Modeling of Irrigation and Related Processes with HYDRUS

1
2
3

Naftali Lazarovitch1, Isaya Kisekka2, Tobias Ebong Oker3, Giuseppe Brunetti4, Thomas

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Wöhling5,6, Li Xianyue7, Li Yong8, Todd H. Skaggs9, Alex Furman10, Salini Sasidharan11, Iael

5

Raij-Hoffman2, and Jiří Šimůnek12

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7

1

8

Institutes for Desert Research, Ben-Gurion University of the Negev, Sde Boker Campus, 8499000,

9Israel.

French Associates Institute for Agriculture and Biotechnology of Drylands, Jacob Blaustein

102

Department of Land Air and Water Resources, University of California Davis, Davis, CA, USA

113

College of Agricultural and Environmental Sciences, University of Georgia, Tifton campus, GA,

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USA

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4

14

Sciences, Vienna, Austria

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5Technische Universität Dresden, Chair of Hydrology, Dresden, Germany

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Lincoln Agritech Ltd, Environmental Research, Hamliton, New Zealand17

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Inner Mongolia Agricultural University, Hohhot, China

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Hohai University, Nanjing, China

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9

U.S. Salinity Laboratory, USDA-ARS, Riverside, CA, USA

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10Technion, Israel Institute of Technology, Israel

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11

Department of Biological & Ecological Engineering, Oregon State University, Corvallis, OR,

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USA

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12

Institute of Soil Physics and Rural Water Management, University of Natural Resources and Life

Department of Environmental Sciences, University of California, Riverside, CA, USA

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25

1

Abstract

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Future agriculture calls for increased input (e.g., water, nutrients, pesticides) use efficiency while29

maintaining or improving productivity, minimizing environmental impacts, and increasing30

profitability. Complete understanding of complex irrigation systems requires laborious, time-

31

consuming, and expensive field investigations, which invariably involve only a limited number of32treatments. On the other hand, fully calibrated process-based models, such as HYDRUS, can33

quickly evaluate different irrigation management strategies without the need for labor-intensive

34

fieldwork and have become valuable research tools for predicting complex and interactive water35

flow and solute transport processes in and below the root zone. HYDRUS codes have been used36

worldwide in several hundreds of studies evaluating various types of irrigation (e.g., sprinkler,

37

furrow, basin, and surface and subsurface drip), their scheduling (e.g., the timing of irrigation and38

its amount), and solute-related factors (e.g., fertigation, chemigation, salinization, and39

sodification).

40

The objective of this manuscript is to review the current modeling capabilities of HYDRUS to41

evaluate various irrigation methods and related processes. The manuscript starts with a section

42

describing governing flow and transport equations solved numerically by the HYDRUS codes, the43corresponding initial and boundary conditions, and related factors such as soil hydraulic properties

44

and root water and nutrient uptake. Modeling of different irrigation techniques is described in45

subsequent sections, followed by sections dealing with solute-related topics such as fertigation,46

chemigation, and salinization/sodification. Topics, including the effects of spatial variability,47

optimization of irrigation systems, and special irrigation methods, are covered in the later sections.

48

The manuscript emphasizes the advantages and opportunities of HYDRUS in describing various

49

processes in the root zone of irrigated plants that support sustainable irrigated agriculture. All the50

project files of the discussed examples and their descriptions are available for download at51

https://www.pc-progress.com/en/Default.aspx?hyd5-AdvancesInAgronomy.

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Keywords: Irrigation, HYDRUS, Drip, Furrow, Sprinkler, Fertigation, Salinity, Chemigation

2

1. Introduction

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General Introduction

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Sustainably meeting the comprehensive dietary needs of each individual on this globe is a58

concern that is growing alongside the world population. The major steps needed to meet this goal59

include decreasing food waste, changing our diets by consuming more plant and less animal

60

protein, and intensifying production on existing cultivated areas while reducing the environmental

61

impact of production. Irrigated agriculture is the primary user of freshwater, accounting for nearly

62

85% of the total global water consumption (Jury and Vaux Jr., 2007) and providing about 40% of63total food production (Sepulcre-Cantó et al., 2007). Irrigation water demand is expected to increase64

in the future due to foreseen alterations in rainfall caused by climate change and increased food65

and biofuel demands. Water scarcity is increasing worldwide, necessitating the development of66

more efficient irrigation practices.

67

Complete understanding of irrigation systems and corresponding complex physical,

68

chemical, and biological interactions in the soil-plant-atmosphere system requires laborious, time-69

consuming, and expensive field investigations. Such investigations invariably involve only a70

limited number of treatments that only provide information about the studied systems and71

conditions (e.g., for a particular climate, plants, soils, and irrigation methods). This information

72

then needs to be extrapolated to other relevant conditions. Numerical models have become

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indispensable tools for evaluating, summarizing, and extrapolating information about vadose zone

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flow and transport processes in the root zones of irrigated systems. Despite their complexity, the75

use of numerical models is increasing thanks to a better understanding of water flow and transport

76

processes, the development and improvement of mathematical methods for solving governing

77

equations, and accelerated development of computers capable of computing different processes

78

simultaneously for short time and small space increments (van Genuchten and Šimůnek, 2004).

79

There is now a growing use of models for the visualization of irrigation (Assouline, 2002;

80

Gärdenäs et al., 2005) and fertilization-related processes (Cote et al., 2003; Gärdenäs et al., 2005;

81

Hanson et al., 2006). Fully calibrated process-based models can quickly evaluate different

82

irrigation management strategies for different climate and soil conditions and various crops

83

without the need for labor-intensive fieldwork, and they have become valuable research tools for3

84

predicting complex and interactive water flow and solute transport processes in and below the root

85

zone.

86
87

HYDRUS Introduction

88

The HYDRUS software packages (Šimůnek et al., 2008b, 2012, 2016b, 2018, 2022ab)89

represent typical examples of such process-based mathematical models. These software packages

90

numerically solve the Richards equation for variably saturated water flow and advection-

91

dispersion equations for heat and solute transport (and many other processes and factors) in one-,92

two-, and three-dimensional variably saturated soil domains. A sink term in the flow and transport

93

equations accounts for plant roots' uptake of water and nutrients. Heat transport occurs due to94

conduction and convection with flowing water. Advective-dispersive transport in the liquid phase,95

as well as diffusion in the gaseous phase, is considered in the solute transport equations, which96

also include provisions for nonlinear nonequilibrium reactions between the solid and liquid phases,

97

linear equilibrium reactions between the liquid and gaseous phases, and various zero-order and98

first-order degradation reactions. In addition, physical nonequilibrium solute transport can be99

accounted for by assuming a two-region, dual-porosity type formulation, which partitions the100

liquid phase into mobile and immobile regions. Attachment/detachment theory, including filtration

101

theory, is additionally included to enable simulations of the transport of viruses, colloids, and/or

102

bacteria.

103

The HYDRUS software packages and its various predecessors, e.g., SWMS-2D, CHAIN-

104

2D, HYDRUS-1D, HYDRUS-2D, and HYDRUS (2D/3D), have a long history that goes back to

105

the early 1990s and have been used in several thousands of studies involving various agricultural,106

industrial, and environmental applications as documented in detail by Šimůnek et al. (2008, 2012,107

2016b). These applications have resulted in several hundreds of manuscripts evaluating various

108

types of irrigation (e.g., sprinkler, furrow, basin, and surface and subsurface drip), their109

management (e.g., the timing of irrigation and its amount), and associated solute-related factors

110(e.g., fertigation chemigation, salinization, and sodification) (Šimůnek et al., 2016b).

111

The last comprehensive review of the HYDRUS software packages was published in the112

Vadose Zone Journal in 2016 (Šimůnek et al., 2016b) when two independent packages for one-113

dimensional (i.e., HYDRUS-1D, version 4) and two- and three-dimensional (i.e., HYDRUS

114(2D/3D), version 3) applications still existed. These two software packages have recently been4

115

merged, resulting in a single HYDRUS (Version 5) software package, consisting of an interactive

116

graphics-based user interface (Šejna et al., 2022) and multiple computational programs for117

simulating water, heat, and solute movement in one-, two-, and three-dimensional variably

118

saturated porous media (Šimůnek et al., 2022ab). Version 5 of HYDRUS (compared to its earlier

119

versions), in addition to providing various new graphical capabilities, such as depth-time graphs120

for various variables of one-dimensional simulations, also includes several new modules that121

greatly expand the numerical capabilities of the software package, some of which are highly122

relevant for agricultural and irrigation applications.

123

While various new modules implemented into HYDRUS 5 have been individually124

described in their corresponding manuscripts (see references given below), they have not125

comprehensively been listed in one text.

126

The Furrow module to simulate the coupled surface-subsurface water flow and solute

127transport in blocked and open-ended furrows has recently been developed by Brunetti et al.

128(2018b). This hybrid Finite Element-Finite Volume pseudo-3D model combines the two-129

dimensional Richards-based Finite-Element solver, HYDRUS-2D, for subsurface flow with a one-130

dimensional Finite-Volume discretization of the zero-inertia and advection equations for overland

131

flow. The model has been successfully validated in synthetic and experimental scenarios by132

Brunetti et al. (2018b). This add-on module has been fully incorporated into the GUI of the133

HYDRUS 5.

134

The PFAS module (Silva et al., 2020) can simulate the fate and transport of135

polyfluoroalkyl substances (PFAS) under dynamic vadose zone conditions. PFAS are a class of136

thousands of related chemicals used for their water-repellent and fire-retardant qualities. They are137

sometimes called “forever chemicals” because they persist in the environment and accumulate in138

people and animals. They have been linked to health problems, including testicular and kidney

139cancers, high cholesterol, and thyroid disease. The contamination of agricultural soils by these140

long-lasting toxins can be traced to sludge from treatment plants spread over the years as fertilizer

141

on farmlands to add nutrients. These chemicals behave differently than traditional organic

142

contaminants. Rather than being present in the gas phase (a process considered in the standard

143

HYDRUS model), these chemicals sorb to the air-water interface. The PFAS module thus replaces

144

the storage term for the air phase with the storage term for the sorption to the air-water interface.

145

It additionally considers the concentration effects on surface tension and viscosity.
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Another new module is the Dynamic Plant Uptake (DPU) module (Brunetti et al., 2019a),

147

which simulates the translocation and transformation of neutral compounds in the soil-plant148

domain. Traditionally, HYDRUS deals only with processes in the subsurface and general plant149

uptake, leaving what happens with water and chemicals once they enter plants unaccounted for.

150

However, plants (fruits or leaves) represent the most common pathway for environmental

151

contaminants to enter the human and animal food chain. Plants can uptake chemicals from152

contaminated environments (e.g., by irrigation water) and bioaccumulate and metabolize them in

153

by‐products, which can be dangerous for human health. The DPU module couples HYDRUS with154

a multicompartment Dynamic Plant Uptake model of Trapp (2007), which accounts for multiple

155

differentiated metabolization pathways in plant tissues, and allows one to simulate the entire

156

pathway from irrigation to soil, from soil to roots, and from roots to stems, fruits, and leaves, while157

considering relevant biochemical reactions in all these compartments.

158

Additional new modules in Version 5, less relevant to irrigation applications, are COSMIC

159(Brunetti et al., 2019b), Particle Tracking (Zhou et al., 2021), Fumigant (Spurlock et al., 2013b), and160

C-Ride (Šimůnek et al., 2006). The COSMIC module (Brunetti et al., 2019b) calculates above-161

ground neutron fluxes using the physically based COsmic-ray Soil Moisture Interaction Code162

(COSMIC) of Shuttleworth et al. (2013). The Particle Tracking module (Zhou et al., 2021) tracks

163

hypothetical particles in the subsurface, and its results can be used to calculate soil water travel

164

times and water age for different locations in the soil profile. The Fumigant module considers

165

additional options and factors related to the transport of fumigants, i.e., chemicals used to sterilize

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soils before planting crops (e.g., tarp removal, temperature-dependent tarp properties, an additional

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injection of fumigants) (Spurlock et al., 2013b). Finally, the C-Ride module (Šimůnek et al., 2006)

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can simulate variably saturated water flow, colloid transport, and colloid-facilitated solute

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transport in porous media.

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Manuscript Objectives

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As discussed above, HYDRUS models have been used in several hundreds of manuscripts

173

evaluating various types of irrigation, their management, and associated solute-related factors.

174

Some of these older applications, mainly furrow and drip irrigation, were reviewed by Šimůnek et

175

al. (2016b). The objective of this manuscript is to further expand on this older review and to176

examine HYDRUS’s current modeling abilities to evaluate various irrigation methods and related6

177

processes. The manuscript begins with a section describing governing flow and transport equations

178

solved numerically by the HYDRUS codes, the corresponding initial and boundary conditions,179

and related factors such as soil hydraulic properties and root water and nutrient uptake. Modeling180of different irrigation techniques is described in subsequent sections, followed by sections dealing

181

with solute-related topics, including fertigation, chemigation, and salinization/sodification. Topics

182

such as the effects of spatial variability, optimization of irrigation systems, and special irrigation

183

methods are covered in the later sections. The manuscript emphasizes the advantages and184

opportunities of HYDRUS in describing various processes in irrigated plant root zones that support

185

sustainable irrigated agriculture.

186

The manuscript provides many examples of HYDRUS simulating various types of

187

irrigation and related processes. The use of various (standard and special) boundary conditions

188

available in the software to simulate different irrigation types is demonstrated throughout the189

manuscript. Different ways of displaying the obtained results are also presented. Finally, the190

manuscript proposes possible directions for software development so that it continues to be191

relevant in optimizing irrigation scheduling in the twenty-first century.

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2. Mathematical and Physical Background

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Governing Flow and Transport Equations

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Water flow and solute transport in soils are described in the HYDRUS models using the196

modified Richards (1) and convection-dispersion (2) equations, respectively (Šimůnek et al.,197

2022ab):198 (h)  h=+ K izA   - S (h) K (h)  K ijAtxi 
x j



(1)

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( b s +  c + ag )   w c    g g   ( qi c )=−  Dij+ aDij−txi 
x j  xi x j 
xi

(2)

200

The Richards equation (1) combines the mass balance equation with the Darcy-

201

Buckingham equation, and it contains the following variables:  is the volumetric water content

202[L3L-3], h is the pressure head or matric potential [L], t is time [T], xi is the spatial coordinate [L],203

K is the unsaturated hydraulic conductivity [LT-1], KijA are components of a dimensionless204

anisotropy tensor KA, and S is the sink/source term [L3L-3T-1], usually accounting for root water205

uptake (transpiration). Alternative formulations to (1) that partition the liquid phase into mobile

206

and immobile (or less mobile) regions (i.e., dual-porosity or dual-permeability models) are also207

available in the HYDRUS code to account for nonequilibrium or preferential flow (e.g., Šimůnek

208

et al., 2003, 2008ab; Šimůnek and van Genuchten, 2016b). This partial differential equation is209

strongly nonlinear, and as a result, only a relatively few simplified analytical solutions can be

210

derived. Most practical applications require a numerical solution, which can be obtained using

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various numerical methods such as finite differences or finite elements.

212

The equation governing the transport of dissolved solutes in the vadose zone (2) is obtained213

by combining the solute mass balance equation with equations defining the total mass of the214chemical (on the left side) and the solute flux densities and solute reactions (on the right side). In

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its most general interpretation, the solute transport equation (2) allows chemicals to reside in all

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three phases of the soil (i.e., gaseous, liquid, and solid) and a broad range of transport mechanisms

217(including advective transport, diffusion, and hydrodynamic dispersion in both the liquid and218

gaseous phases), and it facilitates different types of chemical reactions that lead to losses or gains

219

in the total chemical concentration. Advective-dispersive transport in the liquid phase, as well as8

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diffusion in the gaseous phase, is considered in the solute transport equations, which also include

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provisions for linear and nonlinear (i.e., Freundlich, Langmuir, or Freundlich-Langmuir adsorption

222

isotherms), and equilibrium and nonequilibrium (i.e., one- or two-site sorption models) reactions

223

between the solid and liquid phases, linear equilibrium reactions between the liquid and gaseous

224

phases (i.e., Henry's law), and various zero-order and first-order degradation reactions.

225

In (2), b is the bulk density [ML-3], a is the volumetric air content [L3L-3], s [MM-1], c

226[ML-3], and g [ML-3] are concentrations in the solid, liquid, and gaseous phases, respectively, Dijw

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and Dijg are components of the effective dispersion tensors in the liquid and gaseous phases [L2T-

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1

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volume by reactions or other sources (negative) or sinks (positive) such as plant solute uptake [ML-

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3 -1

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with the storage term for the sorption to the air-water interface, Ae, where Ae is the air-water232

interfacial area ([L2L-3] or [L-1]), and  is the interfacial adsorbed concentration (i.e., mass per unit

233

area of the interface) [ML-2].

], respectively, qi is the volumetric flux density [LT-1], and  is the rate of change of mass per unit

T ]. The PFAS module replaces in (2) the air phase storage term (the third term on the left side)

234

Again, alternative formulations to (2) that partition the liquid phase into mobile and235

immobile (or less mobile) regions (i.e., dual-porosity or dual-permeability models) are also236

available in the standard HYDRUS code to account for physical nonequilibrium or preferential

237

solute transport (e.g., Šimůnek et al., 2003; Šimůnek and van Genuchten, 2016). Alternative

238

formulations are also used in the add-on modules, such as UnsatChem and C-Ride.

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The HYDRUS models numerically solve the governing flow and transport equations,240

meaning that the transport domain and simulation time are divided into small pieces called finite

241

elements or finite differences. Triangles and quadrilaterals are typical spatial finite elements in two

242

dimensions, and tetrahedral, triangular prisms, and hexahedrals in three dimensions (Šimůnek et243al., 2022ab).

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Initial and Boundary Conditions

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The flow and transport equations can be solved, providing that applicable initial and247boundary conditions are specified. Initial conditions characterizing the system's initial state can be248

specified either in terms of water contents or pressure heads for the flow equation and in terms of249(liquid-phase or total) concentrations for the transport equation. When nonequilibrium (either
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physical or chemical nonequilibrium) solute transport models are used, the initial condition also251

needs to be specified for the nonequilibrium phase (i.e., concentrations in the mobile phase or at252

the kinetic sorption sites).

253

One of the main advantages of the HYDRUS codes is their flexibility in defining boundary254

conditions (BC), allowing simulations of various realistic irrigation scenarios. While the codes can

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accommodate standard system-independent boundary conditions, when conditions on boundaries

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are known beforehand, such as the Dirichlet (prescribed boundary pressure heads) and Neumann

257(prescribed boundary fluxes) boundary conditions, they can also accommodate a large number of258

system-dependent boundary conditions, when conditions on boundaries are not known apriori but

259

are the results of interactions between external (e.g., meteorological) conditions and conditions in

260(feedback from) the subsurface system.

261

The soil-air interface, exposed to atmospheric conditions, is an example of a system-262

dependent boundary. While the potential flux across this interface is controlled exclusively by

263

external conditions (e.g., precipitation, irrigation, evaporation), the actual flux also depends on the264(transient) water content conditions in the soil. Soil surface boundary conditions may change from265

prescribed flux to prescribed head type conditions (and vice-versa). This occurs, for example,

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when the precipitation/irrigation rate exceeds the infiltration capacity of the soil, resulting in either

267

surface runoff or accumulation of excess water on top of the soil surface, depending upon the soil

268

conditions. Similarly, the potential evaporation rate calculated from meteorological conditions (the269

evaporative demand of the atmosphere) may exceed the capability of the soil to deliver enough270

water toward the soil surface. In this case, the potential evaporation rate can be significantly

271

reduced to an actual evaporation rate controlled by the soil.

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Other system-dependent boundary conditions, less commonly used with irrigation

273

applications, include the seepage face BC, deep drainage BC, or flow to tile drains. Additionally,274

several boundary conditions implemented in HYDRUS have been specially developed for

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irrigation applications. These include, for example, subsurface drip characteristic function, surface

276

drip with dynamic wetting, triggered irrigation, and a reservoir BC.

