Volume 54, Issue 3 p. 1428-1459
Research Article
Free Access

Modeling Subsurface Hydrology in Floodplains

Cristina M. Evans

Corresponding Author

Cristina M. Evans

School of Earth and Environmental Sciences, University of St Andrews, St Andrews, UK

School of Mathematics and Statistics, University of St Andrews, St Andrews, UK

Correspondence to: C. M. Evans, [email protected]Search for more papers by this author
David G. Dritschel

David G. Dritschel

School of Mathematics and Statistics, University of St Andrews, St Andrews, UK

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Michael B. Singer

Michael B. Singer

School of Earth and Ocean Sciences, Cardiff University, Cardiff, UK

Earth Research Institute, University of California Santa-Barbara, Santa Barbara, CA, USA

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First published: 30 January 2018
Citations: 5

Abstract

Soil-moisture patterns in floodplains are highly dynamic, owing to the complex relationships between soil properties, climatic conditions at the surface, and the position of the water table. Given this complexity, along with climate change scenarios in many regions, there is a need for a model to investigate the implications of different conditions on water availability to riparian vegetation. We present a model, HaughFlow, which is able to predict coupled water movement in the vadose and phreatic zones of hydraulically connected floodplains. Model output was calibrated and evaluated at six sites in Australia to identify key patterns in subsurface hydrology. This study identifies the importance of the capillary fringe in vadose zone hydrology due to its water storage capacity and creation of conductive pathways. Following peaks in water table elevation, water can be stored in the capillary fringe for up to months (depending on the soil properties). This water can provide a critical resource for vegetation that is unable to access the water table. When water table peaks coincide with heavy rainfall events, the capillary fringe can support saturation of the entire soil profile. HaughFlow is used to investigate the water availability to riparian vegetation, producing daily output of water content in the soil over decadal time periods within different depth ranges. These outputs can be summarized to support scientific investigations of plant-water relations, as well as in management applications.

Key Points

  • Floodplains can have complex patterns of soil-moisture due to infiltration/evaporation and subsurface river flux
  • We developed a model that links these processes to assess moisture patterns at chosen depths and floodplain locations
  • This model can be used to assess water availability to vegetation over rooting depths at any distance from the channel

1 Introduction

The vadose zone is a region of unsaturated soil, vertically bounded by the land surface and the water table. This zone is an important pathway controlling water exchange between surface water and groundwater in the hydrological cycle. It buffers hydrologic extremes, such as floods and droughts by storing water and modulating its movement (Harter & Hopmans, 2004). It also provides a critical moisture source for local ecosystem functioning (van Genuchten, 1991). However, the dependence of these ecosystems on groundwater hydrology is poorly understood (Rohde et al., 2017).

Riparian environments have especially complex hydrology due to the joint contribution of vertical processes (precipitation, evaporation, and capillary rise) and lateral processes (subsurface hyporheic flow). With streambed connection, river water feeds into the floodplain's phreatic zone (the saturated zone underlying the vadose zone). This water influx provides a crucial resource for water-limited vegetation (Snyder & Williams, 2000; Williams et al., 2006). Water-stressed riparian vegetation is particularly sensitive to changes in soil-moisture (Sargeant & Singer, 2016; Singer et al., 2014, 2013; Snyder & Williams, 2000; Williams et al., 2006). Hence, it is important to understand how climate is expressed in subsurface hydrology to predict the impact of future climatic trends on riparian ecosystems. Spatial and temporal variations in soil-moisture, driven by direct local climate (i.e., precipitation and evaporation) and indirect nonlocal climate (manifesting as riverine process), should be studied both individually, as decoupled units, and in tandem. Modeling the effects of each process can allow us to decipher patterns of water availability to vegetation, which is especially important in light of the fact that there are open questions about which water sources plants use (Evaristo et al., 2015; Sprenger et al., 2016).

Surface hydrology, i.e., precipitation and river discharge regimes, is extensively both monitored and modeled. Surface models play an essential role in water management schemes (Singh & Woolhiser, 2002). However, as soil-moisture can be costly to measure at high spatial and temporal resolution, the complex relationships between subsurface processes can be more-easily investigated using physically based mathematical models. Generally, we can expect infiltrating precipitation to move downward through the soil and hyporheic flow to supply the water table laterally, but modeling is needed to understand their interplay and temporal legacy on soil-moisture patterns. This interplay is important because it can generate water stores for riparian vegetation with particular water demands and rooting depths (Canham et al., 2012; Singer et al., 2014). The residence times of these stores determine the moisture availability in the soil during drought conditions, yet water content at any particular soil depth is difficult to predict without numerical simulations. Models also provide the flexibility to quickly and economically analyze large time intervals with high spatial and temporal resolution.

A variety of models with varying complexity exist to simulate subsurface hydrology; from data-driven/stochastic to physically/process-based models. These models are created for a range of different purposes and are made available open-source or as commercial products. Some of the most commonly used hydrology models include HydroGeoSphere (Therrien et al., 2010), HYDRUS 3D (Šimůnek et al., 2008), MIKE SHE (Refsgaard & Storm, 1995), MODFLOW (Harbaugh et al., 2000), and ParFlow (Maxwell et al., 2016). The main differences between these models are the formulation of the governing equations, their coupling, the boundary conditions, and their spatial and temporal discretization (Maxwell et al., 2014). Differences also arise in the chosen dimensionality of each equation used; groundwater flow can be simulated in one or two dimensions and the infiltration Richards' equation can be simulated in up to three dimensions.

The scientific community has questioned the increasing number of models being produced in this field. Baird and Wilby (1999) discuss the needless pursuit of new model solutions in ecohydrology when good solutions already exist. The authors agree that it would be ideal to progress with existing models; however, the widely used available models are large and complex, making it hard to gain a full understanding of the limitations and assumptions involved. In developing a model specific to our intended application, we are able to ensure that the model is as accurate and efficient as possible, and is constructed in a way that is suitable for simulating the processes involved.

In this paper, we introduce HaughFlow, a light-weight, flexible model which couples the one-dimensional Richards equation (Richards, 1931) for vertical moisture transport, with the Boussinesq equation (Boussinesq, 1904) for lateral saturated flow perpendicular to the river channel. HaughFlow is a modern, more flexible version of the Pikul et al. (1974) model, with an optimization procedure for calibrating key parameters and capability for a range of output data visualizations. HaughFlow requires minimal inputs, ensuring model application is as simple as possible. The simplicity of the model structure allows us to investigate the roles of each subsurface flow component, and to specifically identify the influence of a shallow hyporheic-dominated water table on patterns of soil-water saturation throughout the vadose zone. HaughFlow assumes lateral hyporheic flow is the dominant driver of water table levels in the riparian corridor; the model is thus applicable to floodplains where the river and groundwater are hydraulically connected. A source/sink term in the Boussinesq equation, as described by Zucker et al. (1973) and Pikul et al. (1974), is used to fully represent the interplay between the vadose and phreatic zones. This term allows the soil-moisture conditions in the vadose zone to influence water table dynamics. A capillary fringe, induced by boundary conditions at the water table, allows for water table contributions to soil-moisture in the vadose zone.

Numerical simulations are presented using the HaughFlow model to investigate the role of hydroclimate in controlling water content and fluxes at all subsurface depths down to the floodplain water table. After evaluating the model against piezometer data, we apply the model over decadal time scales along a floodplain transect in the Murray-Darling basin, Australia, using existing data on climate, soil parameters, and river stage. We obtain daily output from HaughFlow to explore the subsurface moisture and flow responses to seasonally and annually varying boundary conditions. We use these output data to quantify the lateral expression of water table dynamics within the floodplain, as well as to identify the impact of water table fluctuations on deep soil-moisture.

2 Model Structure and Components

HaughFlow, after the Scottish “Haugh” for the flat alluvial land by a river, is a physically based numerical model constructed to be flexible, user-friendly, and with a simplistic composition. The soil and computational parameters can be easily adapted, along with the depth and position of the simulated area, to suit the user's experimental scope and scale of interest. HaughFlow constitutes the minimum processes required to simulate water movement.

The components and structure of HaughFlow are outlined in Figure 1. The structure reflects that of the soil region it simulates, with separate modules defining each of the input fluxes that feed into the vertical Richards and horizontal Boussinesq components. Each component has its own boundary conditions, initial conditions, and spatial step, however they share a common time step. The surface input fluxes comprise precipitation and evapotranspiration, and are incorporated through the upper boundary condition. This is read into the main body of the code that calculates internal vertical infiltration and diffusion. Horizontally, the water table position is determined by calculating the hyporheic exchange flow, and assimilated through a lower boundary coupling with the vertical infiltration equation. An accurate tridiagonal solver along with a predictor-corrector scheme (as in Pikul et al., 1974) is used for the temporal discretization, and centered finite differences are used for the spatial discretization (see Appendix Appendix A for details).

Details are in the caption following the image

Structure of HaughFlow, the model created for this study. Box 1 shows the calculation of the water table or the extent of the phreatic zone, with river stage records as input to the Boussinesq equation and the water table height feeding into the vadose zone calculation. Box 2 is the vadose zone component of the model, calculated using precipitation and evaporation inputs to the Richards equation. Upon completion of the Richards calculation, the source term in the Boussinesq equation is evaluated and fed across. Box 3 shows the section of the model which is looped over each day, Boxes 1 and 2 are looped over each time step.

2.1 Infiltration and Diffusion Component

The vertical transport of moisture in the vadose zone is modeled using the Richards (1931) equation. This equation combines Darcy's law (1856) for vertical unsaturated flow and the principle of conservation of mass, leading to the following evolution equation for the hydraulic pressure head ψ (L),
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0001(1)
where z (L) is the vertical coordinate (measured upward), K (L/T) is the effective hydraulic conductivity, t is time, and C (1/L) is the specific moisture capacity defined below. A substitution of urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0002 is then made to yield positive ζ values above the water table. Applying this substitution to equation 1 gives
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0003(2)
where C is defined by
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0004(3)
and urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0005 is the soil-water content (see below). The head-based form of the Richards equation has been chosen over the saturation-based version because ζ is continuous and differentiable across the water table and thus can simulate vertical water movement in both the vadose and phreatic zones (Diersch & Perrochet, 1999).
Van Genuchten (1980) describes the use of Mualem theory to characterize the soil properties used in the Richards equation. The soil-water content θ can be expressed in terms of the saturated water content θs and the residual water content θr. These terms can be combined to give the fractional water content,
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0006(4)
or urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0007. Mualem theory gives us the following equation describing the relationship between the saturation of the soil Θ and the pressure head term ζ (when urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0008)
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0009(5)
where urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0010 and urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0011. α and λ are parameters relating to the soil-water retention properties of the soil.
Using these in equation 3, we find the specific moisture capacity (C) in terms of ζ
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0012(6)
The expression for the effective hydraulic conductivity in terms of moisture and pressure is given in Van Genuchten (1980)
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0013(7)

When urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0014, the soil is saturated and then Θ = 1, C = 0, and K = Ks (Ks is the saturated hydraulic conductivity). An explanation of the boundary conditions follows (for details of the numerical methods see Appendix A2).

