Solute Transport

Transcription

Solute Transport

Table of Contents
Lecture 1: Vadose zone flow and transport modeling: An overview.
3
Lecture 2: The HYDRUS-1D software for simulating one-dimensional variablysaturated water flow and solute transport.
33
Computer Session 1: HYDRUS-1D: Infiltration of water into a one-dimensional soil
profile.
37
Lecture 3: On the characterization and measurement of the hydraulic properties of
unsaturated porous media.
43
Lecture 4: Application of the finite element method to variably-saturated water flow and
solute transport.
59
Computer Session 2: HYDRUS-1D: Water flow and solute transport in a layered soil
profile.
65
Lecture 5: Inverse modeling.
77
Computer Session 3: HYDRUS-1D: One- or multi-step outflow experiment.
89
Lecture 6a: Application of the finite element method to 2D variably-saturated water flow
and solute transport.
95
Lecture 6b: HYDRUS (2D/3D) software for simulating two- and three-dimensional
variably-saturated water flow and solute transport.
99
Computer Session 4: HYDRUS (2D/3D): Subsurface line source.
109
Computer Session 5: HYDRUS (2D/3D): Furrow infiltration with a solute pulse.
119
Computer Session 6: HYDRUS (2D/3D): Flow and transport in a transect to a stream.125
Computer Session 7: HYDRUS (2D/3D): Three-Dimensional Water Flow and Solute
Transport.
135
Lecture 7: Preferential and Nonequilibrium Flow and Transport.
143
Computer Session 8: HYDRUS-1D: Nonequilibrium Flow and Transport.
155
Lecture 8: Coupled movement of water, vapor, and energy.
161
Computer Session 9: HYDRUS-1D: Coupled movement of water, vapor, and energy.167
Lecture 9: Multicomponent biogeochemical transport modeling using the HYDRUS
computer software packages; Introduction to the HP1 code, which was obtained
by coupling HYDRUS-1D with the PHREEQC biogechemical code.
173
Computer Session 10: Application of HP1 to a simple solute transport problem
involving cation exchange.
187
Lecture 10: Other applications and future plans in HYDRUS development.
195
References
207
1
2
3
Research on variably-saturated water flow and contaminant transport,
analytical and numerical modeling, nonequilibrium transport, preferential
flow, characterization and measurement of the unsaturated soil hydraulic
functions, salinity management, and root-water uptake. Most often
referenced researcher in the field of Soil Physics. Dr. van Genuchten is
probably best known for the theoretical equations he developed for the
nonlinear constitutive relationships between capillary pressure, water
content and the hydraulic conductivity of unsaturated media.
Dr. van Genuchten is a recipient of the Soil Science Society of America’s Don
and Betty Kirkham Soil Physics Award, and fellow of the Soil Science Society
of America, American Society of Agronomy, American Geophysical Union
and American Association for the Advancement of Sciences. Founding Editor
of the Vadose Zone Journal. Currently with the University of Rio de Janeiro,
Brazil.
A soil physicist originally with the George E. Brown, Jr.
Salinity Laboratory, USDA, ARS, Riverside, CA. Received a
B.S. and M.S. in irrigation and drainage from Wageningen
University in The Netherlands, and a Ph.D. in soil physics
from New Mexico State University.
Course Developers – Rien van Genuchten
of Mechanical Engineering
Federal University of Rio de Janeiro, Brazil
2Department
of Environmental Sciences
University of California, Riverside, CA
1Department
Jirka Šimůnek1 and Rien van Genuchten2
Modeling Water Flow and
Contaminant Transport in
Soils and Groundwater
Using the HYDRUS
Computer Software Packages
homogeneous and heterogeneous Condensation.
X COCHEM Flow - Software package simulating 2D water steam flow with
Finite-Element and Finite-Volume applications.
X MESHGEN Plus - FE-mesh generator and open modeling environment for
transport in variably saturated porous media.
X HYDRUS 2D/3D - Software package for simulating water, heat, and solute
Ing.-Software Dlubal, GmbH, Germany
X RFEM, RSTAB - Structural Engineering Software packages, 1995-2009,
Selected Software Projects:
Recently specializes in the development of GUI (Graphical User Interfaces) for
FEM/CFD software packages for Windows. He has more than twenty years of
experience in developing programs for numerical modeling in Fluid Mechanics and
Structural Engineering. His software helps thousands of scientists and engineers from
around the world.
Expertise in numerical modeling of Transonic Flow with homogeneous and
heterogeneous condensation and chemicals in steam through Turbine Cascade (Euler
and Navier-Stokes equations).
A Director and Development Lead of PC-Progress, Software company located in
Prague, Czech Republic. Received B.S. and M.S. from the Charles University of
Prague, Faculty of Mathematics and Physics, Prague, Czechoslovakia, and a PhD.
from the Czech Academy of Sciences, Prague, Czech Republic.
Program Developers – Miroslav Šejna
He has authored and coauthored over 160 peer-reviewed publications and
over 20 book chapters. His numerical HYDRUS models are used by
virtually all scientists, students, and practitioners modeling water flow,
chemical movement, and heat transport through variably saturated soils.
Dr. Simunek is a recipient of the Soil Science Society of America’s Don and
Betty Kirkham Soil Physics Award and serves currently as the past chair of
the Soil Physics (S1) of SSSA. He is an associate editor of Water Resources
Research, Vadose Zone Journal, and Journal of Hydrological Sciences.
Expertise in numerical modeling of subsurface water flow and solute
transport processes, equilibrium and nonequilibrium chemical transport,
multicomponent major ion chemistry, field-scale spatial variability, and
inverse procedures for estimating soil hydraulic and solute transport
parameters.
A Professor of Hydrology with the Department of Environmental
Sciences of the University of California, Riverside. Received an
M.S. in Civil Engineering from the Czech Technical University,
Prague, Czech Republic, and a Ph.D. in Water Management from
the Czech Academy of Sciences, Prague.
Course Developers – Jirka Šimůnek
1
4
X
X
X
X
Preferential and Nonequilibrium Flow and Transport
Computer Session 8: Nonequilibrium Flow and Transport
Coupled Movement of Water, Vapor and Energy
Computer Session 9: Coupled Water, Vapor and Energy
Transport
Biogeochemical Transport - Introduction to HP1 (coupled
HYDRUS-1D and PHREEQC) and UNSATCHEM
Computer Session 10: Application of HP1 to Cation
Exchange
Other Applications, Future Plans
Open Session
Contents - 3
X
X
X
X
X
X
X
X
X
Vadose Zone Flow and Transport Modeling: An Overview
Introduction to HYDRUS-1D, its Functions and Windows
Computer Session 1: Infiltration into 1D Soil Profile
Unsaturated Soil Hydraulic Properties, RETC and Rosetta
Numerical Solutions for 1D Variably-Saturated Flow and
Solute Transport
X Computer Session 2: Transient Water Flow and Solute
Transport in a Layered Soil Profile
X Parameter Estimation and Inverse Modeling
X Computer Session 3: Inverse modeling – One- or Multi-stepOutflow Method
Contents - 1
2Department of Mechanical Engineering,
1Department
Vadose Zone Flow and Transport
Modeling
An Overview
X Other Topics, Open Session
X Computer Session 7: 3D Water Flow and Solute Transport
X Computer Session 6b: Solute Plume Migrating to a Stream
X Computer Session 6a: Water Flow to a Stream
X Computer Session 5: Furrow Irrigation with a Solute Pulse
X Computer Session 4: Infiltration from Subsurface Source
Windows
X Introduction to HYDRUS (2D/3D), its Structure and
X 2D/3D Numerics for Variably-Saturated Flow and Transport
Contents - 2
2
5
Source Zone
Control Planes
Observation wells
X
X
X
X
X
X
X
Hillel (2003)
Fluxes
Heat Exchange and
Fluxes
Nutrient Transport
Soil Respiration
Microbiological
Processes
Effects of Climate
Change
Riparian Systems
Stream-Aquifer
Interactions
∂ ( ρ s ) ∂ (θ c ) ∂ ⎛
∂c
⎞
+
= ⎜ θ D − qc ⎟ − φ
∂t
∂t
∂z ⎝
∂z
⎠
Solute Transport (Convection-Dispersion Equation)
Heat Movement
∂C p (θ )T ∂ ⎡
∂T ⎤
∂qT
= ⎢ λ (θ )
− Cw
− C w ST
⎥
∂t
∂z ⎣
∂z ⎦
∂z
∂θ
∂ ⎡
∂h
⎤
= ⎢ K ( h) − K ( h) ⎥ − S
∂t ∂z ⎣
∂z
⎦
Variably-Saturated Water Flow (Richards Equation)
Industrial Pollution
Municipal Pollution
Landfill Covers
Waste Repositories
Radioactive Waste
Disposal Sites
Remediation
Brine Releases
Contaminant
Plumes
Seepage of
Wastewater from
Land Treatment
Systems
X Ecological Apps
X Carbon Storage and
X
X
X
X
X
X
X
X
X
Governing Equations
Precipitation
Irrigation
Runoff
Evaporation
Transpiration
Root Water Uptake
Capillary Rise
Deep Drainage
Fertigation
Pesticides
Fumigants
Colloids
Pathogens
Industrial and Environmental Applications
Environmental
Applications
X
X
X
X
X
X
X
X
X
X
X
X
X
Agricultural
Applications
3
6
http://www.pc-progress.com/en/Default.aspx
Šimůnek, J., M. Šejna, and M. Th. van Genuchten, The HYDRUS Software
Package for Simulating TwoTwo- and ThreeThree-Dimensional Movement of Water,
Heat, and Multiple Solutes in Variably-Saturated Media, User Manual,
Manual
Version 1.0, PC Progress, Prague, Czech Republic, pp. 161, 2007.
Šimůnek, J., M. Th. van Genuchten, and M. Šejna, The HYDRUS Software
Package for Simulating TwoTwo- and ThreeThree-Dimensional Movement of Water,
Heat, and Multiple Solutes in Variably-Saturated Media, Technical Manual,
Manual
Šimůnek, J., M. Šejna, H. Saito, M. Sakai, and M. Th. van Genuchten, The
HYDRUSHYDRUS-1D Software Package for Simulating the OneOne-Dimensional
Movement of Water, Heat, and Multiple Solutes in Variably-Saturated Media,
Version 4.0, HYDRUS Software Series 1, Department of Environmental
Sciences, University of California Riverside, Riverside, CA, pp. 315, 2008.
Inverse Optimization
Equation Solvers
Root Uptake
Pedotransfer Functions
Soil Hydraulic Properties
HYDRUS – Main Module
HYDRUS - References
Heat Transport
Solute Transport
Water Flow
Input, Output, Meshgen
HYDRUS Graphical Interface
HYDRUS –Modular Structure
Richards equation for variably-saturated water flow
Various models of soil hydraulic properties
Hysteresis
Sink term to account for water uptake by plant roots
gaseous phase
Nonlinear nonequilibrium reactions between the solid and liquid phases
Linear equilibrium reactions between the liquid and gaseous phases
Zero-order production
First-order degradation reactions
Physical nonequilibrium solute transport
USSL - SWMS-3D
Šimůnek et al. [1995]
IGWMC - HYDRUSHYDRUS-2D (2.0)
USSL - CHAIN-2D
Šimůnek and van Genuchten [1994]
MIT:
Celia et al. [1990]
Princeton U.:
van Genuchten [1978]
UCR, PC-Progress – HYDRUS (2D/3D)
USSL - HYDRUS-2D (1.0)
USSL - SWMS-2D
Agr. Univ. in Wageningen:
Feddes et al. [1978]
Vogel [1987] - SWMII
Israel: Neuman [1972] - UNSAT
U. of Arizona: Davis and Neuman [1983]
HYDRUS - History of Development
X
X
X
X
X
X Convective-dispersive transport in the liquid phase, diffusion in the
Solute Transport:
X Conduction and convection with flowing water
Heat Transport:
X
X
X
X
Water Flow:
HYDRUS Software Packages
4
7
X
X
X
X
X
X
X
X
Net Inward
Force
Vapor Molecules
Water Molecules
Liquid surface
Liquid
Liquid wets the solid
Hydrophilic surface
Solid
γ
Liquid is repelled by the solid
Hydrophobic surface
γ
The angle measured from the liquid-solid interface to the
liquid-air interface, when liquid is present in a three-phase
system containing air and solids.
Molecules at fluid interface are exposed to different forces
than within fluids. At the water-air interface, a net inward
force exists because of higher density of water molecules in
water than in air. Within the water there is no net attraction
in any direction. The extra energy of water at the interface is
called surface tension and is defined as energy per unit
surface area or force per unit length
Gas
Contact Angle
X
Variably-Saturated Flow (Richards Eq.)
X Root Water Uptake (water and salinity stress)
X Solutes Transport (decay chains, ADE)
- Nonlinear Sorption
- Chemical Nonequilibrium
- Physical Nonequilibrium
X Heat Transport
X Pedotransfer Functions (hydraulic properties)
X Parameter Estimation
X Interactive Graphics-Based Interface
X HYDRUS (2D/3D) and Additional Modules
The HYDRUS Software Packages
Surface Tension
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.
Šimůnek, 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.
Šimůnek, J., and M. Th. van Genuchten, The CHAIN_2D code for simulating two-dimensional
movement of water flow, heat, and multiple solutes in variably-saturated porous media, Version
1.1, Research Report No 136, U.S. Salinity Laboratory, USDA, ARS, Riverside, California,
205pp., 1994.
Šimůnek, J., K. Huang, and M. Th. van Genuchten, The SWMS_3D code for simulating water
flow and solute transport in three-dimensional variably saturated media. Version 1.0, Research
Report No. 139, U.S. Salinity Laboratory, USDA, ARS, Riverside, California, 155 pp., 1995.
HYDRUS (2D/3D) - History (References)
5
2σ cos γ
ρ gR
0
100
200
300
400
500
0
0.1
0.3
Water Content [-]
0.2
Loam
Sand
Clay
Soil-water characteristic curve
Characterizes the energy status of the soil water
Retention Curve
g – gravitational acceleration
R – capillary radius
H – capillary rise
σ – surface tension
γ – contact angle
ρ – bulk density of water
H=
Laplace Equation:
0.4
0.5
When a small cylindrical glass capillary tube is inserted in a water
reservoir open to atmosphere, water will rise upward in the tube.
Capillary Rise
|Pressure head| [cm]
8
Δ( P + z )
dH
= − Ks
L
dz
Δ(h + z )
dH
= − K (h)
L
dz
H - sum of the matric (h) and gravitational (z) head
q = K (h )
X Unsaturated water flow (Darcy-Buckingham Law)
q = Ks
X Groundwater flow (Darcy’s Law)
Water Flow in Soils
- characterizes the energy status of the soil water
Retention Curve
6
0
100
200
300
400
0
0.1
0.3
Water Content [-]
0.2
Loam
Sand
Clay
0.5
-10
-8
-6
-4
-2
0
2
4
0
0.1
0.2
0.3
Water Content [-]
0.4
Clay
Sand
Loam
0.5
- characterizes resistance of porous media to water flow
500
Characterizes the energy status of the soil water
0.4
- volumetric water content [L3L-3]
- pressure head [L]
- unsaturated hydraulic conductivity [LT-1]
- vertical coordinate positive upward [L]
- time [T]
- root water uptake [T-1]
⎡
⎛ A ∂h
⎞⎤
+ KizA ⎟ ⎥ − S ( h )
⎢ K ( h ) ⎜⎜ Kij
⎟
∂x j
⎢⎣
⎝
⎠ ⎥⎦
θ - volumetric water content [L3L-3]
h - pressure head [L]
K - unsaturated hydraulic conductivity [LT-1]
KijA - components of a anisotropy tensor [-]
xi - spatial coordinates [L]
z - vertical coordinate positive upward [L]
t
- time [T]
S - root water uptake [T-1]
∂θ (h )
∂
=
∂t
∂xi
The governing flow equation for two-dimensional isothermal
Darcian flow in a variably-saturated isotropic rigid porous
medium:
Water Flow - Richards Equation
Hydraulic Conductivity, K(θ)
|Pressure head| [cm]
Retention Curve, θ(h)
h
K
z
t
S
θ
∂θ (h) ∂ ⎡
∂h
⎤
= ⎢ K ( h) + K ( h) ⎥ − S ( h)
∂t
∂z ⎣
∂z
⎦
∂θ (h)
∂q
= − − S ( h)
∂t
∂z
The governing flow equation for one-dimensional isothermal
Darcian flow in a variably-saturated isotropic rigid porous
medium:
Water Flow - Richards Equation
log (Hydraulic Conductivity) [cm/d]
9
7
10
Se =
Se =
Se - effective water content [-]
h < -1/α
h ≥ -1/α
n 1−1/ n
Se =
θ − θr
θs − θr
⎧ ln ( h / h0 ) ⎫
1
erfc ⎨
⎬
2
2σ ⎭
⎩
(1 + α h )
1
⎧ |α h |- n
Se = ⎨
⎩ 1
θs - saturated water content [-]
θr - residual water content [-]
α, n, h0, σ - empirical parameters [L-1], [-], [L], [-]
X Kosugi [1996]:
X van Genuchten [1980]:
X Brooks and Corey [1964]:
Retention Curve
Unsaturated hydraulic conductivity decreases as volumetric water content
decreases:
Cross-sectional area of water flow decreases
Tortuosity increases
Drag forces increase
Thus, the unsaturated hydraulic conductivity is a nonlinear function of θ and h.
Hydraulic Conductivity, K(θ)
0
2
3
4
log (|Pressure Head| [cm])
1
Clay
Sand
Loam
5
2
Se - effective water content [-]
ΚS - saturated hydraulic conductivity [LT-1]
Se =
θ − θr
θs − θr
⎧⎪ 1
⎡ ln ( h / h0 ) σ ⎤ ⎫⎪
K (h) = K s Sel ⎨ erfc ⎢
+
⎥⎬
2σ
2 ⎦ ⎭⎪
⎣
⎩⎪ 2
m
K ( h ) = K s Sel ⎡1 − (1 − Se1/ m ) ⎤
⎢⎣
⎥⎦
K ( h ) = K s Se2 / n + l + 2
θr - residual water content [-]
α, n, h0, σ, l - empirical parameters [L-1], [-], [L], [-], [-]
(Mualem [1976])
X Kosugi [1996]:
(Mualem [1976])
X van Genuchten [1980]:
X Brooks and Corey [1964]:
Hydraulic Conductivity Function
-10
-8
-6
-4
-2
0
2
4
- characterizes resistance of porous media to water flow
Hydraulic Conductivity, K(h)
log (Hydraulic Conductivity [cm/d])
2
8
11
ρref, ρT
- Lognormal distribution model
(Kosugi, 1996)
- Dual-porosity model (Durner, 1994)
New functions:
href, hT
σref, σT
Kref, KT
μref, μT
μref ρT
K = α K K ref
μT ρ ref ref
σT
h = α h href
σ ref ref
pressure heads at temperature T and reference temperature Tref [L]
surface tensions at temperature T and reference temperature Tref
conductivities at temperature T and reference temperature Tref [LT-1]
dynamic viscosities at temperature T and reference temperature Tref
[ML-1T-1]
bulk densities at temperature T and reference temperature Tref [ML-3]
KT =
hT =
Temperature Dependence of Soil Hydraulic Properties
Hydraulic Conductivity Function, K(θ)
- van Genuchten (1980)
- Brooks and Corey (1964)
Old functions:
The RETC program for
quantifying the hydraulic
functions of unsaturated
soils
RETC, version 6.0:
Soil Water Retention Curve, θ(h)
9
12
Soil Water Hysteresis
Se - effective saturation [-]
θ - volumetric water content [L3L-3]
θr - residual water content [L3L-3]
θs - saturated water content [L3L-3]
h - pressure head [L]
αw, αd - empirical parameters [L-1]
n, m – empirical parameters [-]
α w ≈ 2α d
n
Se ( h) = [1 + |α d h| ]− m
n
Se ( h) = [1 + |α w h| ]− m
velocities
X Osmotic gradients in the soil water potential are
negligible
X Fluid density is independent of solute
concentration
X Matrix and fluid compressibilities are relatively
small
X Effect of air phase is neglected
X Darcy’s equation is valid at very low and very high
Richards Equation - Assumptions
for z = 0 or
z=L
for z = 0 or
∂h
=1
∂z
for z = L
Gradient boundary conditions:
⎛ ∂h ⎞
- K ⎜ + 1⎟ = q0 ( z , t )
⎝ ∂z
⎠
z=L
Flux (Neumann type) boundary conditions:
h( z, t ) = h0 ( z, t )
Pressure head (Dirichlet type) boundary conditions:
Boundary Conditions (System-Independent)
functions
X Lack of accurate and cheap methods for
measuring the hydraulic properties
X Extreme heterogeneity of the subsurface
X Inconsistencies between scale at which the
hydraulic and solute transport parameters
are measured, and the scale at which the
models are being applied
X Hysteresis in the soil water retention function
X Extreme nonlinearity of the hydraulic
Richards Equation - Complications
10
13
0
0.025
0.05
Time [days]
0.075
0.1
-55
0.00
-50
-45
-40
-35
-30
-25
-20
0.05
Time [days]
0.10
0.15
X
X
X
X
X
X
X
X
Root Water Uptake (water and salinity stress)
Solutes Transport (decay chains, ADE)
Heat Transport
Pedotransfer Functions (hydraulic properties)
Parameter Estimation
Interactive Graphics-Based Interface
HYDRUS (2D/3D) and Additional Modules
0.20
Stage I and II of infiltration (evaporation)
hA ≤ h ≤ hS
-100
-80
-60
-40
-20
0
Ponding
∂h
-K
-K ≤ E
∂x
Atmospheric boundary condition:
Boundary Conditions (System-Dependent)
1D soil profile
x
S
Lt
α
b
Sp
S
Tp
ΩR
Ta
Feddes et al. [1978]
z
E
P
Drainage
Impermeable layer
Groundwater table
Soil surface
h4
h3 low
h3 high
Pressure Head, h [L]
normalized water uptake distribution [L-1]
potential root water uptake [T-1]
actual root water uptake [T-1]
potential transpiration [LT-1]
stress response function [-]
Ω
b(x,z)
Tp = 1 mm d-1
Tp = 5 mm d-1
h2
h1
S ( z,t ) = α ( h ) S p ( z,t ) = α ( h )b( z )Tp
S p ( z,t ) = b( z )Tp
Root Water Uptake
Tile drain
X Tile drains
if(h<0) => q=0
if(h=0) => q=?
X Seepage face (free draining lysimeter, dike)
0
1
Boundary Conditions (System-Dependent)
Stress Response Function, α [-]
11
14
⎞
⎟
⎠
1
p
p2
⎛ p
*
py
(Vrugt et al., 2001)
e
⎛ p
⎞
p
− ⎜ z z * − z + r x* − x ⎟
Xm
⎝ Zm
⎠
⎛
z ⎞⎛
x ⎞
b ( x, z ) = ⎜ 1 −
⎟ ⎜1 −
⎟
Xm ⎠
⎝ Zm ⎠ ⎝
x
⎛
x ⎞⎛
y ⎞⎛
z ⎞ −⎜⎜⎝ X m x − x + Ym
b ( x, y , z ) = ⎜ 1 −
⎟⎜ 1 − ⎟⎜ 1 −
⎟e
⎝ X m ⎠⎝ Ym ⎠⎝ Z m ⎠
y* − y +
pz *
z −z
Zm
pressure head [L]
osmotic head [L]
pressure head at which water extraction rate is reduced by 50% [L]
ditto for osmotic head [L]
experimental constants [-] (=3)
⎛ h ⎞
⎛ hφ ⎞
1+ ⎜
⎟ 1 + ⎜⎜
⎟⎟
h
⎝ 50 ⎠
⎝ hφ 50 ⎠
p1
⎛ h + hφ
1+ ⎜
⎝ h50
1
1
Spatial Root Distribution Function
h
hf
h50
hφ50
p1, p2
α
α (h, hφ ) =
α (h, hφ ) =
Water and solute stress:
Stress Response Functions
⎞
⎟⎟
⎠
LR
L0
Lm
f
r
LR
normalized water uptake distribution [L-1]
potential root water uptake [T-1]
actual root water uptake [T-1]
potential transpiration [LT-1]
actual transpiration [LT-1]
L0
L0 + ( Lm − L0 )e − rt
rooting depth [L]
initial rooting depth [L]
maximum rooting depth [L]
root growth coefficient (Verhulst-Pearl logistic function)
growth rate [T-1]
f (t ) =
LR (t ) = Lm f (t )
Root Growth
Sp
S
Tp
Ta
α
b
LR
Ta = ∫ S (h, z ) dz = Tp ∫ a ( h, z ) b( z ) dz
LR
Tp = ∫ S p (h, z ) dz
Transpiration Rates
12
15
General Structure of the System of Solutes
X
X Heat Transport
- solution concentration [ML-3]
- adsorbed concentration [MM-1]
- water content [L3L-3]
- soil bulk density [ML-3]
- dispersion coefficient [L2T-1]
- volumetric flux [LT-1]
- rate constant representing reactions [ML-3T-1]
X Explosives: TNT (-> 4HADNT -> 4ADNT -> TAT), RDX HMX
X Pharmaceuticals, hormones: Estrogen (17bEstradiol -> Estrone -> Estriol),
Testosterone
X Chlorinated Hydrocarbons: PCE -> TCE -> c-DCE -> VC -> ethylene
X Pesticides: aldicarb (oxime) -> sulfone (sulfone oxime) -> sulfoxide (sulfoxide
oxime)
X Nitrogen: (NH2)2CO -> NH4+ -> NO2- -> NO3-
X Radionuclides: 238Pu -> 234U -> 230Th -> 226Ra
Transport of single ions
Transport of multiple ions (sequential first-order decay)
HYDRUS – Solute Transport
φ
D
q
θ
ρ
c
s
∂ (θ c ) ∂ ( ρ s ) ∂ ⎛
∂c
⎞
+
= ⎜ θ D − qc ⎟ − φ
∂t
∂t
∂z ⎝
∂z
⎠
One-dimensional chemical transport during transient water
flow in a variably saturated rigid porous medium
Solute Transport - Convection-Dispersion Equation
13
16
subscripts corresponding with the liquid, solid and
gaseous phases, respectively
concentration in liquid, solid, and gaseous phase,
respectively
kε (2, ns )
for ( x, z ) ε Γ D
∂c
ni + qi ni c = qi ni c0
∂x j
for ( x, z ) ε ΓC
∂c
θ Dij
ni = 0
∂x j
for ( x, z ) ε Γ N
SecondSecond-type (Neumann type) boundary conditions
-θ Dij
ThirdThird-type (Cauchy type) boundary conditions
c( x, z, t ) = c0 ( x, z, t )
FirstFirst-type (or Dirichlet type) boundary conditions
Solute Transport - Boundary Conditions
c, s, g
w, s, g
'
'
− μs,k
−1ρ sk −1 - μ g,k −1ag k −1 + γ w ,kθ + γ s ,k ρ + γ g ,k a − Scr ,k
∂θ ck ∂ρ sk ∂ag k
∂ ⎛
∂c ⎞ ∂ ⎛
∂g ⎞ ∂qck
+
+
= ⎜ θ Dkw k ⎟ + ⎜ aDkg k ⎟ ∂t
∂t
∂t
∂z ⎝
∂z ⎠ ∂z ⎝
∂z ⎠ ∂z
'
-( μw,k + μw' ,k )θ ck - ( μs ,k + μs' ,k ) ρ sk - ( μ g ,k + μ g' ,k )ag k + μw,k
−1θ ck −1 +
∂ (θ c ) ∂ ( ρ s ) ∂ ⎛
∂c
⎞
+
= ⎜ θ D − qc ⎟ − φ
∂t
∂t
∂z ⎝
∂z
⎠
Governing Solute Transport Equations
θ 7/3
θ s2
τ=
Millington and Quirk [1961]:
Dd - ionic or molecular diffusion coefficient
in free water [L2T-1]
τ - tortuosity factor [-]
λ - longitudinal dispersivity [L]
θ - water content [L3L-3]
q - Darcy’s flux [LT-1]
θ D = λ | q |+θ Ddτ
Bear [1972]:
Solute Transport – Dispersion Coefficient
volumetric flux [LT-1]
soil bulk density [ML-3]
a
air content [L3L-3]
S
sink term in the water flow equation [T-1]
concentration of the sink term [ML-3]
cr
dispersion coefficients for the liquid and gaseous phase [L2T-1],
Dw, Dg
respectively
k
subscript representing the kth chain number
μw, μs, μg first-order rate constants for solutes in the liquid, solid, and
gaseous phases [T-1], respectively
γw, γs, γg
zero-order rate constants for the liquid [ML-3T-1], solid [T-1], and
gaseous phases [ML-3T-1], respectively
μw', μs', μg' first-order rate constants for solutes in the liquid, solid and
gaseous phases [T-1], respectively; these rate constants provide
connections between the individual chain species.
number of solutes involved in the chain reaction
ns
ρ
qi
Governing Solute Transport Equations
14
17
|q|
q j qi
+ θ Ddτδ ij
qx2
q2
+ DT z + θ Ddτ
|q|
|q|
X
X Heat Transport
θ Dzz = DL
qz2
q2
+ DT x + θ Ddτ
|q|
|q|
qq
θ Dxz = ( DL - DT ) x z
|q|
θ Dxx = DL
Dd - ionic or molecular diffusion coefficient in free water [L2T-1]
τ - tortuosity factor [-]
δij - Kronecker delta function (δij =1 if i=j, and δij =0 otherwise)
DL , DT - longitudinal and transverse dispersivities [L]
θ Dij = DT | q | δ ij + ( DL - DT )
Bear [1972]:
Solute Transport - Dispersion Coefficient
Kd
R
s
c
R = 1+
ρ Kd
θ
- distribution coefficient [L3M-1]
- retardation factor [-]
- solid phase concentration [MM-1]
- liquid phase concentration [ML-3]
⎞
∂Rθ c
∂ ⎛
∂c
=
− qi c ⎟ + φ
⎜⎜ θ Dij
⎟
∂t
∂xi ⎝
∂x j
⎠
s = Kd c
X Linear Adsorption
Convection-Dispersion Equation
Gelhar et al. (1985)
Dispersivity as a Function of Scale
15
18
ks , η , β empirical constants
ksc β
s=
1 +η c β
Liquid - Solid: a generalized nonlinear
(Freundlich-Langmuir) empirical equation
HYDRUS assumes nonequilibrium interactions between the
solution (c) and adsorbed (s) concentrations, and equilibrium
interaction between the solution (c) and gaseous (g)
concentrations of the solute in the soil system.