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Subsurface Drip Characteristic Function

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The infiltration rate of water from a subsurface cavity (dripper) is affected by many factors,

280

including the pressure in the cavity, its size and geometry, and the hydraulic properties of the
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surrounding soil (Lazarovitch et al., 2005). When a specified discharge of a subsurface source282(e.g., a subsurface emitter) is larger than the soil infiltration capacity, the pressure head in the283

source outlet increases and becomes positive. The built-up pressure may significantly reduce the284source discharge rate. HYDRUS implements a special system-dependent boundary condition for285

flow from a subsurface source that uses the drip characteristic function:

Q = Q0 ( hin - hs )

286

c

(3)

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where hs is the pressure head at the source-soil interface [L], often called the back pressure, Q0288[L3T-1] ([L2T-1] in 2D) is the nominal discharge (Optimal Flux in the HYDRUS GUI) of the source

289

for the reference inlet pressure hin [L] (usually being 10 m) and the back pressure equal to zero, Q

290[L3T-1] ([L2T-1] in 2D) is the source discharge for the actual back pressure hs [L], and c [-] is an291

empirical constant (Exponent in the HYDRUS GUI) that reflects the flow characteristics of the292

emitter. Normally, c = 0.5 corresponds to a turbulent flow emitter, and c = 1 to a laminar one. This293

equation states that when hs increases, the pressure difference between the soil and the source inlet

294

decreases, and the discharge rate correspondingly decreases (Lazarovitch et al., 2005).

295
296

Surface Drip With Dynamic Wetting

297

The wetted area's radius for drip irrigation gradually increases as irrigation time increases.

298

The radius of this area for transient conditions can be calculated in HYDRUS as described by299

Gärdenäs et al. (2005). The irrigation flux Q is initially applied to a single boundary node300

representing the dripper with the Neumann (flux) boundary condition. When the pressure head301

required to accommodate the specified irrigation flux Q is larger than zero, the boundary condition302

in this particular node is changed to the Dirichlet (head) boundary condition with a zero pressure

303

head value, and the actual infiltration flux Qa through this node is calculated. The excess flux (Q -304

Qa) is then applied to the neighboring node, again with the specified Neumann boundary condition.

305

This procedure is iteratively repeated until the entire irrigation flux Q is accounted for, thus

306

providing the radius of the wetted area. Since the infiltration flux into the dry soil is larger at the307

start of irrigation, the wetted area continuously increases as irrigation proceeds.

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Triggered Irrigation

312

Irrigation can be either defined externally (by specifying the irrigation rate and the initial313

and final irrigation times in the HYDRUS input) or triggered internally by the model, depending

314

on the conditions in the soil (e.g., Dabach et al., 2013). Irrigation may be triggered in HYDRUS

315

when a user-specified pressure head is reached in a selected observation node defined somewhere316

in the transport domain (likely in the root zone). The irrigation then starts after a user-specified lag

317

period at a user-specified irrigation rate or pressure head at a user-specified boundary, and it lasts318

for a user-specified irrigation time duration. In 1D applications, triggered irrigation with a319

specified irrigation rate is applied at the soil surface, thus likely representing sprinkler irrigation.

320

In 2D applications, irrigation can be triggered at the time-variable head, time-variable flux, or321

atmospheric boundaries, likely representing surface or subsurface drip irrigation or sprinkler

322

irrigation.

323
324

Reservoir Boundary Condition

325

The reservoir boundary condition (Šimůnek et al., 2018) allows users to consider a326

reservoir that is external to the HYDRUS transport domain, while water can be added (injected)

327

to or removed (pumped) from this reservoir. Flow into or out of the reservoir through its interface

328

with the subsurface transport domain depends on the conditions in the transport domain (e.g., the329

position of the groundwater table) and external fluxes (pumping or injection):dVw= Qin − Q p + ( P − E ) S
dt
dVwc= Qin cin − Q p c p + PScr
dt

330

(4)

331

where Vw is the volume of water in the reservoir [L3], Qin is the water flow rate into (or out of) the

332

reservoir from the soil profile across the reservoir wall [L3T-1], Qp is the pumping/injection rate333

[L3T-1], P and E are precipitation and evaporation rates [LT-1], respectively, and S is the surface

334

area of the reservoir [L2]. In the latter equation, c is the solute concentration in the reservoir [ML-

335

3

336

water reservoir [ML-3] (cin is equal to c when water infiltrates into the soil profile and equal to the337

solute concentration in the soil profile when water exfiltrates into the reservoir), cp is the solute338

concentration associated with pumping or injection (equal to c for pumping or to the concentration], cin is the solute concentration associated with the mass transfer between the soil profile and

12

339

of water being injected into the reservoir) [ML-3], and cp is the solute concentration associated with

340

precipitation.

341

Since mass balances of water and solute in the external reservoir are constantly being

342

updated (based on all incoming and outgoing fluxes), the boundary conditions across the transport343

domain are dynamically adjusted depending on the water level in the reservoir. Reservoir boundary344

conditions potentially have many applications, such as dynamically estimating the water level in345

wells (e.g., Sasidharan et al., 2018, 2019), furrows during irrigation (e.g., Šimůnek et al., 2016a;

346

Bristow et al., 2020), or wetlands.

347
348

Soil Hydraulic Properties

349

The numerical solution of the Richards equation (1) requires knowledge of the unsaturated

350

soil hydraulic properties, i.e., the retention curve (h) and the hydraulic conductivity function K(h),351which are, in general, highly nonlinear functions of the pressure head. HYDRUS permits using

352

five different analytical models for the hydraulic properties (Brooks and Corey, 1964; van353

Genuchten, 1980; Vogel and Císlerová, 1988; Kosugi, 1995; Durner, 1994).

354

The soil hydraulic parameters must be either specified by users, selected from a soil catalog

355

listing them for 12 main textural classes, or estimated using pedotransfer functions. The van

356

Genuchten and Brooks and Corey parameters in the soil catalog were taken from Carsel and Parrish

357(1988) and Rawls et al. (1982), respectively. HYDRUS users can also use pedotransfer functions

358(PTFs) based on neural networks (Schaap et al., 2001) to predict van Genuchten's (1980)

359

parameters based on textural information.

360
361

Solute Transport and Reaction Parameters362

The advection-dispersion equation (2), numerically solved by the HYDRUS software, is363

written in a relatively general form so that it can represent many different solutes, such as tracers,364

nitrogen species, pesticides, heavy metals, radionuclides, and many other chemicals. Different

365

solutes are characterized by different transport and reaction parameters, and particular values then366

represent a specific chemical in a given soil environment. When solute transport is selected in the367

HYDRUS software, all transport and reaction parameters are initially set for a tracer, such as a368

chloride or a bromide (i.e., no sorption and no reactions). It is up to HYDRUS users to enter the369

proper parameters (e.g., sorption parameters, degradation parameters, Henry's law constant, etc.)
13

370

for a particular solute. The methods for identifying these parameters are described in detail, e.g.,371

by Skaggs et al. (2002).

372
373

Root Water and Nutrient Uptake

374

The sink term, S, in (1) represents the volume of water removed per unit of time from a375

unit volume of soil due to plant water uptake. The macroscopic approach of Feddes et al. (1978)

376

is used in HYDRUS to account for root water uptake. This approach first distributes potential

377

transpiration across the root zone, depending on the spatial root distribution, to obtain potential

378

root water uptake as a function of depth. Potential root water uptake is then reduced due to various

379

stresses (e.g., saturation and salinity) present in the root zone to obtain actual root water uptake.

380

Integration of actual root water uptake over the entire root zone then gives actual transpiration.

381

Root water uptake can be either non-compensated or compensated, i.e., reduced uptake from a382

stressed part of the root zone is compensated by an increased uptake from a non-stressed part of383

the root zone (Šimůnek and Hopmans, 2009).

384

The root solute uptake model implemented in HYDRUS was developed by Šimůnek and385

Hopmans (2009). A similar macroscopic approach as the one described above for root water uptake

386

is also used for root solute uptake. Root solute uptake is defined as the sum of passive and active

387

root solute uptakes. Passive uptake is simulated by multiplying root water uptake by a dissolved

388

nutrient concentration for concentration values below a priori-defined maximum uptake

389

concentration (cmax). The solute dissolved in water is taken up by plant roots when cmax is large

390(larger than the dissolved concentration c), while no solute is taken up when cmax is equal to zero.

391

Active solute uptake is activated when passive solute uptake is insufficient to satisfy the user-

392

specified potential active nutrient uptake rate. Similar to root water uptake, active root solute393

uptake can be either non-compensated or compensated, with the latter having increased solute394

uptake from parts of the root zone with higher solute concentrations compensating for insufficient

395

uptake from parts of the root zone with low solute concentrations (Šimůnek and Hopmans, 2009).

396

The spatial extent of the root zone can be constant during the HYDRUS simulations, or

397

various approaches can be used to describe the root growth as a function of time. One can use a398

logistic growth function or define an a priori the rooting depth at particular times. Options to399

simulate root growth directly using the HYDRUS software or its various couplings with crop

400

growth or root growth models were reviewed by Šimůnek et al. (2018) and Hartmann et al. (2018).
14

401

3. Sprinkler Irrigation

402
403

Sprinkler irrigation is one of the main methods of irrigation used around the world. In the404

US, it is the dominant irrigation type representing approximately 60% of all irrigation systems,405

with most of them being center pivots. Of all pressurized irrigation systems, sprinkler irrigation,406

in comparison to drip, costs less per unit of irrigated land (Amosson et al., 2021), particularly for407

large areas. There are several forms of sprinkler irrigation, but the two main ones involve water408

application as droplets of varying sizes or as thin streams of water emanating from the sprinkler

409

nozzle. In practice, many sprinkler irrigation devices employ both techniques to varying degrees.

410

The irrigation efficiency of sprinkle irrigation ranges between 60 and 90%, with an average of411

75% (Gilley and Watts, 1977; Waller and Yitayew, 2016). To ensure efficient utilization of applied

412

water, it is important that irrigation devices are properly installed and operated under design413

conditions and that there is a good understanding of soil properties/conditions and anticipated414

water flow within the profile. For the latter, the HYDRUS model (Šimůnek et al., 2008, 2016) has

415

been widely applied to simulate soil water dynamics under irrigation, using various methods (e.g.,

416

drip and surface/flood) and management scenarios. This section discusses the application of417

HYDRUS for sprinkler irrigation modeling.

418
419

Existing HYDRUS Applications for Sprinkler Irrigation

420

Sprinklers are usually designed to apply uniform amounts of water across the entire soil

421

surface or crop canopy. Except for variable rate irrigation systems, in well-implemented sprinkler

422

irrigation systems, there is less concern about the occurrence of vastly uneven water application

423

across a field compared to a system such as drip irrigation. The question of soil water redistribution

424

under sprinkler irrigation has been examined by studies such as Martello et al. (2015) and Oker et425

al. (2021). Martello et al. (2015) used HYDRUS (2D/3D) to simulate the effect of canopy

426interception and stem flow in maize on soil moisture redistribution following sprinkler irrigation427

or rainfall. Even though sprinklers are designed to apply water uniformly across the whole surface,428

canopy interception distorts the default water application pattern, significantly affecting irrigation429

uniformity and efficiency across a field. Oker et al. (2021) used HYDRUS (2D/3D) to simulate

430

soil water distribution under a Low Elevation Spray Application (LESA) sprinkling system and
15

431

compared it to that of a Mobile Drip Irrigation (MDI) nozzle package installed on a center-pivot

432

system.

433

HYDRUS has also been applied to model the interactions between soil salinity dynamics

434

and its management using sprinkler irrigation (Phogat et al., 2018; Wang et al., 2017b; Yang et al.,4352019). In many parts of the world, soil salinity is a growing problem (Hopmans et al., 2021). Yang436

et al. (2019) used HYDRUS-1D to assess the effectiveness of sprinkler irrigation in leaching salts

437

from the root zone of an almond orchard and compared the results against those from drip

438

irrigation. Their study showed that sprinkler irrigation is more efficient than drip for leaching salts

439

from the root zone. Phogat et al. (2018) used HYDRUS-2D to model water and salinity dynamics

440

in sprinkler-irrigated almonds exposed to varying salinity levels at different growth stages. In this441

study, less saline irrigation water was substituted for recycled irrigation water in three442

phenologically different almond growth stages to better understand soil water and salinity443

dynamics under almonds irrigated with waters of varying salinity levels. The study found that: (1)444

irrigation with saline water decreased seasonal almond evapotranspiration by 10% and increased445

salinity by 54%; (2) winter rainfall, or non-saline water substitution, was insufficient to reduce soil

446

salinity below the 1.5 dS/m almond tolerance level and: (3) drainage beyond a 2-m root zone was447

irregular and varied significantly during the growth season.

448

Beyond soil water redistribution and salinity, HYDRUS has also been applied to study

449

other physical phenomena and interactions in the vadose zone. Naghedifar et al. (2018) used450

HYDRUS-1D to assess groundwater return flow from a sprinkler irrigation system and compared451

it with flood irrigation. Their study found that return flow from the sprinkler irrigation system was452

negligible, whereas that from flood irrigation was 13.3%. Wang et al. (2015) coupled HYDRUS-

453

1D with the EPIC crop model to assess the irrigation management in sprinkler-irrigated wheat in

454

China. Alves de Oliveira et al. (2019) used HYDRUS-2D to simulate the transport of atrazine in455

the soil profile under sprinkler irrigation.

456
457

An Example of a HYDRUS-2D Application to Sprinkler Irrigation

458

Center-pivot sprinkler irrigation is one of the most commonly used methods of pressurized459

irrigation for field crop production, and these systems are usually fitted with sprinkler or spray

460

nozzle packages (hereafter, the term “sprinkler” is used to refer to both sprinkler and spray461

nozzles). In operation, a sprinkler mounted on a center pivot travels along a circular path across a
16

462

field, as opposed to a stationary or fixed one. Therefore, each point of the soil surface along the463

sprinkler path is subjected to a varying water application rate as the center pivot advances towards

464

and away from that point. In contrast, the application rate of fixed sprinkler devices is relatively

465

constant. A unique variation of sprinkler technology also used in center-pivot systems, the LEPA

466

bubbler (Low Energy Precision Application) (hereafter, bubbler), is designed to concentrate water467

application on the soil surface area directly underneath the nozzle.

468

A case study of applying HYDRUS (2D/3D) to model soil water redistribution following

469

irrigation using a sprinkler and bubbler, both mounted on a center pivot, is presented below (the470

Furrow examples in Table A). The key assumptions are: (1) both the sprinkler and the bubbler are471

mounted 80 m from the pivot point, and nozzle spacing is 1.5 m; (2) sprinkler and bubbler-wetted472

diameters are equal to 3 and 0.762 m, respectively; and (3) irrigation application depth is 2.54 cm,

473

and pivot travel time is 44 h per revolution.

474

For both devices, soil water redistribution up to a depth of 1.8 m is simulated. The

475

representative application rate of the sprinkler is shown in Figure 1. Two soil types were476

considered: loamy sand and silt loam. Their soil hydraulic properties were taken from the477

HYDRUS database (Carsel and Parrish, 1988). The initial conditions (Figure 2a) represent an

478

unsaturated soil profile, similar to usual field conditions before an irrigation event. The sprinkler

479

irrigation is implemented in HYDRUS-2D as an “atmospheric” boundary condition at the top of480

the model domain (soil surface). The sprinkler application rate, shown in Figure 1a, is implemented

481

using HYDRUS-2D’s “Time Variable” boundary conditions. For simplicity, the graph is482

discretized into 25-time steps, and corresponding application rates are computed. Because of483

approximation errors introduced by the temporal discretization, the estimated irrigation depth is484

3.08 cm. The bubbler is also implemented as an atmospheric boundary condition over two lengths

485

of 38.1 cm each, on the right and left sides of the top of the model domain. Figure 1b shows the486

application rate of the bubbler. A “free drainage” boundary condition is implemented at the bottom

487

of the model domain (depth of 183 cm), and a “no flux” boundary condition is used one the right488

and left sides. Soil water redistribution is simulated for a period of 24 hours (Figure 3). Figure 2

489

shows the initial and final water contents in loamy sand and silt loam in the middle of the domain,490

while Figure 3 shows the final water contents in loamy sand and silt loam at different locations of491

the transport domain.

17

492

Infiltration rates for the loamy sand and silt loam soils are plotted against the sprinkler and493

bubbler application rates (Figure 1). As shown in Figure 1a, the infiltration rate of the loamy sand494

matches the sprinkler application rate at every time step, resulting in no surface runoff. For the495

bubbler, however, the infiltration rate of the loamy sand matches the application rate up to t = 0.01

496h, after which runoff is generated. For the silt loam, the infiltration rates increase in tandem with497

the sprinkler and bubbler application rates till t = 0.07 and 0.01 h, respectively, after which they498

begin to decrease. Subsequently, the difference between the application and infiltration rate is499

equivalent to potential runoff. In general, potential runoff occurs much more quickly for the500

bubbler because it has a significantly higher water application rate than the sprinkler (Figure 1b).

501

Figures Figure 4a and Figure 4b show cumulative infiltration and potential runoff for the loamy

502

sand and silt loam soils following irrigation. As expected, the simulation results show that silt loam503

has the highest potential runoff of 1.85 and 2.08 cm for sprinklers and bubblers, corresponding to504

losses of 60.1 and 81.9%, respectively. The potential runoff for the LEPA application on loamy

505

sand is 1.4 cm (55.1%).

506

In a well-drained and managed field, the potential surface runoff would manifest as ponded507

water, eventually infiltrating into the soil. The two-dimensional module of HYDRUS does not

508

facilitate the accumulation of ponded water on the soil surface. Instead, computed surface runoff

509

is instantaneously removed. However, an option to accumulate ponded water on the soil surface is510

available in HYDRUS-1D. If runoff is assumed to be negligible, implying that all applied water

511

should infiltrate into the soil profile, then both sprinkler and bubbler irrigation can be implemented

512

in HYDRUS-2D as a “time-variable flux” that is lower than the saturated hydraulic conductivity

513

of the soil. A representative application time should then be computed to ensure that only the514

desired irrigation depth is introduced into the model domain. This implies that the actual

515

application of the sprinkler, or bubbler, is not considered but rather translated to a “time-variable516

flux.”

517

18

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

0.1

0.2

0.3

0.4

0.5

0.0

0.1

0.2

Time (h)(a) Sprinkler
518
519
520

0.3

0.4

0.5

Time (h)(b) LEPA

Figure 1. Application rates of a sprinkler (a) and LEPA bubbler (b), both located 80 m from thecenter of a pivot and applying an irrigation depth of 2.54 cm, with corresponding infiltration
rates for loamy sand and silt loam soils.

0

0

40

40

Soil depth (cm)Soil depth (cm)

521

80

80

120

120
160

160

200

200
0.05

0.15

0.25

0.05

0.35

0.15

0.25

0.35

Water content (cm3/cm3)

Water content (cm3/cm3)(a) Time = 0 h522
523

Loamy sandSilt loam(b) Time = 24 h

Figure 2. (a) Initial and (b) final (t = 24 h) water contents in loamy sand and silt loam soils in the
HYDRUS-2D simulation of soil water redistribution following sprinkler irrigation.

524

19

0

40

40

Soil depth (cm)Soil depth (cm)

0

80

120

120

0 cm
38 cm
19 cm
76 cm

160

80

160

200

200
0.05

0.10

0.15

0.20

0.05

Water content (cm3/cm3)

0.15

0.35

Water content (cm3/cm3)(a) Loamy sand
525
526
527
528

0.25

(b) Silt loam

Figure 3. The soil moisture content at t = 24 h at distances of 0, 19.1, 38.1, and 76.2 cm from theedge 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 (Figure 2a).

