HYDRUS Lecture.pdf

Advanced Modeling of Water Flow and 

Solute Transport in the Vadose Zone 

Using HYDRUS models 

Agricultural 

Applications 

Jirka Simunek 

Department of Environmental Sciences 

University of California Riverside 

George E. Brown, Jr. Salinity Laboratory, USDA, ARS 

Riverside, CA 

University of California, Davis 

May 26, 2003 

Industrial and Environmental Applications 

HYDRUS models - Governing Equations 

Source Zone 

Observation wells 

Control Planes 

Variably-Saturated Water Flow (Richards Equation) 

∂θ 

∂ ∂h 

= [ Kh ( ) −Kh ( )] −S 

∂t ∂z ∂z 

Heat Movement 

∂Cp( θ ) T ∂ ∂T ∂qT 

= [ λθ ( ) ] −Cw 

−CwST 

∂t ∂z ∂z ∂z 

Solute Transport 

∂( ρs) ∂( θ c) 

∂ ∂c 

+ = ( θD 

−qc) 

−φ 

∂t ∂t ∂z ∂z 

The HYDRUS Software Packages 

Variably-Saturated Flow (Richards Eq.) 

Root Water Uptake (water, salinity stress) 

Multiple Solutes (decay chains, ADE) 

Nonlinear Sorption 

Two-Site Nonequilibrium Sorption 

Mobile-Immobile Water 

Heat Transport 

Pedotransfer Functions (hydraulic properties) 

Parameter Estimation 

Interactive Graphics-Based Interface 

HYDRUS - References 

Šimunek, J., M. Šejna, and M. Th. van Genuchten, The HYDRUS-1D 

software package for simulating one-dimensional movement of water, heat, 

and multiple solutes in variably saturated media. Version 2.0, IGWMC - TPS 

- 70, International Ground Water Modeling Center, Colorado School of 

Mines, Golden, Colorado, 202pp., 1998. 

Šimunek, J., M. Šejna, and M. Th. van Genuchten, The HYDRUS-2D 

software package for simulating two-dimensional movement of water, heat, 

and multiple solutes in variably saturated media. Version 2.0, IGWMC - TPS 

- 53, International Ground Water Modeling Center, Colorado School of 

Mines, Golden, Colorado, 251pp., 1999. 

E-mail: igwmc@mines.edu 

http://www.mines.edu/igwmc 

http://www.ussl.ars.usda.gov/models/hydrus2d.HTM 

http://www.pc-progress.cz

HYDRUS-2D - History of Development 

Israel: Neuman [1972] - UNSAT 

U. of Arizona: Davis and Neuman [1983] 

Agr. U. in Wageningen: 

Feddes et al. [1978] 

Vogel [1987] - SWMII 

USSL - SWMS-2D 

Šimunek et al. [1992] 

USSL - HYDRUS-2D (1.0) 


Princeton U.: 

van Genuchten [1978] 

MIT: 

Celia et al. [1990] 

USSL - CHAIN-2D 


USSL - HYDRUS-2D (2.0) 


HYDRUS-2D - History (References) 

Neuman, S. P., Finite element computer programs for flow in saturated-unsaturated 

porous media, Second Annual Report, Part 3, Project No. A10-SWC-77, 87 p. 

Hydraulic Engineering Lab., Technion, Haifa, Israel, 1972. 

Davis, L. A., and S. P. Neuman, Documentation and user's guide: UNSAT2 -Variably 

saturated flow model, Final Report, WWL/TM-1791-1, Water, Waste & Land, Inc., Ft. 

Collins, Colorado, 1983. 

van Genuchten, Mass transport in saturated-unsaturated media: One-dimensional 

solution, Research Rep. No. 78-WR-11, Water Resources Program, Princeton Univ., 

Princeton, NJ, 1978. 

Celia, M. A., and E. T. Bouloutas, R. L. Zarba, A general mass-conservative numerical 

solution for the unsaturated flow equation, Water Resour. Res., 26(7), 1483-1496, 1990. 

Vogel, T. SWMII - Numerical model of two-dimensional flow in a variably saturated 

porous medium, Research Report No. 87, Dept. of Hydraulics and Catchment 

Hydrology, Agricultural Univ., Wageningen, The Netherlands, 1987. 

Šimunek, J., T. Vogel, and M. Th. van Genuchten. The SWMS_2D code for simulating 

water flow and solute transport in two-dimensional variably saturated media, Version 

1.1. Research Report No. 126, 169 p., U.S. Salinity Laboratory, USDA, ARS, 

Riverside, California, 1992. 

HYDRUS –Modular Structure 

HYDRUS Graphical Interface 

Input, Output, Meshgen 

Water Flow 

Solute Transport 


HYDRUS – Main Module 

Soil Hydraulic Properties 

Pedotransfer Functions 

Root Uptake 

Equation Solvers 

Inverse Optimization 












Water Flow - Richards’ Equation 

The governing flow equation for two-dimensional isothermal 

Darcian flow in a variably saturated isotropic rigid porous medium: 

∂ θ ∂ 

= 

∂ t ∂ x 

i 

⎡ 

⎢ K ( K 

⎢⎣ 

A 

ij 

∂ h 

∂ x 

j 

+ K 

A 

iz 

θ - volumetric water content [L 3 L -3 ] 

h - pressure head [L] 

K - unsaturated hydraulic conductivity [LT -1 ] 

K ijA - components of a anisotropy tensor [-] 

x i - spatial coordinates [L] 

z - vertical coordinate positive upward [L] 

t - time [T] 

S - root water uptake [T -1 ] 

⎤ 

) ⎥ − S 

⎥⎦ 

Richards Equation - Assumptions 

Effect of air phase is neglected 

Darcy’s equation is valid at very low and very high 

velocities 

Osmotic gradients in the soil water potential are negligible 

Fluid density is independent of solute concentration 

Matrix and fluid compressibilities are relatively small


Retention Curve, θ(h) 

