ARTICLE IN PRESS

Modeling of irrigation and related
processes with HYDRUS
Naftali Lazarovitcha,*, Isaya Kisekkab, Tobias E. Okerc,€ hlinge,f, Li Xianyueg, Li Yongh,
Giuseppe Brunettid, Thomas Woij
Todd H. Skaggs , Alex Furman , Salini Sasidharank, Iael Raij-Hoffmanb,
and Jirí Šimůnekl,*aFrench Associates Institute for Agriculture and Biotechnology of Drylands, Jacob Blaustein Institutes for Desert
Research, Ben-Gurion University of the Negev, Be’er Sheva, Israelb
Department of Land Air and Water Resources, University of California Davis, Davis, CA, United StatescCollege of Agricultural and Environmental Sciences, University of Georgia, Tifton Campus, GA,
United StatesdDepartment of Civil Engineering, University of Calabria, Rende, ItalyeTechnische Universit€at Dresden, Chair of Hydrology, Dresden, GermanyfLincoln Agritech Ltd, Environmental Research, Hamliton, New Zealandg
Inner Mongolia Agricultural University, Hohhot, Chinah
Hohai University, Nanjing, China
i
U.S. Salinity Laboratory, USDA-ARS, Riverside, CA, United StatesjTechnion, Israel Institute of Technology, IsraelkDepartment of Biological & Ecological Engineering, Oregon State University, Corvallis, OR, United StateslDepartment of Environmental Sciences, University of California, Riverside, CA, United States*Corresponding authors: e-mail address: lazarovi@bgu.ac.il; jiri.simunek@ucr.edu

Contents
1. Introduction
1.1 General introduction
1.2 HYDRUS introduction
1.3 Manuscript Objectives
2. Mathematical and physical background
2.1 Governing flow and transport equations
2.2 Initial and boundary conditions
2.3 Soil hydraulic properties
2.4 Solute transport and reaction parameters
2.5 Root water and nutrient uptake
3. Sprinkler irrigation
3.1 Existing HYDRUS applications for sprinkler irrigation
3.2 An example of a HYDRUS-2D application to sprinkler irrigation
4. Furrow irrigation
4.1 Existing HYDRUS applications to furrow irrigation
4.2 An example of the HYDRUS furrow module in multiple modelingscenarios

Advances in Agronomy
ISSN 0065-2113
https://doi.org/10.1016/bs.agron.2023.05.002

Copyright # 2023 Elsevier Inc.
All rights reserved.

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5. Drip irrigation
5.1 Existing HYDRUS applications to drip irrigation
5.2 Surface drip irrigation
5.3 Subsurface drip irrigation
5.4 Inverse problem
5.5 Geometrical description
6. Fertigation
6.1 Soil nitrogen transformation
6.2 Soil nitrogen movement
6.3 Soil fertilizer leaching
6.4 Crop root N uptake
7. Salinity and sodicity
7.1 Crop-water simulations for salinity management
7.2 Salinity under micro-irrigation
7.3 Sodic soil reclamation
7.4 Shallow groundwater
7.5 Recycled and produced water
8. Chemigation
8.1 Historical developments
8.2 Standard modules and their predecessors
8.3 Specialized add-on modules
8.4 HP1 example
9. Spatial variability
9.1 Irrigation system uniformity
9.2 Deterministic spatial variability
9.3 Stochastic spatial variability
9.4 Estimation of soil hydraulic parameters for heterogeneous soils
9.5 Challenges, recommendations, and future opportunities
10. Model calibration
10.1 Uncertainty assessment
10.2 Sensitivity analysis
10.3 Irrigation optimization
11. Special irrigation methods
11.1 Dual-drip
11.2 Pitcher
11.3 Clay pipe
11.4 Capillary wick
11.5 Capillary barrier
11.6 Ring-shaped emitters11.7 Other irrigation methods
12. Summary and outlook
Appendix
References

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ARTICLE IN PRESS
Modeling of irrigation and related processes with HYDRUS

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Abstract
Future agriculture calls for increased input (e.g., water, nutrients, pesticides) use efficiency while maintaining or improving productivity, minimizing environmental impacts,
and increasing profitability. Complete understanding of complex irrigation systems
requires laborious, time-consuming, and expensive field investigations, which invariably
involve only a limited number of treatments. On the other hand, fully calibrated
process-based models, such as HYDRUS, can quickly evaluate different irrigation management strategies without the need for labor-intensive fieldwork and have become
valuable research tools for predicting complex and interactive water flow and solutetransport processes in and below the root zone. HYDRUS codes have been used worldwide in several hundreds of studies evaluating various types of irrigation (e.g., sprinkler,
furrow, basin, and surface and subsurface drip), their scheduling (e.g., the timing ofirrigation and its amount), and solute-related factors (e.g., fertigation, chemigation,salinization, and sodification).
The objective of this manuscript is to review the current modeling capabilities of
HYDRUS to evaluate various irrigation methods and related processes. The manuscript
starts with a section describing governing flow and transport equations solved numerically by the HYDRUS codes, the corresponding initial and boundary conditions, andrelated factors such as soil hydraulic properties and root water and nutrient uptake.
Modeling of different irrigation techniques is described in subsequent sections,followed by sections dealing with solute-related topics such as fertigation, chemigation,and salinization/sodification. Topics, including the effects of spatial variability, optimization of irrigation systems, and special irrigation methods, are covered in the later sections. The manuscript emphasizes the advantages and opportunities of HYDRUS in
describing various processes in the root zone of irrigated plants that support sustainableirrigated agriculture. All the project files of the discussed examples and their descriptions
are available for download at https://www.pc-progress.com/en/Default.aspx?hyd5AdvancesInAgronomy.

1. Introduction
1.1 General introduction
Sustainably meeting the comprehensive dietary needs of each individual on
this globe is a concern that is growing alongside the world population.
The major steps needed to meet this goal include decreasing food waste,changing our diets by consuming more plant and less animal protein, andintensifying production on existing cultivated areas while reducing theenvironmental impact of production. Irrigated agriculture is the primaryuser of freshwater, accounting for nearly 85% of the total global water consumption ( Jury and Vaux Jr., 2007) and providing about 40% of total

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Naftali Lazarovitch et al.

food production (Sepulcre-Cantó et al., 2007). Irrigation water demand is
expected to increase in the future due to foreseen alterations in rainfall
caused by climate change and increased food and biofuel demands. Water
scarcity is increasing worldwide, necessitating the development of more
efficient irrigation practices.
Complete understanding of irrigation systems and corresponding
complex physical, chemical, and biological interactions in the soil-plantatmosphere system requires laborious, time-consuming, and expensive field
investigations. Such investigations invariably involve only a limited numberof treatments that only provide information about the studied systems
and conditions (e.g., for a particular climate, plants, soils, and irrigation
methods). This information then needs to be extrapolated to other relevant
conditions. Numerical models have become indispensable tools for evaluating, summarizing, and extrapolating information about vadose zone flow
and transport processes in the root zones of irrigated systems. Despite theircomplexity, the use of numerical models is increasing thanks to a better
understanding of water flow and transport processes, the development
and improvement of mathematical methods for solving governing equations, and accelerated development of computers capable of computing
different processes simultaneously for short time and small space increments(van Genuchten and Šimůnek, 2004). There is now a growing use of models
for the visualization of irrigation (Assouline, 2002; G€arden€as et al., 2005) andfertilization-related processes (Cote et al., 2003; G€arden€as et al., 2005;
Hanson et al., 2006). Fully calibrated process-based models can quickly evaluate different irrigation management strategies for different climate andsoil conditions and various crops without the need for labor-intensive
fieldwork, and they have become valuable research tools for predicting
complex and interactive water flow and solute transport processes in andbelow the root zone.

1.2 HYDRUS introduction
The HYDRUS software packages (Šimůnek et al., 2008, 2012, 2016b,
2018, 2022a,b) represent typical examples of such process-based mathematical models. These software packages numerically solve the Richards
equation for variably saturated water flow and advection-dispersion equations for heat and solute transport (and many other processes and factors)
in one-, two-, and three-dimensional variably saturated soil domains. A
sink term in the flow and transport equations accounts for plant roots’ uptakeof water and nutrients. Heat transport occurs due to conduction andARTICLE IN PRESS
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convection with flowing water. Advective-dispersive transport in theliquid phase, as well as diffusion in the gaseous phase, is considered in
the solute transport equations, which also include provisions for nonlinear
nonequilibrium reactions between the solid and liquid phases, linear
equilibrium reactions between the liquid and gaseous phases, and various
zero-order and first-order degradation reactions. In addition, physicalnonequilibrium solute transport can be accounted for by assuming a
two-region, dual-porosity type formulation, which partitions the liquid
phase into mobile and immobile regions. Attachment/detachment theory,
including filtration theory, is additionally included to enable simulationsof the transport of viruses, colloids, and/or bacteria.
The HYDRUS software packages and its various predecessors, e.g.,
SWMS-2D, CHAIN-2D, HYDRUS-1D, HYDRUS-2D, and HYDRUS(2D/3D), have a long history that goes back to the early 1990s and have beenused in several thousands of studies involving various agricultural, industrial,and environmental applications as documented in detail by Šimůnek
et al. (2008, 2012, 2016b). These applications have resulted in several hundreds of manuscripts evaluating various types of irrigation (e.g., sprinkler,
furrow, basin, and surface and subsurface drip), their management (e.g.,
the timing of irrigation and its amount), and associated solute-related factors(e.g., fertigation chemigation, salinization, and sodification) (Šimůnek
et al., 2016b).
The last comprehensive review of the HYDRUS software packages
was published in the Vadose Zone Journal in 2016 (Šimůnek et al., 2016b)
when two independent packages for one-dimensional (i.e., HYDRUS-1D,
version 4) and two- and three-dimensional (i.e., HYDRUS (2D/3D),
version 3) applications still existed. These two software packages have recently
been merged, resulting in a single HYDRUS (Version 5) software package,
consisting of an interactive graphics-based user interface (Šejna et al., 2022)
and multiple computational programs for simulating water, heat, and solute
movement in one-, two-, and three-dimensional variably saturated porous
media (Šimůnek et al., 2022a,b). Version 5 of HYDRUS (compared to its earlier versions), in addition to providing various new graphical capabilities, suchas depth-time graphs for various variables of one-dimensional simulations, also
includes several new modules that greatly expand the numerical capabilities ofthe software package, some of which are highly relevant for agricultural andirrigation applications.
While various new modules implemented into HYDRUS 5 have been
individually described in their corresponding manuscripts (see references
given below), they have not comprehensively been listed in one text.

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The Furrow module to simulate the coupled surface-subsurface waterflow and solute transport in blocked and open-ended furrows has recently beendeveloped by Brunetti et al. (2018b). This hybrid Finite Element-Finite
Volume pseudo-3D model combines the two-dimensional Richards-based
Finite-Element solver, HYDRUS-2D, for subsurface flow with a onedimensional Finite-Volume discretization of the zero-inertia and advection
equations for overland flow. The model has been successfully validated
in synthetic and experimental scenarios by Brunetti et al. (2018b). This
add-on module has been fully incorporated into the GUI of the HYDRUS 5.
The PFAS module (Silva et al., 2020) can simulate the fate andtransport of polyfluoroalkyl substances (PFAS) under dynamic vadose zone
conditions. PFAS are a class of thousands of related chemicals used fortheir water-repellent and fire-retardant qualities. They are sometimes called“forever chemicals” because they persist in the environment and accumulate
in people and animals. They have been linked to health problems, including
testicular and kidney cancers, high cholesterol, and thyroid disease. The
contamination of agricultural soils by these long-lasting toxins can be traced
to sludge from treatment plants spread over the years as fertilizer on farmlands to add nutrients. These chemicals behave differently than traditionalorganic contaminants. Rather than being present in the gas phase (a process
considered in the standard HYDRUS model), these chemicals sorb to the
air-water interface. The PFAS module thus replaces the storage term forthe air phase with the storage term for the sorption to the air-water interface.
It additionally considers the concentration effects on surface tension andviscosity.
Another new module is the Dynamic Plant Uptake (DPU) module(Brunetti et al., 2019a), which simulates the translocation and transformationof neutral compounds in the soil-plant domain. Traditionally, HYDRUS
deals only with processes in the subsurface and general plant uptake, leaving
what happens with water and chemicals once they enter plants unaccountedfor. However, plants (fruits or leaves) represent the most common pathway
for environmental contaminants to enter the human and animal food chain.
Plants can uptake chemicals from contaminated environments (e.g., byirrigation water) and bioaccumulate and metabolize them in by-products,
which can be dangerous for human health. The DPU module couples
HYDRUS with a multicompartment Dynamic Plant Uptake model ofTrapp (2007), which accounts for multiple differentiated metabolization
pathways in plant tissues, and allows one to simulate the entire pathway

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from irrigation to soil, from soil to roots, and from roots to stems, fruits,
and leaves, while considering relevant biochemical reactions in all thesecompartments.
Additional new modules in Version 5, less relevant to irrigation
applications, are COSMIC (Brunetti et al., 2019b), Particle Tracking(Zhou et al., 2021), Fumigant (Spurlock et al., 2013b), and C-Ride(Šimůnek et al., 2006). The COSMIC module (Brunetti et al., 2019b) calculates above-ground neutron fluxes using the physically based COsmic-ray
Soil Moisture Interaction Code (COSMIC) of Shuttleworth et al. (2013).
The Particle Tracking module (Zhou et al., 2021) tracks hypotheticalparticles in the subsurface, and its results can be used to calculate soil water
travel times and water age for different locations in the soil profile.
The Fumigant module considers additional options and factors related
to the transport of fumigants, i.e., chemicals used to sterilize soils before
planting crops (e.g., tarp removal, temperature-dependent tarp properties,
an additional injection of fumigants) (Spurlock et al., 2013b). Finally, theC-Ride module (Šimůnek et al., 2006) can simulate variably saturated
water flow, colloid transport, and colloid-facilitated solute transport inporous media.

1.3 Manuscript Objectives
As discussed above, HYDRUS models have been used in several hundredsof manuscripts evaluating various types of irrigation, their management, andassociated solute-related factors. Some of these older applications, mainly
furrow and drip irrigation, were reviewed by Šimůnek et al. (2016b).
The objective of this manuscript is to further expand on this older review
and to examine HYDRUS’s current modeling abilities to evaluate various
irrigation methods and related processes. The manuscript begins with asection describing governing flow and transport equations solved numerically by the HYDRUS codes, the corresponding initial and boundary conditions, and related factors such as soil hydraulic properties and root waterand nutrient uptake. Modeling of different irrigation techniques is describedin subsequent sections, followed by sections dealing with solute-related
topics, including fertigation, chemigation, and salinization/sodification.
Topics such as the effects of spatial variability, optimization of irrigation
systems, and special irrigation methods are covered in the later sections.

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The manuscript emphasizes the advantages and opportunities of HYDRUS
in describing various processes in irrigated plant root zones that support
sustainable irrigated agriculture.
The manuscript provides many examples of HYDRUS simulating
various types of irrigation and related processes. The use of various (standard
and special) boundary conditions available in the software to simulate different irrigation types is demonstrated throughout the manuscript. Different
ways of displaying the obtained results are also presented. Finally, themanuscript proposes possible directions for software development so thatit continues to be relevant in optimizing irrigation scheduling in thetwenty-first century.

2. Mathematical and physical background
2.1 Governing flow and transport equations
Water flow and solute transport in soils are described in the HYDRUS
models using the modified Richards (1) and convection-dispersion (2) equations, respectively (Šimůnek et al., 2022a,b):



∂θðhÞ
∂
A ∂h
A
 SðhÞ
¼
K ðhÞ K ij
+ K iz
∂t
∂xi
∂xi




∂ðq c Þ∂ðρb s + θc + agÞ
∂
∂c
∂
g ∂g
 i ϕ
θDwij
+
aDij
¼
∂xi
∂xj
∂xi
∂xj
∂t
∂xi

(1)
(2)

The Richards Eq. (1) combines the mass balance equation with theDarcy-Buckingham equation, and it contains the following variables: θis the volumetric water content [L3L3], h is the pressure head or matricpotential [L], t is time [T], xi is the spatial coordinate [L], K is the unsaturated
hydraulic conductivity [LT1], KAij are components of a dimensionlessanisotropy tensor KA, and S is the sink/source term [L3L3 T1], usuallyaccounting for root water uptake (transpiration). Alternative formulations
to Eq. (1) that partition the liquid phase into mobile and immobile (or lessmobile) regions (i.e., dual-porosity or dual-permeability models) are alsoavailable in the HYDRUS code to account for nonequilibrium or preferential flow (e.g., Šimůnek et al., 2003, 2008; Šimůnek and van Genuchten,
2008; Šimůnek et al., 2016b). This partial differential equation is strongly
nonlinear, and as a result, only a relatively few simplified analytical solutions
can be derived. Most practical applications require a numerical solution,

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9which can be obtained using various numerical methods such as finite
differences or finite elements.
The equation governing the transport of dissolved solutes in the vadosezone (2) is obtained by combining the solute mass balance equation withequations defining the total mass of the chemical (on the left side) andthe solute flux densities and solute reactions (on the right side). In its mostgeneral interpretation, the solute transport Eq. (2) allows chemicals to residein all three phases of the soil (i.e., gaseous, liquid, and solid) and a broad
range of transport mechanisms (including advective transport, diffusion,and hydrodynamic dispersion in both the liquid and gaseous phases), andit facilitates different types of chemical reactions that lead to losses or gains
in the total chemical concentration. Advective-dispersive transport in theliquid phase, as well as diffusion in the gaseous phase, is considered in thesolute transport equations, which also include provisions for linear andnonlinear (i.e., Freundlich, Langmuir, or Freundlich-Langmuir adsorption
isotherms), and equilibrium and nonequilibrium (i.e., one- or two-site
sorption models) reactions between the solid and liquid phases, linear equilibrium reactions between the liquid and gaseous phases (i.e., Henry’s law),and various zero-order and first-order degradation reactions.
In Eq. (2), ρb is the bulk density [ML3], a is the volumetric air content
3 3[L L ], s [MM1], c [ML3], and g [ML3] are concentrations in the solid,
liquid, and gaseous phases, respectively, Dwij and Dgij are components of theeffective dispersion tensors in the liquid and gaseous phases [L2T1], respectively, qi is the volumetric flux density [LT1], and ϕ is the rate of changeof mass per unit volume by reactions or other sources (negative) or sinks(positive) such as plant solute uptake [ML3 T1]. The PFAS module
replaces in Eq. (2) the air phase storage term (the third term on the left side)with the storage term for the sorption to the air-water interface, AeΓ, where
Ae is the air-water interfacial area ([L2L3] or [L1]), and Γ is the interfacialadsorbed concentration (i.e., mass per unit area of the interface) [ML2].
Again, alternative formulations to Eq. (2) that partition the liquid phase
into mobile and immobile (or less mobile) regions (i.e., dual-porosity or
dual-permeability models) are also available in the standard HYDRUS
code to account for physical nonequilibrium or preferential solute transport(e.g., Šimůnek et al., 2003; Šimůnek and van Genuchten, 2016). Alternative
formulations are also used in the add-on modules, such as UnsatChem and
C-Ride.
The HYDRUS models numerically solve the governing flow and transport equations, meaning that the transport domain and simulation time are

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divided into small pieces called finite elements or finite differences. Triangles
and quadrilaterals are typical spatial finite elements in two dimensions, andtetrahedral, triangular prisms, and hexahedrals in three dimensions (Šimůnek
et al., 2022a,b).

