HYDRUS: MODEL USE, CALIBRATION,
AND VALIDATION
J. Šimůnek, M. Th. van Genuchten, M. Šejna

ABSTRACT. The HYDRUS numerical models are widely used for simulating water flow and solute transport in variably
saturated soils and groundwater. Applications involve a broad range of steady-state or transient water flow, solutetransport, and/or heat transfer problems. They include both short-term, one-dimensional laboratory column flow ortransport simulations, as well as more complex, long-duration, multi-dimensional field studies. The HYDRUS models canbe used for both direct problems when the initial and boundary conditions for all involved processes and corresponding
model parameters are known, as well as inverse problems when some of the parameters need to be calibrated or estimatedfrom observed data. The approach to model calibration and validation may vary widely depending upon the complexity of
the application. Model calibration and inverse parameter estimation can be carried out using a relatively simple, gradient-based, local optimization approach based on the Marquardt-Levenberg method, which is directly implemented into the
HYDRUS codes, or more complex global optimization methods, including genetic algorithms, which need to be run separately from HYDRUS. In this article, we provide a brief overview of the HYDRUS codes, discuss which HYDRUS parameters can be estimated using internally built optimization routines and which type of experimental data can be used for this,and review various calibration approaches that have been used in the literature in combination with the HYDRUS codes.
Keywords. Calibration, HYDRUS-1D, HYDRUS (2D/3D), Numerical model, Optimization methods, Parameter estimation,
Solute transport, Unsaturated soils, Validation, Water flow.

Numerical models are increasingly used for predicting or analyzing water flow and contaminanttransport processes in the subsurface, including
the vadose zone. The objective of this article isto briefly review the history of development of the
HYDRUS-1D and HYDRUS (2D/3D) software packages(Šimůnek et al., 2008b) that may be used for this purpose.
We provide key citations, briefly summarize the theory behind the models, describe key parameters needed to run thecodes, and discuss how these parameters can be obtainedby model calibration. Note that many terms often encountered in the literature related to model calibration, verification, and validation, such as parameter uniqueness, identifiability, stability, and ill-posedness, are described in detailelsewhere (Carrera and Neuman, 1986; Šimůnek and deVos, 1999; Šimůnek and Hopmans, 2002, among others)
and will not be further discussed here.
Over the years, various software packages from thebroad family of HYDRUS codes (e.g., SWMS-2D,
CHAIN_2D, HYDRUS-1D, HYDRUS-2D, HYDRUS

Submitted for review in September 2011 as manuscript number
SW 9401; approved for publication by the Soil & Water Division of
ASABE in April 2012.
The authors are Jiří Šimůnek, Professor and Hydrologist, Departmentof Environmental Sciences, University of California, Riverside, California; Martinus Th. van Genuchten, Professor, Department of MechanicalEngineering, Federal University of Rio de Janeiro, Brazil; and Miroslav
Šejna, Director, PC-Progress s.r.o., Prague, Czech Republic. Corresponding author: Jiří Šimůnek, Department of Environmental Sciences, University of California-Riverside, Riverside, CA 92521; phone: 951-8277854; e-mail: Jiri.Simunek@ucr.edu.

(2D/3D), UNSATCHEM, HP1, and CW2D) have become
widely used tools for evaluating water flow and solutetransport in soils and groundwater. For example,
HYDRUS-1D was downloaded more than five thousand
times in 2010 alone by users from around the world. The
HYDRUS website lists almost one thousand peer-reviewed
journal references in which the HYDRUS programs havebeen used. The codes themselves have been reviewed favorably several times in the literature, the latest one being a
recent review by Yu and Zheng (2010). The number of
HYDRUS-1D users is significantly larger than those of
HYDRUS (2D/3D) since HYDRUS-1D remains in the public domain and as such can be downloaded freely from the
HYDRUS website (www.hydrus2d.com), while HYDRUS(2D/3D) is distributed commercially for a nominal fee.
The HYDRUS-1D software package may be used tosimulate water flow and solute transport (as well as heatand carbon dioxide transport) in variably saturated media,
assuming either a vertical, horizontal, or generally inclined
direction. HYDRUS (2D/3D) similarly simulates waterflow and solute/heat transport in two-dimensional verticalor horizontal planes, in axisymmetrical three-dimensional
domains, or in fully three-dimensional variably saturated
domains. Because the history of development of the
HYDRUS (2D/3D) software package is documented in detail elsewhere (Šimůnek et al., 2008b), we will discuss here
primarily the historical development of HYDRUS-1D(Šimůnek et al., 1998a, 2005, 2008a).
HYDRUS-1D traces its roots to the early work of van
Genuchten (1978, 1987) and his SUMATRA and WORM
models, as well as later work by Vogel (1987) and Kool

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and van Genuchten (1989) and their SWMI and HYDRUS
models, respectively (fig. 1). While Hermitian cubic finite
element numerical schemes were used in SUMATRA andlinear finite elements were used in WORM and the older
HYDRUS code for solution of both the water flow and solute transport equations, SWMI used finite differences tosolve the flow equation. Various features of these four earlymodels were combined first in the DOS-based SWMI_ST
model (Šimůnek, 1993) and later in the Windows-based
HYDRUS-1D simulator (Šimůnek et al., 1998a). Version 1of HYDRUS-1D (Šimůnek et al., 1998a) was the first version that included both physical-nonequilibrium (dualporosity mobile-immobile water) and chemicalnonequilibrium (two-site sorption model) solute transport.
After releasing versions 1 (for 16-bit Windows 3.1) and 2(for 32-bit Windows 95), the next two major updates (versions 3 and 4) were released in 2005 and 2008. While version 3 of HYDRUS-1D included an option to consider dual-porosity water flow and solute transport, version 4
additionally included dual-permeability water flow and solute transport (Šimůnek and van Genuchten, 2008). The lasttwo versions included additional modules applicable tomore complex biogeochemical reactions than the standard
HYDRUS modules. While the standard modules of
HYDRUS-1D can simulate the transport of solutes that areeither fully independent or involved in the sequential firstorder degradation chains, the two new modules can consider mutual interactions between multiple solutes, such as
cation exchange and precipitation/dissolution. Version 3 included the UNSATCHEM module (Šimůnek et al., 1996;
Suarez and Šimůnek, 1997) for simulating carbon dioxidetransport as well as the multi-component transport of major
ions. The UNSATCHEM major ion module was recently
included also in version 2 of HYDRUS (2D/3D) (Šimůnek
et al., 2011). Version 4 of HYDRUS-1D now includes not
only the UNSATCHEM module but also the HP1 program(Jacques and Šimůnek, 2005), which resulted from coupling HYDRUS-1D with the biogeochemical program
PHREEQC (Parkhurst and Appelo, 1999).