(Soil-water characteristic curve) 

- characterizes the energy status of the soil water 

|Pressure head| [cm] 

500 

400 

300 

200 

100 

0 

Loam 

Sand 

Clay 

0 0.1 0.2 0.3 0.4 0.5 

Water Content [-] 


Hydraulic Conductivity Function, K(h) 

- resistance of porous media to water flow 

log (Hydraulic Conductivity [cm/d]) 

4 

2 

0 

-2 

-4 

-6 

-8 

-10 

0 1 2 3 4 5 

log (|Pressure Head| [cm]) 

Loam 

Sand 

Clay 


Hydraulic Conductivity Function, K(θ) 

log (Hydraulic Conductivity [cm/d]) 

4 

2 

0 

-2 

-4 

-6 

Loam 

Sand 

-8 

Clay 

-10 

0 0.1 0.2 0.3 0.4 0.5 


Retention Curve 

Brooks and Corey [1964]: 

van Genuchten [1980]: 

Kosugi [1996]: 

θ s - saturated water content [-] 

θ r - residual water content [-] 

α, n, h 0 , σ - empirical parameters [L -1 ], [-], [L], [-] 

S e - effective water content [-] 

-n 

⎧ | α h| 

h < -1/ α 

S e = ⎨ 

⎩ 1 h ≥ -1/ α 

S 1 

= e 1 1/ 

(1 α h 

n − 

+ ) 

n 

( h h0 

) 

σ 

1 ⎧ln / ⎫ 

Se 

= erfc⎨ ⎬ 

2 ⎩ 2 ⎭ 

θ −θr 

Se 

= 

θ −θ 

s 

r 

Hydraulic Conductivity Function 

Brooks and Corey [1964]: 

Kh ( ) = K s S 

2/ n + l + 2 

e 

Soil Water Retention Curve, 2(h) 

van Genuchten [1980]: 

(Mualem [1976]) 

1/ m 

( ) 

l 

K( h) = K S ⎡ 1− 1− 

S 

⎢⎣ 

S e e 

m 

⎤ 

⎥⎦ 

2 

Kosugi [1996]: 

(Mualem [1976]) 

θ s - saturated water content [-] 

θ r - residual water content [-] 

α, n, h 0 , σ, l - empirical parameters [L -1 ], [-], [L], [-], [-] 

S e - effective water content [-] 

Κ S - saturated hydraulic conductivity [LT -1 ] 

( h h0 

) 

σ 

⎧ 

l ⎪1 

⎡ln / ⎤⎫⎪ 

Kh ( ) = KS 

s e⎨ 

erfc⎢ 

+ σ ⎥⎬ 

⎪⎩2 ⎣ 2 ⎦⎪⎭ 

θ −θr 

Se 

= 

θ −θ 

s 

r 

2

Hydraulic Conductivity Function, K(2) 

Richards Equation - Complications 

Hysteresis in the soil water retention function 

Extreme nonlinearity of the hydraulic functions 

Lack of accurate and cheap methods for measuring 

the hydraulic properties 

Extreme heterogeneity of the subsurface 

Inconsistencies between scale at which the 

hydraulic and solute transport parameters are 

measured, and the scale at which the models are 

being applied 

Soil Water Hysteresis 

Boundary Conditions: System-Independent 

Pressure head (Dirichlet type) boundary conditions: 

h( 

x, z, t) 

= ψ ( x, z, t) 

for ( x, z) 

ε 

Flux (Neumann type) boundary conditions: 

Γ D 

∂h 

- [ K ( K + K )] n = σ ( x, z, t) for ( x, z) 

ε Γ 

∂x 

j 

A 

A 

ij iz i 1 

N 

Gradient boundary conditions: 

A ∂h 

A 

( K ij + K iz ) n i = σ 2 ( x, 

z, 

t) 

for ( x, 

z) 

ε Γ 

∂ x j 

G 

Boundary Conditions: System-Dependent 

Atmospheric boundary condition: 

∂h 

| -K -K| ≤ E hA≤h 

≤hS 

∂x 

Ponding 

0 

-20 

-20 

-25 

-30 

-40 

-35 

Boundary Conditions: System-Dependent 

Seepage face (free draining lysimeter, dike) 

if(h q=0 

if(h=0) => q= 

Tile drains 

1D soil profile 

Soil surface 

Groundwater table 

-60 

-80 

-100 

0 0.025 0.05 0.075 0.1 

Time [days] 

-40 

-45 

-50 

-55 

0.00 0.05 0.10 0.15 0.20 

Time [days] 

Tile drain 

Impermeable layer












Root Water Uptake 

Bresler et al. [1982] 

S( z,t ) = - b ( z) K( )[ h -h( z,t) ] 

1 θ 

Feddes et al. [1978] 

S( z,t ) = - b 2( z) α 1( h( z,t )) T p 

r 

1 

van Genuchten (1987): 

Y / Ym 

1 ( c/ c50) 

= + p 












General structure of the system of solutes: 

Products 

Products 

µ g,1 µ w,1 µ s,1 µ g,2 µ w,2 µ s,2 

A µ w,1 B µ w,2 C ... 

g 1 c 1 s 1 µ s,1 g 2 c 2 s 2 µ s,2 

k g,1 k s,1 

µ g,1 

k g,2 k s,2 

µ g,2 

γ g,1 γ w,1 γ s,1 γ g,2 γ w,2 γ s,2 

Products 

Products 

Typical examples of sequential 

first-order chains: 

Radionuclides [van Genuchten, 1985] 