2.2 Initial and boundary conditions
The flow and transport equations can be solved, providing that applicable
initial and boundary conditions are specified. Initial conditions characterizing the system’s initial state can be specified either in terms of water contents
or pressure heads for the flow equation and in terms of (liquid-phase or total)
concentrations for the transport equation. When nonequilibrium (either
physical or chemical nonequilibrium) solute transport models are used,
the initial condition also needs to be specified for the nonequilibrium phase(i.e., concentrations in the mobile phase or at the kinetic sorption sites).
One of the main advantages of the HYDRUS codes is their flexibility indefining boundary conditions (BC), allowing simulations of various realistic
irrigation scenarios. While the codes can accommodate standard systemindependent boundary conditions, when conditions on boundaries are
known beforehand, such as the Dirichlet (prescribed boundary pressure
heads) and Neumann (prescribed boundary fluxes) boundary conditions,they can also accommodate a large number of system-dependent boundary
conditions, when conditions on boundaries are not known apriori but are
the results of interactions between external (e.g., meteorological) conditions
and conditions in (feedback from) the subsurface system.
The soil-air interface, exposed to atmospheric conditions, is an example
of a system-dependent boundary. While the potential flux across this interface is controlled exclusively by external conditions (e.g., precipitation, irrigation, evaporation), the actual flux also depends on the (transient) watercontent conditions in the soil. Soil surface boundary conditions may change
from prescribed flux to prescribed head type conditions (and vice-versa).
This occurs, for example, when the precipitation/irrigation rate exceeds
the infiltration capacity of the soil, resulting in either surface runoff oraccumulation of excess water on top of the soil surface, depending upon
the soil conditions. Similarly, the potential evaporation rate calculated frommeteorological conditions (the evaporative demand of the atmosphere) may
exceed the capability of the soil to deliver enough water toward the soil
surface. In this case, the potential evaporation rate can be significantly
reduced to an actual evaporation rate controlled by the soil.

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Other system-dependent boundary conditions, less commonly used withirrigation applications, include the seepage face BC, deep drainage BC, orflow to tile drains. Additionally, several boundary conditions implemented
in HYDRUS have been specially developed for irrigation applications.
These include, for example, subsurface drip characteristic function, surface
drip with dynamic wetting, triggered irrigation, and a reservoir BC.
2.2.1 Subsurface drip characteristic function
The infiltration rate of water from a subsurface cavity (dripper) is affected bymany factors, including the pressure in the cavity, its size and geometry, andthe hydraulic properties of the surrounding soil (Lazarovitch et al., 2005).
When a specified discharge of a subsurface source (e.g., a subsurface emitter)is larger than the soil infiltration capacity, the pressure head in the source
outlet increases and becomes positive. The built-up pressure may significantly reduce the source discharge rate. HYDRUS implements a special
system-dependent boundary condition for flow from a subsurface sourcethat uses the drip characteristic function:
Q ¼ Q0 ðhin  hs Þc(3)where hs is the pressure head at the source-soil interface [L], often called
the back pressure, Q0 [L3T1] ([L2T1] in 2D) is the nominal discharge(Optimal Flux in the HYDRUS GUI) of the source for the reference
inlet pressure hin [L] (usually being 10 m) and the back pressure equal to zero,
Q [L3T1] ([L2T1] in 2D) is the source discharge for the actual backpressure hs [L], and c [] is an empirical constant (Exponent in the
HYDRUS GUI) that reflects the flow characteristics of the emitter.
Normally, c ¼ 0.5 corresponds to a turbulent flow emitter, and c ¼ 1 to alaminar one. This equation states that when hs increases, the pressure difference between the soil and the source inlet decreases, and the discharge
rate correspondingly decreases (Lazarovitch et al., 2005).
2.2.2 Surface drip with dynamic wetting
The wetted area’s radius for drip irrigation gradually increases as irrigation
time increases. The radius of this area for transient conditions can be calculated in HYDRUS as described by G€arden€as et al. (2005). The irrigation flux
Q is initially applied to a single boundary node representing the dripper with
the Neumann (flux) boundary condition. When the pressure head required
to accommodate the specified irrigation flux Q is larger than zero, theARTICLE IN PRESS
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boundary condition in this particular node is changed to the Dirichlet (head)boundary condition with a zero pressure head value, and the actual infiltration flux Qa through this node is calculated. The excess flux (Q–Qa) isthen applied to the neighboring node, again with the specified Neumannboundary condition. This procedure is iteratively repeated until the entire
irrigation flux Q is accounted for, thus providing the radius of the wettedarea. Since the infiltration flux into the dry soil is larger at the start ofirrigation, the wetted area continuously increases as irrigation proceeds.
2.2.3 Triggered irrigation
Irrigation can be either defined externally (by specifying the irrigation rate
and the initial and final irrigation times in the HYDRUS input) or triggeredinternally by the model, depending on the conditions in the soil (e.g.,
Dabach et al., 2013). Irrigation may be triggered in HYDRUS when auser-specified pressure head is reached in a selected observation node
defined somewhere in the transport domain (likely in the root zone). The
irrigation then starts after a user-specified lag period at a user-specified
irrigation rate or pressure head at a user-specified boundary, and it lasts
for a user-specified irrigation time duration. In 1D applications, triggered
irrigation with a specified irrigation rate is applied at the soil surface, thus
likely representing sprinkler irrigation. In 2D applications, irrigation canbe triggered at the time-variable head, time-variable flux, or atmosphericboundaries, likely representing surface or subsurface drip irrigation or
sprinkler irrigation.
2.2.4 Reservoir boundary condition
The reservoir boundary condition (Šimůnek et al., 2018) allows users toconsider a reservoir that is external to the HYDRUS transport domain,
while water can be added (injected) to or removed (pumped) from thisreservoir. Flow into or out of the reservoir through its interface with thesubsurface transport domain depends on the conditions in the transportdomain (e.g., the position of the groundwater table) and external fluxes(pumping or injection):dVw
dV c¼ Qin  Qp + ðP  EÞS w ¼ Qin c in  Qp c p + PSc r
dt
dt

(4)

where Vw is the volume of water in the reservoir [L3], Qin is the water flowrate into (or out of ) the reservoir from the soil profile across the reservoir
wall [L3T1], Qp is the pumping/injection rate [L3T1], P and E are

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precipitation and evaporation rates [LT1], respectively, and S is the surface
area of the reservoir [L2]. In the latter equation, c is the solute concentrationin the reservoir [ML3], cin is the solute concentration associated withthe mass transfer between the soil profile and water reservoir [ML3](cin is equal to c when water infiltrates into the soil profile and equal to
the solute concentration in the soil profile when water exfiltrates into thereservoir), cp is the solute concentration associated with pumping or injection (equal to c for pumping or to the concentration of water being injected
into the reservoir) [ML3], and cp is the solute concentration associated with
precipitation.
Since mass balances of water and solute in the external reservoir are
constantly being updated (based on all incoming and outgoing fluxes),
the boundary conditions across the transport domain are dynamicallyadjusted depending on the water level in the reservoir. Reservoir boundary
conditions potentially have many applications, such as dynamically estimating the water level in wells (e.g., Sasidharan et al., 2018, 2019), furrows
during irrigation (e.g., Bristow et al., 2020; Šimůnek et al., 2016a), or
wetlands.

2.3 Soil hydraulic properties
The numerical solution of the Richards Eq. (1) requires knowledge ofthe unsaturated soil hydraulic properties, i.e., the retention curve θ(h) andthe hydraulic conductivity function K(h), which are, in general, highly
nonlinear functions of the pressure head. HYDRUS permits using fivedifferent analytical models for the hydraulic properties (Brooks and Corey,
1964; Durner, 1994; van Genuchten, 1980; Vogel and Cı́slerová, 1988;
Kosugi, 1996).
The soil hydraulic parameters must be either specified by users, selectedfrom a soil catalog listing them for 12 main textural classes, or estimated using
pedotransfer functions. The van Genuchten and Brooks and Corey parameters in the soil catalog were taken from Carsel and Parrish (1988) and Rawls
et al. (1982), respectively. HYDRUS users can also use pedotransfer functions (PTFs) based on neural networks (Schaap et al., 2001) to predict van
Genuchten’s (1980) parameters based on textural information.

2.4 Solute transport and reaction parameters
The advection-dispersion Eq. (2), numerically solved by the HYDRUS
software, is written in a relatively general form so that it can represent many

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different solutes, such as tracers, nitrogen species, pesticides, heavy metals,
radionuclides, and many other chemicals. Different solutes are characterized
by different transport and reaction parameters, and particular values thenrepresent a specific chemical in a given soil environment. When solutetransport is selected in the HYDRUS software, all transport and reaction
parameters are initially set for a tracer, such as a chloride or a bromide(i.e., no sorption and no reactions). It is up to HYDRUS users to enter
the proper parameters (e.g., sorption parameters, degradation parameters,
Henry’s law constant, etc.) for a particular solute. The methods for identifying these parameters are described in detail, e.g., by Skaggs et al. (2002).

2.5 Root water and nutrient uptake
The sink term, S, in Eq. (1) represents the volume of water removed per unitof time from a unit volume of soil due to plant water uptake. The macroscopic approach of Feddes et al. (1978) is used in HYDRUS to account forroot water uptake. This approach first distributes potential transpiration
across the root zone, depending on the spatial root distribution, to obtain
potential root water uptake as a function of depth. Potential root water
uptake is then reduced due to various stresses (e.g., saturation and salinity)present in the root zone to obtain actual root water uptake. Integrationof actual root water uptake over the entire root zone then gives actual
transpiration. Root water uptake can be either non-compensated orcompensated, i.e., reduced uptake from a stressed part of the root zone iscompensated by an increased uptake from a non-stressed part of the rootzone (Šimůnek and Hopmans, 2009).
The root solute uptake model implemented in HYDRUS was developed by Šimůnek and Hopmans (2009). A similar macroscopic approachas the one described above for root water uptake is also used for root
solute uptake. Root solute uptake is defined as the sum of passive and active
root solute uptakes. Passive uptake is simulated by multiplying root water
uptake by a dissolved nutrient concentration for concentration values
below a priori-defined maximum uptake concentration (cmax). The solute
dissolved in water is taken up by plant roots when cmax is large (larger thanthe dissolved concentration c), while no solute is taken up when cmax is equal
to zero. Active solute uptake is activated when passive solute uptake is insufficient to satisfy the user-specified potential active nutrient uptake rate.
Similar to root water uptake, active root solute uptake can be either
non-compensated or compensated, with the latter having increased solute

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Modeling of irrigation and related processes with HYDRUS

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uptake from parts of the root zone with higher solute concentrations compensating for insufficient uptake from parts of the root zone with low soluteconcentrations (Šimůnek and Hopmans, 2009).
The spatial extent of the root zone can be constant during the HYDRUS
simulations, or various approaches can be used to describe the root growth
as a function of time. One can use a logistic growth function or define an apriori the rooting depth at particular times. Options to simulate root growth
directly using the HYDRUS software or its various couplings with crop
growth or root growth models were reviewed by Šimůnek et al. (2018)
and Hartmann et al. (2018).

3. Sprinkler irrigation
Sprinkler irrigation is one of the main methods of irrigation usedaround the world. In the US, it is the dominant irrigation type representing
approximately 60% of all irrigation systems, with most of them being center
pivots. Of all pressurized irrigation systems, sprinkler irrigation, in comparison to drip, costs less per unit of irrigated land (Amosson et al., 2021),
particularly for large areas. There are several forms of sprinkler irrigation,
but the two main ones involve water application as droplets of varying
sizes or as thin streams of water emanating from the sprinkler nozzle. In practice, many sprinkler irrigation devices employ both techniques to varying
degrees. The irrigation efficiency of sprinkle irrigation ranges between
60% and 90%, with an average of 75% (Gilley and Watts, 1977; Waller
and Yitayew, 2016). To ensure efficient utilization of applied water, it isimportant that irrigation devices are properly installed and operated underdesign conditions and that there is a good understanding of soil properties/conditions and anticipated water flow within the profile. For the latter,the HYDRUS model (Šimůnek et al., 2008, 2016b) has been widely applied
to simulate soil water dynamics under irrigation, using various methods(e.g., drip and surface/flood) and management scenarios. This section
discusses the application of HYDRUS for sprinkler irrigation modeling.

3.1 Existing HYDRUS applications for sprinkler irrigation
Sprinklers are usually designed to apply uniform amounts of water across theentire soil surface or crop canopy. Except for variable rate irrigation systems,
in well-implemented sprinkler irrigation systems, there is less concern
about the occurrence of vastly uneven water application across a fieldARTICLE IN PRESS
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Naftali Lazarovitch et al.

compared to a system such as drip irrigation. The question of soil water
redistribution under sprinkler irrigation has been examined by studies such
as Martello et al. (2015) and Oker et al. (2021). Martello et al. (2015) used
HYDRUS (2D/3D) to simulate the effect of canopy interception and stem
flow in maize on soil moisture redistribution following sprinkler irrigation
or rainfall. Even though sprinklers are designed to apply water uniformly
across the whole surface, canopy interception distorts the default water
application pattern, significantly affecting irrigation uniformity and efficiency across a field. Oker et al. (2021) used HYDRUS (2D/3D) to simulate
soil water distribution under a Low Elevation Spray Application (LESA)
sprinkling system and compared it to that of a Mobile Drip Irrigation(MDI) nozzle package installed on a center-pivot system.
HYDRUS has also been applied to model the interactions between soil
salinity dynamics and its management using sprinkler irrigation (Phogat
et al., 2018; Wang et al., 2017b; Yang et al., 2019). In many parts of theworld, soil salinity is a growing problem (Hopmans et al., 2021). Yanget al. (2019) used HYDRUS-1D to assess the effectiveness of sprinkler
irrigation in leaching salts from the root zone of an almond orchard andcompared the results against those from drip irrigation. Their study showedthat sprinkler irrigation is more efficient than drip for leaching salts fromthe root zone. Phogat et al. (2018) used HYDRUS-2D to model waterand salinity dynamics in sprinkler-irrigated almonds exposed to varying
salinity levels at different growth stages. In this study, less saline irrigationwater was substituted for recycled irrigation water in three phenologicallydifferent almond growth stages to better understand soil water and salinity
dynamics under almonds irrigated with waters of varying salinity levels.
The study found that: (1) irrigation with saline water decreased seasonalalmond evapotranspiration by 10% and increased salinity by 54%; (2) winterrainfall, or non-saline water substitution, was insufficient to reduce soil
salinity below the 1.5 dS/m almond tolerance level and: (3) drainage beyond
a 2-m root zone was irregular and varied significantly during the growthseason.
Beyond soil water redistribution and salinity, HYDRUS has also been
applied to study other physical phenomena and interactions in the vadosezone. Naghedifar et al. (2018) used HYDRUS-1D to assess groundwater
return flow from a sprinkler irrigation system and compared it with flood
irrigation. Their study found that return flow from the sprinkler irrigation
system was negligible, whereas that from flood irrigation was 13.3%. Wanget al. (2015) coupled HYDRUS-1D with the EPIC crop model to assess

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the irrigation management in sprinkler-irrigated wheat in China. Alves deOliveira et al. (2019) used HYDRUS-2D to simulate the transport ofatrazine in the soil profile under sprinkler irrigation.

3.2 An example of a HYDRUS-2D application to sprinkler
irrigation
Center-pivot sprinkler irrigation is one of the most commonly used methodsof pressurized irrigation for field crop production, and these systems areusually fitted with sprinkler or spray nozzle packages (hereafter, the term
“sprinkler” is used to refer to both sprinkler and spray nozzles). In operation,a sprinkler mounted on a center pivot travels along a circular path acrossa field, as opposed to a stationary or fixed one. Therefore, each point of
the soil surface along the sprinkler path is subjected to a varying water
application rate as the center pivot advances towards and away from thatpoint. In contrast, the application rate of fixed sprinkler devices is relatively
constant. A unique variation of sprinkler technology also used in centerpivot systems, the LEPA bubbler (Low Energy Precision Application) (hereafter, bubbler), is designed to concentrate water application on the soil
surface area directly underneath the nozzle.
A case study of applying HYDRUS (2D/3D) to model soil water redistribution following irrigation using a sprinkler and bubbler, both mounted
on a center pivot, is presented below (the Sprinkler and Bubbler examples
in Table A). The key assumptions are: (1) both the sprinkler and the bubbler
are mounted 80 m from the pivot point, and nozzle spacing is 1.5 m;(2) sprinkler and bubbler-wetted diameters are equal to 3 and 0.762 m,
respectively; and (3) irrigation application depth is 2.54 cm, and pivot travel
time is 44 h per revolution.
For both devices, soil water redistribution up to a depth of 1.8 m is simulated. The representative application rate of the sprinkler is shown in Fig. 1.
Two soil types were considered: loamy sand and silt loam. Their soil
hydraulic properties were taken from the HYDRUS database (Carsel andParrish, 1988). The initial conditions (Fig. 2A) represent an unsaturated soil
profile, similar to usual field conditions before an irrigation event. The sprinkler irrigation is implemented in HYDRUS-2D as an “atmospheric”
boundary condition at the top of the model domain (soil surface). The sprinkler application rate, shown in Fig. 1A, is implemented using HYDRUS2D’s “Time Variable” boundary conditions. For simplicity, the graph is
discretized into 25-time steps, and corresponding application rates arecomputed. Because of approximation errors introduced by the temporal

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Naftali Lazarovitch et al.

90

20

Flow rate (cm/h)

Flow rate (cm/h)

16
12
8

80

Application rate

70

Infiltration rate - Loamy sand60

Infiltration rate - Silt loam

50
40
30
20

4

10
0
0.0

0.1

0.2

0.3

0.4

0
0.0

0.5

0.1

0.2

0.3

Time (h)Time (h)(a) Sprinkler(b) LEPA

0.4

0.5

0

0

40

40

Soil depth (cm)Soil depth (cm)Fig. 1 Application rates of a sprinkler (A) and LEPA bubbler (B), both located 80 m fromthe center of a pivot and applying an irrigation depth of 2.54 cm, with corresponding
infiltration rates for loamy sand and silt loam soils.

80
120

Silt loam
80
120
160

160
200
0.05

Loamy sand

0.15

0.25

0.35

200
0.05

0.15

0.25

Water content (cm3/cm3)

Water content (cm3/cm3)(a) Time = 0 h( b) Ti me = 24 h

0.35

Fig. 2 (A) Initial and (B) final (t ¼ 24 h) water contents in loamy sand and silt loam soils in
the HYDRUS-2D simulation of soil water redistribution following sprinkler irrigation.

discretization, the estimated irrigation depth is 3.08 cm. The bubbler is also
implemented as an atmospheric boundary condition over two lengths of
38.1 cm each, on the right and left sides of the top of the model domain.
Fig. 1B shows the application rate of the bubbler. A “free drainage” boundary condition is implemented at the bottom of the model domain (depthof 183 cm), and a “no flux” boundary condition is used one the right andleft sides. Soil water redistribution is simulated for a period of 24 h(Fig. 3). Fig. 2 shows the initial and final water contents in loamy sand

ARTICLE IN PRESS
19

0

0

40

40

80
120

0 cm
38 cm
19 cm
76 cm

160
200
0.05

0.10

0.15

0.20

Soil depth (cm)Soil depth (cm)Modeling of irrigation and related processes with HYDRUS

80
120
160
200
0.05

0.15

0.25

Water content (cm3/cm3)

Water content (cm3/cm3)(a) Loamy sand(b) Silt loam

0.35

Fig. 3 The soil moisture content at t ¼ 24 h at distances of 0, 19.1, 38.1, and 76.2 cm fromthe edge of the HYDRUS-2D model domain for loamy sand (A) and silt loam (B) following
irrigation with a LEPA bubbler. Note: The same initial water content is used for both the
LEPA and bubbler sprinkler (Fig. 2A).

and silt loam in the middle of the domain, while Fig. 3 shows the final watercontents in loamy sand and silt loam at different locations of the transportdomain.
Infiltration rates for the loamy sand and silt loam soils are plotted against
the sprinkler and bubbler application rates (Fig. 1). As shown in Fig. 1A, theinfiltration rate of the loamy sand matches the sprinkler application rate at
every time step, resulting in no surface runoff. For the bubbler, however,
the infiltration rate of the loamy sand matches the application rate up to
t ¼ 0.01 h, after which runoff is generated. For the silt loam, the infiltration
rates increase in tandem with the sprinkler and bubbler application rates
till t ¼ 0.07 and 0.01 h, respectively, after which they begin to decrease.
Subsequently, the difference between the application and infiltration rateis equivalent to potential runoff. In general, potential runoff occurs much
more quickly for the bubbler because it has a significantly higher water
application rate than the sprinkler (Fig. 1B). Figures Fig. 4A and B show
cumulative infiltration and potential runoff for the loamy sand and silt
loam soils following irrigation. As expected, the simulation results showthat silt loam has the highest potential runoff of 1.85 and 2.08 cm forsprinklers and bubblers, corresponding to losses of 60.1% and 81.9%,
respectively. The potential runoff for the LEPA application on loamy sandis 1.4 cm (55.1%).