HYDRUS DESCRIPTIONSIMULATED PROCESSESThe standard modules of HYDRUS may be used to sim-

Figure 1. History of development of HYDRUS-1D and related software packages. All of the software packages released after 1996 aresupported by graphical user interfaces.

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ulate the one-dimensional (in HYDRUS-1D) and two- or
three-dimensional (in HYDRUS (2D/3D)) movement ofwater, heat, and multiple solutes in variably saturated media. The codes may be applied to unsaturated, partially saturated, or fully saturated homogeneous or layered media.
The programs use mass-lumped linear finite element
schemes to numerically solve the Richards equation forsaturated-unsaturated flow. The unsaturated soil hydraulicproperties can be described using van Genuchten (1980),
Brooks and Corey (1964), modified van Genuchten (Vogel
and Císlerová, 1988), Durner (1994), and Kosugi (1996)
type analytical functions. The codes further incorporate
hysteresis by assuming that drying scanning curves are
scaled from the main drying curve and wetting scanning
curves are scaled from the main wetting curve. The codes
allow consideration of both equilibrium and nonequilibrium
flow. The governing equations for nonequilibrium flow pertain to dual-porosity or dual-permeability flow regimes, inwhich a fraction of the liquid phase is assumed to be mobile (moving relatively rapidly) and another fraction is assumed to be immobile (moving relatively slowly or not atall) (Šimůnek et al., 2003; Šimůnek and van Genuchten,2008).
Both HYDRUS programs can be used to simulate suchprocesses as precipitation, irrigation, infiltration, evaporation, root water uptake (transpiration), soil water storage,
capillary rise, deep drainage, groundwater recharge, andlateral flow (the latter not in 1D). Both models can also
evaluate surface runoff when the applied flux becomeslarger than the infiltration capacity of the soil, as well as
reduce evaporation and transpiration from their potential toactual values based on the prevailing conditions in the soil
profile and specific properties of the vegetation. The flow
equations additionally include sink terms to account for water uptake by plant roots as a function of both water and salinity stress. The codes can simulate both compensated anduncompensated root water uptake, as well as both active andpassive nutrient root uptake (Jarvis, 1989; Šimůnek andHopmans, 2009).
Heat and contaminant transport processes in the
HYDRUS-1D and HYDRUS (2D/3D) programs are described using Fickian-based advection-dispersion typeequations, which are solved numerically using linear finite
element schemes. The solute transport equations assume
advective-dispersive transport in the liquid phase and diffusion in the gaseous phase. The transport equations further
include provisions for nonlinear and/or nonequilibrium reactions between the solid and liquid phases, linear equilibrium reactions between the liquid and gaseous phases, zeroorder production, and two first-order degradation reactions,
one that is independent of other solutes, and one that provides the coupling between solutes involved in sequential
first-order decay reactions (Šimůnek et al., 2008b). In addition, physical nonequilibrium transport can be accounted
for by assuming several two-region, dual-porosity type
formulations that partition the liquid phase into mobile andimmobile regions. Chemical nonequilibrium transport canbe accounted for by dividing the sorption sites of the solid
phase into sites subject to instantaneous sorption and sites
with kinetic sorption (Selim et al., 1976; van Genuchten,

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1981; van Genuchten and Wagenet, 1989; Šimůnek and van
Genuchten, 2008). Because of the relatively general formulation of the transport and reaction terms, the HYDRUS
codes can be used to simulate the fate and transport ofmany different solutes, including non-adsorbing tracers, radionuclides (e.g., Mallants et al., 2003; Pontedeiro et al.,2010), mineral nitrogen species (Hanson et al., 2006), pesticides (Pot et al., 2005; Dousset et al., 2007), chlorinated aliphatic hydrocarbons (Schaerlaekens et al., 1999; Casey andŠimůnek, 2001), hormones (Casey et al., 2003), antibiotics(Wehrhan et al., 2007), and explosives/propellants (Dontsova
et al., 2006, 2009).
The governing transport equations were additionallymodified to allow consideration of kinetic attachment/detachment processes of solutes to the solid phase,and hence of solutes having a finite size. The attachment/detachment features have been used recently in several studies to simulate the transport of viruses (e.g., Schijvenand Šimůnek, 2002) colloids (Bradford et al., 2002, 2003,
2004), bacteria (Gargiulo et al., 2007a, 2007b), and nanoparticles (Torkzaban et al., 2010).
The HYDRUS codes further include modules for simulating carbon dioxide transport (only in 1D) and major ion
chemistry as adopted from the UNSATCHEM programs ofŠimůnek and Suarez (1994) and Šimůnek et al. (1996), respectively. Gonçalves et al. (2006) and Ramos et al. (2011)
recently demonstrated the applicability of these modules tosimulating multicomponent major ion transport in soil lysimeters irrigated with waters of different quality.
HYDRUS-1D was used in these applications to describe
field measurements of the water content, overall salinity,
the concentration of individual soluble cations, as well as ofthe sodium adsorption ratio (SAR), electric conductivity(EC), and the exchangeable sodium percentage (ESP).
To make the HYDRUS codes as attractive as possible
for use by inexperienced numerical modelers, students,
consulting engineers, and practitioners, much effort went
into the development of flexible graphics-based user interfaces. For this purpose, the HYDRUS packages use Microsoft Windows-based graphical user interfaces (GUIs) to
manage the input data required to run the programs, as well
as for nodal discretization and editing, parameter allocation,
problem execution, and visualization of the results. All spatially distributed parameters, such as those for delineating
various soil horizons, root water uptake distributions, and theinitial and boundary conditions for water, heat, and solute
movement, are specified in graphical environments. The
programs further include a small catalog of unsaturated soil
hydraulic properties (Carsel and Parrish, 1988), as well as
pedotransfer functions based on neural network analyses(Schaap et al., 2001).
The HYDRUS models as such are not limited to any
particular spatial or temporal scale, providing that the governing equations are formulated properly and can be used atthat scale. There have been many successful applications of
HYDRUS-1D at scales ranging from small laboratory soilcolumns to agricultural applications for soil profiles one orseveral meters deep (e.g., Gärdenäs et al., 2005; Hanson etal., 2006), up to soil profiles several hundred meters deep(e.g., Scanlon et al., 2003). HYDRUS (2D/3D) has been