238 

Pu 

234 

U 

230 

Th 

226 

Ra 

c 1 s 1 c 2 s 2 c 3 s 3 c 4 s 4 

Nitrogen [Tillotson et al., 1980] 

g2 N 2 

(NH 2 ) 2 CO NH 

+ 4 NO 

- 2 NO 

- 3 

c 1 s 1 c 2 s 2 c 3 c 4 N 2 O

Typical examples of sequential firstorder 

chains: Pesticides [Wagenet and Hutson, 1987] 

Uninterrupted chain - one reaction path: 

Gas 

Parent Daughter Daughter 

pesticide product 1 product 2 Products 

c 1 s 1 c 2 s 2 c 3 s 3 

Product Product Product 

(aldicarb, oxime) (sulfone, sulfone oxime) (sulfoxide, sulfoxide oxime) 

Interrupted chain - two independent reaction paths: 

Gas 

Gas 

Typical examples of sequential firstorder 

chains: Organic Hydrocarbons 

Dechlorination of chlorinated ethenes 

[Schaerlaekens et al., 1999; Casey and Simunek, 2002] 

t-DCE 

PCE TCE c-DCE VC ethylene 

1,2-DCE 

Parent Daughter Parent 

pesticide 1 product 1 Products pesticide 2 Products 

c 1 s 1 c 2 s 2 c 3 s 3 c 4 s 4 

Product Product Product 

Perchloroethylene 

trichloroethylene dichloroethylene vinylchloride 

Typical Examples of Sequential Firstorder 

Chains: Pharmaceuticals and Explosives 

Pharmaceuticals, hormones (Estrogen, Testosterone): 

Estradiol Estrone Estriol 

Explosives (TNT, RDX HMX) 

4ADNT 

TNT 

2ADNT 

4ADNT 

4ADNT 

TAT 

2,4,6 trinitrotoluene 2-amino-4,6-dinitrotoluene 4-amino-2,6-dinitrotoluene 2,4,6-triminotoulene 

Governing Solute Transport Equations 

∂θc ∂ρs ∂ag ∂ ∂c 

∂ ∂g ∂qc 

+ + = ( θ ) + ( a )- - 

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 

-( µ + µ ) -( µ + µ ) -( µ + µ ) a + + 

−µ 

(2, ) 

k k k w k g k i k 

Dij,k 

Dij,k 

t t t xi x j xi x j xi 

' ' ' ' 

θ 

w,k w,k ck ρ 

s,k s,k sk g,k g,k 

g µ 

k w,k −1θck-1 

' 

' 

s,k −1 

ρ sk −1 -µ g,k −1 a g 

k−1+ γ 

w,kθ+ γ 

s,kρ+ γ 

g,ka−Scr , k 

kε 

ns 

w, s, g subscripts corresponding with the liquid, solid and gaseous phases, 

respectively 

c, s, g concentration in liquid, solid, and gaseous phase, respectively 

2,4DANT; 2,6DANT; 2,4DNT; 2,6DNT 

Governing Solute Transport Equations 

q i i-th component of the volumetric flux 

ρ soil bulk density 

a air content 

S sink term in the water flow equation 

c r concentration of the sink term 

D ijw ,D 

g 

ij dispersion coefficient tensor for the liquid and gaseous phase, respectively 

k subscript representing the kth chain number 

µ w , µ s , µ g first-order rate constants for solutes in the liquid, solid, and gaseous phases, 

respectively 

γ w , γ s , γ g zero-order rate constants for the liquid, solid, and gaseous phases, 

respectively 

µ w ', µ s ', µ g ' first-order rate constants for solutes in the liquid, solid and gaseous phases, 

respectively; these rate constants provide connections between the individual 

chain species. 

number of solutes involved in the chain reaction 

n s 











Interactive Graphics-Based Interface

3 

3 

Interactions Among Phases 

Linear Adsorption 

s = 

∂ Rθ 

c 

∂ t 

Steady-State 

∂ 

R 

c ∂ t 

kc 

∂ 

⎛ 

∂ 

⎞ 

⎜ c 

= θ 

⎟ 

+ φ 

∂ ⎜ 

D − q c 

x 

ij 

∂ x 

i ⎟ 

i ⎝ 

j ⎠ 

2 

∂ c ∂ c 

= D − v 

2 

∂ z ∂ z 

Nonlinear Equilibrium Adsorption 

+ k c 

Equation Model Reference 

s=kc+k 1 

Linear Lapidus and Amundson [1952] 

2 

Lindstrom et al. [1967] 

s=k 1 c k 2 Freundlich Freundlich [1909] 

k c 

Langmuir Langmuir [1918] 

1 

s = 

1 + k c 2 

k 

k c 

Freundlich-Langmuir Sips [1950] 

1 

s = 

1 

k + k 2 c 

k 1 c 

1 1 + k 4 c 

k 3 c 

Double Langmuir Shapiro and Fried [1959] 

s= 

2 

c 

s=k c k 

1 

2 / k 3 Extended Freundlich Sibbesen [1981] 

k c 

Gunary Gunary [191970] 

1 

s = 

1 + k 2 c + k 3 c 

k 2 

s=kc 1 - k 

Fitter-Sutton Fitter and Sutton [1975] 

3 

s=k - + k c k 3 

1 1 k 4 

{ [ ] } Barry Barry [1992] 

1 2 

2 

RT ln( ) Temkin Bache and Williams [1971] 

s = 

k c 

k 1 

s=k 1cexp( -2 

k 2s) Lindstrom et al. [1971] 

van Genuchten et al. [1974] 

s 

= 

c 

modified Kielland Lai and Jurinak [1971] 

sT [ c+ k1( cT −c)exp{ k2( cT−2c)}] 

Interactions Among Phases 

Equilibrium interactions between the solution (c) 

and gaseous (g) concentrations (Henry’s law) 

Nonequilibrium interactions between the solution 

(c) and adsorbed (s) concentrations 

A generalized nonlinear empirical equation 

β 

kc 

d 

s= 

1 + c 

β 

η 

k d , η , β 

empirical constants 












Non-Equilibrium Adsorption Equations 

Nonequilibrium two-site adsorption model 

[1952] 