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Naftali Lazarovitch et al.

4.0

Potential runoff (cm)Infiltration (cm)

4.0

3.0

2.0

1.0

0.0
0.0

0.1

0.2

0.3

0.4

0.5

Loamy sand - Sprinkler
Loamy sand - LEPA
Silt loam - Sprinkler
Silt loam -LEPA

3.0

2.0

1.0

0.0
0.0

0.1

0.2

0.3

0.4

Time (h)Time (h)(a) Cumulative Infiltration(b) Cumulative Runoff

0.5

Fig. 4 Cumulative infiltration (A) and potential surface runoff (B) of loamy sand, loam,
and silty clay loam for irrigation using a sprinkler and a LEPA bubbler.

In a well-drained and managed field, the potential surface runoff
would manifest as ponded water, eventually infiltrating into the soil. The
two-dimensional module of HYDRUS does not facilitate the accumulationof ponded water on the soil surface. Instead, computed surface runoff isinstantaneously removed. However, an option to accumulate ponded wateron the soil surface is available in HYDRUS-1D. If runoff is assumed to be
negligible, implying that all applied water should infiltrate into the soil
profile, then both sprinkler and bubbler irrigation can be implemented in
HYDRUS-2D as a “time-variable flux” that is lower than the saturated
hydraulic conductivity of the soil. A representative application time should
then be computed to ensure that only the desired irrigation depth is introduced into the model domain. This implies that the actual applicationof the sprinkler, or bubbler, is not considered but rather translated to a“time-variable flux.”

4. Furrow irrigation
Despite being replaced by pressurized irrigation in developed countries, surface irrigation systems, such as furrows and basins, remain widespread among farmers in developing regions (FAO, 2011). Water (andsometimes also solutes) is conveyed by gravity flow to sections of a field thatis either leveled (border-dyke irrigation) or prepared with parallel ridges andARTICLE IN PRESS
Modeling of irrigation and related processes with HYDRUS

21

furrows (furrow irrigation). In the process, water moves over dry soil,
generating a front that transports water and nutrients along the field andsimultaneously into the soil via infiltration. When correctly designed, thesesystems can simultaneously minimize deep drainage and leaching below thecrop root zone, maximize water and nutrient distribution to the ridges(where the plants are growing), and uniformly irrigate the field, thus leading
to optimal crop yields (Fahong et al., 2004; Horst et al., 2005; Šimůnek et al.,2016a; Siyal et al., 2012). Yet, the interacting surface and subsurface transport processes involved make their design and implementation a challenging
and complex task.
Ubiquitously, this problem is tackled by using models that can predict
and/or explain the system’s behavior under different operating conditions
at different levels of complexity. Setting aside data-driven models (Mattar
et al., 2015), numerical models provide a theoretical and transferrable
framework to explain and predict physical processes involved in surface
irrigation, which can be used to design and optimize these systems.
Several models have been developed in the literature to simulate the coupledsurface-subsurface transport. Earlier attempts used zero-inertia and kinematic wave approximations of the shallow water equation coupled witha modified Kostiakov function to simulate infiltration (Katopodes andStrelkoff, 1977; Oweis and Walker, 1990; Strelkoff and Katopodes, 1977;
Walker and Humpherys, 1983). Later, the advection-dispersion equation
was added to simulate solute transport (Abbasi et al., 2003b; Perea et al.,2010). Despite their good predictive performance, these models generally
oversimplify subsurface transport processes.
To overcome this limitation, analytical or numerical zero-inertia formulations were coupled with the Richards equation to mechanistically describetransport processes in the subsurface (Ebrahimian et al., 2013; Tabuada et al.,
1995; W€ohling et al., 2004; W€
ohling and Schmitz, 2007; Zerihun et al.,2005; K€ohne et al., 2011). Of particular relevance are the pseudo-3D
models developed by Tabuada et al. (1995) and W€
ohling et al. (2004), which
provide a 1D and 2D description of surface and subsurface water flow
along the channel, respectively. This approach has an important utility as
it allows assessing the irrigation uniformity in different 2D cross-sections
along the furrow. Nevertheless, while accurately simulating water flow,
neither of the above models considered solute transport in surface andsubsurface domains. Furthermore, their use of a Lagrangian approach posessignificant numerical challenges (W€ohling et al., 2006).

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Naftali Lazarovitch et al.

4.1 Existing HYDRUS applications to furrow irrigation
While the HYDRUS model (Šimůnek et al., 2016b) has been quite
widely used to simulate water flow and solute transport under furrow irrigation systems, it has typically been used only to simulate processes in thetwo-dimensional transport domain perpendicular to furrow without considering the flow and transport dynamics in the furrow itself (Bristow et al.,2020; Šimůnek et al., 2016a; Siyal et al., 2012; see also many more references
listed in (Šimůnek et al., 2016b) and on the HYDRUS website).
Recently, Brunetti et al. (2018b) developed a hybrid Finite ElementFinite Volume pseudo-3D model to simulate the coupled surfacesubsurface water flow and solute transport in blocked and open-ended
furrows. The model combines the two-dimensional Richards-based
Finite-Element solver, HYDRUS-2D, with a one-dimensional FiniteVolume discretization of the zero-inertia and advection equations. The
model has been successfully validated in synthetic and experimental scenarios. Its main advantages are its computational efficiency, mass conservation
performance, and the detailed description of water flow, solute transport,
and root water and solute uptake along the entire furrow channel. To
increase its usability and widespread use among scientists and practitioners,
the model has been equipped with a GUI and embodied an add-on modulein the newly released HYDRUS 5 software suite (https://www.pc-progress.
com/en/Default.aspx?h3d2-Furrow). Details about the model and the
GUI can be found in Brunetti et al. (2018b) and Brunetti et al. (2019c).

4.2 An example of the HYDRUS furrow module in multiple
modeling scenarios
Selected modeling scenarios are used to demonstrate the potential applicability of the HYDRUS Furrow module for fertigation designing purposes.
As pointed out by Siyal et al. (2012), there is a need to improve water,
fertilizer, and soil surface management strategies to increase irrigation efficiency and reduce nitrogen losses due to leaching into groundwater.
Most of the existing research has focused on fertilizer placement (Mailhol
et al., 2007; Siyal et al., 2012; Waddell and Weil, 2006) and irrigation
management (Abbasi et al., 2003b; Mailhol et al., 2007), while only a few
studies have investigated the effect of soil surface management to reduceleaching and maximize fertigation efficiency (Bristow et al., 2020;
Šimůnek et al., 2016a; Siyal et al., 2012). In particular, Šimůnek et al.
(2016a) used HYDRUS-2D to examine the effect of compacting theARTICLE IN PRESS
Modeling of irrigation and related processes with HYDRUS

23bottom of the furrow or placing a plastic sheet on the bottom of the furrowon the fertigation uniformity and efficiency. While results reported in thesestudies were promising, numerical simulations neglected the complex
interacting surface-subsurface processes that occur during the water
advance-storage-recession phases that are typical for furrow fertigation.
The analysis was limited to two-dimensional vertical infiltration, which
neglects the differences of the longitudinal distribution of water and solutein the field hat result from furrow fertigation. A similar approach was used,and similar limitations were encountered in the recent work of Pahlevani
et al. (2021).
Therefore, in the present study, the HYDRUS Furrow module is used
to simulate the influence of different soil management strategies on thefertigation of a 100 m-long blocked-end furrow (Table A). A non-reactive
solute is considered in the analysis. The geometric characteristics of the
furrow and input data are reported in Table 1. In particular, the effect ofthe plastic sheet (Sp) and soil compaction (Sc) is analyzed (Fig. 5) against abaseline scenario (S0). The soil hydraulic parameters of loamy sand soil
considered in this simulation are taken from the HYDRUS soil catalog(Carsel and Parrish, 1988) and reported in Table 1. For the Sc scenarios,
the saturated hydraulic conductivity of the top 2 cm of the soil is decreased
to 1/10th of the original uncompacted value. Soil compaction in the field
can be achieved by tractor wheels or with the Eversman v-shaped wheel(Siyal et al., 2012).
The subsurface vertical domain is discretized into 2049 triangularelements and 1078 nodes. The distribution of boundary conditions fordifferent soil management scenarios is shown in Figure. The groundwatertable is assumed far below the domain of interest, and thus a “Free
Drainage” BC is assigned to the bottom nodes, while an “Atmospheric”
boundary condition is set for the furrow ridge. Due to symmetry, thenodes representing the left and right boundaries of the subsurface domain
are set as “no-flow” boundaries because no water flow or solute transport
occurred across these boundaries. A hybrid Dirichlet/Neumann (variable
pressure/zero flux) BC is assigned to the border of the trapezoidal channel
to simulate variations of the water depth in the furrow. Finally, a “Third
Type” Cauchy BC was set on top and bottom of the numerical domain
to simulate the concentration flux along the boundaries. Plastic placementon the bottom of the furrow (Sp) is simulated by setting a no-flow boundary
condition at the bottom of the furrow, so that infiltration can occur only
through the sides of the furrow.

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Naftali Lazarovitch et al.

Table 1 Geometric characteristics, soil hydraulic properties, and other
input parameters used in theoretical validation.
Bottom-end boundary condition
Blocked end

Furrow length, L (m)100

Bottom width, bw (cm)16Side slope, SS ()1.2

Maximum depth, hmax (cm)15

Bed slope, S0 ()0.001

Manning’s coefficient, m (s/m1/3)

0.04

Residual water content, θr (m /m )0.057

Saturated water content, θs (m /m )0.41

Retention function shape parameter, α (1/cm)

0.124

Retention function shape parameter, n ()2.28

Saturated hydraulic conductivity, Ks (cm/s)

0.004

Tortuosity, l ()

0.5

3

3

3

3

Initial water content, θ0 (m /m )0.06

Inflow rate, Qir (l/s)

2Solute injection rate, Qinj (l/s)

0.014

Inflow concentration, Cinj (g/l)

40Longitudinal dispersivity (cm), DL

0.5

Transverse dispersivity (cm), DT

0.1

Irrigation cutoff time, tw (min)30

Solute cutoff time, ts (min)

30

3

3

Figs. 6 and 7 show the simulated distributions of soil water contents andnitrate concentration, respectively, at time t ¼ 5400 s at different locations
along the furrow for the baseline (S0), compacted (Sc), and plastic (Sp)scenarios. Simulation results show that the distribution of water contents
and solute along the furrow in the baseline scenario is not perfectly homogeneous. Water and solute mostly accumulate in the first part of the furrow,
where they percolate toward deeper soil horizons, while their penetration
into the furrow ridge decreases along the furrow. This leads to an

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Modeling of irrigation and related processes with HYDRUS

25

Fig. 5 A schematic of the boundary conditions for different soil surface management
scenarios.

Fig. 6 Simulated distributions of soil water contents at time t ¼ 5400 s at different locations along the furrow (horizontal direction) for the original (S0), compacted (Sc), andplastic (Sp) scenarios.

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Naftali Lazarovitch et al.

Fig. 7 Simulated distributions of solute concentration at time t ¼ 5400 s at different
locations along the furrow (horizontal direction) for the original (S0), compacted (Sc),and plastic (Sp) scenarios.

inhomogeneous fertigation scheme with the upper part of the furrow
receiving more water and solute and thus expected to have higher crop
yields (or larger nutrient losses by leaching) than the lower part of the
furrow, more exposed to potential root water and nutrient stresses. The
situation radically changes when the soil at the bottom of the furrow iscompacted. The reduction of the soil hydraulic conductivity diminishes
infiltration and increases velocities of the waterfront traveling in the furrow,thus leading to a more homogeneous distribution of the irrigation wateralong the field (and thus, potentially, to more uniform crop yield).
Furthermore, the lower soil hydraulic conductivity of the bottom of the
furrow increases the water level in the channel and favors the infiltration
into the ridge.
Interestingly, Fig. 6 shows that, at the end of the recession phase, the final
part of the furrow is slightly moister in the compacted (Sc) scenarios. This ismainly due to the alteration of the infiltration patterns between the sides
and the bottom of the furrow for this management strategy. Nevertheless,solute remains well distributed (Fig. 7). The placement of a plastic sheet
at the bottom of the furrow overcomes this problem and presents the greatest

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27

benefits. The irrigation and fertigation homogeneity increases, similarly aswater and solute infiltration into the furrow ridge. The amount of waterand fertilizer that leaches under the furrow bottom is lower compared toother analyzed scenarios.
The analysis shows how the HYDRUS Furrow module can be used
to design efficient fertigation strategies by providing a comprehensive andrealistic description of overland and subsurface transport processes.

5. Drip irrigation
5.1 Existing HYDRUS applications to drip irrigation
Drip irrigation is the most common method for high-value crops but is also
very common for irrigating staple food crops in water-limited environments. This method has many advantages, including optimal control over
the amount of irrigation water and the concentration of fertilizers given
to the plant, so farmers have widely adopted the method. Previous studies
have found that the same amount of water given at different timings canyield different crop yields (Or and Hanks, 1993). According to Rawlins
and Raats (1975), the high water content in the root zone causes high soil
water pressure head and hydraulic conductivity and, therefore, an increasein water availability to the plant. Stansell and Smittle (1989) argued thathigh water content in the root zone has a direct effect on the rate of transpiration and stomatal conductance, and therefore frequent irrigation will
increase the efficiency of water use, photosynthesis, and yield.
Effective design and management of drip irrigation systems lead to theeffective delivery of water and nutrients to the root zone resulting inhigher water use efficiency and improved yields. Many design and management decisions involve understanding the wetted zone pattern aroundthe emitter (Bresler, 1978; Lubana and Narda, 2001) and its relation to
the root system structure and function. Water distribution in the soil isaffected by many factors, including soil hydraulic characteristics, initialconditions, emitter discharge rate, application frequency, root characteristics, evaporation, and transpiration. HYDRUS (2D/3D) (Šimůnek
et al., 2016b) has been previously used to successfully simulate water flow
for surface and subsurface drip irrigation systems (e.g., Skaggs et al., 2004;
G€arden€as et al., 2005, Kandelous and Šimůnek, 2010; Honari et al., 2017;
Domı́nguez-Niño et al., 2020a,b; see many more references listed on the
HYDRUS website: https://www.pc-progress.com//Documents/Jirka/
Drip_irrigation_papers.pdf).

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In both surface and subsurface drip applications, irrigation is often
modeled using a constant water flux across a boundary with fixed dimensions. However, the soil’s infiltration capacity depends on the initialconditions and the soil hydraulic properties. When the prescribed boundary flux is larger than the soil’s infiltration capacity, the model will force
the water to infiltrate into the soil at a rate higher than this capacity,which will cause the pressure head at the boundary to rise to unphysicalvalues, with consequences on the flow and soil water patterns. To
overcome such problems with unphysical boundary pressure heads,
various system-dependent boundary conditions (e.g., dynamic surface
wetting for surface drip irrigation or drip characteristic function forsubsurface drip irrigation) were implemented in the HYDRUS code
(see Section 2) to facilitate proper physical characterization of the flow
field around the surface and subsurface drippers.
Another system-dependent boundary condition is used to trigger irrigation when certain predetermined conditions in the system are reached(Dabach et al., 2013; M€uller et al., 2016). This condition initiates irrigation
whenever the matric head at a predetermined observation point drops
below a certain pressure head threshold.

5.2 Surface drip irrigation
First, simulations of surface drip irrigation can be made using the boundary
condition accounting for dynamic surface wetting (G€arden€as et al., 2005).
This condition allows the spatial distribution of the discharge rate at the soil
surface to change with time. A ponded zone around this point is created
when water is applied at a constant rate to a point on the soil surface.
The ponded area increases with time at the beginning of irrigation, but later
approaches a constant circular area (Shani et al., 1987). Fig. 8 demonstrates
dynamic wetting of the soil surface for surface drip irrigation of two soils ofdifferent textures (soil hydraulic parameters for loam and sandy loam weretaken from the HYDRUS soil catalog). In this example (the SurfaceDrip
examples in Table A), 4 L were applied at the dripper discharge rate of
2 L/H using the system-dependent boundary condition accounting for
dynamic surface wetting. Fig. 8 shows different responses in the two soils.
While in the soil with higher saturated hydraulic conductivity (i.e., sandyloam), the radius reached a steady state in a relatively short time, in theheavier soil (i.e., loam), this process took much longer, showing the needfor this system-dependent boundary condition.

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Fig. 8 The radius of the saturated zone as a function of time in various soils.

In recent years, low flow rate emitters with flow rates of 0.4–0.6 L/H
have been developed, even with pressure regulation. The effect of the discharge rates on soil water flow and root water uptake by plants was studied
by Assouline (2002) using HYDRUS-2D. The simulated results showedthat lowering the emitter flow rate from 2 L/H to 0.25 L/H increased
the soil water content in the upper layer of the root zone and root water
uptake, diminishing the reduction of potential transpiration. However,these results were greatly affected by the description of the root system,
which was considered uniform between irrigations. Another study foundthat when irrigating the soil with the same amount of water at different flow
rates, no differences were found in the water content distribution after the
redistribution process (Skaggs et al., 2010). It should be noted that thisstudy was done without plants that can change the soil wetting pattern
because of the interaction between soil moisture, emitter flow rate, and rootsystem structure and function.
Further attempts to describe water flow in the soil at low flow rates using
HYDRUS (2D/3D) were made by Lazarovitch et al. (2009). These numerical experiments were performed without the presence of plants, and theconclusions are thus suitable only for the first stages of plant growth, inwhich root water uptake is minimal. The following examples illustrate
how the root structure and response to water stress influences the plant’s
ability to uptake water while surface-drip-irrigated with a continuous low
flow rate (the DripPlant examples in Table A). The simulation duration
was 1 week, and the discharge rate was 0.4 L/H (). The critical stress index(ωc), which accounts for root water uptake compensation in the root water

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Fig. 9 Relative root water uptake as a function of the critical stress index and tworooting depths.

uptake model, was set either equal to 1, i.e., no compensation, or to values
smaller than 1, accounting for root water uptake compensation (i.e., reduced
uptake from one part of the domain is compensated by increased uptake
from other parts). The maximum rooting depth, Zm, was set to 20 or
30 cm. Fig. 9 shows ratios between actual transpiration, calculated by themodel, and potential transpiration for different values of the critical stressindex and two maximum rooting depths. It can be seen that the issue ofcompensation in root water uptake is very important and has implications
for a reliable description of deficit-irrigated systems, particularly forshallow root systems. There are currently insufficient data to better understand the compensation in root water uptake, and more studies are neededin that direction.

5.3 Subsurface drip irrigation
Second, simulations of subsurface drip irrigation can be made with asystem-dependent boundary condition that describes the effect of backpressure on the dripper discharge rate while considering the source characteristics, inlet pressure, and the effects of the soil hydraulic properties(Lazarovitch et al., 2005; Naglic et al., 2014). When a predetermineddischarge of a subsurface emitter is larger than the soil infiltration capacity,
the pressure head in the source outlet increases and becomes positive. The
built-up pressure head may significantly reduce the source discharge rate.
One of the challenges in developing these models is the ability toaccurately describe the uptake of water and nutrients by roots under different growing conditions. The HYDRUS (2D/3D) model can simulate
water flow and the transport of nutrients when the space is two- or

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three-dimensional, as in drip irrigation, or one-dimensional, as in sprinkler
irrigation. This model reduces potential root water uptake due to saturation (i.e., dryness) and salinity stresses (van Genuchten, 1987) and allows
for passive and active nutrient uptake (Šimůnek and Hopmans, 2009). The
root absorption functions have a number of parameters without which
reliable simulations cannot be performed. These parameters cannot be
evaluated directly due to the inability to maintain constant water content
or nutrient concentrations in the root zone during the experiment.