Nonlinear Equilibrium Adsorption
DR - retarded dispersion coefficient [L2T-1]
vR - retarded velocity [LT-1]
∂c
∂ 2c
∂c
R = D 2 −v
∂t
∂z
∂z
2
∂c
∂c
∂c
= DR 2 − vR
∂t
∂z
∂z
X Steady-State Transport (1D)
⎞
∂Rθ c
∂ ⎛
∂c
=
− qi c ⎟ + φ
⎜⎜ θ Dij
⎟
∂t
∂xi ⎝
∂x j
⎠
X Transient Transport (2D)
Convection-Dispersion Equation
k3
k1c
1 + k2c
Temkin
RT
ln( k2 c)
k1
X
X
X
X
X
X
X
X
Root Water Uptake (water and salinity stress)
Solutes Transport (decay chains, ADE)
Heat Transport
modified Kielland
s
c
=
sT [ c + k1 ( cT − c )exp{ k2 ( cT − 2 c )}]
s = k1c exp(-2k2 s)
s=
Lindstrom et al. [1971]
van Genuchten et al. [1974]
Lai and Jurinak [1971]
Barry [1992]
Bache and Williams [1971]
Barry
s = k1{1 - [1+k2 c k3 ]k4 }
s = k1 c - k3
k2
k1c
1+ k2c + k3 c
Gunary [191970]
s=
Fitter and Sutton [1975]
Shapiro and Fried [1959]
Sibbesen [1981]
Sips [1950]
Langmuir
Freundlich-Langmuir
Double Langmuir
Langmuir [1918]
Freundlich
Extended Freundlich
Lapidus and Amundson [1952]
Freundlich [1909]
Linear
Fitter-Sutton
c k2 /k3
Reference
Model
Gunary
s = k1 c
s=
k1 c
1 + k 2 c k3
k1c
kc
s=
+ 3
1 + k2 c 1 + k 4 c
s=
s = k1 c k2
Equation
s = k1c + k2
Nonlinear Equilibrium Adsorption
16
19
∂s k
= α [(1 - f ) K d c - s k ] - μ s ,k s k
∂t
∂
∂
∂c
(θ + f ρ K d )c = (θ D - qc ) ∂t
∂z
∂z
−αρ [(1 - f ) K d c - s k ] - θ μ lc - f ρ K d μ s,ec
Linear sorption:
Two-Site Chemical Nonequilibrium Transport
Leenheer and Ahlrichs [1971]
Enfield et al. [1976]
Šimunek and van Genuchten [1994]
∂s
= α exp(k2 s){k1c exp(-2k2 s) - s}
∂t
∂s
= α ck1 sk2
∂t
FreundlichLangmuir
⎛ k c k3
⎞
∂s
=α ⎜ 1 k - s⎟
3
∂t
⎝ 1+k2 c
⎠
Fava and Eyring [42]
Langmuir
⎛ kc
⎞
∂s
=α ⎜ 1 - s⎟
∂t
⎝ 1+k2c ⎠
Reference
Lapidus and Amundson [1952]
Oddson et al. [1970]
Hornsby and Davidson [1973]
van Genuchten et al. [1974]
Hendricks [1972]
⎛ s −s ⎞
∂s
= α ( sT - s)sinh ⎜ k1 T
⎟
∂t
⎝ sT − si ⎠
Freundlich
Model
Linear
∂s
= α (k1 c k2 - s)
∂t
∂s
= α (k1c + k2 - s)
∂t
Equation
Non-Equilibrium Adsorption Equations
∂s e
∂s
= f
∂t
∂t
Type - 1 sites with instantaneous sorption
Type - 2 sites with kinetic sorption
fraction of exchange sites assumed to be at
equilibrium
Solid
Water
Air
ψaca
Scstr
ψs
kac
kdc
Attached Colloids, Scatt
kdca
Mobile Colloids, Cc
Strained Colloids,
ψsstr
kstr
kaca
Air-Water Interface Colloids, Γc
Colloid, Virus, and Bacteria Transport
f
β
⎡
⎤
∂ sk
kc
= α ⎢(1 - f ) s β - s k ⎥ - μ s s k + (1 - f )γ s
1 +η c
∂t
⎣
⎦
se
sk
s = se + sk
Nonequilibrium Two-Site Adsorption Model
17
20
⎛ d c + x − x0 ⎞
⎟
dc
⎝
⎠
smax − s
s
= 1−
smax
smax
- straining [T-1]
- deposition (attachment) coefficient [T-1]
- entrainment (detachment) coefficient [T-1]
- reduction of attachment coefficient due to blockage of sorption sites
β
[θim + (1 - f ) ρ kd ]
∂cim
= α ( cm - cim ) - [θ im μw,im + (1 - f ) ρ kd μs ,im ]cim
∂t
∂
∂
∂c
(θ m + f ρ kd ) cm = ⎛⎜ θm Dm m - qcm ⎞⎟ - α (cm - cim ) - (θ m μw,m + f ρ kd μ,ms )cm
∂t
∂z ⎝
∂z
⎠
Two-Region Physical Nonequilibrium Transport
kstr
ka
kd
ψ
ψ str = ⎜
∂s
ρ 2 = θ kstrψ str c
∂t
∂s
ρ 1 = θ kaψ t c − kd ρ s1
∂t
ψt =
Straining
Attachment/Detachment
∂c
∂s1
∂s2
∂ 2c
∂c
θ +ρ
+ρ
= θ D 2 −θ v
∂t
∂t
∂t
∂x
∂x
Two-Region Physical Nonequilibrium Transport
X
X
X
X
X
Heat Transport
X
18
21
empirical constant equal to
Henry's Law constant
universal gas constant
absolute temperature
(KHRTA)-1
E
ar, aT
coefficient values at a reference absolute temperature,
TrA, and absolute temperature, TA, respectively
activation energy of the reaction or process
⎡ E (T A - TrA ) ⎤
aT = ar exp ⎢
⎥
A A
⎣ RT Tr ⎦
Most of the diffusion (Dw, Dg), distribution (ks, kg), and reaction
rate (γw, γs , γg , μw', μs', μg', μw , μs , and μg) coefficients are
strongly temperature dependent. HYDRUS assumes that this
dependency can be expressed by an Arrhenius equation
[Stumm and Morgan, 1981].
Temperature Dependence of Transport and
Reaction Coefficients
kg
KH
R
TA
g = kgc
Liquid - Gas: a linear relation (Henry’s Law)
Interaction Among Phases
∂c
∂ 2c
∂c
= DijE
- qiE
∂t
∂xi ∂x j
∂ xi
X Heat Transport
X
R
w
a
⎛ ρ K d ak g ⎞ ∂c ⎛ w
a ⎞ ∂ 2 c qi + qi k g ∂ c
a
+
+
=
D
+
D
k
1
⎜
⎟
ij
g
θ
θ ⎠ ∂t ⎜⎝ ij
θ ⎟⎠ ∂xi ∂x j
θ
∂ xi
⎝
Steady-State (a new retardation factor and effective diffusion
coefficient):
g = kg c
⎞
∂ ( ρ s ) ∂ (θ c ) ∂ ( ag )
∂ ⎛
∂g
w ∂c
+
+
=
+ aDija
- qiwc - qia g ⎟ +φ
⎜⎜ θ Dij
⎟
∂t
∂t
∂t
∂xi ⎝
∂x j
∂x j
⎠
Volatilization
19
22
X
X Heat Transport
volumetric fraction
n, o, g, w subscripts representing solid phase, organic matter, gaseous
phase, and liquid phase, respectively.
θ
C (θ ) = Cnθ n +Coθo +Cwθ + C g a
de Vries [1963]:
liquid phase, respectively
λij(θ)
apparent thermal conductivity of the soil
C(θ), Cw volumetric heat capacities of the porous medium and the
∂T
∂ ⎡
∂T ⎤
∂T
C (θ )
=
⎢λij (θ )
⎥ - Cw qi
∂t ∂xi ⎢⎣
∂x j ⎦⎥
∂xi
Sophocleous [1979]:
Governing Heat Transport Equation
|q|
qi q j
+ λ0 (θ )δ ij
empirical parameters
Average values of selected soil water retention parameters for 12 major
soil textural groups
PTFs by Carsel and Parrish (1988)
b1, b2, b3
λ0 (θ ) = b1 + b2θ w + b3θ w0.5
Chung and Horton [1987]
respectively
λL, λT longitudinal and transverse thermal dispersivities,
plus water) in the absence of flow
λ0(θ) thermal conductivity of the porous medium (solid
λij (θ ) = λT Cw | q | δ ij + (λL - λT )Cw
Thermal Conductivity
20
23
Textural Class
Sand, Silt, Clay %
Same + Bulk Density
SSCBD + θ at 33 kPa
Same + θ at 1500 kPa
TXT
SSC
SSCBD
SSCBD + θ33
SSCBD + θ33 + θ1500
θr
0.053
0.049
0.039
0.061
0.050
0.065
0.063
0.079
0.090
0.117
0.111
0.098
[L3L-3]
0.375
0.390
0.387
0.399
0.489
0.439
0.384
0.442
0.482
0.385
0.481
0.459
θs
[L3L-3 ]
0.035
0.035
0.027
0.011
0.007
0.005
0.021
0.016
0.008
0.033
0.016
0.015
α
[cm-1]
3.18
1.75
1.45
1.47
1.68
1.66
1.33
1.41
1.52
1.21
1.32
1.25
n
[-]
643.
105.
38.2
12.0
43.7
18.3
13.2
8.18
11.1
11.4
9.61
14.8
Ks
[cm d-1]
Soil hydraulic parameters for the analytical functions of van Genuchten (1980) for the
twelve textural classes of the USDA textural triangle obtained with the Rosetta light
program (Schaap et al., 2001).
Sand
Loamy Sand
Sandy Loam
Loam
Silt
Silty Loam
Sandy Clay Loam
Clay Loam
Silty Clay Loam
Sandy Clay
Silty Clay
Clay
Textural class
Textural Class Averages: Rosetta
Input Data
Model
Schaap et al. (2001)
Pedotransfer Functions: Rosetta
X
X
X
X
X
Heat Transport
X Root Water Uptake (water stress)
X
21
24
ij
*
j
i
j
i
2
j =1
nb
+ ∑vˆ j[b*j - b j ]2
wi - weight of a particular measured point
i =1
*
Φ ( β ) = ∑ wi ⎡⎣ qi − qi ( β ) ⎤⎦
n
2
X Heat Transport
X
The problem can be simplified into
the Weighted LeastLeast-Squares Problem
deviations between measured and calculated space-time
variables
differences between independently measured, pj*, and predicted,
pj, soil hydraulic properties
penalty function for deviations between prior knowledge of the
soil hydraulic parameters, bj*, and their final estimates, bj .
i =1
+
[ g *j ( x , ti ) - g j ( x , ti , b )]2 +
3rd term:
2nd term:
1st term:
j =1
n pj
i, j
∑w [ p (θ ) - p (θ , b)]
i =1
nqj
∑w
+ ∑v j
mq
j =1
Φ ( b, q, p ) = ∑v j
mq
Objective Function for Inverse Problems
Formulation of the Inverse Problem
Method:
- Marquardt-Levenberg optimization
Sequence:
- Independently
- Simultaneously
- Sequentially
Parameter Estimation:
- Soil hydraulic parameters
- Solute transport and reaction parameters
- Heat transport parameters
Parameter Estimation with HYDRUS
22
25
*** BLOCK H: NODAL INFORMATION
*********************************
NumNP NumEl
IJ
NumBP NObs
380
342
19
5 0
n Code x
z
h
Conc
Q M B
1 1 0.00 230.00
0.00 0.00E+00 0.00E+00 1 0.00
2 0 0.00 228.00 -145.50 0.00E+00 0.00E+00 1 0.00
3 0 0.00 226.00 -143.40 0.00E+00 0.00E+00 1 0.00
4 0 0.00 224.00 -141.00 0.00E+00 0.00E+00 1 0.00
5 0 0.00 220.00 -135.60 0.00E+00 0.00E+00 1 0.00
6 0 0.00 215.00 -127.70 0.00E+00 0.00E+00 1 0.00
7 0 0.00 210.00 -119.00 0.00E+00 0.00E+00 1 0.00
8 0 0.00 205.00 -109.90 0.00E+00 0.00E+00 1 0.00
9 0 0.00 200.00 -100.50 0.00E+00 0.00E+00 1 0.00
10 0 0.00 190.00 -82.80 0.00E+00 0.00E+00 1 0.00
11 0 0.00 180.00 -71.00 0.00E+00 0.00E+00 2 0.00
12 0 0.00 170.00 -60.30 0.00E+00 0.00E+00 2 0.00
13 0 0.00 160.00 -49.80 0.00E+00 0.00E+00 2 0.00
14 0 0.00 150.00 -39.60 0.00E+00 0.00E+00 2 0.00
15 0 0.00 140.00 -29.50 0.00E+00 0.00E+00 2 0.00
16 0 0.00 130.00 -19.40 0.00E+00 0.00E+00 2 0.00
17 0 0.00 120.00 -9.40 0.00E+00 0.00E+00 2 0.00
18 0 0.00 110.00
0.60 0.00E+00 0.00E+00 2 0.00
19 0 0.00 100.00 10.20 0.00E+00 0.00E+00 2 0.00
20 1 5.00 230.00
0.00 0.00E+00 0.00E+00 1 0.00
21 0 5.00 228.00 -145.50 0.00E+00 0.00E+00 1 0.00
22 0 5.00 226.00 -143.40 0.00E+00 0.00E+00 1 0.00
23 0 5.00 224.00 -141.00 0.00E+00 0.00E+00 1 0.00
Capillary Barrier
Material Distributions
*** BLOCK A: BASIC INFORMATION ******************
Heading
'Example 4 - Infiltration Test '
LUnit TUnit MUnit (indicated units are obligatory for all data)
'cm' 'min' '-'
Kat (0:horizontal plane, 1:axisymmetric vertical flow, 2:vertical )
1
MaxIt TolTh TolH (maximum number of iterations)
20 .0005 0.1
lWAt lChem CheckF ShortF FluxF AtmInF SeepF FreeD
t t f f t f f f f
*** BLOCK B: MATERIAL INFORMATION ***************
NMat NLay hTab1 hTabN NPar
2 2 .001 200. 9
thr ths tha thm Alfa n
Ks Kk thk
.0001 .399 .0001 .399 .0174 1.3757 .0207 .0207 .399
.0001 .339 .0001 .339 .0139 1.6024 .0315 .0315 .339
*** BLOCK C: TIME INFORMATION ********************
dt dtMin dtMax DMul DMul2 MPL
.1 .001 100. 1.33 .33 10
TPrint(1),TPrint(2),...,TPrint(MPL)
(print-time array)
60 180 360 720 1440 2160 2880 4320 5760 7200
*** BLOCK G: SOLUTE TRANSPORT INFORMATION *****
EpsilUpW lArtD PeCr
0.5 t
f
0.
Bulk.d. Difus.
Disper. Adsorp. SinkL1 SinkS1 SinkL0
1.4 0.026 0.50 0.10 0.100 -3.472E-5 -6.9444E-6 0. 0.
1.4 0.026 0.50 0.10 0.100 -3.472E-5 -6.9444E-6 0. 0.
Traditional Input to Hydrological Models
Capillary Barrier
Velocity Vectors
23
26
Plume Movement in
a Transect with Stream
Cut-off Wall
Finite Element Mesh
Cut-off Wall
Solute Plume
24
27
X
X
X
X
X
Heat Transport
X
Water flow in a dual-porosity system allowing for preferential flow
in fractures or macropores while storing water in the matrix.
Root water uptake with compensation.
Spatial root distribution functions of Vrugt et al. (2002).
Soil hydraulic property models of Kosugi (1995) and Durner (1994).
Transport of viruses, colloids, and/or bacteria using an
attachment/detachment model, straining, filtration theory, and
blocking functions.
A constructed wetland module (only in 2D).
The hysteresis model of Lenhard et al. (1991) to eliminate pumping
by keeping track of historical reversal points.
New print management options.
Dynamic, system-dependent boundary conditions.
Flowing particles in two-dimensional applications.
HYDRUS (2D/3D) – New Features
25
28
- variably saturated water flow
- heat transport
- root water uptake
- solute transport
UNSATCHEM (Šimůnek et al., 1996)
- carbon dioxide transport
- major ion chemistry
- cation exchange
- precipitation-dissolution (instantaneous and kinetic)
- complexation
HYDRUS-1D (Šimůnek et al., 1998)
HYDRUS-1D + UNSATCHEM
Completely new GUI based on Hi-End 3D graphics libraries.
MDI architecture – multiple projects and multiple views.
New organization of geometric objects.
Navigator window with an object explorer.
Many new functions improving the user-friendliness, such as dragand-drop and context sensitive pop-up menus.
Improved interactive tools for graphical input.
Saving Cross-Sections and Mesh-Lines for charts within a given
project.
Display Options – all colors, line styles, fonts and other parameters
of graphical objects can be customized.
Extended print options.
Extended information in the Project Manager (including project
preview).
Many additional improvements.
HYDRUS (2D/3D) – New Features
Ca2+, Mg2+, Na+, K+, SO42-, Cl-, NO3-
4
7
3
4 Sorbed species
(exchangeable)
5 CO2-H2O species
6 Silica species
H4SiO4, H3SiO4-, H2SiO42-
PCO2, H2CO3*, CO32-, HCO3-, H+, OH-,
H2O
Ca, Mg, Na, K
CaCO3, CaSO4⋅ 2H2O, MgCO3⋅ 3H2O,
Mg5(CO3)4(OH)2⋅ 4H2O,
Mg2Si3O7.5(OH) ⋅ 3H2O, CaMg(CO3)2
Kinetic reactions: calcite precipitation/dissolution, dolomite dissolution
Activity coefficients: extended Debye-Hückel equations, Pitzer expressions
6
10 CaCO3o, CaHCO3+, CaSO4o, MgCO3o,
MgHCO3+, MgSO4o, NaCO3-, NaHCO3o,
NaSO4-, KSO4-
7
3 Precipitated
species
2 Complexed
species
1 Aqueous
components
Geochemical Modeling
26
29
HYDRUS-1D GUI for HP1
Gonçalves et al. (2006)
Lysimeter Study
(this example considers kinetic precipitation-dissolution
of kaolinite, illite, quartz, calcite, dolomite, gypsum,
hydrotalcite, and sepiolite)
X Long-term transient flow and transport of major
cations (Na+, K+, Ca2+, and Mg2+) and heavy metals
(Cd2+, Zn2+, and Pb2+) in a soil profile.
X Kinetic biodegradation of NTA (biomass, cobalt)
X Infiltration of a hyperalkaline solution in a clay sample
dependent cation exchange complex
X Heavy metal transport in a medium with a pH-
and gibbsite (Al(OH)3)
X Transport with mineral dissolution of amorphous SiO2
subject to multiple cation exchange
X Transport of heavy metals (Zn2+, Pb2+, and Cd2+)
HP1 examples
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
Available chemical reactions:
PHREEQC [Parkhurst and Appelo, 1999]:
Variably Saturated Water Flow
Solute Transport
Heat transport
Root water uptake
HYDRUSimůnek et al., 1998]:
HYDRUS-1D [Šimů
HP1 - Coupled HYDRUS-1D and PHREEQC
27
30
historical fluxes (Scanlon et al., 2003)
X Flow in historical monuments (Ishizaki et al., 2001)
X Flow and transport around land mines (Das et al., 2001; Šimůnek et al., 2001)
X Analyses of Chloride profiles in deep vadose zones to evaluate
al., 2002)
X Multicomponent geochemical transport (Jacques and Šimůnek, 2002)
X Analyses of riparian systems (Whitaker, 2000)
X Fluid flow and chemical migration within the capillary fringe (Silliman et
and Šimůnek, 2002)
X Hill-slope analyses
X Transport of TCE and its degradation products (Scharlaekens et al., 2000; Casey
al., 2000)
X Virus and bacteria transport (Shijven and Šimůnek, 2002, Bradford et al., 2002a,b, Yates et
(Gribb et al., 1996; Kodešová et al., 1998, 1999; Šimůnek et al., 1997, 1998, 1999)
X Stream-aquifer interactions
X Environmental impact of the drawdown of shallow water tables
X Analysis of cone permeameter and tension infiltrometer experiments
NonNon-Agricultural:
HYDRUS - Existing Applications
Experimental Validation of Model
Validity of the processes embedded in a model
Mathematical Verification of Model (algorithm)
Accuracy of mathematical solution
Model Testing
Lake basin recharge analysis (Lee, 2000)
1993; Roth, 1995; Roth and Hammel, 1996; Kasteel et al. 1999; Hammel et al., 1999; Roth et al., 1999;
Vanderborght et al., 1998, 1999)
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,
et al., 1999, Scanlon et al., 2002)
Landfill covers with and without vegetation (Abbaspour et al, 1997; Albright, 1997; Gee
and Stormont, 1997; Kampf and Montenegro, 1997; Heiberger, 1998)
Evaluation of approximate analytical analysis of capillary barriers (Morris
1998)
Capillary barrier at Texas low-level radioactive waste disposal site (Scanlon,
1996; Wilson et al., 2000)
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.,
NonNon-Agricultural:
Agricultural:
Irrigation management (FREP, LINK , Bristow et al., 2002)
Drip irrigation design (FREP, LINK, Bristow et al., 2002)
Sprinkler irrigation design (FREP, LINK, Hansen et al., 2007, 2008)
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
...
28
31
Approximations arise because of
Incorporation of a limited number of processes
Limited understanding of the actual process
Inability to translate observed processes into usable mathematics
(how to quantify things?)
Inconsistency of small-scale heterogeneities with numerical grid
(effective parameters)
A model is a simplified representation of the real system or process
Does the model (i.e., the equations embedded in the
code) correctly represent the actual processes?
Model Validation
Comparison with other codes
Self-consistency (different grids and time steps)
Compare with analytical solutions (steady-state flow)
Compare with linearized solutions (simplified constitutive relationships)
Steady-state solutions
Homogeneous media
Simplified initial and boundary conditions
Verification of parts of the code:
Approximate Tests:
Mass balance errors
Does the computer code (model) provide an accurate
solution of the government PDE’s for different initial and
boundary conditions within the range of possible model
parameter values?
Mathematical Verification
Question: When all nonlinear, coupled, and/or transient
processes are introduced into the model, how do we know
that the computer code gives accurate numerical results?
Mathematical Verification
29
32
33
X HYDRUS
X PROFILE
- flow domain design
- finite element generator
- initial conditions and domain
properties
- water flow and solute transport
calculations
X POSITION - project manager
X HYDRUS1D - major module
HYDRUS-1D - Model Structure
Department of Environmental Sciences, University of California
Riverside, CA
Federal University of Rio de Janeiro, Rio de Janeiro, Brazil
PC-Progress, Ltd., Prague, Czech Republic
Jirka Šimůnek, Rien van Genuchten,
and Miroslav Šejna
The HYDRUS-1D Software for Simulating
One-Dimensional Variably-Saturated
Water Flow and Solute Transport
Data and Project Management
Data Pre-Processing
- Input parameter
- Transport domain design
- Finite element grid generator
- Initial and boundary conditions
Computations
Data Post-Processing
- Graphical output
- ASCII output
X Root Growth - logistic growth function
X Hysteresis - Scott et al. [1983], Kool and Parker [1988]
X Soil Hydraulic Properties
- van Genuchten [1980]
- Brooks and Corey [1964]
- modified van Genuchten type functions [Vogel and Cislerova, 1989]
- dual-porosity model of Durner [1994]
X Richards Equation - saturated-unsaturated water flow
- porous media:
- unsaturated
- partially saturated
- fully saturated
- sink term - water uptake by plant roots
- water stress
- salinity stress
X Water, Solute, and Heat Movement:
Movement
- one-dimensional porous media
HYDRUS-1D - Fortran Application
X
X
X
X
HYDRUS-1D - Functions
1
34
HYDRUS-1D – Major Module
X Scaling Procedure for Heterogeneous Soils
X Nonuniform Soils
X Heat Transport - convection-dispersion equation
- heat conduction
- convection
X Solute Transport - convection-dispersion equation
- liquid, solid, and gaseous phase
- nonlinear adsorption [Freundlich-Langmuir equations]
- nonequilibirum [two-site sorption model, mobile-immobile water]
- Henry’s Law
- convection and dispersion in liquid phase, diffusion in gaseous phase
- zero-order production in all three phases
- first-order degradation in all three phases
- chain reactions
- atmospheric conditions
- free drainage
- horizontal drains
X PostPost-processing unit
- simple x-y graphs for graphical presentation of soil hydraulic properties
and other output results
X PrePre-processing unit
- specification of all parameters needed to successfully run HYDRUS
- small catalog of soil hydraulic properties
- Rosetta – pedotransfer functions based on Neural Networks
application
X Determines which other optional modules are necessary for a particular
X Controls execution of the program
X Main program unit of the system
HYDRUS-1D - Major Module
- vertical direction
- horizontal direction
- generally inclined direction
X Flow and Transport:
- prescribed head and flux
- seepage face
- deep drainage
X Water Flow Boundary Conditions:
solution of the solute transport equation:
- upstream weighting
- artificial dispersion
- performance index
X Three Stabilizing Options to avoid oscillation in the numerical
2
35
Post-processing
Profile Information
Observation Nodes
HYDRUS-1D – Post-processing
Pre-processing
HYDRUS-1D - Major Module
Solute Transport Models
selected input and/or
output data
existing projects
X Locates
Opens
Copies
Deletes
Renames
-- desired projects or
X Manages data of
POSITION - Project Manager
Water Flow Models
HYDRUS-1D - Preprocessing
3
36
0.0
0.1
0.2
0.3
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
50
All Fluxes
100
200
Time [days]
150
Water Contents [-]
0.4
0.5
250
300
50
100
actBot
actRoot
actTop
potRoot
potTop
-300
-200
-100
0
100
200
50
100
Observation Nodes
-40
-20
-2500
-3000
0
20
40
60
200
200
Time [days]
150
Time [days]
150
Cumulative Fluxes
-2000
-1500
-1000
-500
0
HYDRUS-1D
properties of the flow domain
- material distribution
- scaling factors
X Observation Nodes
X Material Layers - parameters which describe the
X Root Uptake Distribution
contents, temperatures, and concentrations
X Initial Conditions for pressure heads, water
250
250
X Discretization of the soil profile into finite elements
PROFILE - Domain Design, Mesh
Generator, Domain Properties
300
300
PROFILE - Domain Design, Mesh Generator,
Domain Properties
4
Computer Session 1
Computer Session 1
The purpose of Computer Sessions 1, 2, and 3 is to give users hands-on experience
with the HYDRUS-1D software package (version 3.0). Three examples are given
to familiarize users with the major parts and modules of HYDRUS-1D (e.g., the
project manager, Profile and Graphics modules), and with the main concepts and
procedures of pre- and post-processing (e.g., domain design, finite element
discretization, initial and boundary conditions specification, and graphical display
of results).