Equation Model Reference 

∂s 

∂t Linear Lapidus and Amundson 

1 2 

Oddson et al. [1970] 

∂ s 

k 2 

= α ( kc -s) 

Freundlich Hornsby and Davidson [1973] 

1 

∂ t van Genuchten et al. [1974] 

∂s 

⎛ ⎞ 

⎜ 

k1c 

= α - s⎟ 

∂t 

⎝1+k 

2c 

⎠ 

Langmuir Hendricks [1972] 

∂s 

⎛ k3 

⎞ 

⎜ 

k1c 

= α - s 

⎟ 

Freundlich- Šimunek and van Genuchten [1994] 

∂t 

k3 

⎝1+k 

2c 

⎠ 

Langmuir 

∂s 

⎛ s - s ⎞ 

⎜ T ⎟ 

Fava and Eyring [42] 

= α ( s -s)sinh 

T ∂ 

⎜ 

k1 

t 

s - ⎟ 

⎝ T si 

⎠ 

∂s 

∂t α exp( k s ){ k c exp( -2 k s ) -s } 

Lindstrom et al. [1971] 

∂s 

∂t = c k 

s k 

α Leenheer and Ahlrichs [1971] 

1 2 

Enfield et al. [1976] 

s 

e k 

s 

k k 

f 

s k= s e k+ sk k 

Type - 1 sites with instantaneous sorption 

Type - 2 sites with kinetic sorption 

e 

∂ sk 

∂ sk 

= f 

∂t 

∂t 

k 

β 

∂s 

⎡ 

k 

k 

ks,k 

c ⎤ 

k k k 

= αk 

⎢( 1- 

f 

k) 

- µ γ 

β 

sk⎥ - 

s,k sk 

+ (1 - f ) 

∂t 

1+ 

η 

k 

⎣ 

kck 

⎦ 

fraction of exchange sites assumed to be at equilibrium 

s,k












Two-Region Physical Nonequilibrium Transport 

∂ 

∂ ∂ 

+f k c = D c m 

( θ m ρ D) m ( θ m m -qc )-α( c -c )-( θµ + fρ k µ ) c 

∂t 

∂ z ∂z m m im l,m D s,m m 

∂cim 

[ θ im + ( 1- f) ρkD] = α( c - c )-[ θ µ + ( 1- f) ρ k µ ] c 

∂t 

m im im l,im D s,im im 

Interaction among phases 

Liquid - Gas: a linear relation 

g = k gc 

k g,k empirical constant equal to (K H RT A ) -1 

K H 

R 

T A 

Henry's Law constant 

universal gas constant 

absolute temperature 

Volatilization 

∂( ρs) ∂( θc) ∂( ) ∂ ∂ ∂ 

+ + ag = ( θD c ) φ 

∂t 

∂t 

∂t 

∂ x ∂x +aD g w 

a w a 

ij 

ij 

-qi 

c-qi 

g + 

∂x Steady-State: 

i 

g = kH 

c 

j 

i 

j 

j 

w 

i 

⎛ ρ k ak ⎞ 

2 q q 

D H c w a a c + 

⎜ 

∂ ⎛ 

⎞ ∂ 

1 + + ⎟ = ⎜D 

+ D k ⎟ - 

θ θ 

∂t 

ij ij H 

⎝ 

⎠ ⎝ θ ⎠ ∂x 

∂x 

θ 

a 

k 

i H 

∂c 

∂x 

i 

Temperature Dependence of Transport 

and Reaction Coefficients 

Most of the diffusion (D w , D g ), distribution (k s , k g ), and reaction 

rate (γ w , γ s , γ g , µ w ', µ s ', µ g ', µ w , µ s , and µ g ) coefficients are 

strongly temperature dependent. HYDRUS_2D assumes that 

this dependency can be expressed by an Arrhenius equation 

[Stumm and Morgan, 1981]. 

A A 

⎡ E( 

T - T r ) ⎤ 

aT 

= ar 

exp⎢ 

A A ⎥ 

⎣ RT T r ⎦ 

a r ,a T coefficient values at a reference absolute temperature, 

T rA , and absolute temperature, T A , respectively 

E activation energy of the reaction or process 

Solute Transport - Dispersion Coefficient 

Bear [1972]: 

q 

j 

q 

θ D ij = D 

T 

| q | δ ij + ( D 

L 

- D 

T 

) 

| q | 

i 

+ θ D 

D d - ionic or molecular diffusion coefficient in free water [L 2 T -1 ] 

τ - tortuosity factor [-] 

δ ij - Kronecker delta function (δ ij =1 if i=j, and δ ij =0 otherwise) 

D L , D T - longitudinal and transverse dispersivities [L] 

2 

2 

q 

x 

q 

z 

θ D xx = D L + DT 

+ θ Dd 

τ 

| q | | q | 

2 

2 

q 

z 

q 

x 

θ D zz = D L + DT 

+ θ Dd 

τ 

| q | | q | 

q 

x 

q 

z 

θ D xz = ( D L - DT) 

| q | 

d 

τ δ 

ij

Solute Transport - Boundary Conditions 

First-type (or Dirichlet type) boundary conditions 

c( x, z, t ) = c 0( x, z, t ) for( x, z ) ε Γ D 

Third-type (Cauchy type) boundary conditions 

∂ c 

- θ Dij 

ni + qinic = qi0 nic0 

for ( x, z ) ε Γ 

∂ x j 

Second-type (Neumann type) boundary conditions 

∂ c 

θ Dij 

ni 

= 0 for ( x, z ) ε Γ 

∂ x j 

C 

N 

Dispersivity as a function of scale 












Governing Heat Transport Equation 

Sophocleous [1979] 

∂T 

C 

t = ∂ ∂T 

x x - ∂ 

( θ) [ λ C q T ij( θ) ] w i 

∂ ∂ i ∂ j ∂ xi 

λ ij (θ) apparent thermal conductivity of the soil 

C(θ), C w volumetric heat capacities of the porous medium and the liquid phase, 

respectively 

de Vries [1963] 

C ( θ ) = C θ + C θ + C θ + C a 

n n o o w g 

θ volumetric fraction 

n, o, g, w subscripts representing solid phase, organic matter, gaseous phase, and 

liquid phase, respectively. 