2.1.1 Phreatic Zone and Lower Boundary

At the water table (subscript wt) urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0015, the pressure head is zero (ψ = 0) (Pikul et al., 1974). Since urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0016 we also have ζ = 0 at urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0017. Conditions describing the phreatic zone, namely C = 0 and K = Ks, hold true for the water table boundary and below, down to an impermeable layer. In this saturated zone, the flux in equation 2, namely
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0018(8)
is zero. Since K = Ks in this zone and ζ = 0 at urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0019, it follows that ζ is a simple linear function of z
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0020(9)

Note, in general, zwt is a function of time.

When the water table is within the model domain, the lower boundary at z = 0 is assumed to be saturated. Using equation 9 for the pressure term in the phreatic zone, the condition at the lower boundary is urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0021. If the water table is deeper than and below the model domain, then the lower boundary is assigned a zero flux boundary condition. (For numerical implementation see Appendix A2.1.)

2.1.2 Upper Boundary

The top boundary z = H (where H is the depth of the modeled soil domain) is assigned a Neumann (flux) boundary condition defined by precipitation (pr) and evapotranspiration (er) fluxes. This top-flux (F in equation 8 evaluated at z = H) is equated to the incoming precipitation minus the surface-moisture-dependent evapotranspiration, urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0022. This study assumes grassland vegetation cover with water extraction only at the surface for simplicity.

Daily records of climate variables including precipitation, temperature, wind speed, and relative humidity are required to calculate this top-flux, along with grassland evapotranspiration parameters provided by Dingman (2015). The precipitation can be included directly from total daily records which are resolved over smaller temporal steps by linear interpolation. Atmospherically controlled evapotranspiration is defined by the Penman-Monteith equation, producing a daily average value (see Appendix Appendix B).

The influx of water at the surface can never exceed the maximum saturation in the soil. Hence, a further condition has been added to allow ponding at the surface of the domain. Ponding can occur if the soil at the surface is saturated and there is a positive top-flux, urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0023. More precisely, ponding occurs if urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0024 at z = H where Θ = 1 and thus ζ = 0. To determine if ponding occurs, first the maximum flux, Fm, which the soil can receive is estimated by
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0025(10)
where urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0026 is calculated using ζ = 0 at z = H and the current value of ζ at the adjacent grid point (see Appendix A2.2). If the incoming flux urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0027 exceeds Fm, then the excess water is stored at the top of the domain as a virtual pond and can be transferred to the top-flux in subsequent time steps when capacity permits. Capacity is made available as water drains or evaporates from the upper regions of the soil. When more water can be infiltrated, the ponded water diminishes. The maximum height of water that can pond above the surface can be altered within HaughFlow to account for different floodplain slopes or surface features affecting surface-ponding capacity—in this paper an estimated levee height is chosen as the maximum ponding depth. (For upper boundary numerical methods see Appendix A2.2.)

2.2 Lateral Flow Component

We use the Boussinesq (1904) equation to simulate lateral hyporheic exchange flow and calculate the position of the water table in the floodplain. The Boussinesq equation is a combination of Darcy's law (1856) for lateral groundwater flow and the continuity equation. It is based on the Dupuit-Forchheimer approximation which states that flow moves horizontally in a shallow groundwater system, driven by the gradient of the water table. The use of this equation for the model setup also simplifies the modeled domain to have a fully penetrating channel at one side and an impermeable horizontal layer below. The equation is presented by Baird and Wilby (1999), with an additional source/sink term
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0028(11)

Here h(x, t) (L) is the height of the water table above the impermeable layer at a given distance x from the river channel at time t. The lateral movement of water from the river is dependent on the saturated hydraulic conductivity, Ks, and the specific yield, sy. Water is added to or drained from the domain by a source/sink term, S(t), which accounts for water transfer from the vadose zone; this allows for cycles of draining and filling. S varies with time and is determined by calculating the materials balance of the water content in the vadose zone with the applied fluxes at the surface (see Appendix A3.1 for details). Dr is an additional drainage term which accounts for seepage into the underlying stratum and/or groundwater extraction, if negative, and regional groundwater contributions if positive.

2.3 Coupling and Model Spin-up

The interaction between the vadose and phreatic zones is twofold; capillary rise draws water up from the water table, and a source/sink term can account for capillary losses and allow water in the vadose zone to assimilate into the water table. This coupling complexity makes the spin-up calculation a particularly important component of the model. Two options for spin-up can be chosen; one fixes the initial water table and surface evapotranspiration and runs the simulation over a given number of days, creating a natural gradient of water movement upward from the water table by capillary rise. This, however, does not create natural patterns of water content in the upper regions of the vadose zone which would exist due to a temporal rainfall legacy. For this reason, a more robust spin-up is applied; the code is run over the first year of data as many times as required until the upper regions of soil are consistent with the remainder of the simulation. The latter method, repeated for 2 years, was the chosen spin-up method for this study. Spinning up the model using this method minimizes the influence of the initialization on the model output. (Further details of the initial conditions used are provided in Appendix A3.)

3 HaughFlow Implementation

HaughFlow requires site data, soil parameters, river stage data, and climate data to run. The site data comprises longitude, latitude, elevation, and distance from the river channel. The soil parameters can be subdivided into surface parameters (used for the infiltration component), and lateral, deeper soil parameters (used for the lateral flow component). The van Genuchten parameters used for the infiltration include the saturated and residual water contents, the saturated hydraulic conductivity, and the aforementioned Mualem parameters α and n. For the lateral exchange, another saturated hydraulic conductivity is used, along with a specific yield and the depth to the impermeable layer from the surface. An optional drainage term can be used to account for deep-percolation losses, water table extraction, or regional contributions. The river stage data are input as daily river stage records with an associated zero gauge elevation, for conversion to river water elevation. Finally, the climate data required consist of daily values needed for the evaporation calculation (the mean station pressure, wind speed, maximum and minimum daily temperature, dew-point temperature, and total precipitation) and for the transpiration calculation (the leaf-area index (LAI), zero-plane displacement, roughness height, top of canopy height, albedo, maximum value of leaf conductance, and a shelter factor).

3.1 Study Site

The Murray-Darling river basin in south-east Australia is one of the largest catchments in Australia (>1 million km2) (Taylor et al., 2013). We applied HaughFlow to sites adjacent to the Namoi River (drainage area of 40,000 km2), located in the north-east of the Murray-Darling catchment (Lamontagne et al., 2015) (see Figure 2 for a site map). The climate is subtropical (Lamontagne et al., 2015), characterized by hot, wet summers (December–February), and mild winters (June–August). The vicinity of the study sites has been extensively studied (CSIRO, 2007; Ivkovic et al., 2009; Lamontagne et al., 2011, 2014, 2015; Taylor et al., 2013), providing ample sources for site descriptions, soil properties, and input data sets. A brief description of the sites follows, with the detail required for understanding the model setup. For a more detailed description of the sites see Lamontagne et al. (2015).

Details are in the caption following the image

Map (modified from Lamontagne et al., 2015) showing the locations of the study sites along the Namoi river, within the Murray-Darling catchment, in south-east Australia.

The Namoi River at the sites was found to be hydrologically connected to the surrounding floodplain by Lamontagne et al. (2014), meaning exchange with the local water table occurs directly at the channel boundary. This connectivity supports the suitability of applying the laterally hyporheic-driven HaughFlow model. Two locations along the Namoi River, at Old Mollee and Yarral East ∼2 km apart, contained three piezometers each, providing time series of water table levels at different distances from the river channel (see Table 1). Piezometer data are monitored by the New South Wales (NSW) Office of Water. The model was applied to evaluate HaughFlow at each of these six sites for time periods of 2.5 and 4 years, depending on data availability.

Table 1. Summary of the Climate, Soil Profile, River Stage, and Piezometer Data Sets Used for the Study
Data Reference name Latitude Longitude Elevation (m AHD)
Soil profile ed199 −30.242 149.664 202
Soil profile ed200 −30.243 149.694 204
Climate Narrabri Airport (957340) −30.317 149.817 230
River stage Old Mollee (419039) −30.255 149.680 197.43
River stage Yarral East (419110) −30.237 149.671 195.47
Piezometer Old Mollee (GW098211) −30.254 149.681 202.25 (50 m to river)
Piezometer Old Mollee (GW098206) −30.254 149.682 204.37 (140 m to river)
Piezometer Old Mollee (GW098207) −30.254 149.684 203.82 (320 m to river)
Piezometer Yarral East (GW098208) −30.237 149.671 202.98 (40 m to river)
Piezometer Yarral East (GW098209) −30.236 149.671 202.84 (110 m to river)
Piezometer Yarral East (GW098210) −30.234 149.671 202.02 (290 m to river)
  • Note. The initial data sets were used as model input while the piezometer data were used for comparison with model output. (NB: the elevation column contains values of ground elevation for the climate and piezometer sites and the elevation of the zero gauge for the river stage readings. The piezometer metadata include the distance of each station from the river channel.)

The input data used to apply the model to this site are summarized in Table 1. CSIRO Division of Soils (Karssies, 2011) provided soil compositions (percentage of sand, silt, and clay from grain size) at different depths for two locations near the study sites, labeled “ed199” and “ed200.” The characteristic soil profiles, spanning 2.6 m of depth, were matched with the sites depending on their elevation; “ed199” properties used for lower elevations (<203 m above the Australian Height Datum, AHD) and “ed200” for higher elevations (>203 m AHD). The soil profiles comprised clays, clay loams, and sandy clay loams.

To generate the average van Genuchten parameters, each soil composition was input to the ROSETTA model (Schaap et al., 2001) (for all values see Appendix Appendix C). The average of each parameter within the soil profile was used as a simplification of the soil's heterogeneity (Table 2). The van Genuchten properties have a high level of uncertainty attributed to both the range of soil values provided as input as well as to the ROSETTA calculation method. These properties are required for HaughFlow's infiltration component and are expected to differ from the deeper hyporheic soil properties due to anisotropy, and also heterogeneity arising from the prevalence of finer surface sediments from river deposition through channel migration and flooding.

Table 2. Average van Genuchten Properties for Two Soil Profiles Named “ed199” and “ed200,” Calculated Using the ROSETTA Model (Schaap et al., 2001)
Property ed199 value ed200 value
θr (m3/m3) 0.08175 0.09066
θs (m3/m3) 0.4240 0.4599
α (1/m) 2.171 1.674
n (no units) 1.289 1.300
Ks (m/d) 0.08725 0.1044

The upper geological layer, the Narrabri Formation, extends to a depth of urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0029 m AHD (Lamontagne et al., 2015). A specific yield of 4.5% was calculated for the alluvium by Rassam et al. (2013) using records of stage heights at nearby Boggabri. The movement of water within this formation was simulated by HaughFlow, with the assumption that the underlying Gunnedah Formation is relatively impermeable with respect to the scale of model simulation. The saturated hydraulic conductivity for the deeper hyporheic component and the drainage term were the most difficult values to measure in the field so they were calibrated using piezometer readings (see section 9).