55(4): 1261-1274

applied similarly to transport domains ranging from lessthan 1 m wide to transects of several tens or hundreds ofmeters wide, and involving both laboratory (e.g., Silliman
et al., 2002) and field-scale applications (Siyal et al., 2009;
Yakirevitch et al., 2010). However, we generally do notrecommend HYDRUS to be used for large threedimensional (i.e., catchment size) applications. The highly
nonlinear Richards equation requires relatively fine spatial
discretization, especially at locations where large hydraulicgradients are expected, such as close to the soil surface
where variable meteorological conditions can cause very
rapid changes in the soil water contents and corresponding
pressure heads. Spatial discretization of even a relatively
small catchment can lead to hundreds of thousands, or evenmillions, of finite element nodes, which may significantly
strain available computational resources. On the other
hand, there are no limitations on the temporal scale, which
can range from a few minutes for small-scale laboratory
studies to hundreds of thousands of years for studies evaluating the effects of the past and current climate(e.g., Scanlon et al., 2003), the effects of the future climatechange (Leterme and Mallants, 2011), or long-term environmental risk analyses of radioactive contaminants entering the subsurface (Pontedeiro et al., 2010).
MODEL CALIBRATION AND INVERSE PARAMETERESTIMATION
Model calibration is generally defined as the process oftuning a model for a particular problem by manipulating
the input parameters (e.g., soil hydraulic parameters) andinitial or boundary conditions within reasonable ranges until the simulated model results closely match the observed
variables (e.g., pressure heads, water contents, concentrations, various fluxes) (Šimůnek and Hopmans, 2002). The
general approach in model calibration is to select an objective function to serve as a measure of the agreement between measured and modeled data, and which directly or
indirectly is related to the adjustable parameters. The bestfit parameters are then obtained by minimizing this objective function. Model calibration can be achieved by trial
and error or by using automated minimization or parameter
estimation techniques. A model is considered successfully
calibrated when it reproduces data within some subjectively
acceptable level of precision (Konikow and Bredehoeft,1992). Model calibration is often referred to also as history
matching. When the ultimate goal of model calibration is
not merely to calibrate a model but rather to optimize unknown parameters in that model, the process is often calledparameter optimization or parameter estimation (Šimůnek
and de Vos, 1999).
Both HYDRUS software packages implement a Marquardt-Levenberg type parameter estimation technique(Marquardt, 1963; Šimůnek and Hopmans, 2002) for inverse estimation of soil hydraulic, solute transport, and/or
heat transport parameters from measured transient or
steady-state flow and/or transport data. For this purpose,the HYDRUS programs are written in such a way that almost any application that can be run in a direct mode
(i.e., when all parameters and initial and boundary conditions are specified and predictions are made) can be run

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equally well in an inverse mode, and thus are effective formodel calibration and parameter estimation applications.
Because of its generality, the inverse option in HYDRUS
has proven to be very popular with many users, leading to alarge number of applications ranging from relatively simple
laboratory experiments, such as one- and multi-step outflow experiments (e.g., Wildenschild et al., 2001) or evaporation experiments (Šimůnek et al., 1998b), to more elaborate field problems involving multiple soil horizons andchemicals (Vrugt et al., 2008; Wöhling et al., 2008). Referto the HYDRUS website (www.pc-progress.com/en/
Default.aspx?hydrus-3d) for many specific examples.
The objective function to be minimized during the parameter estimation process using one of the HYDRUS
codes is written in a relatively general manner to allow
simultaneous consideration of a large number of different
types of measurements and information. The objective function, Φ, is defined as (Šimůnek et al., 1998a):Φ (b, q, p) =

mq

nqj

j =1

i =1

 v j  wi, j q*j (x, ti ) − q j (x, ti , b)mp+

  wi, j  p*j (x,θi ) − p j (x,θi , b) 
vj

j =1

+

n pj

2

nb

2

(1)

i =1

vˆ j b*j (x) − b j (x) 

2

j =1

where the first term on the right side represents deviations
between measured and calculated space-time variables. In
this first term, mq is the number of different sets of measurements, nqj is the number of measurements within a particular measurement set, qj*(x,ti) represents specific measurements at time ti for the jth measurement set at location x,qj(x,ti,b) represents the corresponding model predictions forthe vector of optimized parameters b (e.g., soil hydraulic,heat transport, and/or solute transport and reaction parameters), and vj and wi,j are weights associated with a particularmeasurement set or point, respectively. The first term can
include, for example, observed pressure heads, water contents, temperatures, and/or concentrations at different locations and/or times in the flow domain, or actual or cumulative fluxes versus time across a boundary of specified type.
Resident or flux concentrations, liquid phase or total concentrations, and concentrations at a linear or logarithmic
scale can all be used in the optimization depending on theapplication and data availability.
The second term on the right side of equation 1 represents differences between independently measured and predicted soil hydraulic properties (e.g., retention, θ(h), and/or
hydraulic conductivity, K(θ) or K(h), data) for different soil
horizons (x), while the terms mp, npj, pj*(x,θi), pj(x,θi, b), v̂ j
and wi,j have similar meanings as for the first term but are
now used for the soil hydraulic properties. The last term ofequation 1 represents a penalty function for deviations between prior knowledge of the soil hydraulic parameters,
bj*(x), and their final estimates, bj(x), with nb being thenumber of parameters with prior knowledge and vj repre-

