm{t}-{E}_{\mathrm{uz}}-{T}_{\mathrm{uz}}-\mathrm{R}\mathrm{O}-\mathrm{I}\mathrm{F}+\Delta S $$
(1)
$$ {R}_{\mathrm{net}}={R}_{\mathrm{gross}}-{E}_{\mathrm{gw}}-{T}_{\mathrm{gw}} $$
(2)

Where R gross is gross recharge to the water table, R net is the difference between R gross and evapotranspiration from groundwater (ETgw), Int is canopy interception, E uz is evaporation from the unsaturated zone, E gw is evaporation from groundwater, T uz is transpiration from the unsaturated zone, T gw is transpiration from groundwater, RO is runoff from the land surface, IF is interflow, and ΔS is the change in soil-moisture storage.

Recharge is defined as the water that crosses the water table into the saturated zone (Fig. 1). Evapotranspiration is divided into evaporation (E) lost through soil processes, and transpiration (T) lost through vegetation water use. It is further divided into components of the flux originating from the unsaturated zone (E uz and T uz) and originating from upward flux from the saturated zone or groundwater (E gw and T gw). Note that there are different physical processes driving the evaporation and transpiration components of the water balance, and that the different subscripts are a technical separation of water originating from the unsaturated or saturated zones, based on the given definition of the groundwater control volume.

Fig. 1
figure 1

Conceptual model of recharge and evapotranspiration processes in areas with shallow groundwater. Precipitation (P) falls on the site and is intercepted by vegetation (Int), runs off down slope (RO) or infiltrates into the subsurface (I). Infiltrating water is transpired from the unsaturated zone by vegetation (T uz), evaporated from the soil surface (E uz), moves down slope as interflow (IF) or crosses the water table as gross recharge (R). From within the groundwater control volume, water may flow out from the aquifer (Q), be evaporated from the capillary fringe (E gw) or transpired by vegetation (T gw)

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The groundwater control volume provides a convenient method of quantifying groundwater for water management purposes and modelling with Darcy-type groundwater models such as MODFLOW (Harbaugh et al. 2000; McDonald et al. 1988) or FEFLOW (Diersch 2005). Previous studies separate evapotranspiration into groundwater ET (ETgw, GWET) and vadose or unsaturated zone ET (ETuz, VZET; Shah et al. 2007), or separate soil-based evaporation processes from transpiration, but none have been found that separate all four components.

More complex fully coupled research code models such as HydroGeoSphere (Brunner and Simmons 2011; Therrien et al. 2006) and MIKE SHE (Refsgaard and Storm 1995) simultaneously model saturated and unsaturated groundwater flow and surface-water flow; therefore, the operational control volume includes the unsaturated zone and possibly a small volume above the soil surface. The relationship between control volumes for fully coupled models will depend on the model being used, but care should be taken that modelled and remotely sensed fluxes are defined identically.

Remote sensing data such as reflectance data and thermal infrared land surface temperature data (Li et al. 2013) from satellites such as NASA’s Moderate Resolution Imaging Spectroradiometer (MODIS) and Landsat, and NOAA’s Advanced Very High Resolution Radiometer (NOAA-AVHRR) are increasingly being used in modelling land surface processes. They are presently used for estimating green cover, saturated surfaces, urban space and other masks in hydrological models, and are increasingly being used to estimate recharge and ET specifically for groundwater models, in regions such as semi-arid to arid Botswana and arid Xinjiang Uygur in China (Brunner et al. 2007) and temperate to semi-arid Nebraska Sand Hills, USA (Szilagyi et al. 2011). These remote sensing data have a control volume that includes the land surface plus vegetation and atmospheric processes. There is a mismatch between control volumes, and some kind of representation of the downward (R) and upward (ETgw) fluxes is required. In order to provide an appropriate representation of recharge and ETgw fluxes, the characteristics of these processes are discussed in the following sections.

Recharge

Recharge may be represented as either gross recharge (the volume of water that infiltrates through the unsaturated zone and crosses the water table) or net recharge (gross recharge minus ETgw). Remote sensing data often describes net recharge. Due to the difficulty in separating ETgw and ETuz in remote sensing evaporation signatures, net recharge may simply be approximated by the difference between rainfall and remotely sensed total ET estimates minus runoff (Crosbie et al. 2015). For field estimates of recharge, the water table fluctuation (WTF) method is commonly used to estimate gross recharge (Healy and Cook 2002; Meinzer and Stearns 1929), while the chloride mass balance (CMB) method provides estimates of net recharge where net recharge is positive (Anderson 1945; Wood 1999). Where solute transport is of interest, for example in salinity problems (Bauer et al. 2006; Jolly et al. 1993), gross recharge must be used to maintain a solute balance. It is critical that recharge is explicitly defined in the reporting of modelling results to avoid double accounting in the water balance.