Records of daily meteorological data for sites around the world are provided as part of the Global Summary of the Day (GSOD) (NCDC, NOAA). This source was used to obtain climate records at Narrabri Airport, which is within 17 km from each of the two study locations (Old Mollee and Yarral East). Stream gauges were colocated with these locations, giving daily river stage records corresponding to each site. Model simulations were conducted to evaluate HaughFlow over the common period of piezometer, climate, and stage data for each location; 1 July 2011 to 30 June 2015 for Old Mollee and 1 January 2013 to 30 June 2015 for Yarral East.

The model inputs are plotted in Figure 3; the surface flux at Old Mollee 50 m in Figure 3a, and the river water elevation above the AHD at both locations in Figure 3b. Old Mollee, the upstream location, has water elevations generally >2 m higher than those at Yarral East. The stage data are owned and maintained by the NSW Department of Industry, Skills, and Regional Development (DPI Water).

Details are in the caption following the image

Plots of the surface and lateral inputs to the model. (a) A time series of precipitation minus the actual evapotranspiration at Old Mollee 50 m. (NB: the actual evapotranspiration may vary between sites as it is dependent on the preexisting water at the surface of the soil.) (b) Water surface elevation in the river channel at the two sites along the Namoi River; Old Mollee, and Yarral East (further downstream). AHD stands for the Australian Height Datum.

3.2 Calibration and Sensitivity Analysis

Model calibration was conducted to constrain the two remaining model parameters pertaining to the lateral component of the simulation; the saturated hydraulic conductivity (Ks) and the drainage term (Dr). The two soil parameters control different aspects of the lateral flow signature. Ks controls the fluctuations/variability of the water table levels, so the optimal value was determined by evaluating the water table variability. Dr controls the vertical shift in the mean water table elevation, i.e., it is a vertical permeability or regional infiltration. This variable was calibrated by obtaining the difference between the running average of the modeled and observed values.

The running average ( urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0030) was taken over 31 days, i.e.,
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0031(12)
for each water table value Yi. The variability (v) was calculated by taking the difference of each value from the running average, i.e.,
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0032(13)
where n is the number of values in the data set. The difference between the observed and modeled variability ( urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0033) was used to calibrate Ks. The relationship between the two is roughly directly proportional and a result of 0 would be produced by the best fit hydraulic conductivity. The difference in running average (yd) was calculated by
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0034(14)

The relationship between yd and Dr is also roughly linear and a yd of 0 would be produced by the best fitting Dr.

Full details of the calibration methods and results are provided in Appendix E1. Calibrations were conducted based on two model setups (coupled or uncoupled), two time-frames (2013 or 2014), and at each of the six sites, producing 24 pairs of calibrated parameters. In general, coupled-model calibrations at sites further from the river channel calibrated higher hydraulic conductivities.

The calibration plots for the lowest and highest calibrated hydraulic conductivities are shown in Figure 4. The uncoupled 2014 Yarral East 40 m calibration plots are shown in Figures 4a and 4b, and the coupled 2013 Yarral East 290 m calibration plots are shown in Figures 4c and 4d. Figures 4a and 4c are plots of the water table variability for the iterated range of Ks values, and Figures 4b and 4d are plots of the average differences for the iterated range of drainage values.

Details are in the caption following the image

(a and b) Uncoupled 2014 Yarral East 40 m calibration, and (c and d) coupled 2013 Yarral East 290 m calibration. (a and c) The difference in variation between observed and modeled values plotted against the hydraulic conductivity values chosen as input. (A drainage of 0 m/d was used for these simulations.) (b and d) The average difference in running average between observed and modeled values plotted against drainage values. (The hydraulic conductivity calibrated in the associated left-hand plot was used for all drainage calibration simulations.) Simulations did not include vadose zone dynamics.

The rough linearity between the hydraulic conductivity and the difference in variation, and between the drainage and the average difference in running average, confirms that the dominant effect of these properties on the water table has been captured. High hydraulic conductivity values represent high connectivity with the river channel and lead to high variability in the water table. Conversely, low hydraulic conductivity values smooth the river channel signal in the water table. Positive drainage terms shift the water table values down and vice versa.

Figure 4 also depicts the sensitivity of the model to the lateral flow soil parameters. Looking at the units and values in each of the Figure 4 plots, we can see that the model is much more sensitive to the drainage rate than the hydraulic conductivity. This is because the drainage rate is subtracted directly from the Boussinesq equation 11, while the hydraulic conductivity is tempered by the specific yield and the shape of the water table in the floodplain. The numerical sensitivity is described in Appendix Appendix D. Each of the 24 calibrated pairs of parameters were then applied to each of the six sites and evaluation methods were used to determine the most effective calibration configuration (see Appendixes E2 and E3).

3.3 Evaluation Methods

Moriasi et al. (2007) provide guidance for effective model evaluation using statistical tests. Based on these guidelines the model evaluation comprises: a graphical comparison, a standard regression, the Nash-Sutcliffe efficiency (NSE), and the percentage bias (PBIAS).

A standard regression is used to calculate the Pearson's coefficient of determination (R2), which is a measure of the fit to the linear relationship between the modeled and observed values. A result of 1 is a perfect fit. This classic model evaluation formula is calculated by
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0035(15)
where each field-measured value (observed) is given as Yobs and each model-simulated value is Ysim. Each ∑ represents the sum over all n values.
The Nash-Sutcliffe efficiency is a ratio of the residual variance (differences between modeled and observed values) to the data variance (spread of the observed values). It is defined by
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0036(16)
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0037 represents the mean observed value. The resultant NSE values range from urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0038 to 1 with values urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0039 representing good model-prediction performance (Moriasi et al., 2007). The final test of model performance is percentage bias. This calculates the average tendency of the modeled values to be larger or smaller than their corresponding observed values. The equation for PBIAS is
urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0040(17)

The output is a percentage value, with 0% being optimal. Positive results indicate an underestimation of the values in the model simulation, and negative values an overestimation.

4 Results

The results comprise a model evaluation and a full examination of the model output. The evaluation allows us to determine the level of correlation between the observed and simulated patterns and the output allows us to understand the contribution of each flux within the system to general trends.

4.1 Evaluation

The results of each of the 24 calibrated pairs of parameters and a full discussion of their performance are provided in Appendixes E2 and E3. The results recognize the need to calibrate floodplain models based on the furthest available sites from the river. This is due to the potential to underestimate floodplain connectivity (with low lateral hydraulic conductivity values) at sites nearer the river. Furthermore, the need for fully coupled vadose-phreatic model to fully capture soil-moisture dynamics is exemplified by the distinct improvement in model performance when calibrating based on coupled-model setups. The coupled, 2013, Yarral East 290 m calibration has the best overall performance, with the highest hydraulic conductivity value of any calibration configuration, Ks=5.756 m/d, and a very small drainage term, Dr = 0.01568 m. These parameters are used for the simulations throughout the remainder of the study.

Time series of the model and piezometer output are plotted in Figure 5. Model output at all sites for this calibration provides a good general fit to piezometer data. The Old Mollee 320 m site appears to perform worst, followed by the Old Mollee 140 m. At these sites, the highest peaks (i.e., at the beginning of 2012) are underestimated, and gentle rises are overestimated (i.e., at the beginning of 2014). The underestimation of the peaks is discussed further in section 13 The overestimation of the gentle rises is because of a higher hydraulic conductivity in the lateral flow component than is required at these sites.

Details are in the caption following the image

Model and piezometer output for all the sites, including the Yarral 320 m site (bottom left) of which the year 2013 was used for the calibration.

The model outputs appear to provide a generally good representation of water dynamics, but the evaluation parameters can help us quantify the extent of the fit. The detailed evaluation results chosen calibration are listed in Table 3. Overall, the results are very good. The Pearson's R2 values, which are a measure of the linear fit between the model and piezometer data sets, are all >0.7 (with a value of 1 representing a perfect fit). The NSE is a ratio of the residual to the data variance (a value of 1 indicates equal variability in each), hence if the variability of the model output is within the scope of the piezometer variability then the NSE value shows good model performance. Hence, Old Mollee 320 m, with visibly less variability in model than piezometer data, has the poorest NSE value. Nevertheless, all NSE values were >0.5.

Table 3. Results of the Statistical Tests Carried out on the Simulated Model Output Data Compared With the Observed Field-Measured Data
Data R2 NSE PBIAS Days
Old Mollee 50 m 0.974 0.855 −0.145% 1,462
Old Mollee 140 m 0.817 0.694 −0.093% 1,456
Old Mollee 320 m 0.735 0.520 −0.063% 1,461
Yarral East 40 m 0.849 0.575 −0.005% 912
Yarral East 110 m 0.811 0.566 −0.002% 912
Yarral East 290 m 0.946 0.939 −0.002% 912
Best/worst values 1/0 1/– urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0041 0/ urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0042
  • Note. The Pearson's R2, Nash-Sutcliffe Efficiency, percentage bias, and number of days used in the calibration are provided for each of the sites.

The PBIAS values are a measure of the offset of the model data to the piezometer data. The results are all slightly negative, indicating an overestimation of model values, with the best results (closest to 0%) at the sites furthest from the channel and becoming poorer closer to the river. The Yarral East 110 and 290 m sites (Figures 5d and 5f) have the smallest PBIAS values. This can be attributed to the model values being slightly above or slightly below piezometer readings in roughly equal measures, hence the offset (drainage term) is optimal at these sites. The modulus of the PBIAS values at all sites were <0.15%. Overall, as expected, the Yarral East 290 m site performs best. The perceived worst performer in the visual comparison (Old Mollee 320 m) is substantiated by the results in Table 3.

4.2 Model Output

Time series of soil saturation Θ within a vertical slice of the floodplain, three distances from the river channel, and across two different locations are exhibited in Figure 6. To understand the contribution of each flux (i.e., precipitation and lateral flow) to the soil, we can study soil-water patterns related to the following three categories: the phreatic zone (Θ = 1), the vadose zone ( urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0045), and interactions between the two zones. In the phreatic zone, the most apparent trend is that of reduced variability in the water table with distance from the river channel. In hydraulically connected floodplains, the water table is driven by river stage fluctuations that propagate and dissipate through the floodplain. Hence, the water table is less connected to the river channel with distance, making it less sensitive to changes in river water elevation and having the effect of smoothing the phreatic signal to remove extremes. Along with a dampening of the water table signal, there is a time delay, on the order of weeks. Peaks and troughs observed closer to the channel take time to propagate to the further sites, so rising and falling limbs are more gradual in the time series. There is also a very gradual decrease in average water table levels in the model output due to the positive drainage term.