The following three examples are considered in Computer Sessions 1, 2, and 3,
respectively:
I.
Direct Problem: Infiltration into a one-dimensional soil profile (Computer
Session 1)
A. Water flow
B. Solute transport
C. Possible additional modifications
II.
Direct Problem: Water flow and solute transport in a multilayered soil profile
(Computer Session 2)
III. Inverse Problem: One-step outflow method (Computer Session 3)
The first example represents the direct problem of infiltration into a 1-meter deep
loamy soil profile. The one-dimensional profile is discretized using 101 nodes.
Infiltration is run for one day. Ponded infiltration is initiated with a 1-cm constant
pressure head at the soil surface, while free drainage is used at the bottom of the
soil profile. The example is divided into three parts: (A) first, only water flow is
considered, after which (B) solute transport is added. Several other modifications
are suggested in part (C). These include (1) a longer simulation time, (2)
accounting for solute retardation, (3) using a two-layered soil profile, and (4)
implementing an alternative spatial discretization. Users in this example become
familiar with most dialog windows of the main module, and get an introduction
into using the external graphical Profile module with which one specifies initial
conditions, selects observation nodes, and so on.
37
Computer Session 1
A. Infiltration of Water into a One-Dimensional Soil
Profile
Project Manager
Button "New"
Name: Infiltr1
Description: Infiltration of water into soil profile
Button "OK"
Main Processes
Heading: Infiltration of water into soil profile
Button "Next"
Geometry Information
Button "Next"
Time Information
Final Time: 1
Initial Time Step: 0.0001
Minimum Time Step: 0.000001
Button "Next"
Print Information
Number of Print Times: 12
Button "Select Print Times"
Button "Next"
Water Flow - Iteration Criteria
Button "Next"
Water Flow - Soil Hydraulic Model
Button "Next"
Water Flow - Soil Hydraulic Parameters
Catalog of Soil Hydraulic Properties: Loam
Button "Next"
38
Computer Session 1
Water Flow - Boundary Conditions
Upper Boundary Condition: Constant Pressure Head
Lower Boundary Condition: Free Drainage
Button "Next"
Soil Profile - Graphical Editor
Menu: Conditions->Initial Conditions->Pressure Head
or Toolbar: red arrow
Button "Edit condition", select with Mouse the first node and specify 1
cm pressure head.
Menu: Conditions->Observation Points
Button "Insert", Insert nodes at 20, 40, 60, 80, and 100 cm
Menu: File->Save Data
Menu: File->Exit
Soil Profile - Summary
Button "Next"
Execute HYDRUS
OUTPUT:
Observation Points
Profile Information
Water Flow - Boundary Fluxes and Heads
Run Time Information
Mass Balance Information
Observatio n No des: Pressure H eads
Pro file Informatio n: Pressure H ead
20
0
0
-20
-20
N1
-40
N2
N3
-60
-60
N4
-80
-80
-40
N5
-100
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
-100
-100
Time [days]
-80
-60
-40
h [cm]
Close Project
39
-20
0
20
Computer Session 1
B. Infiltration of Water and Solute into
a One-Dimensional Soil Profile
Project Manager
Click on Infiltr1
Button "Copy"
New Name: Infiltr2
Description: Infiltration of Water and Solute into Soil Profile
Button "OK", "Open"
Main Processes
Check "Solute Transport"
Button "OK"
Solute Transport - General Information
Button "Next"
Solute Transport - Transport Parameters
Disp. = 1 cm
Button "Next"
Solute Transport - Reaction Parameters
Button "Next"
Upper Boundary Condition: 1
Lower Boundary Condition: Zero Gradient
Button "Next"
Execute HYDRUS
OUTPUT:
Observation Points
Profile Information
Solute Transport - Boundary Actual and Cumulative Fluxes
40
Computer Session 1
C. Possible Modifications
1. Longer simulation time:
Project Manager
Click on Infiltr2
Button "Copy"
New Name: Infiltr3
Button "OK", "Open"
Time Information:
Final Time: 2.5 d
Print Information
Button "Default"
Button "Next"
2. Retardation:
Kd = 0.5
3. Two Soil Horizons:
Number of Soil Materials: 2
1. line - Silt
Kd = 0
Button "Edit condition", select with Mouse the lower 50 cm and specify
Material 2.
Menu: File->Exit
41
Computer Session 1
4. Different Spatial Discretization:
Menu: Conditions->Profile Discretization
or Toolbar: ladder
Button "Insert Fixed", at 50 cm
Button "Density", at 50 cm 0.5, at the soil surface 0.3
Button "Edit condition", select with Mouse the first node and specify 1
cm pressure head.
Button "Insert", Insert nodes at 20, 40, 60, 80, and 100 cm
Menu: File->Exit
Observation N odes: W ater C ontent
Pro file Information: Pressure Head
0.50
0
0.45
-20
0.40
-40
N1
0.35
N2
-60
0.30
N3
-80
0.25
N4
-100
-100
N5
0.20
-80
-60
-40
-20
0
0.0
20
0.5
1.0
1.5
Time [days]
h [cm]
42
2.0
2.5
43
2Department
1Department
Rien van Genuchten1 and Jirka Šimůnek2
On the Characterization and Measurement
of the Hydraulic Properties of
Unsaturated Porous Media
- volumetric water content [L3L-3]
- pressure head [L]
- unsaturated hydraulic conductivity [LT-1]
- vertical coordinate positive upward [L]
- time [T]
- root water uptake [T-1]
Hydraulic Conductivity Function, K(θ)
Hydraulic Conductivity Function, K(h) or K(θ)
h
K
z
t
S
θ
∂θ (h) ∂ ⎡
∂h
⎤
= ⎢ K(θ ) +K(θ )⎥
∂t
∂z ⎣
∂z
⎦
Richards Equation for Variably-Saturated Flow
1
44
(Kool et al., 1985; Šimůnek and van Genuchten, 1996)
Parameter Optimization Methods

Horizontal Infiltration (Bruce and Klute, 1956)
Sorptivity Methods (Dirksen, 1975)
One-Step/Multi-Step Outflow Method (Passioura, 1975)
Hot-Air Method (Arya et al., 1975)
Evaporation Method (Boels et al., 1978)
Direct Transient Methods

Using Darcy’s law: q=-K(h) (dh/dz-1)
Long-Column Method
Centrifuge Methods
Laboratory Methods
SteadySteady-State Methods
Unsaturated Hydraulic Conductivity
Soil Water Hysteresis
Tempe Pressure Cell
X Ongoing/Future Research
X Structured Media
X The UNSODA Database
X The Rosetta PTF code
X Pedotransfer Functions
X Statistical Pore-Size Distribution Models
X Direct Measurements of Hydraulic Properties
Outline
2
45
The supply
pressure head
hwet = h2-h1<0
Schematic of Tension Infiltrometer
Pete Shouse’s
Tempe Cell Setup at US Salinity Laboratory
- Water retention data (Pore
Pore--Size Distribution Models)
Models
(Brooks & Corey, van Genuchten, Kosugi, Vogel)
- Pore-scale network models
(Celia & Reeves, Tuller and Or)
- Particle-size distribution (shape similarity)
(Arya & Paris, Haverkamp, Rajkai et. al.)
- Pedotransfer functions
Estimate K(θ) from more easily measured data:
Alternative: Indirect Methods
Direct measurements of the hydraulic conductivity is timeconsuming, costly, and generally very approximate
- Extreme nonlinearity of K(θ)
- Soil heterogeneity
- Instrumental limitations
Direct Measurement Methods
Soil Hydraulic Properties, θ(h) and K(θ)

Russo et al. (1991)
Abbaspour et al. (1996)
Šimůnek et al. (1998, 2000)
...
Parameter Optimization Methods

Instantaneous Profile Methods (Watson, 1966)
Unit-Gradient Methods (Sisson et al., 1980)
Plane-of-Zero-Flux Method
Sorptivity Methods (Clothier and White, 1981)
Constant Head Permeameters (Reynolds et al., 1983)
Tension Infiltrometers (White and Perroux, 1988)
Direct Methods
Field Methods
Unsaturated Hydraulic Conductivity
3
46
λ
VG vs BC Retention Functions
⎛h ⎞
θ(h) =θr + (θs −θr )⎜ b ⎟
⎝ h⎠
Brooks and Corey (1964)
θ (h) = θr + θ s θ rn m
[1 + |α h| ]
VG vs BC Retention Functions
VG
BC
van Genuchten (1980)
4
47
- Fatt and Dykstra
- Childs and Collis-George
- characterizes the energy status of the soil water
Retention Curve
Specific models:
- Purcell
- Burdine
- Mualem
Approach and Assumptions:
- Soil consists of bundle of cylindrical pores, with certain psd
- Pores are either full or empty, depending on pore radius, r
- Use law of capillarity: h ≈ r-1
- Apply Poiseuille’s law to each individual pore: qi ≈ r4
- Integrate over all pore sizes
- Equate to Darcy’s law for total system
Statistical Pore-Size Distribution Models
2σ cos γ
ρ gR
π R 4 ΔP
8 Lν
A tube of radius 2R will thus have 16 times as much water flowing
through it per unit time as a tube of radius R.
For a given hydrostatic pressure difference Δ P across a length L of
cylindrical capillary, the volume of water flowing per unit time Q
will be proportional to the fourth power of the radius.
Poiseuille's law:
Q=
Water Flow in Capillary Tubes
g – gravitational acceleration
R – capillary radius
H – capillary rise
σ – surface tension
γ – contact angle
ρ – bulk density of water
H=
Laplace Equation:
When a small cylindrical glass capillary tube is inserted in a water
reservoir open to atmosphere, water will rise upward in the tube.
Capillary Rise
5
48
l
e
0
∫
Se
⎡1 + α h ⎤
⎣
⎦
n m
θs − θr
⎡ 1 dx ⎤
⎢∫
⎥
⎣ 0 h( x ) ⎦
Fitted θ(h) and Predicted Kr(h)
2
θ (h ) − θ r
θs − θr
2
K r ( Se ) = Sel ⎡⎣1 − (1 − Se1/ m ) ⎤⎦
into Mualem’s model gives for m=1-1/n
θ (h) = θ r +
⎤
⎥
⎥⎦
Se ( h ) =
dx
h( x )
Substituting van Genuchten’
Genuchten’s (1980) equation
where Se is relative saturation:
⎡
K ( Se ) = K s S ⎢
⎢⎣
Mualem’
Mualem’s (1976) model:
van Genuchten-Mualem Approach
2
θs - θr
n m
[1 + |α h| ]
m = 1 − 1/ n
θ − θr
θs − θr
Unknown Soil Hydraulic Parameters: θr , θs , α, n, l, Ks
Se =
K r ( Se ) = Sel [1 − (1 − Se1 / m )]2
θ ( h) = θ r +
VG Functions for Soil Hydraulic Properties
6
49
Problem for fine
textured soils
van Genuchten-Mualem Model
Hysteresis
Vogel et al. (1988, 2001) – introduced small hs (- 2 cm)
Modified VGM model
7
50
VG
VGM
Effect on Infiltration Calculations (n=1.09)
hs=-2 cm
Modified VGM model
θ (h) − θ r
=
θ s −θr
K ( Se ) = K s SeA+2+ 2 / λ
S e ( h) =
(α h > 1)
(α h ≤ 1)
2
Se
- effective water content
θr, θs - residual and saturated water contents
α, n, m (= 1 - 1/n), l and λ - empirical parameters
- saturated hydraulic conductivity
Ks
(α h) − λ
1
Brooks and Corey (1964):
m
K ( Se ) = K s Sel ⎡⎢1 − (1 - Se1/ m ) ⎤⎥
⎣
⎦
Se (h ) = [1 + (α h )n ]-m
van Genuchten (1980):
Soil Hydraulic Property Models
Modified VGM model for Beit Netofa Clay
8
- residual and saturated water contents
- empirical parameters
i =1
i
∑w
k
1
k
The hydraulic characteristics contain 4+2k unknown parameters: θr , θs , αi , ni , l, and Ks.
Of these, θr, θs, and Ks have a clear physical meaning, whereas αi, ni and l are essentially
empirical parameters determining the shape of the retention and hydraulic conductivity
functions [van Genuchten, 1980].
k
- number of overlapping subregions
- weighting factors for the sub-curves
wi
αi, ni, mi (= 1 - 1/ni), and l - empirical parameters of the sub-curves.
⎤⎞
⎦ ⎟⎠
2
ni mi
(1+ αi h )
2
⎛
1/ mi mi
⎡
⎜ ∑ wiα i ⎣1- (1- S ei )
⎛ k
l ⎞ ⎝ i =1
K (θ ) = K s ⎜ ∑ wi S ei ⎟
2
⎝ i =1
⎠
⎛ k
⎞
⎜ ∑ wiα i ⎟
⎝ i =1
⎠
θ (h) - θ r
Se ( h ) =
=
θs - θr
θr , θs - residual and saturated water contents, respectively
Se
Durner (1994):
Se
θr, θs
h0, σ, and l
Ks
⎡ ln ( h / h0 )
⎤ ⎪⎫
⎪⎧ 1
K (h) = K s Sel ⎨ erfc ⎢
+ σ ⎥⎬
2σ
⎪⎩ 2
⎣
⎦ ⎪⎭
⎧ ln ( h / h0 ) ⎫
θ (h) − θ r 1
Se ( h ) =
= erfc ⎨
⎬
2
θs − θr
2σ ⎭
⎩
10-2
10-7
10-6
10-5
10-4
10-3
10-2
10-1
10-1
100
101
102
Soil Water Pressure Head (-mm )
103
-1
0
2
3
Log(|Pressure Head [cm]|)
1
4
Fracture
Matrix
Total
5
-10
-8
-6
-4
-2
0
-1
0
1
2
3
104
4
Fracture
Matrix
Total
5
Example of composite retention (left) and hydraulic conductivity (right)
functions (θr=0.00, θs=0.50, α1=0.01 cm-1, n1=1.50, l=0.5, Ks=1 cm d-1, w1=0.975,
w2=0.025, α2=1.00 cm-1, n2=5.00).
0
0.1
0.2
0.3
0.4
0.5
0.6
Durner (1994):
Wate r Conte nt [-]
Lognormal Distribution Model (Kosugi, 1996):
Observed Bimodal Hydraulic Conductivity
Hydraulic Conductivity (m m/sec)
Log(C onductivity [cm/days])
51
9
52
Two Approaches:
- Predict specific retention values
- Predict soil hydraulic parameters
X Chemical Properties (EC, pH, SAR, …)
X Clay Mineralogy
X Soil Structure
X Organic Matter Content
X Porosity
X Bulk Density
X Soil Texture (class or particle-size distribution)
Predict the hydraulic properties from
more easily measured data:
Pedotransfer Functions
Average values of selected soil water retention parameters for 12 major
soil textural groups
10
53
Input data
Hierarchica
l Models
Predicted
parameters +
uncertainties
hydraulic conductivity
X Prediction of: water retention, Ks , and the unsaturated
predictions
X Bootstrap: generate confidence intervals of the
predictions
X Neural networks: to provide the most accurate
SSCBD + θ at 33 kPa
Same + θ at 1500 kPa
SSCBD + θ33
SSCBD + θ33 + θ1500
Sand, Silt, Clay %
SSC
Same + Bulk Density
Textural Class
TXT
SSCBD
Input Data
Model
Schaap et al. (2001)
Rosetta (Schaap et al., 2001)
X Hierarchy: try to match various levels of data availability
Hierarchical Neural-Network Bootstrap Approach
11
54
SSC:
BD:
θ33, θ1500
θr
-
0.066
0.086
0.094
0.121
0.387
θs
-
0.143
0.178
0.581
0.605
0.600
-
0.203
0.238
0.265
0.417
0.577
-
0.452
0.473
0.495
0.599
0.760
3
0.012
0.072
0.070
0.060
0.041
0.039
3
RMSEw
Log α Log n cm /cm
Water retention
R2
Sand, silt, clay percentages
Bulk density
Water content at 33 and 1500 kPa
Textural Class
SSC
SSCBD
SSCBDθ33
SSCBDθ33θ1500
H1
H2
H3
H4
H5
Direct fit to data
Input
Model
Rosetta’s Performance
-
0.427
0.461
0.535
0.640
0.647
-
0.739
0.717
0.666
0.586
0.581
Saturated Conductivity
R2
RMSEs
(-)
Log Ks
40
80
60
40
Sand [%]
60
L
0
20
20
40
60
0
80
100
Silt [%]
0
100
S
20
lS
Clay [%]
40
80
scL
sC
60
sL
80
C
L
cL
40
0
Sand [%]
60
100
60
sicL
40
20
siL
siC
20
Si
0
80
100
Silt [%]
Unsaturated Conductivity
(N=235)
Rosetta’s Class-Average PTFs
0
100
20
Clay [%]
80
100
Retention
(N=2134)
Rosetta’s Calibration Data
12
55
2
Drip Irrigation (Skaggs et al., 2004)
m
K = K s SeL ⎡⎢1 − (1 − Se1/ m ) ⎤⎥
⎣
⎦
Pore-Connectivity Parameter L
http://www.ussl.ars.usda.gov/models.htm
Drip Irrigation (Skaggs et al., 2004)
• MS-ACCESS
• Flexible Queries
• Graphics support
• Downloadable
UNSODA 2 Unsaturated Soil Hydraulic Database
13
0.33
0.021
0.023
α
1.4
n
1.3
Ks (cm hr-1)
-0.93
l
-70
-60
-50
-40
-30
-20
-10
0
0
10
20
10
20
Predicted
30
40
DISTANCE (cm)
50
60
40
50
60 0
10
20
Predicted
30
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
40
DISTANCE (cm)
Volumetric Water Content
Observed
30
-70
-60
-50
-40
-30
-20
-10
0
0
10
20
Observed
30
40
DISTANCE (cm)
50
60 0
10
20
Predicted
30
40
DISTANCE (cm)
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
60 0
20
DISTANCE (cm)
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
50
10
40
0
50
50
Trial 2: 10 hour irrigation, 40 L/m applied water
Time = 10.75 hr
-70
-60
-50
-40
-30
-20
-10
0
Time = 5.5 hr
Observed
30
DISTANCE (cm)
Time = 28 hr
θs
θr
Hydraulic properties
estimated using Rosetta
pedotransfer function
(sand, silt, and clay, bulk
density, 1/3 and 15 bar
water content)
DEPTH (cm)
DEPTH (cm)
HYDRUS
DEPTH (cm)
56
60
60
14
-70
-60
-50
-40
-30
-20
-10
0
0
10
20
10
20
Predicted
30
40
-70
-60
-50
-40
-30
-20
-10
0
0
10
40
50
60 0
10
20
Predicted
30
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
40
DISTANCE (cm)
Observed
30
DISTANCE (cm)
20
50
50
60
60
-70
-60
-50
-40
-30
-20
-10
10
20
Observed
30
40
DISTANCE (cm)
50
60 0
10
20
Predicted
30
0
0
0.1
0.2
0.3
0
-20 -30
DEPTH (cm)
-10
-20 -30
DEPTH (cm)
DISTANCE=10
-10
-40
-40
-50
0
0.1
0.2
0.3
-50
0.1
0.1
0
0.2
0.2
0
0.3
0.3
DISTANCE = 0
0
0
10
20
30
DISTANCE (cm)
DEPTH=-20
10
20
30
DISTANCE (cm)
DEPTH=-10
40
40
Time = 5.5 hr
40
DISTANCE (cm)
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
60 0
0
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
50
0
40
DISTANCE (cm)
Time = 16 hr
Observed
30
DISTANCE (cm)
Time = 39 hr
DEPTH (cm)
DEPTH (cm)
DEPTH (cm)
Time = 31 hr
WATER C ON TEN T
WATER C ON TEN T
57
50
60
15
58
-40
0
-10
-20 -30
DEPTH (cm)
-40
-50
0
0
0
10
20
30
DISTANCE (cm)
DEPTH=-20
10
20
30
DISTANCE (cm)
DEPTH=-10
40
40
X
X
X
X
X
X
X
X
X
X
X
Hysteresis
Dry end effects; residual saturation, θr
Dynamic effects; non-equilibrium flow
Air entrapment (θs versus porosity)
Swelling soils; effects of chemistry
Description near saturation
Second order continuity in θ(h) (n → 1.0)
Structured media; preferential flow
Scale Issues (upscaling; effective properties)
Required Accuracy (flux vs profile controlled inf.)
...
Hydraulic Properties - Challenges
0
0.1
0.1
DISTANCE=10
0.2
0
0.2
-50
0.3
-20 -30
DEPTH (cm)
0.3
-10
0.1
0.1
0
0.2
0.2
0
0.3
0.3
DISTANCE = 0
Time = 28 hr
WATER C ON TEN T
WATER C ON TEN T
Continue data mining (UNSODA)
Laboratory versus field data
Effects of chemistry and clay mineralogy
- NRCS soil characterization database
Effects of soil structure
Generic versus site-specific PTF’s
…

Future Plans – Rosetta
16
59
where:
Δt
⎜K
Δ x⎝
θ ij +1,k +1 - θ ij = 1 ⎛
xi +1 - xi −1
Δ xi = xi +1 - xi
Δ xi −1 = xi - xi −1
2
j +1,k
j +1,k
j +1,k
+ K i-j +11,k
K i +1 + K i
Ki
j+1,k
j +1,k
K i+1/ 2 =
K i-1/ 2 =
2
2
Δ t = t j +1 - t j
j +1, k +1
- h ij +1,k +1
− h ij-1+1,k +1 ⎞ K ij++1/1,2k - K ij-1/+1,2k
h
- K ij-1/+1,2k i
- S ij
⎟+
Δ xi
Δ xi −1
Δx
⎠
j +1, k +1
h i +1
Δ x=
j +1, k
i +1/ 2
Final finite difference scheme:
∂θ ∂ ⎡ ⎛ ∂h
∂q
⎞⎤
=
K ⎜ +cos α ⎟⎥ - S = − - S
∂t ∂x ⎢⎣ ⎝ ∂x
∂x
⎠⎦
Richards Equation:
Application of Finite Element Method
to 1D Variably-Saturated Flow
2Department
1Department
Application of Finite Element Method to
1D Variably-Saturated Water Flow
and Solute Transport
∂h
−K
∂x
− hi
Δxi
j +1
− K i +j +11/ 2
xi-1
xi
xi+1
hi −j 1
hi
j
hi +j 1
Δt
hi j−+11
hi
j +1
hi j++11
ti+1
Δx
has the form:
The symmetrical
tridiagonal matrix [Pw]
[ Pw ] j +1,k{h} j +1,k +1 = {Fw }
[Pw ]=
e2
0
0
0
0
d2
1
e1
1
Matrix1D
Form:
Variably-Saturated Flow
d
e
0
d3
e2
0
.
.
0
eN −3
e3
0
.
.
0
0
eN − 2
d N −2
.
.
eN −1
d N −1
eN − 2
Application of Finite Element Method to 1D
Variably-Saturated Flow
h
j +1
i +1
q j +1 − q j +1
= − i +1/ 2 i −1/ 2 - Si j
Δx
qij++11/ 2 = − K i +j +11/ 2
q = −K
Δt
θi j +1,k +1 - θi j
ti
dN
eN −1
0
0
0
0
Δxi-1
Δxi
∂θ
∂ ⎡ ⎛ ∂h
∂q
⎞⎤
= ⎢ K ⎜ + cos α ⎟ ⎥ - S = − - S
∂t ∂x ⎣ ⎝ ∂x
∂x
⎠⎦
Final finite difference scheme:
Richards Equation:
1
60
Δt
θ i j +1,k +1 − θ i j +1,k + θ i j +1,k - θ i j
j +1, k
i
hi
+
θ i j +1,k - θ i j
Δt
- h ij+1,k θ i j +1,k - θ i j
+
Δt
Δt
j+1,k +1
Δt
θ i j +1,k +1 − θ i j +1,k
=C
=
=
=
Δt
θ Nj +1,k +1 - θ Nj
=N −1
j +1,k
2 ( q Nj +1 - q N1/ 2 )
- S Nj
Δx
KN
j +1,k
2Δt
Δ x N −1
j +1,k
+ K N1
2 Δ x N −1
j +1,k
KN
Δx
( θ j +1,k - θ i j ) 2Δt N
Δ x N −1 j
j +1
S N - qN
2
-
+
j +1,k
hN
j +1,k
+ K N1
2
j +1,k
CN
2Δt
j +1,k
CN
qN is the prescribed soil surface boundary flux
fN =
dN =
Δ x N −1
Expanding the time derivative on the left hand [Celia et al., 1990], and
using the discretized form of Darcy's law for qN-1/2 leads to:
∂θ
∂q
=- -S
∂t
∂x
The mass balance equation instead of Darcy's law is discretized.