Thermal Conductivity 

j i 

λ ij ( θ ) = λT C w |q| δ ij + ( λ L -λT ) C q q 

w + λ0( θ ) δ ij 

|q| 

λ 0 (θ) thermal conductivity of the porous medium (solid plus water) 

in the absence of flow 

λ L , λ T longitudinal and transverse thermal dispersivities, 

respectively 

Chung and Horton [1987] 

b 1 , b 2 , b 3 

( ) = b + b + b 

05 . 

λ 0 θ 1 2θ w 3θ 

w 

empirical parameters. 












Pedotransfer Functions: Rosetta 

Schaap et al. (2001) 

Model 

Input Data 

TXT 

Textural Class 

SSC Sand, Silt, Clay % 

SSCBD 

Same + Bulk Density 

SSCBD + 2 33 

SSCBD + 2 at 33 kPa 

SSCBD + 2 33 + 2 1500 Same + 2 at 1500 kPa 

Estimation of Soil Hydraulic Properties With 

Artificial Neural Networks 

Log Suction [cm] 

4 

3 

2 

1 

0 

Sand 

Retention 

Clay 

0.2 0.4 0.6 

Water Content [cm 3 /cm 3 ] 

Sand Clay 

Sand % 79 28 

Silt % 13 29 

Clay % 8 43 

Bulk d. 1.45 1.44 

Probability 

Saturated Conductivity 

0.25 

0.2 

0.15 

Sand 

0.1 Clay 

0.05 

Log K(h) [cm/day] 

0 

Log(K s ) [cm/day ] 

Unsaturated Conductivity 

2 

0 

-2 

-4 

-6 

Clay 

1 2 3 4 

Sand 

1 2 3 4 

Log Suction [cm] 












Parameter Estimation in HYDRUS 

Parameter Estimation: 

- Soil hydraulic parameters 

- Solute transport and reaction parameters 

- Heat transport parameters 

Sequence: 

- Independently 

- Simultaneously 

- Sequentially 

Method: 

- Marquardt-Levenberg optimization 

Objective Function 

Φ ( b , q , p ) = 

m q n q j 

* 

∑ v j ∑ w i, j [ q 

j 

j = 1 i = 1 

m p n p j 

* 

∑ v j ∑ w i, j 

[ p 

j 

j = 1 i = 1 

n b 

* 2 

∑ vˆ 

j [ b j - b j ] 

j = 1 

( x, t i) 

- q 

( θ ) - 

i 

( x, t i , b ) ] + 

p ( θ , b ) ] + 

1st term: deviations between measured and calculated space-time 

variables 

2nd term: differences between independently measured, p j* , and 

predicted , p j , soil hydraulic properties 

3rd term: penalty function for deviations between prior knowledge 

of the soil hydraulic parameters, b j* , and their final 

estimates, b j . 

j 

j 

i 

2 

2 












Furrow Irrigation – Pressure Heads 

Dike – Pressure Heads 

Dike - Velocity Vectors 

Finite Element Mesh – Cut-off Wall 

Solute Plume Under Dam 

Capillary Barrier - Material Distributions

Capillary Barrier - Velocity Vectors 

Mesh Generator 

Tunnel - Spectral Color Maps 

Tunnel - Velocity Vectors 

Plume Movement in a Transect with Stream 

HYDRUS - Existing Applications 

Agricultural: 

Irrigation management (FREP, LINK , Bristow et al., 2002) 

Drip irrigation design (FREP, LINK, Bristow et al., 2002) 

Sprinkler irrigation design (FREP, LINK) 

Tile drainage design and performance (Mohanty et al., 1998, do Vos et al., 2000) 

Studies of root and crop growth (Vrugt et al., 2001, 2002) 

Salinization and reclamation processes (Šimůnek and Suarez, 1998) 

Nitrogen dynamics and leaching (Ventrella et al., 2001; Jacques et al., 2002) 

Transport of pesticides and degradation products (Wang et al., 1998) 

Non-point source pollution 

Seasonal simulation of water flow and plant response 

. . .


Non-Agricultural: 

Leaching from radioactive waste sites at the Nevada test Site (DRI, DOE) 

Flow around nuclear subsidence craters at the Nevada test site (Pohll et al., 1996; 

Wilson et al., 2000) 

Capillary barrier at Texas low-level radioactive waste disposal site (Scanlon, 1998) 

Evaluation of approximate analytical analysis of capillary barriers (Morris and 

Stormont, 1997; Kampf and Montenegro, 1997; Heiberger, 1998) 

Landfill covers with and without vegetation (Abbaspour et al, 1997; Albright, 1997; Gee et al., 

1999, Scanlon et al., 2002) 

Risk analysis of contaminant plumes from landfills 

Seepage of wastewater from land treatment systems 

Tunnel design - flow around buried objects (Knight, 1999) 

Highway design - road construction - seepage (de Haan, 2002) 

Stochastic analyses of solute transport in heterogeneous media (Tseng and Jury, 1993; 

Roth, 1995; Roth and Hammel, 1996; Kasteel et al. 1999; Hammel et al., 1999; Roth et al., 1999; Vanderborght et 

al., 1998, 1999) 

Lake basin recharge analysis (Lee, 2000) 


Non-Agricultural: 