Details are in the caption following the image

Saturation plots showing the upper 10 m of soil across the time series of sites at Old Mollee (left column), (a) 50, (c) 140, and (e) 320 m from the river channel, and at Yarral East (right column), (b) 40, (d) 110, and (f) 290 m from the river channel. The elevation of the land at each site is (a) 202.25, (c) 204.37, (e) 203.82 m above the Australian Height Datum (AHD) (left column), and (b) 202.98, (d) 202.84, and (f) 202.02 m AHD (right column).

Soil-moisture patterns in the upper part of the vadose zone are dominated by the surface flux. The climate inputs were the same for all six sites, so saturation patterns in the upper part are consistent. Soil saturation near the surface fluctuates, as heavy rains and evapotranspiration lead to saturated and dry soil-moisture extremes, respectively. The temporal legacy of individual rainfall events is exhibited in the angle of its infiltration front as it progresses downward (see labeled infiltration on Figure 6e). Moving deeper into the soil, moisture fluctuations dampen and base soil saturation gradually increases. The increase in saturation is a result of slow infiltration rates, accumulating rainfall contributions, and the influence of the capillary fringe.

The capillary fringe below contributes soil-moisture to a significant depth interval of the vadose zone. The extent of this depth is influenced by the soil properties. This contribution is more visible at the near-stream sites (Figure 6a Mollee 50 m and Figure 6b Yarral 40 m), where capillary contributions from a higher water table interact more readily with the soil-moisture extremes from rainfall events. The capillary fringe moisture contribution is also more prominent at the further upstream Old Mollee sites, which have water table values at higher elevations and hence higher capillary fringes, than at Yarral East sites. It must be noted, however, that the floodplain surface elevation varies across the sites, which also affects the location of the water table with respect to the soil surface.

At Old Mollee 50 m (Figure 6a) in early and mid-2012, the moisture in the capillary fringe connects with the infiltrating precipitation creating a region of very high saturation that reaches the surface. This underground connectivity, combined with high variability in water table levels, leads to a higher risk of flooding near the river channel—irrespective of overbank conditions. Spikes in the water table due to capillary fringe connection with infiltrating water also occur in early 2013 and 2014 but do not rise as far as the soil surface. Using this knowledge of increased connectivity in the vadose zone when the capillary fringe meets infiltrating water, we can look back at the model and piezometer comparisons (Figure 5), specifically at Mollee 140 m (c) and Mollee 320 m (e), and see that the under-prediction of water table peaks in early 2012 may be due to inaccuracies in the bulk soil properties used in the infiltration calculation (equation 2). A higher saturated hydraulic conductivity in the vadose zone, for example, would lead to faster infiltration of rainfall events allowing them to connect with associated peaks in the capillary fringe. This exemplifies the importance of the relationship between the vadose and phreatic zones for understanding patterns of water change in each.

4.3 Decadal Simulation

Applying HaughFlow to a decadal time series allows us to investigate the response of soil-moisture distributions to climatic fluctuations. Climate data from the Inverell Research Centre (station ID 94541099999, GSOD) provided a decadal time frame for the inputs. The Inverell Research Centre is located at −29.78, 151.08 and has continuous data available from the beginning of 2003. Figure 7 shows the results of the longer input series at the Old Mollee 140 m site from 2004 to 2017. Comparing the surface flux (Figure 7a) and river stage (Figure 7c) inputs with the soil saturation output (Figure 7b) allows us to fully examine the soil-moisture profiles and how each process drives changes. During the Australian summer months (December–February), patterns of heavy rainfall and strong evaporative drying are seen in Figure 7a, and replicated in Figure 7b where they diffuse diagonally downward in time within the soil. River stage variations in Figure 7c are replicated in the water table (Figure 7b) with a time delay of approximately 24 days (which emerges directly from the Boussinesq equation, 140 m/Ks), and a reduced amplitude.

Details are in the caption following the image

(a) Plots showing the surface flux, (b) saturation profile, and (c) river stage time series for Old Mollee 140 m from 2004 to 2017. A particularly dry period in the soil is annotated with a white dashed line, and asterisks mark two similar peaks in the river stage and water table plots

Comparing similar river stage peaks in late 2004 and late 2011 (asterisks in Figures 7b and 7c), the second peak is much more pronounced in the water table than the first. The only difference between the two being that soil-moisture levels were much higher deep in the soil preceding the second peak. Hence, river stage peaks are more pronounced in the water table when preceded by rainfall events that have infiltrated deep into the soil. There is a visible period of dryness in the soil, particularly for midrange depths from 2006 to mid-2010 (white dashed line in Figure 7b). Looking at Figure 7c, this dryness seems to be triggered by low river stage values from late 2005 to late 2008. Low rainfall in 2009 exacerbates this drying. This dry period matches the time frame of the southeast Australian “Millenium” drought (2001–2009), during which time groundwater storage was found to be particularly depleted from the end of 2006 until the beginning of 2010 (Van Dijk et al., 2013). Soil-moisture throughout the soil recovered in 2011 and 2012, which were very wet years due to a combination of high rainfall events and high river stage peaks.

The power of this model is its ability to quantify the water availability to rooting vegetation, in particular within dynamic and sensitive riparian environments, at different distances from the river. Plant species, depending on their rooting depths, may access different potential water reservoirs in the floodplain. To compare water availability for different plants at different locations, time series of water content was produced over depth intervals. The soil-moisture was rescaled from saturation Θ (values range from 0 to 1), to a raw value of actual water content θ within a meter of soil (values range from the residual water content θr, to the saturated water content θs). Using the water content allows us to quantify the actual water available to vegetation rather than merely how filled the pore spaces are. In Figure 8, (a) Old Mollee 50 m, (b) Old Mollee 140 m, and (c) Old Mollee 320 m are compared. The plots show the water content in 1 m intervals down to 5 m. The saturated and residual water content for a 1 m depth range are provided as references for each plot.

Details are in the caption following the image

The water content in 1 m depth intervals (θ) down to 5 m for (a) Old Mollee 50 m, (b) Old Mollee 140 m, and (c) Old Mollee 320 m. The shades of blue in the plot become darker with depth into the soil.

The uppermost depth interval (0–1 m) at all the sites has the highest variability and contains the lowest overall levels of daily water content. Deeper-rooting vegetation has access to more water overall. The “0 m to 1 m” depth range is almost identical among all the three sites, yet differences become more pronounced moving deeper into the soil. The water content across all depths is high at Old Mollee 50 m, with near-complete saturation maintained within the 4–5 m depth. Saturation in this depth range indicates that the water table has risen to 4 m depth. The dry period identified between 2006 and mid-2010 in Figure 7 can be seen more clearly in the Old Mollee 140 and 320 m plots (Figures 8b and 8c). This is because the baseline water table depth is below 5 m for Old Mollee 140 and 320 m (see Figures 6 and 7b). So these depths are meters away from both the soil surface and the water table, making them more sensitive to periods of drought. When we compare the peaks in the 4–5 m water content in late 2010, late 2011, and early 2011 they decline more gradually in sites furthest from the river. This is because the water table further from the river is less variable. As water table levels are maintained for longer, the capillary fringe prolongs the decline to baseline water content levels. At the Old Mollee 320 m site, the capillary fringe allows water to be retained for up to months at 4–5 m deep. Hence, the water content just above the water table, further from the river is actually a much more reliable supply than above the water table nearer the river, which is more prone to daily fluctuation.

To further investigate the influence of the capillary fringe, we can observe the sharp difference in water content in the peaks between 3–4 and 4–5 m depths in Figure 8c, indicating the extent of the capillary fringe. Hence, the draw from the capillary fringe is not indefinite. Conversely, changes in water content moving down from the surface are more gradual, since water can continue infiltrating over time. Figure 9 allows us to examine these moisture patterns in a different way. Each column in the figure represents one of the Old Mollee sites, with histograms to the right representing sites further from the river.

Details are in the caption following the image

Histograms of the soil saturation in 1 m depth intervals (up to 5 m) for (the first column: a, d, g, j, and m) Old Mollee 50 m, (the second column: b, e, h, k, and n) Old Mollee 140 m, and (the third column: c, f, i, l, and o) Old Mollee 320 m. Plots further down in the figure represent soil saturation deeper in the soil.

The shape of the histograms indicates the mean and variance in water availability for each depth interval. Bell-shaped curves (i.e., all three 0–1 m histograms: Figures 9a–9c) represent depths that are sometimes exposed to extremes of wet and dry water conditions. Right-skewed distributions (i.e., the 2–3 m soil depth at Old Mollee 50 m and the 4–5 m depth at Old Mollee 320 m: Figures 9g and 9o) have a consistent lower threshold of water availability. The left-skewed distribution in Figure 9m portrays a depth with both high and consistent water availability. The highest water contents (all urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0046) are located at 2–3, 3–4, and 4–5 m at the Old Mollee 50 m site and at 4–5 m at the Old Mollee 320 m site. The higher content at the Old Mollee 50 m site is because the water table is higher at this site (at ∼4 m). The higher content at the deepest Old Mollee 320 m site is because of the low variability in the water table located a few meters below (at ∼7 m deep). Hence, the water availability is higher and more reliable nearer the water table and further from the river channel. Depending on the water requirements and rooting depths of a particular plant species, plots like these could be used to analyze current, or plan future riparian plantations. Furthermore, climate projections can be used to create these plots for future scenarios to investigate the water balance, soil-moisture, and vegetative water availability for a range of scientific and management applications.

5 Discussion

In this paper, we developed a new model with the capability to analyze coupled soil-moisture and water table fluctuations. Our model, HaughFlow, enables simple simulation of subsurface water fluxes in floodplains, through dynamic coupling of lateral hyporheic flow and rainfall infiltration, based on existing theoretical frameworks. We have coded this model in a manner that enables straightforward calibration of water table fluctuations and allows for decadal simulations of the impact of climate or even climate change on subsurface moisture. Water availability in catchments like the Murray-Darling basin are under increasing pressure from agricultural production and human use (Pittock & Connell, 2010). Having already experienced prolonged periods of drought, the majority of climate change scenarios foresee further water scarcity in the Murray-Darling region (Pittock & Connell, 2010). HaughFlow could be used to assess the impacts to water availability at multiple depths, and help inform water and ecological management plans.

Our application of HaughFlow to sites in southeast Australia identified a best approach for model calibration. The model performed best when calibrated using piezometer sites furthest from the river. This is due to the propensity for otherwise underestimating floodplain connectivity with a low lateral hydraulic conductivity value. We also showed the utility of the coupled-model approach, which significantly improves model performance. Further work is necessary to calibrate the unsaturated model component using high resolution data on soil-moisture.

The model outputs demonstrate the importance of the interdependence of the vadose and phreatic zones. In the upward direction, the capillary fringe contributes to soil-moisture patterns and connectivity in the vadose zone; and downward, the temporal legacy of infiltrating water in the vadose zone manifests as increased water content stored deeper in the soil. High antecedent moisture conditions in the vadose zone can facilitate connection between rising and infiltrating waters, raising the water table. Hence, the capillary fringe is an important exchange pathway between the vadose and phreatic zones. The vadose zone provides a significant store of water, particularly within the capillary fringe, which replicates fluctuations in the water table, and retains high levels of water saturation after peaks in the water table. This provides a critical moisture resource for shallow-rooting plants that are unable to reach the water table, which is particularly important further from the river channel.