529
4.0

Potential runoff (cm)Infiltration (cm)

4.0

3.0

2.0

1.0

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

3.0

2.0

1.0

0.0

0.0
0.0

0.1

0.2

0.3

0.4

0.0

0.5

0.1

0.2

0.3

0.4

0.5

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

Figure 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.
530

20

4. Furrow Irrigation

531
532
533

Despite being replaced by pressurized irrigation in developed countries, surface irrigation

534

systems, such as furrows and basins, remain widespread among farmers in developing regions

535(FAO, 2011). Water (and sometimes also solutes) is conveyed by gravity flow to sections of a field

536

that is either leveled (border-dyke irrigation) or prepared with parallel ridges and furrows (furrow537

irrigation). In the process, water moves over dry soil, generating a front that transports water and538

nutrients along the field and simultaneously into the soil via infiltration. When correctly designed,539

these systems can simultaneously minimize deep drainage and leaching below the crop root zone,

540

maximize water and nutrient distribution to the ridges (where the plants are growing), and541

uniformly irrigate the field, thus leading to optimal crop yields (Fahong et al., 2004; Horst et al.,542

2005; Siyal et al., 2012; Šimůnek et al., 2016a). Yet, the interacting surface and subsurface

543

transport processes involved make their design and implementation a challenging and complex

544

task.

545

Ubiquitously, this problem is tackled by using models that can predict and/or explain the546

system's behavior under different operating conditions at different levels of complexity. Setting

547

aside data-driven models (Mattar et al., 2015), numerical models provide a theoretical and548transferrable framework to explain and predict physical processes involved in surface irrigation,

549

which can be used to design and optimize these systems. Several models have been developed in550

the literature to simulate the coupled surface-subsurface transport. Earlier attempts used zero-

551

inertia and kinematic wave approximations of the shallow water equation coupled with a modified

552

Kostiakov function to simulate infiltration (Katopodes and Strelkoff, 1977; Oweis and Walker,

553

1990; Strelkoff and Katopodes, 1977; Walker and Humpherys, 1983). Later, the advection-554

dispersion equation was added to simulate solute transport (Abbasi et al., 2003b; Perea et al.,5552010). Despite their good predictive performance, these models generally oversimplify subsurface

556

transport processes.

557

To overcome this limitation, analytical or numerical zero-inertia formulations were558

coupled with the Richards equation to mechanistically describe transport processes in the559

subsurface (Ebrahimian et al., 2013; Tabuada et al., 1995; Wöhling et al., 2004; Wöhling and560

Schmitz, 2007; Zerihun et al., 2005; Köhne et al., 2011). Of particular relevance are the pseudo-

561

3D models developed by Tabuada et al. (1995) and Wöhling et al. (2004), which provide a 1D and
21

562

2D description of surface and subsurface water flow along the channel, respectively. This

563

approach has an important utility as it allows assessing the irrigation uniformity in different 2D

564

cross-sections along the furrow. Nevertheless, while accurately simulating water flow, neither of565

the above models considered solute transport in surface and subsurface domains. Furthermore,566

their use of a Lagrangian approach poses significant numerical challenges (Wöhling et al., 2006).

567
568

Existing HYDRUS Applications to Furrow Irrigation

569

While the HYDRUS model (Šimůnek et al., 2016b) has been quite widely used to simulate

570

water flow and solute transport under furrow irrigation systems, it has typically been used only to571

simulate processes in the two-dimensional transport domain perpendicular to furrow without

572

considering the flow and transport dynamics in the furrow itself (Bristow et al., 2020; Šimůnek et573al., 2016a; Siyal et al., 2012; see also many more references listed in (Šimůnek et al., 2016b) and574

on the HYDRUS website).

575

Recently, Brunetti et al. (2018b) developed a hybrid Finite Element-Finite Volume pseudo-

576

3D model to simulate the coupled surface-subsurface water flow and solute transport in blocked

577

and open-ended furrows. The model combines the two-dimensional Richards-based Finite-

578

Element solver, HYDRUS-2D, with a one-dimensional Finite-Volume discretization of the zero-579

inertia and advection equations. The model has been successfully validated in synthetic and580

experimental scenarios. Its main advantages are its computational efficiency, mass conservation

581

performance, and the detailed description of water flow, solute transport, and root water and solute582

uptake along the entire furrow channel. To increase its usability and widespread use among583

scientists and practitioners, the model has been equipped with a GUI and embodied an add-on

584

module585

progress.com/en/Default.aspx?h3d2-Furrow). Details about the model and the GUI can be found586

in Brunetti et al. (2018b) and Brunetti et al. (2019c).

in

the

newly

released

HYDRUS

5

softwaresuite(https://www.pc-

587
588

An Example of the HYDRUS Furrow Module in Multiple Modeling Scenarios

589

Selected modeling scenarios are used to demonstrate the potential applicability of the590

HYDRUS Furrow module for fertigation designing purposes. As pointed out by Siyal et al. (2012),

591

there is a need to improve water, fertilizer, and soil surface management strategies to increase592

irrigation efficiency and reduce nitrogen losses due to leaching into groundwater. Most of the
22

593

existing research has focused on fertilizer placement (Mailhol et al., 2007; Siyal et al., 2012;

594

Waddell and Weil, 2006) and irrigation management (Abbasi et al., 2003b; Mailhol et al., 2007),

595

while only a few studies have investigated the effect of soil surface management to reduce leaching596

and maximize fertigation efficiency (Bristow et al., 2020; Šimůnek et al., 2016a; Siyal et al., 2012).

597

In particular, Šimůnek et al. (2016a) used HYDRUS-2D to examine the effect of compacting the598

bottom of the furrow or placing a plastic sheet on the bottom of the furrow on the fertigation599

uniformity and efficiency. While results reported in these studies were promising, numerical

600

simulations neglected the complex interacting surface-subsurface processes that occur during the601water advance-storage-recession phases that are typical for furrow fertigation. The analysis was602

limited to two-dimensional vertical infiltration, which neglects the differences of the longitudinal

603

distribution of water and solute in the field hat result from furrow fertigation. A similar approach604

was used, and similar limitations were encountered in the recent work of Pahlevani et al. (2021).

605

Therefore, in the present study, the HYDRUS Furrow module is used to simulate the606

influence of different soil management strategies on the fertigation of a 100 m-long blocked-end

607

furrow (Table A). A non-reactive solute is considered in the analysis. The geometric characteristics

608

of the furrow and input data are reported in Table 1. In particular, the effect of the plastic sheet

609(Sp) and soil compaction (Sc) is analyzed (Figure 5) against a baseline scenario (S0). The soil

610

hydraulic parameters of loamy sand soil considered in this simulation are taken from the HYDRUS

611

soil catalog (Carsel and Parrish, 1988) and reported in Table 1. For the Sc scenarios, the saturated

612

hydraulic conductivity of the top 2 cm of the soil is decreased to 1/10th of the original uncompacted

613

value. Soil compaction in the field can be achieved by tractor wheels or with the Eversman v-

614

shaped wheel (Siyal et al., 2012).

615

23

616
617
618

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

619

The subsurface vertical domain is discretized into 2049 triangular elements and 1078

620

nodes. The distribution of boundary conditions for different soil management scenarios is shown621

in Figure . The groundwater table is assumed far below the domain of interest, and thus a ‘Free

622

Drainage’ BC is assigned to the bottom nodes, while an ‘Atmospheric’ boundary condition is set

623

for the furrow ridge. Due to symmetry, the nodes representing the left and right boundaries of the624

subsurface domain are set as ‘no-flow’ boundaries because no water flow or solute transport

625

occurred across these boundaries. A hybrid Dirichlet/Neumann (variable pressure/zero flux) BC is626

assigned to the border of the trapezoidal channel to simulate variations of the water depth in the627

furrow. Finally, a ‘Third Type’ Cauchy BC was set on top and bottom of the numerical domain to628

simulate the concentration flux along the boundaries. Plastic placement on the bottom of the furrow

629(Sp) is simulated by setting a no-flow boundary condition at the bottom of the furrow, so that630

infiltration can occur only through the sides of the furrow.

631
632
633
634

24

635

Table 1. Geometric characteristics, soil hydraulic properties, and other input parameters used in636

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 (m3/m3)

0.057

Saturated water content, θs (m3/m3)

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

Initial water content, θ0 (m3/m3)

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

637
638

Figure 6 and Figure 7 show the simulated distributions of soil water contents and nitrate639

concentration, respectively, at time t = 5,400 s at different locations along the furrow for the640baseline (S0), compacted (Sc), and plastic (Sp) scenarios. Simulation results show that the

641

distribution of water contents and solute along the furrow in the baseline scenario is not perfectly642

homogeneous. Water and solute mostly accumulate in the first part of the furrow, where they643

percolate towards deeper soil horizons, while their penetration into the furrow ridge decreases
25

644

along the furrow. This leads to an inhomogeneous fertigation scheme with the upper part of the645

furrow receiving more water and solute and thus expected to have higher crop yields (or larger

646

nutrient losses by leaching) than the lower part of the furrow, more exposed to potential root water647

and nutrient stresses. The situation radically changes when the soil at the bottom of the furrow is648

compacted. The reduction of the soil hydraulic conductivity diminishes infiltration and increases

649

velocities of the waterfront traveling in the furrow, thus leading to a more homogeneous

650

distribution of the irrigation water along the field (and thus, potentially, to more uniform crop

651

yield). Furthermore, the lower soil hydraulic conductivity of the bottom of the furrow increases

652

the water level in the channel and favors the infiltration into the ridge.

653

Interestingly, Figure shows that, at the end of the recession phase, the final part of the654

furrow is slightly moister in the compacted (Sc) scenarios. This is mainly due to the alteration of655

the infiltration patterns between the sides and the bottom of the furrow for this management656

strategy. Nevertheless, solute remains well distributed (Figure 7). The placement of a plastic sheet

657

at the bottom of the furrow overcomes this problem and presents the greatest benefits. The

658

irrigation and fertigation homogeneity increases, similarly as water and solute infiltration into the659

furrow ridge. The amount of water and fertilizer that leaches under the furrow bottom is lower

660

compared to other analyzed scenarios.

661

The analysis shows how the HYDRUS Furrow module can be used to design efficient662

fertigation strategies by providing a comprehensive and realistic description of overland and663

subsurface transport processes.

664

26

665

666
667
668
669

Figure 6. Simulated distributions of soil water contents at time t=5,400 s at different locations
along the furrow (horizontal direction) for the original (S0), compacted (Sc), and plastic (Sp)scenarios.

670
671
672
673
674

Figure 7. Simulated distributions of solute concentration at time t=5,400 s at different locations
along the furrow (horizontal direction) for the original (S0), compacted (Sc), and plastic (Sp)scenarios.
27

675

5. Drip irrigation

676
677

Existing HYDRUS Applications to Drip Irrigation

678

Drip irrigation is the most common method for high-value crops but is also very common

679

for irrigating staple food crops in water-limited environments. This method has many advantages,

680

including optimal control over the amount of irrigation water and the concentration of fertilizers

681

given to the plant, so farmers have widely adopted the method. Previous studies have found that682

the same amount of water given at different timings can yield different crop yields (Or and Hanks,

683

1993). According to Rawlins and Raats (1975), the high water content in the root zone causes high684

soil water pressure head and hydraulic conductivity and, therefore, an increase in water availability

685

to the plant. Stansell and Smittle (1989) argued that high water content in the root zone has a direct686

effect on the rate of transpiration and stomatal conductance, and therefore frequent irrigation will687

increase the efficiency of water use, photosynthesis, and yield.

688

Effective design and management of drip irrigation systems lead to the effective delivery

689

of water and nutrients to the root zone resulting in higher water use efficiency and improved yields.

690

Many design and management decisions involve understanding the wetted zone pattern around the691

emitter (Bresler, 1978; Lubana and Narda, 2001) and its relation to the root system structure and692

function. Water distribution in the soil is affected by many factors, including soil hydraulic693

characteristics, initial conditions, emitter discharge rate, application frequency, root694

characteristics, evaporation, and transpiration. HYDRUS (2D/3D) (Šimůnek et al., 2016) has been

695

previously used to successfully simulate water flow for surface and subsurface drip irrigation

696

systems (e.g., Skaggs et al., 2004; Gärdenäs et al., 2005, Kandelous and Šimůnek, 2010; Honari et697al., 2017; Domínguez-Niño et al., 2020ab; see many more references listed on the HYDRUS

698

website: https://www.pc-progress.com//Documents/Jirka/Drip_irrigation_papers.pdf).

699

In both surface and subsurface drip applications, irrigation is often modeled using a

700

constant water flux across a boundary with fixed dimensions. However, the soil's infiltration

701

capacity depends on the initial conditions and the soil hydraulic properties. When the prescribed702

boundary flux is larger than the soil's infiltration capacity, the model will force the water to703

infiltrate into the soil at a rate higher than this capacity, which will cause the pressure head at the704boundary to rise to unphysical values, with consequences on the flow and soil water patterns. To

705

overcome such problems with unphysical boundary pressure heads, various system-dependent
28

706

boundary conditions (e.g., dynamic surface wetting for surface drip irrigation or drip characteristic707

function for subsurface drip irrigation) were implemented in the HYDRUS code (see Section 2) to708

facilitate proper physical characterization of the flow field around the surface and subsurface709

drippers.

710

Another system-dependent boundary condition is used to trigger irrigation when certain

711

predetermined conditions in the system are reached (Dabach et al., 2013; Müller et al., 2016). This

712

condition initiates irrigation whenever the matric head at a predetermined observation point drops

713

below a certain pressure head threshold.

714
715

Surface Drip Irrigation

716

First, simulations of surface drip irrigation can be made using the boundary condition

717

accounting for dynamic surface wetting (Gärdenäs et al., 2005). This condition allows the spatial

718

distribution of the discharge rate at the soil surface to change with time. A ponded zone around719

this point is created when water is applied at a constant rate to a point on the soil surface. The

720

ponded area increases with time at the beginning of irrigation, but later approaches a constant721

circular area (Shani et al., 1987). Figure 8 demonstrates dynamic wetting of the soil surface for722

surface drip irrigation of two soils of different textures (soil hydraulic parameters for loam and723

sandy loam were taken from the HYDRUS soil catalog). In this example (the SurfaceDrip

724

examples in Table A), 4 L were applied at the dripper discharge rate of 2 L/H using the system-

725

dependent boundary condition accounting for dynamic surface wetting. Figure 8 shows different

726

responses in the two soils. While in the soil with higher saturated hydraulic conductivity (i.e.,

727

sandy loam), the radius reached a steady state in a relatively short time, in the heavier soil (i.e.,728loam), this process took much longer, showing the need for this system-dependent boundary729

condition.

730

29

731
732

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

733
734

In recent years, low flow rate emitters with flow rates of 0.4-0.6 L/H have been developed,

735

even with pressure regulation. The effect of the discharge rates on soil water flow and root water736

uptake by plants was studied by Assouline (2002) using HYDRUS-2D. The simulated results

737

showed that lowering the emitter flow rate from 2 L/H to 0.25 L/H increased the soil water content

738

in the upper layer of the root zone and root water uptake, diminishing the reduction of potential

739

transpiration. However, these results were greatly affected by the description of the root system,

740

which was considered uniform between irrigations. Another study found that when irrigating the741

soil with the same amount of water at different flow rates, no differences were found in the water742

content distribution after the redistribution process (Skaggs et al., 2010). It should be noted that743

this study was done without plants that can change the soil wetting pattern because of the744interaction between soil moisture, emitter flow rate, and root system structure and function.

745

Further attempts to describe water flow in the soil at low flow rates using HYDRUS

746(2D/3D) were made by Lazarovitch et al. (2009). These numerical experiments were performed

747

without the presence of plants, and the conclusions are thus suitable only for the first stages of748

plant growth, in which root water uptake is minimal. The following examples illustrate how the749root structure and response to water stress influences the plant’s ability to uptake water while

750

surface-drip-irrigated with a continuous low flow rate (the DripPlant examples in Table A). The
30

751

simulation duration was one week, and the discharge rate was 0.4 L/H (). The critical stress index

752(c), which accounts for root water uptake compensation in the root water uptake model, was set

753

either equal to 1, i.e., no compensation, or to values smaller than 1, accounting for root water754

uptake compensation (i.e., reduced uptake from one part of the domain is compensated by755

increased uptake from other parts). The maximum rooting depth, Zm, was set to 20 or 30 cm.

756

Figure 9 shows ratios between actual transpiration, calculated by the model, and potential

757

transpiration for different values of the critical stress index and two maximum rooting depths. It

758

can be seen that the issue of compensation in root water uptake is very important and has

759

implications for a reliable description of deficit-irrigated systems, particularly for shallow root760

systems. There are currently insufficient data to better understand the compensation in root water761

uptake, and more studies are needed in that direction.

762

763
764

Figure 9. Relative root water uptake as a function of the critical stress index and two rooting

765

depths.

766
767

Subsurface Drip Irrigation

768

Second, simulations of subsurface drip irrigation can be made with a system-dependent769

boundary condition that describes the effect of back pressure on the dripper discharge rate while

770

considering the source characteristics, inlet pressure, and the effects of the soil hydraulic properties
31

771

(Lazarovitch et al., 2005; Naglic et al., 2014). When a predetermined discharge of a subsurface

772

emitter is larger than the soil infiltration capacity, the pressure head in the source outlet increases

773

and becomes positive. The built-up pressure head may significantly reduce the source discharge

774

rate.

775

One of the challenges in developing these models is the ability to accurately describe the776

uptake of water and nutrients by roots under different growing conditions. The HYDRUS (2D/3D)

777

model can simulate water flow and the transport of nutrients when the space is two- or three-778

dimensional, as in drip irrigation, or one-dimensional, as in sprinkler irrigation. This model reduces779

potential root water uptake due to saturation (i.e., dryness) and salinity stresses (van Genuchten,

7801987) and allows for passive and active nutrient uptake (Šimůnek and Hopmans, 2009). The root

781

absorption functions have a number of parameters without which reliable simulations cannot be

782

performed. These parameters cannot be evaluated directly due to the inability to maintain constant783

water content or nutrient concentrations in the root zone during the experiment.

784
785

Inverse Problem

786

Another option for evaluating the parameters is to solve the inverse flow and transport

787

problem that includes the uptake of water and nutrients by roots (Vrugt et al., 2001a,b; Hupet et788

al., 2002, 2003). In this approach, the plant traits are estimated by assuming a deterministic model789

and calculating the parameters of the selected model. The assumption, in this case, is that the790

characteristics of the plant can be described using a deterministic model that contains a relatively

791

small number of parameters typical of each plant. Estimation of plant properties is obtained by792

optimizing the deviations between results calculated by numerically solving the flow and transport793

equations and measured flow variables, such as pressure heads, water contents, fluxes, or nutrient

794

concentrations, at relevant times and space locations.

795

A calibrated numerical model can then be used to optimize system variables, leading to an

796

optimal irrigation and fertilization regime that will maximize profit for the farmer and preserve the797

environment. Questions regarding the optimal emitter flow rate or spacing between the emitters798

can be answered with the help of the numerical model.

799
800
801
32

802

Geometrical Description

803

In the inverse problem method, the model has to be executed several times, and for the804high accuracy of the results, fine spatial and temporal discretization is required. For drip irrigation

805

problems, it is almost impossible to run the optimization problems on a regular computer. In this806

situation, there is a constant attempt to find symmetry lines in the system to reduce the807

computational load and streamline the process. One symmetry line exists in the center of a808

subsurface emitter if the soil and the root zone look similar on both sides of the emitter. Another809

line of symmetry can be found at half the distance between emitters. A line of symmetry is810

equivalent to a no-flow boundary since water contents and pressure heads are the same on both811

sides of the hypothetical line of symmetry, resulting in zero pressure head gradients perpendicular812

to this boundary. Kandelous et al. (2011) simulated subsurface drip irrigation as a two-dimensional813(i.e., infinite line source), axisymmetrical two-dimensional (i.e., a point source), and three-814

dimensional (i.e., a full description of the drip line geometry) problem, and discussed when and815

under what conditions each of these approaches should be used.

816

Figure 10 shows the simulated results for two alternative ways of describing water flow817

around a subsurface emitter. Water flow is simulated in a fully three-dimensional domain (Figure

81810A) or a two-dimensional domain exhibiting radial symmetry around the vertical axis (Figure

819

10B). Figure 10C shows the soil water contents at three locations 5, 15, and 25 cm away (radially)

820

from the dripper. These simulations (the Drip3D examples in Table A) depict two cycles of821

subsurface drip irrigation and redistribution. Initially, there is irrigation for an entire day, and then822

there is a cessation of irrigation for four days. It is clear that the water content response with time823

is almost identical in both domains, and the line of symmetry between emitters can be used. The

824

simulation duration (the CPU time) for a three-dimensional domain was 679 s, and for an

825

axisymmetric domain, 11.8 s. This significant time reduction (orders of magnitude) allows for826

faster evaluation of complex problems involving entire growing seasons and the use of the inverse

827

problem to estimate parameters.