Discretization gives:
Implementation of the Upper Flux Boundary Condition:
Δt
θ i j +1,k +1 - θ i j
The massmass-conservative method proposed by Celia et al. [1990], in which
θj+1,k+1 is expanded in a truncated Taylor series with respect to h about
the expansion point hj+1,k, is used in the time difference scheme:
j+1
⎛ h j+1 - h Nj+-11
⎞ Δ x N −1
j+1
+ 1 ⎟⎟ q N = - K Nj+-11/ 2 ⎜⎜ N
2
⎝ Δ x N −1
⎠
⎛ θ Nj+1 - θ Nj
⎞
+ S Nj ⎟⎟
⎜⎜
Δt
⎝
⎠
⎛ j+1 - j+1
⎞
⎛ j+1 - hij-+11
⎞
+ 1 ⎟ Δ xi
- K ij++11/ 2 ⎜ hi+1 hi + 1 ⎟ Δ xi -1 - K ij-+11/ 2 ⎜ hi
Δ
Δ
xi
xi -1
⎝
⎠
⎝
⎠
=
Δ xi −1 + Δ xi
⎛ h j+1 - h1j+1
⎞
j+1
+ 1 ⎟⎟
q1 = - K 1j++11/ 2 ⎜⎜ 2
Δ xi
⎝
⎠
Computation of Nodal Fluxes:
qi
i
+ K i+j +11,k
2Δ x
j +1,k
Ki
Δ x j +1,k j +1,k Δ x j +1,k
K j +1,k - K ij−+11,k
(θ
- θ j ) + i+1
- S ij Δ x
Ci
hi
i
i
2
Δt
Δt
ei = -
j ,k
j ,k
Δ x j +1,k
K +1 + K i +1
K j +1,k + K i-j 1+1,k
+ i+1
+ i
Ci
Δt
2 Δ xi
2 Δ xi −1
fi =
di =
The diagonal entries di and above-diagonal entries ei of the matrix [Pw],
and the entries fi of vector {Fw}, are given by:
2
61
no k=k+1 goto 2
yes continue
1
1
c
i-1
i-1
ci-1
i
i
φ1
ci
Δxi
i+1
φ2
i+1
ci+1
x
x
Basics of Finite Element Method
5) j=j+1, k=1, h j+1,1 = h j,k+1 goto 2
4) Tolerance Criteria:
abs(h j+1,k+1- h j+1,k)<Tol_h
abs(θ j+1,k+1- θ j+1,k)<Tol_θ
3) Gaussian elimination - h j+1,k+1, q j+1,k+1
2) Derivation of the system of linearized algebraic equations
using h j+1,k, q j+1,k, K j+1,k, C j+1,k
1) First time step: h j+1,1 = hinit, j=1, k=1
j - time step
k - iteration
Iterative Process: Picard linearization
⎡ ∂θ Rc ∂ ⎛ ∂c
⎤
⎞
+ ⎜ E - Bc ⎟ + Fc +G ⎥ φn dx = 0
∂t
∂x ⎝ ∂x
⎠
⎦
L
⎛ ∂c
⎞ ∂φ
− ∫ ⎜ E - Bc ⎟ n dx - qsLφn ( L) + qs 0φn (0) = 0
∂x
⎠ ∂x
0⎝
qs0 and qsL are solute fluxes across the lower and upper boundaries
∫ ⎢⎣-
L
⎡ ∂θ Rc
⎤
+ Fc +G ⎥ φn dx t
∂
⎦
0
Integrating by parts the terms containing spatial derivatives
0
∫ ⎢⎣-
L
Galerkin method:
to 1D Solute Transport
Discretization 1:
1) Starts with a prescribed initial time increment, Δt, and is automatically adjusted at
each time level
2) Time increments cannot become less than a preselected minimum time step, Δtmin,
nor exceed a maximum time step, Δtmax
3) If the number of iterations necessary to reach convergence is <3, the time increment
for the next time step is increased by multiplying Δt by a predetermined constant >1
(usually between 1.1 and 1.5). If the number of iterations is >7, Δt for the next time
level is multiplied by a constant <1 (usually between 0.3 and 0.9).
4) If the number of iterations at any time level becomes greater than a prescribed
maximum (usually between 10 and 50), the iterative process for that time level is
terminated. The time step is subsequently reset to Δt/3, and the iterative process
restarted.
Three different time discretizations:
1) associated with the numerical solution
2) associated with the implementation of boundary conditions
3) which provide printed output of the simulation results
Time Control:
3
62
∂
∂c
u' =
0
L
dx - ∫
∂φ n
∂x
u = φn
n
a
a
b
b
∂c ∂qc
∂
θD −
∂x ∂x
∂x
∂c
v = θD − qc
∂x
v' =
2
2
0
a
n
1
φ ma dx = L
2
a!b!
(a + b + 1)!
2
3
1
Δx
12
1
2
1
3θ1 R1 + θ 2 R2
θ1 R1 + θ 2 R2
2
θ1 R1 + θ 2 R2
θ1 R1 + 3θ 2 R2
Δx
(θR )1 + Δx (θR )2
= (θR )1 ∫ φ dx + (θR )2 ∫ φ φ 2 dx =
4
12
1
1
2
2
Δx
Δx
2 2
2
= (θR )1 ∫ φ1 φ 2 dx + (θR )2 ∫ φ1 φ 2 dx = (θR )1 + (θR )2
12
12
1
1
2
2
Δx
(θR )1 + Δx (θR )2
= (θR )1 ∫ φ1 φ 22 dx + (θR )2 ∫ φ 23 dx =
12
4
1
1
1
Qnm = ∫θR φ n φ m dx = (θR )l ∫ φ l φ n φ m dx
e
L
∫φ
e
Qnm
= ∫θ Rφmφn dx =
n = 2, m = 2
n = 1, m = 2
n = 1, m = 1
Expansion:
b
∂qc
∂c
∂c
−
− qcφ n
φ n dx = θD φ n
∂x
∂x a
∂x a
L
L
∂φ n
∂c ∂ φ n
∫0 θD ∂x ∂x dx − ∫0 qc ∂x dx
b
∫ uv' = uv a − ∫ u ' v
b
0
∫ ∂x θD ∂x φ
L
Solute transport:
Integration per partes:
partes:
⎛
m
∂φm
⎞ ∂φ
- Bcmφm ⎟ n dx - qsLφn ( L) + qs 0φn (0) = 0
∂x
⎠ ∂x
2
1
2
3θ1 R1 + θ 2 R2
θ1 R1 + θ 2 R2
1
2
f ne = ∫Gφn dx =
Δx
6
G1 + 2G2
2G1 + G2
F1 + 3F3
F1 + F3
B1 + 2 B2
- B1 - 2 B3
2 B1 + B2
E1 + E2
- E1 - E2
θ1 R1 + θ 2 R2
θ1 R1 + 3θ 2 R2
- 2 B1 - B2
Δx 3F1 + F2
12 F1 + F2
d φn
1
φm =
dx
6
e
Snm
3 = ∫Fφmφn dx =
1
Δx
12
E1 + E2
d φm d φ n
1
dx =
dx dx
2Δ x - E1 - E2
e
S nm
2 = ∫B
1
2
e
Snm
1 = ∫E
1
2
e
Qnm
= ∫θ Rφmφn dx =
Contribution from particular elements:
d ([Q1] {c})
d {c}
+ [Q 2]
+ [ S ]{c} = { f }
dt
dt
In matrix forms:
e 0
c( x, t ) = cm (t )φm ( x )
⎡ ∂θ R1cm
⎤
∂c
φm - θ R 2 m φm - Fcmφm + G ⎥ φn dx t
t
∂
∂
⎣
⎦
∑ ∫ ⎜⎝ Ec
Le
e 0
Le
∑ ∫ ⎢-
Substituting c':
4
63
where
(
(
)
)
1
1
2
[Q ] j+1 + [Q ] j+ε + ε [ S ] j+1
Δt
1
[T ] =
[Q1 ] j + [Q2 ] j+ε - ( 1 - ε ) [ S ] j
Δt
j+
j
{R } = ε { f } 1 + ( 1 - ε ) { f }
[ Ps ] =
[ P s ] {c } j+1 = [T ]{c } j + {R}
j and j+1 indicate previous and actual time level and Δt is time step.
Can be rewritten:
ε [ S ] j+1{c } j+1 + (1 - ε )[ S ] j {c } j = ε { f } j+1 + (1 - ε ){ f } j
[Q1 ] j+1{c } j+1 - [Q1 ] j {c } j
{c } j+1 - {c } j
+
+ [Q2 ] j + ε
Δt
Δt
The time derivatives are discretized by means of finite differences:
to 1D Solute Transport - Time Discretization:
5
64
Computer Session 2
Computer Session 2
I. Water Flow and Solute Transport in a Layered Soil Profile
Diederik Jacques, Jirka Šimůnek, and Rien van Genuchten
In this computer session with HYDRUS-1D we consider water flow and the transport of tracers
and adsorbing chemicals through a Podzol soil profile. The computer session is divided into two
parts. In the first part (II) only water flow is simulated, while in the second part (I) solute
transport is additionally considered. Examples in the first part of this computer session involve
both steady-state and transient variably saturated flow in a 1-m deep multi-layered soil profile.
Transient flow is induced by atmospheric boundary conditions. No root water uptake is
considered, thus restricting the atmospheric boundary conditions to daily values of precipitation
and evaporation. The example is divided into two parts:
A. Initial conditions for transient water flow example
B. Transient water flow (atmospheric conditions) example
Soil hydraulic and physical parameters (Table 1) of the dry Spodosol located at the “Kattenbos”
site near Lommel, Belgium were taken from Seuntjens (2000, Tables 3.1 and 7.1).
Table 1 Soil hydraulic and other properties of six soil horizons.
Horizon
Depth
(cm)
A
E
Bh1
Bh2
BC
C1
C2
0–7
7 – 19
19 – 24
24 – 28
28 – 50
50 – 75
75 – 100
ρ
(g cm ³)
1.31
1.59
1.3
1.38
1.41
1.52
1.56
Organic
Carbon
(%)
2.75
0.75
4.92
3.77
0.89
0.12
0.08
θr
0.065
0.035
0.042
0.044
0.039
0.030
0.021
θs
0.48
0.42
0.47
0.46
0.46
0.42
0.39
Α
-1
(cm )
0.016
0.015
0.016
0.028
0.023
0.021
0.021
n
(-)
1.94
3.21
1.52
2.01
2.99
2.99
2.99
Ks
-1
(cm d )
95.04
311.04
38.88
864
1209.6
1209.6
1209.6
Part A:
The steady-state flow example corresponds with experimental conditions in a lysimeter
experiment described in Seuntjens (2000). The initial condition is defined assuming a constant
flux of 0.12 cm day-1 and a free-drainage lower boundary condition. The flux corresponds to the
long term (1972-1981) actual infiltration rate (precipitation - actual evapotranspiration).
Part B:
The upper boundary condition now involves daily precipitation and evaporation fluxes defined
using meteorological data from the Brogel station weather (Belgium) for 1972. Some input data
are summarized in the “HYDRUS-Course-Data.xls“ file.
Reference
Seuntjens, P., 2000. Reactive solute transport in heterogeneous porous media. Cadmium
leaching in acid sandy soil. PhD, University of Antwerp, 236 p.
65
Computer Session 2
A. Steady-State Water Flow in a Layered Soil Profile
Project Manager
Button "New"
Name: LSP-W1
Description: Steady-State Water Flow (q=0.12 cm/d) in a Layered Soil
Profile
Button "OK"
Button "Open"
Main Processes
Heading: Calculate steady-state conditions
Button "Next"
Length Units: cm
Depth of the Soil Profile: 100 cm
Button "Next"
Time Information
Time Units: Days
Final Time: 100
Maximum Time Step: 0.5
Button "Next"
Print Information
Button "OK"
Button "Next"
Water Flow – Iteration Criteria
Button "Next"
Water Flow – Soil Hydraulic Model
Button "Next"
66
Computer Session 2
Water Flow – Soil Hydraulic Parameters
Copy the soil hydraulic parameters from the Excel file (units are cm and
day)
Button "Next"
Water Flow – Boundary Conditions
Upper Boundary Condition: Constant Flux
Button "Next"
Water Flow – Constant Boundary Fluxes
Upper Boundary Flux: -0.12 cm/day
Button "Next"
HYDRUS-1D Guide:
Button "OK"
Profile Information – Graphical Editor
Button "Edit condition"
Select with the Mouse: nodes from 8 to 19 cm; specify Material 2
nodes from 20 to 24 cm; specify Material 3
Specify initial pressure head of –100 cm
Include observation points at 50 and 100 cm
Save and Exit
Execute HYDRUS-1D
OUTPUT:
Observation Points
Profile Information
67
Computer Session 2
B. Transient Water Flow in a Layered Soil Profile
Project Manager
Select the LSP-W1 project
Button "Copy”
Name: LSP-W2
Description: Transient Water Flow in a Layered Soil Profile
Button "OK"
Button "Open"
Main Processes
Heading: Transient Water Flow in a Layered Soil Profile
Button “OK"
Time Information
Final Time: 360
Check Time-Variable Boundary Conditions
Number of Time-Variable Boundary Records: 360
Button "Next"
Print Information
Button "Default"
Button "OK"
Upper Boundary Condition: Atmospheric BC with Surface Run Off
Button "Next"
Variable Boundary Conditions
Open the Excel file with meteorological variables
Select and Copy Atmospheric Boundary Conditions
Paste copied values
Button "OK"
68
Computer Session 2
Soil Profile Summary
Open the NOD_INF.OUT file from the project LSP-W1.h1d using MS
Excel
Select and copy the last pressure head profile
Paste pressure head profile in h column to define initial conditions.
Execute HYDRUS-1D
OUTPUT:
Observation Points
Profile Information
Water Flow – Boundary Fluxes and Heads
Close Project
Observation Nodes: Pressure Heads
Actual Surface Flux
-120
1.0
0.5
-140
0.0
-160
-0.5
-180
-1.0
-200
-1.5
-220
-2.0
-240
-2.5
-260
-3.0
0
50
100
150
200
250
300
350
0
400
50
100
150
200
250
Time [days]
Time [days]
Profile Information: W ater C ontent
0
-20
-40
-60
-80
-100
0.00
0.05
0.10
0.15
0.20
Theta [-]
69
0.25
0.30
0.35
300
350
400
Computer Session 2
Computer Session 4
II. Solute Transport in a Layered Soil Profile
Diederik Jacques, Jirka Šimůnek, and Rien van Genuchten
In the second part of the Computer Session 4 we will use the project “LSP-W2” created in the
first part of the Computer Session 4 and assume that there is a spill of a chemical on the first day
of simulation at the soil surface. The example is divided into three parts, each of increasing
complexity:
A. Tracer Transport
B. Reactive Chemical Transport
C. Transport of PCE and its Daughter Product
For the first run we assume that a nonreactive chemical is spilled on the soil surface. The
second and third runs consider the transport of a reactive chemical and that of PCE and its
degradation products, respectively. PCE degrades to sequentially form trichloroethylene (TCE),
cis-1,2-dichloroethylene (cis-DCE), trans-1,2-dichloroethylene (trans-DCE), 1,1dichloroethylene (1,1-DCE), vinyl chloride (VC) (after Schaerlaekens et al., 1999). VC
eventually degrades to ethylene (ETH) which is environmentally acceptable and does not cause
direct health effects. HYDRUS-1D can not consider diverging and converging branches.
Consequently, all DCE species must be lumped into a single constituent. Some of the input data
are again given in the “HYDRUS-Course-Data.xls“ file.
Figure: Perchloroethylene (PCE) degradation pathway (picture from Schaerlaekens et al., 1999).
References:
Schaerlaekens, J., D. Mallants, J. Šimůnek, M. Th. van Genuchten, and J. Feyen,
Numerical simulation of transport and sequential biodegradation of chlorinated aliphatic
hydrocarbons using CHAIN_2D, Hydrological Processes, 13, 2847-2859, 1999.
70
Computer Session 2
A. Tracer Transport
Project Manager
Select the LSP-W2 project
Button "Copy”
Name: LSP-S1
Description: Transport of Tracer
Button "OK"
Button "Open"
Main Processes
Heading: Transport of Tracer
Check Solute Transport
Button "Next"
Button "Next"
Copy Bulk Densities from the Excel File (1.31, 1.59, 1.3, 1.38, 1.41, 1.52,
1.56)
Dispersivity = 1 cm
Diffusion Coefficient in liquid phase is 1 cm2/d
Button "Next"
Button "Next"
Upper Boundary Condition: Concentration Flux BC
Lower Boundary Condition: Zero Gradient
Button "Next"
Time Variable Boundary Conditions
Precipitation on day 1: 1 cm/d
Evaporation on day 1: 0 cm/d
cTop on day 1: 1.0
Button "Next"
71
Computer Session 2
HYDRUS-1D Guide:
Button "Next"
Execute HYDRUS-1D
OUTPUT:
Observation Points
Profile Information
Solute Transport Fluxes
Close Project
Observation N odes: Concentration
0.30
0.25
0.20
N1
0.15
N2
0.10
0.05
0.00
0
50
100
150
200
250
300
350
400
Time [days]
Profile Information: Concentration
0
-20
-40
-60
-80
-100
0.0
0.1
0.2
0.3
Conc [mmol/cm3]
72
0.4
Computer Session 2
B. Reactive Solute Transport
Project Manager
Select the LSP-S1 project
Button "Copy”
Name: LSP-S2
Description: Transport of Reactive Solute
Button "OK"
Button "Open"
Distribution coefficient Kd: 0.784 in all layers
Degradation constant SinkWater1’: 0.075 in all layers
Execute HYDRUS-1D
OUTPUT:
Observation Points
Profile Information
Profile Information: C oncentration
0
-20
-40
-60
-80
-100
0.00
0.05
0.10
Conc [mmol/cm3]
73
0.15
Computer Session 2
C. Transport of PCE and its Daughter Products
Project Manager
Select the LSP-S2 project
Button "Copy”
Name: LSP-S3
Description: Transport PCE and its Daughter Products
Button "OK"
Button "Open"
Main Processes
Heading: Transport PCE and its Daughter Products
Button “OK"
Number of Solutes: 5
Copy Bulk Densities from the Excel File (1.31, 1.59, 1.3, 1.38, 1.41, 1.52,
1.56)
Dispersivity = 1 cm
Diffusion Coefficient = 1 cm2/day
Solute 1: Kd=0.784, SinkWater1*=0.075 in all layers
Solute 5: Kd=0.000, SinkWater1*=1e-6 in all layers
Execute HYDRUS-1D
OUTPUT:
Observation Points
Profile Information
74
Computer Session 2
Observation N odes: C oncentration - 1
0.00012
0.0030
0.00010
0.0025
0.00008
0.0020
N1
0.0015
N1
0.00004
0.0010
N2
0.00002
0.0005
0.00006
N2
0.00000
0.0000
0
50
100
150
200
250
300
350
400
0
50
100
Time [days]
150
200
250
300
350
400
Time [days]
0.0035
0.025
0.0030
0.020
0.0025
N1
0.015
0.0020
0.0015
N1
0.0010
N2
N2
0.010
0.005
0.0005
0.0000
0.000
0
50
100
150
200
250
300
350
400
0
50
100
Time [days]
150
200
250
300
350
400
Time [days]
Observation Nodes: C oncentration - 5
0.06
0.05
N1
0.04
N2
0.03
0.02
0.01
0.00
0
50
100
150
200
250
300
350
400
Time [days]
Transport of (1) PCE, (2) trichloroethylene (TCE),
(3) dichloroethylene (DCE), (4) vinyl chloride (VC), and
(5) ethylene (ETH)
75
Computer Session 2
76
77
Input data files
Experiment
START
Numerical
Simulation
Stop
ok?
yes
Nonlinear
Optimization
no
New Parameters
Initial parameters
Parametric model
for soil hydraulic
functions
Cumulative Outflow and
Soil water pressure heads
Boundary &
initial
conditions
Analysis Structure and Flowchart
with contribution of many others
(Jan Hopmans, Mirek Šejna, Ole Wendroth, Norbert Wypler, Nobuo Toride, Feike
Leij, Frank Casey, Mitchy Inoue, and others)
2Department
1Department
Inverse Modeling
Pc
Se
?
Kr
P c = P c (S e )
K r = K r (S e )
Se
Out
Out
Out = ?
θ s , θ r , k, α, n, m, l = ?
In
In
In
Laboratory Experiments:
One-Step Outflow
Multi-step Outflow
Evaporation Experiment
Wetting fluid
0.57 cm
3.95 cm
6.0 cm
T2(w)
Pnw
T3(nw)
T1
One- and Multi-step Outflow Experiments
“Gray-Box” Technique
Inverse Problem
“Black-Box” Tec hni qu e
Fo rward Probl em
“W hi te-Box” Te chnique
Inverse Modeling
1
78
]
[
]
m - number of optimized parameters
q*{q1*, q2*,..., qn*} - vector of length n incorporating observations
(pressure heads h, water contents θ, cumulative infiltration
rates Q, ...)
q(β) {q1, q2,..., qn} is a vector of corresponding model predictions
which is dependent on optimized parameters
n - number of observations
β{β1, β2,..., βm} - vector of optimized parameters
T
⎧ 1
⎫
L( β ) = (2π ) −n / 2 det−1/ 2 V exp⎨− q* − q(β ) V −1 q* − q(β ) ⎬
⎩ 2
⎭
[
When measurement errors follow a multivariate normal
distribution with zero mean and covariance matrix V, the
likelihood function,
function L(b), can be written as [Bard, 1974]
Feddes et al. (1988)
Ciollaro and Romano (1995)
Santini et al. (1995)
Šimunek et al. (1998)
Evaporation Method:
Kool et al. (1985) and Parker et al. (1985) - onestep method
Russo (1988) - influence of parametric form
Toorman et al. (1992) - uniqueness problems
van Dam et al. (1992, 1994) - multistep method
Eching and Hopmans (1993, 1994) - one- and multi-step with h
OneOne- and MultiMulti-step Methods:
Inverse Methods - Laboratory:
Φ( β ) = q* − q β
bg
T
bg
V −1 q * − q β
If all the elements of covariance matrix V are known, then the
values of unknown parameters b must minimize the following
equation:
∂ ln L( β )
=0
∂β
The maximum of the likelihood function must satisfy the set of b
likelihood equations
Gribb (1996), Gribb et al. (1998), Kodešová et al. (1998), Šimůnek et al.
(1998)
X MultipleMultiple-Step Extraction Experiment:
Experiment: Inoue et al. (1998)
X Cone Penetrometer:
Penetrometer:
Šimůnek and van Genuchten (1996, 1997), Šimůnek et al. (1998a,b)
X Instantaneous Profile Method:
Method: Dane and Hruska (1983)
X Ponded Infiltration:
Infiltration:
Russo et al. (1991)
Bohne et al. (1992)
X Tension Disc Infiltrometer:
Infiltrometer:
Inverse Methods - Field
2
79
]
[
]
T
- vector of optimized parameters with the prior information (e.g., θr, θs, α, n,
and Ks)
- vector of predicted optimized parameters
- covariance matrix for parameters β
H ij ( β ) =
Newton method: Ri = Hi-1 :
Steepest descent method: Ri = I
∂ 2Φ
∂β i∂β j
vi - direction vector
Ri - positive definite matrix
ρi - scalar that insures that the iteration step is acceptable
β i +1 = β i + ρ i vi = β i − ρ i Ri pi
Gradient methods:
methods:
Solution of the Inverse Problem
Vβ
β
β*
∧
∧
T
⎡
⎤
⎤
−1 ⎡
Φ ( β ) = q * − q (β ) V −1 q * − q (β ) + ⎢ β * − β ⎥ V β ⎢ β * − β ⎥
⎣
⎦
⎣
⎦
[
Inclusion of the prior information leads to the maximization of the following
equation:
p* ( β ) = cL ( β ) p0 ( β )
Any information about the distribution of the fitted parameters known before the
inversion, can be included into the parameter identification procedure by
multiplying the likelihood function by the prior pdf, p0(b), which summarizes the
prior information.
information Estimates which make use of the prior information are known as
Bayesian estimates,
estimates and they lead to the maximizing of the posterior pdf, p*(b),
given by:
i =1
n
i
∑ w [q
i
*
The method represents a compromise between the inverse-Hessian
method and the steepest descend method by switching from the latter
method used when far from the minimum to the former as the
minimum is approached. This is accomplished by multiplying the
diagonal in the Hessian matrix (or its approximation N), sometimes
called the curvature matrix, with ( 1 + λ ), where λ is a positive scalar.
When λ is large, then the matrix is diagonally dominant resulting in
the steepest descend method. On the other hand, when λ is zero, the
inverse-Hessian method will result.
MarquardtMarquardt-Levenberg method:
J - Jacobian matrix
H ≈ N = JTJ
GaussGauss-Newton method:
method:
Solution of the Inverse Problem
2
]
− q i (β )
wi - weight of a particular measured point
Φ(β ) =
When the covariance matrix V is diagonal and all elements
of matrix Vβ are equal to zero , i.e., the measurement errors
are noncorrelated and no prior information about the
optimized parameters exists, the problem simplifies into the
weighted leastleast-squares problem
3
80
et al., 1996, 1997,
1998)
MultipleMultiple-Step Extraction Experiment (Inoue et al.,
1998, Šimůnek et al., 1998)
Cone Penetrometer (Gribb, 1996; Kodešová et al.,
1998, 1999; Šimůnek et al., 1998)
Root uptake analyses (Vrugt et al., 2001, 2002)
Heat pulse probe analyses (Hopmans et al., 2002;
Mortensen et al., 2003; Saito et al., 2007)
Drip irrigation (Lazarovic et al., 2003)
Tension disc infiltrometer (Šimůnek
Parameter Estimation in HYDRUS-2D
Method:
- Marquardt-Levenberg optimization
Sequence:
- Independently
- Simultaneously
- Sequentially
Parameter Estimation:
- Soil hydraulic parameters
- Solute transport and reaction parameters
- Heat transport parameters
Parameter Estimation in HYDRUS

mq
3rd term:
2nd term:
1st term:
ij
*
j
i
j
i
2
[ g*j ( x, ti ) - g j ( x, ti , b)] +
i =1
nb
j =1
2
+ ∑vˆ j[b*j - bj ]
deviations between measured and calculated spacetime variables
differences between independently measured, pj*, and
predicted , pj, soil hydraulic properties
penalty function for deviations between prior
knowledge of the soil hydraulic parameters, bj*, and
their final estimates, bj.
j =1
npj
i, j
∑w [ p (θ ) - p (θ , b)] +
i =1
nqj
∑w
+ ∑v j
mq
j =1
Φ (b, q, p) = ∑v j
2
One-step outflow method (Wildenschield et al., 2000)
Multi-step outflow method (Wildenschield et al., 2000)
Evaporation method (Šimůnek et al., 1998)
Horizontal infiltration (Šimůnek et al., 2000)
Horizontal infiltration and redistribution (hysteresis) (Šimůnek et al.,
1998)
Dynamic effects during one- and multistep experiments (Wildenschield
et al., 2000)
Tension upward infiltration (Šimůnek et al., 2000, Young et al., 2002)
Simultaneous estimation of soil hydraulic and solute transport
parameters from infiltration experiment (Inoue et al., 2000)
Water and chloride transport, field experiments (Ventrella et al., 2000,
Jacques et al., 2001)
Nonequilibrium solute transport (Šimůnek et al., 2000, 2002)
Nonlinear solute transport (Šimůnek et al., 2002)
Cadmium nonlinear transport (Seuntjens et al., 2001)
Transport of chlorinated hydrocarbons subject to sequential
transformation reactions (Casey and Šimůnek, 2001)
Nonequilibrium transport with flow interruption (Šimůnek et al., 2000)
Transport of 17b-estrodial (Casey et al., 2003)
Objective Function
Solute Transport:
Water Flow:
Parameter Estimation in HYDRUS-1D
4
81
The three unknown parameters α, n, and θr were
estimated by numerical inversion of the observed
cumulative outflow data and the measured water
content at the pressure head of -150 m.
At the end of the experiment, the soil was
resaturated and the saturated hydraulic
conductivity of the soil and porous plate were
measured with a falling head method.
After resaturating, the pneumatic pressure was
increased instantaneously to 10 m and cumulative
outflow was recorded with time.
An undisturbed core sample 3.95 cm long and 5.4
cm diameter was equilibrated at zero tension in a
Tempe pressure cell.
Kool et al. [1985]:
Wetting fluid
0.57 cm
3.95 cm
6.0 cm
T2(w)
One-Step Outflow Method
}
Pnw
T3(nw)
T1
1st term: space-time variables:
- pressure heads
at different locations and/or time
- water contents
- concentrations
- actual fluxes across boundaries
- cumulative fluxes across boundaries
2nd term: soil hydraulic properties:
- retention data, θ(h)
- hydraulic conductivity data, K(θ) or K(h)
- diffusivity data, D(θ) or D(h)
3rd term: prior knowledge of the soil hydraulic parameters:
- θr , θs , α, n, Ks , and l
Objective Function
i i
⎞
i =1
nj
⎟
⎟
ni ⎟
∑
i =1
⎠
nj
∑q n ⎟
1
n jσ 2j
-0.7
0.001
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
Kool et al. [1985]
0.01
1
Time [hours]
0.1
10
3. Input information: General input
vj =
2. Weighting by variances,
variances σj2:
⎛
⎜
min ⎜ q j ;
⎜
⎜
⎝
vj =
qj
100
1. Weighting by measurements means,
means qj :
Weighting in the Objective Function
5
0.01
0.1
1
10
100
0.1
0.15
0.25
0.3
Water Content [-]
0.2
0.35
0.4
log D iffusivity [m 2/s]
-10
-9
-8
-7
-6
-5
-4
0.1
0.3
Water Content [-]
0.2
0.4
Hydraulic conductivity close to saturation: steady-state infiltration using
tension disc infiltrometers
Objective function: tensiometer readings and the total water
volume at the end of the experiment
Analysis: a) Modified Wind’s method [Wendroth et al., 1993]
b) Parameter estimation [Šimůnek et al., 1998]
- Two samples: height=10 cm, inside diameter=10 cm, saturated with
deionized water
- Five tensiometer (0.6-cm o.d.; 6-cm length) - 1,3,5,7,9 cm
- Initial pressure head: -15.4 cm in the middle of the sample
- Two-rate experiment: 1.2 cm/d and 0.2 cm/d
- Measurements: every 30 min and every 4 h
- Terminated: upper tensiometer -650 cm
Evaporation Method
The predicted and measured retention curve, as well as the comparison
comparison of
the diffusivity curve obtained by parameter estimation methods against
against the
values calculated independently by the method of Passioura [1976].