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senting pre-assigned weights.
The Marquardt-Levenberg parameter estimation approach implemented in the HYDRUS codes assumes thatthe covariance (weighting) matrices, which provide information about the measurement accuracy as well as possible
correlation between measurement errors and/or the parameters, are diagonal (Šimůnek and Hopmans, 2002). The
weighting coefficients, vj, may be used to minimize differences in weighting between different data types because ofdifferent absolute values and numbers of data involved.
They can either be specified as an input parameter orweighted proportionally to the means or variances of individual data sets. In addition, the inverse option as currently
implemented in HYDRUS can be used to estimate a rangeof soil hydraulic parameters, heat transport parameters,
and/or solute transport and reaction parameters. However,
the inverse option cannot be used to estimate other variables or parameters, such as initial and boundary conditions
or root water uptake parameters. In that case, other (external) optimization tools need to be used.
Finally, we note that the Marquardt-Levenberg methodin the HYDRUS programs is a local optimization gradientmethod that requires initial estimates of the unknown parameters to be optimized. The behavior of the objectivefunction, Φ, in the neighborhood of this initial estimate isused to select a direction vector, from which updated valuesof the unknown parameter vector b in equation 1 are determined. Depending on the problem being considered (i.e.,the magnitude of the measurement errors, the number of
optimized parameters, and the type of measurements), theobjective function may be topographically very complex
without having a well-defined global minimum, or having
several local minima in the parameter space (Šimůnek andHopmans, 2002). The minimization can then be very sensitive to the initial values of the optimized parameters. Depending on the initial estimate, the final solution of the calibration may then not be the global minimum, but instead a
local minimum. Consequently, it is generally recommended
to repeat the minimization problem with different initial estimates of the optimized parameters and then select thoseparameter values that minimized the objective function. Alternatively, more robust global minimization techniques
could be used, as will be discussed later.

ESTIMATION OF PARAMETERS
USING HYDRUS
SOIL HYDRAULIC PARAMETERS
Variably saturated flow models, such as the HYDRUS
codes, need as input the unsaturated soil hydraulic properties comprising the water retention curve (or soil moisturecharacteristic) and the hydraulic conductivity function.
While these properties can be measured both in the laboratory and in the field, standard protocols using classicalsteady-state methods are generally very elaborate, timeconsuming, and hence expensive. Alternative and much
faster transient methods, usually combined with numericalinversion of the flow process, have been studied extensively throughout the 1980s, 1990s, and up to the present (vanTRANSACTIONS OF THE ASABE

Genuchten et al., 1999; Hopmans et al., 2002b). Because of
their inverse options, the HYDRUS codes have been usedin many of these studies. HYDRUS-1D example applications include one-step and multi-step outflow experiments(Wildenschild et al., 2001; Bitterlich et al., 2004; Laloy et al.,2010; Schelle et al., 2010), evaporation experiments(Šimůnek et al., 1998b; Schindler et al., 2010), horizontal
infiltration (Šimůnek et al., 2000; Jacques et al., 2012), upward infiltration (Šimůnek et al., 2001; Young et al., 2002),
and centrifuge experiments (Šimůnek and Nimmo, 2005;
Nakajima and Stadler, 2006; van den Berg et al., 2009),
among others. HYDRUS (2D/3D) examples include experiments with tension disc permeameters (Šimůnek and van
Genuchten, 1996, 1997), a modified cone penetrometer(Gribb et al., 1998; Kodešová et al., 1998; Šimůnek et al.,1999a), and a multistep soil-water extraction method (Inoueet al., 1998). The physical principles of most of these methods are discussed at length in several sections of the SSSA
Methods monograph by Dane and Topp (2002). Later, we
briefly discuss a case study in which the inverse option of
HYDRUS is used to estimate the unsaturated soil hydraulicproperties from an evaporation experiment carried out on asmall laboratory soil core.
All of the above studies assume equilibrium flow processes in combination with relatively standard functions forthe unsaturated soil hydraulic properties. A large number ofstudies also exist in which parameters of various nonequilibrium flow and transport models, including especially selected dual-porosity or dual-permeability flow formulations, have been estimated using the HYDRUS codes(e.g., Pot et al., 2005; Dousset et al., 2007). Refer to Köhne
et al. (2009a, 2009b) for a detailed review of these applications. We note that the HYDRUS models implement only
one single overall approach to modeling preferential andnonequilibrium water flow (i.e., the flow is consistently described using the Richards equation or various modifications thereof), while many other approaches have been developed in the literature and implemented in other models(for a review of these approaches, see Šimůnek et al., 2003,
including those discussed in this special collection. It is still
an open question which approach is more suitable to different conditions (e.g., Köhne et al., 2009a, 2009b).
SOLUTE TRANSPORT AND REACTION PARAMETERS
When, in addition to water flow, solute transport is also
simulated, the models need various solute transport(e.g., dispersivities, diffusion coefficients in the liquid andgaseous phases) and reaction (sorption and degradation) parameters as input. These transport and reaction parameters
are often obtained using laboratory column studies that are
carried out under steady-state water flow conditions. One
standard laboratory procedure for obtaining some of theseparameters is to carry out miscible displacement experiments in short columns and analyze the observed effluent
curves using analytical solutions of the advectiondispersion equation (e.g., Skaggs et al., 2002). A large
number of such analytical solutions for different initial andboundary conditions, as well as for one- and multidimensional applications, have been assembled in the
STANMOD software package (Šimůnek et al., 1999b;55(4): 1261-1274