828
829

33

A

B

-100.000

0.121 0.124 0.130 0.137 0.143 0.149 0.156 0.163 0.169 0.176 0.182 0.191

830

Project drip_3D
Results, Water Content, Time 5 - 2.5 days

-88.000-82.500-77.000-71.500-66.000-60.500-55.000-49.500-44.000-38.500-37.457

Pressure Head - h[cm], Min=-100.000, Max=-37.457

Water Content - th[-], Min=0.121, Max=0.191

Project drip_3D_axi
Results, Pressure Head, Time 5 - 2.5 days

C

831
832
833
834
835

Figure 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 from the dripper)
as a function of time (C)34

6. Fertigation

836
837
838

The HYDRUS software has been used over the years in many studies evaluating the fate839

and transport of various chemicals used in irrigated agriculture, including nitrogen species. Due to840

its flexible boundary conditions and well-designed graphical user interface, the HYDRUS model841

has been widely used to capture soil nitrogen dynamics, especially for different irrigation and842

fertilization strategies (Chen et al., 2020bc; Mekala and Nambi, 2017ab; Phogat et al., 2013). It843

has been used, for example, to quantify processes such as N leaching, transformation (e.g.,844

mineralization, nitrification, denitrification, volatilization, and assimilation), and root uptake under845

different management conditions (e.g., mulching, tillage), different irrigation scenarios (e.g.,846

furrow or drip irrigation, different irrigation rates), and different planting systems (e.g.,847

monoculture or intercropping systems). This section reviews these HYDRUS applications.

848
849

Soil Nitrogen Transformation

850

Nitrogen, one of the most abundant elements in the atmosphere, plays a critical role in plant851

photosynthesis and primary production in natural terrestrial ecosystems (Makino, 2010; Sorrell et852

al., 2013). In general, biogeochemical transformations of nitrogen in the soil-plant-atmosphere

853

continuum mainly include internal and external cycles. The external cycle refers to nitrogen854

transport among different ecosystems. The internal cycle refers to nitrogen transformations among

855

different pools in the same ecosystem. These transformations can include mineralization (i.e.,856

nitrogen fixation), assimilation, nitrification, denitrification, and volatilization (Figure 11). High857

crop yield and sustainable agricultural ecosystems can be achieved only when the internal nitrogen858

cycle is well understood (Subbarao et al., 2012).

859

35

860
861
862
863

Figure 11. Soil nitrogen transformations in the internal nitrogen cycle (Nelson et al., 2016). Notethat assimilatory pathways are in orange and dissimilatory pathways in blue.

864

The HYDRUS model has been widely used to simulate nitrogen transformations under865

different environmental conditions, such as different soil textures, different irrigation methods, or866

different mulching scenarios (Azad et al., 2020; Salehi et al., 2017; Pahlevani et al., 2021;

867

Castaldelli et al., 2018). Apparent differences in nitrogen transformations can be found in different

868

scenarios and conditions. For example, the nitrification rate in sandy soils is faster than in clay and869

sandy loam soils, while nitrogen assimilation in clays is significantly higher than in the other two

870

soils. Mekala and Nambi (2017ab) evaluated the impact of different soil textures on nitrogen871

transformations using the HYDRUS model. They found that nitrification in sandy soil is872

significantly higher than in sandy loam soil. On the other hand, denitrification can become a873

dominant nitrogen transformation process in sandy loam soils, especially at saturations higher than874

96%. Berlin et al. (2015) illustrated that nitrification in soils with high clay content was lower than875

in soils with low clay content and that the nitrogen peak displayed hysteresis. The reason may be

876

attributed to the difference in soil organic matter of different soil textures. Usually, the assimilation877

rate (i.e., the transformation from inorganic nitrogen to organic nitrogen) increases in response to878

an increase in soil organic matter, which results in a reduction in the soil NH4-N concentration and879

a corresponding nitrification rate.

880

Nitrification is an important process of releasing energy in the nitrogen cycle, and it

881

includes autotrophic nitrification and heterotrophic nitrification (Langergraber and Šimůnek,8822005). Autotrophic nitrification refers to the process of oxidizing soil NH4-N to NO3-N by883

autotrophic microorganisms. Heterotrophic nitrification refers to transforming nitrogen into
36

884

oxidized nitrogen using organic carbon as a carbon source under the mediation of heterotrophic

885

microorganisms. There are also apparent differences in nitrification among differently irrigated886

fields, especially between dry and flooded fields (Halvorson et al., 2016; Eltarabily et al., 2019;

887

Mo'allim et al., 2018). In general, denitrification is the primary transformation process in flooded

888

fields (Becker et al., 2007), in which high saturation in the surface soil layer results in an anaerobic889

soil environment. Meanwhile, anaerobic bacteria use NO3-N as the final electron acceptor for890

respiration in an anaerobic soil environment to reduce NO3-N to N2O and N2. Volatilization of891

nitrogen gases (N2O and N2) produced by denitrification is the main cause of low nitrogen use

892

efficiency in agricultural production. However, nitrification is still stronger than denitrification in893

flooded fields with good drainage. For example, Li et al. (2018a) evaluated nitrogen transformation894

mechanisms in different soil layers of paddy farmland using the HYDRUS model and found that895

nitrification was about 25% higher than denitrification. Similarly, nitrification is the main

896

transformation process in dry farming fields due to their high soil pore oxygen content. For

897

example, Chen et al. (2020b) found using the HYDRUS model that average nitrification in a drip-

898

irrigated corn field reached 31.2 kg ha-1. Also, the nitrification rate gradually increased with a899

decrease in the irrigation depth (Zeng et al., 2017). In general, soil nitrification and denitrification

900

are mainly affected by soil's physical and chemical properties, e.g., soil temperature, soil moisture,901

and soil pH (Mekala and Nambi, 2017ab; Herman et al., 2007; Scherger et al., 2021).

902

Nitrogen transformations also show apparent differences under different mulching903

treatments due to differences in mulching ratios. Compared with no mulching, plastic film

904

mulching can increase soil temperature, promote the activity of soil microorganisms, accelerate

905

the nitrogen denitrification rate, and increase N2O emissions (Lee et al., 2022). Furthermore, the906

nitrification rate increases in response to a decrease in the mulching ratio (Chen et al., 2020b).

907

The HYDRUS model is also often used to evaluate the mechanism of nitrogen908

transformation in fields with different tillage technologies (Colombani et al., 2020; Shafeeq et al.,909

2020; Miranda-Velez et al., 2022). According to simulation results, N2O emissions in the no-till

910

field were significantly limited compared with the conventional tillage field (Yoo et al., 2017;911

Tellez-Rio et al., 2015; Chatskikh et al., 2007). This phenomenon is mainly related to different

912

soil structures under different tillage technologies. Conventional tillage technology can decrease913

soil porosity while significantly increasing oxygen content in soil pores. Thus, the nitration rate914

can be promoted in conventional tillage fields. Compared with a conventional tillage field, the
37

915

surface soil bulk density increased due to soil compaction in the no-tillage field, resulting in a916

reduction in N2O emissions.

917

Mineralization can also be accurately quantified using the HYDRUS model. For example,

918

Shekhar et al. (2021) compared mineralization rates under different soil moisture conditions and919

found that mineralization under moderate and severe water stress treatments decreased by 44.9%

920

and 36.5% compared to a no-stress treatment, respectively. Organic macromolecular compounds

921(such as proteins, amino sugars, and nucleic acids) are decomposed into biomonomers via

922hydrolysis of microbial enzymes, and the biomonomers are then further ammoniated to ammonium

923

nitrogen. Therefore, soil microbial activity is an important factor affecting nitrogen mineralization.

924

In the study of Shekhar et al. (2021), the transformation of soil organic nitrogen to inorganic

925

nitrogen was inhibited due to low soil microbial activity under water stress conditions.

926
927

Soil Nitrogen Movement928

It is crucial to accurately capture the nitrogen transport to formulate reasonable fertilization

929

application strategies, improve nitrogen use efficiency, avoid non-point source pollution, and930

promote sustainable agricultural development (e.g., Mekala et al., 2017b). Iqbal et al. (2016)931

compared the distribution characteristics of soil nitrogen in differently sized furrow-irrigated fields

932

and found that soil nitrogen accumulation significantly increased in the field with a 2:1 ridge-ditch

933

ratio compared to the 1:1 ridge-ditch ratio. Azad et al. (2018) evaluated differences in soil nitrogen934

dynamics under different irrigation rates and illustrated that the leaching amount of soil NO3-N

935

could be effectively decreased during the fertilization application period when the irrigation rate936

is 0.8 L h-1. Tao et al. (2021) studied the effect of an improved underground drainage system on937

soil NH4-N and soil NO3-N dynamics using HYDRUS. They reported that the largest decrease in938

soil nitrogen mass consistently occurred in the filter profile, and soil NH4-N in the 0-20 cm soil

939

layer and soil NO3-N in the 60-80 cm soil layer increased with the prolongation of irrigation

940

duration. Karandish and Šimůnek (2017) evaluated the effects of 176 different water and nitrogen941

management strategies on soil water and nitrogen dynamics and showed that the 200 kg ha -1 N-

942

fertilizer application is the optimal amount, which can reduce nitrogen leaching by 12-99%.

943

The HYDRUS model has also been commonly used to evaluate soil nitrogen transport

944

under different planting systems. For instance, Chen et al. (2020c) revealed the distribution

945

characteristics of soil nitrogen in the corn-tomato intercropping field and found that soil NO3-N
38

946

concentrations in the tomato's root zone were significantly higher than in the corn's root zone.

947

Moreover, the soil solute flux in the horizontal direction occurred mainly from the tomato's root948

zone to the corn's root zone. Doltra and Muñoz (2010) studied the soil NO3-N distribution in the949

sweet pepper-cauliflower-Swiss chard rotation field and illustrated that the surface soil NO3-N

950

concentration first increased and then decreased with the crop growth.

951

Since the HYDRUS model has flexible upper boundary conditions to represent different

952

mulching scenarios, the model can accurately capture the soil nitrogen dynamics under different

953

mulching scenarios. Chen et al. (2020b) compared differences in the soil nitrogen distribution

954

under biodegradable film mulching (BM), polyethylene film mulching (PM), and non-mulching

955(NM) scenarios. They showed no significant differences in soil NO3-N concentrations under BM

956

and PM in the early crop growth stage. However, soil NO3-N concentrations under BM were higher

957

than under PM 80 days after sowing due to an increase in the disintegrated area of the958

biodegradable film. Overall, the HYDRUS model can capture the soil nitrogen dynamics in959

different crop growth stages at the field scale and accurately quantify the nitrogen distribution

960

characteristics for a long time series in large-scale regions (Nasta et al., 2021).

961
962

Soil Fertilizer Leaching

963

Fertilizers have been applied in large quantities as one of the effective ways to increase964

crop yields. However, irrational fertilizer applications cause waste and lead to non-point source

965

pollution of agricultural soils (Guo et al., 2010). Nitrogen leaching can significantly damage

966

surrounding environments, including surface and groundwater bodies, affecting ecological967

security (Bouraoui and Grizzetti, 2011). Recently, many researchers have used the HYDRUS

968

model to carry out high-precision simulations of N transport and transformations under different

969

soil textures and agronomic measures (Pare et al., 2006; Zhou et al., 2006), providing effective

970

guidance and helping to optimize fertilization schemes and enhance the N use efficiency of crops

971

under different conditions to ensure food-ecological security (Azad et al., 2018; Guo et al., 2013;

972

Siyal et al., 2012).

973

Most previous studies have applied HYDRUS models to simulate water-N dynamics under974

specific soil conditions (Clement et al., 2021; Miranda-Velez et al., 2022; Singh et al., 2019). Soil

975

texture varies significantly from region to region and is an important factor that directly affects the976water-N transport processes and the amount of N leaching from the soil. Bi et al. (2003) used the
39

977

HYDRUS model to simulate the vertical transport of water and NO3-N in two soils (yellow moist

978soil and aeolian sandy soil) in the Huang-Huai-Hai Plain of China. Based on this, Rees et al. (2020)

979

constructed six reclaimed coarse-textured soil profiles consisting of two cover soils (peat-mineral980

mix and forest floor mineral mix) and simulated nutrient leaching from these six profiles, using981

the simplicity of HYDRUS customizable soil profile properties of each layer. Results revealed that982

the peat-mineral mix retained 44% of the initial inorganic N within the surface 20 cm of the983

reclaimed soil profiles after heavy rainfall, while 84% of the inorganic N was leached from the984

forest floor mineral mix.

985

Taking advantage of the HYDRUS models' ability to customize soil profiles using different

986

soil characteristics, soil N leaching under different long-term conditions could be simulated to987

provide data to support the development of the best irrigation and fertilization scenarios. Azad et988

al. (2018) used a two-stage optimization approach and a calibrated HYDRUS model to develop an

989

optimal water and fertilizer management program for maize fields for the soil reclaimed for990

growing crops. Azad et al. (2018) simultaneously optimized the irrigation flow rate, application

991

starting time, and fertilizer application duration to minimize NO3-N leaching in a fertigation cycle.

992

Correspondingly, the fertilizer amount at each crop growth stage was optimized throughout the993

growth season by comparing multi-scheme simulations based on the optimized irrigation rate.

994

HYDRUS-simulated results provided insight into how cover crops affect N leaching losses,

995

helping optimize the design of a soil cropping system to minimize N losses.

996

Anthropogenic agronomic measures significantly affect root zone leaching of soil NO3-N

997(Arbat et al., 2013; Shaygan et al., 2018ab). Shafeeq et al. (2020) investigated the effects of several

998

conservation agriculture practices (conventional tillage, permanent broad beds (PBB), zero tillage

999(ZT), PBB with residues, and ZT with residues) on the spatial-temporal distribution of soil

1000

moisture and NO3-N during the wheat growth stages, inversely optimized the relevant reaction

1001

parameters (i.e., nitrification rates, and the distribution coefficient), and predicted daily variations

1002

of NO3-N in the soil profile. Based on the HYDRUS simulation results, this study recommended

1003

using PBB with residues for wheat cultivation in the maize-wheat cropping system to improve1004

water and N use efficiency and reduce water and N losses.

1005

To address the problem of water and nutrient leaching below the root zone in light-textured

1006

soils due to the position of subsurface drip lines and/or poor management of the irrigation system,

1007

El-Nesr et al. (2014) used the HYDRUS model to evaluate the effectiveness of three anti-leaching
40

1008

techniques (a physical barrier, a dual-drip system with concurrent irrigation, a dual-drip system

1009

with sequential irrigation) applied in bare and vegetated soils. Results showed that the dual-drip

1010

system with sequential or concurrent irrigation is very efficient in limiting the downward leaching1011

of nutrients and that these systems can be applied in actual production to reduce the risk of1012

environmental pollution.

1013

Since current regional-scale groundwater models, such as MODFLOW, tend to simulate1014

the fate of solutes in the saturated zone by greatly simplifying the near-surface hydrologic

1015

processes, their simulations often lead to poor results. Lyu et al. (2019) coupled the one-1016

dimensional vadose zone flow model HYDRUS-1D and the groundwater model MODFLOW to1017

provide a feasible method for simulating the fate and transport of salt and N in the subsurface and1018

for assessing the long-term effects of irrigation with reclaimed water. The HYDRUS model can

1019

be coupled with other models to simulate various biogeochemical processes under different

1020

conditions, providing an effective tool for improving simulation accuracy and rationality.

1021

Figure 12 demonstrates soil NO3-N leaching in sandy loam for surface drip irrigation with1022

three different irrigation depths (15, 22.5, and 30 mm; the N application duration of 5 days), three1023

different durations of fertilizer application (5, 10, and 15 days; the irrigation depth of 30 mm), and1024

the same N-fertilizer application (2.1 mg cm-2) (the NLeach example in Table A). In this example,1025

the longitudinal dispersivity (DL) was considered to be 50 cm within the 0-100 cm soil layer, and1026

the transverse dispersivity (DT) was assumed to be one-tenth of DL (Cote et al., 2003). The

1027

molecular diffusion coefficients of NH4-N and NO3-N in free water were set to be 0.064 and 0.068

1028

cm2 h-1, respectively (Cote et al., 2003; Nakamura et al., 2004). The distribution coefficient Kd for1029

NH4-N and NO3-N were assumed to be 3.5 and 0 cm3g-1, respectively (Hanson et al., 2006). This

1030

study also assumed that NH4-N transforms directly into NO3-N since nitrification of NO2-N to

1031

NO3-N is relatively fast. The soil bulk density is 1.5 g cm-3, and the nitrification rate1032

(transformation of NH4-N to NO3-N) in the liquid and solid phases in the sandy loam were set to1033

0.12 and 0.28 d-1, respectively (Castaldelli et al., 2018). These values lead to an approximate

1034

retardation factor for NH4-N of about 22.9 and a half-life of about 3-6 d. NH4-N was assumed to1035

be initially present in the top 10 cm of the soil profile and additionally applied with the irrigation

1036

water. Both NH4-N and NO3-N were assumed to be passively taken up by plant roots (rooting

1037

depth of 80 cm and rooting radius of 40 cm). The third-type Cauchy boundary condition was used1038

for solute transport along all boundaries with specified water fluxes (at the top boundary), while
41

1039

the second-type Neumann boundary condition was used at the bottom boundary. In the HYDRUS

1040

model, the N-fertilizer application mass equals the product of the irrigation water volume (L3) and1041

the solute concentration (M L-3) in irrigation water. Thus, the solute concentration of irrigation1042

water (input value) was determined by dividing the N-fertilizer application mass by the irrigation

1043

water volume. In this example, the first topdressing mass and the irrigation water depth are 0.631044

mg cm-2 and 3 cm3 cm-2, and thus the solute concentration of irrigation water is 0.21 mg cm-3.

1045

Since NH4-N was greatly retarded in the soil profile, Figure 13 only shows actual and cumulative1046

solute fluxes for NO3-N for different scenarios. The simulation results showed that soil nitrogen

1047

leaching increased in response to an increase in the irrigation depth, while it decreased with an

1048

increase in the duration of the fertilizer application.

1049

1050
1051
1052
1053
1054
1055

Figure 13. Actual and cumulative NO3-N leaching at a depth of 40 cm in scenarios with a)different irrigation depths (I = 15, 22.5, and 30 mm; the N application duration of 5 days) and b)
different durations of the fertilizer application (D = 5, 10, and 15 days, the irrigation depth of 30mm) and the same N-fertilizer application of 2.1 mg cm-2.

1056

Crop Root N Uptake

1057

Optimization of N fertilizer applications is needed to prevent the negative impacts of these1058

fertilizers on the environment. Such optimization requires considering complex interactions1059

between root water and N uptake and their implications for crop production (Azad et al., 2020;

1060

Groenveld et al., 2021). The transfer of N from the soil solution into the plant can be described1061

using active or passive mechanisms, which refer to the mass flows of N into roots with water taken

1062

up for transpiration or against energy gradients, respectively (Groenveld et al., 2021). The passive

42

1063

and active root N uptake functions implemented in the HYDRUS models have been widely used1064

to optimize N fertilizer applications in agricultural production.

1065

All principles and parameter settings of passive/active root nutrient uptake implemented in

1066

HYDRUS are described in detail by Šimůnek and Hopmans (2009) and in the model's technical1067

and user manuals (Šimůnek et al., 2022ab; Šejna et al., 2022). Root passive N uptake is simulated1068

by multiplying water uptake (compensated or uncompensated) by the dissolved N concentration1069

for concentration values below an a priori-defined maximum concentration (cmax, the maximum

1070

allowed solution concentration for uptake). All nutrients dissolved in water are taken up by plant1071

roots when cmax is large (larger than the dissolved concentration c), while no nutrients are taken up

1072

when cmax=0, with only active uptake remaining in this case. The maximum solution concentration1073

for passive root uptake, cmax, thus controls the relative proportion of passive root water uptake to1074

total uptake. Using this flexible formulation, uptake mechanisms can vary between specific1075

nutrients. For example, Na uptake can be excluded by setting cmax=0, passive Ca uptake can be1076

limited by defining a finite cmax value, or all of the soil solution available P or N can be allowed to1077

be taken up passively by setting cmax to a very large value.