Pressure Head [m ]
1000
-800
-600
-400
-200
0
0
100
150
Time [hours]
50
200
0
0.0
-0.8
-0.6
-0.4
-0.2
100
150
Time [hours]
50
200
0
0.5
1
1.5
2
2.5
Time [d]
Measured 1
Fitted 1
Measured 2
Fitted 2
Measured 3
Fitted 3
Measured 4
Fitted 4
Measured 5
Fitted 5
Experiment I
-700
-600
-500
-400
-300
-200
-100
0
3
3.5
4
4.5
0
-100
-200
-300
-400
-500
-600
-700
-800
-900
-10
Evaporation Method
-8
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
-4
Depth [cm]
-6
-2
0
The experimental setup consisted of a 6-cm soil column in a Tempe pressure cell
modified to accommodate a microtensiometer-transducer system. A tensiometer
was installed with the cup centered 3 cm below the soil column surface. Soil
sample was saturated from the bottom and subsequently equilibrated to an initial
soil water pressure head of -25 cm at the soil column surface. Pressure was
applied in steps of 100, 200, 400, and 700 cm pressure head at 0, 12, 48, and 106
hours.
Pressure Head [cm]
Kool et al. [1985]
Pressure Head [cm]
Multi-Step Outflow Method
Cumulative Outflow
[cm]
P re s s u re He a d [c m ]
82
6
0.15
0.2
0.25
0.3
0.5
1.0
2.0
2.5
3.0
-4
-3
-2
-1
0
1
2
0.0
1.0
1.5
2.0
log h [cm]
All Tensiometers
Tensiometer 1
Tensiometer 2
Tensiometer 3
Tensiometer 4
Tensiometer 5
Fitted Wind Points
Wind Method
Experiment II
0.15
0.2
0.25
0.3
0.35
0.4
0.45
2.5
3.0
-5
-4
-3
-2
-1
0
1
2
Evaporation Method
log h [cm]
1.5
All tensiometers
Tensiometer 1
Tensiometer 2
Tensiometer 3
Tensiometer 4
Tensiometer 5
Fitted Wind's Points
Wind Method
Experiment I
W ater C o n ten t [-]
0.35
W a te r C o n te n t [-]
0.0
1.0
log h [cm]
1.5
0.5
1.0
log h [cm]
1.5
All tensiometers
Tensiometer 1
Tensiometer 2
Tensiometer 3
Tensiometer 4
Tensiometer 5
Wind Method
Tension Disc Infiltration
0.5
All Tensiometers
Tensiometer 1
Tensiometer 2
Tensiometer 3
Tensiometer 4
Tensiometer 5
Wind Method
Tension Disc Infiltration
2.0
2.0
2.5
2.5
3.0
3.0
0
-600
-500
-400
-300
-200
-100
0
2
6
Time [d]
4
8
Measured 1
Fitted 1
Measured 2
Fitted 2
Measured 3
Fitted 3
Measured 4
Fitted 4
Measured 5
10
0
-200
-400
-600
-800
-1000
-10
-8
t = 0.042 d
t = 0.25 d
t = 1.257 d
t = 5.502 d
t = 8.008 d
t = 9.494 d
Measured
-6
-4
Depth [cm]
-2
0
10
100
1000
Time [min]
10000
100000
data from George Vachaud [1968], soil column: 60 cm long, i.d. 9 cm, silt
locations: 5, 9.5,12.5, 15.5 18.5, 21.5, 25.5, 28.5, 31.5, 35.5 cm
γ-ray attenuation technique, hysteresis
0.0
0.1
0.2
0.3
0.4
0.5
Horizontal Infiltration and Redistribution
Experiment II
P re s s u re h e a d [c m ]
Evaporation Method
P r e s s u r e H e a d [c m ]
Evaporation Method
lo g K [c m /d ]
lo g K [c m /d ]
83
7
0
2
4
6
10
12
Time [days]
8
14
16
18
20
Flow interruption techniques are often used
to
elucidate
rate-limited
sorption
processes.
Experimental conditions:
- data taken from Fortin et al. (1997)
- 15-cm long repacked soil column (loamy
sand)
- saturated water content - 0.47 cm3cm-3
- flow rate of 0.674 cm h-1
- bromide and the herbicide simazine
dissolved in a 0.01 M CaSO4 solution at
concentrations of 50 mg l-1 and 0.025 μg l-1,
respectively
- flow stopped for 185 h after 5 pore
volumes of input.
Optimization: the two-site sorption model
- dispersivity (λ=1.08±0.246 cm)
- adsorption coefficient (kd=1.49±2.57 cm3g1)
- fraction of equilibrium sorption sites
(f=0.328±0.564)
- mass transfer coefficient (αmt=0.00128
±0.00406 h-1) were optimized
0
2
6
8
Pore Volume [-]
4
10
Optimization II
Optimization I
Optimization I - dispersivity
estimated from the bromide effluent
data
Optimization II - fits dispersivity
simultaneously
0
0.2
0.4
0.6
0.8
1
1.2
Laboratory Transport Subject to Flow Interruption
This slide demonstrates the option of the HYDRUS-1D model to optimized
solute transport parameters for non-equilibrium solute transport, in this case a
two-site kinetic sorption. Although it is based on a real soil column study, we
used generated breakthrough curve.
0
5
10
15
20
Relative C oncentration [-]
12
0
2
4
0
5
10
Time [d]
Optimized
Measured
15
20
25
30-cm long and 5-cm inner diameter laboratory
soil column
repacked coarse-textured soil (Tottori sand)
infiltration
experiment,
simultaneously
increasing the solute concentration (from 0.02
-1
to 0.1 mol l NaCl) and the infiltration rate
(from 0.00032 to 0.0026 cm s-1)
pressure heads, h, and bulk soil electrical
conductivities, ECa, were measured using
automated mini-tensiometer and four-electrode
sensors, respectively, at the 23 cm depth

sequential estimate soil hydraulic and solute
transport parameters
simultaneously estimate soil hydraulic and
solute transport parameters
Optimization:

0
1000
Measured h
Measured ECa
Optimized
2000
Time, t [s]
3000
0
4000
0.1
0.2
0.3
0.4
0.5
0.6
Measured and optimized pressure
heads and bulk electrical
conductivities (data from Inoue et
al., 2000).
-10
-14
-18
-22
-26
Transient Laboratory Experiment with
Simultaneous Water Flow and Solute Transport
Selim et al. (1987)
Optimization:
- dispersion coefficient (D)
- Freundlich coefficients (kd and β)
6
8
10
- 10.75-cm long soil column
- saturated with a 0.005 M CaCl2 solution
- 14.26 pore volume pulse (t=358.05 h) of 0.005 M MgCl2 solution
- followed by the original CaCl2 solution
- flow rate was equal to 6.495 cm/d
Non-Linear Solute Transport
C oncentration [mmolc /L]
Non-Equilibrium Solute Transport
Pressure Head, h [cm]
84
Electrical C onductivity, EC a [dS/m]
8
85
-1.15±0.178
-0.816±0.220
0.221±0.024
0.207±0.020
0.00347
0.00170
Objective F. Φ
----------------------------------------------------------------------Mean Pore Water Velocity, v [cm/s]
0.01
λ =D /v =0.2210.024 cm
Dispersion coefficient D obtained
by inverse optimization (line)
and from analysis of steady-state
data (symbols) (data from Inoue
et al., 2000).
0.001
0.0001
0.001
0.01
0.1
0.1
Transport of Chlorinated Hydrocarbons Subject
to Sequential Transformation Reactions
Soil hydraulic and solute transport
parameters and their confidence intervals
obtained using sequential and
simultaneous optimizations.
λ (cm)
l (-)
-----------------------------------------------------------------------Parameter
Sequential Opt.
Simultaneous
Opt.
-----------------------------------------------------------------------θr (cm3cm-3)
0.0265±0.0286
0.0206±0.0198
θs (cm3cm-3)
0.310±0.0386
0.310±0.0248
-1
Ks (cm s )
0.138±0.038
0.137±0.024
n (-)
2.014±0.318
1.969±0.186
α (cm-1)
0.0446±0.0388
0.0570±0.0322
2
Transient Laboratory Experiment with
Simultaneous Water Flow and Solute Transport
D ispe rsio n C o e ffic ie nt, D [c m /s]
two columns (21.4 mm diameter and 124 mm length)
packed with 40-mesh iron filings (Fisher Scientific) or 40-mesh Fisher iron filings plated
with 1.78 % copper
a pulse of one pore volume of 42 mg L-1 TCE
three velocities of 12.4 (denoted as fast), 6.2 (intermediate), or 3.1 (slow) mm min-1
high performance liquid chromatography (HPLC) system
HPLC was equipped with a Beckman 128 diode-array flow through photo detector
capable of separating out TCE and its daughter products (1,1-dichloroethylene;
1,2-cis-dichloroethylene; 1,2-trans-dichloroethylene; vinyl chloride; and ethylene) and
determining their concentrations in the effluent
equilibrium solute transport model
nonequilibrium solute transport model with a two-site sorption concept
3. Tension disc infiltrometer
- Šimůnek and M. Th. van Genuchten [1996, 1997]
- Šimůnek et al. [1998a, 1998b]
2. Modified cone penetrometer
- Gribb et al. [1998]
- Kodešová et al. [1998]
1. Multistep tension extraction
- Inoue et al. [1998]
Examples for HYDRUS-2D

Optimization:

Miscible-displacement experiments involving dissolved trichloroethylene (TCE) undergoing
reduction/transformations in the presence of zero-valent metal porous media (i.e., iron
or copper coated iron filings) to produce ethylene
Casey and Šimůnek (2001):
Transport of Chlorinated Hydrocarbons Subject
to Sequential Transformation Reactions
9
T e n sio m e te r
N e u tro n p ro b e
m e a s u re m e n t
z
S o il w a te r
e x tra c tio n
r
T1
B u re tte
T3
T2
vacuu
m
k
W a te r
ta n k
T4
Inoue et al. [1998]
0.6
0
-50
-150
Pressure Head [cm]
-100
-200
Numerical Inversion - Fitted Kcer
Numerical Inversion - Fitted Kcer + 0(h)
Soil Samples
Instantaneous Profile Method
-250
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
0
-50
-150
Pressure Head [cm]
-100
Instantaneous Profile Method
-200
Numerical Inversion - Fitted Kcer + 0(h)
Numerical Inversion - Fitted Kcer
-250
Estimated soil water retention and hydraulic conductivity functions for
the two optimizations of multi-step extraction data compared with
independently determined retention and unsaturated hydraulic
conductivity data.
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
Inoue et al. [1998]
Multistep Extraction Experiment
Water Content [-]
0
20
Fitted
40
Measured
Time [h]
60
80
100
120
0
-300
-250
-200
-150
-100
-50
0
20
40
Time [h]
60
Measured T1
Fitted T1
Measured T2
Fitted T2
Measured T3
Fitted T3
80
120
addition of extension tubes
Internally threaded to allow
Water source
(tank)
Screen
Tensiometer rings
Signal conditioner
and A/D converted
Bleed and fill port
for tensiometers
Laptop
computer
Cone Penetrometer (Gribb et al., 1998)
100
Inoue et al. [1998]
Comparison of measured and optimized cumulative
extraction and pressure head values
0
100
200
300
400
500
C u m u la tiv e E x tra c tio n
V o lu m e [m l]
P ressu re H ead [cm ]
H ydraulic C onductivity [cm/h]
86
10
0
200
-60
-50
-40
0
Run 6
Run 8
Run 10
Run 12
200
Time [s]
400
Upper Tensiometer
600
800
0
-2000
-4000
-6000
-30
Cumulative Infiltration
-8000
-20
-12000
-10000
Lower Tensiometer
Optimization
800
0
2000
4000
6000
8000
10000
12000
14000
C umulative Infiltration [ml]
-10
0
600
Upper Tensiometer
Lower Tensiometer
Cumulative Infiltration
Time [s]
400
Cone Penetrometer
-60
-50
-40
-30
-20
-10
0
Measured Data
Cumulativ e Infiltration [ml]
0
100
200
300
Measured
Runs 1
Runs 2
Runs 3
Runs 4
Runs 5
500
Time [s]
400
600
700
Hysteresis
900
800
No Hysteresis
Prediction of Redistribution
Tension Disc Infiltrometer
-60
-50
-40
-30
-20
-10
0
Cone Penetrometer
Pressure H ead [cm]
Cone Penetrometer
Pressure H ead [cm]
Pressure H ead [cm]
87
11
88
0
5
10
15
20
25
0
1000
-11.5 cm
2000
3000 4000
Time [s]
-9.0
-6.0
5000
-3.0
6000
Fitted
Measured
-1.0
-0.1
7000
Measured and optimized cumulative infiltration curves for a
tension disc infiltrometer
experiment carried out on a
sandy soil in the Sahel region.
X
X
X
X
- evaluating the design and performance of different
experimental approaches,
- optimally analyzing data collected with different laboratory
and field instruments.
Parameter estimation technique couples experimental work
with numerical modeling.
Optimized parameters are obtained by similar numerical
models for which they are needed as input.
Soil water retention curve and hydraulic conductivity
function are obtained from a single experiment.
Parameter estimation procedure provides a confidence
intervals of the optimized parameters.
X Parameter estimation approaches provide unique tools for:
Summary
Cum ulative Infiltr ation [cm ]
The small breaks in the cumulative infiltration curve were caused by brief interruptions to
resupply the infiltrometer with water, and to adjust the tension for a new time interval
- Crusted soil system in the Sahel region of Africa (Šimůnek et al., 1998)
- Sandy subsoil
- Tension disc diameter of 25 cm
- Supply tensions of 11.5, 9, 6, 3, 1, and 0.1 cm
-1
-0.5
0
0.5
1
1.5
log(|h| [cm ])
2
Wooding's Analysis
2.5
Numerical Optimization
3
Šimůnek, J., and J. W. Hopmans, Parameter Optimization and Nonlinear
Fitting, In: Methods of Soil Analysis, Part 1, Physical Methods, Chapter
1.7, Eds. J. H. Dane and G. C. Topp, Third edition, SSSA, Madison, WI,
2001.
Hopmans, J. W., J. Šimůnek, N. Romano, and W. Durner, Inverse Modeling
of Transient Water Flow, In: Methods of Soil Analysis, Part 1, Physical
Methods, Chapter 3.6.2, Eds. J. H. Dane and G. C. Topp, Third edition,
SSSA, Madison, WI, 2001.
Šimůnek, J., D. Jacques, J. W. Hopmans, M. Inoue, M. Flury, and M. Th.
van Genuchten, Solute Transport During Variably-Saturated Flow Inverse Methods, In: Methods of Soil Analysis, Part 1, Physical Methods,
Chapter 6.6, Eds. J. H. Dane and G. C. Topp, Third edition, SSSA,
Madison, WI, 2001.
Methods of Soil Analysis, Part 1, Physical Methods, Eds. J.
H. Dane and G. C. Topp, Third edition, SSSA,
Madison, WI, 2001
References
Unsaturated hydraulic conductivities calculated using Wooding's
analytical solution, and the complete function obtained with
numerical inversion
0.000
0.001
0.002
0.003
0.004
0.005
Hydr aulic Conductivity [cm /s ]
12
Computer Session 3
Computer Session 3
The example in Computer Session 3 considers the inverse solution of a onestep outflow experiment. Data presented by Kool et al. [1985], and used in
example 6 of the HYDRUS-1D manual (p. 102), are used in the analysis.
Three hydraulic parameters will be estimated by numerical inversion of the
observed cumulative outflow and the measured water content at a pressure
head of -150 cm.
Pnw
T2(w)
T3(nw)
3.95 cm
0.57 cm
6.0 cm
T1
Wetting fluid
Since water exits the soil column across a ceramic plate, the flow problem
involves a two-layered system. The profile, consists of a 3.95-cm long soil
sample and a 0.57-cm thick ceramic plate, and is discretized using 50 nodes,
with five nodes associated with the ceramic plate.
Only a few nodes are needed for the ceramic plate since the plate remains
saturated during the entire experiment, thus causing the flow process in the
plate to be linear.
Outflow is initiated using a pressure head of -10 m imposed on the lower
boundary.
89
Computer Session 3
A. Inverse Modeling - One-Step Outflow Method
Project Manager
Button "New"
Name: Onestep
Description: Onestep Outflow Method
Button "OK"
Main Processes
Heading: Onestep Outflow Method
Check "Inverse Solution"
Inverse Solution
Check "Soil Hydraulic Parameters"
Check "No Internal Weighting"
Max. Number of Iteration:
20
Number of Data Points:
10
Number of Soil Materials:
2
Number of Layers for Mass Balances: 2
Depth of the Soil Profile:
4.52
Time Information
Time Units: Hours
Final time:
Initial Time Step:
Minimum Time Step:
Maximum Time Step:
100
0.001
0.0001
10
Print Information
Uncheck "Screen Output"
Number of Print Times:
Button "Select Print Times":
11
0.017, 0.033, 0.05, 0.167, 0.5, 1.33,
2.75, 5.417, 10, 15, 100
90
Computer Session 3
Upper Limit of the Tension Interval: 15000
Initial Estimate:
θr=0.15, θs=0.388, α=0.025, n=1.5, Ks=5.4,
l= 0.5
Fitted:
Qr, Alpha, n
Second Material: θr=0., θs=1., α=1e-20, n=1.001, Ks=0.003, l= 0.5
Lower Boundary Condition: Constant Pressure Head
Water Flow - Constant BC
Upper Boundary Flux: 0
Data for Inverse Solution
0.017
0.033
0.05
0.167
0.5
1.033
2.75
5.417
100
-15000
-0.0786
-0.1616
-0.2097
-0.3408
-0.4456
-0.498
-0.5614
-0.5937
-0.6824
0.157
0
0
0
0
0
0
0
0
0
5
2
2
2
2
2
2
2
2
2
1
91
1
1
1
1
1
1
1
1
1
1
Computer Session 3
Menu: Options->Grid - Height: 0.05
Menu: Condition->Profile Discretization
Button "Number": 50
Button "Insert Fixed" at 3.95 cm
Button "Density": deselect "Use upper", upper density =0.1 at 3.95 cm
Menu: Condition->Initial Condition->Pressure Head
Select entire profile: Top value=-2, Bottom value=2.52
Deselect "Use top value for both"
Lowest node = -1000 cm
Menu: Condition->Material Distribution
Select the ceramic plate and specify "Material Index"=2
Ditto for "subregions"
Observation Points?
Execute HYDRUS
OUTPUT:
Water Flow - Boundary Fluxes and Heads
Cumulative Bottom Flux
Inverse Solution Information
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
0.001
0.01
0.1
1
Time [hours]
92
10
100
Computer Session 3
Exercises:
Multistep Outflow Experiments (SGP97 project):
Height of the soil sample: 5.9 cm
Thickness of the ceramic: 0.5 cm
Conductivity of the ceramic, boundary conditions, and output data
depend on the sample (see the Excel file).
Cum. Bottom Flux
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
0
500000
1e+006
1.5e+006
Time [sec]
Bottom Pressure Head
0
-100
-200
-300
-400
-500
-600
0
500000
1e+006
Time [sec]
93
1.5e+006
94
95
)
⎛
⎞
∂φ
∑e ∫ ⎜ -KKizA ∂xn - Sφn ⎟ d Ω
i
⎝
⎠
Ωe
Ωe represents the domain occupied by element e
Γe is a boundary segment of element e
e
∑
(
∂θ
∂h ' ∂φn
dΩ =
φn + KK ijA
∂t
∂x j ∂xi
⎛ A ∂ h’
⎞
A
∫ K ⎜⎝ Kij ∂ x j + Kiz ⎟⎠ niφn d Γ +
Γe
e Ωe
∑∫
Applying Green's first identity and replacing h by h’
⎧⎪ ∂θ ∂ ⎡ ⎛
⎫⎪
⎞⎤
A ∂h
A
∫Ω ⎨ ∂t - ∂xi ⎢⎢ K ⎜⎜ Kij ∂x j + Kiz ⎟⎟⎥⎥ + S ⎬ φn d Ω = 0
⎪⎩
⎪⎭
⎠⎦
⎣ ⎝
The Galerkin method:
2D Variably-Saturated Flow
2Department of Mechanical Engineering
1Department
to 2D Variably-Saturated Water Flow
e
e
2
κ
K K xzA bn + K zzA cn
(
Ωe
)
κ=1
for two-dimensional problem
κ = 2 π r for axisymmetric problem
e
=∑
A
ij
A
i
n
j
m
e Ωe
n
nm
e
κ
e
∑ ∫φ d Ω = δ ∑ 3 A
e
Ωe
e
Dn = ∑Sl ∫φlφn d Ω = ∑
Γe
κ
12
Ae (3S + Sn )
e
Qn = -∑σ 1l ∫φlφn d Γ = −∑σ n λn
e
Fnm = δ nm
A
∂φ ∂φ
dΩ =
∂x ∂x
K [ K xx bm bn +K xz (cm bn + bm c n ) +K zz cn cm ]
A
l
∂φ
Bn = ∑ Kl KizA ∫ φl n d Ω =
∂xi
e
Ωe
κ
e
l
d {θ }
+ [ A]{h} = {Q} - {B} - {D}
dt
∑K K ∫ φ
[F ]
Anm =
= ∑ 4A
In matrix form:
Examples of the unstructured triangular finite element grids for regular (left) and
irregular (right) two-dimensional transport domains.
Discretization Using Finite Elements
1
96
⎝
e
=∑
κ
4 Ae
A
A
A
K [ K xx bm bn +K xz (cm bn +bm cn )+K zz cn cm ]
xx
bn bm
b c
b c
c c ⎞
+ K xz n m + K zx m n + K zz n m ⎟ ( K1 + K 2 + K
2A 2A
2A 2A
2A 2A
2A 2A ⎠
Δtj
{θ } j+1 - {θ } j
= [F ]
Δtj
{θ }kj++11 - {θ }kj+1
+[F ]
Δtj
{θ }kj+1 - {θ } j
Δtj
= [ F ] [C ] j+1
Δtj
{h }kj++11 - {h }kj+1
+[F ]
Δtj
{θ }kj+1 - {θ } j
Cnm=δnmCn, Cn - nodal value of the soil water capacity
[F ]
{θ } j+1 - {θ } j
k+1, k - current and previous iteration levels, respectively
[F ]
The "mass"mass-conservative" method of Celia et al. [1990]:
e
= ∑⎛⎜ K
A
) =
3
= ∑⎛⎜ K xx ∂φn ∂φm + K xz ∂φn ∂φm + K zx ∂φm ∂φn + K zz ∂φn ∂φm ⎞⎟ Kl ∫ φ ld Ω =
∂x ∂x
∂x ∂z
∂x ∂z
∂z ∂z ⎠ Ωe
e ⎝
e Ωe
∂N I
c
= I
∂y
2A
a !b! c!
(a +b+ c + 2)!
∂φn ∂φm
∂φ ∂φ
d Ω = ∑K l K ijA n m ∫ φ ld Ω =
∂xi ∂x j
∂xi ∂x j Ωe
e
∂N I bI
=
∂x 2 A
N 2b N 3c d Ω = 2 A
Anm = ∑ ∫ KK ijA
Ω
a
1
∫N
d {θ }
+ [ A]{h} = {Q} - {B} - {D}
dt
Δ tj
{θ } j+1 - {θ } j
+ [ A ] j+1 {h } j+1 = {Q } j - {B } j+1 - {D } j
Ω
or in a matrix form:
e Ωe
∑ ∫[(-θ R
[Q]
∂c’
niφn d Γ = 0
∂x j
d{c}
+ [ S ]{c}+{ f } = -{Q D}
dt
e Γ eN
+ ∑ ∫ θ Dij
∂c’
∂c’
∂c’ ∂φn
- qi
+ Fc’ +G )φn - θ Dij
]d Ω
∂t
∂xi
∂x j ∂xi
results in the following system of time-dependent differential equations
n=1
Application of Green's theorem to the second derivatives and substitution of c
by c’
N
c’ ( x, y, z, t ) = ∑φn ( x, y, z )cn (t )
n
The Galerkin method:
⎡
⎤
∂c ∂c ∂ ⎛
∂c ⎞
∫ ⎢-θ R -qi + ⎜ θ Dij
⎟⎟+Fc+G ⎥φ d Ω = 0
⎜
∂t ∂xi ∂xi ⎝
∂x j ⎠
⎢⎣
⎥⎦
2D Solute Transport
j+1, j - current and previous time levels
Δtj=tj+1-tj - time step
[F ]
Integration in time is achieved by discretizing the time domain into a
sequence of finite intervals and replacing the time derivatives by finite
differences. An implicit (backward) finite difference scheme is used for
both saturated and unsaturated conditions:
[F ]
Time discretization:
2
97
Ωe
e
12
κ Ae
(3θ R +θ n Rn )δ nm
−
κ
(3q z + q zn) + Ae (3F + F n + F m )(1 + δ nm ) −
24
60
κ cm
e
Ωe
e
f n = ∑ Gl ∫ φlφn d Ω = ∑
12
κ Ae
n
( 3G + G )
⎡b mb nθ D xx + (b mc n + c mb n)θ D xz + c m c nθ D zz ⎤ }
⎦
4 Ae ⎣
κ
(3 q x + q xn ) −
d{c}
+ [ S ]{c}+ { f } = -{Q D}
dt
+t
{c} j +1 − {c} j
+ ε [ S ] j +1 {c} j +1 + (1 − ε ) [ S ] j {c} j + ε { f } j +1 + (1 − ε ){ f } j = 0
w
3 1 2
⎛ uL ⎞ 2 D
⎟−
⎝ 2 D ⎠ uL
α iw = coth ⎜
u, D, L - flow velocity, dispersion coefficient and length associated with
side i. The weighing functions φu ensure that relatively more weight is
placed on the flow velocities of nodes located at the upstream side of an
element
Li - weighting functions
Christie et al. [1976]:
αiw - weighing factor associated with the size of the element opposite to node i
φ3u = L3 − 3α 2w L1 L3 + 3α1w L2 L3
w
1
φ = L2 − 3α L3 L2 + 3α L L
u
2
Crie =
ωs - performance index [-]
Three stabilizing options
- upstream weighing
- Perrochet and Berod [1993]
The grid Courant number:
qi Δxi
θ Dii
Pe ⋅ Cr ≤ ωs = 2
qi Δt
θ RΔxi
Peie =
Numerical solutions of the transport equation often exhibit oscillatory behavior
and/or excessive numerical dispersion near relatively sharp concentration fronts.