Šimůnek et al., 2008b), which is widely used for thesetypes of transport studies (see van Genuchten et al., 2012).
Unfortunately, analytical solutions generally do not exist
for nonlinear transport problems, such as for nonlinear
sorption, colloid attachment with surface blocking, or whencomplex reaction pathways are present, in which case numerical solutions need to be employed. A large number ofexamples again exist in which the HYDRUS models havebeen used to identify solute transport and reaction parameters. A detailed review of various HYDRUS-1D applications to the analysis of nonlinear solute transport experiments, including transient flow conditions, is given byŠimůnek et al. (2002). Additional examples include thetransport of interacting nitrogen species (Bolado-Rodríguez
et al., 2010), pesticides (Pot et al., 2005; De Wilde et al.,2009, 2010; Cheyns et al., 2010), chlorinated aliphatic hydrocarbons (Casey and Šimůnek, 2001), hormones (Casey etal., 2003, 2004, 2005; Sangsupan et al., 2006; Fan et al.,2007), and explosives/propellants (Dontsova et al., 2006,
2009; Alavi et al., 2011). Refer to the HYDRUS website(www.pc-progress.com/en/Default.aspx?hydrus-3d) for additional examples.
HEAT TRANSPORT PARAMETERS
The governing equation describing heat transport insoils and other porous media contains several parameters
representing the thermal properties of the medium involved. These properties include thermal volumetric capacities of individual soil constituents (e.g., solid mineralphase, organic matter), the thermal dispersivity, and thethermal conductivity function (dependent on water content). All of these parameters can be estimated from measured soil temperatures using one of the HYDRUS codes.
Examples in which soil thermal parameters have been estimated using HYDRUS include studies by Hopmans et al.
(2002a), Mortensen et al. (2006), Saito et al. (2007), andSakai et al. (2009).
OTHER PARAMETERS
Several other model parameters, such as those describing root water uptake and root distributions (e.g., Vrugt etal., 2001a, 2001b), have also been estimated using the
HYDRUS codes. However, since in general this cannot bedone directly using the standard HYDRUS inverse option,either the HYDRUS source code needs to be modified orsome external optimization tool must be employed. We will
not here further discuss particular applications of the modified HYDRUS codes. However, in the next section, we
briefly discuss external tools that have been used in combination with HYDRUS to optimize various HYDRUS model
parameters and to calibrate the HYDRUS codes.

ESTIMATION OF PARAMETERS USING
EXTERNAL OPTIMIZATION TOOLSMODEL-INDEPENDENT OPTIMIZATION TOOLS
Two general, model-independent, parameter optimization packages that have been popularly used in subsurfacehydrology are PEST (Doherty, 2005) and UCODE (Poeter

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et al., 2005). While initially developed for the groundwaterflow model MODFLOW, both codes have since been usedwith other models, including HYDRUS. PEST and
UCODE provide users with great flexibility in choosing
which parameters to optimize and which variables to use in
the objective function. As such, they can also be used with
HYDRUS to optimize parameters and to use variables andinformation that would not have been possible with thestandard HYDRUS inversion option. Still, we note thatboth PEST and UCODE use a gradient-type minimization
method similar to the one used by the HYDRUS models
and thus are equally prone to finding local rather than global minima.
Example applications of the PEST program with
HYDRUS are studies by Singh et al. (2010) and Singh andWallender (2011), who evaluated soil hydraulic parametersfrom field data sets at different levels of soil water salinity,and by Huang et al. (2011), who analyzed infiltration anddrainage processes in multi-layered coarse soils. The
UCODE software was used in combination with
HYDRUS-1D by Dahiya et al. (2007) to simultaneously
optimize the soil hydraulic parameters of several soil layers, as well as the potential evaporation rate. Recently,
Jacques et al. (2012) documented the use of UCODE to optimize soil hydraulic and solute transport parameters, aswell as cation exchange capacity (CEC) and the Gapon exchange constants from observed water content, Cl concentration, and total aqueous and sorbed concentration profilesof three major cations (Ca, Mg, and K) during water absorption into a horizontal column. While the standard
HYDRUS-1D inverse option was used to estimate soil hydraulic and solute transport parameters, HP1 was used toestimate CEC and the Gapon exchange constants.
GENETIC ALGORITHMS
Local gradient-based optimization algorithms, such as
the Levenberg-Marquardt scheme (Marquardt, 1963), generally work best when only a limited number of parametersneed to be identified. An alternative is needed when a larger
set of parameters is involved, such as when optimizing soil
hydraulic parameters for multiple horizons (Vrugt et al.,2001b). While many different global optimization algorithms, such as simulated annealing, stochastic approximation methods, evolution algorithms, multiple-level singlelinkage methods, interval arithmetic techniques, taboo search
schemes, subenergy tunneling, and non-Lopschitzian terminal
repellers, have been developed in the past (e.g., see references in Oreskes et al., 1994; Barhen et al., 1997), only a
few of these have been used in combination with the
HYDRUS models.
Genetic algorithms provide a powerful tool for parameter identification when the number of fitting parameters islarge (Bäck, 1996). Genetic algorithms were developed inevolution theory based on the concepts of natural selection
and genetics. Vrugt et al. (2001a, 2001b) used a genetic algorithm to simultaneously identify soil hydraulic, root water uptake, and root distribution parameters, which would
be very difficult, if not impossible, with the standard parameter approaches implemented in HYDRUS. Although

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genetic algorithms are effective in determining the general
region with the global minimum (if it exists), they are notnecessarily the most efficient in finding the exact optimum
location. For this reason, Vrugt et al. (2001a) only used a
genetic algorithm to identify the region where the global
minimum of the objective function existed, and then used
the results of the genetic algorithm as initial values in a
simplex optimization to determine the local minimum within the global minimum region. Zhang et al. (2010) similarly
used genetic algorithms to infer the effective soil hydraulicproperties from field evaporation experiments.
SCEM-UA, AMALGAM, AND OTHERGLOBAL OPTIMIZATION TOOLSThe SCEM-UA (Shuffled Complex Evolution MetropolisUniversity of Amsterdam/Arizona) algorithm (Vrugt et al.,2003b), which is based on the Markov chain Monte Carlo(MCMC) method, is likely the most commonly used external global optimization method employed with HYDRUS.
This method was successfully used to identify soil hydraulic parameters using HYDRUS by Vrugt et al. (2003a,
2004), Schoups et al. (2005), and Mertens et al. (2006),
among many others.
The number of global optimization tools that are being
developed is still increasing rapidly. For example, Wöhling
et al. (2008) used HYDRUS to estimate the soil hydraulicparameters of multiple soil horizons using three global optimization algorithms: the Nondominated Sorting Genetic
Algorithm (NSGA-II), the Multi-objective Shuffled Complex Evolution Metropolis algorithm (MOSCEM-UA), and
AMALGAM, the latter being A Multi-ALgorithm Genetically Adaptive search Method for multi-objective estimation of soil hydraulic parameters. A recent review of many
new developments, including the use of methods suitable
for massively parallel computing, is given by Vrugt et al.
(2008).