The review papers previously mentioned all conceptualise recharge as a single time-varying inflow into the groundwater store. It is difficult to find examples of where recharge estimates are related to depth to water table (DTWT); however, a few studies do indicate that where groundwater is shallow, recharge does change as a function of water table depth. These studies include field measurements (Benyon et al. 2006; Crosbie 2003; Sophocleous 1992), remote sensing measurements (Crosbie et al. 2015; Szilagyi et al. 2013) and numerical modelling (Smerdon et al. 2008, 2010; Carrera‐Hernández et al. 2011).

For the Smerdon et al. (2008) study of the Boreal Forest of Canada, the Smerdon et al. (2010) study of the Okanagan Basin, Canada, the Crosbie et al. (2015) study of the Mediterranean climate South East region of South Australia, and the Szilagyi et al. (2013) study of Nebraska, the depth dependence is a result of reporting net recharge, i.e. the difference between gross recharge and ETgw. Net recharge is characteristically depth dependent due to the influence of ETgw. However, in the Crosbie (2003) and Crosbie et al. (2005) studies of the humid-subtropical Tomago Sand Beds, Australia, and the Carrera‐Hernández et al. (2011) study of Aspen harvesting in the Canadian Boreal Plains, even gross recharge was found to be a function of DTWT.

Crosbie (2003) used aggregated monthly recharge from high-frequency recharge time-series derived from the water table fluctuation method at seven piezometers in the Tomago Sandbeds over a 2-year period to describe the relationship between recharge and DTWT. Details of the calculations are in Crosbie et al. (2005). This relationship between recharge and DTWT shows recharge of zero when the water table is near the surface, increasing to a maximum recharge at a DTWT between 0.5 and 1.25 m, before stabilising at a lower rate below 2.0 m (Fig. 2a). The shape of the curve can be characterised by rejected infiltration for very shallow water tables, followed by a maximum rate of recharge due to minimal evapotranspiration of the water as it moves through the very thin unsaturated zone. With shallow groundwater, rather than infiltrating precipitation constantly replenishing soil moisture after depletion by ET (Shah et al. 2007), the antecedent moisture conditions that are consistently approaching field capacity provide ideal conditions for maximum rates of recharge. At greater water table depths, recharge as a percentage of rainfall is relatively constant with DTWT. This same relationship has also been shown using long-term average data from the Limestone Coast region of South Australia (Mediterranean climate) for around 400 monitoring bores (Fig. 2b) (Crosbie et al. 2015; Crosbie and Davies 2013).

Fig. 2
figure 2

a Gross recharge in the Tomago Sand Beds (Australia) showing dependence on DTWT, after Crosbie (2003), with data from Crosbie et al. (2005). b A similar relationship can be found in the Crosbie and Davies (2013) study of the Limestone Coast, a Mediterranean climate region of South Australia, which is also described in Crosbie et al. (2015), with permission from the Goyder Institute for Water Research

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At a catchment scale, this relationship may also be deduced from water balance studies. It was observed that during the Australian Millenium Drought of 1997–2008, in the Mediterranean climate regions of south Western Australia (Hughes et al. 2012; Petrone et al. 2010) and south eastern Australia (Petheram et al. 2011), that catchments with low relief and moderate rainfall showed significantly more reduction in runoff than higher-relief high-rainfall catchments. The studies suggested that the relatively shallow groundwater levels in these catchments resulted in increased runoff during pre-drought conditions due to a reduced storage capacity in the unsaturated zone. Although this level of detail in the recharge function may not be required for models with larger spatial and temporal scales, it should not be ignored where quantification of recharge to shallow groundwater is required.

The factors that affect groundwater recharge include climate, particularly precipitation and potential evapotranspiration (PET), vegetation cover, soil texture, macropores and preferential pathways, soil moisture, surface topography and depth to groundwater or bedrock. A summary of these factors and their impact on groundwater recharge is given in Table 1.

Table 1 Factors affecting groundwater recharge functions
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Evaporation

Shallow water tables also increase the rate of groundwater evaporation. The shape of the relationship between soil evaporation and DTWT has previously been described in soil physics literature (Gardner 1958; Gardner and Fireman 1958; Philip 1957; Talsma 1963). This function has also been observed in more recent modelling and field studies (Shah et al. 2007; Soylu et al. 2011).