5.4 Inverse problem
Another option for evaluating the parameters is to solve the inverse flow
and transport problem that includes the uptake of water and nutrients byroots (Vrugt et al., 2001a,b; Hupet et al., 2002, 2003). In this approach,the plant traits are estimated by assuming a deterministic model and calculating the parameters of the selected model. The assumption, in this case, isthat the characteristics of the plant can be described using a deterministic
model that contains a relatively small number of parameters typical ofeach plant. Estimation of plant properties is obtained by optimizing the deviations between results calculated by numerically solving the flow and transport equations and measured flow variables, such as pressure heads, watercontents, fluxes, or nutrient concentrations, at relevant times and spacelocations.
A calibrated numerical model can then be used to optimize systemvariables, leading to an optimal irrigation and fertilization regime that will
maximize profit for the farmer and preserve the environment. Questions
regarding the optimal emitter flow rate or spacing between the emitterscan be answered with the help of the numerical model.

5.5 Geometrical description
In the inverse problem method, the model has to be executed several times,
and for the high accuracy of the results, fine spatial and temporal discretization is required. For drip irrigation problems, it is almost impossible
to run the optimization problems on a regular computer. In this situation,
there is a constant attempt to find symmetry lines in the system to reducethe computational load and streamline the process. One symmetry line exists
in the center of a subsurface emitter if the soil and the root zone look similar
on both sides of the emitter. Another line of symmetry can be found at half
the distance between emitters. A line of symmetry is equivalent to a no-flow

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boundary since water contents and pressure heads are the same on both sidesof the hypothetical line of symmetry, resulting in zero pressure head gradients perpendicular to this boundary. Kandelous et al. (2011) simulated
subsurface drip irrigation as a two-dimensional (i.e., infinite line source),
axisymmetric two-dimensional (i.e., a point source), and three-dimensional(i.e., a full description of the drip line geometry) problem, and discussed
when and under what conditions each of these approaches should be used.
Fig. 10 shows the simulated results for two alternative ways of describing
water flow around a subsurface emitter. Water flow is simulated in a
fully three-dimensional domain (Fig. 10A) or a two-dimensional domain
exhibiting radial symmetry around the vertical axis (Fig. 10B). Fig. 10C
shows the soil water contents at three locations 5, 15, and 25 cm away (radially) from the dripper. These simulations (the Drip3D examples in Table A)

Fig. 10 The spatial distribution of water contents in a three-dimensional domain (A), an
axisymmetric domain (B), and at three observation points (5, 15, and 25 cm away fromthe dripper) as a function of time (C).

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depict two cycles of subsurface drip irrigation and redistribution. Initially,
there is irrigation for an entire day, and then there is a cessation of irrigationfor 4 days. It is clear that the water content response with time is almost
identical in both domains, and the line of symmetry between emitters canbe used. The simulation duration (the CPU time) for a three-dimensional
domain was 679 s, and for an axisymmetric domain, 11.8 s. This significant
time reduction (orders of magnitude) allows for faster evaluation of complex
problems involving entire growing seasons and the use of the inverse
problem to estimate parameters.

6. Fertigation
The HYDRUS software has been used over the years in many studies
evaluating the fate and transport of various chemicals used in irrigated agriculture, including nitrogen species. Due to its flexible boundary conditions
and well-designed graphical user interface, the HYDRUS model has been
widely used to capture soil nitrogen dynamics, especially for different
irrigation and fertilization strategies (Chen et al., 2020b,c; Mekala andNambi, 2017; Mekala et al., 2017; Phogat et al., 2013). It has been used,
for example, to quantify processes such as N leaching, transformation(e.g., mineralization, nitrification, denitrification, volatilization, and assimilation), and root uptake under different management conditions (e.g.,mulching, tillage), different irrigation scenarios (e.g., furrow or drip irrigation, different irrigation rates), and different planting systems (e.g., monoculture or intercropping systems). This section reviews these HYDRUS
applications.

6.1 Soil nitrogen transformation
Nitrogen, one of the most abundant elements in the atmosphere, plays a critical role in plant photosynthesis and primary production in natural terrestrial
ecosystems (Makino, 2010; Sorrell et al., 2013). In general, biogeochemicaltransformations of nitrogen in the soil-plant-atmosphere continuum mainly
include internal and external cycles. The external cycle refers to nitrogentransport among different ecosystems. The internal cycle refers to nitrogentransformations among different pools in the same ecosystem. These transformations can include mineralization (i.e., nitrogen fixation), assimilation,
nitrification, denitrification, and volatilization (Fig. 11). High crop yieldand sustainable agricultural ecosystems can be achieved only when theinternal nitrogen cycle is well understood (Subbarao et al., 2012).

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Fig. 11 Soil nitrogen transformations in the internal nitrogen cycle (adopted based on
Nelson et al., 2016).

The HYDRUS model has been widely used to simulate nitrogen transformations under different environmental conditions, such as different soil
textures, different irrigation methods, or different mulching scenarios (Azadet al., 2020; Salehi et al., 2017; Pahlevani et al., 2021; Castaldelli et al., 2018).
Apparent differences in nitrogen transformations can be found in different
scenarios and conditions. For example, the nitrification rate in sandy soils
is faster than in clay and sandy loam soils, while nitrogen assimilation in clays
is significantly higher than in the other two soils. Mekala and Nambi (2017),
Mekala et al. (2017) evaluated the impact of different soil textures on
nitrogen transformations using the HYDRUS model. They found that nitrification in sandy soil is significantly higher than in sandy loam soil. On theother hand, denitrification can become a dominant nitrogen transformation
process in sandy loam soils, especially at saturations higher than 96%. Berlin
et al. (2015) illustrated that nitrification in soils with high clay content was
lower than in soils with low clay content and that the nitrogen peak

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displayed hysteresis. The reason may be attributed to the difference insoil organic matter of different soil textures. Usually, the assimilation rate(i.e., the transformation from inorganic nitrogen to organic nitrogen)increases in response to an increase in soil organic matter, which results
in a reduction in the soil NH4-N concentration and a corresponding
nitrification rate.
Nitrification is an important process of releasing energy in the nitrogen
cycle, and it includes autotrophic nitrification and heterotrophic nitrification (Langergraber and Šimůnek, 2005). Autotrophic nitrification refers
to the process of oxidizing soil NH4-N to NO3-N by autotrophic microorganisms. Heterotrophic nitrification refers to transforming nitrogen into
oxidized nitrogen using organic carbon as a carbon source under the mediation of heterotrophic microorganisms. There are also apparent differencesin nitrification among differently irrigated fields, especially between dry andflooded fields (Eltarabily et al., 2019; Halvorson et al., 2016; Mo’allim et al.,2018). In general, denitrification is the primary transformation process inflooded fields (Becker et al., 2007), in which high saturation in the surface
soil layer results in an anaerobic soil environment. Meanwhile, anaerobic
bacteria use NO3-N as the final electron acceptor for respiration in ananaerobic soil environment to reduce NO3-N to N2O and N2.
Volatilization of nitrogen gases (N2O and N2) produced by denitrificationis the main cause of low nitrogen use efficiency in agricultural production.
However, nitrification is still stronger than denitrification in flooded fields
with good drainage. For example, Li et al. (2018a) evaluated nitrogen transformation mechanisms in different soil layers of paddy farmland using the
HYDRUS model and found that nitrification was about 25% higher thandenitrification. Similarly, nitrification is the main transformation process
in dry farming fields due to their high soil pore oxygen content. For example, Chen et al. (2020b) found using the HYDRUS model that averagenitrification in a drip-irrigated corn field reached 31.2 kg ha1. Also, thenitrification rate gradually increased with a decrease in the irrigation depth(Zeng et al., 2017). In general, soil nitrification and denitrification are
mainly affected by soil’s physical and chemical properties, e.g., soil temperature, soil moisture, and soil pH (Mekala and Nambi, 2017; Mekala et al.,2017; Herman et al., 2007; Scherger et al., 2021).
Nitrogen transformations also show apparent differences under different
mulching treatments due to differences in mulching ratios. Compared withno mulching, plastic film mulching can increase soil temperature, promote
the activity of soil microorganisms, accelerate the nitrogen denitrification

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rate, and increase N2O emissions (Lee et al., 2022). Furthermore, thenitrification rate increases in response to a decrease in the mulching ratio(Chen et al., 2020b).
The HYDRUS model is also often used to evaluate the mechanismof nitrogen transformation in fields with different tillage technologies(Colombani et al., 2020; Shafeeq et al., 2020; Miranda-Velez et al.,2022). According to simulation results, N2O emissions in the no-till field
were significantly limited compared with the conventional tillage field(Yoo et al., 2016; Tellez-Rio et al., 2015; Chatskikh et al., 2008). This phenomenon is mainly related to different soil structures under different tillage
technologies. Conventional tillage technology can decrease soil porosity
while significantly increasing oxygen content in soil pores. Thus, thenitration rate can be promoted in conventional tillage fields. Compared witha conventional tillage field, the surface soil bulk density increased due tosoil compaction in the no-tillage field, resulting in a reduction in N2Oemissions.
Mineralization can also be accurately quantified using the HYDRUS
model. For example, Shekhar et al. (2021) compared mineralization rates
under different soil moisture conditions and found that mineralization undermoderate and severe water stress treatments decreased by 44.9% and 36.5%
compared to a no-stress treatment, respectively. Organic macromolecularcompounds (such as proteins, amino sugars, and nucleic acids) aredecomposed into biomonomers via hydrolysis of microbial enzymes, andthe biomonomers are then further ammoniated to ammonium nitrogen.
Therefore, soil microbial activity is an important factor affecting nitrogenmineralization. In the study of Shekhar et al. (2021), the transformationof soil organic nitrogen to inorganic nitrogen was inhibited due to low soil
microbial activity under water stress conditions.

6.2 Soil nitrogen movement
It is crucial to accurately capture the nitrogen transport to formulate reasonable fertilization application strategies, improve nitrogen use efficiency,
avoid non-point source pollution, and promote sustainable agriculturaldevelopment (e.g., Mekala and Nambi, 2017). Iqbal et al. (2016) compared the distribution characteristics of soil nitrogen in differently sized
furrow-irrigated fields and found that soil nitrogen accumulation significantly increased in the field with a 2:1 ridge-ditch ratio compared to the

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1:1 ridge-ditch ratio. Azad et al. (2018) evaluated differences in soil nitrogendynamics under different irrigation rates and illustrated that the leachingamount of soil NO3-N could be effectively decreased during the fertilization application period when the irrigation rate is 0.8 L h1. Tao et al.
(2021) studied the effect of an improved underground drainage systemon soil NH4-N and soil NO3-N dynamics using HYDRUS. They
reported that the largest decrease in soil nitrogen mass consistently
occurred in the filter profile, and soil NH4-N in the 0–20 cm soil layerand soil NO3-N in the 60–80 cm soil layer increased with the prolongationof irrigation duration. Karandish and Šimůnek (2017) evaluated the effectsof 176 different water and nitrogen management strategies on soil waterand nitrogen dynamics and showed that the 200 kg ha1 N-fertilizer application is the optimal amount, which can reduce nitrogen leaching by
12–99%.
The HYDRUS model has also been commonly used to evaluate soil
nitrogen transport under different planting systems. For instance, Chenet al. (2020c) revealed the distribution characteristics of soil nitrogen in
the corn-tomato intercropping field and found that soil NO3-N concentrations in the tomato’s root zone were significantly higher than in the corn’sroot zone. Moreover, the soil solute flux in the horizontal direction
occurred mainly from the tomato’s root zone to the corn’s root zone.
Doltra and Muñoz (2010) studied the soil NO3-N distribution in the sweet
pepper-cauliflower-Swiss chard rotation field and illustrated that the surface
soil NO3-N concentration first increased and then decreased with the crop
growth.
Since the HYDRUS model has flexible upper boundary conditions torepresent different mulching scenarios, the model can accurately capture
the soil nitrogen dynamics under different mulching scenarios. Chenet al. (2020b) compared differences in the soil nitrogen distribution underbiodegradable film mulching (BM), polyethylene film mulching (PM),and non-mulching (NM) scenarios. They showed no significant differencesin soil NO3-N concentrations under BM and PM in the early crop growthstage. However, soil NO3-N concentrations under BM were higher thanunder PM 80 days after sowing due to an increase in the disintegrated areaof the biodegradable film. Overall, the HYDRUS model can capture thesoil nitrogen dynamics in different crop growth stages at the field scale
and accurately quantify the nitrogen distribution characteristics for a longtime series in large-scale regions (Nasta et al., 2021).

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6.3 Soil fertilizer leaching
Fertilizers have been applied in large quantities as one of the effective ways toincrease crop yields. However, irrational fertilizer applications cause waste
and lead to non-point source pollution of agricultural soils (Guo et al.,2010). Nitrogen leaching can significantly damage surrounding environments, including surface and groundwater bodies, affecting ecologicalsecurity (Bouraoui and Grizzetti, 2011). Recently, many researchers
have used the HYDRUS model to carry out high-precision simulationsof N transport and transformations under different soil textures andagronomic measures (Pare et al., 2006; Zhou et al., 2006), providing effective guidance and helping to optimize fertilization schemes and enhance
the N use efficiency of crops under different conditions to ensure
food-ecological security (Azad et al., 2018; Guo et al., 2013; Siyal
et al., 2012).
Most previous studies have applied HYDRUS models to simulate
water-N dynamics under specific soil conditions (Clement et al., 2021;
Miranda-Velez et al., 2022; Singh et al., 2019). Soil texture varies significantly from region to region and is an important factor that directly affects
the water-N transport processes and the amount of N leaching from thesoil. Bi et al. (2003) used the HYDRUS model to simulate the verticaltransport of water and NO3-N in two soils (yellow moist soil and eolian
sandy soil) in the Huang-Huai-Hai Plain of China. Based on this,Rees et al. (2020) constructed six reclaimed coarse-textured soil profiles
consisting of two cover soils (peat-mineral mix and forest floor mineralmix) and simulated nutrient leaching from these six profiles, using the simplicity of HYDRUS customizable soil profile properties of each layer.
Results revealed that the peat-mineral mix retained 44% of the initial
inorganic N within the surface 20 cm of the reclaimed soil profiles after
heavy rainfall, while 84% of the inorganic N was leached from the forest
floor mineral mix.
Taking advantage of the HYDRUS models’ ability to customize
soil profiles using different soil characteristics, soil N leaching under different long-term conditions could be simulated to provide data to support
the development of the best irrigation and fertilization scenarios. Azadet al. (2018) used a two-stage optimization approach and a calibrated
HYDRUS model to develop an optimal water and fertilizer management
program for maize fields for the soil reclaimed for growing crops. Azadet al. (2018) simultaneously optimized the irrigation flow rate, application
starting time, and fertilizer application duration to minimize NO3-N

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leaching in a fertigation cycle. Correspondingly, the fertilizer amount at
each crop growth stage was optimized throughout the growth season bycomparing multi-scheme simulations based on the optimized irrigation rate.
HYDRUS-simulated results provided insight into how cover crops affect N
leaching losses, helping optimize the design of a soil cropping system tominimize N losses.
Anthropogenic agronomic measures significantly affect root zone
leaching of soil NO3-N (Arbat et al., 2013; Shaygan et al., 2018a,b).
Shafeeq et al. (2020) investigated the effects of several conservation
agriculture practices (conventional tillage, permanent broad beds (PBB),
zero tillage (ZT), PBB with residues, and ZT with residues) on thespatial-temporal distribution of soil moisture and NO3-N during the wheat
growth stages, inversely optimized the relevant reaction parameters (i.e.,
nitrification rates, and the distribution coefficient), and predicted daily
variations of NO3-N in the soil profile. Based on the HYDRUS simulation
results, this study recommended using PBB with residues for wheat cultivation in the maize-wheat cropping system to improve water and N use
efficiency and reduce water and N losses.
To address the problem of water and nutrient leaching below the rootzone in light-textured soils due to the position of subsurface drip lines
and/or poor management of the irrigation system, El-Nesr et al. (2014) usedthe HYDRUS model to evaluate the effectiveness of three anti-leaching
techniques (a physical barrier, a dual-drip system with concurrent irrigation,a dual-drip system with sequential irrigation) applied in bare and vegetatedsoils. Results showed that the dual-drip system with sequential or concurrent irrigation is very efficient in limiting the downward leaching of
nutrients and that these systems can be applied in actual production to reduce
the risk of environmental pollution.
Since current regional-scale groundwater models, such as MODFLOW,
tend to simulate the fate of solutes in the saturated zone by greatly simplifying the near-surface hydrologic processes, their simulations often lead topoor results. Lyu et al. (2019) coupled the one-dimensional vadose zone
flow model HYDRUS-1D and the groundwater model MODFLOW toprovide a feasible method for simulating the fate and transport of salt
and N in the subsurface and for assessing the long-term effects of irrigation
with reclaimed water. The HYDRUS model can be coupled with othermodels to simulate various biogeochemical processes under different conditions, providing an effective tool for improving simulation accuracy andrationality.

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Fig. 12 demonstrates soil NO3-N leaching in sandy loam for surface drip
irrigation with three different irrigation depths (15, 22.5, and 30 mm; the N
application duration of 5 days), three different durations of fertilizer application (5, 10, and 15 days; the irrigation depth of 30 mm), and the same
N-fertilizer application (2.1 mg cm2) (the NLeach example in Table A).
In this example, the longitudinal dispersivity (DL) was considered to be
50 cm within the 0–100 cm soil layer, and the transverse dispersivity (DT)was assumed to be one-tenth of DL (Cote et al., 2003). The molecular
diffusion coefficients of NH4-N and NO3-N in free water were set to be
0.064 and 0.068 cm2 h1, respectively (Cote et al., 2003; Nakamura
et al., 2004). The distribution coefficient Kd for NH4-N and NO3-N wereassumed to be 3.5 and 0 cm3g1, respectively (Hanson et al., 2006). This
study also assumed that NH4-N transforms directly into NO3-N sincenitrification of NO2-N to NO3-N is relatively fast. The soil bulk density
is 1.5 g cm3, and the nitrification rate (transformation of NH4-N to
NO3-N) in the liquid and solid phases in the sandy loam were set to 0.12
and 0.28 d1, respectively (Castaldelli et al., 2018). These values leadto an approximate retardation factor for NH4-N of about 22.9 and ahalf-life of about 3–6 d. NH4-N was assumed to be initially present in
the top 10 cm of the soil profile and additionally applied with the irrigationwater. Both NH4-N and NO3-N were assumed to be passively taken up by
plant roots (rooting depth of 80 cm and rooting radius of 40 cm). The
third-type Cauchy boundary condition was used for solute transport along
all boundaries with specified water fluxes (at the top boundary), while
the second-type Neumann boundary condition was used at the bottom

Fig. 12 Actual and cumulative NO3-N leaching at a depth of 40 cm in scenarios with(A) different irrigation depths (I ¼ 15, 22.5, and 30 mm; the N application duration of
5 days) and (B) different durations of the fertilizer application (D ¼ 5, 10, and 15 days,
the irrigation depth of 30 mm) and the same N-fertilizer application of 2.1 mg cm2.

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41boundary. In the HYDRUS model, the N-fertilizer application mass equals
the product of the irrigation water volume (L3) and the solute concentration(M L3) in irrigation water. Thus, the solute concentration of irrigationwater (input value) was determined by dividing the N-fertilizer application
mass by the irrigation water volume. In this example, the first topdressing
mass and the irrigation water depth are 0.63 mg cm2 and 3 cm3 cm2,and thus the solute concentration of irrigation water is 0.21 mg cm3.
Since NH4-N was greatly retarded in the soil profile, Fig. 12 only shows
actual and cumulative solute fluxes for NO3-N for different scenarios.
The simulation results showed that soil nitrogen leaching increased in
response to an increase in the irrigation depth, while it decreased with anincrease in the duration of the fertilizer application.