Stream-aquifer interactions 

Environmental impact of the drawdown of shallow water tables 

Analysis of cone permeameter and tension infiltrometer experiments (Gribb et al., 

1996; Kodesova et al., 1998, 1999; Šimůnek et al., 1997, 1998, 1999) 

Virus and bacteria transport (Shijven and Šimůnek, 2002, Bradford et al., 2002a,b, Yates et al., 2000) 

Hill-slope analyses 

Transport of TCE and its degradation products (Scharlaekens et al., 2000; Casey and 

Simunek, 2002) 

Multicomponent geochemical transport (Jacques and Šimůnek, 2002) 

Analyses of riparian systems (Whitaker, 2000) 

Fluid flow and chemical migration within the capillary fringe (Silliman et al., 2002) 

Flow in historical monuments (Ishizaki et al., 2001) 

Flow and transport around land mines (Das et al., 2001; Šimůnek et al., 2001) 

Analyses of Chloride profiles in deep vadose zones to evaluate historical fluxes 

(Scanlon et al., 2003) 

Current and Future Development 

Coupled movement of water and energy, 

including vapor transport 

Modified Richards Equation: 

∂θ 

∂ ⎡ ∂ 

Lh 

( ) ∂ 

K h K 

Lh 

( h) K 

LT 

( h) 

T ∂ 

K h ∂ T ⎤ 

= + + + 

vh 

+ KvT 

− S 

∂t ∂z ⎢ 

⎣ ∂z ∂z ∂z ∂z 

⎥ 

⎦ 

K Lh - hydraulic conductivity for liquid phase fluxes due to gradient in h 

K LT - hydraulic conductivity for liquid phase fluxes due to gradient in T 

K vh - isothermal vapor hydraulic conductivity 

K vT - thermal vapor hydraulic conductivity 

Energy Transport: 

∂ C T p ∂ 

v 

T 

l v v 

L ∂ ⎡ ∂ ⎤ ∂ 

0 

λ( θ ) C qT ∂ q ∂ 

+ = − qT 

w 

−CwST − L0 

−Cv 

∂t ∂t ∂z ⎣ 

⎢ ∂z ⎦ 

⎥ ∂z ∂z ∂z 

(1) (2) (3) (4) (5) 

(1) Soil heat flow by conduction 

(2) Convection of sensible heat by water flow 

(3) Heat removed by root water uptake 

(4) Transfer of latent heat by diffusion of water vapor 

(5) Transfer of sensible heat by diffusion of water vapor 

Coupled Movement of Water and Energy 

Coupled movement of water and energy 

-1000 -500 0 

0 1000 2000 3000 4000 5000 

0 

5 

0 yrs 

Central 

10 1 kyr 

5 kyr High Plains 

15 9 kyr 

Field 

20 

Amargosa Desert 

0 

10 

20 

0 yrs 10 kyr 

30 

5 kyr 16 kyr 

40 Field Lab 

50 

1 kyr 

5 kyr 

9 kyr 

Measured 

0 2000 4000 6000 8000 10000 

5 kyr 

10 kyr 

16 kyr 

Measured 

Depth [cm] 

10 

8 

6 

4 

2 

T=0 

t=0.25 d 

t=1 

t=5 

t=25 

10 

8 

6 

4 

2 

10 

T=0 

t=0.25 d 

t=1 8 

t=5 

t=25 

6 

4 

2 

T=0 

t=0.25 d 

t=1 

t=5 

t=25 

10 

8 

6 

4 

2 

T=0 

t=0.25 d 

t=1 

t=5 

t=25 

0 

5 

10 

15 

20 

Eagle Flat 

0 yrs 

5 kyr 

12 kyr 

Field 

Field - OP 

0 2000 4000 6000 8000 10000 

5 kyr 

12 kyr 

Measured 

0 

0 

0 

0 

0 0.05 0.1 0.15 0.2 0 0.02 0.04 0.06 0 10 20 30 0 2 4 6 8 10 


Total Flux [cm/d] 

Temperature [C] 

Concentration [-] 

Hueco Bolson 

0 

5 

10 0 yrs 10 kyr 

15 

5 kyr 13 kyr 

Field 

20 

0 5000 10000 15000 20000 

5 kyr 

10 kyr 

13 kyr 

Measured 

Simulated matric potentials and chloride 

concentrations from wet initial conditions 

(pluvial period) to different times of 

upward flow. (Scanlon et al. 2003) 

Total flux=water flux+vapor flux

Coupled movement of water and energy, 

freezing/thawing cycle 

Modified Richards Equation: 

∂θ 

ρi 

∂θi 

∂ ⎡ ∂h ∂T ∂h ∂T 

⎤ 

+ = K 

Lh 

( h) + K 

Lh 

( h) + K 

LT 

( h) 

+ Kvh + KvT 

− S 

∂t ρ 

w 

∂t ∂z ⎢ 

⎣ ∂z ∂z ∂z ∂z 

⎥ 

⎦ 

K Lh - hydraulic conductivity for liquid phase fluxes due to gradient in h 

K LT - hydraulic conductivity for liquid phase fluxes due to gradient in T 

K vh - isothermal vapor hydraulic conductivity 

- thermal vapor hydraulic conductivity 

K vT 

Energy Transport: 

∂C T p v i 

T 

l v v 

L θ 

0 

L 

f 

ρ 

∂ ⎡ ∂ ⎤ 

+ − 

i 

= λ( θ ) −C ∂qT q qT 

w 

−CwST − L 0 

−C 

∂ 

v 

∂t ∂t ∂t ∂z ⎢ 

⎣ ∂z ⎥ 

⎦ ∂z ∂z ∂z 

(6) (1) (2) (3) (4) (5) 

