We begin our introduction to the SHUD model with a conceptual description of water movement in a watershed. Figure  depicts processes that are incorporated in the model. Catchment hydrology is driven by atmospheric inputs, including rainfall, snowfall, and irrigation. The land surface hydrology first interacts with vegetation through plant-canopy interception and throughfall. When precipitation exceeds the interception capacity, throughfall or excess precipitation falls to the land surface. Snowfall accumulation and melting is an important land-surface process and a major source of soil moisture and recharge in many temperate or mountainous climate regions. SHUD captures the partitioning of excess precipitation into surface runoff and infiltration into the soil.

Vertically, the aquifer is divided into two coupled layers based on its saturation status: the top unsaturated layer (or vadose layer) is constrained to 1-D vertical flow, and the saturated groundwater layer admits 2-D flow. These layers overlie an impermeable or low-permeable layer such as bedrock or an effective depth of circulation where deeper flows are small and unlikely to contribute to baseflow. The vertical fluxes within the unsaturated zone include infiltration and exfiltration. Deep percolation or recharge to groundwater is fully coupled to soil moisture dynamics. SHUD accounts for conditions when the groundwater table reaches or exceeds the land surface, decreasing infiltration, allowing exfiltration as local ponding and runoff. The model also accounts for upward capillary flow from a shallow water table depending on soil moisture and vegetation conditions. Lateral (2-D) groundwater flow represents the basic mechanism for baseflow to streams or rivers.

Surface runoff or overland flow is generated by excess rainfall and ponding and is represented as 2-D shallow water flow in SHUD. Surface runoff complements baseflow runoff as the dominant source of streams or rivers. SHUD also allows reversing flows from the channel to the hillslope as surface inundation or channel losses to groundwater. This may occur when the river stage rises above bankfull storage during flooding events or in an arid region where the river recharges the local groundwater.

Evaporation generally represents the largest water loss from the catchment with four components: evapotranspiration (ET) from interception storage, surface ponding, soil moisture, and shallow groundwater. Transpiration occurs only when vegetation is present and could draw from the saturated groundwater when the groundwater level is high enough. Direct evaporation occurs from interception, ponding water, and soil moisture.

Following the above description, several assumptions and simplifications are made in the SHUD model:

In the default configuration, the watershed boundary is generally handled as a closed domain, in which precipitation and evapotranspiration are the major vertical fluxes, and river flow is the major lateral flow into and out of the domain. It is a common water balance assumption of ΔS=P-E-Q, where ΔS, P, E, and Q are total water storage, precipitation, evapotranspiration, and discharge, respectively. Modifications to these conditions can be applied to realize additional flows such as pumping wells, irrigation, basin diversions, and so forth. The model is able to further support boundary conditions as needed for various research questions.

The notation used in this section is summarized in the list of symbols in the Appendix.

Figure  depicts the geometric structure of the discrete cells in SHUD. The watershed domain is discretized using an irregular unstructured triangular network (Delaunay triangles) generated with imposed spatial constraints by triangle . Non-Delaunay triangulations are also permitted, so any GIS software and advanced programming language (R, MATLAB, and Python) could potentially generate the triangular domain. A prismatic control volume is formed by the vertical extension of the Delaunay triangles to produce three layers: land surface layer, unsaturated zone, and groundwater layer. The modeler is responsible for defining the aquifer depth (from the land surface to the impervious bedrock) based on measurements or terrestrial characteristics. The thickness of the unsaturated zone (Dus) is determined by the difference between the land surface elevation (zsf) and groundwater table (zgw) above datum, i.e., Dus=zsf-zgw. When the groundwater table reaches the land surface (zgw>zsf), the thickness of unsaturated zone →0.

Figure depicts the exchange of water between the rivers and hillslope cells. Within each river channel, there are two longitudinal fluxes and two lateral fluxes: upstream (Qup), downstream (Qdn), overland (Qsf), and groundwater (Qsub).