6.4 Crop root N uptake
Optimization of N fertilizer applications is needed to prevent the negative
impacts of these fertilizers on the environment. Such optimization requiresconsidering complex interactions between root water and N uptake andtheir implications for crop production (Azad et al., 2020; Groenveld
et al., 2021). The transfer of N from the soil solution into the plant canbe described using active or passive mechanisms, which refer to the massflows of N into roots with water taken up for transpiration or against energy
gradients, respectively (Groenveld et al., 2021). The passive and active
root N uptake functions implemented in the HYDRUS models have been
widely used to optimize N fertilizer applications in agricultural production.
All principles and parameter settings of passive/active root nutrient
uptake implemented in HYDRUS are described in detail by Šimůnek
and Hopmans (2009) and in the model’s technical and user manuals(Šimůnek et al., 2022a,b; Šejna et al., 2022). Root passive N uptake issimulated by multiplying water uptake (compensated or uncompensated)by the dissolved N concentration for concentration values below an apriori-defined maximum concentration (cmax, the maximum allowed solution concentration for uptake). All nutrients dissolved in water are taken up
by plant roots when cmax is large (larger than the dissolved concentration c),while no nutrients are taken up when cmax ¼ 0, with only active uptake
remaining in this case. The maximum solution concentration for passiveroot uptake, cmax, thus controls the relative proportion of passive root water
uptake to total uptake. Using this flexible formulation, uptake mechanisms
can vary between specific nutrients. For example, Na uptake can be

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excluded by setting cmax ¼ 0, passive Ca uptake can be limited by defining afinite cmax value, or all of the soil solution available P or N can be allowed tobe taken up passively by setting cmax to a very large value.
Ranjbar et al. (2019) calibrated and validated the HYDRUS model
parameters for root passive N uptake using data from two maize experiments
over the 2015 and 2016 growing seasons. Simulations showed thatHYDRUS was reasonably accurate in estimating total N uptake at harvest,
although it could not simulate cumulative N uptake during various growthstages with high accuracy. The authors hypothesized that some neglected
processes, such as active N uptake, caused an underestimation of the N
uptake rate during the vegetative stage when the corn reached its maximum
growth rate. Azad et al. (2019) used the HYDRUS model (water-solutetransport, root water/N uptake, and root growth) to simulate N uptake
in a sandy clay/sandy loam soils under a surface drip irrigation system andcalibrated various parameters to optimize nitrogen use efficiency. Azadet al. (2019) compared root N uptake for two fertilizer application frequencies with the maize N requirements during various maize growth stages.
Results showed that reducing the number of irrigation fertilizer applications
in sandy clay loam soils increased the NO3-N uptake by plants. However,fewer irrigation-fertilization events in sandy loam soils decreased theNO3-N uptake.
The HYDRUS model has also yielded meaningful results about active
root N uptake, another important process. When optimizing N fertigation,
Groenveld et al. (2021) considered the effects of water and N stress on theplant size and corresponding feedback on potential transpiration and N
uptake. To accomplish this, Groenveld et al. (2021) modified HYDRUS
to consider the reduction of potential transpiration due to N uptake limitation. The model was then calibrated and validated using a dataset consistingof three NO3-N concentrations and six irrigation levels, i.e., using 18 treatments to fertilize cucumber plants grown in the evaporator. The model was
validated by testing its ability to accurately reduce potential N uptake andtranspiration in the presence of water and N deficiency.
Li et al. (2015c) observed and simulated (using HYDRUS-1D) N transport and transformations in an experimental direct-seeded-rice (DSR) fieldin the Taihu Lake Basin of eastern China for two consecutive seasons.
The calibrated and validated HYDRUS model indicated that NH4-N
was the predominant form (74% of N uptake on average) for crop N uptake,and NH4-N uptake represented 32.1% and 30.8% of total N input over the

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2008 and 2009 seasons, respectively. Similarly, Sharmiladevi and Ravikumar(2021) also improved the accuracy of root N uptake simulation by the
HYDRUS model through intensive field experimentation, where various
fluxes of urea, NH4-N, and NO3-N under drip irrigation were quantified
to develop the N fertilizer application schedule for the entire rice growth
period. In soil improvement, Mazloomi and Jalali (2019) studied NH4-N
uptake and transport in sandy loam soils for different application rates(2%, 4%, and 8%) of vermiculite, nano clay, and zeolite amendments.
The HYDRUS model indicated that the application of inorganic amendments to sandy loam soils reduced NH4-N transport and increased theeffective utilization of N fertilizer in agricultural systems. HYDRUS simulations helped to identify the best fertigation management to reduce N
leaching and increase root uptake. It is necessary to investigate different
fertigation strategies for different soils and growing seasons to increase theefficiency of N uptake by plants.
The example used above to assess N leaching (the NLeach example in
Table A) also provides information about root NH4-N and NO3-N uptake
for different simulated scenarios, i.e., different irrigation depths (15, 22.5,
and 30 mm) and different durations of fertilizer application (5, 10, and15 days) with the same N-fertilizer application (2.1 mg cm2) (Fig. 13).
The maximum allowed solution concentration (cmax) for passive uptake
for NH4-N and NO3-N were set as 1.0 and 0.4 mg cm3, respectively(Chen et al., 2020b). In general, root NO3-N uptake was significantly higher
than NH4-N uptake in different scenarios during the simulation period since
NH4-N was relatively quickly nitrified into NO3-N. Moreover, in different
irrigation depth scenarios, root NO3-N or NH4-N uptake increased in
response to an increase in the irrigation depth. However, an interesting
result was obtained in the scenario with different durations of the fertilizer
application. The highest root NO3-N or NH4-N uptake occurred forthe application duration of 10 d, which showed that soil nitrogen could
be easily taken up by plant roots in this scenario.
Another example (the NUptake example in Table A) is used to demonstrate the modeling of active and passive crop nitrogen uptake and sprinkler
irrigation. This example includes three different N-fertilizer applicationlevels (2.8, 2.1, and 1.4 mg cm2) and three different soil textures (sandy,
sandy loam, and loam). In this example, the transformation of NH4-N
to NO3-N was neglected, and NO3-N was directly applied with theirrigation water. The dispersivity (D) was considered to be 10 cm, theARTICLE IN PRESS
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Naftali Lazarovitch et al.

0.0

a)
0.04

0.5

0.03

1.0

0.02

1.5

0.01

2.0

I_30
0.00

0

10

I_22.520

I_15
30

Root uptake for NO3-N (mg cm-2)

Root uptake for NH4-N (mg cm-2)

0.05

2.5
40

DAS
0.0

b)
0.04

0.5

0.03

1.0

0.02

1.5

0.01

2.0

D_5
0.00

0

10

D_10
20

D_15
30

2.5
40

Root uptake for NO3-N (mg cm-2)

Root uptake for NH4-N (mg cm-2)

0.05

DAS
Fig. 13 Root NH4-N and NO3-N uptake in scenarios with (A) different irrigation depths(I ¼ 15, 22.5, and 30 mm, with an N application duration of 5 days) and (B) different durations of the fertilizer application (D ¼ 5, 10, and 15 days, with an irrigation depth of
30 mm) and the same N-fertilizer application of 2.1 mg cm2.

maximum allowed solution concentration (cmax) for passive uptake
was set as 0.4 mg cm3, and the potential active solute uptake rate wasset as 0.1 mg cm2 day1. The Michaelis-Menten constant was set to
0.5 mg cm3. It was also assumed that potential solute uptake was reduced
due to reduced root water uptake. A concentration flux boundary condition
was set at the top and bottom boundary, and 20% of the N-fertilizer application was assumed to be initially present in the top 10 cm of the soil profile.

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Root nitrogen uptake rate (mg cm-2 d-1)

0.08

2.5

a)

NF_2.8

NF_2.1

NF_1.4
2.0

0.06
1.5
1.0

0.04

0.5
0.02
0.0
0.00

0

20

40

60

80

100

120

Cum. Root nitrogen uptake (mg cm-2)

45

Modeling of irrigation and related processes with HYDRUS(0.5)

Root nitrogen uptake rate (mg cm-2 d-1)

0.08

4.0

b)

Sand

Sandy loam

loam
3.0

0.06

2.0
0.04
1.0
0.02

0.00

0.0

0

20

40

60

80

100

120

Cum. Root nitrogen uptake (mg cm-2)

DAS

(1.0)

DAS

Fig. 14 Actual and cumulative root nitrogen uptake rate in scenarios with (A) different
N-fertilizer application levels (2.8, 2.1, and 1.4 mg cm2; sand) and (B) different soil
textures (sand, sandy loam, and loam, with an N application of 2.8 mg cm2).

The results showed that the crop nitrogen uptake rate increased
after fertilization applications, while the cumulative crop nitrogen uptake
decreased in response to a decrease in nitrogen-fertilizer application levels(Fig. 14). Additionally, an apparent difference in the crop nitrogen uptake
rate was also found for different soil textures. In general, cumulative crop
nitrogen uptake is different in different soil textures, and it increases fromsand to sandy loam and loam.

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7. Salinity and sodicity
A major challenge for irrigated agriculture, particularly in arid andsemi-arid regions, is irrigation-induced salinization of land and water
resources (U. S. Salinity Laboratory Staff, 1954; Shainberg and Shalhevet,
1984; Wallender and Tanji, 2012). Many irrigation projects, both ancientand modern, have struggled or failed over time due to the degradationof soils and waters ( Jacobsen and Adams, 1958; Ghassemi et al., 1995;
Hopmans et al., 2021). The pervasiveness of soil salinity and sodicity inirrigated agriculture is such that salt management has been identified as a
key motivation for early fundamental studies in soil science (Letey, 1984).
HYDRUS was initially developed at the USDA-ARS U.S. Salinity
Laboratory (Šimůnek et al., 2008), and it has always had strong capabilities
for modeling salinity under irrigation. HYDRUS has been widely used toevaluate the impacts of irrigation and salinity management on soil and water
quality and crop yields.
Two approaches are available to model irrigation in salt-affected
lands. The simpler method is to evaluate soil salinity in terms of the total
salt concentration of the soil solution. The concentration c in Eq. (2) istreated as a non-reactive tracer, equal to the total aqueous salt concentration. With this approach, it is possible to simulate salt additions, accumulation, and leaching in the soil profile, as well as to evaluate the impactof salinity on crop productivity via the HYDRUS root water uptake
features.
More detailed assessments of irrigation and salinity are possible using the
HYDRUS UNSATCHEM module (Šimůnek and Suarez, 1993; Suarez
and Šimůnek, 1993; Šimůnek and Suarez, 1994; Šimůnek et al., 1996;
Gonçalves et al., 2006; Ramos et al., 2011). The module includes capabilities for simulating multiphase transport of CO2, transport of major ions
(Ca2+, Mg2+, Na+, K+, SO24 , and Cl ), major ion chemistry including
complexation, cation exchange, precipitation and dissolution (e.g., ofcalcite, gypsum, and dolomite), and water quality effects on hydraulicconductivity.
A search of “HYDRUS” and “salinity” on Google Scholar locateshundreds of journal articles and other texts documenting HYDRUS applications to soil salinity problems using both approaches discussed above. Here
we offer a selective overview of representative projects.

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7.1 Crop-water simulations for salinity management
All irrigation waters contain some amount of salt. As crops grow and extract
water from the soil, salt is left behind in the root zone, where it willaccumulate over time unless the applied irrigation and/or rain exceeds cropwater requirements by an amount that is sufficient to flush salts into thedeeper subsurface and maintain the rootzone salt balance. The fraction of
applied water that passes below the root zone is known as the leaching
fraction (LF); the LF required to keep root zone salinity at acceptable levels
for a given crop is called the leaching requirement (LR). While LF andLR are valuable management concepts, their utility in operational management has been questioned (e.g., Corwin et al., 2007; Letey et al., 2011;
Oster et al., 2012). Traditional LR analyses have been criticized for overestimating the impact of salinity on crop growth, thereby encouraging
over-irrigation. Limitations identified in LR analyses include steady-stateand one-dimensional flow assumptions and neglect of salt precipitationand dissolution reactions.
An alternative is to simulate a growing season(s) using HYDRUS, evaluating the impacts of water quality and irrigation practices on crop water
uptake using the HYDRUS solute transport and root water uptake features.
Among others, Corwin et al. (2007) found that differences in leachingrequirements derived from transient and steady-state models were relatively minor, whereas accounting for salt precipitation led to significantly
different recommendations. Similarly, Oster et al. (2012) found that withhigher salinity irrigation waters, yields predicted using UNSATCHEM
were greater than yields predicted with other models due to
UNSATCHEM’s ability to account for salt precipitation, as well as thenonlinear relationship between salt concentration and osmotic potential.
As an illustration, consider the following scenario, which is based on
one assessed by Oster et al. (2012) and Skaggs et al. (2014). Forage corn
is grown over a 20-week season. Multiple seasons having the sametime-varying atmospheric boundary conditions are simulated back-toback. By the final season, the simulated system has approached aquasi-steady state with respect to seasonal water uptake and dynamics.
Four model setups, each for two irrigation regimes, are evaluated(the Salinity examples in Table A). The first is a HYDRUS simulation that
does not include salinity. Second is a HYDRUS simulation with an irrigation water salinity of 3 dS/m. The last two are UNSATCHEM simulations

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Table 2 Computed seasonal relative yields (RY) and leaching fractions (LF).
Model
RY (%)
LF (%)—————— Irr ¼ 0.9 Tp ——————HYDRUS, no salinity

900

HYDRUS, ECiw ¼ 3 dS/m

77

14

UNSATCHEM, ECiw  3 dS/m,chloride

79

12

UNSATCHEM, ECiw  3 dS/m,
sulfate828—————— Irr ¼ 1.1 Tp ——————HYDRUS, no salinity

100

9

HYDRUS, ECiw ¼ 3 dS/m

88

20

UNSATCHEM, ECiw  3 dS/m,chloride90

19

UNSATCHEM, ECiw  3 dS/m, 93
sulfate15

ECiw ¼ irrigation water salinity; Irr ¼ seasonal irrigation; Tp ¼ seasonal potential transpiration.

with irrigation water salinities of  3 dS/m but differing in chemical compositions, one a chloride-dominated water and the other having higher
levels of sulfate and alkalinity.
Table 2 gives model results for two simulated irrigation regimes, one a
deficit irrigation scheme in which seasonal applied water (Irr) is less than
the seasonal potential transpiration (Tp), Irr/Tp ¼ 0.9, and the other in whichapplied water is in excess of potential transpiration, Irr/Tp ¼ 1.1. The presented results are the final seasonal leaching fraction (LF), evaluated as theratio of cumulative water flux at the bottom of the root zone to the seasonal
applied water, and the final seasonal relative yield, evaluated as the ratio of
seasonal actual and potential transpiration.
The results in Table 2 demonstrate that relationships between applied
water, water quality, leaching fraction, and crop yield are straightforward
only when salinity is neglected (as is done with classical LF and LR assessments). Among the tested models featuring salinity, a range of outcomes
was obtained. The UNSATCHEM model with chloride water generated
higher yields and lower leaching fractions than the HYDRUS model withsalinity because, in the former simulation, calcite precipitation in the upperARTICLE IN PRESS
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part of the root zone removed salt from the solution and reduced the effective root zone salinity. Similarly, the UNSATCHEM simulation withhigher sulfate and alkalinity produced the highest yields and lowest leaching
because significant amounts of calcite and gypsum were precipitated in
the root zone. If the simulations were done with higher irrigation watersalinities, the differences between model outcomes would become more
pronounced. Fig. 15 illustrates the root zone salinity profiles at the midpointof the final simulated season for each model.

7.2 Salinity under micro-irrigation
Micro-irrigation systems (drip, micro-sprinkler, etc.) create relatively complex, spatially varying patterns of water and salinity in irrigated soils (e.g.,
Burt and Isbell, 2005). HYDRUS (2D/3D) has been used in many studies
investigating leaching and water requirements in micro-irrigated systems.
Among others, Hanson et al. (2008) showed that a conventional or waterbalance approach for estimating leaching fractions predicted little or noleaching when applied water was less than the potential evapotranspiration,
whereas field data and HYDRUS modeling showed considerable leachingaround the drip lines. Raij et al. (2016) used HYDRUS (2D/3D) and
UNSATCHEM to study spatial leaching patterns in soil lysimeters andFig. 15 Rootzone salinity profiles at the midpoint of the growing season for two irrigation levels (Irr ¼ 0.9 Tp and 1.1 Tp) and three modeling scenarios (the standard HYDRUS
and the UNSATCHEM model with chloride- and sulfate-dominated water). The plot on
the right shows the root uptake density distribution.

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Naftali Lazarovitch et al.

different methods for evaluating leaching fractions. Yang et al. (2019) compared leaching in 1D (sprinkler-irrigated) and 2D (drip-irrigated) systems,
finding that leaching fractions were higher under drip than under sprinkler,
and that sprinkler was a more efficient means for maintaining salt balance.

7.3 Sodic soil reclamation
Irrigated soils (particularly heavy-textured clay soils) with a high exchangeable sodium percentage usually possess undesirable physical characteristics
such as low permeability and surface crusting. Reclamation of sodic soils
generally involves replacing sodium on the exchange surfaces with calcium.
Typically, calcium is introduced to the soil through the addition of amendments such as gypsum, although native soil minerals can also be a source.
Standard formulas exist for calculating the required calcium or amendment
needed to reduce the exchangeable sodium percentage to desirable levels,
typically less than 10% (e.g., James et al., 1982; Loveday, 1984). Sodic soil
reclamation is relatively complicated owing to the interactions of sodicity,salinity, and permeability; one must take care not to worsen an infiltration
problem by introducing an amendment solution with a low electrolyte concentration. Further, sodic soil reclamation can be expensive due to the costof amendments.
Many researchers have used UNSATCHEM to study sodic soil reclamation with the goal of improving efficiency and reducing costs (e.g., Šimůnek
and Suarez, 1997; Kaledhonkar et al., 2006; Reading et al., 2012; Wanget al., 2016a; Shaygan et al., 2018a,b). Among others, Suarez (2001) used
UNSATCHEM to investigate alternative reclamation strategies and concluded that for soils possessing native calcite, “green manuring” (i.e., incorporating fresh organic matter into the soil to boost soil CO2) can be used toreclaim soil without gypsum additions and with less water. More recently,
Phogat et al. (2020a) assessed the impact of annual gypsum additions onmaintaining acceptable sodicity and salinity levels.

7.4 Shallow groundwater
In addition to irrigation water additions and native soil minerals, root zone
salinization can occur due to the upward movement of shallow, saline
groundwater. HYDRUS has been used in a number of studies to evaluate
the impact of shallow groundwater. For example, Forkutsa et al. (2009)
modeled irrigated cotton production as affected by salinity and shallow
groundwater in the Aral Sea Basin. Karimov et al. (2014) assessed the impact

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of salinity and shallow groundwater on winter wheat production in the
Fergana Valley (Central Asia). Li et al. (2015a) modeled soil water andsalt dynamics as influenced by shallow groundwater in the Heihe River
Basin. Narjary et al. (2021) evaluated the effects of water table depth andevaporative flux on soil salinity.

7.5 Recycled and produced water
The disposal of recycled or produced water to irrigate agricultural crops is
another interesting application of the HYDRUS and UNSATCHEM
models (e.g., Mallants et al., 2017; Phogat et al., 2020a,b, 2021).
Produced water is water that comes out of the wells during fracking operations, coal seam gas production, or crude oil production. Produced watercontains soluble and non-soluble oil/organics, suspended solids, dissolved
solids, and various chemicals used in the production process. The application of produced water as irrigation water (possibly after blending withfresh water) is associated with many problems, such as salinity and sodicity
risks and associated changes in soil physical and chemical properties (e.g.,
reduction of hydraulic conductivity), which can be assessed using the
HYDRUS model and its modules (e.g., UNSATCHEM).