DISCUSSION
MODEL PARAMETERS AND CALIBRATIONThe HYDRUS codes are physically based models and assuch require little or no calibration when all required input
parameters (i.e., soil hydraulic parameters for water flow,
solute transport and reaction parameters for solutetransport, and thermal parameters for heat transport) areexperimentally determined. Several examples can be foundin the literature where the HYDRUS models have beensuccessfully applied to laboratory and field experiments using parameters that were all determined independently ofthe modeling itself and hence were not calibrated. Verycomprehensive studies, in which all required input parameters were determined independently, were carried out by
Gonçalves et al. (2006) and Ramos et al. (2011). They determined soil hydraulic and solute transport parameters onlaboratory columns and reaction parameters (Gapon selectivity coefficients) from chemical analysis, and then used
HYDRUS to successfully simulate water contents, soluteconcentrations of individual cations, and various integral
variables (such as SAR, EC, and ESP) in multiple lysime-

TRANSACTIONS OF THE ASABE

ters irrigated with waters of different qualities.
However, these were relatively unique studies wheremuch effort went into accurate measurement of all parameters. In most applications, one or more parameters requiredby the model may be missing. HYDRUS offers several options for determining the missing parameters for such conditions, with calibration being only one option. For example, when users do not have estimates of the soil hydraulicparameters, they can either use a catalog of average parameters for 12 textural classes of the USDA textural triangle(Carsel and Parrish, 1988) or pedotransfer functions as derived by Schaap et al. (2001) using their Rosetta program.
Rosetta is a widely used computer program for estimating
soil hydraulic parameters from soil textural properties using
hierarchical pedotransfer functions. HYDRUS additionally
includes a small database with parameters of the water
stress response functions that reduce the potential root water uptake rate as a result of water and salinity stress.
When transient laboratory or field data exist, these canbe used in combination with Rosetta-type pedotransfer
functions or other information, including independently determined flow and transport parameters, to calibrate the
HYDRUS models. The pedotransfer-derived or independently determined parameters can be then used as initial
estimates in the model calibration, or as prior informationin the Bayesian sense such that the model can penalize forany deviations from this prior information (e.g., Šimůnek
and Hopmans, 2002).
In addition to the final values of the optimized parameters, the parameter estimation modules in HYDRUS also
report estimates of the 95% confidence intervals, as well as
mutual correlations between the parameters (see Šimůnek
and Hopmans, 2002). Any type of uncertainty, such as in themeasured data, the specified initial and boundary conditions, or the overall conceptual model, has an adverse effectand will be reflected in these two statistical measures. Boththe confidence intervals for optimized parameters and theirmutual correlations increase with input uncertainties. However, because of the nonlinearity of the overall flow problem, it is impossible to make exact predictions about how
uncertainty in the input would propagate into modelingoutput or uncertainty in the calibrated parameters without adetailed analysis of the particular problem being investigated. Both statistical measures can be obtained from analysesof the objective function in the vicinity of the final optimized values. Both indicate how well the optimized parameters are estimated, or if they can be optimized at all. Strong
correlations between the optimized parameters indicate thatthe parameters cannot be simultaneously optimized and that
one of the parameters should be independently determined
and fixed during the optimization process. Alternatively, perhaps additional data could be included in the objective function to diminish correlation. Hence, the correlation matrix
may be used to identify nonadjustable parameters because of
their high correlation with other fitting parameters. Interdependence of the optimized parameters often also causes
slow convergence and non-uniqueness during the inversion
process and increased parameter uncertainty.

55(4): 1261-1274

For highly nonlinear problems, such as many variably
saturated flow studies, it is sometimes difficult to estimate
a priori what information to collect and which parameters
to optimize in order to obtain a well-posed parameter estimation solution. It is not always immediately obvious
whether or not a given type of experiment, or a given dataset, will result in a well-posed inverse problem, or how
many parameters can be uniquely estimated. Since it is notalways clear what may cause a non-unique or unstable solution, inverse problems always require an in-depth analysis to determine whether or not the problem is well-posed
and, if not, what causes the ill-posedness (Šimůnek and van
Genuchten, 1996; Šimůnek and Hopmans, 2002).
While the HYDRUS codes have been used in several
studies to identify soil hydraulic parameters for multiple
soil horizons (e.g., Jacques et al., 2002; Dahiya et al., 2007;
Wöhling et al., 2008; Huang et al., 2011), an importantquestion remains: whether the models can also be used toestimate effective soil hydraulic properties for the entire
soil profile, assuming its homogeneity (e.g., Jacques et al.,
2002; Durner et al., 2008; Schelle et al., 2012). For example, in attempts to estimate effective soil hydraulic properties of lysimeters with layered soils, Durner et al. (2008)
found that although boundary fluxes of the lysimeters could
be matched closely using effective properties, internal
states could not be described correctly. Furthermore, the effective properties depended on the boundary conditions andthe sequence of soil layers, and hence no unique effective
properties could be found. Similar conclusions about the
non-uniqueness of effective soil hydraulic properties andtheir dependence on boundary conditions were obtained instudies by Zhu and Mohanty (2002), Mohanty and Zhu(2007), Vereecken et al. (2006), and Zhu et al. (2007),
among others.
HISTORICAL VALIDATION
If one accepts the notion that models can be formallyvalidated (e.g., see the discussions by Konikow and
Bredehoeft, 1992, and Oreskes et al., 1994, among others),
the optimized model parameters are often validated using a
two-step calibration scheme or split sampling. The existingdata set is then divided into two parts, with the first (calibration) part being used to calibrate the model and to estimate all necessary parameters, while the second (validationor verification) part of the data set serves to compare predicted and measured data using the parameters found during the calibration (Šimůnek and de Vos, 1999). Although
successful performance of the model during the second partdoes not constitute a rigorous validation of the model, it isoften accepted that the model is validated if it predicts thesystem response during this second phase within acceptable
limits (Anderson and Woessner, 1991). A two-step calibration process of this type is sometimes called historical validation. Many examples exist of historical validation of the
HYDRUS codes with parameters estimated using model
calibration. An early example is given by Šimůnek and deVos (1999), while more recent applications are provided by
Luo and Sophocleous (2010), Wang et al. (2010), and Zhao
et al. (2010), among others.