1078

Ranjbar et al. (2019) calibrated and validated the HYDRUS model parameters for root

1079

passive N uptake using data from two maize experiments over the 2015 and 2016 growing seasons.

1080

Simulations showed that HYDRUS was reasonably accurate in estimating total N uptake at1081

harvest, although it could not simulate cumulative N uptake during various growth stages with

1082

high accuracy. The authors hypothesized that some neglected processes, such as active N uptake,1083

caused an underestimation of the N uptake rate during the vegetative stage when the corn reached1084

its maximum growth rate. Azad et al. (2019) used the HYDRUS model (water-solute transport,

1085

root water/N uptake, and root growth) to simulate N uptake in a sandy clay/sandy loam soils under1086

a surface drip irrigation system and calibrated various parameters to optimize nitrogen use

1087

efficiency. Azad et al. (2019) compared root N uptake for two fertilizer application frequencies

1088with the maize N requirements during various maize growth stages. Results showed that reducing1089

the number of irrigation fertilizer applications in sandy clay loam soils increased the NO3-N uptake

1090

by plants. However, fewer irrigation-fertilization events in sandy loam soils decreased the NO3-N

1091

uptake.

1092

The HYDRUS model has also yielded meaningful results about active root N uptake,1093

another important process. When optimizing N fertigation, Groenveld et al. (2021) considered the
43

1094

effects of water and N stress on the plant size and corresponding feedback on potential

1095

transpiration and N uptake. To accomplish this, Groenveld et al. (2021) modified HYDRUS to1096

consider the reduction of potential transpiration due to N uptake limitation. The model was then1097

calibrated and validated using a dataset consisting of three NO3-N concentrations and six irrigation

1098

levels, i.e., using 18 treatments to fertilize cucumber plants grown in the evaporator. The model1099

was validated by testing its ability to accurately reduce potential N uptake and transpiration in the1100

presence of water and N deficiency.

1101

Li et al. (2015c) observed and simulated (using HYDRUS-1D) N transport and1102

transformations in an experimental direct-seeded-rice (DSR) field in the Taihu Lake Basin of1103

eastern China for two consecutive seasons. The calibrated and validated HYDRUS model

1104

indicated that NH4-N was the predominant form (74% of N uptake on average) for crop N uptake,1105

and NH4-N uptake represented 32.1% and 30.8% of total N input over the 2008 and 2009 seasons,1106

respectively. Similarly, Sharmiladevi and Ravikumar (2021) also improved the accuracy of root N

1107

uptake simulation by the HYDRUS model through intensive field experimentation, where various

1108

fluxes of urea, NH4-N, and NO3-N under drip irrigation were quantified to develop the N fertilizer

1109

application schedule for the entire rice growth period. In soil improvement, Mazloomi and Jalali

1110(2019) studied NH4-N uptake and transport in sandy loam soils for different application rates (2%,1111

4%, and 8%) of vermiculite, nano clay, and zeolite amendments. The HYDRUS model indicated1112

that the application of inorganic amendments to sandy loam soils reduced NH4-N transport and1113

increased the effective utilization of N fertilizer in agricultural systems. HYDRUS simulations

1114

helped to identify the best fertigation management to reduce N leaching and increase root uptake.

1115

It is necessary to investigate different fertigation strategies for different soils and growing seasons1116

to increase the efficiency of N uptake by plants.

1117

The example used above to assess N leaching (the NLeach example in Table A) also

1118

provides information about root NH4-N and NO3-N uptake for different simulated scenarios, i.e.,1119

different irrigation depths (15, 22.5, and 30 mm) and different durations of fertilizer application1120(5, 10, and 15 days) with the same N-fertilizer application (2.1 mg cm-2) (Figure 14). The

1121

maximum allowed solution concentration (cmax) for passive uptake for NH4-N and NO3-N were1122

set as 1.0 and 0.4 mg cm-3, respectively (Chen et al., 2020b). In general, root NO3-N uptake was1123

significantly higher than NH4-N uptake in different scenarios during the simulation period since1124

NH4-N was relatively quickly nitrified into NO3-N. Moreover, in different irrigation depth
44

1125

scenarios, root NO3-N or NH4-N uptake increased in response to an increase in the irrigation depth.

1126

However, an interesting result was obtained in the scenario with different durations of the fertilizer

1127

application. The highest root NO3-N or NH4-N uptake occurred for the application duration of 101128d, which showed that soil nitrogen could be easily taken up by plant roots in this scenario.

1129
0.0

a)
0.04

0.5

0.03

1.0

0.02

1.5

0.01

2.0

I_30

I_22.5

I_15

0.00

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

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

0.05

2.5
0

10

20

30

40

DAS

1130

0.0

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

b)

0.04

0.5

0.03

1.0

0.02

1.5

0.01

2.0

D_5

D_10

D_15

0.00

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

0.05

2.5
0

10

20

30

40

1131

DAS

1132
1133
1134
1135
1136

Figure 14. Root NH4-N and NO3-N uptake in scenarios with a) different irrigation depths (I = 15,
22.5, and 30 mm, with an N application duration of 5 days) and b) different durations of thefertilizer application (D = 5, 10, and 15 days, with an irrigation depth of 30 mm) and the same Nfertilizer application of 2.1 mg cm-2.

1137

Another example (the NUptake example in Table A) is used to demonstrate the modeling1138

of active and passive crop nitrogen uptake and sprinkler irrigation. This example includes three
45

1139

different N-fertilizer application levels (2.8, 2.1, and 1.4 mg cm-2) and three different soil textures1140(sandy, sandy loam, and loam). In this example, the transformation of NH4-N to NO3-N was

1141

neglected, and NO3-N was directly applied with the irrigation water. The dispersivity (D) was

1142

considered to be 10 cm, the maximum allowed solution concentration (cmax) for passive uptake

1143

was set as 0.4 mg cm-3, and the potential active solute uptake rate was set as 0.1 mg cm-2 day-1.

1144

The Michaelis-Menten constant was set to 0.5 mg cm-3. It was also assumed that potential solute

1145

uptake was reduced due to reduced root water uptake. A concentration flux boundary condition1146

was set at the top and bottom boundary, and 20% of the N-fertilizer application was assumed to1147

be initially present in the top 10 cm of the soil profile. The results showed that the crop nitrogen1148

uptake rate increased after fertilization applications, while the cumulative crop nitrogen uptake

1149

decreased in response to a decrease in nitrogen-fertilizer application levels (Figure 15).

1150

Additionally, an apparent difference in the crop nitrogen uptake rate was also found for different

1151

soil textures. In general, cumulative crop nitrogen uptake is different in different soil textures, and1152

it increases from sand to sandy loam and loam.

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
0.04

1.0
0.5

0.02
0.0
0.00

(0.5)
0

1154

Cum. Root nitrogen uptake (mg cm-2)

1153

20

40

60

DAS

46

80

100

120

4.0

b)

Sand

Sandy loam

loam
3.0

0.06
2.0
0.04
1.0
0.02
0.0

0.00

(1.0)
0

1155
1156
1157
1158
1159
1160

Cum. Root nitrogen uptake (mg cm-2)

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

0.08

20

40

60

80

100

120

DAS

Figure 15. Actual and cumulative root nitrogen uptake rate in scenarios with a) different Nfertilizer application levels (2.8, 2.1, and 1.4 mg cm-2; sand) and b) different soil textures (sand,sandy loam, and loam, with an N application of 2.8 mg cm-2).

47

7. Salinity and Sodicity

1161
1162
1163

A major challenge for irrigated agriculture, particularly in arid and semi-arid regions, is1164

irrigation-induced salinization of land and water resources (U. S. Salinity Laboratory Staff, 1954;

1165

Shainberg and Shalhevet, 1984; Wallender and Tanji, 2012). Many irrigation projects, both ancient1166

and modern, have struggled or failed over time due to the degradation of soils and waters (Jacobsen

1167

and Adams, 1958; Ghassemi et al., 1995; Hopmans et al., 2021). The pervasiveness of soil salinity

1168

and sodicity in irrigated agriculture is such that salt management has been identified as a key

1169

motivation for early fundamental studies in soil science (Letey, 1984). HYDRUS was initially1170

developed at the USDA-ARS U.S. Salinity Laboratory (Šimůnek et al., 2008), and it has always

1171

had strong capabilities for modeling salinity under irrigation. HYDRUS has been widely used to1172

evaluate the impacts of irrigation and salinity management on soil and water quality and crop

1173

yields.

1174

Two approaches are available to model irrigation in salt-affected lands. The simpler

1175

method is to evaluate soil salinity in terms of the total salt concentration of the soil solution. The

1176

concentration c in Eq. (2) is treated as a non-reactive tracer, equal to the total aqueous salt

1177

concentration. With this approach, it is possible to simulate salt additions, accumulation, and1178

leaching in the soil profile, as well as to evaluate the impact of salinity on crop productivity via

1179

the HYDRUS root water uptake features.

1180

More detailed assessments of irrigation and salinity are possible using the HYDRUS

1181

UNSATCHEM module (Šimůnek and Suarez, 1993; Suarez and Šimůnek, 1993; Šimůnek and1182

Suarez, 1994; Šimůnek et al., 1996; Gonçalves et al., 2006; Ramos et al., 2011). The module1183

includes capabilities for simulating multiphase transport of CO2, transport of major ions (Ca2+,1184−
Mg2+, Na+, K+, SO24 , and Cl ), major ion chemistry including complexation, cation exchange,

1185

precipitation and dissolution (e.g., of calcite, gypsum, and dolomite), and water quality effects on

1186

hydraulic conductivity.

1187

A search of “HYDRUS” and “salinity” on Google Scholar locates hundreds of journal

1188

articles and other texts documenting HYDRUS applications to soil salinity problems using both1189

approaches discussed above. Here we offer a selective overview of representative projects.

1190
1191
48

1192

Crop-Water Simulations for Salinity Management

1193

All irrigation waters contain some amount of salt. As crops grow and extract water from1194

the soil, salt is left behind in the root zone, where it will accumulate over time unless the applied

1195

irrigation and/or rain exceeds crop water requirements by an amount that is sufficient to flush salts

1196

into the deeper subsurface and maintain the rootzone salt balance. The fraction of applied water

1197

that passes below the root zone is known as the leaching fraction (LF); the LF required to keep

1198

root zone salinity at acceptable levels for a given crop is called the leaching requirement (LR).

1199

While LF and LR are valuable management concepts, their utility in operational management has

1200

been questioned (e.g., Corwin et al., 2007; Letey et al., 2011; Oster et al., 2012). Traditional LR

1201

analyses have been criticized for overestimating the impact of salinity on crop growth, thereby1202

encouraging over-irrigation. Limitations identified in LR analyses include steady-state and one-1203

dimensional flow assumptions and neglect of salt precipitation and dissolution reactions.

1204

An alternative is to simulate a growing season(s) using HYDRUS, evaluating the impacts

1205

of water quality and irrigation practices on crop water uptake using the HYDRUS solute transport1206

and root water uptake features. Among others, Corwin et al. (2007) found that differences in

1207

leaching requirements derived from transient and steady-state models were relatively minor,

1208

whereas accounting for salt precipitation led to significantly different recommendations. Similarly,

1209

Oster et al. (2012) found that with higher salinity irrigation waters, yields predicted using

1210

UNSATCHEM were greater than yields predicted with other models due to UNSATCHEM’s

1211

ability to account for salt precipitation, as well as the nonlinear relationship between salt1212

concentration and osmotic potential. As an illustration, consider the following scenario, which is1213

based on one assessed by Oster et al. (2012) and Skaggs et al. (2014). Forage corn is grown over1214

a 20-week season. Multiple seasons having the same time-varying atmospheric boundary1215

conditions are simulated back-to-back. By the final season, the simulated system has approached1216

a quasi-steady state with respect to seasonal water uptake and dynamics.

1217

Four model setups, each for two irrigation regimes, are evaluated (the Salinity examples in

1218

Table A). The first is a HYDRUS simulation that does not include salinity. Second is a HYDRUS

1219

simulation with an irrigation water salinity of 3 dS/m. The last two are UNSATCHEM simulations

1220

with irrigation water salinities of ≈ 3 dS/m but differing in chemical compositions, one a chloride-

1221

dominated water and the other having higher levels of sulfate and alkalinity.

49

1222

Table 2 gives model results for two simulated irrigation regimes, one a deficit irrigation

1223

scheme in which seasonal applied water (Irr) is less than the seasonal potential transpiration (Tp),1224

Irr/Tp = 0.9, and the other in which applied water is in excess of potential transpiration, Irr/Tp =1225

1.1. The presented results are the final seasonal leaching fraction (LF), evaluated as the ratio of

1226

cumulative water flux at the bottom of the root zone to the seasonal applied water, and the final

1227

seasonal relative yield, evaluated as the ratio of seasonal actual and potential transpiration.

1228

The results in Table 2 demonstrate that relationships between applied water, water quality,1229

leaching fraction, and crop yield are straightforward only when salinity is neglected (as is done

1230

with classical LF and LR assessments). Among the tested models featuring salinity, a range of1231

outcomes was obtained. The UNSATCHEM model with chloride water generated higher yields

1232

and lower leaching fractions than the HYDRUS model with salinity because, in the former

1233

simulation, calcite precipitation in the upper part of the root zone removed salt from the solution1234

and reduced the effective root zone salinity. Similarly, the UNSATCHEM simulation with higher

1235

sulfate and alkalinity produced the highest yields and lowest leaching because significant amounts

1236

of calcite and gypsum were precipitated in the root zone. If the simulations were done with higher

1237

irrigation water salinities, the differences between model outcomes would become more1238

pronounced. Figure 16 illustrates the root zone salinity profiles at the midpoint of the final

1239

simulated season for each model.

1240
Table 2. Computed seasonal relative yields (RY) and leaching fractions (LF).
Model

RY (%)

LF (%)-------- Irr = 0.9 Tp -------HYDRUS, no salinity

900

HYDRUS, ECiw = 3 dS/m

77

14

UNSATCHEM, ECiw ≈ 3 dS/m, chloride

79

12

UNSATCHEM, ECiw ≈ 3 dS/m, sulfate

828-------- Irr = 1.1 Tp -------HYDRUS, no salinity

100

9

HYDRUS, ECiw = 3 dS/m

88

20

UNSATCHEM, ECiw ≈ 3 dS/m, chloride

90

19

50

UNSATCHEM, ECiw ≈ 3 dS/m, sulfate

9315

ECiw = irrigation water salinity; Irr = seasonal irrigation; Tp = seasonalpotential transpiration
1241

1242
1243
1244
1245
1246
1247

Figure 16. Rootzone salinity profiles at the midpoint of the growing season for two irrigation levels(Irr = 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.

1248

Salinity Under Micro-Irrigation

1249

Micro-irrigation systems (drip, micro-sprinkler, etc.) create relatively complex, spatially

1250

varying patterns of water and salinity in irrigated soils (e.g., Burt and Isbell, 2005). HYDRUS

1251(2D/3D) has been used in many studies investigating leaching and water requirements in micro-

1252

irrigated systems. Among others, Hanson et al. (2008) showed that a conventional or water balance1253

approach for estimating leaching fractions predicted little or no leaching when applied water was

1254

less than the potential evapotranspiration, whereas field data and HYDRUS modeling showed

1255

considerable leaching around the drip lines. Raij et al. (2016) used HYDRUS (2D/3D) and

1256

UNSATCHEM to study spatial leaching patterns in soil lysimeters and different methods for

1257

evaluating leaching fractions. Yang et al. (2019) compared leaching in 1D (sprinkler-irrigated) and

1258

2D (drip-irrigated) systems, finding that leaching fractions were higher under drip than under1259

sprinkler, and that sprinkler was a more efficient means for maintaining salt balance.
51

1260

Sodic Soil Reclamation

1261

Irrigated soils (particularly heavy-textured clay soils) with a high exchangeable sodium

1262

percentage usually possess undesirable physical characteristics such as low permeability and1263

surface crusting. Reclamation of sodic soils generally involves replacing sodium on the exchange

1264

surfaces with calcium. Typically, calcium is introduced to the soil through the addition of

1265

amendments such as gypsum, although native soil minerals can also be a source. Standard formulas

1266

exist for calculating the required calcium or amendment needed to reduce the exchangeable sodium

1267

percentage to desirable levels, typically less than 10% (e.g., James et al., 1982; Loveday, 1984).

1268

Sodic soil reclamation is relatively complicated owing to the interactions of sodicity, salinity, and1269

permeability; one must take care not to worsen an infiltration problem by introducing an

1270

amendment solution with a low electrolyte concentration. Further, sodic soil reclamation can be

1271

expensive due to the cost of amendments.

1272

Many researchers have used UNSATCHEM to study sodic soil reclamation with the goal1273

of improving efficiency and reducing costs (e.g., Šimůnek and Suarez, 1997; Kaledhonkar et al.,1274

2006; Reading et al., 2012; Wang et al., 2016a; Shaygan et al., 2018ab). Among others, Suarez

1275(2001) used UNSATCHEM to investigate alternative reclamation strategies and concluded that1276

for soils possessing native calcite, “green manuring” (i.e., incorporating fresh organic matter into1277

the soil to boost soil CO2) can be used to reclaim soil without gypsum additions and with less1278

water. More recently, Phogat et al. (2020a) assessed the impact of annual gypsum additions on

1279

maintaining acceptable sodicity and salinity levels.

1280
1281

Shallow Groundwater1282

In addition to irrigation water additions and native soil minerals, root zone salinization can

1283

occur due to the upward movement of shallow, saline groundwater. HYDRUS has been used in a

1284

number of studies to evaluate the impact of shallow groundwater. For example, Forkutsa et al.

1285(2009) modeled irrigated cotton production as affected by salinity and shallow groundwater in the1286

Aral Sea Basin. Karimov (2014) assessed the impact of salinity and shallow groundwater on winter1287

wheat production in the Fergana Valley (Central Asia). Li et al. (2015a) modeled soil water and1288

salt dynamics as influenced by shallow groundwater in the Heihe River Basin. Narjary et al. (2020)

1289

evaluated the effects of water table depth and evaporative flux on soil salinity.

1290
52

1291

Recycled and Produced Water

1292

The disposal of recycled or produced water to irrigate agricultural crops is another interesting

1293

application of the HYDRUS and UNSATCHEM models (e.g., Mallants et al., 2017; Phogat et al.,1294

2020ab, 2021). Produced water is water that comes out of the wells during fracking operations,

1295

coal seam gas production, or crude oil production. Produced water contains soluble and non-1296

soluble oil/organics, suspended solids, dissolved solids, and various chemicals used in the

1297

production process. The application of produced water as irrigation water (possibly after blending

1298

with fresh water) is associated with many problems, such as salinity and sodicity risks and1299

associated changes in soil physical and chemical properties (e.g., reduction of hydraulic1300

conductivity), which can be assessed using the HYDRUS model and its modules (e.g.,

1301

UNSATCHEM).

1302

53

8. Chemigation

1303
1304
1305

Modern irrigation systems are used, besides delivering water, also for transporting

1306

agrochemicals (e.g., pesticides and herbicides) to benefit plant health. The chemicals are injected1307

into the irrigation system (Chemigation) and transported directly into the plant environment (Taha,

13082022). Each agrochemical has its own chemistry and unique interactions with the soil matrix. Each

1309

solute can be involved in several processes, such as transport, adsorption, degradation,

1310

volatilization, or even interactions with other solutes (Spurlock et al., 2013a). Besides the ability

1311

to improve the scheduling of chemical applications to allow for their maximum utilization,

1312

numerical models also can track the fate of applied solutes and their metabolites in the root zone1313

and their transport below the root zone into groundwater, which in general could be harmful to the1314

environment.