Groundwater evaporation is maximised where the water table is at or near the surface and decays exponentially with depth (Fig. 3). This conceptualisation, or a simplification thereof, is used in many groundwater flow models such as MODFLOW EVT and ETS1 packages (Banta 2000; Harbaugh 2005; Harbaugh et al. 2000).

Fig. 3
figure 3

Dependence of groundwater evaporation rate on depth to water table for five example soil types (Talsma 1963), with permission from Wageningen University

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Models with physical representation of the unsaturated zone using Richards equation reproduce this relationship between evaporation and DTWT through their use of the van Genuchten (1980), Brooks and Corey (1964) or Campbell (1974) equations relating pressure, saturation and hydraulic conductivty: for example, HYDRUS (Šimůnek et al. 2003; Vogel et al. 2000), HydroGeoSphere (Therrien et al. 2006), MIKE-SHE (Doummar et al. 2012), and WAVES (Doble et al. 2015; Zhang and Dawes 1998).

Evaporation from the soil surface will lead to salt accumulation and salinisation where groundwater is saline (Peck 1978a, b; Peck and Hatton 2003). Secondary or dryland salinity is of concern in Australia (Jolly et al. 1993; Walker et al. 1994, 1998; Wood 1924), Canada and the United States (Miller et al. 1981), Thailand, South Africa and Argentina (Pannell and Ewing 2006). Allowance should be made for factors that limit evaporation from the soil surface such as mulch, vegetation cover or salt deposits (Benoit and Kirkham 1963; Gardner 1958). A summary of the processes impacting groundwater evaporation is given in Table 2.

Table 2 Factors affecting groundwater evaporation functions
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Transpiration

Transpiration is also known to be a function of DTWT (Nichols 1994; Smith et al. 1998). Similar to groundwater evaporation, transpiration also reduces to zero below an extinction depth, which is a function of the capillary fringe thickness and plant rooting depth. In contrast to bare soils, though, water tables near the surface create anoxic conditions, which decrease rates of transpiration (Amlin and Rood 2001). Some exceptions exist where vegetation has adapted to inundated and saline environments (Bell 1999). For all but obligate wetland species, transpiration is zero for a water table at or near the soil surface (Baird et al. 2005).

Sapflow measurement of groundwater transpiration (T gw) from willow (Salix spp.) and cottonwood (Populus fremontii) in semi-arid California and Arizona (USA) was used to develop functional curves for the RIP-ET package in MODFLOW (Fig. 4; Baird et al. 2005). Coupled groundwater–energy–plant growth models produce similar curves in the Mediterranean climate of southern Australia (Doble et al. 2015). The shape of the transpiration function depends heavily on the characteristics of the site, as presented in Table 3.

Fig. 4
figure 4

Mean daily transpiration canopy flux (cm/day) curves for five plant functional groups (obligate wetland, shallow rooted riparian, deep rooted riparian, trans-riparian and bare ground) during summer months. Positive numbers denote standing water (Baird et al. 2005); with permission from Springer

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Table 3 Factors affecting groundwater transpiration functions
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Both the maximum transpiration rate and the extinction depth are influenced by the factors described in Table 3. The use of parameters for plant functional groups (riparian, dryland, tropical savannah etc.) may be used to represent the relationship between transpiration and DTWT in regional models, in other areas species-specific information may be required. While vegetation root distribution tends to be concentrated near the surface, deep roots are ecologically significant, with a small proportion of deep roots providing a large percentage of water uptake during dry periods (Canadell et al. 1996).

The temporal characteristics of the groundwater model are also critical for representation of transpiration. Vegetation will respond to hydrologic stimuli such as changes in DTWT, solute concentration and climatic conditions, through root and leaf growth or decline and death (Doody et al. 2015). Care should therefore be taken when upscaling plant behaviour to a hydrogeological time-scale that the vegetation life cycle and adaptations are accounted for.

Net recharge

Net recharge is defined as gross recharge minus ET gw , and is worthy of further consideration as it is becoming more frequently used due to recharge estimation from remote sensing data. The shape of the relationship between net recharge and DTWT will be a combination of those of recharge, evaporation and transpiration with DTWT and is climate dependent. In water-limited environments, net recharge will generally be negative (ET) for shallow water tables, and positive (recharge) as DTWT increases. The relationships between recharge, E gw, T gw and net recharge are dependent on local geology, hydrogeology, vegetation and climate, and examples are shown in the figures by Sanford (2002) and Doble et al. (2015; Fig. 5a,b respectively).