The hydrologic model uses the method of moments to reduce the partial differential equations (PDEs) into ordinary differential equations (ODEs) and solve the ODE system using a globally implicit numerical solver. The state variables include water height on the land surface (Ysf), soil moisture storage (Yus), groundwater depth (Ygw), and river stage (Yriv). The initial value problem for these ODEs is formulated as dYdt=f(t,Y),Yt0=Y0, where the discrete state vector is denoted by Y, Y=YsfYusYgwYriv. Y0 contains the initial conditions, and f(t,Y) denotes the equations governing the hydrologic flow, which are described in this section.

The system of ODEs describing the hydrologic processes are fully coupled and solved simultaneously at each time step (Δt=tn-tn-1) using CVODE, a stiff solver based on Newton–Krylov iteration . In brief, the CVODE solver calculates Y(tn), given Y(tn-1) and dYdt|tn-1. The technical description of the CVODE solver can be found in the literature . The governing equations in SHUD are provided in Table .

Figure depicts the workflow within the SHUD model. The explicit model time step (MTS) Δt=tn-tn-1 is user specified, typically varying from 1 min to 1 h. Within the MTS, the relevant gradient law determines the fluxes and hence all state values.

The fluxes of ET and interception change relatively slowly within short periods (such as 1 h) so that full coupling of ET with soil water is not necessary for this model. Instead, the interception, ET, and snow calculations are solved explicitly at the MTS, while the calculation of Ysf, Yus, Ygw, and Yriv uses the implicit time step (ITS).

The CVODE solver determines the ITS automatically based on both the specified tolerances and the error function of Y and dY. The initial ITS equates to the explicit MTS. Within the ITS, dYs is calculated based on Y from the last MTS. If the CVODE solver converges with the current value of the ITS, it returns the updated Y. Otherwise, a convergence failure occurs that forces an ITS reduction.

The mathematical model underlying SHUD consists of five components: vegetation and evapotranspiration, land surface, unsaturated layer, saturated layer, and river channel. These are described in the following subsections.

The V-catchment (VC) experiment is a standard test case for numerical hydrologic models to validate their performance for overland flow along a hillslope and in the presence of a river channel . The VC domain consists of two inclined planes draining into a sloping channel (Fig. ). Both hillslopes are 800m×1000 m with Manning's roughness n=0.015. The river channel between the hillslopes is 20 m wide and 1000 m in length with n=0.15. The slope from the ridge to the river channel is 0.05 (in the x direction), and the longitudinal slope (in the y direction) is 0.02.

Rainfall in the VC begins at time zero at a constant rate of 18 mm h-1 and stops after 90 min, producing 27 mm of accumulated precipitation. Since this experiment excludes the evaporation and infiltration, the total outflow from lateral boundaries and the river outlet must be the same as the total precipitation (following conservation of mass).

Figure illustrates the discharge from the side plane to the river channel and at the river outlet. The specific discharge (the volume discharge divided by the total area of the catchment) increases with precipitation until it reaches the maximum discharge rate, which is equal to the precipitation rate. Discharges along lateral boundaries and from the river outlet reach the maximum discharge rate but at different times; namely, the discharge rate from the side plane reaches the maximum value earlier than in the river outlet. The dots are discharge digitalized from with WebPlotDigitizer (https://automeris.io/WebPlotDigitizer/, last access: 19 December 2019). The results suggest SHUD can correctly capture the processes in overland flow and channel routing, although flow from the river outlet occurs earlier than the prediction in . Both the fluxes from side plane and outlet meet the maximum flow rate, which is the same magnitude of precipitation after a short period of rainfall. The flux rates start decreasing after precipitation stops. The accumulated volume of flux confirms the correct mass balance of both fluxes.