1315
1316

Historical Developments

1317

Modeling of solute transport in soil systems has been practiced for several decades, initially1318

using the ADE's analytical solutions and later numerical solutions. The analytical solutions

1319

typically require complete decoupling of water flow and additional simplifications such as1320

uniformity of soil properties and are, therefore, very limited. In most cases, they are also limited

1321

to strict linearity, zero- or first-order reactions, and full decoupling of the solutions for different

1322

solutes (as well as the decoupling of the water flow from the solute transport). A large number of1323

the ADE's analytical solutions (e.g., van Genuchten, 1985; van Genuchten and Wierenga, 1976;

1324

van Genuchten and Wagenet, 1989) and programs evaluating these solutions (e.g., CFITM,

1325

CFITIM, CXTFIT, 3DADE, N3DADE, and CHAIN) were implemented in the public-domain

1326

Windows-based STANMOD (STudio of ANalytical MODels) computer software package

1327(Šimůnek et al., 1999), which can be considered to as belonging to the HYDRUS family of1328

software packages.

1329

Numerical solutions are more flexible in many aspects mentioned above and, therefore,

1330

gained popularity over the years, with HYDRUS playing a major role in that popularity. The vast1331

majority of numerical applications of solutions to water flow and solute transport in soil systems

1332

were of an agricultural orientation (e.g., tracking the fate of fertilizers, pesticides, or salinity), but

1333

numerous applications for environmental problems soon followed, looking at nutrient leaching to
54

1334

groundwater, contaminant transport from landfills, and more. In the following, we present several

1335

examples of such applications.

1336
1337

Standard Modules and their Predecessors

1338Šimůnek et al. (1992) were among the first to add solute transport capabilities to the1339

numerical solutions of the Richards equation (then SWMII, a predecessor of HYDRUS). The ADE

1340

used at the time included advection, dispersion, and a very limited range of reactivity processes

1341(linear adsorption (i.e., constant retardation factor) and zero- and first-order reactions in both the1342

liquid and solid (sorbed) phases). The code gained increasing popularity starting mostly at the end

1343

of the 1990s, with the advancement of personal computation and the stability of the code as it

1344

developed from SWMII to SWMS_2D to the early HYDRUS packages in one or more dimensions

1345(Šimůnek et al., 2008). In that relatively limited approach, linear sorption (assuming equilibrium)1346

is considered in (2) using s = kd c , and the sink/source term  includes zero (k0) and first (k1c)1347

order reactions, and parallel terms for the solid (sorbed) phase. Note that in most cases, sorption is1348

mathematically represented as a retardation factor that comes in front of the left-hand side of (2)1349

and is constant for linear sorption.

1350

Examples of early applications include De Vos et al. (2000), who used SWMS_2D to1351

model nitrate leaching (i.e., as an inert solute). Persicani (1996) used HYDRUS (and three other1352

models) to check the sensitivity of pesticide leaching in soil profiles. Vanderborght et al. (1998)

1353

modeled advection-dispersion (ignoring diffusion) in heterogeneous soils and compared their1354

results to the particle tracking approach. Yang et al. (1996) considered linear adsorption in their1355

2D stochastic simulation framework (with the hydraulic conductivity and the adsorption constant

1356

being the stochastic variables). Boateng and Cawlfield (1999) conducted an extensive sensitivity

1357

analysis of the Richards-ADE system to contaminant transport, considering most parameters of1358

the Richards equation (with van Gunuchten-Mualem parameters), considering advection,1359

dispersion, first-order sorption, and zero- and first-order reactions. These and many other earlier

1360

references to HYDRUS packages and their predecessors can be found here.

1361

An example of a relatively advanced application of the classic HYDRUS model for solute

1362

transport is seen in Hanson et al. (2006). They considered drip irrigation in two geometrical1363

configurations (one of which is axisymmetric), where fertilizers (urea, ammonium, and nitrate)
55

1364

were applied with the water (fertigation). In their model, they considered sequential first-order1365

reactions (hydrolysis of urea and nitrification) and linear adsorption (of ammonium).

1366

Denitrification was not considered (as in the experiment associated with the model, it was

1367

negligible), and volatilization and gas transfer were ignored. This represents a classic agronomic

1368

study that looks into the fate of fertilizer in the subsurface. However, for more complex cases when1369

oxygen may be absent at times or in regions of the subsurface, such a modeling approach that relies

1370

only on a sequence of first-order reactions may be insufficient.

1371

In later versions of HYDRUS and other codes that solve similar problems, more

1372

sophisticated reactions evolved over the years. Clearly, the space is too limited to review them all,1373

and we will only mention several, referring the reader to the HYDRUS manual and website for1374

further elaboration and presentation of specific equations. For example, Kodešová et al. (2004)

1375

studied chlorotoluron transport in a soil profile and modeled it using both Freundlich and Langmuir1376

isotherms. Pal et al. (2014) used HYDRUS-1D to model phenol breakthrough experiments in soil

1377

columns using the three classical sorption isotherms (linear, Freundlich, and Langmuir). Pot et al.

1378(2005) and Köhne et al. (2006) used HYDRUS-1D to model herbicide transport using the dual-

1379

permeability model with two-site non-equilibrium sorption. Chotpantarat et al. (2011) studied the1380

transport of several heavy metals in soils and used HYDRUS-1D to model non-equilibrium

1381

sorption, considering Langmuir isotherm and two-site sorption. Finally, Köhne et al. (2009ab)

1382

provided a comprehensive review of HYDRUS applications involving transient water flow and1383

tracer and pesticide transport in structured soils.

1384
1385

Specialized Add-On Modules

1386

The next level of complexity in modeling solute transport is using a higher complexity of1387

reactions, which typically involve more than a single reactive species and coupling of the different

1388

reactions. In the HYDRUS framework, several specific tools were developed over the years for1389

specific applications and environments (e.g., UNSATCHEM, Suarez and Šimůnek, 1997).

1390

Goldberg et al. (2007) provide a comprehensive overview of both existing models and their1391

applications, mostly related to surface complexation. Later on, a more generic solution to the need1392

emerged by coupling the PhreeqC code (Parkhurst and Appelo, 1999) with HYDRUS (Jacques et1393

al., 2006) to form HP1. For example, Jacques et al. (2008a) studied aqueous phase complexation1394

and ion exchange (competitive sorption) for several major ions and heavy metals, including
56

1395

cadmium and zinc. The use of PhreeqC in HP1 allows great flexibility in modeling complex

1396

biogeochemical processes. For example, Ben Moshe et al. (2021) used this flexibility to model

1397

carbon and nitrogen processes in the context of soil aquifer treatment using multi-Monod-type1398

reactions.

1399

An interesting development in solute transport codes, HP1 included, is their use in

1400

modeling the transport of non-solutes. This is increasingly practiced to study colloid and1401

nanoparticle transport, considering processes such as attachment and detachment, straining, and1402

more. Makselon et al. (2017) provide an example of such a model considering silver nanoparticles.

1403

Many references involving HYDRUS applications to simulate the fate and transport of various

1404

types

1405

progress.com//Documents/Jirka/Particles_Transport_Papers.pdf).

ofparticlesarelistedonthe

HYDRUS

website(i.e.,

https://www.pc-

1406
1407

HP1 Example

1408

The following example is used to demonstrate in more detail the ability of the HP1 code to1409

track more complex biogeochemical processes beyond the capability of the conventional ADE

1410

with first-order reactions (The HP1 example in Table A). Jacques et al. (2008b) demonstrated the1411

modeling of uranium transport in soil following fertilizer applications. Their model considered

1412

annual fertilizer applications with inorganic phosphate contaminated by elements from the1413

uranium decay chain. The soil profile comprised seven (7) layers to a total depth of about 1 m. The

1414

numerical solution considered 14 different species (C, Ca, Cl, F, Mg, N(V), Na, P, S(VI), and1415

U(VI), O, and H) and a liquid-phase charge (to be balanced by the solid phase surface charge).

1416

Three types of geochemical equilibrium reactions were considered, including aqueous speciation

1417

reactions, cation exchange reactions on organic matter, and surface complexation reactions on Fe

1418

oxides. The reactions were strongly pH sensitive (i.e., protons, hydroxides, and surface hydrolysis

1419

were also considered). Carbon dioxide was not considered, as the pH range in the system was 3.5–1420

4.5. Most reactions (including surface complexation ones) involve more than a single species;

1421

therefore, first-order reactions cannot be used. The results of this model showed a very good fit to1422experimental data, which could not be achieved using the more conventional models.

1423

A different example (Liu, 2020) looked at the root zone's total carbon and nitrogen cycle

1424(specifically, the goal was to explore these processes under varying irrigation frequencies). She

1425

considered not only the classic nitrification and denitrification but also mineralization,
57

1426

decomposition of soil organic matter, dissolved organic carbon, and nitrogen, as well as microbial

1427

biomass and its growth or decay, and oxygen transfer between the liquid and gas phases. The

1428

considered processes are shown in Figure 17, including the differentiation of the process direction1429

for oxic and anoxic conditions.

1430

1431
1432

Figure 17. Schematic of biochemical cycling in the root zone and microbial communities.

1433
1434

Figure 18 shows a sample of the results obtained by this complex model, looking at

1435

nitrification rates for three different irrigation regimes (see captions). Above all, this model

1436

considers oxygen levels in the soil, and accordingly, the reaction rates and the microbial dynamics

1437

are adjusted. One can easily note the influence of water quality and irrigation frequency on the1438

biochemical processes. For example, nitrification rates in the upper 30 cm are higher in treatment

1439

TL than TH, primarily because of the higher aeration allowed between irrigation events. At the1440

same time, FH rates are lower than TH because the concentration of organic nitrogen and carbon

1441

is lower.

58

1442
1443

Figure 18. Spatio-temporal distributions of nitrification (i.e., ammonium transformation to nitrate)1444

rates in different fertigation regimes (weekly cycles, for TH and FH irrigation on days 2, 4, and 7;

1445

for TL irrigation on days 3 and 7; water quantities adjusted; TH and TL use treated wastewater

1446

and FH uses freshwater).

1447
1448

The flexibility that is associated with HP1 seems endless. Nevertheless, the number of reported

1449

applications, especially in the agricultural context, is relatively small. While no statistical

1450

significance test was conducted, apparently, most of the applications of HP1 are in more

1451

environmentally related research. The reason for this is not clear, but we suspect that this is related,1452

in part, to the complexity of the model and the lack of a graphical user interface, even if minimal,

1453

dedicated to specific scientific environments.

1454

59

9. Spatial Variability

1455
1456
1457

Efficient irrigation management decisions require a good understanding of temporal

1458

variations in root water uptake and spatial variations in soil hydraulic properties, which inherently

1459

impact water dynamics, solute transport, root distribution, and plant growth (Dabach et al., 2015).

1460

The soil-plant system of irrigated crops can be simulated by considering various model

1461

complexities, such as defining 1D, 2D, or 3D transport domains, describing homogeneous, layered,1462

or fully heterogeneous soils in a single simulation, or by running a series of simulations

1463

representing measured or estimated variability in soil hydraulic properties. Additionally, spatial

1464

variation in irrigation inputs will cause water content heterogeneity in the irrigated field. Therefore,

1465

understanding different levels of spatial variability in the irrigated root zone can help with various

1466

decisions, such as selecting sensor locations for optimal triggered irrigation scheduling, providing

1467

information on the expected levels of variability in the measured data, or optimizing irrigation

1468

application methods that rely on non-uniform applications or involving different soil textures or

1469

materials. This section provides an overview of different levels of spatial variability, how they1470

have been addressed using HYDRUS, and additional examples and guidance on how to apply them1471

in various scenarios.

1472
1473

Irrigation System Uniformity1474

The HYDRUS code can be used to study the effects of different levels of irrigation

1475

uniformity on drainage and solute leaching. Irrigation uniformity levels can be simulated by

1476

running Monte Carlo simulations with discharge rates sampled from a selected distribution

1477

associated with the desired irrigation uniformity. Such an approach can be applied to several

1478

drippers in a multi-dimensional simulation (Raij et al., 2018), to individual simulations, each with1479

a discharge in a defined distribution, or a series of simulations representing variable emitter

1480

discharge rates along an irrigation line due to pressure head loss (Guan et al., 2019; Wang et al.,1481

2016b). In a 3D study, it was found that lower irrigation uniformity in drip-irrigated lysimeters1482

caused higher drainage variability than the observed variability in chloride concentrations (Raij et1483

al., 2018). On the other hand, in 2D studies, it was found that increased irrigation uniformity

1484

decreased drainage amounts and nitrate loads (Guan et al., 2019, Wang et al., 2016b). Figure 191485

demonstrates how various levels of irrigation uniformity, i.e., with a coefficient of variation (CV)
60

1486

of 2%, 4.4%, and 9%, affect soil water contents at different locations relative to a dripper, as

1487

simulated in 500 runs. The simulation domain was the same as in the Drip Plant project (Section

1488

5 - Drip Irrigation, Table A). Sandy loam and loamy soils were considered (Figures 19A and C

1489

and 19B and D, respectively). Daily irrigation events from a surface irrigation area with a radius

1490

of 20 cm were simulated for four days. The irrigation discharge rate was randomly sampled 500

1491

times from a normal distribution with the same mean but varying standard deviations, according1492

to the selected CV (Figure 19E). The Matlab code used to create these simulations can be found1493

in this link. Water contents in three observation nodes (Figure 19F) varied in 500 simulations

1494

because of different imposed discharge rates. As expected, the higher the CV of the irrigation

1495

discharge rate, the higher the variability in the soil water content, especially during irrigation

1496

events (Figures 19A and B). As a result, values of the water content CV during the irrigation events

1497

were higher than the irrigation CV (Figures 19C and D).

1498

1499
1500
1501
1502

Figure 19. Water contents (A, B) and water content coefficient of variation (CV) (C, D) in thethree observation nodes (F) (x = 4.3, y = 6.3 cm; x = 8.5, y = 13.5 cm; x = 13.9, y = 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
61

1503
1504

(CV = 2%, 4.4%, and 9%) of irrigation uniformity (E). Irrigation fluxes for three levels ofheterogeneity (E) and the simulation domain with three observation nodes (F).

1505
1506

Deterministic Spatial Variability

1507

Vertical heterogeneity due to soil layering, for example, in alluvial soils, can be simulated

1508

in HYDRUS by defining each soil layer's hydraulic parameters and assuming horizontal

1509

homogeneity (Botros et al., 2012. If enough data is available, the depth of each layer can vary in1510

any dimension accordingly (Bednorz et al., 2016). Botros et al. (2012) simulated water flow and

1511

nitrogen transport in an irrigated nectarine field with a 3D domain with horizontal soil layers of1512

varying depths in the east-west direction but homogeneous in the north-south direction. Water

1513

fluxes below the root zone and nitrate concentrations in the deep vadose zone, simulated using the1514

layered approach, were comparable to stochastic simulations using either a Miller-Miller approach

1515

or a measured spatial correlation of the van Genuchten hydraulic parameters (Botros et al., 2012).

1516

3D transport domains can be used to simulate water contents and solute distributions in complex

1517

irrigated systems such as a drip-irrigated cotton field with partial plastic mulch on the growing

1518

beds (Li et al., 2018b) or rainfed banana plants grown on Andisol with a complex canopy geometry

1519(Sansoulet et al., 2008). In addition, simulations using 3D domains can help explain processes of1520

the wet bulb and salinity overlap that are underestimated in 2D conventional models assuming

1521

symmetry (Kandelous et al., 2011). When 2D or 3D domains cannot be used due to computational

1522

constraints, 1D simulations can be carried out at the field or regional scale, and the results can be1523

ensembled to include and describe the spatial differences. For example, soil properties, weather,1524

slope, and land use maps can be used to create unique combinations of 1D simulations, which can

1525then be integrated spatially to describe water and solute fluxes, as well as solute and water content

1526

distributions in the landscape (Turkeltaub et al., 2018; Henri et al., 2020, 2021; Qiu et al., 2020;

1527

Sprenger et al., 2016; Sela et al., 2012; Lai and Ren, 2016).

1528

Figure 20 presents two examples of layered soils where the first has a sandy loam layer in1529

a loam domain (Figure 20A, the Drip3D2L1 example in Table A), and the second has a loam layer

1530

in a sandy loam domain (Figure 20B, the Drip3D2L2 example in Table A ). These simulations use

1531

the same boundary conditions as the Drip3D examples in Section 5 - Drip Irrigation (Table A).

1532

Following an irrigation event during day 1, the volumetric water content is lower in the sandy loam

1533

layer than in the loam layer. This approach has been used to describe naturally occurring
62

1534

variability. In addition, induced soil profile variabilities, such as capillary barriers utilized to1535

modify and control soil water distribution or irrigation using properties of porous materials, have

1536

been simulated with HYDRUS (2D/3D) for optimizing such systems (e.g., Ben-Gal et al., 2004;

1537

Siyal and Skaggs, 2009; Ityel et al., 2011; Sao et al., 2021).

1538
1539
1540
1541

Figure 20. 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 withtime (0–5 days).

1542
1543

Stochastic Spatial Variability

1544

The effects of the spatial variability caused by the randomness of specific soil hydraulic1545

properties, such as saturated hydraulic conductivities or porosities, can be simulated using the1546

HYDRUS code by first generating the spatial distribution of this parameter (using a theoretical

1547(Guan et al., 2019; Wang et al., 2016b; Martinez et al., 2014) or measured (Hu et al., 2007)1548

probability distribution of this property) and then running sequential simulations with a range of

1549

this parameter’s values. Theoretical distributions can be generated by defining a mean and
63

1550

coefficient of variation (CV) or standard deviation for the variable of interest (Guan et al., 2019;

1551

Martinez et al., 2014; Bossa and Diekkrueger, 2014) by generating a range of textures and bulk

1552

densities as inputs to Rosetta (Wang et al., 2016b), or by using means and standard deviations

1553

reported by Rosetta3 (Zhang and Schaap, 2017, Raij-Hoffman et al., 2022). Such simulations can

1554

optimize irrigation amounts to minimize leaching, not at a specific point in the field but as a

1555

probability density function (Hu et al., 2007). Additionally, different levels of hydraulic1556

parameters’ heterogeneity can be studied, and their effects on water fluxes and solute loads can be

1557

described (Guan et al., 2019).

1558

Figure 21 presents the average and 95% confidence intervals of water contents during two

1559

irrigation events at an increasing lateral distance from a sub-surface dripper obtained from 187

1560

simulations. Simulations were carried out for a homogeneous soil profile with varying Ks values

1561

for two soil textures reported in Rosetta3. The mean and standard deviation of Ks (on a log scale

1562

with base 10) were used to randomly sample Ks values for each simulation. These simulations use

1563

the same boundary conditions as the Drop3D1 example in Section 5 - Drip Irrigation (Table A).

1564

The confidence bounds for time-variable water contents at each location in each soil demonstrate1565

the uncertainty associated with using pedotransfer functions when only soil texture is known.

1566

While the uncertainty of the water content remains high during the two irrigation events, the

1567

variability of the flux at the bottom boundary decreases with time in this example. Considering

1568

measured or estimated variability in soil properties or irrigation systems can provide information

1569

on their effect on the uncertainty of different model outputs (in this case, water content and bottom

1570

boundary fluxes). This information can be used to evaluate model-to-measurement fit and a choice1571

of type, method, and frequency of measurements.

64

1572
1573
1574
1575
1576

Figure 21. Average and 95% confidence intervals for 187 simulations with varying Ks values forloam and sandy loam soils. Water contents at observation points located at depths of 20 cm(vertical) (dripper depth) and 5 cm (A), 15 cm (B), and 25 cm (C) away (horizontal) from thedripper. Time-dependent water fluxes at the bottom boundary (D).

1577
1578

In addition to non-dimensional parameter uncertainty (as discussed above), the measured

1579

or assumed spatial correlations of any selected hydraulic parameter can be considered in HYDRUS

1580

simulations. Correlation lengths in different directions can be used to generate a spatially1581

correlated parameter space (Botros et al., 2009). Significant differences in 3D N fluxes were1582

observed in simulations considering the spatial correlation of soil hydraulic properties, as opposed1583

to those using uncorrelated scaling factors or deterministic layered heterogeneities. On the other1584

hand, water fluxes in the field with spatially correlated soil hydraulic properties were similar to1585

those in the deterministic layered profile and different from those in the field with uncorrelated

1586

scaling factors. This was due to the nature of the highly horizontally correlated alluvial soil

1587

presented in Botros et al. (2009, 2012).