Water table fluctuations became more delayed and less variable with distance from the river. At the furthest sites, delays in the propagation of river stage peaks are on the order of weeks, and the reduced variability means storage in the capillary fringe can maintain high water content for up to months. This storage potentially provides a valuable water resource to deeply rooting plants, particularly later in the dry season.

Differences between depths in the vadose zone were explored further using decadal data sets. Seasonal patterns in the surface flux were clearly visible in the upper regions of soil, i.e., the high rainfall and strong evapotranspiration rates during the summer months. As a result of the “Millenium” drought (Van Dijk et al., 2013), an extended dry period was observed in the soil, and its causes easily identified by comparison with the two input data sets (Figure 7). Critical depths and locations in the floodplain that were most sensitive to this dry period were also determined. Changes in soil-moisture patterns are more gradual with movement down from the soil surface as water infiltrates continuously. Conversely, with movement up from the water table there is a jump down in water content, as the extent of the capillary fringe is reached (seen in Figure 8). Overall, higher and more consistent water contents occurred in the soil regions that were both closer to the water table and further from the river channel.

Water content over different depth ranges allow HaughFlow to be used for a range of scientific and management applications relating to water and vegetation. If the rooting depths are known for a particular riparian species, then knowledge of the soil-water saturation over that depth can elucidate changes in the plant's behavior. Of course, root-water uptake is not as simple as this, as is it challenging to accurately represent root-water uptake with depth (Feddes et al., 2001) and plants can grow to adapt to their water availability (Canham et al., 2012). For example, river red gum trees in the Murray-Darling Basin were found to adapt well to the “Millenium” drought (Doody et al., 2015). So for most effective results, HaughFlow should be used alongside knowledge of plant physiology.

The combination of different processes working in tandem within the riparian vadose and phreatic zones make floodplains highly dynamic in terms of hydrology and water availability to vegetation. To better understand the relative contributions of each process, further work could be done to decouple their interlocking signals within the floodplain. One such way would be by delineating the capillary fringe. This is hard to do precisely, due to dynamic baseline soil-moisture conditions, no clear moisture threshold for the fringe, and unknown contributions from water infiltration. The incorporation of stable isotope tracers in HaughFlow could be used to establish water sourcing (Gat, 1996; Kendall & McDonnell, 2012). This would also make the model more powerful for use in vegetative water-use analysis, as plants take-up and store the stable isotopes from water at their rooting depths (Ehleringer & Dawson, 1992).

Acknowledgments

We would like to thank Sébastien Lamontagne and Andrew Taylor from the Commonwealth Scientific and Industrial Research Organisation (CSIRO) Land and Water in Australia for their provision of and advice with the groundwater monitoring and soil parameter data for the Murray-Darling basin. The authors would also like to thank the editor Jean Bahr and three anonymous reviewers for their detailed and constructive evaluation of the work which was very helpful and encouraging. Evans would also like to acknowledge the generous funding provided by the Natural Environment Research Council (NERC) and the University of St Andrews 600 Fund, without which this work would not have been possible. Singer was supported by funding from NSF EAR #700555. The sources of each data input are provided throughout the paper. The data output files are hosted online by the NERC Environmental Information Data Centre (Evans et al., 2018).

    Appendix A: Numerical Methods

    The numerical methods for the Richards equation and the Boussinesq equation are explained in the following subsections. A schematic of the physical and computational domains is shown in Figure A1.

    The lateral domain extends from the river channel (x = W, grid point i = nx) to the internal floodplain (x = 0, grid point i = 0). The computational grid points in the vertical domain are located at midlayers. Across both systems the vertical domain begins at the impermeable layer (z = 0, grid point urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0047) up to the soil surface (z = H, grid point urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0048). Richards equation can be applied to any position along the lateral domain; in Figure A1 this is displayed as the midpoint on the lateral axis for simplicity. The spatial steps are urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0049 and urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0050.

    The boundary conditions in the model consist of:
    1. A prescribed flux (defined by the precipitation and surface evapotranspiration) at z = H in the Richards equation.
    2. A specified hydraulic head ( urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0051) at the impermeable lower boundary z = 0 in the Richards equation, or a zero flux condition if there is no water table within the modeled domain.
    3. A prescribed river water elevation at x = W in the Boussinesq equation.
    4. A zero flux condition at the far-channel extent of the Boussinesq equation.

    The two equations are coupled through the water table. Specifically, the water table height urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0052 provides the lower boundary condition in the Richards equation, while the flux at the water table boundary arising from the vadose zone provides the source term S in the Boussinesq equation. Details are given below.

    A1. Initialization

    The initialization of the water content throughout the vadose and phreatic zones is described herein. In equilibrium, with a constant level of water hr in the river channel, and no external influences (including no surface flux), the water table will eventually equilibrate laterally with the adjacent floodplain. This means the water level will become uniform throughout ( urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0053) and there will be no vertical flux. Hence, the zero flux solution urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0054 in equation 9, with urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0055 can be applied throughout the domain.

    A2. Richards Equation

    Vertical water movement is simulated using the Richards equation (equation 2) along with three equations for Θ (5), C (6), and K (7). For numerical simplicity, the following substitutions are applied to the three equations when urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0056
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0057(A1)
    These lead to the following equivalent forms of equations 5-7
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0058(A2)

    When urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0059, the soil is saturated; there these functions simplify to the constants Θ = 1, C = 0, and K = Ks. These constants apply to the grid points below the water table.

    Following Pikul et al. (1974), a tridiagonal formula along with a predictor-corrector scheme was used to evolve equation 2 in time. The main difference from Pikul et al. (1974) is in placing grid points at midlayer depths, as this simplifies the upper boundary condition. We also use ζ in place of ψ.

    The predictor step from urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0060 to urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0061 at each interior vertical grid point, i = 2,…, urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0062, for equation 2 uses the following discretization:
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0063(A3)
    where the grid-averaged conductivity term is defined by urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0064. Using the constants
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0065(A4)
    we can rearrange this equation into tridiagonal form:
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0066(A5)
    With the solution for urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0067, we can use equations 6 and 7 to obtain values for urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0068 and urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0069 for i = 1,…, nz. From urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0070, we obtain urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0071 as before. Using urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0072, we can then update the flux term urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0073. The corrector step from urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0074 to urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0075 for the Richards equation is then calculated from
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0076(A6)
    which we can rearrange as
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0077(A7)

    When the water table is above the highest grid point in the domain, then flooding occurs. For flooding, no infiltration calculation is required and the surface flux is assigned at the water table boundary, urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0078. Evaporative fluxes can then diminish the water through the sink term in the lateral flow calculation until a vadose zone is formed (see Appendix A3.1 for more details on the calculation of the sink term).

    A2.1. Lower Boundary

    At the lowest grid point, i = 1, we make use of the bottom boundary condition urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0079. Taking ζ to be a linear function of z near z = 0 then leads to urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0080. This results in the following predictor and corrector equations:
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0081(A8)
    and
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0082(A9)
    When the water table is below the lowest half-grid point ( urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0083), the domain is considered to be unsaturated everywhere. In this case, a zero flux condition is applied at z = 0. For this case, the predictor step is
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0084(A10)
    while the corrector step is
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0085(A11)

    A2.2. Upper Boundary

    The upper boundary flux is calculated using the aforementioned daily precipitation rate (rp) minus the evapotranspiration rate (re) (which is multiplied by the water content Θ at the top boundary so it can never exceed the available water). Thus, the surface flux urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0086. Here urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0087 is used in lieu of urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0088 to avoid overshoots in extrapolation.

    We use this given surface flux to simplify the equations near the upper boundary (i = nz). The predictor step is
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0089(A12)
    while the corrector step is
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0090(A13)
    The model is able to simulate surface ponding both by precipitation rates exceeding the infiltration capacity and by the water table rising above the surface. The infiltration excess is evaluated by estimating the maximum flux possible at each time step. Equation 10 is discretized numerically as
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0091(A14)
    where the pressure term ( urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0092) is located at urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0093. If the input flux at the surface in the predictor or corrector step is greater than this maximum flux, urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0094, then we add the excess, urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0095, to the incoming flux at the next time step and set the current flux to the maximum, F = Fm. (NB: our unit of time is 1 day; otherwise the excess would need to be divided by the length of the day to give a flux.) The ponding has not been limited in these simulations.

    A3. Boussinesq Equation

    We also use a predictor-corrector formulation for the Boussinesq equation, following Pikul et al. (1974) but using the variable urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0096 to solve equation 11 for all internal horizontal points, i, equally-spaced in x. Equation 11 with the b substitution is
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0097(A15)
    The discretized predictor version of the equation is
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0098(A16)
    The source/sink term Sn is kept constant throughout the time step. Using the constant
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0099(A17)
    we can rearrange the predictor step into the tridiagonal formula
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0100(A18)
    The corrector step is
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0101(A19)
    which we can rearrange as
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0102(A20)

    At the near-river boundary, i = nx, a Dirichlet boundary condition is used to specify the height of the water in the river channel as urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0103. The predictor and corrector steps for the river boundary calculation urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0104 are the same as given in (A18) and (A20), after replacing urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0105 by hr and urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0106 by urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0107.

    The horizontal spatial extent of the model is chosen large enough so that the far-river boundary does not interfere with the water table dynamics in the area of interest near the channel. For this reason the far-river boundary, at x = 0 or i = 0 is specified to have a zero flux (or Neumann) boundary condition. For accuracy, near x = 0, b is expanded in a quadratic polynomial. The quadratic Taylor/MacLaurin Series expansion of h is
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0108(A21)
    where h0 is the unknown value of h at x = 0. Zero flux requires that at x = 0,
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0109(A22)
    We obtain h0 from the known values of h at grid points i = 1 and 2, giving
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0110(A23)

    This term can then be used to replace h0 when i = 1 in equations A18 and A20.

    A3.1. Evaluating the Source/Sink Term

    The source/sink term, S, is calculated by calculating the materials balance of the water in the domain. This calculation is carried out at the vertical evaluation site along the x axis. As a default this is half way between the river channel and the inner-floodplain boundary. This simplification incurs some error because the extent of the vadose zone is defined by the x- dependent position of the water table. A more accurate solution would calculate the x-dependent materials balance by calculating S(x, t) throughout the lateral domain. This however would be more computationally intensive.