1267

CASE STUDY: INVERSE ANALYSIS OF
AN EVAPORATION EXPERIMENT
A large number of references are given in this article of
selected HYDRUS applications to a range of laboratory andfield-scale studies. Hundreds of additional examples can befound on the HYDRUS website (www.pc-progress.com/en/
Default.aspx?h1d-references). In particular, we would liketo direct readers to a recent study by Wöhling et al. (2008)
showing a detailed analysis of flow data from a complex
multilayered soil system using HYDRUS-1D in combination with three different multi-objective optimization algorithms. Here we briefly summarize a more modest application, in which the parameter estimation option of
HYDRUS-1D was used to analyze data involving a laboratory evaporation experiment carried out by Wendroth et al.
(1993). The example is actually one of the test cases directly distributed with the HYDRUS-1D installation files.
This example shows how the parameter estimation
method can be used to extract the most information from anevaporation experiment. The method, also known as
Wind’s method (Wind, 1968; Arya, 2002), involves evaporation from the top of an initially saturated soil core, which
is closed at the bottom, while water loss from the column isbeing monitored as a function of time by repeatedly weighing the core. Measurements are simultaneously made of thepressure head using tensiometers inserted at two or more
depths in the column. At the end of the experiments, thesample is over-dried and average transient water contentsof the core during the experiment are back-calculated byadding incremental evaporative losses between successive
measurements to the final mean water content. The evaporation rate, mean water content, and tensiometer data provide the necessary information to obtain estimates of thewater retention and hydraulic conductivity functions using
the standard Wind method or various modifications and extensions thereof (Wendroth et al., 1993; Arya, 2002).
The data for our example were obtained using an undisturbed soil core having a height of 10 cm and inside
diameter of 10 cm (Wendroth et al., 1993; Šimůnek et al.,1998b). The soil had a bulk density of 1.59 g cm-3 andsand, silt, and clay fractions of 7.4%, 79.3%, and 13.3%,
respectively. Five tensiometers with ceramic cups having
a length of 6 cm and outside diameter of 0.6 cm were inserted horizontally through drill holes into the soil core atlocations 1, 3, 5, 7, and 9 cm below the surface. An initialpressure head of -15.4 cm was measured in the middle ofthe soil sample. Before starting the evaporation experiment, several tension infiltrometer experiments were also
carried out on the core to estimate the hydraulic conductivity at pressure heads of -1, -5, and -10 cm.
Evaporation losses versus time during the experiments
were obtained by weighing the core immediately after taking the tensiometer readings. The evaporation rate at the
beginning of the experiments was artificially increased toapproximately 1.2 cm d-1 by using a fan to blow air awayfrom the soil surface at room temperature (Wendroth et al.,1993). Once the gradient between the tensiometers reacheda value between 1.5 and 2.5 m m-1, the top of the soil sample was covered to prevent further evaporation. After re-1268

establishing hydraulic equilibrium in the samples, evaporation was allowed to proceed without the fan at a rate of approximately 0.2 cm d-1. Measurements were taken every
30 min during the initial high-evaporation rate period andevery 4 h during the second stage of evaporation. The experiment was terminated when the upper tensiometer recorded a pressure head value of -650 cm. Evaporation losses
between consecutive measurements were used to define theupper flow boundary condition for the numerical simulations.
This laboratory experiment was first analyzed using themodified Wind method, as described in detail by Wendrothet al. (1993). The soil hydraulic parameters were obtainedby simultaneously fitting the van Genuchten (1980) soil
hydraulic parameters to all of the 179 measured θ(h) and94 K(h) data pairs, augmented with the three independently
measured tension infiltrometer K(h) data points. The fitting
was accomplished using the RETC code of van Genuchten
et al. (1991). The hydraulic parameters consisted of the residual (θr) and saturated (θr) water contents, the van
Genuchten shape factors (α and n), and the saturated hydraulic conductivity (Ks). The RETC fitted parameters arelisted in table 1. Data obtained using Wind’s method andthe fitted soil hydraulic functions (the thick solid lines) are
shown in figure 2.
The soil hydraulic parameters were also estimated fromthe evaporation experiment using the parameter inversion
option of HYDRUS-1D. Parameter estimation was accomplished using only the first term of equation 1, in which bj ={θr, θs, α, n, Ks} represented the vector of unknown soil
hydraulic parameters, while the vector qj*(x,ti) consisted ofpressure heads measured at either one or several tensiometer
locations (x), as well as the total amount of water in the soilcore measured at the end of the experiment. The weighing
coefficients wi,j in equation 1 were all fixed at 1, thus assuming that variances of the errors in the tensiometer readings were all the same. Furthermore, the weighting coefficients vj were fixed at (Šimůnek and Hopmans, 2002):
vj =1
nqj σ 2qj

(2)

which implies that the objective function Φ in this example
is essentially the average weighted squared deviation between the measured and fitted values normalized by themeasurement variances (σ2). Since the final water volume
is only one number and the variance cannot be defined, theweight of that extra data point was also fixed at 1.
Table 1 lists the optimized soil hydraulic parameters, toTable 1. Soil hydraulic parameters obtained from an evaporation experiment using parameter estimation and Wind’s method.
Ks
Method ofαθs(cm-1)nR2
Analysisθr(cm d-1)
Parameter
0.013
0.327 0.0273 1.22
88.6
0.9987
estimation (0-0.03)[a] (0.311- (0.026- (1.20- (73.20.343) 0.029) 1.24)
104.)
Wind’s
0.00450.321 0.0249 1.23
73.3
0.992
method[a]
Values in parentheses indicate 95% confidence intervals of the
optimized parameters.