Fig. 5
figure 5

a Example of the relationships between recharge, evapotranspiration (ET) and depth to water table (DTWT) (Sanford 2002) with permission from Springer. Recharge is gross recharge, groundwater discharge by ET represents ETgw, total evapotranspiration represents ETgw + ETuz, and the recharge–discharge characteristic function is net recharge (gross recharge; ETgw). b Long-term average depth to water table (DTWT) vs. water balance flux curves generated from WAVES modelling for temperate Mount Gambier, Australia, native vegetation on a silty loam, showing overstory transpiration from both soil and groundwater (OS_T), overstory interception (OS_i), evaporation from the unsaturated zone (Soil_E), runoff (Q), gross recharge (Gross_R), evapotranspiration directly from groundwater (ETGW) and net recharge (net_R). After Doble et al. (2015), with permission from the Goyder Institute for Water Research

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Maxwell and Kollet (2008) plot a form of net recharge, precipitation minus ET (P – ET) against DTWT and propose that net recharge is controlled by temperature where the water table is less than 1 m in depth, groundwater (depth) where DTWT is between 1–6 m, and precipitation at DTWT greater than 6 m. These depth intervals correspond with the depths at which soil-based evaporation is likely to dominate and climate is a driver (<1 m), although DTWT is likely to still have an impact here, the depths at which transpiration is dominant and net recharge is a function of DTWT (1 – 6 m) and where DTWT is below the influence of vegetation and net recharge is controlled by gross recharge (>6 m).

Smerdon et al. (2010) indicated the importance of seasonality in relationships between net recharge and DTWT for the Okanagan Basin in western Canada, modelled using MIKE-SHE. Negative net recharge (ET) was predicted for water tables less than 2 m deep during spring, summer and fall, but winter showed only positive net recharge. Maximum net recharge was highest in spring and fall. Relationships will vary depending on climatic and meteorological conditions of the study site.

Methods for modelling recharge and ET

There are numerous methods for modelling recharge and ET within a catchment-scale groundwater model. Three basic approaches include (1) using a Darcy-based groundwater model with physical or emulated representations of recharge and ET boundary conditions, (2) using groundwater models coupled with 1-D unsaturated models, and (3) using fully coupled saturated–unsaturated models. The modelling methods, advantages and disadvantages, example models and case studies are described in Table 4.

Table 4 Methods for modelling recharge and evapotranspiration
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Recharge and ET as a boundary condition

Saturated groundwater flow models provide the simplest means of modelling groundwater recharge and evapotranspiration. These models tend to have faster computational times, and can therefore be more easily applied to regional or continental problems, long time scales and probabilistic risk-analysis modelling. It may be easier to facilitate data assimilation into simpler models, so that observations drive the model outputs. The more linear functions that are associated with this type of model can lead to better model convergence; however as the models are more empirical than physically based, predictions for climatic conditions or land use changes outside of those used for calibration may be compromised. Local scale and small temporal scale results (monthly or seasonal) may not be as accurate as more physically based models.

As an example, the core MODFLOW model (Harbaugh et al. 2000; McDonald et al. 1988) represents recharge as a single time-varying value for each model cell using the recharge (RCH) package, and ET as a linear or piecewise function of groundwater depth using the evapotranspiration (EVT) or segmented evapotranspiration (ETS) packages (Banta 2000). Recently, many other representations of recharge and ET processes have been added to the MODFLOW suite in the form of additional packages and processes.

Coupled saturated–unsaturated groundwater flow models

A saturated groundwater model coupled with a 1-D unsaturated zone model using physically based equations can provide a good conceptualisation of recharge and evapotranspiration processes and is a compromise between the faster, simpler saturated models and slower, more complex fully coupled, physically based recharge. The coupling can involve representations of the subsurface—saturated and unsaturated zones, surface-water processes and often land surface and atmospheric components.

Physically based models can provide better estimates of recharge and ET at monthly or seasonal timescales than water balance models. The coupled model can be easily tailored to the questions that it is intended to address. It is possible to select or develop an unsaturated model that has functions specific to the site which may not be available from commercially available models such as the impact of salinity on transpiration, or the effects of changing CO2 levels on plant growth. The groundwater flow facility is maintained through the use of the groundwater model, while a 1-D unsaturated zone model is computationally less intensive than a 3-D unsaturated zone model. Modelling platforms are available to automate linkages between groundwater and unsaturated model zones.

Fully coupled models

Fully coupled models are valuable for modelling sites that are small in spatial and temporal scale, and where more complex processes are involved, for example recharge into hillslope catchments where interflow and recharge rejection are an important part of the water balance. These codes are well suited to developing a better understanding of groundwater/surface-water interactions, and saturated–unsaturated soil processes, as the soil and water are treated as a single store rather than separated into saturated and unsaturated components. However, obtaining separate outputs from the water balance can require extra processing due to this ‘one water’ approach.