8. Chemigation
Modern irrigation systems are used, besides delivering water, also fortransporting agrochemicals (e.g., pesticides and herbicides) to benefit plant
health. The chemicals are injected into the irrigation system (Chemigation)and transported directly into the plant environment (Taha, 2022). Eachagrochemical has its own chemistry and unique interactions with the soil
matrix. Each solute can be involved in several processes, such as transport,
adsorption, degradation, volatilization, or even interactions with othersolutes (Spurlock et al., 2013a). Besides the ability to improve the schedulingof chemical applications to allow for their maximum utilization, numericalmodels also can track the fate of applied solutes and their metabolites in theroot zone and their transport below the root zone into groundwater, whichin general could be harmful to the environment.

8.1 Historical developments
Modeling of solute transport in soil systems has been practiced for several
decades, initially using the ADE’s analytical solutions and later numerical

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solutions. The analytical solutions typically require complete decouplingof water flow and additional simplifications such as uniformity of soil properties and are, therefore, very limited. In most cases, they are also limited tostrict linearity, zero- or first-order reactions, and full decoupling of thesolutions for different solutes (as well as the decoupling of the water flowfrom the solute transport). A large number of the ADE’s analytical solutions(e.g., van Genuchten, 1985; van Genuchten and Wierenga, 1976; van
Genuchten and Wagenet, 1989) and programs evaluating these solutions(e.g., CFITM, CFITIM, CXTFIT, 3DADE, N3DADE, and CHAIN)
were implemented in the public-domain Windows-based STANMOD(STudio of ANalytical MODels) computer software package (Šimůnek
et al., 1999), which can be considered to as belonging to the HYDRUS
family of software packages.
Numerical solutions are more flexible in many aspects mentioned above
and, therefore, gained popularity over the years, with HYDRUS playing amajor role in that popularity. The vast majority of numerical applicationsof solutions to water flow and solute transport in soil systems were of an
agricultural orientation (e.g., tracking the fate of fertilizers, pesticides, or
salinity), but numerous applications for environmental problems soon
followed, looking at nutrient leaching to groundwater, contaminanttransport from landfills, and more. In the following, we present several
examples of such applications.

8.2 Standard modules and their predecessors
Šimůnek et al. (1992) were among the first to add solute transport capabilities to the numerical solutions of the Richards equation (then SWMII, apredecessor of HYDRUS). The ADE used at the time included advection,
dispersion, and a very limited range of reactivity processes (linear adsorption(i.e., constant retardation factor) and zero- and first-order reactions in boththe liquid and solid (sorbed) phases). The code gained increasing popularity
starting mostly at the end of the 1990s, with the advancement of personal
computation and the stability of the code as it developed from SWMII to
SWMS_2D to the early HYDRUS packages in one or more dimensions(Šimůnek et al., 2008). In that relatively limited approach, linear sorption(assuming equilibrium) is considered in Eq. (2) using s ¼ kdc, and thesink/source term ϕ includes zero (k0) and first (k1c) order reactions, andparallel terms for the solid (sorbed) phase. Note that in most cases, sorption
is mathematically represented as a retardation factor that comes in frontof the left-hand side of (2) and is constant for linear sorption.

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Examples of early applications include De Vos et al. (2000), who used
SWMS_2D to model nitrate leaching (i.e., as an inert solute). Persicani(1996) used HYDRUS (and three other models) to check the sensitivityof pesticide leaching in soil profiles. Vanderborght et al. (1998) modeled
advection-dispersion (ignoring diffusion) in heterogeneous soils and compared their results to the particle tracking approach. Yang et al. (1996)
considered linear adsorption in their 2D stochastic simulation framework(with the hydraulic conductivity and the adsorption constant being
the stochastic variables). Boateng and Cawlfield (1999) conducted an extensive sensitivity analysis of the Richards-ADE system to contaminanttransport, considering most parameters of the Richards equation (withvan Gunuchten-Mualem parameters), considering advection, dispersion,
first-order sorption, and zero- and first-order reactions. These and manyother earlier references to HYDRUS packages and their predecessors canbe found here.
An example of a relatively advanced application of the classic HYDRUS
model for solute transport is seen in Hanson et al. (2006). They considereddrip irrigation in two geometrical configurations (one of which is axisymmetric), where fertilizers (urea, ammonium, and nitrate) were applied withthe water (fertigation). In their model, they considered sequential first-order
reactions (hydrolysis of urea and nitrification) and linear adsorption (ofammonium). Denitrification was not considered (as in the experimentassociated with the model, it was negligible), and volatilization and gas transfer were ignored. This represents a classic agronomic study that looks into
the fate of fertilizer in the subsurface. However, for more complex cases
when oxygen may be absent at times or in regions of the subsurface, sucha modeling approach that relies only on a sequence of first-order reactions
may be insufficient.
In later versions of HYDRUS and other codes that solve similar problems, more sophisticated reactions evolved over the years. Clearly, the spaceis too limited to review them all, and we will only mention several, referring
the reader to the HYDRUS manual and website for further elaborationand presentation of specific equations. For example, Kodešová et al.
(2004) studied chlorotoluron transport in a soil profile and modeled itusing both Freundlich and Langmuir isotherms. Pal et al. (2014) used
HYDRUS-1D to model phenol breakthrough experiments in soil columns
using the three classical sorption isotherms (linear, Freundlich, andLangmuir). Pot et al. (2005) and K€
ohne et al. (2006) used HYDRUS-1D
to model herbicide transport using the dual-permeability model with
two-site non-equilibrium sorption. Chotpantarat et al. (2011) studied theARTICLE IN PRESS
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Naftali Lazarovitch et al.

transport of several heavy metals in soils and used HYDRUS-1D to model
non-equilibrium sorption, considering Langmuir isotherm and two-site
sorption. Finally, K€
ohne et al. (2009a,b) provided a comprehensive review
of HYDRUS applications involving transient water flow and tracer andpesticide transport in structured soils.

8.3 Specialized add-on modules
The next level of complexity in modeling solute transport is using a higher
complexity of reactions, which typically involve more than a single reactive
species and coupling of the different reactions. In the HYDRUS framework,
several specific tools were developed over the years for specific applications
and environments (e.g., UNSATCHEM, Suarez and Šimůnek, 1997).
Goldberg et al. (2007) provide a comprehensive overview of both existingmodels and their applications, mostly related to surface complexation.
Later on, a more generic solution to the need emerged by coupling thePhreeqC code (Parkhurst and Appelo, 1999) with HYDRUS ( Jacqueset al., 2006) to form HP1. For example, Jacques et al. (2008a) studied
aqueous phase complexation and ion exchange (competitive sorption) for
several major ions and heavy metals, including cadmium and zinc. The
use of PhreeqC in HP1 allows great flexibility in modeling complex biogeochemical processes. For example, Ben Moshe et al. (2021) used this flexibility to model carbon and nitrogen processes in the context of soil aquifer
treatment using multi-Monod-type reactions.
An interesting development in solute transport codes, HP1 included, istheir use in modeling the transport of non-solutes. This is increasingly
practiced to study colloid and nanoparticle transport, considering processes
such as attachment and detachment, straining, and more. Makselon et al.
(2017) provide an example of such a model considering silver nanoparticles.
Many references involving HYDRUS applications to simulate the fate
and transport of various types of particles are listed on the HYDRUS
website (i.e., https://www.pc-progress.com//Documents/Jirka/Particles_
Transport_Papers.pdf).

8.4 HP1 example
The following example is used to demonstrate in more detail the abilityof the HP1 code to track more complex biogeochemical processes beyond
the capability of the conventional ADE with first-order reactions (TheHP1 example in Table A). Jacques et al. (2008b) demonstrated the modeling

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of uranium transport in soil following fertilizer applications. Their model
considered annual fertilizer applications with inorganic phosphate contaminated by elements from the uranium decay chain. The soil profile comprised seven (7) layers to a total depth of about 1 m. The numericalsolution considered 14 different species (C, Ca, Cl, F, Mg, N(V), Na, P,
S(VI), and U(VI), O, and H) and a liquid-phase charge (to be balancedby the solid phase surface charge). Three types of geochemical equilibrium
reactions were considered, including aqueous speciation reactions, cationexchange reactions on organic matter, and surface complexation reactions
on Fe oxides. The reactions were strongly pH sensitive (i.e., protons,hydroxides, and surface hydrolysis were also considered). Carbon dioxide
was not considered, as the pH range in the system was 3.5–4.5. Most
reactions (including surface complexation ones) involve more than a single
species; therefore, first-order reactions cannot be used. The results of thismodel showed a very good fit to experimental data, which could not be
achieved using the more conventional models.
A different example (Liu, 2020) looked at the root zone’s total carbon
and nitrogen cycle (specifically, the goal was to explore these processes
under varying irrigation frequencies). She considered not only the classic
nitrification and denitrification but also mineralization, decompositionof soil organic matter, dissolved organic carbon, and nitrogen, as well as
microbial biomass and its growth or decay, and oxygen transfer between
the liquid and gas phases. The considered processes are shown in Fig. 16,
including the differentiation of the process direction for oxic and anoxic
conditions.
Fig. 17 shows a sample of the results obtained by this complex model,
looking at nitrification rates for three different irrigation regimes (see captions). Above all, this model considers oxygen levels in the soil, and accordingly, the reaction rates and the microbial dynamics are adjusted. One
can easily note the influence of water quality and irrigation frequency on
the biochemical processes. For example, nitrification rates in the upper
30 cm are higher in treatment TL than TH, primarily because of the higher
aeration allowed between irrigation events. At the same time, FH rates arelower than TH because the concentration of organic nitrogen and carbon
is lower.
The flexibility that is associated with HP1 seems endless. Nevertheless,the number of reported applications, especially in the agricultural context, isrelatively small. While no statistical significance test was conducted, apparently, most of the applications of HP1 are in more environmentally related

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Fig. 16 Schematic of biochemical cycling in the root zone and microbial communities.

Flooding

Flooding

0
Reaction rate
(10–1 mmol dm–3 d–1)(a) TH

Depth (cm)

–30

1.238
1.111
1.027
0.950
0.883
0.821
0.763
0.706
0.646
0.586
0.523
0.461
0.406
0.360
0.322
0.290
0.264
0.242
0.229
0.215
0.201
0.187
0.175
0.163
0.153
0.144
0.136
0.130
0.125
0.120
0.114
0.108
0.099

–60
–90
(i) July(ii) August(i) July(ii) August–120
0Depth (cm)–30(b) TL

–60
–90
–120
0

(c) FH

Depth (cm)

–30
–60
–90
(i) July(ii) August

–120
0

1

2

3

4

Time (d)

5

6

7

0

1

2

3

4

5

6

7

Time (d)Fig. 17 Spatio-temporal distributions of nitrification (i.e., ammonium transformation tonitrate) rates in different fertigation regimes (weekly cycles, for TH and FH irrigation ondays 2, 4, and 7; for TL irrigation on days 3 and 7; water quantities adjusted; TH and TL
use treated wastewater and FH uses freshwater).

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research. The reason for this is not clear, but we suspect that this is related,in part, to the complexity of the model and the lack of a graphical userinterface, even if minimal, dedicated to specific scientific environments.

9. Spatial variability
Efficient irrigation management decisions require a good understanding of temporal variations in root water uptake and spatial variations insoil hydraulic properties, which inherently impact water dynamics, solutetransport, root distribution, and plant growth (Dabach et al., 2015). The
soil-plant system of irrigated crops can be simulated by considering various
model complexities, such as defining 1D, 2D, or 3D transport domains,
describing homogeneous, layered, or fully heterogeneous soils in a single
simulation, or by running a series of simulations representing measured or
estimated variability in soil hydraulic properties. Additionally, spatial variation in irrigation inputs will cause water content heterogeneity in theirrigated field. Therefore, understanding different levels of spatial variability in the irrigated root zone can help with various decisions, such as
selecting sensor locations for optimal triggered irrigation scheduling,
providing information on the expected levels of variability in themeasured data, or optimizing irrigation application methods that rely on
non-uniform applications or involving different soil textures or materials.
This section provides an overview of different levels of spatial variability,
how they have been addressed using HYDRUS, and additional examples
and guidance on how to apply them in various scenarios.

9.1 Irrigation system uniformity
The HYDRUS code can be used to study the effects of different levels of
irrigation uniformity on drainage and solute leaching. Irrigation uniformity
levels can be simulated by running Monte Carlo simulations with discharge
rates sampled from a selected distribution associated with the desired
irrigation uniformity. Such an approach can be applied to several drippers
in a multi-dimensional simulation (Raij et al., 2018), to individual simulations, each with a discharge in a defined distribution, or a series of simulations representing variable emitter discharge rates along an irrigation line due
to pressure head loss (Guan et al., 2019; Wang et al., 2016b). In a 3D study,
it was found that lower irrigation uniformity in drip-irrigated lysimeterscaused higher drainage variability than the observed variability in chloride

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Fig. 18 Water contents (A, B) and water content coefficient of variation (CV) (C, D) in thethree observation nodes (F) (x ¼ 4.3, y ¼ 6.3 cm; x ¼ 8.5, y ¼ 13.5 cm; x ¼ 13.9, y ¼ 23.7 cm)
during four irrigation events in a loam soil (A and C) and a sandy loam soil (B and D) for
three levels (CV ¼ 2%, 4.4%, and 9%) of irrigation uniformity (E). Irrigation fluxes for threelevels of heterogeneity (E) and the simulation domain with three observation nodes (F).

concentrations (Raij et al., 2018). On the other hand, in 2D studies, it was
found that increased irrigation uniformity decreased drainage amounts andnitrate loads (Guan et al., 2019; Wang et al., 2016b). Fig. 18 demonstrates
how various levels of irrigation uniformity, i.e., with a coefficient ofvariation (CV) of 2%, 4.4%, and 9%, affect soil water contents at different
locations relative to a dripper, as simulated in 500 runs. The simulation
domain was the same as in the Drip Plant project (Section 5—Drip irrigation, Table A). Sandy loam and loamy soils were considered (Figs. 18A
and C and 18B and D, respectively). Daily irrigation events from a surface
irrigation area with a radius of 20 cm were simulated for four days. The
irrigation discharge rate was randomly sampled 500 times from a normal
distribution with the same mean but varying standard deviations, according

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to the selected CV (Fig. 18E). The Matlab code used to create thesesimulations can be found in this link. Water contents in three observation
nodes (Fig. 18F) varied in 500 simulations because of different imposed
discharge rates. As expected, the higher the CV of the irrigation discharge
rate, the higher the variability in the soil water content, especially during
irrigation events (Figs. 18A and B). As a result, values of the water content
CV during the irrigation events were higher than the irrigation CV(Figs. 18C and D).

9.2 Deterministic spatial variability
Vertical heterogeneity due to soil layering, for example, in alluvial soils,
can be simulated in HYDRUS by defining each soil layer’s hydraulicparameters and assuming horizontal homogeneity (Botros et al., 2012. Ifenough data is available, the depth of each layer can vary in any dimension
accordingly (Bednorz et al., 2016). Botros et al. (2012) simulated waterflow and nitrogen transport in an irrigated nectarine field with a 3D domain
with horizontal soil layers of varying depths in the east-west direction but
homogeneous in the north-south direction. Water fluxes below the rootzone and nitrate concentrations in the deep vadose zone, simulated using
the layered approach, were comparable to stochastic simulations using
either a Miller-Miller approach or a measured spatial correlation of thevan Genuchten hydraulic parameters (Botros et al., 2012). 3D transportdomains can be used to simulate water contents and solute distributions
in complex irrigated systems such as a drip-irrigated cotton field with partial
plastic mulch on the growing beds (Li et al., 2018b) or rainfed banana
plants grown on Andisol with a complex canopy geometry (Sansoulet
et al., 2008). In addition, simulations using 3D domains can help explain
processes of the wet bulb and salinity overlap that are underestimated in
2D conventional models assuming symmetry (Kandelous et al., 2011).
When 2D or 3D domains cannot be used due to computational constraints,
1D simulations can be carried out at the field or regional scale, and the results
can be ensembled to include and describe the spatial differences. For example, soil properties, weather, slope, and land use maps can be used to create
unique combinations of 1D simulations, which can then be integrated
spatially to describe water and solute fluxes, as well as solute and watercontent distributions in the landscape (Turkeltaub et al., 2018; Henriet al., 2020, 2021; Qiu et al., 2020; Sprenger et al., 2016; Sela et al.,2012; Lai and Ren, 2016).

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Fig. 19 presents two examples of layered soils where the first has a sandy
loam layer in a loam domain (Fig. 19A, the Drip3D2L1 example in Table A),
and the second has a loam layer in a sandy loam domain (Fig. 19B, theDrip3D2L2 example in Table A). These simulations use the same boundary
conditions as the Drip3D examples in Section 5—Drip irrigation (Table A).
Following an irrigation event during day 1, the volumetric water content islower in the sandy loam layer than in the loam layer. This approach has beenused to describe naturally occurring variability. In addition, induced soil
profile variabilities, such as capillary barriers utilized to modify and control
soil water distribution or irrigation using properties of porous materials,
have been simulated with HYDRUS (2D/3D) for optimizing such systems(e.g., Ben-Gal et al., 2004; Siyal and Skaggs, 2009; Ityel et al., 2011; Saoet al., 2021).

9.3 Stochastic spatial variability
The effects of the spatial variability caused by the randomness of specific soil
hydraulic properties, such as saturated hydraulic conductivities or porosities,

Fig. 19 The flow domain of two-layered systems (the left panel; A:loam with a sandy
loam layer, B: sandy loam with a loam layer) and corresponding soil water content
distributions with time (0–5 days).