TRANSACTIONS OF THE ASABE

0.35

0

a)

a)

0.3

0.25

Pressure Head [cm]Water Content [-]-100

All tensiometers
Tensiometer 1
Tensiometer 2
Tensiometer 3
Tensiometer 4
Tensiometer 5
Fitted W ind's Points
W ind Method

0.2

-200
-300

Measured 1
Fitted 1
Measured 2
Fitted 2
Measured 3
Fitted 3
Measured 4
Fitted 4
Measured 5
Fitted 5

-400
-500
-600

0.15
0.5

1.0

1.5

2.0

2.5

3.0

-700

log h [cm]

0

2

0.5

1

1.5

2

b)
1

3

3.5

4

4.5

-900

-700

All Tensiometers
Tensiometer 1
Tensiometer 2
Tensiometer 3
Tensiometer 4
Tensiometer 5
Fitted W ind's Points
W ind Method
Tension Disc Infiltration

-1
-2
-3
-4
0.0

0.5

1.0

1.5

log h [cm]

2.0

2.5

3.0

Figure 2. (a) Water retention and (b) hydraulic conductivity functions
determined with inverse parameter estimation and Wind’s method.
2gether with R values of the regressions between the predicted and measured values, as obtained by numerical inversion and Wind’s method. Figure 2 further comparesplots of the resulting soil water retention and hydraulicconductivity functions. Notice the excellent agreement between the water retention curves obtained by parameter optimization and the θ(h) data points determined with Wind’s
method, or their complete analytical descriptions using
RETC (fig. 2a). The soil-water retention functions obtainedwith Wind’s method and via parameter optimization were
almost identical and indistinguishable from each other(fig. 2a). The estimated unsaturated hydraulic conductivity
functions obtained with the two methods (fig. 2b) weresimilarly very close.
Figure 2 also shows results of inversions when the objective function was alternatively defined in terms of dataobtained with only one tensiometer. Results remained relatively close to those obtained when readings from all tensiometers were used simultaneously. All optimizations gave
similar results for the soil hydraulic properties within therange of measured pressure heads (-20 to -700 cm). Extrapolation beyond this range involved considerable uncertainty
because of high correlation between the parameters θr and n.
This uncertainty could be greatly reduced by including an independently measured water content at or below the wilting
point in the objective function (Šimůnek et al., 1998b).
Measured and fitted tensiometer readings within the soilcore are shown in figure 3. The largest deviations wereabout 5 and 20 cm for the first (high) and second (low)evaporation rates, respectively, with most deviations being

55(4): 1261-1274b)t = 0.043 d
t = 0.334 d
t = 1.674 d
t = 2.667 d
t = 3.507 d
t = 4.167 d
Measured-800

0Pressure Head [cm]log K [cm/d]

2.5Time [d]

-600
-500
-400
-300
-200
-100
0
-10

-8

-6

z [cm]-4-2

0

Figure 3. Measured and fitted tensiometer readings as a function of(a) time and (b) depth.

much lower. The analyses in this example were carried out
without accounting for the effects of hysteresis, which
could have slightly affected the results during redistribution.
In addition to the parameter estimation analysis briefly
described above, Šimůnek et al. (1998b) carried out detailed
analyses of the evaporation method in terms of the identifiability of various soil hydraulic parameters. A sensitivity
analysis and a response surface analysis using numericallygenerated data showed that the optimization method was mostsensitive to the soil hydraulic parameters n and θs, and least toθr. Pressure heads measured close to the soil surface were
found to be more valuable for the parameter estimation analysis than those measured at lower locations, mostly because of
the wider resolution of tensiometer readings near the surface.
One other important conclusion was that, for the invoked experimental setup (five tensiometers in a 20 cm long core), one
tensiometer reading (especially if obtained from the upper
part of the core) would have been sufficient to obtain accurate
estimates of the soil hydraulic functions within the range ofmeasurements. Refer to Šimůnek et al. (1998b) for additional
details about the experiments and parameter optimization
analysis, and especially about the sensitivity and response
surface analyses.

1269

FUTURE DEVELOPMENTS
As reflected by the many references listed in this article
and listed on the HYDRUS website, the HYDRUS software packages have been used over the years for a large
number of applications. Feedback from users has been extremely important in identifying particular strengths andweaknesses of the codes, and for defining additional processes or features that should be included in the models.
Feedback is also continuously obtained from several discussion forums at the HYDRUS website, where users can
submit questions or suggestions about the models. The
website also provides tutorials of the HYDRUS codes, including downloadable videos, in which the tutorials are
performed step by step, thus allowing software users toteach themselves interactively. All of this is generating valuable feedback that is continuously being used to improvethe codes. It is also motivating the recent and planned inclusion of additional features, such as virus and bacteriatransport, more elaborate biogeochemical processes, computationally more effective approaches for coupling of vadose zone processes with existing larger-scale groundwaterflow models, and overland flow. More sophisticated GUIs
are also continuously being implemented, especially for themulti-dimensional codes, in an attempt to make the codes
as attractive as possible for non-expert users. As such, we
believe that the codes and associated manuals and tutorials
are serving an important role in the transfer of vadose zone
R&D technologies to both the scientific community andpractitioners.
The above comments also pertain to the calibration andparameter estimation capabilities of the codes. For the majority of laboratory experiments involving variably saturated water flow and transport of various solutes or particles
in homogeneous soil columns, the minimization approach
implemented in the HYDRUS codes seems to be sufficientat present. For such conditions, the Marquardt-Levenberg
method has many significant advantages over global minimization techniques. Its main advantage is computational
efficiency because the Marquardt-Levenberg approach requires far fewer evaluations of the direct model compared
to global minimization techniques (tens, compared to tensof thousands), thus providing results relatively quickly (forone-dimensional problems generally within a few seconds).
The iterative inverse approach is also fully automated,while global searches still often require direct involvementof users. Additionally, the Marquardt-Levenberg method
provides confidence intervals for, and correlations between,
the optimized parameters, thus facilitating immediate insight into the optimization problem.
A major disadvantage of the Marquardt-Levenberg
method is that the method searches only for a local minimum of the objective function. This will be an issue forsome field-scale problems when, for example, multiple soil
horizons or uncertain initial and boundary conditions are
involved. The objective function may then be spatially very
complex and have many local minima. Gradient-based minimization approaches, such as the Marquardt-Levenberg
method, may well fail to locate the global minimum forsuch conditions, thus requiring the use of a global minimi-

1270

zation approach. For this reason, we are also planning to
implement in HYDRUS some of the global minimization
approaches, such as the MOSCEM-UA and/or AMALGAM
methods, as reported by Vrugt et al. (2008), while at thesame time attempting to reduce computational demands bytaking advantage of the multicore structure of recent desktop processors and parallelizing the minimization problem.

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