To verify correct performance of the numerical method, we calculate the bias of mass balance in the model and assess the differences among input, output, and storage change in the system (Eq. ). For this simulation the bias ends up being ∼0.2 %. 33ΔS=P-Q-E34ΔS^=ΔSic+Δysf+Δyus+Δygw+Δyriv35Bias=ΔS^-ΔSΔS×100%

Vauclin's laboratory experiment is to assess groundwater table change and soil moisture in the unsaturated layer under precipitation or irrigation. The experiment was conducted in a sandbox with dimensions of 3 m long ×2 m deep ×0.05 m wide (see Fig. ). The box was full of uniform sand particles with measured hydraulic parameters: the saturated hydraulic conductivity was 35 cm h-1, and porosity was 0.33 m3 m-3. The left and bottom of the sandbox were impervious layers, and the top and the right side were open. The hydraulic head was 0.65 m constantly on the right boundary. Constant irrigation (1.48 cm h-1) was applied over the first 50 cm of the top left of the sandbox, while the rest of the top was covered to avoid water loss via evaporation.

The experiment's initial condition is an equilibrium water table with a uniform hydraulic head. Irrigation was initiated at t=0. The groundwater table was then measured at 2, 4, 6, and 8 h at several locations along the length of the box. also uses a 2-D (vertical and horizontal) numeric model to simulate the soil moisture and groundwater table. The maximum bias between measurement and simulation was 0.52 m, according to the digitalized value of Fig. 10.

Besides the parameters specified in , additional information is needed in the SHUD model, including the α and β in the van Genuchten equation and residual water content (θr). Therefore, we use a calibration tool to estimate the representative values of these parameters. The use of calibration in this simulation is reasonable because the model, inevitably, simplifies the real hydraulic processes. The calibration thus nudges the parameters to representative values that approach or fit the true natural processes. The calibrated values are θr=0.001 m3 m-3, α=0.3 ,and β=5.2. The simulated results in our modeling and literature show a modest error between the simulations and measurements.

This error is likely due to (1) the need for a detailed aquifer layer description of soil layers or (2) possible invalidity of the vertical and horizontal flow assumptions in the SHUD model.

SHUD simulated the groundwater table at all four measurement points (see Fig. b). The maximum bias between simulation and Vauclin's observations is 5.5 cm, with R2=0.99, which is comparable to the bias 5.2 cm of numerical simulation in . By adding more parameters into the calibration, the bias in the simulation decreased to 3 cm. Indeed, as discussed earlier, the simplifications employed by SHUD for the unsaturated and saturated zone mainly benefits computational efficiency while limiting applicability where fine-scale soil profile information is required.

These simulations, compared against Vauclin's experiment, generally validate the algorithm for infiltration, recharge, and lateral groundwater flow. A more reliable vertical flow within the unsaturated layer requires multiple layers, which is planned in the next version of SHUD.

The Cache Creek Watershed (CCW) is a headwater catchment with area 196.4 km2 in the Sacramento Watershed in northern California (Fig. a, b and c). The elevation ranges from 450 to 1800 m, with a 0.38 m m-1 average slope, where the steep implies a particular challenge for numeric models.

Based on NLDAS-2 from 2000 to 2017, the mean temperature and precipitation were 12.8 ∘C and ∼817 mm, respectively, in this catchment. Precipitation varies seasonally; winter and spring are the primary wet seasons (Fig. ).

Table lists the geospatial and forcing data supporting hydrologic modeling in the CCW. The DEM is 30 m resolution raster data from National Elevation Dataset (NED) . Forcing data, including precipitation, temperature, relative humidity, wind speed, and net radiation, are from NLDAS-2 (, https://ldas.gsfc.nasa.gov/nldas/v2/forcing, last access: 19 December 2019). Our simulation in the CCW covers the period from 2000 to 2007. Because of the Mediterranean climate in this region, the simulation starts in summer to ensure adequate time before the October start to the water year. In our experiment, the first year (1 July 2000 to 30 June 2001) is the spin-up period, the following 2 years (1 July 2001 to 30 June 2003) are the calibration period, and the period from 1 July 2003 to 1 July 2007 is for validation.

