Hydrologic modeling is an essential strategy for understanding and predicting natural flows, particularly where observations are lacking in either space or time or where complex terrain leads to a disconnect in the characteristic time and space scales of overland and groundwater flow. However, significant difficulties remain for the development of efficient and extensible modeling systems that operate robustly across complex regions. This paper introduces the Simulator for Hydrologic Unstructured Domains (SHUD), an integrated, multiprocess, multiscale, flexible-time-step model, in which hydrologic processes are fully coupled using the finite volume method. SHUD integrates overland flow, snow accumulation/melt, evapotranspiration, subsurface flow, groundwater flow, and river routing, thus allowing physical processes in general watersheds to be realistically captured. SHUD incorporates one-dimensional unsaturated flow, two-dimensional groundwater flow, and a fully connected river channel network with hillslopes supporting overland flow and baseflow.

The paper introduces the design of SHUD, from the conceptual and mathematical description of hydrologic processes in a watershed to the model's computational structures. To demonstrate and validate the model performance, we employ three hydrologic experiments: the V-catchment experiment, Vauclin's experiment, and a model study of the Cache Creek Watershed in northern California.
Ongoing applications of the SHUD model include hydrologic analyses of hillslope to regional scales (1 m

The complexity of today's environmental issues, the multidisciplinary nature of scientific and resource management questions, and the diversity and incompleteness of available observational data have all led to the need for models as a means of synthesis. When models are computationally efficient and physically consistent, they become important tools for extrapolation across observations and systems. They help us better understand the physical history of a given system and make decisions about the future in light of socioeconomic, hydrologic, or climatological change. The datasets produced through modeling can assist with decisions for infrastructural planning, water resource management, flood protection, contamination mitigation, and other relevant concerns. Nonetheless, models of varying complexity are available, and the required model complexity (resolution, scale, and coupled states/fluxes) for a given problem depends on the particular research or management purpose, the questions to be answered, and the data available.

Nonetheless, environmental managers, policymakers, and stakeholders have a growing demand for high-resolution and detailed information about hydrologic flows at fine temporal–spatial resolution across watersheds. This need reflects the ever-increasing importance of detailed long-term predictions and projections for ecological systems, agricultural development, and food security under future climate change. Global climate modeling, typically performed with a general circulation model, also requires information on soil moisture and groundwater fluctuations, which are related to streamflow and reservoir management

In hydrology, lumped models

Communities have become more sophisticated about their choice of models

The Simulator for Hydrologic Unstructured Domains (SHUD) is a multiprocess, multiscale model where major hydrologic processes are fully coupled using the finite volume method (FVM). SHUD encapsulates the strategy for the synthesis of multi-state distributed hydrologic models using the integral representation of the underlying physical process equations and state variables. As an intellectual descendant of Penn State Integrated Hydrologic Model (PIHM), the SHUD model is a continuation of 16 years of PIHM model development in hydrology and related fields since the release of its first PIHM version

The conceptual structure of the

The family tree of PIHM and SHUD. PIHM and SHUD share the same fundamental conceptual model but use different realization. The PIHMgis and rSHUD

SHUD's design is based on a concise representation of a watershed and river basin's hydrodynamics, which allows for interactions among major physical processes operating simultaneously but with the flexibility to add or drop state-process-constitutive relations depending on the objectives of the numerical experiment.

As a distributed hydrologic model, the computational domain of the SHUD model is discretized using an unstructured triangular irregular network (e.g., Delaunay triangles) generated with constraints (geometric and parametric). A local prismatic control volume is formed by the vertical projection of the Delaunay triangles forming each layer of the model. Given a set of constraints (river network, watershed boundary, elevation, and hydraulic properties), an

The objective of this paper is to introduce the design of SHUD, from the fundamental conceptual model of hydrology to governing hydrologic equations in a watershed to computational structures describing hydrologic processes.
Section 2 describes the conceptual design and equations used in the model.
In Sect. 3, we employ three hydrologic experiments to demonstrate the simulation and capacity of the model. The three applications presented here are (1) the V-catchment experiment, (2) the Vauclin experiment

The conceptual schematic of hydrologic processes in the SHUD model.

We begin our introduction to the SHUD model with a conceptual description of water movement in a watershed. Figure

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

The hydraulic gradient is vertical within the soil column and is controlled by gravity and capillary potential. This assumption is invalid for microscale soil water movement but useful when the model grid spacing ranges from meters to kilometers

The evaporative fluxes that occur due to ET from rivers are assumed to be small and can be approximated by the evapotranspiration from the riparian or hyporheic area of the model where the shallow water table and high soil moisture conditions exist.