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can be simulated using the HYDRUS code by first generating the spatial
distribution of this parameter (using a theoretical (Guan et al., 2019;
Wang et al., 2016b; Martinez et al., 2014) or measured (Hu et al., 2007)
probability distribution of this property) and then running sequential simulations with a range of this parameter’s values. Theoretical distributions
can be generated by defining a mean and coefficient of variation (CV)or standard deviation for the variable of interest (Guan et al., 2019;
Martinez et al., 2014; Bossa and Diekkr€
uger, 2014) by generating a rangeof textures and bulk densities as inputs to Rosetta (Wang et al., 2016b), orby using means and standard deviations reported by Rosetta3 (Zhang
and Schaap, 2017; Raij-Hoffman et al., 2022). Such simulations can optimize irrigation amounts to minimize leaching, not at a specific point in thefield but as a probability density function (Hu et al., 2007). Additionally,
different levels of hydraulic parameters’ heterogeneity can be studied, andtheir effects on water fluxes and solute loads can be described (Guanet al., 2019).
Fig. 20 presents the average and 95% confidence intervals of water contents during two irrigation events at an increasing lateral distance from a
sub-surface dripper obtained from 187 simulations. Simulations were carriedout for a homogeneous soil profile with varying Ks values for two soil
textures reported in Rosetta3. The mean and standard deviation of Ks(on a log scale with base 10) were used to randomly sample Ks values foreach simulation. These simulations use the same boundary conditions as
the Drop3D1 example in Section 5—Drip irrigation (Table A). The confidence bounds for time-variable water contents at each location in each soil
demonstrate the uncertainty associated with using pedotransfer functions
when only soil texture is known. While the uncertainty of the water content
remains high during the two irrigation events, the variability of the flux atthe bottom boundary decreases with time in this example. Considering
measured or estimated variability in soil properties or irrigation systems
can provide information on their effect on the uncertainty of different
model outputs (in this case, water content and bottom boundary fluxes).
This information can be used to evaluate model-to-measurement fit anda choice of type, method, and frequency of measurements.
In addition to non-dimensional parameter uncertainty (as discussed
above), the measured or assumed spatial correlations of any selected hydraulic parameter can be considered in HYDRUS simulations. Correlation
lengths in different directions can be used to generate a spatially correlatedparameter space (Botros et al., 2009). Significant differences in 3D N fluxes

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Fig. 20 Average and 95% confidence intervals for 187 simulations with varying Ks
values for loam and sandy loam soils. Water contents at observation points locatedat depths of 20 cm (vertical) (dripper depth) and 5 cm (A), 15 cm (B), and 25 cm(C) away (horizontal) from the dripper. Time-dependent water fluxes at the bottom
boundary (D).

were observed in simulations considering the spatial correlation of soil
hydraulic properties, as opposed to those using uncorrelated scaling factors
or deterministic layered heterogeneities. On the other hand, water fluxes in
the field with spatially correlated soil hydraulic properties were similar to
those in the deterministic layered profile and different from those in thefield with uncorrelated scaling factors. This was due to the nature of
the highly horizontally correlated alluvial soil presented in Botros et al.
(2009, 2012).
An additional option offered by HYDRUS is to generate stochastic
distributions of hydraulic conductivity (αK) and pressure head (αh) scaling
factors using the Miller-Miller similitude approximation (Miller andMiller, 1956), where these two scaling factors are correlated. The stochastic
distribution requires a mean Ks, a standard deviation of log10 value of αK, andhorizontal (X) and vertical (Z) correlation lengths. Additional numerical

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experiments were performed to explore the influence of stochastic
heterogeneity during an irrigation event using the flow domain in the
DripPlant1 example in Section 5—Drip irrigation (Table A) with a sandy
loam soil for a homogeneous soil profile, and heterogeneous soil profiles
with a standard deviation of log10(αK) ¼ 0.5 cm and (a) a vertical correlation
length Z ¼ 5 cm, (b) a horizontal correlation length X ¼ 10 cm, and(c) horizontal and vertical correlation lengths X ¼ 10 cm and Z ¼ 5 cm,
respectively. Fig. 21A presents representative water content soil profiles
for a homogeneous domain, and heterogeneous domains with vertical,
horizontal, and both vertical and horizontal heterogeneities. Fig. 21B
presents water contents at three observation nodes and the fluxes at thebottom boundary. The results demonstrate the significant variability ofwater contents in heterogeneous domains compared to the homogeneous
one, with higher variability of both water content and bottom boundary
fluxes for the vertical and horizontal heterogeneity than the combined
heterogeneities. Similarly, water fluxes at the bottom boundary were morevariable when only one directional heterogeneity was simulated rather thanboth directions combined. Examples of horizontal heterogeneity can befound in highly layered alluvial soils, and vertical heterogeneity is foundin cracked soils with/or soils with preferential flow paths. In either case,the extent of the spatial correlations will depend on the soil profile’s specificproperties. These results demonstrate the importance of considering heterogeneity, especially for contaminant flow and transport studies. Scaling factors
can also be generated externally and imported into HYDRUS domains. An
example can be found in Raij et al. (2018) or Botros et al. (2009), where
three-dimensional distributions of scaling factors for unsaturated hydraulicconductivity were externally generated and imported into HYDRUS-3D.

9.4 Estimation of soil hydraulic parameters
for heterogeneous soils
Various laboratory and field methods and in-situ soil sensors and devices areused to estimate soil hydraulic properties, and water flow and solute transport dynamics during irrigation. However, identifying optimum locations of
sensors for a given spatial heterogeneity is critical for efficient irrigationwater management. For example, the wetting front resulting from drip
irrigation will be affected by the spatial heterogeneity of the soil between
the dripper (with and without geotextile membrane) and the tensiometer.
As a result, the wetting front will not reach tensiometers located at thesame distance from the dripper at the same time. Dabach et al. (2013) used

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Fig. 21 Water content distributions in the 16th hour in a representative homogeneous
profile and three heterogeneous profiles with a standard deviation of log10(αK) ¼
0.5 cm, a vertical correlation length Z ¼ 5 cm, a horizontal correlation length
X ¼ 10 cm, and vertical and horizontal correlation lengths Z ¼ 5 cm and X ¼ 10 cm, andconsidering the Miller-Miller similitude (A). Water contents and corresponding 95%
confidence intervals in three observation nodes resulting from 20 simulations (B).

a system-dependent boundary condition in HYDRUS that triggers irrigation of a fixed duration when a specified soil water matric head thresholdis reached at a predetermined point in the soil profile. This boundary condition represents a demand-based irrigation system that directly depends onplant water uptake. HYDRUS simulations showed that the optimal location

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of tensiometer placement is near the subsurface dripper, resulting in a low
CV of matric head measurements and applied irrigation water, high sensitivity to irrigation, and lower drainage below the root zone (Dabach
et al., 2015). Zeng et al. (2012) compared the centrifugal, tensiometer,
soil core, disc infiltration, and inverse solution methods to estimate soil
hydraulic parameters for a drip-irrigated heterogeneous gravel soil. The
two-dimensional HYDRUS inverse simulation produced the best result
even though a slight overestimation of soil moisture was present, which
revealed a problem of underestimating water movement with the disc
infiltration approach compared to the soil core method.
Similarly, Abbasi et al. (2003a) used the Levenberg–Marquardt optimization algorithm implemented in HYDRUS to estimate soil hydraulicand solute transport parameters of several soil horizons below experimental
furrows. Their results showed only minor improvements in model predictions with optimized parameters. A two-step method, i.e., sequential estimation of the soil hydraulic parameters followed by the estimation of thetransport parameters, was used to improve the model predictions further.
Despite implementing a scaling procedure for the unsaturated soil hydraulicproperties, variability in the optimized longitudinal dispersivity values forthe layered soil profiles was found to be higher than for the homogeneous
soil profile due to the variations in soil water retention curves of the different
soil horizons.
The Miller–Miller scaling approach implemented in HYDRUS wasused by Weiherm€uller et al. (2006) to assess the variability in solute breakthrough curves in suction cups. Weiherm€uller et al. (2006) demonstrated
that mean pore water velocities and dispersivities ranged by a factor of
1.6 and 1.5, respectively, for different heterogeneous structures. In addition,
the suction cup sampling area, deformation of streamlines, and the flow
channels were also found to have an influence, and numerical simulations
indicated that about 20 suction cups were required for calculating a mean
breakthrough curve in the chosen heterogeneous flow field (Weiherm€uller
et al., 2006).
Similarly, lysimeters are widely used for closing water mass balances bymonitoring the storage in the soil profile, drainage, and solute leaching at
the bottom. Raij et al. (2018) investigated the influence of local heterogeneity in soil hydraulic conductivities and dripper discharge rates on drainage
and chloride concentrations using 100 HYDRUS three-dimensional simulations per treatment. Different random spatial realizations of the conductivity scaling factor were generated, assuming a log-normal distribution

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(Lazarovitch et al., 2006). Their result demonstrated that irrigation caused
higher variability in drainage amounts than soil heterogeneity, while soil
heterogeneity caused higher variability in drainage concentrations than
irrigation heterogeneity.
Most of the studies mentioned above used numerical simulations to demonstrate the influence of spatial heterogeneity on sensors. Domı́nguez-Niñoet al. (2020a,b) investigated the sensor-to-sensor variability due to uncertainties in the measurement process plus the natural variability in the actual
soil water dynamics by comparing 57 capacitance-type soil moisture sensors
in the field and HYDRUS-3D virtual sensors under drip irrigation.
However, the HYDRUS-3D model was used to simulate soil water dynamics only in a homogeneous soil and uniform root distribution and did notaccount for heterogeneous soil and root distributions, macropores, andsoil irregularities. The results demonstrated that the HYDRUS-3D simulations successfully reproduced the soil water dynamics and patterns of daily
amplitude, but compared to the sensors, simulations were less variable,
underestimated root water uptake, and overestimated the influx of irrigationwater. Overall, these results demonstrated that various soil sensors could
be too sensitive to local variations in soil texture and the presence ofgravel, stones, roots, macropores, or small compacted soil parts.
Understanding field wetting patterns during irrigation and fertigation
can provide helpful information for efficiently managing these systems. In
the past decade, advanced, minimally invasive geophysical tools, such as
electromagnetic induction, cosmic ray sensors, ground penetrating radar,and electrical resistivity tomography (ERT), have been used for 2D mappingof wetting patterns and solute distributions for various irrigation systems(Algeo et al., 2016; Cassiani et al., 2015; Coppola et al., 2016; Han et al.,2016; Hardie et al., 2018). For example, Hardie et al. (2018) compared
the drip irrigation wetting patterns and nitrate distributions using ERT,
dye tracer, and HYDRUS-2D modeling. The results showed some variability in observed, measured, and predicted infiltration patterns and nitratetransport across the three approaches due to the preferential flow processes
in the field. However, simultaneous ERT, dye tracer, and modelingstudies can be used to understand water flow and solute transport in an
irrigation field.

9.5 Challenges, recommendations, and future opportunities
All the processes described above can be simulated simultaneously, anddifferent levels of variability can be considered simultaneously to study

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complex processes (e.g., irrigation, soil hydraulic parameters, and scaling factors, Guan et al., 2019; Raij et al., 2018). The global problem of non-point
source pollution from irrigated agriculture needs to be assessed at theregional scale, even though the processes occur at the field/local scale.
One approach is to use an ensemble of one-dimensional HYDRUS
simulations, including all combinations of soils and crops occurring in thearea of interest (Henri et al., 2020; Turkeltaub et al., 2018). Then, eachcombination of water and solute loading is assigned to various spatial
locations and either fed into a groundwater model or used as potential
groundwater loadings. Qiu et al. (2020) studied landscape-level ecosystemservices, including agricultural landscapes, using a biophysical model
coupled with HYDRUS-1D and found that “time” components were more
important than “space” drivers in such landscapes.
Emerging technologies, such as electrical resistivity tomography(Hardie et al., 2018; Peddinti et al., 2020), are increasing the opportunity
to use spatially distributed data at larger scales than traditional measurements(i.e., Skaggs et al., 2004). Therefore, combining field monitoring, virtual
monitoring, and validation of observed sensor data using representative
HYDRUS simulations can provide great insight into local heterogeneity.
Future studies should include a combination of multiple sensors for collecting a range of parameters and site-specific soil characterization to better
understand the effects of spatial heterogeneity on irrigation management.
Such efforts will also provide additional opportunities to fill the data gaps
further and refine the model.

10. Model calibration
Numerical models are mainly developed to describe and predict the
behavior of physical systems. When the deviation between model predictions and actual observations (e.g., soil moisture patterns or water depths
in the furrow channel) is acceptable, then the model can complement or evenreplace real experiments. The model’s predictive accuracy depends on theadequacy of the model structure, as well as on a proper (and physicallyrealistic) definition of model parameters. The process by which the model
structure is selected and parameters are adjusted to minimize the distance
between model predictions and measurements is broadly defined as model
calibration. This procedure is generally accomplished by combining themodel with iterative statistical and numerical optimization techniques.
HYDRUS is internally coupled with the gradient-based LevenbergMarquardt search method (Marquardt, 1963), which solves a nonlinear least

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squares optimization problem to calibrate various model parameters usingtransient observations. The algorithm is robust when the response surfaceof the calibration problem is unimodal, which is generally the case forwell-posed inverse problems. Being supported by the HYDRUS
GUI, the algorithm has been used in several irrigation-related studies toinversely estimate soil hydraulic and/or transport parameters. This has been
generally accomplished by comparing simulated and observed soil watercontents at different depths in drip-irrigated fields (Bufon et al., 2012;
Chen et al., 2021; Gonçalves et al., 2015; Li et al., 2015b; Nayebloie
et al., 2022; Patel and Rajput, 2008; Qi et al., 2018; Rana et al., 2022;
Selim et al., 2013; Zhang et al., 2018), furrow channels (e.g., Abbasi
et al., 2004; Dialameh et al., 2022; Ebrahimian et al., 2013; Ranjbar
et al., 2019; Zhang et al., 2013), or other irrigated areas (e.g., Arbat et al.,2008; Bethune et al., 2008; Deb et al., 2013; Er-Raki et al., 2021; Houet al., 2017; Li, 2020; Mokari et al., 2019; Rai et al., 2019; Rezayati
et al., 2020; Saefuddin et al., 2019; Salehi et al., 2017; Tan et al., 2014).
The main limitation of the Levenberg-Marquardt algorithm is its local
search capability, which makes its use questionable for high-dimensional
inverse problems that use different types of measurements. In these circumstances, the response surface tends to be complex, noisy, and characterizedby multiple local minima. To address this problem, several studies
have externally coupled HYDRUS with global optimization algorithms
such as the Particle Swarm, Genetic, and Shuffled Complex Evolution
Optimization algorithms (Azad et al., 2018, 2019; Brunetti et al., 2018a;
Erazo-Mesa et al., 2022; Kumar et al., 2022; Peddinti et al., 2018, 2020;
Schneider et al., 2013; Vrugt et al., 2008; Zhou et al., 2007, 2012) or with
the PEST calibration suite (Carlos et al., 2022; Liu et al., 2021, 2022).
Despite requiring external tools, advanced programming skills, and significantly more computational resources, the results of these studies showthat the use of global optimization algorithms can improve the HYDRUS
calibration procedure and better diagnose model structural inadequacies
when estimated parameters assume physically unrealistic values.

10.1 Uncertainty assessment
Uncertainty analysis complements model calibration and represents avaluable tool for assessing the model’s structural adequacy and predictive accuracy. Despite being frequently applied in combination with HYDRUS for
vadose zone hydrological modeling, its use in irrigation-related studies

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remains rather limited. Brunetti et al. (2018a) coupled HYDRUS-1D
and the Global Sensitivity Analysis-Generalized Likelihood Uncertainty
Estimation technique (GSA-GLUE) to calibrate and assess the uncertaintyof the soil hydraulic and thermal parameters of an irrigated extensive
Green Roof (GR). Recently, the Bayesian inference was used to statisticallycompare one conceptual and three HYDRUS-based mechanistic parameterizations (e.g., unimodal, bimodal, and dual porosity) of an irrigatedGR at the laboratory scale (Brunetti et al., 2020). In another study,
Brunetti et al. (2021) coupled HYDRUS-1D and the multimodal Nested
Sampling algorithm (Feroz et al., 2009) to reproduce observations fromthree GRs irrigated with wastewater and simultaneously assess the uncertainty of inversely estimated soil hydraulic and solute transport parameters.
The Bayesian inference was also applied by Guo et al. (2020), who used
two surrogate-based uncertainty assessment techniques to calibrate the
HYDRUS model against measurements from field irrigation experiments.
Albeit limited, the results of these studies highlight the beneficial role ofuncertainty assessment in combination with HYDRUS to (1) assess how
the model’s predictive uncertainty affects model outcomes, which is of
crucial importance when delivering model results to decision-makers,(2) verify whether the informational content of calibration measurements
is sufficient to constrain model predictions, or update the prior belief ofthe modeler, and (3) compare multiple model parameterizations and avoid
the use of data-unjustified complex model structures that tend to overfit
measurements.

10.2 Sensitivity analysis
Along with uncertainty assessment, sensitivity analysis is a fundamental
tool for model diagnostics, as it can be used for both predictive and explanatory purposes. In the first case, it can complement model calibration anduncertainty assessment by providing a statistical basis to quantify how different model parameters affect, alone and/or jointly, the fitting accuracy.
Uninfluential parameters can then be excluded from the calibration, thus
reducing the dimensionality of the inverse problem. When used for explanatory intentions, it can identify factors that most influence a certain variableof interest (e.g., root water uptake, nutrients leaching, etc.), thus guiding
the experimental design (e.g., specific complementary laboratory measurements). Sensitivity analyses are generally divided into two groups: local
one-factor-at-a-time (LSA) and global sensitivity analysis (GSA).

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By varying only one factor at a time, LSA is easy to implement andcomputationally cheap. It has been used in several studies in combination
with HYDRUS to assess the influence of soil hydraulic and transportparameters on water flow and nutrient transport in irrigated fields(Beyene et al., 2018; Geza et al., 2021; Jimenez-Martı́nez et al., 2009;
Kadyampakeni et al., 2018; Reading et al., 2012; Rocha et al., 2006; Sunet al., 2022; Xu et al., 2019; Yang et al., 2017). However, LSA neglects
the effects of parameters’ interaction, which is known to play an importantrole in environmental models. To overcome this limitation, GSA explores alarge portion of the parameters’ space to identify combined or individual
parameter effects. This frequently requires thousands of model executions
to achieve stable estimates of the sensitivity indices, thus significantly
increasing the analysis’s computational burden. Techniques such as theMorris screening scheme (Morris, 1991), the Sobol’ sequence (Sobol’,2001), or the E-FAST method (Saltelli et al., 1999) have been successfully
coupled with HYDRUS in multiple studies (Brunetti et al., 2018a; Li et al.,2012; Liang et al., 2017; Liu et al., 2022; Peddinti et al., 2018; Skaggs et al.,2014; Tao et al., 2021; Zhou et al., 2012).
The literature review suggests that GSA should be preferred when thecomputational cost of a single HYDRUS execution is low (e.g., seconds),
as it provides statistically meaningful information about the most influential
parameters and their interaction, which can be used to prioritize experiments (i.e., explanatory setting) or reduce the dimensionality of the inverse
problem (i.e., predictive setting). Instead, LSA should be used when themodel is computationally intensive to assess its sensitivity to small changes
in specific parameters’ values. Ideally, for predictive purposes, this shouldbe performed around the global optimum identified during the calibration
procedure to check the model sensitivity near its most probable values. A
prognostic use before calibration could lead to misleading conclusions if
the LSA starting point lies in low-fitting regions. Similarly, for explanatory
purposes, it is advisable to perform the LSA in regions of the parameters’
space identified as probable based on the literature review, prior information,and the modeler’s expertise.

10.3 Irrigation optimization
Model calibration, uncertainty assessment, and sensitivity analysis are all
fundamental steps to obtain a model that can robustly predict the physicalbehavior of the system under investigation. Once this objective is achieved,

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the model can be used for multiple purposes, among which the most important in irrigation management is the optimization of the irrigation system.
This includes establishing the irrigation schedule (e.g., water/nutrients
injection timing and duration) and defining locations and types of sensors(e.g., tensiometer, soil moisture sensors) and actuators (e.g., pumps, valves).
The latter are key components of real-time systems, which are becoming
very popular due to their capability to dynamically adapt the irrigation
strategy depending on soil status. In this direction, the HYDRUS model
includes a system-dependent boundary condition, which initiates irrigation
whenever the matric head at a predetermined location drops below acertain threshold (Dabach et al., 2013). This enables the modeler to designand optimize a real-time sensor (i.e., tensiometer)—actuator (e.g., valve)irrigation system tailored to the specific soil.
In any case, regardless of the irrigation system type, the irrigation strategy
should be designed to maximize crop yield while simultaneously minimizing
the environmental footprint and installation and maintenance costs.
This leads to a complex multi-objective optimization problem that can besolved by coupling the model with various numerical optimization
techniques. For example, Kandelous et al. (2012) coupled HYDRUS-2D
with the multi-objective optimization algorithm AMALGAM (Vrugt andRobinson, 2007) to derive the Pareto front and explore tradeoffs between
water applications, irrigation system parameters, and crop transpiration, andto evaluate the best management practices for subsurface drip irrigation systems in alfalfa. Similarly, Roy et al. (2019) obtained the Pareto front by combining HYDRUS-2D with an evolutionary multi-objective optimization
procedure to minimize water utilization and maximize crop yield prediction. Both studies prove that coupling HYDRUS with multi-objective
search techniques can provide decision-makers with multiple tradeoffs forphysically based irrigation strategies. However, more research is needed
to demonstrate the benefits of this type of analysis and include the economic
costs as an additional objective function, which would lead to a 3D
Pareto surface.
Despite being a valuable tool, the multi-objective optimization of theirrigation system can be computationally intensive and requires specificprogramming and numerical skills that are not widespread among irrigation managers. Hence, a huge portion of the literature body has simplified
the multi-objective optimization problem by focusing only on using
HYDRUS to optimize a single objective function (e.g., water and nutrients use efficiency, root water uptake, etc.) (Akhtar et al., 2013; Brunetti

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et al., 2018a; Fontanet et al., 2020, 2022; Garcı́a Morillo et al., 2017;
Karandish and Šimůnek, 2018; Lena et al., 2022; M€uller et al., 2016;
Ning et al., 2021; Qin et al., 2016; Seidel et al., 2015; Xu et al., 2019;
Yi et al., 2022). The results of these studies are encouraging and showthat the mechanistic framework provided by HYDRUS enables themodeler to obtain irrigation management strategies that are hydrologicallyfunctional, agriculturally efficient, and, more importantly, physicallyrealistic. Nevertheless, the optimization procedure can still be improvedby: (1) performing multi-objective optimization searches, in which theeconomic costs are included, as this is a crucial choice criterion in realapplications, (2) propagating the effects of model uncertainty in the optimization procedure, as this would provide more honest predictions todecision-makers, and (3) taking into account dynamic changes in model
structure (e.g., tillage effects on soil hydraulic properties) over mediumto-long term simulation periods.