The grid Peclet number:
φ1u = L1 − 3α 3w L2 L1 + 3α 2w L3 L1
Oscillatory Behavior:
2D Solute Transport
j, j+1 - previous and current time levels, respectively
Δt - time increment
[G ]{c} j+1 = {g}
ε -time-weighing factor
Equation can be rewritten in the form:
1
[G] = [Q] j+ε +ε[S ] j+1
Δt
1
{g}= [Q] j+ε{c} j − (1- ε )[S ] j{c} j - ε{ f } j+1 - (1- ε ){ f } j
Δt
[Q ] j +ε
To minimize the problems with numerical oscillations. The flux term of transport
equation is weighted using the nonlinear functions φnu [Yeh and Tripathi, 1990]
2D Solute Transport
24
κ bm
[Q ]
The time derivatives are discretized by means of finite differences. A first-order
approx. of the time derivatives:
Time discretization:
2D Solute Transport
Upstream Weighted Formulation:
e
= ∑{−
⎡
⎤
∂φ
∂φ ∂φ
Snm = ∑ ⎢( −qi )l ∫ φlφn m d Ω − (θ Dij ) ∫ φl n m d Ω + Fl ∫ φlφnφm d Ω ⎥ =
l
∂
∂
x
x
x
e ⎣
⎢
i
i
j
Ωe
Ωe
Ωe
⎦⎥
e
Qnm = ∑ (-θ R ) l ∫φlφnφm d Ω = −∑
2D Solute Transport
3
98
Memory requirement
Round-off errors
Solution approximation
Number of steps
Fixed
N2
X*N2
yes
no
Direct
Variable
N1.5
X*N1.5
no
yes
Iterative
Comparison of direct and iterative methods
Direct methods:
Gaussian elimination, LU decomposition
Iterative methods: Gauss Seidel, alternating direction implicit
(ADI), strongly implicit procedures (SIP),
successive over-relaxation (SOR), conjugate
gradients, ORTHOMIN.
Matrix Equation Solvers:
2D Water Flow and Solute Transport
4
99
Riverside, CA
and Miroslav Šejna
The HYDRUS (2D/3D) Software for
Simulating Two- and Three-Dimensional
Variably-Saturated Water Flow
Data and Project Management
Data Pre-Processing
- Input parameter
- Transport domain design
- Finite element grid generator
- Initial conditions
- Boundary conditions
- Domain Properties
Computations
Data Post-Processing
- Graphical output
- ASCII output
X
X
X
Manages data of existing projects
Provides information about existing projects
- geometry type
- considered processes
- existence of results
- size
- when modified
Locates
Opens
Copies
Deletes
Renames
-- desired projects or selected input and/or output data
Project Manager
X
X
X
X
HYDRUS (2D/3D) - Functions
1
100
Navigator
Bars
Opens
Copies
Deletes
Renames
- desired projects
XProvides Information
Project Type
Processes considered
Existence of results
Project size
Last modification
XLocates
Project
Manager
The Edit Bar is very dynamic since it changes
depending upon the process being carried out.
The Edit Bar is by default located on the right
side of the HYDRUS main window.
Edit Bar
Output data include various Results.
Data are organized in a tree-like structure.
♦ A View Tab to specify what and how information will be
displayed in the View window, and
♦ A Sections Tab to show various Sections
Domain Geometry
Flow Parameters
FE-Mesh
Domain Properties
Initial and Boundary Conditions
Auxiliary Objects
The Navigator Bar is by default located on the left side of the
HYDRUS main window. The Navigator Bar has three Tabs:
Tabs
♦ A Data Tab to allow quick access to all input and output
data. Input data include:
Navigator Bar
2
101
♦ Time Layer Toolbar
♦ GUI Toolbar
♦ View Toolbar
♦ Tools Toolbar
♦ Standard Toolbar
Toolbars
Edit
Bars
File
Menus
Edit
Bars
Edit
View
Insert
3
102
Results
Tools
Options
Internal holes
Internal curves
X Boundary objects
- lines
- polylines
- splines
- arcs
- circles
- fixed points
X Boundary curves
X Graphical input of a flow domain
Domain Design
Calculation
Menus
Windows
Boundary curves
- Internal holes
- Internal curves
Boundary objects
- lines, polylines
- splines,
- arcs, circles
- fixed points
Domain Design – Boundary Curves
Tabs: Geometry, FE-Mesh, Domain Properties, Initial
Conditions, Boundary Conditions, Results
View Window
4
103
- isolines - contour maps
- spectral color maps
- velocity vectors
- animation of both contour and spectral maps
X Contour and spectral maps may be drawn for
- pressure heads
- water contents
- velocities
- concentrations
- temperatures
X Graphs of all variables at the boundaries, as well as along
any selected cross-section
X Presents results of the simulation by means of
Graphical Output
2) Mesh refinement - inserting new points in all triangles
which do not fulfill a certain smoothness criterion
3) Remeshing - implementation of Delaunay’s retriangulation
for the purpose of eliminating all nodes surrounded by more than six
triangles, as well as for avoiding extreme angles
4) Smoothing - smoothing of the mesh by solving a set of
coupled elliptic equations in a recursive algorithm
5) Convexity check - correction of possible errors which may
appear during smoothing of the finite element mesh
domain into triangles with vertices at given boundary nodes
triangular finite element mesh:
1) Fundamental triangulation - discretization of the flow
X Discretization of transport domain into unstructured
X Discretization of boundary curves
Meshgen –Mesh Generator
transport
Capillary Barrier
Material Distributions
of the flow domain
- material distribution
- scaling factors
- anisotropy parameters
X Observation Nodes
X Drains
X Root Uptake Distribution
X Material Layers - parameters which describe the properties
temperatures, and concentrations
X Initial Conditions for pressure heads, water contents,
X Boundary Conditions for water flow, heat and solute
Boundary and Domain Properties
5
104
Cut-off Wall
Solute Plume
Capillary Barrier
Velocity Vectors
Plume Movement in
a Transect with Stream
Cut-off Wall
Finite Element Mesh
6
105
Flowing Particles in Two-Dimensional Applications
7
106
- Scott et al. [1983], Kool and Parker [1988]
- Lenhard et al. [1991] – hysteresis without pumping
X Hysteresis
- van Genuchten [1980]
- Brooks and Correy [1964]
- modified van Genuchten type functions
[Vogel and Cislerova, 1989]
- Durner [1994]
- Kosugi [1994]
X Soil Hydraulic Properties :
HYDRUS (2D/3D) - Fortran Application
Actual and Cumulative Fluxes Across
Internal Meshlines
X Solute Transport - convection-dispersion equation
- liquid, solid, and gaseous phase
- adsorption
- linear, Freundlich, Langmuir isotherms
- nonequilibrium
chemical - two-site sorption model
physical - mobile-immobile water
- Henry’s Law
- convection and dispersion in liquid phase,
- diffusion in gaseous phase
- zero-order production in all three phases
- first-order degradation in all three phases
- chain reactions
- attachment/detachment and straining
transport of colloids, viruses, and bacteria
- filtration theory
X Nonequilibrium Water Flow – MobileMobile-immobile concept
- unsaturated
- partially saturated
- fully saturated
- sink term - water uptake by plant roots
- compensated
- uncompensated
- water stress
- salinity stress
- porous media:
X Richards Equation - saturatedsaturated-unsaturated water flow
- two-dimensional in variably-saturated porous media
- axisymmetrical three-dimensional
- three-dimensional
X Water, Solute, and Heat Movement:
8
107
with time
Switch the boundary condition from variable pressure head to zero flux
(e.g., disc permeameter)
Switch the boundary conditions from time-variable pressure head to zero
flux when the specified nodal pressure head is negative (e.g., above the
water table)
As above, except that an atmospheric boundary condition is assigned to
nodes with negative calculated pressure heads
As above, except that a seepage face boundary condition is assigned to
nodes with negative calculated pressure heads
Treat the time-variable flux boundary conditions similarly as atmospheric,
i.e., with limiting pressure heads (hCritS and hCritA)
Apply atmospheric boundary conditions on non-active seepage face
Snow accumulation on top of the soil surface when temperatures are
negative
Old: only Atmospheric and Seepage Face boundary conditions
Interpolate variable pressure head and flux boundary conditions smoothly
Dynamic, System-Dependent Boundary Conditions
anisotropy
X Scaling Procedure for Heterogeneous Soils in 2D
X Transport domain delineated by irregular
boundaries
X Nonuniform Soils;
Soils an arbitrary degree of local
X Heat Transport - convection-dispersion equation
- heat conduction
- convection
- atmospheric conditions
- free drainage
- horizontal drains
- One-Dimensional Vertical Infiltration; data from Warrick et al. [1971]
- Cone Penetrometer Infiltration Test; data from Gribb et al. [1998]
- In-Situ Multistep Extraction Experiment; data from Inoue et al. [1998]
- Water Flow and Nutrient Transport in a Layered Soil; data from de Vos [1997]
X Experimental Validation
- Water Flow in a Cropped Field Soil Profile; intercode comparison [Feddes et
al., 1978]
- Column Infiltration Test; intercode comparison [Davis and Neuman, 1983]
- Two-Dimensional Horizontal Infiltration; comparison with published results of
Rubin
[ 1968] and Zyvolovski at al. [1976]
- Steady Downward Unsaturated Flow Around Tunnel; comparison with 2D
analytical
solution [Philip et al., 1989]
- Two-Dimensional Solute Transport; comparison with 2D analytical solution
[Cleary
and Ungs, 1978]
X Mathematical Verification
HYDRUS-2D - Testing
X Flow and Transport:
- vertical plane
- horizontal plane
- three-dimensional region exhibiting radial symmetry
X Water Flow Boundary Conditions:
Conditions
- prescribed head and flux
- seepage face
- deep drainage
solute transport equation:
- upstream weighting
- performance index
X Three Stabilizing Options to avoid oscillation in the numerical solution of the
9
108
Computer Session 4-7
The purpose of Computer Sessions 4 through 7 is to give HYDRUS (2D/3D)
users hands-on experience with the software package. Four examples are given
to familiarize users with the major parts and modules of HYDRUS (e.g., the
Graphical User Interface, the Project Manager, FE-Mesh generation,
specification of Domain properties, and Initial and Boundary conditions, and
Graphical Output), and with the main concepts and procedures of pre- and
post-processing (e.g., domain design, boundary and domain discretization,
initial and boundary conditions specification, and graphical display of results).
The following four examples are considered out in Computer Sessions 4
through 7:
I.
Infiltration from a subsurface source into a vertical plane (Computer
Session 4)
A. Water flow
B. Solute transport
II. Furrow irrigation with a solute pulse (Computer Session 5)
III. Flow and transport along a transect to a stream (Computer Session 6)
A. Steady-state water flow
B. Water and contaminant source at the surface
C. Plume movement towards the stream
IV. Three-Dimensional Water Flow and Solute Transport (Computer
Session 7)
109
Computer Session 4
Subsurface Line Source
The example in this computer session considers a subsurface line source
(e.g. drip irrigation) of water (first without and then with a solute) in a
vertical cross-section. The (x, z) transport domain is 75 x 100 cm2, with the
source located 20 cm below the soil surface on the left boundary of the
transport domain. Infiltration is initiated with a variable flux boundary
condition and is maintained for 1 day, with the duration of the solute pulse
being 0.1 days; with 2 cycles per week. An unstructured finite element mesh
is generated using the Meshgen program. The example is again divided into
two parts: first only water flow is considered, after which solute transport is
added. This example will familiarize users with the basic concepts of
transport domain design in the graphical environment of HYDRUS, with
boundaries and domain discretization, and with the graphical display of
results using contour and spectral maps.
(75, 100)
(0, 100)
(0, 80)
(0, 0)
(75, 0)
110
Computer Session 4
A. Infiltration of Water From a Subsurface Source
Project Manager (File->Project Manager)
Button "New"
New Project (or File->New Project)
Name: Source1
Description: Infiltration of Water from a Subsurface Source
Working Directory: Temporary – is deleted after closing the project
Button "Next"
Geometry Information (Edit->Domain Geometry->Geometry Information)
Type of Geometry: 2D Vertical Plane
Domain Definition: General
Units: cm
Initial Workspace: Xmin=-25 cm, Xmax=100 cm, Zmin=-25 cm, Zmax=125 cm
(to accommodate the transport domain)
Button "Next"
Main Processes (Edit->Flow and Transport Parameters->Main Processes)
Check Box: Water Flow
Button "Next"
Time Information (Edit->Flow and Transport Parameters->Time Information)
Time Units: days
Final Time: 7
Initial Time Step:
0.0001
Maximum Time Step: 5
Time Variable BC:
Check
Number of Time-Variable BC: 4
Button "Next"
Output Information (Edit->Flow and Transport Parameters->Output Information)
Print Options:
Check T-Level Information
Check Screen Output
Check Press Enter at the End
Print Times: Count: 14
Update
Print Times: 0.1, 0.25, 0.5, 0.75, 1, 1.5, 2, 3.5, 3.6, 3.75, 4, 4.5, 5.5, 7 d
Button "Next"
Water Flow - Iteration Criteria (Edit->Flow and Transport Parameters->Water Flow
Parameters->Output Information)
111
Computer Session 4
Leave default values as follows:
Maximum Number of Iterations: 10
Water Content Tolerance: 0.001
Pressure Head Tolerance: 1
Lower Optimal Iteration Range: 3
Upper Optimal Iteration Range: 7
Lower Time Step Multiplication Factor: 1.3
Upper Time Step Multiplication Factor: 0.7
Lower Limit of the Tension Interval: 0.0001
Initial Condition: In the Pressure Head
Button "Next"
Water Flow - Soil Hydraulic Model (Edit->Flow and Transport Parameters->Water
Flow Parameters ->Soil Hydraulic Model)
Radio button - van Genuchten-Mualem
Radio button - No hysteresis
Button "Next"
Water Flow - Soil Hydraulic Parameters (Edit->Flow and Transport Parameters>Water Flow Parameters ->Soil Hydraulic Parameters)
Leave default values for loam
Button "Next"
Variable Boundary Conditions (Edit->Flow and Transport Parameters->Variable
Boundary Conditions)
Time Transp
Var.Fl1 (variable flux)
1.0
0
-60 (drip discharge distributed over the circumference of
the drip)
3.5
0
0
4.5
0
-60
7
0
0
Button "Next"
FE-Mesh - FE-Mesh Generator (Edit->FE-Mesh->FE-Mesh Generator)
Radio button - Meshgen
Button "Next"
FE-Mesh - FE-Mesh Parameters (Edit->FE-Mesh->FE-Mesh Parameters)
Targeted FE – Size – Unselect Automatic and specify TS = 5 cm
Button "OK"
Definition of the Transport Geometry
Click on Grid and Work Plane Setting at the toolbar (or Tools->Grid and Work Plane)
Grid Point Spacing – Distance w = 1 cm, Distance h = 1 cm
112
Computer Session 4
Click on Snap to Grid at the toolbar (or Tools->Snap to Grid)
a) Outer Boundary
Select the Line-Polyline command from the Edit Bar (or Insert->Domain Geometry>Lines->Polylines->Graphically)
Nodes coordinates: (0,79), (0,0), (75,0), (75,100), (0,100),(0,81)
b) Drip
Zoom at the source.
Select the Arc via Three Points command from the Edit Bar (or Insert->Domain
Geometry->Lines->Arc->Graphically->Three Points) and specify coordinates of three
points: (0,81), (1,80), (0,79)
View All (View->View All).
Define the Base Surface
Domain Geometry->Surface->Graphically and click at the outer boundary
Alternatively select the Surface via Boundaries command from the Edit Bar and click at
the outer boundary
Define FE-Mesh
Insert->FE-Mesh Refinement->Graphically: a dialog appears in which specify Finite
Element size S=0.5 cm
After clicking OK, select three nodes defining the drip at the left side.
Click on the Insert Mesh Refinement at the Edit Bar, click New, and specify Finite
Element Size = 2 cm. Assign this refinement to the node at the top left corner.
Click Generate FE-Mesh from the Edit Bar (or Edit->FE-Mesh->Generate FE-Mesh)
Specify Initial Condition:
On the Navigator Bar click on Initial Conditions – Pressure Head (or Insert->Initial
Conditions->Pressure Head)
Select the entire transport domain
Click on the Set Value command at the Edit Bar, and set equal to -400 cm (Pressure
Head Value).
Water Flow Boundary Conditions:
On the Navigator Bar click on Boundary Conditions – Water Flow (or Insert->Boundary
Conditions->Constant Head)
Zoom on source: (0,80)
a) Select Variable Flux 1 from the Edit Bar and assign it to the arc
Click on View All at the toolbar (or View->View All)
b) Select Free Drainage from the Edit Bar and assign t points at the bottom of the soil
profile
113
Computer Session 4
Observation Nodes
On the Navigator Bar click on Domain Properties – Observation Nodes (or Insert>Domain Properties->Observation Nodes)
Click on the Insert command on the Edit Bar and specify 5 points arbitrarily in the
transport domain between source and drain
Menu: File->Save (or from Toolbar)
Menu: Calculation->Run HYDRUS (or from Toolbar)
(Execution time on 3 GHz PC – 18 s)
OUTPUT:
Results – Other Information: Observation Points (from the Navigator Bar, or Results>Observation Points from menu)
Pressure Heads
Water Contents
Results – Other Information: Boundary Fluxes (from the Navigator Bar, or Results>Boundary Information->Boundary Fluxes from menu)
Variable Boundary Flux
Free Drainage Boundary Flux
Results – Other Information: Cumulative Fluxes (from the Navigator Bar, or Results>Boundary Information->Cumulative Fluxes from menu)
Results – Other Information: Mass Balance Information (from the Navigator Bar, or
Results->Mass Balance Information from menu)
Results – Graphical Display: Pressure Heads (from the Navigator Bar, or Results>Display Quantity->Pressure Heads from menu)
Use Listbox Time Layer or Slidebar on the Edit Bar to view results for different
print times
Check Flow Animation
Select Boundary Line Chart from the Edit Bar and draw pressure heads for one
vertical column
Select Cross Section Chart and draw pressure heads through the middle of the
column
Select different display modes using Options->Graph Type
Results – Graphical Display: Water Contents (from the Navigator Bar, or Results>Display Quantity->Water Contents from menu)
Results – Graphical Display: Velocity Vectors (from the Navigator Bar, or Results>Display Quantity->Velocity Vectors from menu)
114
Computer Session 4
B. Infiltration of Water and Solute From a
Subsurface Source
Close the Source1 Project (click Save Project at the Toolbar or File->Save)
Select the Source1 project
Button "Copy"
Name: Source2
Description: Infiltration of Water and Solute from a Subsurface Source
Button "OK"
Check Box: Solute Transport
Button "OK"
Solute Transport – General Info (Edit->Flow and Transport Parameters->Solute
Transport Parameters->General Information)
Leave default values
Button "Next"
Solute Transport - Solute Transport Parameters (Edit->Flow and Transport
Parameters->Solute Transport Parameters->Solute Transport Parameters)
Leave the default values
Bulk Density = 1.5 cm3/g
Disp.L = 2 cm
Disp.T = 0.2 cm
Diff.=0
Button "Next"
Solute Transport - Transport Parameters (Edit->Flow and Transport Parameters>Solute Transport Parameters->Solute Reaction Parameters)
Leave the default values for tracer
Note that cBnd in Boundary Conditions is equal to 1 (this is boundary
concentration)
Button "Next"
Variable Boundary Conditions (Edit->Flow and Transport Parameters->Variable
Click on Time 1 and click add line
Click on Time 4.5 and click add line
Time Transp
Var.Fl.1
cValue1
0.1
0
-60
1
1.0
0
-60
0
115
Computer Session 4
3.5
0
3.6
0
4.5
0
7
0
Button "Next"
0
-60
-60
0
0
1
0
0
Import the final pressure head profile from Source1 as the initial condition for Source1
(Edit->Initial Conditions->Import)
Find project Source1
Select Pressure Head and click OK
On the Navigator Bar click Initial Condition.
OUTPUT:
Results – Other Information: Solute Fluxes (from the Navigator Bar, or Results>Boundary Information->Solute Fluxes from menu)
Free Drainage Boundary Flux
Results – Graphical Display: Concentrations (from the Navigator Bar, or Results>Display Quantity->Concentrations from menu)
Click with the right mouse button on the color scale and from the pop-up menu click on
Min/Max Global in Time. See how the display changed.
116
Computer Session 4
Observation Nodes: Concentration
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
1
2
3
4
Time [days]
117
5
6
7
118
Computer Session 5
Furrow infiltration with a solute pulse
The third example considers alternate furrow irrigation into a soil profile with a
subsurface drain. Water infiltration is evaluated for 100 days, with a solute pulse being
added to the irrigation water during the first 50 days. The soil profile is 1 m deep with
furrows 3 m apart; the drain is located in the middle between the two furrows at a depth
of 75 cm. Alternate furrow irrigation is initiated by ponding the left furrow;
mathematically this is accomplished using a constant pressure head boundary condition.
The drain is represented by a circle to which a seepage face boundary condition is
applied. Users become in this example more familiar with the basic concepts of
transport domain design in the graphical environment of HYDRUS, including how to
numerically define boundary objects, and again with boundaries and domain
discretization. Initial and boundary conditions are specified, and graphical displays of
the results using contour and spectrum maps, including animation, are provided, for a
more complex transport domain than in the previous example.
Button "New"
Name: Furrow
Description: Furrow irrigation with solute pulse
Button "Next"
Units: cm
Initial Workspace: Xmin=-50 cm, Xmax=350 cm, Zmin=-50 cm, Zmax=150 cm
(to accommodate the transport domain)
Button "Next"
Check Box: Water Flow and Solute Transport
Button "Next"
Time Units: days
Final Time: 100
Initial Time Step:
0.01
Button "Next"
119
Computer Session 5
Print Options:
Check Screen Output
Update
Print Times: 0.5, 1, 2.5, 5 10, 20, 30, 40, 50, 50.5, 51, 52.5, 55, 60, 70, 80, 90, 100
Button "Next"
Button "Next"
Button "Next"
Leave default values for silt
Explore Catalog of Soil Hydraulic Properties and Neural Network Predictions
Button "Next"
Solute Transport – General Info (Edit->Flow and Transport Parameters->Solute
Transport Parameters->General Information)
Leave default values except
Select GFE with artificial dispersion
Pulse Duration = 50 d
Button "Next"
Solute Transport - Solute Transport Parameters
Leave the default values
120
Computer Session 5
Disp.L = 1 cm
Disp.T = 0.1 cm
Diff.W=10 cm2/d
Button "Next"
Leave the default values for tracer
Note that cBnd in Boundary Conditions is equal to 1 (this is boundary
concentration)
Button "Next"
Button "Next"
Targeted FE – Size – Unselect Automatic and specify TS = 10 cm
Button "OK"
a) Outer Boundary
b) Drain
Select the Circle via Center and Radius command from the Edit Bar (or Insert->Domain
Geometry->Lines->Circle->Graphically)
Specify Coordinates of the Center X=150 cm, Z = 25 cm numerically on the Edit Bar
121
Computer Session 5
Click Apply
Specify Parameter R = 5 cm
Click Apply
Click Stop
the outer boundary
Select the Opening via Boundaries command at the Edit Bar and click on the circle. This
will specify that the inside of the circle is not part of the transport domain.
Define FE-Mesh
Element size S=2.5 cm
After clicking OK, select two nodes at the bottom of the furrow at the left side.
Select the entire transport domain
Click on the Set Value command at the Edit Bar, check Equilibrium from the lowest
located nodal point, and set equal to 50 cm (Bottom Pressure Head Value).
Zoom in on the left furrow.
a) Select Constant Head from the Edit Bar, select bottom of the left furrow and 3 nodes
on the side, specify 12 cm with Equilibrium from the lowest located nodal point.
Zoom on the drain.
b) Select "Seepage face": nodes along the drain.
Default View.
Observation Nodes
On the Navigator Bar click on Domain Properties – Observation Nodes (or Insert>Domain Properties->Observation Nodes)
Click on the Insert command on the Edit Bar and specify 5 points arbitrarily in the
transport domain between source and drain
122
Computer Session 5
OUTPUT:
Results – Other Information: Observation Points (from the Navigator Bar, or Results>Observation Points from menu)
Pressure Heads
Water Contents
Results – Other Information: Boundary Fluxes (from the Navigator Bar, or Results>Boundary Information->Boundary Fluxes from menu)
Constant Boundary Flux
Seepage Face Boundary Flux
Results – Other Information: Cumulative Fluxes (from the Navigator Bar, or Results>Boundary Information->Cumulative Fluxes from menu)
print times
Results – Graphical Display: Water Contents (from the Navigator Bar, or Results>Display Quantity->Water Contents from menu)
123
Computer Session 5
124
Computer Session 6
Flow and transport in a transect to a stream
The most complicated fourth example considers water flow and solute transport in a
vertical transect with a stream. The transport domain is relatively complex and consists
of objects formed by polylines and splines. The problem, divided into three parts, also
demonstrates how results of a previous simulation can be used in follow-up
calculations with different boundary conditions or having additional features. At first
(A), steady state water flow in the transect towards the stream is calculated. Second
(B), a source (e.g., simulating water drainage from waste disposal site) is added to the
soil surface about 30 m to the left of the stream for a duration of 100 d. Finally (C), the
contaminant source is assumed to be removed after 100 days. Transport of the 100-day
solute pulse through the unsaturated zone into groundwater and to the stream is
subsequently followed for 1100 days.
A. Steady-state water flow
B. Water and contaminant source at the surface
C. Plume movement towards a stream
We believe that by carrying out these four examples, HYDRUS users will obtain the
basic skills necessary to solve their own two-dimensional problems. We wish you all the
luck and patience needed in this endeavor.
125
Computer Session 6
A. Water Flow to a Stream 1
Button "New"
Name: Plume1
Description: Water flow to a stream - 1
Button "Next"
Units: cm
Initial Workspace: Xmin=-100 cm, Xmax=5100 cm, Zmin=-50 cm, Zmax=550
cm (to accommodate the transport domain)
Button "Next"
Button "Next"
Time Units: days
Final Time: 100
Initial Time Step:
0.0001
Button "Next"
Print Options:
Check Screen Output
Update
Print Times: 1 5 10 25 50 100
Button "Next"
126
Computer Session 6
Button "Next"
Button "Next"
Button "Next"
Button "Next"
Tab Main: Targeted FE – Size –Automatic with TS = 25 cm
Tab Stretching: Stretching Factor = 3
Button "OK"
View->View Stretching: In Z-direction: 5
Define Outer Boundary
Select the Spline command from the Edit Bar (or Insert->Domain Geometry->Lines>Splines->Graphically)
127
Computer Session 6
*
Notice that units in this figure are in meters, and thus have to be converted to cm
the outer boundary
Define FE-Mesh
Element size S=10.0 cm.
Alternatively select FE-Mesh from the Navigator Bar and Insert Mesh Refinement from
the Edit Bar and specify Finite Element size S=10.0 cm.
After clicking OK, select all nodes at the top of the transport domain.
Water Flow Initial Conditions:
a) Select the entire transport domain between x=0 cm and 4700 cm.
located nodal point, set equal to 400 cm (Bottom Pressure Head Value), and check
Slope in the x-direction = -2.8o.
b) Select the entire transport domain between x=4600 cm and 5000 cm.
located nodal point, set equal to 175 cm (Bottom Pressure Head Value), and check
Slope in the x-direction = 2.4o.
128
Computer Session 6
a) Select Constant Head from the Edit Bar, select the left side boundary, and specify
400 cm with Equilibrium from the lowest located nodal point.
b) Select Constant Head from the Edit Bar, select the right side boundary, and specify
190 cm with Equilibrium from the lowest located nodal point.
c) Zoom on the stream. Select Constant Head from the Edit Bar, select all nodes with
the z-coordinate smaller than 175 cm, and specify 80 cm with Equilibrium from the
lowest located nodal point.
d) Zoom on the slope left of the stream. Select Seepage Face from the Edit Bar and
select all nodes with the z-coordinate smaller than 300 cm,
Default View.