1588

An additional option offered by HYDRUS is to generate stochastic distributions of1589

hydraulic conductivity (K) and pressure head (h) scaling factors using the Miller-Miller
65

1590

similitude approximation (Miller and Miller, 1956), where these two scaling factors are correlated.

1591

The stochastic distribution requires a mean Ks, a standard deviation of log10 value of K, and1592

horizontal (X) and vertical (Z) correlation lengths. Additional numerical experiments were1593

performed to explore the influence of stochastic heterogeneity during an irrigation event using the1594

flow domain in the DripPlant1 example in Section 5 - Drip Irrigation (Table A) with a sandy loam

1595

soil for a homogeneous soil profile, and heterogeneous soil profiles with a standard deviation of

1596

log10(K) = 0.5 cm and a) a vertical correlation length Z = 5 cm, b) a horizontal correlation length1597

X = 10 cm, and c) horizontal and vertical correlation lengths X = 10 cm and Z = 5 cm, respectively.

1598

Figure 22A presents representative water content soil profiles for a homogeneous domain, and

1599

heterogeneous domains with vertical, horizontal, and both vertical and horizontal heterogeneities.

1600

Figure 22B presents water contents at three observation nodes and the fluxes at the bottom

1601boundary. The results demonstrate the significant variability of water contents in heterogeneous

1602

domains compared to the homogeneous one, with higher variability of both water content and1603

bottom boundary fluxes for the vertical and horizontal heterogeneity than the combined

1604

heterogeneities. Similarly, water fluxes at the bottom boundary were more variable when only one

1605

directional heterogeneity was simulated rather than both directions combined. Examples of1606

horizontal heterogeneity can be found in highly layered alluvial soils, and vertical heterogeneity is1607

found in cracked soils with/or soils with preferential flow paths. In either case, the extent of the1608

spatial correlations will depend on the soil profile’s specific properties. These results demonstrate1609

the importance of considering heterogeneity, especially for contaminant flow and transport studies.

1610

Scaling factors can also be generated externally and imported into HYDRUS domains. An example

1611

can be found in Raij et al. (2018) or Botros et al. (2009), where three-dimensional distributions of1612

scaling factors for unsaturated hydraulic conductivity were externally generated and imported into1613

HYDRUS-3D.

1614

66

1615
1616
1617
1618
1619
1620
1621

Figure 22. Water content distributions in the 16th hour in a representative homogeneous profile andthree heterogeneous profiles with a standard deviation of log10(K) = 0.5 cm, a vertical correlation
length Z = 5 cm, a horizontal correlation length X = 10 cm, and vertical and horizontal correlation
lengths Z = 5 cm and X = 10 cm, and considering the Miller-Miller similitude (A). Water contents
and corresponding 95% confidence intervals in three observation nodes resulting from 20
simulations (B).

1622
67

1623

Estimation of Soil Hydraulic Parameters for Heterogeneous Soils

1624

Various laboratory and field methods and in-situ soil sensors and devices are used to1625

estimate soil hydraulic properties, and water flow and solute transport dynamics during irrigation.

1626

However, identifying optimum locations of sensors for a given spatial heterogeneity is critical for

1627

efficient irrigation water management. For example, the wetting front resulting from drip irrigation

1628

will be affected by the spatial heterogeneity of the soil between the dripper (with and without

1629

geotextile membrane) and the tensiometer. As a result, the wetting front will not reach tensiometers

1630

located at the same distance from the dripper at the same time. Dabach et al. (2013) used a system-

1631

dependent boundary condition in HYDRUS that triggers irrigation of a fixed duration when a1632

specified soil water matric head threshold is reached at a predetermined point in the soil profile.

1633

This boundary condition represents a demand-based irrigation system that directly depends on

1634

plant water uptake. HYDRUS simulations showed that the optimal location of tensiometer

1635

placement is near the subsurface dripper, resulting in a low CV of matric head measurements and1636

applied irrigation water, high sensitivity to irrigation, and lower drainage below the root zone

1637

(Dabach et al., 2015). Zeng et al. (2012) compared the centrifugal, tensiometer, soil core, disc1638

infiltration, and inverse solution methods to estimate soil hydraulic parameters for a drip-irrigated1639

heterogeneous gravel soil. The two-dimensional HYDRUS inverse simulation produced the best

1640

result even though a slight overestimation of soil moisture was present, which revealed a problem

1641

of underestimating water movement with the disc infiltration approach compared to the soil core1642

method.

1643

Similarly, Abbasi et al. (2003a) used the Levenberg–Marquardt optimization algorithm

1644

implemented in HYDRUS to estimate soil hydraulic and solute transport parameters of several soil

1645

horizons below experimental furrows. Their results showed only minor improvements in model

1646

predictions with optimized parameters. A two-step method, i.e., sequential estimation of the soil

1647

hydraulic parameters followed by the estimation of the transport parameters, was used to improve1648

the model predictions further. Despite implementing a scaling procedure for the unsaturated soil

1649

hydraulic properties, variability in the optimized longitudinal dispersivity values for the layered1650

soil profiles was found to be higher than for the homogeneous soil profile due to the variations in

1651

soil water retention curves of the different soil horizons.

1652

The Miller–Miller scaling approach implemented in HYDRUS was used by Weihermüller

1653

et al. (2006) to assess the variability in solute breakthrough curves in suction cups. Weihermüller
68

1654

et al. (2006) demonstrated that mean pore water velocities and dispersivities ranged by a factor of

1655

1.6 and 1.5, respectively, for different heterogeneous structures. In addition, the suction cup

1656

sampling area, deformation of streamlines, and the flow channels were also found to have an

1657

influence, and numerical simulations indicated that about 20 suction cups were required for

1658

calculating a mean breakthrough curve in the chosen heterogeneous flow field (Weihermüller et1659

al., 2006).

1660

Similarly, lysimeters are widely used for closing water mass balances by monitoring the1661

storage in the soil profile, drainage, and solute leaching at the bottom. Raij et al. (2017)1662

investigated the influence of local heterogeneity in soil hydraulic conductivities and dripper

1663

discharge rates on drainage and chloride concentrations using 100 HYDRUS three-dimensional

1664

simulations per treatment. Different random spatial realizations of the conductivity scaling factor

1665

were generated, assuming a log-normal distribution (Lazarovitch et al., 2006). Their result

1666

demonstrated that irrigation caused higher variability in drainage amounts than soil heterogeneity,

1667

while soil heterogeneity caused higher variability in drainage concentrations than irrigation

1668

heterogeneity.

1669

Most of the studies mentioned above used numerical simulations to demonstrate the1670

influence of spatial heterogeneity on sensors. Domínguez-Niño et al. (2020ab) investigated the1671

sensor-to-sensor variability due to uncertainties in the measurement process plus the natural

1672

variability in the actual soil water dynamics by comparing 57 capacitance-type soil moisture1673

sensors in the field and HYDRUS-3D virtual sensors under drip irrigation. However, the1674

HYDRUS-3D model was used to simulate soil water dynamics only in a homogeneous soil and

1675

uniform root distribution and did not account for heterogeneous soil and root distributions,

1676

macropores, and soil irregularities. The results demonstrated that the HYDRUS-3D simulations

1677

successfully reproduced the soil water dynamics and patterns of daily amplitude, but compared to1678

the sensors, simulations were less variable, underestimated root water uptake, and overestimated1679

the influx of irrigation water. Overall, these results demonstrated that various soil sensors could

1680

be too sensitive to local variations in soil texture and the presence of gravel, stones, roots,

1681

macropores, or small compacted soil parts.

1682

Understanding field wetting patterns during irrigation and fertigation can provide helpful

1683

information for efficiently managing these systems. In the past decade, advanced, minimally

1684

invasive geophysical tools, such as electromagnetic induction, cosmic ray sensors, ground
69

1685

penetrating radar, and electrical resistivity tomography (ERT), have been used for 2D mapping of

1686

wetting patterns and solute distributions for various irrigation systems (Coppola et al. 2016; Han

1687

et al. 2016; Algeo et al. 2016; Cassiani et al. 2015; Hardie et al. 2018). For example, Hardie et al.

1688(2018) compared the drip irrigation wetting patterns and nitrate distributions using ERT, dye

1689

tracer, and HYDRUS-2D modeling. The results showed some variability in observed, measured,1690

and predicted infiltration patterns and nitrate transport across the three approaches due to the1691

preferential flow processes in the field. However, simultaneous ERT, dye tracer, and modeling

1692

studies can be used to understand water flow and solute transport in an irrigation field.

1693
1694

Challenges, Recommendations, and Future Opportunities

1695

All the processes described above can be simulated simultaneously, and different levels of

1696

variability can be considered simultaneously to study complex processes (e.g., irrigation, soil

1697

hydraulic parameters, and scaling factors, Guan et al., 2019; Raij et al., 2018). The global problem

1698

of non-point source pollution from irrigated agriculture needs to be assessed at the regional scale,1699

even though the processes occur at the field/local scale. One approach is to use an ensemble of1700

one-dimensional HYDRUS simulations, including all combinations of soils and crops occurring1701

in the area of interest (Henri et al., 2020; Turkeltaub et al., 2018). Then, each combination of water

1702

and solute loading is assigned to various spatial locations and either fed into a groundwater model

1703

or used as potential groundwater loadings. Qiu et al. (2020) studied landscape-level ecosystem

1704

services, including agricultural landscapes, using a biophysical model coupled with HYDRUS-1D

1705

and found that "time" components were more important than "space" drivers in such landscapes.

1706

Emerging technologies, such as electrical resistivity tomography (Hardie et al., 2018, Peddinti et1707

al., 2020), are increasing the opportunity to use spatially distributed data at larger scales than1708

traditional measurements (i.e., Skaggs et al., 2004). Therefore, combining field monitoring, virtual1709

monitoring, and validation of observed sensor data using representative HYDRUS simulations can

1710provide great insight into local heterogeneity. Future studies should include a combination of1711

multiple sensors for collecting a range of parameters and site-specific soil characterization to better

1712

understand the effects of spatial heterogeneity on irrigation management. Such efforts will also

1713

provide additional opportunities to fill the data gaps further and refine the model.

1714
70

1715

10. Model Calibration

1716
1717

Numerical models are mainly developed to describe and predict the behavior of physical1718

systems. When the deviation between model predictions and actual observations (e.g., soil

1719

moisture patterns or water depths in the furrow channel) is acceptable, then the model can

1720

complement or even replace real experiments. The model’s predictive accuracy depends on the1721adequacy of the model structure, as well as on a proper (and physically realistic) definition of

1722

model parameters. The process by which the model structure is selected and parameters are

1723

adjusted to minimize the distance between model predictions and measurements is broadly defined1724

as model calibration. This procedure is generally accomplished by combining the model with

1725

iterative statistical and numerical optimization techniques.

1726

HYDRUS is internally coupled with the gradient-based Levenberg-Marquardt search1727

method (Marquardt, 1963), which solves a nonlinear least squares optimization problem to1728

calibrate various model parameters using transient observations. The algorithm is robust when the1729

response surface of the calibration problem is unimodal, which is generally the case for well-posed1730

inverse problems. Being supported by the HYDRUS GUI, the algorithm has been used in several

1731

irrigation-related studies to inversely estimate soil hydraulic and/or transport parameters. This has

1732

been generally accomplished by comparing simulated and observed soil water contents at different

1733

depths in drip-irrigated fields (Bufon et al., 2012; Chen et al., 2021; González et al., 2015; Li et1734

al., 2015b; Nayebloie et al., 2022; Patel and Rajput, 2008; Qi et al., 2018; Rana et al., 2022; Selim

1735

et al., 2013; Zhang et al., 2018), furrow channels (e.g., Abbasi et al., 2004; Dialameh et al., 2022;

1736

Ebrahimian et al., 2013; Ranjbar et al., 2019; Zhang et al., 2013), or other irrigated areas (e.g.,

1737

Amin Salehi et al., 2017; Arbat et al., 2008; Bethune et al., 2008; Deb et al., 2013; Er-Raki et al.,1738

2021; Hou et al., 2017; Li, 2020; Mokari et al., 2019; Rai et al., 2019; Rezayati et al., 2020;

1739

Saefuddin et al., 2019; Tan et al., 2014).

1740

The main limitation of the Levenberg-Marquardt algorithm is its local search capability,1741

which makes its use questionable for high-dimensional inverse problems that use different types

1742

of measurements. In these circumstances, the response surface tends to be complex, noisy, and1743

characterized by multiple local minima. To address this problem, several studies have externally1744

coupled HYDRUS with global optimization algorithms such as the Particle Swarm, Genetic, and1745

Shuffled Complex Evolution Optimization algorithms (Azad et al., 2018, 2019; Brunetti et al.,
71

1746

2018; Erazo-Mesa et al., 2022; Kumar et al., 2022; Peddinti et al., 2018, 2020; Schneider et al.,1747

2013; Vrugt et al., 2001a, 2001b; Zhou et al., 2007, 2012) or with the PEST calibration suite

1748(Carlos et al., 2022; Liu et al., 2021, 2022). Despite requiring external tools, advanced

1749

programming skills, and significantly more computational resources, the results of these studies

1750

show that the use of global optimization algorithms can improve the HYDRUS calibration

1751

procedure and better diagnose model structural inadequacies when estimated parameters assume

1752

physically unrealistic values.

1753
1754

Uncertainty Assessment

1755

Uncertainty analysis complements model calibration and represents a valuable tool for

1756

assessing the model's structural adequacy and predictive accuracy. Despite being frequently

1757

applied in combination with HYDRUS for vadose zone hydrological modeling, its use in

1758

irrigation-related studies remains rather limited. Brunetti et al. (2018) coupled HYDRUS-1D and1759

the Global Sensitivity Analysis-Generalized Likelihood Uncertainty Estimation technique (GSA-

1760

GLUE) to calibrate and assess the uncertainty of the soil hydraulic and thermal parameters of an1761

irrigated extensive Green Roof (GR). Recently, the Bayesian inference was used to statistically1762

compare one conceptual and three HYDRUS-based mechanistic parameterizations (e.g.,1763

unimodal, bimodal, and dual porosity) of an irrigated GR at the laboratory scale (Brunetti et al.,

17642020). In another study, Brunetti et al. (2021) coupled HYDRUS-1D and the multimodal Nested

1765

Sampling algorithm (Feroz et al., 2009) to reproduce observations from three GRs irrigated with1766

wastewater and simultaneously assess the uncertainty of inversely estimated soil hydraulic and1767

solute transport parameters. The Bayesian inference was also applied by Guo et al. (2020), who

1768

used two surrogate-based uncertainty assessment techniques to calibrate the HYDRUS model

1769

against measurements from field irrigation experiments. Albeit limited, the results of these studies

1770

highlight the beneficial role of uncertainty assessment in combination with HYDRUS to 1) assess

1771

how the model’s predictive uncertainty affects model outcomes, which is of crucial importance

1772

when delivering model results to decision-makers, 2) verify whether the informational content of

1773

calibration measurements is sufficient to constrain model predictions, or update the prior belief of1774

the modeler, and 3) compare multiple model parameterizations and avoid the use of data-

1775

unjustified complex model structures that tend to overfit measurements.

1776
72

1777

Sensitivity Analysis

1778

Along with uncertainty assessment, sensitivity analysis is a fundamental tool for model

1779

diagnostics, as it can be used for both predictive and explanatory purposes. In the first case, it can

1780

complement model calibration and uncertainty assessment by providing a statistical basis to1781

quantify how different model parameters affect, alone and/or jointly, the fitting accuracy.

1782

Uninfluential parameters can then be excluded from the calibration, thus reducing the1783

dimensionality of the inverse problem. When used for explanatory intentions, it can identify factors

1784

that most influence a certain variable of interest (e.g., root water uptake, nutrients leaching, etc.),

1785

thus guiding the experimental design (e.g., specific complementary laboratory measurements).

1786

Sensitivity analyses are generally divided into two groups: local one-factor-at-a-time (LSA) and1787

global sensitivity analysis (GSA).

1788

By varying only one factor at a time, LSA is easy to implement and computationally cheap.

1789

It has been used in several studies in combination with HYDRUS to assess the influence of soil

1790

hydraulic and transport parameters on water flow and nutrient transport in irrigated fields (Beyene

1791

et al., 2018; Geza et al., 2021; Jiménez-Martínez et al., 2009; Kadyampakeni et al., 2018; Reading1792

et al., 2012; Rocha et al., 2006; Sun et al., 2022; Xu et al., 2019; Yang et al., 2017). However, LSA

1793

neglects the effects of parameters’ interaction, which is known to play an important role in1794

environmental models. To overcome this limitation, GSA explores a large portion of the1795parameters’ space to identify combined or individual parameter effects. This frequently requires

1796

thousands of model executions to achieve stable estimates of the sensitivity indices, thus1797

significantly increasing the analysis’s computational burden. Techniques such as the Morris

1798

screening scheme (Morris, 1991), the Sobol’ sequence (Sobol′, 2001), or the E-FAST method

1799(Saltelli et al., 1999) have been successfully coupled with HYDRUS in multiple studies (Brunetti

1800

et al., 2018; Li et al., 2012; Skaggs et al., 2014; Liang et al., 2017; Liu et al., 2022; Peddinti et al.,1801

2018; Tao et al., 2021; Zhou et al., 2012).

1802

The literature review suggests that GSA should be preferred when the computational cost1803

of a single HYDRUS execution is low (e.g., seconds), as it provides statistically meaningful

1804

information about the most influential parameters and their interaction, which can be used to1805

prioritize experiments (i.e., explanatory setting) or reduce the dimensionality of the inverse

1806

problem (i.e., predictive setting). Instead, LSA should be used when the model is computationally1807

intensive to assess its sensitivity to small changes in specific parameters’ values. Ideally, for
73

1808

predictive purposes, this should be performed around the global optimum identified during the1809calibration procedure to check the model sensitivity near its most probable values. A prognostic

1810

use before calibration could lead to misleading conclusions if the LSA starting point lies in low-

1811

fitting regions. Similarly, for explanatory purposes, it is advisable to perform the LSA in regions

1812of the parameters’ space identified as probable based on the literature review, prior information,1813

and the modeler’s expertise.

1814
1815

Irrigation Optimization

1816

Model calibration, uncertainty assessment, and sensitivity analysis are all fundamental1817

steps to obtain a model that can robustly predict the physical behavior of the system under1818

investigation. Once this objective is achieved, the model can be used for multiple purposes, among

1819

which the most important in irrigation management is the optimization of the irrigation system.

1820

This includes establishing the irrigation schedule (e.g., water/nutrients injection timing and1821

duration) and defining locations and types of sensors (e.g., tensiometer, soil moisture sensors) and1822

actuators (e.g., pumps, valves). The latter are key components of real-time systems, which are1823

becoming very popular due to their capability to dynamically adapt the irrigation strategy

1824

depending on soil status. In this direction, the HYDRUS model includes a system-dependent

1825

boundary condition, which initiates irrigation whenever the matric head at a predetermined1826

location drops below a certain threshold (Dabach et al., 2013). This enables the modeler to design

1827

and optimize a real-time sensor (i.e., tensiometer) – actuator (e.g., valve) irrigation system tailored1828

to the specific soil.

1829

In any case, regardless of the irrigation system type, the irrigation strategy should be

1830

designed to maximize crop yield while simultaneously minimizing the environmental footprint

1831

and installation and maintenance costs. This leads to a complex multi-objective optimization

1832

problem that can be solved by coupling the model with various numerical optimization techniques.

1833

For example, Kandelous et al. (2012) coupled HYDRUS-2D with the multi-objective optimization1834

algorithm AMALGAM (Vrugt and Robinson, 2007) to derive the Pareto front and explore

1835

tradeoffs between water applications, irrigation system parameters, and crop transpiration, and to1836

evaluate the best management practices for subsurface drip irrigation systems in alfalfa. Similarly,

1837

Roy et al. (2019) obtained the Pareto front by combining HYDRUS-2D with an evolutionary multi-1838

objective optimization procedure to minimize water utilization and maximize crop yield
74

1839

prediction. Both studies prove that coupling HYDRUS with multi-objective search techniques can1840

provide decision-makers with multiple tradeoffs for physically based irrigation strategies.