    The materials balance calculation is defined by
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0111(A24)
    where the change in vadose water content is calculated from
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0112(A25)
    in which urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0113 is the nondimensional soil saturation at the end of time step n at grid point i. The change in surface water input is
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0114(A26)
    with Fn representing the incoming flux at time step n. So the flux Fwt at the top of the water table into the phreatic zone is
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0115(A27)
    Finally, the source/sink term is the phreatic zone influx scaled by the capacity or specific yield sy of the soil:
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0116(A28)
    Details are in the caption following the image

    Physical and computational domains used in the Richards equation (vertical) and in the Boussinesq equation (horizontal). NB: if there is no water table/phreatic zone in the modeled domain, the bottom boundary becomes a zero flux condition.

    Appendix B: Evapotranspiration Calculation

    B1. Evapotranspiration Equation

    The Penman-Monteith equation (Monteith, 1965; Penman, 1948) calculates the evapotranspiration rate as a function of the amount of energy incident on a region, the mass transfer gradient (wind and relative humidity), and the canopy conductance (for transpiration). The full equation is given by
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0117(B1)

    Here Δ is the slope of the saturation-vapor versus temperature curve (kPa K−1). The net radiation is calculated using Rns and Rnl which are the net shortwave and incoming longwave radiation, respectively (MJ m−2 d−1).

    In the second part of the numerator, γ is the psychrometric constant (kPa K−1). The remainder of this part calculates the mass transfer gradient using: KE, a coefficient that reflects the efficiency of vertical transport of water vapor by the turbulent eddies of the wind (mkm−1 kPa−1), ρw which is the mass density of water (999.97 kg m−3), λv which is the latent heat of water vaporization (2.47 MJ kg−1), va the wind speed (km d−1), urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0118 the saturation-vapor pressure at the air temperature (kPa) and, Wa which is the relative humidity of the air.

    Finally, in the denominator, Cat and Ccan are the atmospheric and canopy conductances, respectively. Equation B1 is based on the assumptions that there is no ground-heat conduction or water-advected energy, and heat-energy storage remains constant (Dingman, 2015). Appendix B2 describes the methods used to calculate each of the climate parameters from equation B1, and Appendix B3 describes the calculation of the transpiration parameters.

    B2. Climate Parameters

    To calculate daily evaporation using equation B1, the following parameters are required: Δ, Rns, Rnl, γ, KE, ρw, λv, va, urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0119, and Wa. These can be calculated using the latitude ( urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0120 in decimal degrees) and altitude (zalt in m) of the site along with the following climate values for a given day nday of that year's total number of days nyear (365 or 366): the maximum temperature (Tmax in °C), the minimum temperature (Tmin in °C), the dewpoint temperature (Tdew in °C), the atmospheric pressure at the station (P in kPa), the mean pressure at sea level (P0 in kPa), and the average wind speed (va in km d−1). The equations presented below have been sourced from Dingman (2015) unless stated otherwise.

    The slope of the relationship between saturation-vapor pressure and temperature, Δ, can be calculated in kPa K−1 using the following equation:
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0121(B2)
    where Tmean is the average daily temperature in °C.
    The methods for calculating the solar radiation have been taken from Allen et al. (1998). The net radiation incident on the surface in MJ m−2 d−1 is the sum of the net shortwave radiation, S, and incoming longwave radiation, L. First, we need the extraterrestrial radiation (Ra in MJ m−2 d−1), which is calculated using only the location of the site
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0122(B3)
    In this equation, Gsc is a solar constant (0.0820 MJ m−2 min−1), dr is the inverse relative distance between the Earth and Sun, calculated by
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0123(B4)
    and δ is the solar declination
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0124(B5)
    ωs is the sunset hour angle, defined by the following equation:
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0125(B6)
    where urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0126 is the latitude in radians.
    The incoming solar radiation (Rs in MJ m−2 d−1) can be inferred from this extraterrestrial radiation, along with the maximum and minimum temperature readings, using the Hargreages and Samani [1982] radiation formula
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0127(B7)
    which is a function of kRs, an adjustment coefficient. Allen (1997) produced the following relation for kRs
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0128(B8)
    where kRa is 0.17 for inland regions and P0 is the mean pressure at sea level.
    The net solar radiation is then found by
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0129(B9)
    where a is the albedo. A typical grassland albedo is 0.107 (Dingman, 2015).
    The clear-sky solar radiation, Rso can be calculated using the extraterrestrial radiation (Rs) and the site elevation above sea level (zalt, m)
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0130(B10)
    We are then able to calculate the incoming longwave radiation using
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0131(B11)
    where σ is the Stefan-Boltzmann constant (4.903 × 10−9 MJ K−4 m−2 d−1), and ea is the actual vapor pressure in kPa (calculated later in this section).
    In order to calculate the psychrometric constant, γ (kPa K−1), the latent heat of vaporization is required λv (MJ kg−1). This value is dependent on the temperature (°C) by the following relation:
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0132(B12)
    We can then use the heat capacity of air ( urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0133) along with the atmospheric pressure, P in kPa, to calculate the psychrometric constant by
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0134(B13)

    MWratio is the ratio of the molecular mass of water vapor to dry air (0.622).

    KE is a coefficient that describes the ability of the wind to move water vapor by turbulent eddies. It can be calculated in mkm−1 kPa−1 for the liquid-vapor transition by
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0135(B14)
    where ρa is the density of air (kg m−3), ρw is the density of water (kg m−3), zm is the height at which wind speed and air vapor pressure are measured (m), zd is the zero-place displacement (m), and z0 is the roughness height of the surface (m).
    The density of air can be calculated using the ideal gas law
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0136(B15)
    where Rspecific is the specific gas constant for dry air (287.058 Jkg−1 K−1). For a bare soil, the zero-plane displacement is 0 and the roughness height can be taken as 0.001 m (Garratt & Hicks, 1973 cited in Haghighi and Or, 2015). We can assume the wind speed and air vapor pressure readings were taken from a height of 2 m.
    Using the methods presented by Allen et al. (1998), the saturation-vapor pressure ( urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0137 in kPa) is related to the air temperature by the following equation:
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0138(B16)
    The mean saturation-vapor pressure which is required for the evaporation equation can then be calculated by
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0139(B17)
    In order to calculate the relative humidity from the known parameters we need the actual vapor pressure ( urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0140). The fractional relative humidity can then be calculated by (Allen et al., 1998)
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0141(B18)

    B3. Transpiration Parameters

    The transpiration parameters needed for equation B1 are Cat and Ccan. The atmospheric conductance can be calculated by
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0142(B19)
    where u is the wind speed, zd is the displacement height, z0 is the roughness height, and zm can be taken as 2 m above the top of the vegetation canopy, zveg. The values for grassland taken from Dingman (2015) are urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0143, and urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0144. The canopy conductance is calculated using
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0145(B20)
    where ks is a shelter factor, LAI is the leaf-area index, and Cleaf is the leaf conductance. Dingman (2015) recommends a shelter factor (which accounts for some plants being sheltered from the sun and wind) of 0.5 for vegetated land. Grassland has a LAI of 0.7 to 2.6 and maximum leaf conductance of 50 m/s (Dingman, 2015). In the present study, we use the average grassland LAI of 1.65.

    Appendix C: van Genuchten Parameters

    The sand, silt, and clay percentages and van Genuchten parameters for the full soil profiles (“ed199” and “ed200”) used in the study are provided in Tables C1 and C2. The sand, silt, and clay percentages were used to calculate the van Genuchten parameters using the ROSETTA model. The soil profile averages used as input to the model are provided in Table 2.

    Table C1. Soil Profile ed199 Sand, Silt, and Clay Content With Respective van Genuchten Parameter (to Four Decimal Places)s Calculated Using the ROSETTA Model (Schaap et al., 2001)
    Depth (m) Sand (%) Silt (%) Clay (%) Texture class θr (m3/m3) θs (m3/m3) a (1/m) n (N/A) Ks (m/d)
    0.01 39.9 24.1 36.0 Clay loam 0.0820 0.4240 1.9643 1.3110 0.0537
    0.05 42.8 22.4 34.8 Clay loam 0.0802 0.4174 2.0864 1.3065 0.0602
    0.15 44.8 20.7 34.5 Clay loam 0.0794 0.4133 2.2060 1.2978 0.0667
    0.35 29.9 23.2 46.9 Clay 0.0908 0.4545 2.0249 1.2581 0.1085
    0.75 26.3 25.4 48.3 Clay 0.0922 0.4635 1.8828 1.2633 0.1288
    1.25 39.9 20.9 39.2 Clay loam 0.0841 0.4264 2.1928 1.2779 0.0638
    1.75 64.0 11.0 25.0 Sandy clay loam 0.0659 0.3806 2.7536 1.3077 0.1472
    2.55 45.3 20.0 34.7 Sandy clay loam 0.0794 0.4125 2.2537 1.2930 0.0691
    Table C2. Soil Profile ed200 Sand, Silt, and Clay Content With Respective van Genuchten Parameters (to Four Decimal Places) Calculated Using the ROSETTA Model (Schaap et al., 2001)
    Depth (m) Sand (%) Silt (%) Clay (%) Texture class θr (m3/m3) θs (m3/m3) a (1/m) n (N/A) Ks (m/d)
    0.01 17.8 31.8 50.4 Clay 0.0965 0.4833 1.6293 1.2835 0.1597
    0.05 31.0 28.0 41.0 Clay 0.0875 0.4466 1.7045 1.3104 0.0691
    0.15 29.4 28.3 42.3 Clay 0.0887 0.4512 1.6939 1.3056 0.0778
    0.35 25.7 29.7 44.6 Clay 0.0911 0.4612 1.6410 1.3014 0.0961
    0.65 29.1 30.0 40.9 Clay 0.0881 0.4505 1.5926 1.3225 0.0751
    0.75 30.5 30.1 39.4 Clay loam 0.0867 0.4462 1.5751 1.3314 0.0686
    1.25 29.9 27.7 42.4 Clay 0.0886 0.4502 1.7290 1.3017 0.0767
    2.55 16.6 27.3 56.1 Clay 0.0981 0.4903 1.8281 1.2434 0.2125

    Appendix D: Numerical Sensitivity

    The spatial step differs between the vertical infiltration and lateral exchange equations. For the (vertical) Richards equation, we consider an idealized domain of total soil depth H down to an impermeable layer divided into
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0146(D1)
    equal layers in z (nearest integer is implied). For the (horizontal) Boussinesq equation, we consider a domain width W to be twice the distance from the river channel to the vertical soil profile being modeled, divided into
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0147(D2)
    equal intervals in x. Here N is the number of grid points used to resolve one characteristic diffusion length. By trial and error, N = 60 has been adopted in the results presented.
    The time step is required to be the same in both equations to allow alternation of the calculations of each equation at each step. As a result, to accurately resolve the diffusive processes in both of the equations, the time step was chosen as the minimum value of the following options
    urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0148(D3)
    where c = 0.5, urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0149, and urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0150. The first and last terms ensure the diffusion processes are accurately resolved, while the middle term ensures that the vertical conduction is resolved. The choice of time step balances accuracy and efficiency. To demonstrate the accuracy, simulations were run at half time steps ( urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0151) which were found to have a negligible impact on the results. Moreover, the uncertainties involved in the input data and parameters were orders of magnitude higher than the differences resulting from time step alterations.