(1) Soil heat flow by conduction 

(2) Convection of sensible heat by water flow 

(3) Heat removed by root water uptake 

(4) Transfer of latent heat by diffusion of water vapor 

(5) Transfer of sensible heat by diffusion of water vapor 

(6) Freezing/thawing term 

Coupled Movement of Water and Energy, 

Freezing/thawing Cycle 

Apparent Capacity [Jm -3 K -1 ] 

Apparent Capacity [Jm -3 K -1 ] 

1.00E+12 

1.00E+11 

1.00E+10 

1.00E+09 

1.00E+08 

1.00E+07 

1.00E+06 

0.02 

1.00E+12 

1.00E+11 

1.00E+10 

1.00E+09 

1.00E+08 

1.00E+07 

1.00E+06 

0.50 

0.00 

0.00 

-0.02 -0.04 -0.06 

Temperature [ o C] 

-0.50 -1.00 

Temperature [ o C] 

-0.08 

Silty clay 

Loam 

Sand 

-1.50 

Silty clay 

Loam 

Sand 

-0.10 

-2.00 

Apparent heat capacity 

for different textures 

Coupled movement of water and energy 

Freezing/thawing cycle 

Silty Clay 

Depths (cm): 

0, 0.5. 1, 2, 3.5, 5, 10 

4 

2 

0 

-2 

0 

-20000 

-40000 

Heterogeneity, 

Layering 

-4 

-60000 

-6 

-0.5 0.0 0.5 1.0 1.5 2.0 

Time [days] 

-80000 

-0.5 0.0 0.5 1.0 1.5 2.0 

Time [d ays] 

0.36 

0.34 

0.32 

0.30 

0.28 

0.26 

0.24 

-0.5 0.0 0.5 1.0 1.5 2.0 

Time [days] 

1.5 

1.4 

1.3 

1.2 

1.1 

1.0 

0.9 

0.8 

0.7 

-0.5 0.0 0.5 1.0 1.5 2.0 

Time [d ays] 

Nonequilibrium and Preferential Flow and Transport 

Dual-porosity approach (Richards eq. for water, mobileimmobile 

concept for solute) 

Dual-porosity approach (mobile-immobile concept for both 

water and solute) 

Dual-permeability approach (two overlapping porous media, one 

for matrix flow, one for preferential flow) [Gerke and van 

Genuchten, 1993] 

Kinematic wave approach for flow in macropores [Jarvis, 1991] 

Simplified first-order approach [Ross and Smettem, 2000] 

∂ θ 

θe 

−θ 

= f( θθ , 

e) = 

∂t 

τ 

Dual-porosity hydraulic property models [Durner, 1994] 

Design experiments that would provide parameters for above 

models

Dual-permeability Approach 

Gerke and van Genuchten [1993]: (two overlapping porous 

media, one for matrix flow, one for preferential flow) 

Flux [cm/d] 

60 

50 

40 

30 

20 

10 

0 

Fracture Flux 

Matrix Flux 

Mass Transfer 

Mass Transfer (-25%) 

Mass Transfer (+25%) 

0 0.02 0.04 0.06 0.08 0.1 

Time [d] 

Depth [cm] 

0 

5 

10 

15 

20 

t = 0.01 d 

25 

t = 0.04 d 

t = 0.08 d 

30 

t = 0.08 d (-25%) 

35 

t = 0.08 d (+25%) 

40 

0.25 0.3 0.35 0.4 0.45 0.5 


Depth [cm] 

0 

5 

t = 0.01 d 

10 

t = 0.04 d 

t = 0.08 d 

15 

t = 0.08 d (-25%) 

t = 0.08 d (+25%) 

20 

25 

30 

35 

40 

0 0.005 0.01 0.015 0.02 0.025 


Infiltration and mass exchange fluxes (a), water contents in the matrix 

(b) and fracture (c) domains 

Multicomponent solute transport: 

Coupling HYDRUS and PHREEQC [Parkhurst and 

Appelo, 1999]) 

Available chemical reactions: 

Aqueous complexation 

Redox reactions 

Ion exchange (Gains-Thomas) 

Surface complexation – diffuse double-layer model and nonelectrostatic 

surface complexation model 

Precipitation/dissolution 

Chemical kinetics 

Biological reactions 

Verification of HYDRUS-PHREEQC 

Transport and Cation Exchange (major ions and heavy metals): 

(cations - Ca, Mg, Na, K, Cd, Pb, Zn; anions – Cl, Br, Al) 

a) Initially the 8-cm column contains a solution (with heavy metals) in equilibrium with the 

cation exchanger. 

b) The column is then flushed with three pore volumes of solution without heavy metals. 

Parameters: 

q=2 cm/d, λ=0.2 cm, CEC=11 mmol/cell. 

Initial concentrations: Al=0.5, Br=11.9, K=2, Na=6, Mg=0.75, Cd=0.09, Pb=0.1, Zn=0.25 

mmol/L. 

Boundary concentration: Al= 0.1, Br=3.7, Cl=10, Ca=5, Mg=1 mmol/L. 

Species and Complexes: Al 3+ , Al(OH) 2+ , Al(OH) 2+ , Al(OH) 3 , Al(OH) 4- , Br - ,Cl - , Ca 2+ , 

Ca(OH) + , Cd 2+ , Cd(OH) + , Cd(OH) 2 , Cd(OH) 3- , Cd(OH) 2- 

4 , CdCl + , 

CdCl 2 , CdCl 3- , K + , KOH, Na + ,NaOH, Mg 2+ , Mg(OH) + , Pb 2+ , 

Pb(OH) + , Pb(OH) 2 , Pb(OH) 3- , Pb(OH) 2- 

4 ,PbCl + , PbCl 2 , PbCl 3- , 

PbCl 2- 

4 , Zn 2+ , Zn(OH) + , Zn(OH) 2 , Zn(OH) 3- , Zn(OH) 2- 

4 , ZnCl + , 

ZnCl 2 , ZnCl 3- , ZnCl 2 

4 

Exchange Species: AlX 3 , AlOHX 2 , CaX 2 , CdX 2 , KX, NaX, MgX 2 , PbX 2 , ZnX 2 