Again, fully coupled models have a better predictive capability outside of calibration conditions and at a sub-annual timescale, and there are potentially more types of observations that may be used in calibration such as vegetation greenness indices and remotely sensed soil-moisture data. The long computational times may prohibit probabilistic modelling in larger catchments or for long timescales. Because of this, highly complex models are not generally used in modelling for risk-based water resources management and decision-making. Data-model merging may also be more difficult with more complex models.

A note on model complexity

There are currently two well-justified schools of thought on model complexity. One is that higher complexity is better, and that a thorough, automated calibration will result in better predictive capabilities, even if data are not available for all parameters. The other is that simplified models, with ‘just enough’ functionality, are better as they allow for better model interrogation through uncertainty analysis and therefore better understanding of model and system behaviour and sensitivities.

With reasonable data sets for calibration and realistic bounds for parameters where no data are available, more complex, physically based groundwater models can provide more robust predictions than simple models. Although it is less straightforward, methods have been developed for optimisation and uncertainty analysis on highly non-linear models. The use of surrogate modes and the null-space Monte Carlo method for parameter estimation and uncertainty analysis of a groundwater model is compared against a formal Bayesian approach in Keating et al. (2010).

However, there can be a tendency to use more complex models than are justified for the problem being addressed, particularly since highly complex models often appear more credible to stakeholders. The benefit of increasing model complexity to improve error metrics by 1–2 % is questionable when compared with minimising run times enough to interrogate the model performance, uncertainty and sensitivity to different processes and parameters. Complex models may have over 50 parameters, but in data poor regions, only enough information to form reasonable bounds for five of these. Similarly, model sensitivity analyses often show that only five to ten of these 50 parameters have a significant impact on the model results (Peeters et al. 2014).

Paradoxically, the level of complexity required is only known after a model has been developed and a sensitivity analysis has been undertaken. Appropriate complexity could be attained by incremental increases in model parameters and processes, developing a complex model and using a model emulator for uncertainty analysis, or at least dedicating an adequate proportion of model development effort to develop an optimal conceptualisation. Ultimately, of course, the appropriate model selection will depend on the site conditions and the questions that the modelling is intended to answer.

Emulation modelling

Where recharge and ET relationships are well understood, they can be represented by statistical, empirical or simplified biophysical (lower-fidelity) relationships, linked with groundwater flow models. This method has the potential to maintain adequate representations for recharge and ET, while reducing computational effort. Emulation modelling—also known as substitution modelling, metamodelling or reduced modelling—involves training the model emulator by running a more complex, physically based model with various parameter realisations several hundreds or thousands of times, and approximating the function between each of the training points. The emulator may then be used for predictive modelling, and more powerfully, for uncertainty analysis to better understand system function or in a risk analysis framework (Keating et al. 2010).

Examples of emulation modelling of recharge and ET processes can be found in studies involving: the unconfined Chalk in the Berkshire region of England (UK) by Ireson and Butler (2013); recharge to the Mediterranean climate Gnangara Mound, north of Perth, Australia, by Brown et al. (2014); the semi-arid Murray River, Australia, by Doble et al. (2006); and the Mediterranean climate Limestone Coast, southern Australia, by Doble et al. (2015). General information for using emulation modelling in the water resources sector can be found in O’Hagan (2006), Castelletti et al. (2012), Razavi et al. (2012) and Asher et al. (2015).

The recharge and ET functions that are used in emulation modelling may be tailored to the model purpose, spatial and temporal scale and the site characteristics. Understanding the physical processes of recharge and ET processes is paramount to effective emulation modelling. More detailed modelling on a fine scale may be required to understand the local nuances of these processes.

Considerations for representation of recharge and ET

In many groundwater models, ET is represented as a function of DTWT. This facilitates the use of Cauchy (head dependent) boundary conditions at the surface of the saturated groundwater model. Conventionally, recharge is represented by a Neumann (variable flux) surface boundary condition, independent of depth to groundwater. While this simplifies the model algorithms, in shallow groundwater it may not necessarily be a valid assumption. When saturated groundwater models are coupled with 1-D unsaturated zone models such as HYDRUS or WAVES, recharge and ET are controlled by the lower boundary condition of the 1-D unsaturated model. Where groundwater is deep, a free draining (Neumann) lower boundary condition will be adequate to represent recharge and ET processes. For shallow groundwater, a variable head (Dirichlet) lower boundary is necessary to provide accurate estimations of recharge and evapotranspiration—for example, Lu et al. (2011) and Naylor et al. (2015) used either variable head or free draining boundary conditions in HYDRUS-1D to model groundwater recharge in the semi-arid to semi-humid Hebei Plain, China and the humid-continental Great Lakes region of the USA respectively.