    Appendix E: Calibration Details

    The calibration methods and a discussion of the results for each of the 24 settings are provided in Appendix E1. The evaluation results for each set of paired parameters are listed in Appendix E2. Finally, Appendix E3 provides a discussion of the best practice for calibration identified by performance trends.

    E1. Methods and Results

    Two codes were created to automate the calibration process. The first (in Fortran 90) calculates the calibration parameters (difference in variability and difference in running average) for the available time period, by comparing the simulation output and piezometer data. The second code, written in Python, controls the iterative process of calculating the optimal Ks and Dr values. The Python code reads the initial Ks and Dr values and uses the Fortran code to generate the calibration parameters. It then reruns the simulation for incremental values of Ks until the sign of the difference in variability changes. A linear interpolation of the Ks values either side of a 0 difference in variability is then taken. This process is repeated with Ks values closer and closer to 0 until the difference from the previous Ks value is within a user-chosen threshold (see below). Then the optimal Ks value is used to conduct the same incremental changes and interpolations for the Ds value.

    The years 2013 and 2014 were used as calibration periods. The 2 years were chosen because they have data available for all sites and are two distinct hydrological years, allowing a range of comparisons to be made. The initial Ks and Dr values used for all the calibrations were 1.0 and 0.0, respectively. Increments of Ks and Dr were set to initially be 1.0 and 0.01, respectively, and threshold differences were 0.005 and 0.00005, respectively. This means the Ks solution will be accurate up to three decimal places, as the calibration will run until the last two Ks solutions will be within 0.005 of each other, either side of a “0” difference in variability. Similarly, the Dr solution will be accurate up to five decimal places. Note that a Ks value of 0 is unphysical, so if urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0152 is required, then the value will step down by factors of 10 until the sign of vd changes.

    For each calibration period, the simulation was run from the previous 1 July. This meant that the spin-up, along with the initial half-year of data would make the water table levels perfectly representative at any distance from the river channel, no matter the Ks and Dr values at the start of the calibration period. The number of grid divisions in x (nx) was kept constant throughout the calibration and chosen to be the solution to equation D2 when Ks = 1. This meant that for solutions below Ks = 1 the nx was less than the equation-recommended amount. However, since the N = 60 value in the equation was chosen as a very conservative value, when a select few outputs were compared with higher nx counterparts, the differences were well below the Ks calibrated accuracy, so deemed insignificant. For reference, the nx values used were: 212 for Old Mollee 50 m, 568 for Old Mollee 140 m, 1,325 for Old Mollee 320 m, 168 for Yarral East 40 m, 462 for Yarral East 110 m, and 1,231 for Yarral East 290 m.

    The calibration was also run based on two different model setups; simulations based solely on the uncoupled lateral flow component and coupled simulations that included the vadose zone dynamics. The uncoupled version used no source/sink term or capillary fringe to represent water exchange with the vadose zone. This comparison was made both because the vadose zone component is not calibrated so may incorporate an unknown error, and also to indicate the effect of including interzone exchange. The six sites, two years and two model setups led to 24 different calibration combinations, the results of which are provided in Table E1.

    In general, the further the site is located from the river, the higher the calibrated hydraulic conductivity (with the exception of the Old Mollee 50 m site). As the lateral hydraulic conductivity determines how quickly input variations translate across the floodplain, this finding is logical. Sites nearer the river do not require as high Ks values for variations to reach them when compared with similar variations much further away from the river channel. The overall difference between the 2013 and 2014 calibrations is not pronounced. In 8 out of the 12 sites, the 2013 calibrated hydraulic conductivity is higher than 2014 (but only by up to 0.4 m/d). The higher 2013 Ks values may be because the river stage inputs are more variable in 2013 so require a higher Ks to translate this variability into the floodplain. The exceptions are Old Mollee 50 m in both cases, Yarral East 110 m uncalibrated, and Old Mollee 320 m calibrated. Hence the differences are not pronounced or consistent enough to be certain of a pattern.

    The uncoupled calibrations yielded much lower hydraulic conductivities and lower drainage terms than the coupled calibrations. This is because the source/sink term dampens the variability of the water table by drawing water into the unsaturated zone to lower peaks and infiltrating water from the unsaturated zone to reduce trough sharpness. Hence, a higher hydraulic conductivity is needed than without the unsaturated zone coupling, to maintain the variability. This higher hydraulic conductivity leads to a higher overall water table, as water propagates quickly into the floodplain so is able to maintain its elevation. Hence a higher drainage term than without the coupling is needed, to shift the water table elevations downward into the right range of values. NB: the drainage term is subtracted from the Boussinesq equation, therefore a higher drainage term leads to lower water table elevations.

    E2. Evaluation

    The exhaustive lists of evaluation parameters (R2, NSE, and PBIAS) based on each of the paired calibrated parameters applied to each of the sites based (24 different combinations run on each of the six sites) are provided in Tables E2–E4, respectively.

    E3. Results Discussion

    For easy comparison, the evaluation parameters were combined to distinguish the relative performance of different sites, years and model setups in Table E5. As the evaluation parameters are all on different scales, the performance of a given calibration cannot be judged by summing the results across all evaluation parameters. The level of accuracy according to the R2 value is determined by its magnitude, hence the R2 values are best compared by summing the results for each calibration option. Again by summing, the NSE results are compared for each calibration. Finally, the modulus of the PBIAS values can be summed for comparison but, contrary to the other two parameters, the lowest values indicate the best performance. Table E5 shows the sum of each evaluation parameter for each calibration category, along with the number of NSE values < 0 and the overall rank of each calibration category. This allows us to understand which calibration options perform best and to what extent they perform well. Along with the comparative evaluation results, the best and worst possible values for each calibration setting are shown for reference.

    In Table E5, the three sections allow us to compare: the performance of each site, the two calibration years, and the coupled or uncoupled-model setups. The maximum number of available days was used for evaluation at each of the sites but the sample size varied depending on the completeness of the piezometer data set. The number of days used for each evaluation was 1,462 days for Old Mollee 50 m, 1,456 days for Old Mollee 140 m, 1,461 days at Old Mollee 320 m, and 912 days for each of the Yarral East sites. The two locations (Old Mollee and Yarral East) were not compared due to distinct differences in data set size. However, the different distances from the river within each of the locations have a similar number of days available so can be compared fairly. Clearly, the sites furthest from the river performed best.

    Looking at the detailed results in Appendix E2, the sites closer to the river (especially the closest site Yarral East 40 m) are less sensitive to the calibration parameters, so lower values of hydraulic conductivity in the automated calibration will suffice for a good match. Hence, the further, more sensitive sites produce the highest hydraulic conductivities and calibrations that perform best overall. Since these were calibrated to have the highest hydraulic conductivities, we can conclude that calibrating based on sites closer to the river results in an under-estimation of the floodplain connectivity. Therefore, the furthest sites available should be used for calibration.

    The Old Mollee 50 m site calibrations, which were exceptions in many of the trends identified during the calibration process (see section 9), performs poorly in the NSE and PBIAS tests. However, unexpectedly, the R2 result is favorable. Little can be concluded from the interyear comparison. 2013 performs better in the R2 and PBIAS results but worse in NSE, and both years produce the same number of simulations with NSE values < 0. Moreover, looking at the full results in Appendix E2, improved performance does not match up with any trends seen in the calibrated Ks values.

    From Table E5, we can see that the coupled calibrations perform far better across most of the categories than the uncoupled calibrations. The NSE sum is the exception, but there are far fewer results with NSE values < 0 in the coupled calibrations. The higher overall NSE sum is attributed to disproportionately poor results from the Old Mollee 50 m output based on the furthest site calibrations. Hence, the coupled calibrations perform best, with Old Mollee 50 m being the exception. Therefore, despite the fact that the infiltration component of the model remains uncalibrated, including the dynamics of the unsaturated zone is crucial to the model's performance. This indicates the great importance of the interzone exchange and relationship between the two zones in capturing the patterns of water table movement.

    The evaluation parameters for the six best calibrations, which are also the only calibrations that have no NSE values < 0, are summarized in Table E6. We can see that all the best sites are all based on the coupled calibration, starting with the furthest sites.