The hydrologic characteristics, including all physical parameters in soil, land-use, and terrain, are homogeneous within each cell. This is a typical assumption in distributed models, as these models still need discretized domains.

At present SHUD requires all geographic, topographic, and hydraulic parameters do not change in time.

Finally, SHUD uses a simplified representation of the geometry of the river networks due to the limitation of such data. This assumption is due to the inherent challenges in measuring the geometry of the river cross section everywhere along with the stream network.

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

The three layers of the SHUD model and fluxes between layers.

Figure

A depiction of the interaction between cells and the river network in
SHUD showing

Figure

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
(

The system of ODEs describing the hydrologic processes are fully coupled and solved simultaneously at each time step (

The governing equations in the SHUD model.

The flowchart demonstrates calculation of variables and time step control in the SHUD model. The hydrologic processes are simulated in each finite volume cell, then the state variables (

Figure

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

The CVODE solver determines the ITS automatically based on both the specified tolerances and the error function of

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.

Interception refers to the direct water loss of precipitation when vegetation cover exists and is treated as a simple storage

The three conditions for the interception calculation within the imaginary canopy

The interception is equal to the deficit of interception – the difference between interception capacity (

The total actual evapotranspiration (AET) consists of three parts: evaporation from interception (

Transpiration also has two potential sources: soil moisture and groundwater from the groundwater table and root depth for the land-use class. Once the groundwater table is higher than the root zone depth, vegetation uses groundwater, and soil moisture stress for transpiration is equal to one (

Water balance on the land surface is given by

Infiltration is estimated using Richards equation:

The infiltration rate is a function of soil saturation ratio (

As discussed above, the horizontal flow in the vadose zone is neglected compared to the dominant vertical flow. There are three processes controlling the water in vadose zone: infiltration (

When the bottom of the vegetation root zone is below the groundwater table, then

The water balance of groundwater is controlled by Eq. (

The horizontal groundwater flux

In Eq. (

The effective horizontal conductivity captures the effect of spatially varying conductivity on the saturated flow
when the groundwater level
rises

Effective conductivity for horizontal groundwater flow changes along the changing groundwater level. When the groundwater level is higher than macropore depth, groundwater flow increases due to the contribution of horizontal macropores.

The water balance in river channels is described by

The topological relationship between cells and river channels is shown in Fig.

The downstream channel flux

The upstream flux

The overland flow between river segment and associated hillslope cell (

The groundwater exchange between river segment and hillslope cell is described by

In this section, we present the results of using SHUD for three applications: first, we use the V-catchment experiment

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 tilted V-catchment:

Rainfall in the VC begins at time zero at a constant rate of 18 mm h

Comparison of overland flow and outflow at the outlet of the V-catchment from the SHUD modeling versus

Figure

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.

Vauclin's laboratory experiment

A schematic of

The experiment's initial condition is an equilibrium water table with a uniform hydraulic head. Irrigation was initiated at

Besides the parameters specified in

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.

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)

The location and terrestrial and hydrologic description of the Cache Creek in California. The red diamond in the map is the USGS gage station (11451100) used for calibration and validation.

Based on NLDAS-2

The monthly precipitation and temperature in Cache Creek based on NLDAS-2 data from 2000 to 2018. The blue ribbon is monthly precipitation in meters per month; the red line is monthly mean temperature, while the blue shaded region depicts the minimum and maximum temperature.

The basic data sources used to build the model domain of the Cache Creek Watershed.

Table

The unstructured domain of the CCW (Fig.

Figure

The hydrograph in Cache Creek (simulation versus observation) in the calibration (1 July 2001 to 30 June 2003) and validation periods (1 July 2003 to 30 June 2007).

Within the calibration period, Nash–Sutcliffe efficiency (NSE;

Figure

The monthly water balance trends in Cache Creek Watershed from 1 July 2001 to 30 June 2007.

We use the groundwater distribution (Fig.

The features of the SHUD model are as follows:

SHUD is a physically based process, spatially distributed catchment model. The model applies national geospatial data resources to simulate surface and subsurface flow in gaged or ungaged catchments. SHUD represents the spatial heterogeneity that influences the hydrology of the region based on national soil data and superficial geology. Several other groups have used PIHM, a SHUD ancestor to couple processes from biochemistry, reaction transport, landscape, geomorphology, limnology, and other related research areas.