Whatever method is used to conceptualise and model recharge and ET, there are some key points that should be considered. Thorough planning in the conceptualisation stage of modelling, including a rigorous problem description, will improve the way that recharge and ET are represented in groundwater models.

Depth dependence of ET and recharge functions

In water resources management, it is often the water budget that is of interest; therefore, model input and output volumes, rather than groundwater heads, are important. The depth dependence of ET, net recharge and in some cases gross recharge, can therefore sometimes result in a seemingly circular argument between recharge and ET parameter inputs and the resulting recharge and ET model outputs. However, this depth dependence provides a self-correcting environment for the water table in unconfined aquifers with shallow groundwater, which may lead to improved estimations of groundwater head, especially at a break of slope in the land surface (Doble et al. 2006). Depth dependence does, however, force the model to solve non-linear functions for ET and net recharge, which can increase problems with model convergence. In particular, rewetting of cells during iterations can lead to instability and non-convergence. Convergence may be improved by changing solvers or solver parameters, smoothing parameters across boundaries with large changes, reducing grid sizes, or conducting a preliminary run with a simplified version of the model (alternative steady state or transient, all confined layers, rewetting off, ET represented by a constant flux) then use the final head outputs as initial conditions for the original model.

While it is best practice for any groundwater model, it is imperative that models of shallow groundwater systems should be calibrated using flux observations in addition to the conventional piezometric head observations (Sanford 2002). These flux observations might be in the form of spatial estimations of recharge and ET from field measurements or remote sensing observations or measurements of baseflow from gauged rivers and drains. In particular, the use of remote sensing data has great potential here to improve model calibration in data-poor regions.

Representation of the ET surface

While the estimation of maximum ET and extinction depth (or equivalent soil and vegetation parameters for fully coupled land surface models) is critical, even more important in regional groundwater models is the estimation of the evapotranspiration surface. For larger model cell sizes, the ability to accurately represent the proportion of the cell in which the water table exceeds the extinction depth becomes difficult (Fig. 6), which can lead to errors in estimation of ET rates (Ajami et al. 2011; Kambhammettu et al. 2014; Kuniansky et al. 2009).

Fig. 6
figure 6

Representing ET extinction depth in a regional groundwater model: schematic of a single model cell. When cell dimensions are large, the proportion of cell undergoing ET will not represent the proportion of cell undergoing ET at a fine scale. In this example, approximately 2/3 of the cell has the water table (WT) above the extinction depth (d ext ). When the average land-surface elevation (surf), water table and extinction depth are used, 100 % of the cell has the water table above the extinction depth, therefore is experiencing ET

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There are several methods for reducing this ET error associated with scale. One option is to reduce the size of the model cells, particularly in areas of the model where there is a large variation in elevation such as around rivers and surface-water bodies. Grid refinement in MODFLOW is possible with the Local Grid Refinement (LGR) package or unstructured grid (USG) process. The RIP-ET package allows fractions of cells to be covered by different riparian vegetation subgroups, and the land surface elevation to change within a cell for different plant functional type subgroups, although the groundwater is at a constant elevation within the cell (Maddock III et al. 2012). While the unstructured grid version of MODFLOW, MODFLOW-USG (Panday et al. 2013) currently only supports the RCH and EVT packages, it may provide a means of increasing cell discretisation adjacent rivers and in low-lying areas to improve representation of the evapotranspiration surface. Finite element models such as FEFLOW or HydroGeoSphere, also allow for grid refinement around areas of interest and can be defined using mesh generators such as Algomesh (Merrick 2015), GridBuilder (McLaren 2004), EasyMesh (Niceno 2002) or Triangle (Shewchuk 1996, 2002). It may be possible to calculate cell size as a function of surface slope to obtain more efficient mesh designs.

Where the size and number of cells is limited by the required computational effort, or where elevation information is available at a much finer scale than desired cell sizes, statistical representations of the land surface can be used within a single cell. Petheram et al. (2003) used a sub-grid representation of the land surface to calculate groundwater discharge using the 1-D flow model FLOWTUBE. Peeters et al. (2013) used hypsometric curves to more accurately represent the land surface within the continental scale Australian Water Resources Assessment (AWRA) model. To the authors’ knowledge, this process has not yet been formally included in groundwater flow model codes.