OUTPUT:
print times
Results – Graphical Display: Velocity Vectors (from the Navigator Bar, or Results>Display Quantity->Velocity Vectorss from menu)
129
Computer Session 6
B. Water Flow and Solute Transport to a
Stream 2
Add the source at the soil surface:
Close the Plume1 Project (click Save Project at the Toolbar or File->Save)
Select the Plume1 project
Button "Copy"
Name: Plume2
Description: Water flow and solute transport to a stream - 2
Button "OK"
Button "OK"
Solute Transport - General Information (Edit->Flow and Transport Parameters>Solute Transport Parameters->General Information)
Select GFE with artificial dispersion
Button "Next"
Solute Transport - Transport Parameters (Edit->Flow and Transport Parameters>Solute Transport Parameters-> Solute Transport Parameters)
Diffus. W. = 3
Disp.L = 10
Disp.T = 1
Button "Next"
Solute Transport - Reaction Parameters (Edit->Flow and Transport Parameters>Solute Transport Parameters-> Solute Reaction Parameters)
CBound1=0
Cbound2=1
Button "Next"
Water Flow Initial Condition:
Import the final pressure head profile from Plume1 as the initial condition for Plume2
Find project Plume1
Water Flow and Solute Transport Boundary Conditions:
130
Computer Session 6
a) On the Navigator Bar click on Boundary Conditions – Water Flow.
Zoom on the soil surface with x=16-17 m.
Select Constant Head from the Edit Bar, select the top four nodes between x=16 and
17 m, and specify h=0 cm.
b) On the Navigator Bar click on Boundary Conditions – Solute Transport.
Select Third-Type from the Edit Bar, select the top nodes between x=16 and 17
m, specify Pointer to the Vector of Boundary Conditions = 2.
Menu: View->View All (or from Toolbar)
OUTPUT:
print times
Click with the right mouse button on the color scale and from the pop-up menu click on
Min/Max Global in Time. See how the display changed.
131
Computer Session 6
C. Water Flow and Solute Transport to a
Stream 3
Change boundary condition after 100 d of simulation:
Close the Plume2 Project (click Save Project at the Toolbar or File->Save)
Button "Copy"
Name: Plume3
Description: Water flow and solute transport to a stream - 3
Button "OK"
Time Units: days
Initial time: 100
Final time: 1200
Button "Next"
Print Options:
Check Screen Output
Update
Default
Button "Next"
Water Flow and Solute Transport Initial Condition:
Import the final pressure head profile from Plume2 as the initial condition for Plume3
Find project Plume2
Select Pressure Head and Concentrations and click OK
Water Flow and Solute Transport Boundary Conditions:
On the Navigator Bar click on Boundary Conditions – Water Flow.
Select Constant Flux from the Edit bar and assigned it to all nodes at the soil surface
between the seepage face and the left side, and between the stream and the right side; and
specify a flux=0.05 cm/d.
132
Computer Session 6
Observation Nodes
On the Navigator Bar click on Domain Properties – Observation Nodes.
Click on Insert at the Edit Bar and specify 5 points arbitrarily between the source and the
stream.
Menu: View->View All (or from Toolbar)
OUTPUT:
Menu: Post-Processing ->Observation Points: Concentrations
Menu: Post-Processing ->Time Information: Peclet Numbers
Menu: Post-Processing ->Boundary Information->Solute Fluxes: Constant Boundary
Flux
Menu: Post-Processing ->Mass Balances Information
Menu: Post-Processing -> Graphical Display of Results
Concentrations
133
134
HYDRUS (2D/3D) Computer Session
Three-Dimensional Water Flow and Solute Transport
This tutorial considers water flow and solute transport in a simple three-dimensional
transport domain. The transport domain is a relatively simple hexahedral domain with
a slope in the X-direction. Dimensions of the transport domain are 1000 * 250 * 200
cm and there is a groundwater 100 cm below the soil surface. There is a source of
water and contaminant at the soil surface. The problem is divided into two parts. In the
first part, the geometry of the transport domain and its discretization is defined and
initial and boundary conditions are specified. In the second part, final pressure head
profile from the first run is imported as an initial condition, and pulse of solute is
added into the surface source. The example thus again demonstrates how results of a
previous simulation can be used in follow-up calculations with different boundary
conditions or having additional features. Users will learn how to define a simple threedimensional transport domain and how to use Sections when defining initial and
boundary conditions. Users will also learn various ways of viewing transport domain
and simulation results.
135
A. Three-Dimensional Water Flow
Button "New"
Name: 3DTest1
Description: 3D HYDRUS short course example - water flow
Button "Next"
Type of Geometry: 3D-Layered
Domain Definition: Hexahedral
Units: cm
Initial Workspace: Xmin = 0 cm, Xmax = 1000 cm, Ymin = 0 cm, Ymax=250 cm,
Zmin = 0 cm, Zmax=200 cm (to accommodate the transport domain)
Button "Next"
Hexahedral Domain Definition Information (Edit->Domain Geometry->Geometry
Definition)
Dimension: Lx = 1000 cm, Ly = 250 cm, Lz = 200 cm
Slope: Alpha = - 5o, Beta = 0
Button "Next"
Button "Next"
Time Units: days
Final Time: 5
Initial Time Step:
0.0001
Number of Time-Variable Boundary Records = 1
Button "Next"
Print Options:
Check Screen Output
Update
136
Print Times: 0.25, 0.5, 0.75, 1, 1.5, 2, 2.5, 3, 4, 5
Button "Next"
Button "Next"
Button "Next"
Button "Next"
Time-Variable Boundary Conditions (Edit->Flow and Transport Parameters->Variable
Time = 5 d
Transp = 0
Var.H-2 = 100
Var.H-3 = 100
Button "Next"
137
Hexahedral Domain Spatial Discretization (Edit->FE-Mesh->FE-Mesh Parameters)
Horizontal Discretization in X
Count = 39
Entries in the x column: 0, 25, 50, 75, 100, 125, 150, 170, 185, 195, 200,
205, 210, 220, 235, 250, 265, 280, 290, 295, 300, 305, 315, 330, 350, 375,
400, 450, 500, 550, 600, 650, 700, 750, 800, 850, 900, 950, 1000
Horizontal Discretization in Y
Count = 18
Entries in the y column: 0, 10, 20, 30, 40, 45, 50, 55, 60, 70, 85, 100, 125,
150, 175, 200, 225, 250.
Horizontal Discretization in Z
Count = 22
Entries in the z column: 200, 197.5, 195, 192.5, 190, 187, 184, 180, 175,
170, 165, 158, 150, 140, 125, 110, 95, 80, 65, 50, 25, 0
Button "Next"
Default Domain Properties (Edit->Domain Properties->Default Domain Properties)
Button "Next"
138
Water Flow Initial Conditions:
Select the entire transport domain.
Edit Bar: Click on Set Values
In the “Water Flow Initial Condition” dialog select:
Equilibrium from the lowest located nodal point
Slope in X – direction = -5o
Bottom Pressure Head Value: 100 cm
Boundary conditions:
Tool Bar: View Commands (
Direction->In Y-direction)
Tool Bar: Perspective view (
): In Y-direction (or from Menu: View->View in
) (or from Menu: View->Perspective)
Select the first column of nodes on the left and on the Edit Bar select “Variable Head 2”
boundary condition.
Select the last column of nodes on the right and on the Edit Bar select “Variable Head 3”
boundary condition.
Navigator Bar: Select the “Section” Tab and select “D2_001 Mesh Layer, Z=200 cm”
Section
Tool Bar: View Commands ( ): In Reverse Z-direction (or from Menu: View->View in
Direction->In Reverse Z-direction).
Tool Bar: Zoom by Rectangle (
cm) and Y=(0-100 cm)
) and zoom on area of approximately X=(150 - 350
Select nodes between X=(200-300 cm) and Y=(0-50 cm).
From the Edit Bar select the “Constant Flux” boundary condition and in the “Constant
Flux BC” dialog specify Flux value of 10 cm/d.
139
Observation Nodes (Tab Domain Properties or Insert->Domain Properties->Observation
Nodes)
Navigator Bar: Select the “Section” Tab and select “D1_001 Shell” Section
Tool Bar: View Commands (
Direction->In Y-direction)
): In Y-direction (or from Menu: View->View in
Edit Bar: Insert Observation Nodes approximately as follows:
OUTPUT:
print times
Results – Other Information: Observation Nodes (from the Navigator Bar, or
Results->Observation Nodes from menu)
140
B. Three-Dimensional Water Flow and Solute
Transport
Close the 3DTest1 Project (click Save Project at the Toolbar or File->Save)
Button "Copy"
Name: 3DTest1
Description: 3D HYDRUS short course example - water flow and solute transport
Button "3DTest2"
Button "Next"
Final Time: 50
Initial Time Step:
0.01
Button "Next"
Print Options:
Update
Print Times: 1, 2, 5, 10, 20, 30, 40, 50
Button "OK"
Solute Transport - General Information (Edit->Flow and Transport Parameters>Solute Transport Parameters->General Information)
Pulse Duration = 5 days
Button "Next"
Solute Transport - Transport Parameters (Edit->Flow and Transport Parameters>Solute Transport Parameters-> Solute Transport Parameters)
Disp.L = 10
Disp.T = 1
Button "Next"
Time-Variable Boundary Conditions (Edit->Flow and Transport Parameters->Variable
Time = 50 d
Import the final pressure head profile from Source1 as the initial condition for Source1
141
Find project 3DTest1
(Execution time on 3 GHz PC – 10 min)
Check out various output options
142
143
Fractured Rock
Riverside, CA
Jirka Šimůnek and Rien van Genuchten
Modeling Nonequilibrium and
Preferential Flow and Transport
with HYDRUS
Pot et al. (2005)
Photo of the Soil Structure at the Column Scale
Heterogeneity, Layering
1
144
Pot et al. (2005)
- same models for each region
- different models for each region
X Dual-Permeability Approach
- to solute transport only
- to both water flow and solute transport
X Dual-Porosity Approaches
- mono-porosity hydraulic property models
- dual-porosity hydraulic property models
X Uniform Flow Models
Hierarchical System of Models in HYDRUS-1D:
Nonequilibrium and Preferential Flow and Transport
Diameter : 5/14 cm
Height : 15/30 cm
Laboratory Column Experiments
Mobile
c
θ
s= sk + se
sk
se
Chemical Nonequilibrium
Two-Site Sorption Model
a)
θ = θim + θmo
θ = θim + θmo
Solute
Immob. Mobile
Solute
Immob. Mobile
Water
Immob. Mobile
Water
c)
d)
Fast
Fast
θ = θM + θF
Slow
Solute
Slow
Water
Fast
Solute
Water
Slow
θ = θM ,im + θM ,mo + θF
Im. Slow Fast
e)
a) Uniform Flow
b) Mobile-Immobile Water
c) Dual-Porosity
d) Dual-Permeability
e) Dual-Permeability with MIM in the Matrix Domain
θ
Solute
Water
b)
Physical Nonequilibrium Solute Transport Models
θ = θim + θmo
Immob.
Solute
Water
Physical Nonequilibrium
Mobile-Immobile Water
(Dual-Porosity Model)
Traditional Nonequilibrium Solute Transport Models
2
145
b) + c)
e)
d) Dual-Permeability
e) Dual-Permeability with MIM
d)
a)
b)
c)
d)
e)
sk
a)
sk
c
θ
s2k
s1k
c)
c
θ
sime
d)
cim
θim
cmo
θmo
Immob. Mob.
smok
smoe
smk
sme
e)
cm
θm
cf
θf
Slow Fast
One-Site Kinetic Model
Two-Site Model (kinetic and instantaneous sorption)
Two Kinetic Sites Model
Dual-Porosity with One Kinetic Site Model
Dual-permeability with Two-Site Model
c
θ
se
b)
sfk
sfe
Chemical Nonequilibrium Solute Transport Models
a) Uniform Flow
b) Mobile-Immobile Water
c) Dual-Porosity
a)
Physical Nonequilibrium Solute Transport Models
HYDRUS GUI - Water Flow
HYDRUS GUI - Water Flow
3
146
n -m
θ (h) − θ r
=
θ s −θr
K ( Se ) = K s SeA+2+ 2 / λ
S e ( h) =
)
⎤
⎦⎥
(α h > 1)
(α h ≤ 1)
Se
θr, θs - residual and saturated water contents
α, n, m (= 1 - 1/n), l and λ - empirical parameters
Ks
(α h) − λ
1
Brooks and Corey (1964):
K ( Se ) = K s Sel ⎡⎢1 − (1 - S
⎣
2
1/ m m
e
Se (h ) = [1 + (α h ) ]
van Genuchten (1980):
with mono-porosity hydraulic property models
Uniform Flow Models
Se
θr, θs
h0, σ, and l
Ks
- residual and saturated water contents
- empirical parameters
l
e
2
⎧ ln ( h / h0 ) ⎫
θ (h) − θ r 1
= erfc ⎨
⎬
2
θs − θr
2σ ⎭
⎩
⎧⎪ 1
⎡ ln ( h / h0 )
⎤ ⎫⎪
K (h) = K s S ⎨ erfc ⎢
+ σ ⎥⎬
2σ
⎣
⎦ ⎭⎪
⎩⎪ 2
Se ( h ) =
Lognormal Distribution Model (Kosugi, 1996):
with mono-porosity hydraulic property models
Uniform Flow Models
∂ ( ρ s ) ∂ (θ c ) ∂ ⎛
∂c
⎞
+
= ⎜ θ D − qc ⎟ − φ
∂t
∂t
∂z ⎝
∂z
⎠
Solute Transport (Convection–Dispersion Equation)
∂θ ( h) ∂ ⎡
∂h
⎤
= ⎢ K ( h) − K ( h) ⎥ − S ( h)
∂t
∂z ⎣
∂z
⎦
Uniform Flow Models
Solute
Water
4
147
i =1
∑ wi
k
1
k
The hydraulic characteristics contain 4+2k unknown parameters: θr , θs , αi , ni , l, and Ks.
Of these, θr, θs, and Ks have a clear physical meaning, whereas αi, ni and l are essentially
empirical parameters determining the shape of the retention and hydraulic conductivity
functions [van Genuchten, 1980].
k
- number of overlapping subregions
- weighting factors for the sub-curves
wi
αi, ni, mi (= 1 - 1/ni), and l - empirical parameters of the sub-curves.
⎤⎞
⎦ ⎟⎠
2
n mi
(1+ αi h i )
θr , θs - residual and saturated water contents, respectively
Se
θ (h) - θ r
=
θs - θr
⎛
1/ mi mi
⎡
⎜ ∑ wiα i ⎣1- (1- S ei )
⎛ k
⎞
K (θ ) = K s ⎜ ∑ wi S eli ⎟ ⎝ i =1
2
⎝ i =1
⎠
⎛ k
⎞
⎜ ∑ wiα i ⎟
⎝ i =1
⎠
Se ( h ) =
-1
0
2
3
1
4
Fracture
Matrix
Total
5
-10
-8
-6
-4
-2
0
-1
0
1
2
3
4
Fracture
Matrix
Total
5
Water
θ
θ = θ im + θ m
MobileMobile-Immobile Water
Solute
concept is applied to
Imob. Mobile
Solute Transport
Uniform Water Flow
Only solute transport is nonequilibrium
Dual-Porosity Approaches
Example of composite retention (left) and hydraulic conductivity (right)
functions (θr=0.00, θs=0.50, α1=0.01 cm-1, n1=1.50, l=0.5, Ks=1 cm d-1, w1=0.975,
w2=0.025, α2=1.00 cm-1, n2=5.00).
0
0.1
0.2
0.3
0.4
0.5
0.6
with multi-porosity hydraulic property models
Durner (1994):
Uniform Flow Models - Durner (1994)
with multi-porosity hydraulic property models
Wate r Conte nt [-]
Uniform Flow Models
Log(C onductivity [cm/days])
5
∂cim
∂s
+ (1 − f ) ρ im = α ( cmo - cim ) - φim
∂t
∂t
∂θim cim
∂s
+ (1 − f ) ρ im = α (cmo - cim ) - φim + Γ s
∂t
∂t
∂θ mo cmo ∂f ρ smo ∂ ⎛
∂c
⎞
+
= ⎜ θ mo Dmo mo - qcmo ⎟ - α ( cmo - cim ) - φmo − Γ s
∂t
∂t
∂z ⎝
∂z
⎠
∂θ mo ∂ ⎡
∂h
⎤
= ⎢ K ( h) − K ( h) ⎥ − Smo − Γ w
∂t
∂z ⎣
∂z
⎦
∂θ im
=
− Sim + Γ w
∂t
Both water flow and solute transport are nonequilibrium
θim
∂θ mo cmo ∂f ρ smo ∂ ⎛
∂c
⎞
+
= ⎜ θ mo Dmo mo - qcmo ⎟ - α (cmo - cim ) - φmo
∂t
∂t
∂z ⎝
∂z
⎠
∂θ ( h) ∂ ⎡
∂h
⎤
= ⎢ K ( h) − K ( h) ⎥ − S ( h)
∂t
∂z ⎣
∂z
⎦
0
60
50
40
30
20
10
0
0.15
Mobile Water Content [-]
0.1
0.2
0.25
0.02
60
50
40
30
20
0.06
t = 7200 s
Immobile Water Content [-]
0.04
t = 1800 s
t = 3600 s
t = 5400 s
t=0
0.08
60
50
40
30
20
10
0
0.1
0.2
0.3
t=0
t = 1800 s
t = 3600 s
t = 5400 s
t = 7200 s
Total Water Content [-]
Water content profiles in the fracture domain, matrix
domain, and both domains combined.
0.05
t=0
t = 1800 s
t = 3600 s
t = 5400 s
t = 7200 s
0
10
0
θ = θim + θmo
Imob. Mobile
Solute
Water
Imob. Mobile
as well as to
Solute Transport
MobileMobile-Immobile Water
concept is applied to
Water Flow
Only solute transport is nonequilibrium
Depth [cm]
Depth [cm]
148
Depth [cm]
6
149
60
50
40
30
20
10
0
0.2
0.4
0.6
0.8
Relative Concentration [-]
1
60
50
40
0
0.4
0.6
0.8
0.2
t = 7200 s
t = 3600 s
t = 5400 s
t=0
t = 1800 s
1
0
60
50
40
30
20
10
0
0.2
0.4
0.6
0.8
t = 7200 s
t = 3600 s
t = 5400 s
t=0
t = 1800 s
Concentration profiles in the fracture domain, matrix
domain, and both domains combined.
0
t = 5400 s
t = 7200 s
t = 1800 s
t = 3600 s
t=0
30
20
10
0
Depth [cm]
Slow
Solute
Fast
Water
Slow
Fast
θ = θ M + θ F = (1 − w)θ m + wθ f
Terms:
Matrix – Fracture
Micropores – Macropores
Intra-porosity – Inter-porosity
Fast and slow moving domains
for both water flow and solute
transport
+fρ
=
∂h f
∂ ⎛
+ Kf
⎜Kf
∂z ⎝
∂z
∂s f
=
⎞
Γw
⎟− Sf ∂t
w
⎠
∂θ m ∂ ⎛
∂h
Γ
⎞
= ⎜ K m m + K m ⎟ − Sm + w
∂t
∂z ⎝
∂z
-w
1
⎠
∂θ f
∂c f ⎞ ∂qc f
Γ
∂ ⎛
-φf − s
⎜θ f D f
⎟∂t
∂t ∂z ⎝
∂z ⎠ ∂z
w
∂θ m cm
∂s
∂c ⎞ ∂qcm
Γ
∂ ⎛
+ (1 − f ) ρ m = ⎜ θ m Dm m ⎟ - φm − s
∂t
∂t
∂z ⎝
∂z ⎠ ∂z
1− w
∂θ f c f
Solute Transport:
Water Flow:
Gerke and van Genuchten (1993)
Two overlapping porous media, one for matrix flow,
one for preferential flow.
Dual-Permeability Approaches
1
Depth [cm]
Depth [cm]
7
150
0
0.02
0.06
Time [d]
0.04
0.08
Matrix Flux
Mass Transfer
Fracture Flux
0.1
40
0.25
35
30
25
20
15
10
5
b) 0
0.3
0.35
0.4
Water Content [-]
0.45
0.5
t = 0.08 d
t=0
t = 0.01 d
t = 0.04 d
c)
40
35
30
25
20
15
10
5
0
0
0.005
0.01
0.015
Water Content [-]
0.02
water content in the immobile zone
first-order rate coefficient
effective fluid saturations of the mobile and immobile regions,
respectively.
Compared to assuming a pressure head based driving force, the dual-porosity
model based on this mass transfer equation requires significantly fewer
parameters since one does not need to know the retention function for the
matrix region explicitly, but only its residual and saturated water contents.
Semo , Seim
θim
ω
∂θ
Γ w = im = ω ⎡⎣ Semo − Seim ⎤⎦
∂t
0.025
t=0
t = 0.01 d
t = 0.04 d
t = 0.08 d
The mass transfer rate for water between the fracture and matrix regions
can be proportional to the difference in effective saturations of the two
regions (e.g. Phillip, 1968; Šimůnek et al., 2001) using the first-order rate
equation:
Water Mass Exchange
Infiltration and mass exchange fluxes (a), water contents in
the matrix (b) and fracture (c) domains
0
10
20
30
40
50
60
Depth [cm]
a)
Flux [cm/d]
(two overlapping porous media, one for matrix flow, one for preferential flow)
Gerke and van Genuchten [1993]:
Dual-Permeability Approach
Depth [cm]
d
2
β
Ka γ w
K a ( h ) = 0.5 ⎡⎣ K a ( h f ) + K a ( hm ) ⎤⎦
Ka fracture-matrix interface using a simple arithmetic average involving
both hf and hm as follows:
αw =
αw first-order mass transfer coefficient
Γ w = α w (h f - hm )
The rate of exchange of water between the fracture and matrix regions,
can also be assumed to be proportional to the difference in pressure
heads between the two pore regions (Gerke and van Genuchten, 1993a):
Water Mass Exchange
8
151
d
2
β
Da
Pot et al. (2005)
Bromide and Isoproturon BTCs
Da effective diffusion coefficient which represents the diffusion properties
of the fracture-matrix interface as well as other parameters
αs =
c* equal to cf for Γw>0 and cm for Γw<0
αs first-order solute mass transfer coefficient (T-1) of the form:
Γ s = α s (1 − wm )θm (c f - cm ) + Γ w c *
The transfer rate, Γs, for solutes between the fracture and matrix
regions is usually given as the sum of diffusive and convective fluxes,
and can be written as (Gerke and van Genuchten, 1996):
Solute Mass Exchange
Bromide BTCs measured and calculated using four different physical transport models
(CDE, MIM, DP and DP-MIM). Relative concentrations of the effluents are presented
against time.
Bromide BTCs (Pot et al., 2005)
9
0
5
10
Time [d]
15
ω =10.
ω =0.5
ω =0.1
20
0
0
5
10
Time [d]
15
c)
b)
20
0
5
10
Time [d]
α=10
α=0.5
α=0.1
15
20
0
0.2
0.4
0.6
0.8
1
0
5
Time [d]
10
fem=1
fem=0.7
fem=0.4
fem=0.1
15
Breakthrough curves calculated using the DualDual-Porosity Model with One Kinetic
Site for a 10-cm long soil column and the following parameters: solute pulse
duration = 10 d, q = 3 cm/d, θ = 0.5, θmo = 0.3, θim = 0.2, λmo = 1 cm, Kd = 1 cm3/g,
ρb= 1.5 g/cm3, fmo = 0.6, α = 0.1 d-1, fem = 0.4, αch = 0.1, 0.5, 10 d-1 (left), and αch = 0.1
d-1 and fem = 1.0, 0.7, 0.4, and 0.1 (right).
0
0.2
0.4
0.4
0.2
0.6
a)
0.8
1
Effects of Mass Transfer and Fraction of Equilibrium Sorption
Sites in the Dual-Porosity Model with One Kinetic Site
multiple permeability preferential flow are needed to
describe transport processes for different fluxes
X Macropores dominate flow at high velocities, when soils
are close to saturation
X Mesopores contribute to flow at high velocities, and
dominate transport at lower velocities
X No flow domains often develop in soils
X Models considering multiple porosity domain and
needed to describe transport processes in field soils
X Complex, highly flexible numerical models are often
equilibrium processes depending on water fluxes
X Undisturbed soils often display contrasted physical non-
Conclusions
0.6
0.8
1
Breakthrough curves calculated using the MobileMobile-Immobile Water Model for a 10cm long soil column and the following parameters: q = 3 cm/d, θ = 0.5, θmo = 0.3, θim
= 0.2, λmo = 1 cm, Kd = 1 cm3/g, ρb= 1.5 g/cm3, fmo = 0.6, ωmim = 0.1, 0.5, 10 d-1 (left),
and ωmim = 0.5 d-1 and a) fmo = 0.4, θmo = 0.2, θim = 0.3, b) fmo = 0.6, θmo = 0.3, θim =
0.2, and c) fmo = 0.8, θmo = 0.4, θim = 0.1 (right).
0
0.2
0.4
0.6
Concentration [-]
0.8
1
Concentration [-]
Effects of Mass Transfer and Immobile Water
Content in the MIM Model
Isoproturon BTCs measured and calculated
using four different physical transport
models. Relative concentrations of the
effluents are presented against time.
Concentration [-]
Isoproturon BTCs (Pot et al., 2005)
Concentration [-]
152
20
10
153
0.8
1
0
2
4
Time [d]
6
8
ω=0.5
ω=0.1
10
model applications for structured soils: b) Pesticide
transport, J. Contam. Hydrology, Special Issue “Flow
Domains”, 104(1-4), 36-60, 2009.
X Köhne, J. M., S. Köhne, and J. Šimůnek, A review of
model applications for structured soils: a) Water
flow and tracer transport, J. Contam. Hydrology,
Special Issue “Flow Domains”, 104(1-4), 4-35, 2009 .
X Köhne, J. M., S. Köhne, and J. Šimůnek, A review of
Recent Preferential Flow and
Solute Transport Reviews
Breakthrough curves calculated using the DualDual-Permeability Model for a 10-cm
long soil column and the following parameters: qm = 3 cm/d, qf = 30 cm/d, θ = θm =
θf = 0.5, w=0.1, λm = λf = 1 cm, Kdm = Kdf = 1 cm3/g, ρb= 1.5 g/cm3, ωdp = 0, 0.1, 0.5 d1. Matrix, fracture and total breakthrough curves are represented by thin, medium
and thick lines, respectively.
0
0.2
0.4
0.6
ω=0
Effects of Mass Transfer in the DualPermeability Model
Concentrations [-]
X
X
X
X
X
X
X
transfer term for variably saturated dual-permeability models, Water Resour. Res., 40,
doi:10.1029/2004WR00385, 2004.
Köhne, J. M., S. Köhne, B. P. Mohanty, and J. Šimůnek, Inverse mobile-immobile modeling of transport
during transient flow: Effect of between-domain transfer and initial soil moisture, Vadose Zone Journal,
3(4), 1309-1321, 2004.
Kodešová, R., J. Kozák, J. Šimůnek, and O. Vacek, Field and numerical study of chlorotoluron transport in
the soil profile: Comparison of single and dual-permeability model, Plant, Soil and Environment, 51(6),
2005.
Pot, V., J. Šimůnek, P. Benoit, Y. Coquet, A. Yra and M.-J. Martínez-Cordón, Impact of rainfall intensity on
the transport of two herbicides in undisturbed grassed filter strip soil cores. J. of Contaminant Hydrology, 81,
63-88, 2005.
Haws, N. W., P. S. C. Rao, and J. Šimůnek, Single-porosity and dual-porosity modeling of water flow and
solute transport in subsurface-drained fields using effective field-scale parameters, J. of Hydrology, 313(3-4),
257-273, 2005.
Köhne, S., B. Lennartz, J. M. Köhne, and J. Šimůnek, Bromide transport at a tile-drained field site:
experiment, one- and two-dimensional equilibrium and non-equilibrium numerical modeling, J. Hydrology,
321(1-4), 390-408, 2006.
Köhne, J. M., S. Köhne, and J. Šimůnek, Multi-process herbicide transport in structured soil columns:
Experiment and model analysis, J. Contam. Hydrology, 85, 1-32, 2006.
Dousset, S., M. Thevenot, V. Pot, J. Šimůnek, and F. Andreux, Evaluating equilibrium and non-equilibrium
transport of bromide and isoproturon in disturbed and undisturbed soil columns, J. Contam. Hydrol., 94,
261-276, 2007.