    Table E1. Calibration Results for the Uncoupled and Coupled Code, for the Years 2013 and 2014, and for all Six Sites, Using the Automated Calibration Codes
    Coupled? Year Site Ks (m d−1) vd Dr (m) yd
    No 2013 Old Mollee 50 m 1.973 9.94 × 10−4 0.11010 <1 × 10−7
    No 2013 Old Mollee 140 m 0.937 1.28 × 10−4 0.00680 −1.40 × 10−6
    No 2013 Old Mollee 320 m 2.592 1.64 × 10−5 0.00204 −1.80 × 10−5
    No 2013 Yarral East 40 m 0.347 2.60 × 10−4 −0.00020 <1 × 10−7
    No 2013 Yarral East 110 m 0.864 5.00 × 10−5 −0.00076 >−1 × 10−7
    No 2013 Yarral East 290 m 2.935 7.40 × 10−6 0.00045 −2.90 × 10−6
    No 2014 Old Mollee 50 m 2.265 6.39 × 10−4 0.12957 1.00 × 10−7
    No 2014 Old Mollee 140 m 0.869 8.09 × 10−5 0.00901 −2.00 × 10−7
    No 2014 Old Mollee 320 m 2.412 2.32 × 10−5 0.00359 −2.69 × 10−5
    No 2014 Yarral East 40 m 0.298 2.82 × 10−5 −0.00138 1.00 × 10−7
    No 2014 Yarral East 110 m 1.014 6.69 × 10−3 0.00025 −1.00 × 10−7
    No 2014 Yarral East 290 m 2.550 2.07 × 10−5 0.00198 −1.47 × 10−5
    Yes 2013 Old Mollee 50 m 4.558 7.33 × 10−4 0.26775 2.00 × 10−7
    Yes 2013 Old Mollee 140 m 2.279 −4.39 × 10−4 0.02968 >−1 × 10−7
    Yes 2013 Old Mollee 320 m 4.898 −6.20 × 10−4 0.01707 −3.90 × 10−6
    Yes 2013 Yarral East 40 m 0.778 8.85 × 10−5 0.01352 −6.61 × 10−5
    Yes 2013 Yarral East 110 m 2.047 −1.99 × 10−4 0.01244 −2.03 × 10−5
    Yes 2013 Yarral East 290 m 5.756 −6.85 × 10−4 0.01568 −2.94 × 10−5
    Yes 2014 Old Mollee 50 m 5.498 4.19 × 10−4 0.32570 −1.00 × 10−7
    Yes 2014 Old Mollee 140 m 2.117 −2.43 × 10−4 0.02996 −6.90 × 10−6
    Yes 2014 Old Mollee 320 m 5.068 −1.75 × 10−4 0.01537 −4.00 × 10−7
    Yes 2014 Yarral East 40 m 0.692 6.43 × 10−5 0.00684 1.00 × 10−7
    Yes 2014 Yarral East 110 m 1.802 −1.37 × 10−4 0.01064 −8.00 × 10−7
    Yes 2014 Yarral East 290 m 5.355 −1.98 × 10−4 0.01314 −1.10 × 10−6
    • Note. The hydraulic conductivity, Ks, the difference in variability vd ( urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0153), the drainage term Dr and the average difference in running average yd (equation 14) are listed for each calibration.
    Table E2. R2 Evaluation Results for Each of the 24 Pairs of Calibrated Parameters (Each Row) Applied to Each of the Six Sites (Each Column)
    Coupled? Year Site OM50 OM140 OM320 YE40 YE110 YE290
    No 2013 OM50 0.884 0.757 0.660 0.868 0.884 0.909
    No 2013 OM140 0.744 0.579 0.039 0.870 0.913 0.280
    No 2013 OM320 0.922 0.803 0.450 0.864 0.878 0.887
    No 2013 YE40 0.489 0.144 0.059 0.837 0.761 0.841
    No 2013 YE110 0.725 0.558 0.009 0.870 0.912 0.020
    No 2013 YE290 0.935 0.813 0.520 0.862 0.869 0.911
    No 2014 OM50 0.903 0.773 0.698 0.866 0.879 0.919
    No 2014 OM140 0.727 0.551 0.115 0.869 0.909 0.518
    No 2014 OM320 0.914 0.795 0.402 0.865 0.883 0.872
    No 2014 YE40 0.445 0.087 0.581 0.826 0.707 0.703
    No 2014 YE110 0.762 0.611 0.016 0.871 0.917 0.178
    No 2014 YE290 0.921 0.801 0.441 0.864 0.879 0.883
    Yes 2013 OM50 0.964 0.807 0.779 0.853 0.836 0.950
    Yes 2013 OM140 0.906 0.779 0.395 0.866 0.885 0.082
    Yes 2013 OM320 0.968 0.821 0.690 0.852 0.826 0.953
    Yes 2013 YE40 0.700 0.511 0.232 0.868 0.899 0.575
    Yes 2013 YE110 0.892 0.769 0.271 0.868 0.893 0.770
    Yes 2013 YE290 0.974 0.817 0.735 0.849 0.811 0.946
    Yes 2014 OM50 0.972 0.805 0.779 0.850 0.821 0.953
    Yes 2014 OM140 0.896 0.769 0.351 0.867 0.889 0.002
    Yes 2014 OM320 0.969 0.821 0.702 0.851 0.823 0.950
    Yes 2014 YE40 0.672 0.465 0.197 0.865 0.893 0.616
    Yes 2014 YE110 0.873 0.749 0.182 0.869 0.900 0.682
    Yes 2014 YE290 0.971 0.820 0.683 0.850 0.818 0.951
    Days available 1,462 1,456 1,461 912 912 912
    • Note. Each row represents the model performance for a calibration pair (Ks and Dr) applied to each of the six sites. The calibration settings are listed in the left-hand columns. For example the last entry in the first row is the R2 result for the Yarral East 290 m site when the Ks and Dr terms were calculated based on the uncoupled 2013 Old Mollee 50 m calibration. The worst results are when urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0043 the best results when urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0044.
    Table E3. NSE Evaluation Results for Each of the 24 Pairs of Calibrated Parameters (Each Row) Applied to Each of the Six Sites (Each Column)
    Coupled? Year Site OM50 OM140 OM320 YE40 YE110 YE290
    No 2013 OM50 0.874 −4.150 −369.228 0.566 −10.255 −1257.869
    No 2013 OM140 0.572 0.210 −0.566 0.761 0.578 −2.513
    No 2013 OM320 0.779 0.486 −0.431 0.665 0.660 −4.655
    No 2013 YE40 0.147 −2.646 −20.534 0.671 −6.461 −76.631
    No 2013 YE110 0.505 −0.374 −7.908 0.747 −0.538 −40.778
    No 2013 YE290 0.792 0.500 −0.474 0.650 0.632 −4.787
    No 2014 OM50 0.897 −4.604 −401.029 0.535 −11.428 −1350.808
    No 2014 OM140 0.565 0.272 −0.208 0.768 0.706 −2.796
    No 2014 OM320 0.771 0.487 −0.288 0.674 0.689 −3.867
    No 2014 YE40 0.033 −4.094 −79.232 0.599 −10.269 −132.294
    No 2014 YE110 0.560 −0.106 −4.991 0.745 −0.020 −27.649
    No 2014 YE290 0.776 0.480 −0.467 0.667 0.659 −4.862
    Yes 2013 OM50 0.963 −5.476 −439.233 0.418 −14.156 −1468.827
    Yes 2013 OM140 0.810 0.605 −4.468 0.682 0.623 −17.842
    Yes 2013 OM320 0.851 0.691 0.431 0.593 0.613 0.925
    Yes 2013 YE40 0.562 0.359 −3.199 0.778 0.888 −44.862
    Yes 2013 YE110 0.763 0.561 0.148 0.695 0.822 0.693
    Yes 2013 YE290 0.855 0.694 0.520 0.575 0.566 0.939
    Yes 2014 OM50 0.972 −5.693 −451.689 0.390 −14.810 −1510.581
    Yes 2014 OM140 0.802 0.580 −5.541 0.690 0.600 −24.139
    Yes 2014 OM320 0.851 0.687 0.483 0.589 0.603 0.946
    Yes 2014 YE40 0.490 0.055 −0.722 0.773 0.316 −3.007
    Yes 2014 YE110 0.737 0.515 0.146 0.709 0.826 0.291
    Yes 2014 YE290 0.851 0.681 0.504 0.583 0.586 0.893
    Days available 1,462 1,456 1,461 912 912 912
    • Note. The row title gives the calibration settings and the column title gives the simulation site. NSE < 0 indicates poor results while NSE > 0.5 indicates favorable results.
    Table E4. PBIAS Evaluation Results for Each of the 24 Pairs of Calibrated Parameters (Each Row) Applied to Each of the Six Sites (Each Column)
    Coupled? Year Site OM50 OM140 OM320 YE40 YE110 YE290
    No 2013 OM50 −0.016 0.963 8.196 0.083 0.678 6.748
    No 2013 OM140 −0.164 −0.205 −0.276 −0.022 −0.118 −0.242
    No 2013 OM320 −0.157 −0.181 −0.371 −0.014 −0.072 −0.448
    No 2013 YE40 −0.242 −0.736 −1.934 −0.085 −0.584 −1.633
    No 2013 YE110 −0.188 −0.392 −1.196 −0.039 −0.246 −1.214
    No 2013 YE290 −0.158 −0.184 −0.392 −0.014 −0.072 −0.454
    No 2014 OM50 −0.010 1.010 8.541 0.088 0.712 6.993
    No 2014 OM140 −0.158 −0.160 0.002 −0.018 −0.092 0.149
    No 2014 OM320 −0.157 −0.173 −0.325 −0.014 −0.068 −0.414
    No 2014 YE40 −0.267 −0.903 2.381 −0.105 −0.675 −2.166
    No 2014 YE110 −0.180 −0.330 −0.952 −0.032 −0.197 −1.003
    No 2014 YE290 −0.158 −0.182 −0.378 −0.015 −0.073 −0.456
    Yes 2013 OM50 0.002 1.100 8.940 0.097 0.783 7.292
    Yes 2013 OM140 −0.127 0.061 0.926 0.006 0.086 0.804
    Yes 2013 OM320 −0.144 −0.084 0.111 −0.004 0.002 0.031
    Yes 2013 YE40 −0.144 −0.046 0.724 −0.009 −0.022 −1.206
    Yes 2013 YE110 −0.147 −0.095 0.097 −0.008 −0.020 −0.034
    Yes 2013 YE290 −0.145 −0.093 0.063 −0.005 −0.002 −0.002
    Yes 2014 OM50 0.005 1.120 9.066 0.099 0.799 7.395
    Yes 2014 OM140 −0.125 0.076 1.023 0.008 0.095 0.927
    Yes 2014 OM320 −0.145 −0.092 0.070 −0.005 −0.003 0.003
    Yes 2014 YE40 −0.169 −0.236 −0.271 −0.027 −0.155 0.018
    Yes 2014 YE110 −0.150 −0.114 0.015 −0.010 −0.035 −0.110
    Yes 2014 YE290 −0.146 −0.102 0.008 −0.006 −0.009 −0.045
    Days available 1,462 1,456 1,461 912 912 912
    • Note. The row title gives the calibration settings and the column title gives the simulation site. Values are expressed as a percentage with best values being when |PBIAS| < 0.1 and worst values when |PBIAS| > 1.
    Table E5. Sum of the Evaluation Parameters for Each of the Calibration Options for Comparison (NB: the Highest Values Indicate Best Performance Except for the PBIAS Values, for Which the Modulus of the Values is Summed and the Lowest Values Indicate Best Performance)
    Calibration setting urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0154 ∑ NSE # NSE < 0 ∑ |PBIAS| Rank
    OM50 20.370 −7314.222 16 70.734 5
    OM140 14.802 −48.252 8 5.870 3
    OM320 19.761 4.232 4 3.088 1
    YE40 13.973 −378.281 12 14.702 6
    YE110 15.166 −72.902 8 6.804 4
    YE290 19.924 2.814 4 3.161 2
    Best/worst values 24/0 24/– urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0155 0/24 0/ urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0156 1/6
    2013 51.919 −3779.443 26 52.142 1
    2014 52.078 −4027.169 26 52.219 2
    Best/worst values 72/0 72/– urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0157 0/72 0/ urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0158 1/2
    Uncoupled 49.275 −3828.136 36 58.218 2
    Coupled 54.722 −3978.475 16 46.142 1
    Best/worst values 72/0 72/− urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0159 0/72 0/ urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0160 1/2
    • Note. The following abbreviations are used in the table Old Mollee (OM), Yarral East (YE), 50 m (50), etc.
    Table E6. Sum of the Evaluation Parameters for all Six Sites Based on Each of the Best Six Calibrations
    Category urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0161 ∑ NSE # NSE < 0 ∑ |PBIAS| Rank
    Coupled 2013 YE290 5.132 4.150 0 0.310 1
    Coupled 2014 OM320 5.116 4.158 0 0.318 2
    Coupled 2014 YE290 5.094 4.098 0 0.315 3
    Coupled 2013 OM320 5.110 4.103 0 0.377 4
    Coupled 2013 YE110 4.464 3.682 0 0.401 5
    Coupled 2014 YE110 4.255 3.222 0 0.435 6
    Best/worst values 6/0 6/− urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0162 0/6 0/ urn:x-wiley:00431397:media:wrcr23101:wrcr23101-math-0163 1/24
    • Note. The highest values indicate best performance except for the PBIAS values, for which the modulus of the values is summed and the lowest values indicate best performance.