SHUD is a fully coupled hydrologic model, where the conservative hydrologic fluxes are calculated within the same time step. The state variables are the height of ponding water on the land surface, soil moisture, groundwater level, and river stage, while fluxes are infiltration, overland flow, groundwater recharge, lateral groundwater flow, river discharge, and exchange between river and hillslope cells.

The global ODE system in SHUD is solved with a state-of-the-art parallel ODE solver, known as CVODE

SHUD permits adaptable temporal and spatial resolution. The spatial resolution of the model varies from centimeters to kilometers based on modeling requirements computing resources. The internal time step of the iteration is adjustable and adaptive; it can export the status of a catchment at time intervals from minutes to days. The flexible spatial and temporal resolution of the model makes it valuable for coupling with other systems.

SHUD can estimate either a long-term hydrologic yield or a single-event flood.

SHUD is an open-source model, available on GitHub

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.

SHUD implements a redesign of the calculation of water exchange between hillslope and river. The PIHM defines the river channel as adjacent to bank cells – namely, the river channel shares the edges with bank cells. This design leads to sink problems in cells that share one node with a starting river channel that can reduce simulation performance.

Although the mathematical equations underlying SHUD are generally the same as PIHM, the numerical formulation of processes, the coupling strategy, and input/output structures have been greatly enhanced. Common computations in different functions within various processes are extracted and defined as shared inline functions, maintaining calculation consistency and facilitating code updates. The elimination of redundant variables and functions also advances consistency and efficiency.

SHUD adds mass-balance control within the calculation of each layer of cells and river channels, important for accurate and efficient long-term and microscale hydrologic modeling.

The internal data structure and external input/output formats have been redesigned for efficiency and user-friendly formats supporting ASCII and binary. The binary format is particularly important for efficient writing and postprocessing over numerous model domains.

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

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

supports OpenMP parallel computation;

utilizes object-oriented programming (C++);

supports human-readable input/output files and filenames;

exposes unified functions to handle the time-series data (TSD) (in a standardized spreadsheet format), including forcing, leaf area index, roughness length, boundary conditions, and melting factor;

exports model initial condition at specific intervals that facilitate warm starts of continuous simulation;

automatically checks the range of physical parameters and forcing data;

adds a debug mode that monitors potential errors in parameters and memory operations; and

includes a series of R codes for pre- and postprocessing data, visualization, and data analysis (that will be discussed in future work).

The Simulator for Hydrologic Unstructured Domains (SHUD) is a multiprocess, multiscale, and multitemporal model that integrates major hydrologic processes and solves the physical equations with the finite volume method. The governing equations are solved within an unstructured mesh domain consisting of triangular cells. The variables used for the surface, vadose layer, groundwater, and river routing are fully coupled together with a fine time step. The SHUD uses the 1-D unsaturated flow and 2-D groundwater flow. River channels connect with hillslope via overland flow and baseflow. The model, while using distributed terrestrial characteristics (from climate, land use, soil, and geology) and preserving their heterogeneity, supports efficient performance through parallel computation.

SHUD is a robust integrated modeling system that has the potential for providing scientists with new insights into their domains of interest and will benefit the development of coupling approaches and architectures that can incorporate scientific principles. The SHUD modeling system can be used for applications in (1) hydrologic studies from hillslope scale to regional scale; area of the model domain ranges from 1 m

The source code of the SHUD model

LS contributed to conceptualization, investigation, methodology, software, validation, visualization, writing of the original draft, and editing. PU contributed to supervision, investigation, writing of the original draft, and editing. CD contributed to supervision, investigation, writing of the original draft, and editing.

Paul Ullrich is a member of the editorial board of the journal.

Author Lele Shu was supported by the California Energy Commission grant “Advanced Statistical-Dynamical Downscaling Methods and Products for California Electrical System” project (award no. EPC-16-063). Coauthor Paul Ullrich was supported by the U.S. Department of Energy Regional and Global Climate Modeling Program (RGCM) “An Integrated Evaluation of the Simulated Hydroclimate System of the Continental US” project (award no. DE-SC0016605) and the National Institute of Food and Agriculture, U.S. Department of Agriculture, California Agricultural Experiment Station hatch project accession no. 1016611.

This research has been supported by the U.S. Department of Energy, Office of Science (grant no. DE-SC0016605), and the California Energy Commission (grant no. EPC-16-063).

This paper was edited by Andrew Wickert and reviewed by two anonymous referees.