Transport and cation 

exchange of major cations 

and heavy metals 

Concentration (mol/l) 


0.01 

0.008 

0.006 

0.004 

0.002 

0 

0.005 

0.004 

0.003 

0.002 

0.001 

0 

Na 

Al 

Cl 

Ca 

0 3 6 9 12 15 

Time (days) 

ZnX 2 

CdX 2 

CaX 2 

0 3 6 9 12 15 

Time (days) 


8E-004 

6E-004 

4E-004 

2E-004 

0E+000 

Relative mass balance errors (%) 

1.5 

1.2 

0.9 

0.6 

0.3 

0 

Pb 

Cd 

Zn 

0 3 6 9 12 15 

Time (days) 

Cl 

Ca 

Al 

Cd 

Zn 

0 3 6 9 12 15 

Time (days) 


Kinetic biodegradation of NTA (nitrylotriacetate), cell growth, 

complexation with Co, and kinetic sorption 

Processes: 

Bacterially mediated degradation of an organic substrate 

Bacterial cell growth and death 

Aqueous speciation including metal-ligand complexation 

Kinetic sorption of Co and CoNTA 

Convective dispersive transport 

Parameters: L=10m, θ s =0.4 m, ρ=1.5 kg/m 3 , v=1 m/h, , λ=0.05 m. 

Iinitial Cond.: O 2 =3.125e-5, Na=0.001, Cl=0.001 mol/L, Biomass=0.000136g/L 

Boundary Cond.: O 2 =3.125e-5, Co=5.23e-6, NTA=5.23e-6, Na=0.001, Cl=0.001 

mol/L 


Kinetic biodegradation of NTA, cell growth, complexation with Co, and kinetic sorption 

Rate equations: 

NTA degradation: 

RHNTA = −qm 

X 

2− 

[ HNTA ] [ O2 

] 

2− 

K 

s 

+ [ HNTA ] Ka 

+ [ O2 

] 

m 

2− 

Biomass production: 

R 

cell 

= −YR 

2 − 

HNTA 

Kinetic sorption (Co 2+ ,CoNTA - ): 

⎛ Z 

[ ] ⎟ ⎞ 

R 

⎜ 

Z 

= −k 

m 

Z − 

⎝ K 

d ⎠ 

X m –biomass 

K s , K a – half-saturation constants 

Z – species concentration 

− bX 

m 

Concentrations [mol/g] 

1.6E-09 

1.2E-09 

8.0E-10 

4.0E-10 

0.0E+00 

Sorbed Co - PHREEQC 

Sorbed CoNta 

Sorbed Co - HYDRUS 

Sorbed CoNta 

Biomass 

Biomass 

0 10 20 30 40 50 60 70 80 

Time [hours] 

4.0E-04 

3.0E-04 

2.0E-04 

1.0E-04 

0.0E+00 

Biomass [g/L]


Kinetic biodegradation of NTA, cell growth, complexation with Co, and kinetic sorption 

Concentration [mol/kg] 

0.000004 

0.0000035 

0.000003 

0.0000025 

0.000002 

0.0000015 

0.000001 

0.0000005 

0 

CoNTA 

HNTA 

Co 

CoNTA 

HNTA 

Co 

pH 

pH 

0 10 20 30 40 50 60 70 80 

Time [hours] 

8 

7 

6 

5 

4 

3 

2 

1 

0 

Ph 

Constructed Wetlands 

12 Components: 

Dissolved oxygen O 2 

Organic matter: readily biodegradable, slowly biodegradable, inert 

Nitrogen: NH 4+ , NO 2- , NO 3- , N 2 

Inorganic phosphorus 

Heterotrophic micro-organisms 

Autotropic micro-organisms: Nitrosomonas & Nitrobacter 

9 Processes: 

Heterotrophic Organisms 

Hydrolysis 

Aerobic growth of heterotrophs on readily biodegradable OM 

NO 3 -growth of heterotrophs on readily biodegradable OM 

NO 2 -growth of heterotrophs on readily biodegradable OM 

Lysis 

Nitrosomonas 

Aerobic growth of N.somonas on NH 4 

Lysis of n-somonas 

Aerobic growth of N.bacter on NO 2 

Lysis of N.bacter 

Nitrobacter 

Overland Flow 

Kinematic wave equation: 

∂h ∂Q + = qxt ( , ) Q= αh 

∂t 

∂x 

h - unit storage of water (or mean depth), 

Q -discharge per unit width, 

q(x,t) - rate of local input, or lateral inflows (precipitation - infiltration) 

Manning hydraulic resistance law: 

1/2 

S 

α = 1.49 and m = 5/ 3 

n 

n - Manning’s roughness coefficient for overland flow 

S -slope 

m 

100*0.5 m 

HYDRUS-3D - Preview 

HYDRUS-3D - Preview 

HYDRUS Web Site – FAQ

HYDRUS Web Site – Tutorials 

HYDRUS Web Site – Discussion Forum 

HYDRUS-2D – User Manual 

David Rassam, Jirka Šimůnek and 

Rien Van Genuchten 

Introductory examples 

1. A Journey Through HYDRUS Windows 

1.1. Pre-Processing 

1.2. Post-Processing 

2. HYDRUS Output Files 

3. Root Water Uptake 

4. Example Application 

5. Inverse Solution 

6. Trouble Shooting 

Appendices: 

I. Soil Hydraulic Properties 

II. Concept Related to Modelling Evaporation 

III. Root Water Uptake 

IV. Scaling Factors 

V. Inverse Solution 

VI. Alphabetical Index for HYDRUS Windows

HYDRUS Lecture.pdf

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