1841

However, more research is needed to demonstrate the benefits of this type of analysis and include

1842

the economic costs as an additional objective function, which would lead to a 3D Pareto surface.

1843

Despite being a valuable tool, the multi-objective optimization of the irrigation system can1844

be computationally intensive and requires specific programming and numerical skills that are not1845

widespread among irrigation managers. Hence, a huge portion of the literature body has simplified

1846

the multi-objective optimization problem by focusing only on using HYDRUS to optimize a single

1847

objective function (e.g., water and nutrients use efficiency, root water uptake, etc.) (Akhtar et al.,1848

2013; Brunetti et al., 2018; Fontanet et al., 2020, 2022; García Morillo et al., 2017; Karandish and1849Šimůnek, 2018; Lena et al., 2022; Müller et al., 2016; Ning et al., 2021; Qin et al., 2016; Seidel et1850

al., 2015; Xu et al., 2019; Yi et al., 2022). The results of these studies are encouraging and show

1851

that the mechanistic framework provided by HYDRUS enables the modeler to obtain irrigation

1852

management strategies that are hydrologically functional, agriculturally efficient, and, more1853

importantly, physically realistic. Nevertheless, the optimization procedure can still be improved

1854

by: 1) performing multi-objective optimization searches, in which the economic costs are included,

1855

as this is a crucial choice criterion in real applications, 2) propagating the effects of model

1856

uncertainty in the optimization procedure, as this would provide more honest predictions to1857

decision-makers, and 3) taking into account dynamic changes in model structure (e.g., tillage

1858

effects on soil hydraulic properties) over medium-to-long term simulation periods.

1859

75

11. Special Irrigation Methods

1860
1861
1862

Over the last few years, the HYDRUS software has also been used to evaluate multiple

1863

traditional and/or experimental special irrigation methods. These applications include dual-drip

1864

irrigation (El-Nesr et al., 2014), pitcher irrigation (Siyal et al., 2009), irrigation using clay pipes

1865(Siyal and Skaggs, 2009; Siyal et al., 2013), capillary wick irrigation (Kamal et al., 2019; Al-

1866

Mayahi et al., 2020), irrigation from ring-shaped emitters (Saefuddin et al., 2019; Noguchi et al.,18672021), irrigation systems with a capillary barrier (Ityel et al., 2011; Wongkaew et al., 2018; Al-

1868

Mayahi et al., 2020; Noguchi et al., 2021; and Sao et al., 2021), vertical line source irrigation (Fan1869

et al., 2018ab, 2021, 2022; Kanda et al., 2020ab), film hole irrigation (Hou et al., 2014; Fan et al.,1870

2019ab, 2021a; Chen et al., 2020a; Jie et al., 2020), and many others (e.g., Li et al., 2013; Ji et al.,1871

2015; Wang et al., 2017a; Zhou et al., 2017; Fan et al., 2018c; Zemni et al., 2022; and Pan et al.,18722021).

1873
1874

Dual-Drip

1875

El-Nesr et al. (2014) used HYDRUS (2D/3D) to evaluate three novel technologies

1876

enhancing a spatial distribution of water and solutes in the root zone while limiting downward

1877

leaching. These technologies included (a) a physical barrier under a dripper and a dual-drip system

1878

with (b) concurrent and (c) sequential irrigation. Numerical simulations showed that applying a

1879

dual-drip system or installing a physical barrier could significantly alter the wetting pattern and1880

spatial distribution of applied solutes. The simulations also showed that the physical barrier and1881

the dual-drip systems are more suitable for coarse-textured soils than fine-textured soil. A dual-1882

drip irrigation system with two emitters that can be operated sequentially allows controlling which

1883

solute is retained in the root zone and which one is leached by simply altering the operation of the1884

two drippers.

1885
1886

Pitcher

1887

Siyal et al. (2009) used HYDRUS-2D to evaluate the functioning of pitcher irrigation, an1888

ancient but very efficient irrigation system used in many arid and semiarid regions, such as in

1889

Northern Africa, Iran, China, India, Pakistan, Brazil, Indonesia, and Zimbabwe (Figure 23; see1890

also references in Siyal et al., 2009). Siyal et al. (2009) investigated the water content distributions
76

1891

around small, medium, and large pitchers buried in fine- and coarse-textured soils and with1892

different saturated hydraulic conductivities of the pitcher material. Water content distributions

1893

simulated by HYDRUS-2D were found to be in close agreement with the experimental results,

1894

indicating that HYDRUS-2D is a suitable tool for investigating and designing pitcher irrigation

1895

systems. Simulated and experimental results also showed that a large pitcher, double the size of a1896

smaller one but with half the hydraulic conductivity, will produce approximately the same wetting

1897

front as the smaller pitcher. Figure 24 illustrates the impact of the pitcher wall’s saturated hydraulic1898

conductivity, for large pitchers, on simulated infiltration into an initially dry loam soil (the Pitcher

1899

example in Table A).

1900

1901
1902
1903
1904

Figure 23. Photographs illustrating pitcher irrigation and the axisymmetric finite element meshdiagram used by Siyal et al. (2009) to simulate water infiltration from a buried pitcher (Photo
credits: Altaf Siyal).

77

1905
1906
1907
1908

Figure 24. Simulated wetting of initially dry loam soil under pitcher irrigation for differing
pitcher wall saturated hydraulic conductivities.

1909

Clay Pipe1910

HYDRUS (2D/3D) was used by Siyal and Skaggs (2009) and Siyal et al. (2013) to evaluate

1911

soil wetting patterns and solute transport under porous clay pipe subsurface irrigation, respectively.

1912

Subsurface irrigation using porous or perforated clay pipes, similar to pitcher irrigation discussed

1913

above, has been practiced since ancient times. It can still represent an efficient, water-saving

1914

irrigation method for many less developed countries in arid and semiarid regions. Siyal and Skaggs

1915(2009) successfully compared experimental and simulated soil wetting patterns for subsurface pipe

1916

systems operating at different water pressures. They also simulated the irrigation system's function

1917

in different soil textures (sand, sandy loam, loam, silt), as well as in layered soils (sandy loam over1918

loam or loam over sandy loam). Siyal et al. (2013) expanded their study by additionally simulating

1919

salt accumulation near the soil surface from a subsurface clay pipe irrigation system during the

78

1920

growing season of okra. Their results showed that proper management of salt accumulation is vital

1921

for sustainable crop production whenever subsurface irrigation systems are being implemented.

1922
1923

Capillary Wick

1924

Capillary wick irrigation systems were analyzed using HYDRUS models by Kamal et al.

1925(2019) and Al-Mayahi et al. (2020). Low-pressure wick irrigation systems represent energy-

1926

efficient systems that can play an important role in smallholder greenhouse crop production or1927

home gardens by ensuring higher water use efficiency than most traditional approaches. Al-

1928

Mayahi et al. (2020) extended the analysis of the wick irrigation system by also considering a

1929

capillary barrier to prevent the leaching of irrigation water from the root zone.

1930
1931

Capillary Barrier

1932

The functioning of a capillary barrier under various irrigation systems has been studied

1933

using HYDRUS-1D and HYDRUS (2D/3D) by Ityel et al. (2011), Wongkaew et al. (2018), Al-

1934

Mayahi et al. (2020), Noguchi et al. (2021), and Sao et al. (2021). While Wongkaew et al. (2018)1935

considered surface-applied irrigation (sprinkler), Al-Mayahi et al. (2020) considered capillary

1936

wick irrigation, Noguchi et al. (2021) considered subsurface drip irrigation (using ring-shape1937

emitters), and Sao et al. (2021) considered uniform, line-source, and plant-targeted surface

1938

irrigation. All these studies showed that if properly designed, the capillary barrier can keep water

1939

in the root zone by preventing its leaching to deeper soil layers.

1940
1941

Ring-Shaped Emitters

1942

Saefuddin et al. (2019) used HYDRUS (2D/3D) to evaluate the functioning of a subsurface

1943

drip irrigation system using a ring-shaped emitter. A ring-shaped emitter (made from a standard

1944

rubber hose) is a low-cost subsurface irrigation system developed and introduced for subsurface

1945

irrigation in Indonesia for small-scale farmers who have scarce water resources. The three-1946

dimensional version of HYDRUS (2D/3D) was used to evaluate the performance of the ring-shape1947

emitter, including multiple alternative designs with different numbers of holes (in the rubber hose)1948

and changes in the covering method. Numerical results were successfully compared against

1949

experimental data collected during experiments with the original ring-shaped emitter.

1950
79

1951

Other Irrigation Methods

1952
1953

Vertical Line Source Irrigation

1954

Fan et al. (2018ab, 2021b, 2022) used the HYDRUS-2D software and laboratory

1955

experiments to evaluate the performance of the vertical line source (moistube) irrigation system.

1956

Vertical line source irrigation is a water-saving method developed for deep-rooted fruit trees in the1957

arid regions of China to enhance direct water and nutrient delivery to the root zone while reducing

1958

soil evaporation and improving water and nutrient use efficiency. Fan et al. (2018ab) performed

1959

multiple HYDRUS-2D simulations to identify the main influencing factors (e.g., soil texture,1960

initial water content, pressure head, moistube length, and buried depth) for this irrigation system

1961

on the soil wetting pattern.

1962
1963

Moistube Irrigation

1964

HYDRUS (2D/3D) was used in a study by Kanda et al. (2020ab) to evaluate the1965

performance of moistube irrigation (MTI), a subsurface irrigation technology that uses a

1966

semipermeable membrane to emit water continuously into the soil in response to soil water

1967

potential and applied pressure. Kanda et al. (2020a) evaluated how soil texture (loamy sand and1968

sandy clay loam) influences the soil water dynamics of MTI using laboratory experiments and1969

numerical simulations. Kanda et al. (2020a) evaluated how the moistube placement depth

1970

influences the soil water dynamics and root water uptake of cowpea. The goal was to determine

1971

the optimal depth of placement of moistube tapes in cowpea production to minimize water losses

1972

via drainage and soil evaporation and thereby improve water use efficiency.

1973
1974

Film Hole Irrigation

1975

Film hole irrigation (FHI) was investigated using the radially symmetrical option of the1976

HYDRUS (2D/3D) software by Hou et al. (2014), Chen et al. (2020a), Fan et al. (2019ab, 2021a),

1977

Jie et al. (2020), and Pan et al. (2021). This irrigation method is used mainly in the arid regions of

1978

China, where plastic film mulch is widely used to prevent evaporation and weeds growth. Water

1979

flows on the plastic film and infiltrates through crop holes and special irrigation holes. Film hole1980

irrigation is thus similar to point source irrigation. Compared to traditional surface irrigation

1981

methods, film hole irrigation significantly improves water use efficiency. According to Fan et al.
80

1982

(2021a), there are two types of film hole irrigation, i.e., film hole free infiltration (FHFI) and film

1983

hole interference infiltration (FHII), depending on whether the soil wetting fronts under the two

1984neighboring film holes are independent or interfere with each other during FHI. In general,

1985

HYDRUS proved to be a robust and useful tool in evaluating this type of irrigation.

1986
1987

Additional Irrigation Methods

1988

Many other irrigation techniques have been numerically evaluated using HYDRUS

1989(2D/3D), mainly in the literature from China. For example, negative-pressure irrigation has been

1990

studied by Ji et al. (2015), Zhou et al. (2017), and Wang et al. (2017a), bubbled root irrigation by

1991

Li et al. (2013), buried diffuser irrigation (a special type of subsurface drip irrigation method) by

1992

Zemni et al. (2022), and horizontal moistube-irrigation by Fan et al. (2018c). HYDRUS (2D/3D)

1993

proved to be an ideal tool for evaluating the performance of these various irrigation techniques.

1994

81

1995

12. Summary and Outlook

1996
1997

The use of HYDRUS is widespread today in many areas related to irrigation. In this work,

1998

we have reviewed the software's capabilities and demonstrated them using numerous examples.

1999

The model is still being continuously developed, and new features that better describe processes

2000

in the irrigated root zone are being added. HYDRUS can help design irrigation systems and2001

determine the optimal irrigation and fertilization regimes. Since the model can examine flow and2002

transport processes in and under the root zone, it can also be used to examine the sustainability of

2003

various irrigation regimes. Correct use of the model may optimize applications of water and other2004

agrochemicals, such as fertilizers and pesticides. Future agriculture will have to look closely at the2005

transport of agrochemicals through the vadose zone and into groundwater.

2006

Computational power advances every day, but simultaneously, there is a need to evaluate

2007

many processes using a very high spatial and temporal resolution. While using finer spatial and2008

temporal discretizations may reduce numerical errors, the computational demand may, at the same

2009

time, increase dramatically. The number of discretization nodes usually increases by a power of2010

two when the problem's dimensionality increases. While personal computers can perform complex

2011

simulations, the calculation duration may be too long, reducing our ability to estimate model

2012

parameters or optimize irrigation scheduling. Parallel calculation tools can take advantage of the2013

existing capacity of computers with multiple cores or processors and significantly speed up time-

2014

consuming simulations, especially those requiring many finite elements. Examples of such tools

2015

are the HyPar module (Šimůnek et al., 2016) or the ParSWMS model (Hardelauf et al., 2007;

2016

Gohardoust et al., 2021). Today it is also possible to perform calculations in the cloud, which can

2017significantly speed up the simulations and allow HYDRUS users to carry out comprehensive

2018

stochastic simulations (e.g., Sasidharan et al., 2019).

2019

Another alternative and promising approach is using emulators to perform computationally2020

intensive tasks. An example of such an emulator can be found in Neuro-Drip (Hinnell el al., 2010).

2021

This study combines an artificial neural network with a statistical description of the spatio-

2022

temporal distribution of added water from a single drip emitter to provide easily accessible, rapid

2023

illustrations of the spatial and temporal subsurface wetting patterns. These cheap-to-run surrogate

2024

models can be trained to rapidly emulate the response of the mechanistic model to changes in a

2025

finite number of parameters, thus enabling the modeler to perform complex model calibration and
82

2026

irrigation optimization tasks on standard personal computers. However, the applicability of such

2027

surrogate tools is limited to a relatively low number of parameters (e.g., 10) as the training phase2028

becomes computationally expensive in higher dimensions. Another possibility is to hybridize

2029

HYDRUS with machine learning methods to reduce the computational cost of expensive parts of2030

the code (e.g., reaction). Surrogate and hybrid modeling for physically based irrigation analysis is2031

certainly a topic worthy of future investigation.

2032

The HYDRUS ability to simulate crop growth is still relatively limited. The standard

2033

HYDRUS modules account for crop growth only externally by inputting potential evaporation and2034

transpiration fluxes (or the leaf area index in 1D to separate potential evapotranspiration into

2035

potential evaporation and transpiration). HYDRUS also allows users to externally prescribe a time-2036

variable rooting depth (and width) using the logistic growth function or in a tabulated form. The

2037

HYDRUS models do not allow the rooting zone's spatial extent to change actively due to certain

2038

environmental conditions and stresses (e.g., Hartman et al., 2017). While there have been attempts

2039

to provide HYDRUS the ability to simulate crop growth (e.g., Javaux et al., 2013; Zhou et al.,2040

2012; Wang et al., 2015; Han et al., 2015; Hartmann et al., 2017; Shelia et al., 2018; Groenveld et2041

al., 2021), none of these implementations is currently available in the HYDRUS software, and2042

only the models of Hartmann et al. (2017) are directly available from the HYDRUS website as

2043

non-standard HYDRUS modules (i.e., not fully supported by the HYDRUS GUI). Alternative

2044

approaches (e.g., de Jong van Lier et al., 2009, 2013; Couvreur et al., 2012) to the macroscopic

2045

model of Feddes et al. (1978) to account for various root water uptake stresses would also be a

2046

desirable new feature of the HYDRUS models.

2047

The environmental conditions of the irrigated root zone can change rapidly due to irrigation

2048

and soil drying. Soil water contents, temperatures, and salts concentrations and their composition2049

can affect soil hydraulic and solute transport (e.g., degradation) parameters. Since the model2050

dynamically and spatially calculates the environmental variables, it can simultaneously update

2051

model parameters that depend on them. Some of these connections are already implemented in

2052

HYDRUS, such as the dependence of the hydraulic conductivity on the chemical composition of2053

the soil solution (e.g., sodium adsorption ratio, SAR, and electric conductivity, EC) (Suarez and2054Šimůnek, 1997), the dependence of soil hydraulic properties (via viscosity and surface tension) on

2055

solution concentrations (e.g., Silva et al., 2020), or the dependence of solute reaction parameters2056(i.e., degradation) on changes in water contents and temperatures. More complex interactions
83

2057

between the environmental variables and model parameters can be considered using the HP1

2058

model (e.g., Jacques et al., 2008ab). Introducing additional relationships between environmental2059

factors and model parameters (e.g., clogging, i.e., a reduction of the hydraulic conductivity as a2060

function of cumulative infiltration of water and/or solute/bacteria) would further increase the2061

applicability of the HYDRUS models.

2062

Stable water isotopes are increasingly used as environmental tracers to investigate flow

2063

processes in the unsaturated zone in both bare and planted soils (e.g., Stumpp et al., 2012), to

2064

quantify water and nutrient travel times, and to identify water sources for root water uptake and2065

the age of these sources (e.g., Zhou et al., 2021, 2022). Only the 1D computational version of the2066

HYDRUS model can currently simulate the fate and transport of stable water isotopes subject to2067

no (Stumpp et al., 2012) or constant fractionation during evaporation. This contrasts with the2068

standard formulation of solute transport in HYDRUS during evaporation, where solutes

2069

concentrate at and near the soil surface when water evaporates. However, ignoring the evaporative2070

enrichment, as done in this 1D model, leads to underestimating 2H and 18O concentrations in the2071

topsoil, which may be more significant in regions with higher evaporative losses (Sprenger et al.,2072

2018). This evaporative enrichment, as well as its dependence on environmental factors (e.g.,2073

temperature and tension), has been recently implemented into the computational module of2074

HYDRUS-1D by Zhou et al. (2021, 2022), but it has not yet been included in the HYDRUS GUI

2075

or its higher dimensional modules. The ability to simulate the fate of environmental tracers (and2076

related issues, such as travel times and water age) would be another desirable feature of future2077

HYDRUS releases.

2078

Modern agriculture often uses various types of mulching to reduce evaporative losses and2079

prevent weed growth. While traditional mulching has been done using plastic films (e.g., Li et al.,2080

2015, 2017), more environmentally friendly alternatives, such as biodegradable (e.g., Chen et al.,

2081

2019, 2020) or sprayable (e.g., Filipović et al., 2020) mulches, are being intensively investigated.

2082

While HYDRUS has been used in many of these applications, its ability to fully describe dynamic

2083

boundary conditions associated with the application of biodegradable and sprayable mulches and2084

their temporarily changing properties is still rather limited, and further developments are needed.

2085

While considerable efforts have gone into developing the HYDRUS models to account for

2086

irrigation practices, we also understand that model development, calibration, and validation are

2087

never-ending processes. Today, the model can consider various physical, chemical, and biological
84

2088

processes and their interrelationships. While the model development for irrigation practices is

2089

progressing considerably, the progress in sensing the agricultural field is limited. Although there2090

are many point sensors that describe the state of water (water content or potential) in the field in a

2091

temporal manner, the spatial variation in the soil-plant system still limits their use in model2092

calibration. Undoubtedly, the development of non-intrusive methods can improve the spatial

2093

description of the water status in the soil. Even today, there are only a very few methods that can

2094

continuously quantify the chemical composition of soil water, and thus, the model calibration is2095

done with the help of a general but continuous measure of the bulk electrical conductivity or tests

2096

that usually require human intervention (soil sampling, suction caps). Regarding biological tests,2097

the situation is even more complex, and today, very few methods can evaluate the root system

2098

structure and functions under field conditions.

2099
2100

85

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