Spatial and temporal scale

The spatial and temporal scale of the problem being investigated will dictate how recharge and ET are conceptualised in groundwater models. Seasonal and long-term average estimates of recharge may have to consider other water inputs, such as ponded runoff which later infiltrates into the soil and recharges groundwater. Courser model discretisation may require runoff to be added to the infiltration term in the water budget for the same reason. More finely discretised models, however, may require the routing of runoff from one cell into an adjacent cell as recharge, and the coupling of a surface-water model with the groundwater model or use of a fully coupled model is justified. For deeper groundwater, there is also a significant lag time between infiltration at the surface and recharge at the water table (Hvorslev 1951).

Seasonal changes are reflected in recharge rates, soil evaporation and vegetation transpiration. Season impacts the antecedent soil-moisture conditions, therefore altering the proportion of infiltration that becomes recharge at the water table (Castillo et al. 2003). For longer timescales, such as calculating an annual average recharge for 10 or more years of continuous data, the impact of assuming no change in soil-moisture storage from the start to end dates is low. This assumption, however, is not valid for monthly or seasonal estimations of recharge. For shorter temporal intervals, soil moisture also has a greater influence on estimates of recharge using water balance methods with remotely sensed ET.

Season governs whether precipitation is in the form of rainfall or snow, and recharge from snowmelt will be delayed from the original precipitation. Long-term climatic change will not only affect the annual average precipitation rate, but may also change the intensity of precipitation events, influencing the proportion of precipitation that is recharged (Barron et al. 2012; Crosbie et al. 2012).

Physically based recharge estimation methods are likely to provide more accurate predictions of daily–monthly variations in recharge and enable prediction of climate change impacts on the seasonality of recharge (Assefa and Woodbury 2013). They will also be more likely to provide adequate predictions for different climate conditions than those under which the model was calibrated. This may not be the case for simple, empirical models such as recharge as a percentage of rainfall.

The rate of change of DTWT will affect the plant response and plant water use. In water-limited environments, when groundwater is drawn down slowly, vegetation may grow deeper roots and continue to use groundwater as the major water source. When drawdown is rapid and sustained such as from the commencement of pumping from a bore, vegetation root growth may not be rapid enough, causing plants to die and groundwater transpiration to cease (Froend and Sommer 2010).

Remote sensing data

Two major growth areas in groundwater modelling recently are the use of uncertainty modelling and risk analysis, and the incorporation of remote sensing data into spatially variable estimations of recharge and evapotranspiration (ET). Field-derived ET data from flux towers should be used where possible for groundwater model calibration or inverse uncertainty analysis; however, this information is at a point scale, and due to the expense associated with obtaining it, coverage can be limited. In data-poor regions, remote sensing may provide a source of data for model calibration (Carroll et al. 2015), through pilot point calibration techniques (Doherty et al. 2010), as an estimate of uncertainty in recharge and ET model inputs, or as a source of data for assimilation into groundwater models (Pauwels and De Lannoy 2009). Assimilation of remote sensing data into land surface models is an active area of research (Pauwels et al. 2001), but there is very little information on assimilating remote sensing estimates of ET and recharge into groundwater models. Use of remote sensing data in groundwater models include the aforementioned work in Botswana and China (Brunner et al. 2007) and Nebraska (Szilagyi et al. 2011).

Estimates of ET are available through a number of independent data sources, including reflectance data (Nagler et al. 2005) and land surface temperature (Kalma et al. 2008). Remote sensing derived information may be used for predicting antecedent soil-moisture conditions used in recharge estimation or detecting shallow groundwater (Jackson 2002). At a global scale, estimating changes in the groundwater store, and therefore inferring groundwater recharge, can be made using information from the NASA GRACE satellite (Reager and Famiglietti 2013), although there is still work required to improve estimates at a sub-continental scale.

Remotely sensed ET still needs to be calibrated against point–scale field data (Nagler et al. 2015), and the errors and uncertainty in field based methods such as eddy covariance towers and sap flow sensors range from 5 to 30 % (Glenn et al. 2011). This is similar to the error and uncertainty estimations from remotely sensed ET data derived using thermal and vegetation index methods, of around 10–30 % (Glenn et al. 2011). Aggregation of remotely sensed ET data to monthly or longer averages improves its accuracy compared with field estimates, but development of improved spatial scaling methods are required (Kalma et al. 2008).