The unstructured domain of the CCW (Fig. d) is built with rSHUD . The domain consists of 1147 triangular cells, with a mean area of 0.17 km2. The total length of the river network is 126.5 km and consists of 103 river reaches in which the highest order of stream is 4. With a calibrated parameter set, the SHUD model took 5 h to simulate 18 years (2000–2017) in the CCW, with a nonparallel configuration (OpenMP is disabled on Mac Pro 2013 Xeon 2.7 GHz, 32 GB RAM).

Figure  reveals the comparison of simulated discharge against the observed discharge at the gage station of USGS 11451100 (https://waterdata.usgs.gov/ca/nwis/uv/?site_no=11451100, last access: 19 December 2019). The calibration procedure exploits the covariance matrix adaptation–evolution strategy (CMA-ES) to calibrate automatically . The calibration program assigns 72 children in each generation and keeps the best child as the seed for the next generation, with limited perturbations. The perturbation for the next generation is generated from the covariance matrix of the previous generation. After 23 generations, the calibration tool identifies a locally optimal parameter set.

Within the calibration period, Nash–Sutcliffe efficiency (NSE; ), Kling-Gupta efficiency (KGE; ), and R2 are 0.72, 0.83, and 0.72, respectively (Fig. ). The goodness-of-fit in the validation period is less than calibration period (as expected), with NSE =0.66, KGE =0.67, and R2=0.65. Although the SHUD model captures the flood peaks after rainfall events, the magnitude of high flow in the hydrograph is less than the gage data. There are two potential causes of this bias: (1) underestimated precipitation intensity from NLDAS-2 data or (2) overfitting in the calibration, as the NSE tends to capture the mean value of the observational data rather than the extremes.

Figure  represents the monthly water balance in CCW, in which the PET is 3 times the annual precipitation, but the actual evapotranspiration (AET) is only 27 % of the precipitation. This result emerges because the summer is the peak of PET, while winter is the peak of precipitation and water availability. The AET is subjected to PET and water availability, so the maximum of AET occurs in early summer. The runoff ratio is about 73 %.

We use the groundwater distribution (Fig. ) to demonstrate the spatial distribution of hydrologic metrics calculated from the SHUD model. Figure  illustrates the annual mean groundwater table in the validation period. Because the model fixes a 30 m aquifer, the results represent the groundwater within this 30 m aquifer only. The groundwater table and elevation along the green line on the map are extracted and plotted in Figure b. The gray ribbon is the 30 m aquifer, and the blue line is the location where groundwater storage is larger than zero. The light blue polygons (with the right axis for scale) are the groundwater storage along the cross section. The groundwater levels tend to follow the terrain, with groundwater accumulated in the valley or along relatively flat flood plains. In the CCW, groundwater does not stay on steep slopes suggesting the high conductivity of upland slopes.

As a descendant of PIHM, SHUD inherits many fundamental ideas and the conceptual structure from PIHM, including the solution of hydrologic variables using CVODE. The code has been completely rewritten in a new programming language, with a new discretization and corresponding improvements to the underlying algorithms, adapting new mathematical schemes and flexible input/output data formats. Although SHUD is forked from PIHM, SHUD still inherits the use of CVODE for solving the ODE system but modernizes and extends PIHM's technical and scientific capabilities. The major differences are as follows:

SHUD is written in C++, an object-oriented programming language with functionality to avoid memory leaks from C. Most of the functions in SHUD do not exist in PIHM, which are newly defined because of the brand-new design of the SHUD model. Only a few functions related to physical equations, such as Manning's equation and van Genuchten's equation, are shared in SHUD and PIHM. So although SHUD and PIHM follow a similar fundamental perceptual model, they follow different mathematical and computational strategies.

We now briefly summarize the technical model improvements and technical capabilities of the model compared to PIHM. This elaboration of the relevant technical features aims to assist future developers and advanced users with model coupling. Compared with PIHM, SHUD does the following:

supports compatibility with the implicit Sundial/CVODE solver version 5.0 (the most recent version at the time of writing) and above;