11. Special irrigation methods
Over the last few years, the HYDRUS software has also been used toevaluate multiple traditional and/or experimental special irrigation methods.
These applications include dual-drip irrigation (El-Nesr et al., 2014), pitcher
irrigation (Siyal et al., 2009), irrigation using clay pipes (Siyal and Skaggs,
2009; Siyal et al., 2013), capillary wick irrigation (Al-Mayahi et al., 2020;
Kamal et al., 2019), irrigation from ring-shaped emitters (Saefuddin et al.,2019; Noguchi et al., 2021), irrigation systems with a capillary barrier(Al-Mayahi et al., 2020; Ityel et al., 2011; Noguchi et al., 2021; Saoet al., 2021 and Wongkaew et al., 2018), vertical line source irrigation(Fan et al., 2018a,b, 2021a,b, 2022; Kanda et al., 2020a,b), film hole irrigation (Chen et al., 2020a; Fan et al., 2019a,b, 2021a; Hou et al., 2014; Jieet al., 2020), and many others (e.g., Fan et al., 2018c; Ji et al., 2015; Liet al., 2013; Pan et al., 2021; Wang et al., 2017a; Zemni et al., 2022;
Zhou et al., 2017).

11.1 Dual-drip
El-Nesr et al. (2014) used HYDRUS (2D/3D) to evaluate three novel
technologies enhancing a spatial distribution of water and solutes in the rootzone while limiting downward leaching. These technologies included (a) aphysical barrier under a dripper and a dual-drip system with (b) concurrentand (c) sequential irrigation. Numerical simulations showed that applying a

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dual-drip system or installing a physical barrier could significantly alter thewetting pattern and spatial distribution of applied solutes. The simulations
also showed that the physical barrier and the dual-drip systems are moresuitable for coarse-textured soils than fine-textured soil. A dual-drip irrigation system with two emitters that can be operated sequentially allows
controlling which solute is retained in the root zone and which one isleached by simply altering the operation of the two drippers.

11.2 Pitcher
Siyal et al. (2009) used HYDRUS-2D to evaluate the functioning of pitcher
irrigation, an ancient but very efficient irrigation system used in many arid
and semiarid regions, such as in Northern Africa, Iran, China, India,
Pakistan, Brazil, Indonesia, and Zimbabwe (Fig. 22; see also referencesin Siyal et al., 2009). Siyal et al. (2009) investigated the water content
distributions around small, medium, and large pitchers buried in fine- andcoarse-textured soils and with different saturated hydraulic conductivities of the pitcher material. Water content distributions simulated by
HYDRUS-2D were found to be in close agreement with the experimental
results, indicating that HYDRUS-2D is a suitable tool for investigating
and designing pitcher irrigation systems. Simulated and experimental results

Fig. 22 Photographs illustrating pitcher irrigation and the axisymmetric finite element
mesh diagram used by Siyal et al. (2009) to simulate water infiltration from a buried
pitcher (Photo credits: Altaf Siyal).

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Fig. 23 Simulated wetting of initially dry loam soil under pitcher irrigation for differing
pitcher wall saturated hydraulic conductivities.

also showed that a large pitcher, double the size of a smaller one but with half
the hydraulic conductivity, will produce approximately the same wetting
front as the smaller pitcher. Fig. 23 illustrates the impact of the pitcher
wall’s saturated hydraulic conductivity, for large pitchers, on simulated infiltration into an initially dry loam soil (the Pitcher example in Table A).

11.3 Clay pipe
HYDRUS (2D/3D) was used by Siyal and Skaggs (2009) and Siyal et al.
(2013) to evaluate soil wetting patterns and solute transport under porous
clay pipe subsurface irrigation, respectively. Subsurface irrigation using
porous or perforated clay pipes, similar to pitcher irrigation discussed above,
has been practiced since ancient times. It can still represent an efficient,
water-saving irrigation method for many less developed countries in arid
and semiarid regions. Siyal and Skaggs (2009) successfully compared experimental and simulated soil wetting patterns for subsurface pipe systems

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operating at different water pressures. They also simulated the irrigation system’s function in different soil textures (sand, sandy loam, loam, silt), as wellas in layered soils (sandy loam over loam or loam over sandy loam).
Siyal et al. (2013) expanded their study by additionally simulating salt accumulation near the soil surface from a subsurface clay pipe irrigation system
during the growing season of okra. Their results showed that proper
management of salt accumulation is vital for sustainable crop production
whenever subsurface irrigation systems are being implemented.

11.4 Capillary wick
Capillary wick irrigation systems were analyzed using HYDRUS models byKamal et al. (2019) and Al-Mayahi et al. (2020). Low-pressure wick irrigation systems represent energy-efficient systems that can play an importantrole in smallholder greenhouse crop production or home gardens by ensuring higher water use efficiency than most traditional approaches. Al-Mayahi
et al. (2020) extended the analysis of the wick irrigation system by also
considering a capillary barrier to prevent the leaching of irrigation waterfrom the root zone.

11.5 Capillary barrier
The functioning of a capillary barrier under various irrigation systems has
been studied using HYDRUS-1D and HYDRUS (2D/3D) by Ityel et al.
(2011), Wongkaew et al. (2018), Al-Mayahi et al. (2020), Noguchi et al.
(2021), and Sao et al. (2021). While Wongkaew et al. (2018) consideredsurface-applied irrigation (sprinkler), Al-Mayahi et al. (2020) consideredcapillary wick irrigation, Noguchi et al. (2021) considered subsurface drip
irrigation (using ring-shape emitters), and Sao et al. (2021) considered
uniform, line-source, and plant-targeted surface irrigation. All these studies
showed that if properly designed, the capillary barrier can keep water inthe root zone by preventing its leaching to deeper soil layers.

11.6 Ring-shaped emitters
Saefuddin et al. (2019) used HYDRUS (2D/3D) to evaluate the functioningof a subsurface drip irrigation system using a ring-shaped emitter. A
ring-shaped emitter (made from a standard rubber hose) is a low-cost subsurface irrigation system developed and introduced for subsurface irrigationin Indonesia for small-scale farmers who have scarce water resources. The
three-dimensional version of HYDRUS (2D/3D) was used to evaluate

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the performance of the ring-shape emitter, including multiple alternative
designs with different numbers of holes (in the rubber hose) and changes
in the covering method. Numerical results were successfully comparedagainst experimental data collected during experiments with the originalring-shaped emitter.

11.7 Other irrigation methods
11.7.1 Vertical line source irrigation
Fan et al. (2018a,b, 2021b, 2022) used the HYDRUS-2D software andlaboratory experiments to evaluate the performance of the vertical line
source (moistube) irrigation system. Vertical line source irrigation is awater-saving method developed for deep-rooted fruit trees in the arid
regions of China to enhance direct water and nutrient delivery to the rootzone while reducing soil evaporation and improving water and nutrient use
efficiency. Fan et al. (2018a,b) performed multiple HYDRUS-2D simulations to identify the main influencing factors (e.g., soil texture, initial watercontent, pressure head, moistube length, and buried depth) for this irrigation
system on the soil wetting pattern.
11.7.1.1 Moistube Irrigation

HYDRUS (2D/3D) was used in a study by Kanda et al. (2020a,b) to evaluate the performance of moistube irrigation (MTI), a subsurface irrigation
technology that uses a semipermeable membrane to emit water continuously
into the soil in response to soil water potential and applied pressure. Kanda
et al. (2020a) evaluated how soil texture (loamy sand and sandy clay loam)
influences the soil water dynamics of MTI using laboratory experiments andnumerical simulations. Kanda et al. (2020a) evaluated how the moistube
placement depth influences the soil water dynamics and root water uptakeof cowpea. The goal was to determine the optimal depth of placement ofmoistube tapes in cowpea production to minimize water losses via drainage
and soil evaporation and thereby improve water use efficiency.
11.7.2 Film hole irrigation
Film hole irrigation (FHI) was investigated using the radially symmetrical
option of the HYDRUS (2D/3D) software by Hou et al. (2014), Chenet al. (2020a), Fan et al. (2019a,b, 2021a), Jie et al. (2020), and Pan et al.
(2021). This irrigation method is used mainly in the arid regions ofChina, where plastic film mulch is widely used to prevent evaporation
and weeds growth. Water flows on the plastic film and infiltrates throughARTICLE IN PRESS
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77crop holes and special irrigation holes. Film hole irrigation is thus similar
to point source irrigation. Compared to traditional surface irrigation
methods, film hole irrigation significantly improves water use efficiency.
According to Fan et al. (2021a), there are two types of film hole irrigation,i.e., film hole free infiltration (FHFI) and film hole interference infiltration(FHII), depending on whether the soil wetting fronts under the twoneighboring film holes are independent or interfere with each other during
FHI. In general, HYDRUS proved to be a robust and useful tool in
evaluating this type of irrigation.
11.7.3 Additional irrigation methods
Many other irrigation techniques have been numerically evaluated using
HYDRUS (2D/3D), mainly in the literature from China. For example,
negative-pressure irrigation has been studied by Ji et al. (2015), Zhou
et al. (2017), and Wang et al. (2017a), bubbled root irrigation by Li et al.
(2013), buried diffuser irrigation (a special type of subsurface drip irrigation
method) by Zemni et al. (2022), and horizontal moistube-irrigation by
Fan et al. (2018c). HYDRUS (2D/3D) proved to be an ideal tool forevaluating the performance of these various irrigation techniques.

12. Summary and outlook
The use of HYDRUS is widespread today in many areas related toirrigation. In this work, we have reviewed the software’s capabilities anddemonstrated them using numerous examples. The model is still being continuously developed, and new features that better describe processes in theirrigated root zone are being added. HYDRUS can help design irrigation
systems and determine the optimal irrigation and fertilization regimes.
Since the model can examine flow and transport processes in and under
the root zone, it can also be used to examine the sustainability of various
irrigation regimes. Correct use of the model may optimize applications ofwater and other agrochemicals, such as fertilizers and pesticides. Future
agriculture will have to look closely at the transport of agrochemicals
through the vadose zone and into groundwater.
Computational power advances every day, but simultaneously, there is aneed to evaluate many processes using a very high spatial and temporal
resolution. While using finer spatial and temporal discretizations may reduce
numerical errors, the computational demand may, at the same time, increase
dramatically. The number of discretization nodes usually increases by a

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power of two when the problem’s dimensionality increases. While personal
computers can perform complex simulations, the calculation duration may
be too long, reducing our ability to estimate model parameters or optimize
irrigation scheduling. Parallel calculation tools can take advantage of theexisting capacity of computers with multiple cores or processors and significantly speed up time-consuming simulations, especially those requiring
many finite elements. Examples of such tools are the HyPar module(Šimůnek et al., 2016b) or the ParSWMS model (Gohardoust et al.,2021; Hardelauf et al., 2007). Today it is also possible to perform calculationsin the cloud, which can significantly speed up the simulations and allow
HYDRUS users to carry out comprehensive stochastic simulations (e.g.,
Sasidharan et al., 2019).
Another alternative and promising approach is using emulators to
perform computationally intensive tasks. An example of such an emulator
can be found in Neuro-Drip (Hinnell et al., 2010). This study combines
an artificial neural network with a statistical description of the spatiotemporal distribution of added water from a single drip emitter to provide
easily accessible, rapid illustrations of the spatial and temporal subsurface
wetting patterns. These cheap-to-run surrogate models can be trained to rapidly emulate the response of the mechanistic model to changes in a finite
number of parameters, thus enabling the modeler to perform complex model
calibration and irrigation optimization tasks on standard personal computers.
However, the applicability of such surrogate tools is limited to a relatively
low number of parameters (e.g., 10) as the training phase becomes computationally expensive in higher dimensions. Another possibility is to hybridize
HYDRUS with machine learning methods to reduce the computational costof expensive parts of the code (e.g., reaction). Surrogate and hybrid modelingfor physically based irrigation analysis is certainly a topic worthy of future
investigation.
The HYDRUS ability to simulate crop growth is still relatively limited.
The standard HYDRUS modules account for crop growth only externallyby inputting potential evaporation and transpiration fluxes (or the leaf
area index in 1D to separate potential evapotranspiration into potential
evaporation and transpiration). HYDRUS also allows users to externallyprescribe a time-variable rooting depth (and width) using the logistic growth
function or in a tabulated form. The HYDRUS models do not allow therooting zone’s spatial extent to change actively due to certain environmental
conditions and stresses (e.g., Hartmann et al., 2018). While there havebeen attempts to provide HYDRUS the ability to simulate crop growth

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79(e.g., Groenveld et al., 2021; Han et al., 2015; Hartmann et al., 2017; Javaux
et al., 2013; Shelia et al., 2018; Wang et al., 2015; Zhou et al., 2012), noneof these implementations is currently available in the HYDRUS software,and only the models of Hartmann et al. (2017) are directly available fromthe HYDRUS website as non-standard HYDRUS modules (i.e., not fully
supported by the HYDRUS GUI). Alternative approaches (e.g., Couvreur
et al., 2012, de Jong van Lier et al., 2009, 2013) to the macroscopic modelof Feddes et al. (1978) to account for various root water uptake stresses
would also be a desirable new feature of the HYDRUS models.
The environmental conditions of the irrigated root zone can change
rapidly due to irrigation and soil drying. Soil water contents, temperatures,and salts concentrations and their composition can affect soil hydraulic andsolute transport (e.g., degradation) parameters. Since the model dynamicallyand spatially calculates the environmental variables, it can simultaneously
update model parameters that depend on them. Some of these connections
are already implemented in HYDRUS, such as the dependence of thehydraulic conductivity on the chemical composition of the soil solution(e.g., sodium adsorption ratio, SAR, and electric conductivity, EC)(Suarez and Šimůnek, 1997), the dependence of soil hydraulic properties(via viscosity and surface tension) on solution concentrations (e.g., Silva
et al., 2020), or the dependence of solute reaction parameters (i.e., degradation) on changes in water contents and temperatures. More complex interactions between the environmental variables and model parameters can beconsidered using the HP1 model (e.g., Jacques et al., 2008a,b). Introducing
additional relationships between environmental factors and model parameters (e.g., clogging, i.e., a reduction of the hydraulic conductivity as a function of cumulative infiltration of water and/or solute/bacteria) would
further increase the applicability of the HYDRUS models.
Stable water isotopes are increasingly used as environmental tracers toinvestigate flow processes in the unsaturated zone in both bare and planted
soils (e.g., Stumpp et al., 2012), to quantify water and nutrient travel times,
and to identify water sources for root water uptake and the age of thesesources (e.g., Zhou et al., 2021, 2022). Only the 1D computational versionof the HYDRUS model can currently simulate the fate and transport ofstable water isotopes subject to no (Stumpp et al., 2012) or constant fractionation during evaporation. This contrasts with the standard formulationof solute transport in HYDRUS during evaporation, where solutes concentrate at and near the soil surface when water evaporates. However,ignoring the evaporative enrichment, as done in this 1D model, leads toARTICLE IN PRESS
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underestimating 2H and 18O concentrations in the topsoil, which may
be more significant in regions with higher evaporative losses (Sprenger
et al., 2018). This evaporative enrichment, as well as its dependence on
environmental factors (e.g., temperature and tension), has been recently
implemented into the computational module of HYDRUS-1D by
Zhou et al. (2021, 2022), but it has not yet been included in the
HYDRUS GUI or its higher dimensional modules. The ability to simulate
the fate of environmental tracers (and related issues, such as travel times
and water age) would be another desirable feature of future HYDRUS
releases.
Modern agriculture often uses various types of mulching to reduce evaporative losses and prevent weed growth. While traditional mulching has
been done using plastic films (e.g., Li et al., 2015b, 2018b), more environmentally friendly alternatives, such as biodegradable (e.g., Chen et al., 2019,
Chen et al., 2020b,c) or sprayable (e.g., Filipovic et al., 2020) mulches, arebeing intensively investigated. While HYDRUS has been used in many ofthese applications, its ability to fully describe dynamic boundary conditions
associated with the application of biodegradable and sprayable mulches andtheir temporarily changing properties is still rather limited, and further
developments are needed.
While considerable efforts have gone into developing the HYDRUS
models to account for irrigation practices, we also understand that model
development, calibration, and validation are never-ending processes.
Today, the model can consider various physical, chemical, and biologicalprocesses and their interrelationships. While the model development for
irrigation practices is progressing considerably, the progress in sensing the
agricultural field is limited. Although there are many point sensors that
describe the state of water (water content or potential) in the field in a
temporal manner, the spatial variation in the soil-plant system still
limits their use in model calibration. Undoubtedly, the development of
non-intrusive methods can improve the spatial description of the water
status in the soil. Even today, there are only a very few methods that cancontinuously quantify the chemical composition of soil water, and thus,
the model calibration is done with the help of a general but continuous
measure of the bulk electrical conductivity or tests that usually require
human intervention (soil sampling, suction cups). Regarding biologicaltests, the situation is even more complex, and today, very few methods
can evaluate the root system structure and functions under field conditions.

Appendix
Table A List of HYDRUS examples.
Section Name

Dim.

Description

Version3

Sprinkler

2D

Sprinkler irrigation on a center pivot (1: loamy sand, 2: sandy loam)

h3d3

3

Bubbler2D

Bubbler irrigation on a center pivot (1: loamy sand, 2: sandy loam)

h3d3

4

Furrow

3D

Pseudo-3D furrow irrigation for three different soil surface management scenarios

hyd5

5

Drip

2D-radial

Surface drip irrigation with dynamic surface wetting (1: loam, 2: sandy loam)

h3d3

5

DripPlant

2D-radial

Surface drip irrigation with root water uptake (RWU) and two rooting depths LR

h3d3(1: uncompensated RWU (ωc ¼ 1), LR ¼ 30 cm, 2: compensated RWU (ωc ¼ 0.575), LR ¼ 20 cm)
5

Drip3D3D, 2D-radial Subsurface drip irrigation (1: axisymmetric domain, 2: 3D domain)

h3d3

6

NLeach2D

Nitrogen leaching in sandy loam for surface drip irrigation

h3d2

6

NUptake

1D

Active and passive crop nitrogen uptake for sprinkler irrigation with different irrigation depths, h1ddifferent durations of fertilizer application, different N-fertilizer application levels, and threedifferent soil textures (1: sand, 2: sandy loam, 3: loam)

7

Salinity

1D

Salinity simulations with HYDRUS and UNSATCHEM for two irrigation regimes

h1d

8

C + N_Cycles 1D

Modeling of the root zone’s total carbon and nitrogen cycle for different irrigation regimes
using HP1h1d9Drip3D2L

2D-radial

The same as the Drip3D example above, but with two soil layers (1: a sandy loam layer in aloam domain, 2: a loam layer in a sandy loam domain)

h2d2

10

Pitcher

2D-radial

Pitcher irrigation

hyd5

h1d—HYDRUS-1D, version 4.x; h3d3—HYDRUS (2D/3D), version 3.x; hyd5—HYDRUS, version 5.x.

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