Vegetation index-based estimates of ET usually only reflect the transpiration component, and evaporation from the soil surface is not included. Improvements in remotely sensed soil-moisture estimates will potentially improve estimates of ET by improving the algorithms used to convert potential ET (PET) to actual ET (AET). Thermal–based estimates of ET (Kalma et al. 2008) include both vegetation transpiration and soil evaporation components of ET and is independent of the PET to AET conversion process; however the spatial resolution of thermal estimates of ET are generally coarser than reflectance data.

Uncertainty analysis

Including measures of uncertainty for recharge and ET estimations reflect the confidence in both a model’s ability to predict these parts of the water balance, and in the currently available input data used to produce these predictions. Sources of uncertainty include local and global climate models (GCMs; Crosbie et al. 2011), landuse mapping and classification (Eckhardt et al. 2003), soil mapping and classification (Schaap et al. 1998), accurate water table and land surface estimations, functional vegetation responses and the conceptual groundwater model itself. A systematic analysis of the contribution of groundwater conceptual models to uncertainty is presented in Rojas et al. (2010, 2008).

Where field observations of groundwater head or flux are available, inverse uncertainty estimation may be used to determine a range for each recharge and ET parameter that will produce the observed outputs. This can decrease the parameter space required for an emulator model to reproduce the outputs of a more complex model and provide probability distributions or likely ranges for each input parameter. Where observations of groundwater model outputs are not available, expert elicitation or multiple observations of recharge and ET input data may be used to define feasible parameter spaces to use in forward uncertainty propagation to predict probablility distributions for groundwater model outputs.

Presentation of recharge and ET data as probability distributions for groundwater model inputs provides significantly more information to the model user or client and enables model outputs to be easily incorporated into risk analysis and water management planning (Merrick 2000; Raiber et al. 2015). The large number of data points provided by remote sensing data has the potential to assist in this process.

Summary and future research opportunities

Simple representations of recharge and ET in groundwater models have been appropriate in the past, particularly in data poor regions; however, the availability of continuously improving, remotely sensed estimates of ET and recharge mean that a more physically based conceptualisation of recharge and ET may be warranted and potentially lead to improvements in model outputs and confidence. This paper has shown that recharge and ET can both be depth dependent and that this depth dependence can result in additional calibration requirements, particularly estimates of groundwater fluxes such as baseflow to streams. It is critical that recharge and ET are explicitly defined (gross recharge vs net recharge, groundwater ET vs total ET) in the reporting of modelling results to avoid double accounting in the water balance.

There are many options for representing recharge and ET processes in groundwater models, ranging from the basic boundary condition functions to complex fully coupled surface-unsaturated–saturated models. Model emulators enable the behaviour of recharge and ET from complex models to be preserved, while reducing computational effort and model run times. This is particularly important for risk or uncertainty analysis, which is becoming a standard aspect of groundwater modelling.

In whichever manner recharge and ET are modelled, representation of the land surface is critical for accurate estimations of ET. The spatial and temporal scale of the questions being addressed by the model will influence the way in which recharge and ET are represented, through vegetation responses, initial soil-moisture conditions, lag times and interactions between model cells. The use of remote sensing in model parameterisation and calibration is critical for improving recharge and ET in data-poor regions, particularly with respect to the spatial and temporal distributions of these fluxes. Use of risk or uncertainty analysis for estimating recharge and ET or using them as groundwater model inputs is justifiably becoming standard practice. Forward uncertainty analysis to estimate probability bounds for predictive estimates and inverse uncertainty analysis to estimate likely bounds for parameter inputs provide far more information and are more scientifically robust than single predictions and parameter estimations.

Future research opportunities to improve the representation of recharge and ET in groundwater models include:

  • Improvements in constraining estimates of recharge and ET using remote sensing of ET and soil moisture. This field of research is likely to grow and evolve as new remote sensing products become available and improve in accuracy and in temporal and spatial scales.

  • Inclusion of remote sensing estimates of recharge and ET directly into groundwater models, through calibration processes or direct assimilation. This is a growing area of research for land surface models, but there are very few examples in the groundwater modelling literature.

  • Better representation of recharge and ET in terms of risk and uncertainty. While uncertainty analysis is common for hydrological model outputs such as streamflow forecasting and conceptual uncertainty estimates of ET from ensemble global climate models, this has not often translated to recharge and ET estimates in groundwater models. This may be particularly useful for understanding risks for groundwater dependent communities and ecosystems.

  • Using uncertainty analysis to prioritise data acquisition and improvement. Analysis of model and remote sensing estimates may give insight into the most effective locations to calibrate model (and remote sensing algorithm) predictions with field-based measurements, gaining the largest model confidence benefit from further field data collection.