X Köhne, J. M., B. Mohanty, J. Šimůnek, and H. H. Gerke, Numerical evaluation of a second-order water
zeolite/iron pellets, Water Resour. Res., 40, doi:10.1029/2003WR002445, 2004.
X Zhang, P., J. Šimůnek, and R. S. Bowman, Nonideal transport of solute and colloidal tracers through reactive
transport in the vadose zone: review and case study, Journal of Hydrology, 272, 14-35, 2003.
X Šimůnek, J., N. J. Jarvis, M. Th. van Genuchten, and A. Gärdenäs, Nonequilibrium and preferential flow and
Preferential Flow and Solute Transport References
Program and examples are posted at:
http://www.pc-progress.com/en/Default.aspx?h1d-library
Šimůnek, J. and M. Th. van Genuchten, Modeling
nonequilibrium flow and transport with HYDRUS, Special
Issue “Vadose Zone Modeling”, Vadose Zone Journal, 7(2),
782-797, 2008.
Šimůnek, J., M. Šejna, H. Saito, M. Sakai, and M. Th. van
Genuchten, The HYDRUS-1D Software Package for
Simulating the Movement of Water, Heat, and Multiple
Solutes in Variably Saturated Media, Version 4.0, HYDRUS
Software Series 3, Department of Environmental Sciences,
University of California Riverside, Riverside, California, USA,
pp. 315, 2008.
References
11
154
HYDRUS-1D Computer Session
Nonequilibrium Water Flow and Solute Transport
In this computer session with HYDRUS-1D we demonstrate the capability of HYDRUS-1D to
simulate nonequilibrium water flow and solute transport using the dual-porosity model. The
dual-porosity model is demonstrated using the ponded infiltration into a 60-cm deep soil profile.
The soil hydraulic parameters of the macropore (mobile) domain are taken as follows: θr=0.0,
θs=0.200, α=0.041 cm-1, n=1.964, l=0.5, Ks=0.000722 cm s-1, while the (immobile) matrix
domain is assumed to have a saturated water content, θsim, of 0.15. Initial conditions are set equal
to a pressure head of –150 cm. We assume that water mass transfer is proportional to the gradient
of effective saturations in the two domains, with the mass transfer constant ω set at 0.00001 s-1.
For simplicity we consider only convective solute mass transfer between the two pore regions
(i.e. no diffusive transfer), with the dispersivity again fixed at 2 cm.
Results Discussion: While for ponded surface conditions water in the fracture domain quickly
reached full saturation (see the figure below), the water content of the matrix increased only
gradually with time. Consequently, the total water content, defined as the sum of the water
contents of both the fracture and matrix domains, also increased only gradually. The total water
content would be the quantity measured with most field water content measurement devices,
such as a TDR or neutron probe. Pressure head measurements using tensiometers are, on the
other hand, often dominated by the wetter fracture domain that reaches equilibrium relatively
quickly. The dual-porosity model can therefore explain often observed nonequilibrium between
pressure heads and water contents. Similar nonequilibrium profiles as for the water content are
also obtained for the solute concentration (see the modeling results).
References:
Šimůnek, J., N. J. Jarvis, M. Th. van Genuchten, and A. Gärdenäs, Review and comparison of
models for describing non-equilibrium and preferential flow and transport in the vadose
zone, Journal of Hydrology, 272, 14-35, 2003.
Šimůnek, J., M. Šejna, H. Saito, M. Sakai, and M. Th. van Genuchten, The HYDRUS-1D
Software Package for Simulating the Movement of Water, Heat, and Multiple Solutes in
Variably Saturated Media, Version 4.0, HYDRUS Software Series 3, Department of
Environmental Sciences, University of California Riverside, Riverside, California, USA, pp.
315, 2008.
155
0
0
10
10
20
20
Depth [cm]
Depth [cm]
30
t=0
40
30
t=0
40
t = 1800 s
t = 1800 s
t = 3600 s
t = 3600 s
50
50
t = 5400 s
t = 5400 s
t = 7200 s
t = 7200 s
60
0
0.05
0.1
0.15
0.2
60
0.02
0.25
Mobile Water Content [-]
0.04
0.05
0.06
0.07
0.08
Immobile Water Content [-]
Fracture Domain
Matrix Domain
0
0
10
10
20
20
Depth [cm]
Depth [cm]
0.03
30
40
30
40
t=0
t=0
t = 1800 s
t = 1800 s
t = 3600 s
50
t = 3600 s
50
t = 5400 s
t = 5400 s
t = 7200 s
t = 7200 s
60
60
0
0.05
0.1
0.15
0.2
0.25
0.3
0
Total Water Content [-]
0.000002
0.000004
0.000006
0.000008
0.00001
Mass Transfer [1/s]
Both Domains
Water content profiles in the fracture (mobile) domain (top left), the matrix (immobile) domain
(top right), and both domains combined (bottom left), as well as the water mass transfer term
(bottom right) as calculated using the dual-porosity model.
156
Nonequilibrium Water Flow and Solute Transport
Project Manager
Button "New"
Name: Nonequil
Description: Nonequilibrium Water Flow and Solute Transport
Button "OK"
Button "Open"
Main Processes
Heading: Nonequilibrium Water Flow and Solute Transport
Radio Button: General Solute Transport
Button "Next"
Length Units: cm
Decline from Vertical Axes: 1
Button "Next"
Time Information
Time Units: Seconds
Final Time: 7200
Number of Time-Variable Boundary Records: 1
Button "Next"
Print Information
Check T-Level Information, Every n time steps: 1
Check Print at Regular Time Interval, Time Interval: 100
Check Screen Output
Print Times: 1800
3600 5400 7200
Button "OK"
Button "Next"
157
Button "Next"
Radio button – Dual-porosity (mobile-immobile water content mass transfer)
Button "Next"
Residual water content in the mobile zone, Qr = 0.0
Saturated water content in the mobile zone, Qs = 0.20
Alpha = 0.041
n = 1.964
Ks = 0.000722
l = 0.5
Residual water content in the immobile zone, QrIm = 0
Saturated water content in the immobile zone, QsIm = 0.15
Mass transfer coefficient, Omega = 1.e-05
Button "Next"
Upper Boundary Condition: Variable Pressure Head
Initial Conditions: In the Pressure Head
Button "Next"
Solute Transport – General Information
Leave default values, except for
Radio Button: Dual-Porosity (Mobile-Immobile Water) Model (Physical
Nonequilibrium)
Button "Next"
Leave default values for tracer, except
Disp. = 2 cm
Frac = 1 (fraction of sorption sites at equilibrium with the solution)
ThImob = 0 (immobile water content)
Button "Next"
Solute Transport - Transport and Reaction Parameters
Leave default values for tracer
Button "Next"
Solute Transport – Boundary Conditions
158
Lower Boundary Condition: Zero Concentration Gradient
Button "Next"
Time-Variable Boundary Conditions
Time
hTop [cm]
cTop
7200
1
1
Button "Next"
cBot
0
HYDRUS-1D Guide: Do you want to run Profile Application
Button "OK"
from the tool bar)
Conditions->Profile Discretization (or
Click the “Number” command from the Edit Bar and specify 61 nodes.
Conditions->Initial Conditions->Pressure Head (or
from the tool bar)
Select with the Mouse the entire soil profile
Specify initial water content of -150 cm
Include observation points at 10, 20, and 30 cm
Save and Exit
Execute HYDRUS-1D
OUTPUT:
Observation Points
Profile Information
159
160
161
KLh
KLT
Kvh
KvT
- hydraulic conductivity for liquid phase fluxes due to gradient of h [L T-1]
- hydraulic conductivity for liquid phase fluxes due to gradient of T [L2 T-1 K-1]
- isothermal vapor hydraulic conductivity [L T-1]
- thermal vapor hydraulic conductivity [L2 K-1 T-1]
∂θ ∂ ⎡
∂h
∂T ⎤
KTh
=
+ K Lh + KTT
−S
∂t ∂z ⎢⎣
∂z
∂z ⎥⎦
∂θ ∂qliquid ∂qvapor
=
+
−S
∂t
∂z
∂z
∂θ ∂ ⎡
∂h
∂T
∂h
∂T ⎤
K Lh (h) + K Lh (h) + K LT (h)
=
+ K vh
+ K vT
−S
∂t ∂z ⎢⎣
∂z
∂z
∂z
∂z ⎥⎦
∂θ ∂ ⎡
∂h
∂T
=
( K Lh + K vh ) + K Lh + ( K LT + K vT ) ⎤⎥ − S
∂t ∂z ⎢⎣
∂z
∂z ⎦
Modified Richards equation:
Liquid Water & Water Vapor Flow
Riverside, CA
Tokyo University of Agriculture and Technology,
Fuchu, Tokyo, Japan
Jirka Šimůnek, Hirotaka Saito,
And Rien van Genuchten
Simulating Coupled Movement of
Water, Vapor, and Energy
Using HYDRUS
D
ρw
ρ vs
Mg
Hr
RT
(2)
(3)
K vT =
ρw
D
dρ vs
dT
(4)
ηHr
Thermal hydraulic conductivity
⎛
1 dγ ⎞
⎟
K LT (T ) = K Lh (h)⎜⎜ hGwT
γ 0 dT ⎟⎠
⎝
Thermal vapor hydraulic conductivity
Thermal vapor flux
(Chung and Horton, 1987)
λ0 (θ ) = b1 + b2θ + b3θ 0.5
λ (θ ) = λ0 (θ ) + β Cw q l
4.18
1.80
(2.5-0.02369T)ρw
Cv [MJm-3K-1]
L0 [MJm-3]
Value
Cw [MJm-3K-1]
Parameter
Volumetric heat capacity
Conduction of sensible heat
Transferred by convection of liquid water
Transferred by convection of water vapor
Latent heat transported by water vapor
∂T
+ Cw TqL + Cv Tqv + L0 qv
∂z
Apparent thermal conductivity
(1)
(2)
(3)
(4)
q h = −λ(θ )
(1)
Heat Transport
K vh =
Isothermal vapor hydraulic conductivity
]
m 2
K Lh (h) = K S Sel 1 − (1 − Se1/ m )
[
Isothermal hydraulic conductivity
Isothermal vapor flux
qvapor = qvh + qvT
∂h
∂T
= −K vh(h)
− K vT (T )
∂z
∂z
Thermal flux
∂h
∂T
− K Lh (h) − K LT (T )
∂z
∂z
Isothermal flux
q Liquid = qLh + qLT = −K Lh (h)
Liquid Water & Water Vapor Flow
1
(1)
(2)
(3)
(4)
(5)
Soil heat flow by conduction
Convection of sensible heat by water flow
Heat removed by root water uptake
Transfer of latent heat by diffusion of water vapor
Transfer of sensible heat by diffusion of water vapor
∂θ v ∂ ⎡
∂T ⎤
∂q T
∂q
∂q T
= ⎢λ (θ ) ⎥ − Cw l − Cw ST − L0 v − Cv v
∂t
∂z ⎣
∂z ⎦
∂z
∂z
∂z
(1)
(2)
(3)
(4)
(5)
Boundary
conditions
Surface heat, G
Rn = H + LE + G
Estimation of each component using
available models and meteorological
data
Sensible heat, H
Latent heat, LE
Net radiation, Rn
Surface Energy Balance
∂t
+ L0
0.1
0.15
0.2
0
0
0.02
0.04
Total Flux [cm/d]
0.06 0
0
2
4
6
8
10
10
20
Temperature [C]
D e p th [cm]
t=25
t=5
t=1
t=0.25 d
T=0
30
0
2
4
6
8
10
0
2
4
6
Concentration [-]
5
327
10
15
20
25
30
35
329
331
DOY
333
335
337
339
⎛ ⎛ t − 13 ⎞ ⎞
Ta = Ta + At ⋅ cos⎜ 2π ⎜
⎟⎟
⎝ ⎝ 24 ⎠ ⎠
maximum and minimum data
0
327
10
20
30
40
50
60
70
80
329
331
DOY
333
335
337
339
⎛ ⎛ t − 5 ⎞⎞
Hr = Hr + Ar ⋅ cos⎜ 2π ⎜
⎟⎟
⎝ ⎝ 24 ⎠ ⎠
X Approximation of continuous hourly change using daily
Meteorological Variables
Total flux=water flux+vapor flux
Water Content [-]
0
0.05
2
2
0
4
t=25
t=25
6
t=5
t=1
t=5
8
t=0.25 d
t=1
T=0
t=0.25 d
10
T=0
4
6
8
10
o
∂C pT
8
t=25
t=5
t=1
10
t=0.25 d
T=0
Coupled movement of water, vapor and energy
Air temperature [ C]
Heat Transport
Relative humidity [%]
162
2
50 cm
Surface heat flux
Observation nodes
Zero pressure and temperature gradients @ bottom
Node in every 2 mm, leading to the
total of 251 nodes.
12 cm
7 cm
2 cm
Surface water flux (e.g., irrigation)
HYDRUS-1D Simulation
of meteorological data
X Solves energy and water
balance equations at soilatmosphere interface for
boundary conditions
X Numerically solves water and
heat transport models
simultaneously
X Approximates hourly changes
0 0
10
0.1
0.2
0.3
0.4
0.5
1
2
3
10
10
Pressure Head [cm]
[
]
θ s − θr
⎧
⎪ θr +
n m
θl = ⎨
1 + αh
⎪θ
⎩ s
10
drainage
imbibition
VG Model
4
h≥0
h<0
10
0.4447
0.393
1.534
b2 (C&H, 1987)
0.243
0.5
1.3822
b3 (C&H, 1987)
b1 (C&H, 1987)
l
n
0.0277
θs [-]
α [cm-1]
0.0117
34.2
Loam
θr [-]
¼ vG and Mualem model
¼ Clay fraction: 8.8 %
Parameter
Ks [cm/day]
X Arlington fine sandy loam
Soil Properties
Saito et al. (2006)
¼ Light irrigation on DOY 334 (0.549 cm) and 335 (0.199 cm)
depths every 20 and 40 min. respectively
¼ Soil temperature and water content measured at 2, 7, and 12 cm
December 5 (DOY 339), 1995
X Measurements: November 23 (DOY 327) through
Station in Riverside, California
X University of California Agricultural Experimental
X Public domain
X User friendly interface
Modified version
Validation - Study Site
Implementation in HYDRUS-1D
Water Content [-]
163
3
0
0.2
0.4
0.6
0.8
1
327
329
331
DOY
333
335
-20
327
-10
0
10
20
30
329
Net Rad.
Latent
Sensible
Surface
331
333
DOY
335
337
339
0
Saito et al. (2006)
337
339
1
2
3
4
5
6
7
⎛
⎞
6014 .79
− 7.92495 ⋅ 10 −3 (Ta + 273.16 )⎟⎟
exp⎜⎜ 31.3716 −
(Ta + 273.16 )
⎝
⎠ ⋅H
r
(Ta + 273.16 )
Heat Fluxes @ Surface
Vapor pressure [kPa]
−3
Vapor density [g/m3]
ρ va = 10
Heat Flux [MJ/m2/day]
-10
310
0
10
20
30
40
311
312
313
314
315
DOY
316
317
331
329
331
depth = 12cm
0
327
5
10
15
20
25
30
35
329
Observed
Simulated
depth = 2cm
0
327
5
10
15
20
25
30
35
DOY
333
DOY
333
335
335
337
337
339
339
331
329
331
depth = 12cm
0
327
0.1
0.2
0.3
329
Observed
Simulated
depth = 2cm
0
327
0.1
0.2
0.3
335
335
319
337
337
320
339
339
Saito et al. (2006)
DOY
333
DOY
333
318
Measured
Net Radiation
Short Rad
Long.Rad.
Rn = Rns + Rnl = (1− a)St + (ε sε aσTa4 − ε sσTs4 )
Prediction Performance
2
Net Radiation
Radiation [MJ/m /day]
Vapor Density
Temperature [oC]
Temperature [oC]
Volumetric Water Content [-]
Volumetric Water Content [-]
164
4
20
o
25
30
-15
-10
-5
0
Temperature [ C]
-25
0
0.1
-20
15
Flux [cm/day]
0
-50
10
-0.1
0.2
Isotherm liquid
Thermal liquid
Isothermal vapor
Thermal vapor
-40
-30
-20
-10
0
-25
-0.2
-20
-15
-10
Depth [cm]
0.1
-25
-0.2
-20
-15
-10
0.1
0.2
Saito et al. (2006)
Flux [cm/day]
0
Evapotranspiration [cm/d]
0
1/1/1999
0.2
0.4
0.6
0.8
1
1.2
3/2/1999
5/1/1999
Time [d]
6/30/1999
8/29/1999
10/28/1999
12/27/1999
Hargreaves
Penman-Montheith
Measured Bare Lysimeter
Measured Vegetated Lysimeter
Ra - extraterrestrial radiation in the same units as ETp [e.g., mm d-1 or J m-2s-1]
Tm - daily mean air temperature [oC]
TR - temperature range between the mean daily maximum and minimum [oC]
ETp = 0.0023Ra (Tm + 17.8 ) TR
0.15
-0.1
DOY=330.0 (midnight)
Water Content [-]
0.05
DOY 329.5
DOY 330
0
-5
Hargreaves Equation
Depth [cm]
-5
DOY=329.5 (noon)
Depth [cm]
0
Depth [cm]
Water and Vapor Fluxes
- reference crop evapotranspiration [mm d-1]
- net radiation at crop surface [MJ m-2d-1]
- soil heat flux [MJ m-2d-1]
- average temperature [oC]
- windspeed measured at 2m height [m s-1]
- vapour pressure deficit [kPa]
- slope vapour pressure curve [kPa oC-1]
- psychrometric constant [kPa oC-1]
- conversion factor
t < 0.264d, t > 0.736d
Hourly values between 06 a.m. and 18-24 p.m.
represent 1% of the total
daily value and a
sinusoidal shape is
followed during the rest
of the day
(Fayer, 2000)
0
0.5
1
1.5
2
2.5
3
0
0.2
0.6
Time [d]
0.4
⎛ 2π t π ⎞
Tp (t ) = 2.75Tp sin ⎜
− ⎟ t ∈ (0.264d, 0.736d)
⎝ 1day 2 ⎠
Tp (t ) = 0.24Tp
Daily Variations in Evaporation, and
Transpiration Rates
900
γ
ETo
Rn
G
T
U2
(ea-ed)
Δ
ET0 =
900
U 2 (ea - ed )
T + 273
Δ + γ (1 + 0.34U 2 )
0.408 Δ( Rn - G ) + γ
0.8
The Penman-Monteith combination method for calculating
of potential evapotranspiration [FAO, 1990]
Penman-Monteith Combination Equation
E, T [cm/d]
165
1
5
166
- hydraulic conductivity for liquid phase fluxes due to gradient in h
- hydraulic conductivity for liquid phase fluxes due to gradient in T
- isothermal vapor hydraulic conductivity
- thermal vapor hydraulic conductivity
(1)
(2)
(3)
(4)
(5)
(6)
+ L0
Soil heat flow by conduction
Convection of sensible heat by water flow
Heat removed by root water uptake
Transfer of latent heat by diffusion of water vapor
Transfer of sensible heat by diffusion of water vapor
Freezing/thawing term
Silty Clay
Depths (cm):
0, 0.5. 1, 2, 3.5, 5, 10
0.24
- 0.5
0.26
0.28
0.30
0.32
0.34
0.36
-6
- 0.5
-4
-2
0
2
4
0.0
0.0
0.5
1 .0
1.0
Time [d ays]
0.5
T ime [d ays]
1.5
1.5
2.0
2.0
0
0.7
- 0.5
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
- 8000 0
-0 .5
- 6000 0
- 4000 0
- 2000 0
0.0
0.0
0.5
1.0
1.0
T ime [d ays]
0.5
T ime [d ays]
1.5
1.5
2.0
2.0
∂θ v
∂θ
∂q T
∂q
∂q T
∂ ⎡
∂T ⎤
− L f ρ i i = ⎢λ (θ ) ⎥ − Cw l − Cw ST − L0 v − Cv v
∂t
∂t ∂z ⎣
∂z ⎦
∂z
∂z
∂z
(6)
(1)
(2)
(3)
(4)
(5)
Coupled movement of water and energy,
freezing/thawing cycle
∂t
∂C pT
Energy Transport:
KLh
KLT
Kvh
KvT
∂θ ρi ∂θ i ∂ ⎡
∂h
∂T
∂h
∂T ⎤
+
=
+ K vh
+ K vT
−S
K Lh (h) + K Lh (h) + K LT (h)
∂t ρ w ∂t ∂z ⎢⎣
∂z
∂z
∂z
∂z ⎥⎦
Modified Richards Equation:
1.00E+06
0.50
1.00E+07
1.00E+08
1.00E+09
1.00E+10
1.00E+11
1.00E+12
1.00E+06
0.02
1.00E+07
1.00E+08
1.00E+09
1.00E+10
1.00E+11
1.00E+12
0.00
0.00
-0.04
o
-0.06
o
-1.00
Temperature [ C]
-0.50
Temperature [ C]
-0.02
-1.50
Silty clay
Loam
Sand
-0.08
Sand
Silty clay
Loam
-2.00
-0.10
∂CaT ∂C pT
∂θ
=
− L f ρi i
∂t
∂t
∂t
dθi
Ca = C p − L f ρi
dT
Apparent heat capacity
for different textures
-1
-3
-1
-3
Apparent Capacity [Jm K ]
Apparent Capacity [Jm K ]
6
Coupled Water, Vapor and Heat Transport
In this computer session with HYDRUS-1D we demonstrate capabilities of HYDRUS-1D to
simulate coupled water, vapor and heat transport. Water contents, total fluxes, temperatures and
concentration profiles are calculated for a 10-cm long soil sample with zero water fluxes at both
the top and bottom boundaries, and with a specified temperature gradient along the sample
(Nasar and Horton, 1992).
Results Discussion: Increasing temperatures from the top to the bottom of the sample cause
vapor flow from the warmer bottom end of the sample toward the colder end. Water evaporates
at the warmer end, flows upward as vapor and condensates at the colder end. Water contents
correspondingly decrease at the warmer end, and increase at the colder bottom. As a
consequence of changing water contents, a pressure head gradient develops in the sample,
leading to water flow in a direction opposite to vapor flow. A steady-state is eventually reached
when upward vapor flow fully balances downward liquid flow. Since water evaporates at the
bottom of the sample and condensates at the top, solute becomes more concentrated near the
bottom and more diluted near the top. Also, the concentration profile should eventually reach
steady-state, although at a much later time, when the downward advective solute flux balances
the upward diffusive flux.
References:
Nassar, I. N., and R. Horton, Simultaneous transfer of heat, water, and solute in porous media: I.
Theoretical development, Soil Science Society of America Journal, 56, 1350-1356, 1992.
Šimůnek, J., M. Šejna, H. Saito, M. Sakai, and M. Th. van Genuchten, The HYDRUS-1D
Software Package for Simulating the Movement of Water, Heat, and Multiple Solutes in
Variably Saturated Media, Version 4.0, HYDRUS Software Series 3, Department of
Environmental Sciences, University of California Riverside, Riverside, California, USA, pp.
315, 2008.
167
10
10
T=0
t=0.25 d
t=1
t=5
t=25
Depth [cm]
8
6
T=0
t=0.25 d
t=1
t=5
t=25
8
6
4
4
2
2
a
b
0
0
0
0.05
0.1
0.15
0.2
0
0.02
Water Content [-]
10
0.04
0.06
Total Flux [cm/d]
10
c
d
T=0
t=0.25 d
8
8
t=1
Depth [cm]
t=5
6
6
4
4
T=0
t=0.25 d
t=1
t=5
t=25
2
t=25
2
0
0
0
10
20
30
Temperature [C]
0
2
4
6
8
10
Concentration [-]
Water content (a), total flux (b), temperature (c), and solute concentration (d) distributions in a 10-cm
long vertical soil sample with zero water fluxes across the top and bottom boundaries, and with
temperature increasing from top to bottom.
168
Coupled Water, Vapor and Heat Transport
Project Manager
Button "New"
Name: Coupled
Description: Coupled Water, Vapor and Heat Transport
Button "OK"
Button "Open"
Main Processes
Heading: Coupled Water, Vapor and Heat Transport
Check Box: Vapor Flow
Radio Button: General Solute Transport
Check Box: Heat Transport
Button "Next"
Length Units: cm
Decline from Vertical Axes: 0 (horizontal flow)
Button "Next"
Time Information
Time Units: Days
Final Time: 25
Button "Next"
Print Information
Check Screen Output
Print Times: 0.25
0.5
1
Button "OK"
Button "Next"
2
3
Water Content Tolerance: 5.e-06
Button "Next"
169
4
5
10
14
25
Keep default values as follows:
Button "Next"
Residual water content, Qr = 0.03
Saturated water content, Qs = 0.499
Alpha = 0.036
n = 1.56
Ks = 33.7
l = 0.5
Check Box: Temperature Dependence
Button "Next"
Lower Boundary Condition: Constant Flux
Initial Conditions: In the Water Content
Button "Next"
Water Flow – Constant Boundary Fluxes
Upper Boundary Flux: 0 (no flux)
Lower Boundary Flux: 0 (no flux)
Button "Next"
Solute Transport – General Information
Leave default values
Button "Next"
Leave default values for tracer, except
Disp. = 1 cm
Frac = 1 (fraction of sorption sites at equilibrium with the solution)
ThImob = 0 (immobile water content)
Button "Next"
Solute Transport - Transport and Reaction Parameters
Leave default values for tracer
Button "Next"
Solute Transport – Boundary Conditions
170
Boundary Condition = 0
Lower Boundary Condition: Concentration Flux BC
Boundary Condition = 0
Button "Next"
Heat Transport – Heat Transport Parameters
Temperature Amplitude: 0
Button "Next"
Heat Transport – Boundary Conditions
Upper Boundary Condition: Temperature BC
Boundary Value = 10 (cold end)
Lower Boundary Condition: Temperature BC
Boundary Value = 25 (warm end)
Button "Next"
HYDRUS-1D Guide: Do you want to run Profile Application
Button "OK"
from the tool bar)
Conditions->Profile Discretization (or
Click the “Number” command from the Edit Bar and specify 51 nodes.
from the tool bar)
Conditions->Initial Conditions->Water Content (or
Specify initial water content of 0.134
Conditions->Initial Conditions->Concentration (or
from the tool bar)
Specify initial concentration of 1.0
Include observation points at 0, 2, 4, 6, 8, and 10 cm
Save and Exit
Execute HYDRUS-1D
OUTPUT:
Observation Points
Profile Information
171
172
173
of Environmental Sciences, University of California Riverside, CA
and Disposal Department, SCK•CEN, Mol, Belgium
3Institute for Sanitary Engineering and Water Pollution Control, BOKU - University of
Natural Resources and Applied Life Sciences, Vienna, Austria
4Department of Mechanical Engineering,
- more freedom in designing particular chemical systems
- much wider possible applications
- e.g., DYNAMIX, HYDROGEOCHEM, MULTIFLO, CRUNCH,
and OS3D/GIMRT
X General Models
- limited in the number of species that can be incorporated
- restricted to problems having a prescribed chemical system
- much easier to use
- more computationally efficient than general models
- LEACHM (Wagenet and Hutson, 1987)
- UNSATCHEM (Šimůnek and Suarez, 1994; Šimůnek et al., 1996)
- reclamation models (Dutt et al., 1972; Tanji et al., 1972)
X Models with Specific Chemistry
Introduction - BioGeoChemical Models
2Waste
1Department
Jirka Šimůnek1, Diederik Jacques2, Günter Langergraber3,
Rien van Genuchten4, and Dirk Mallants2
Multicomponent Biogeochemical Transport
Modeling Using the HYDRUS Computer
Software Packages
with Specific Chemistry
Models
Šimůnek, 2005]
3. HP1 (HYDRUS-1D-PHREEQC) [Jacques and
XGeneral
module) [Langergraber and Šimůnek, 2005]
2. HYDRUS-CW2D (constructed wetlands
chemistry module) [Šimůnek et al., 2005]
1. HYDRUS-1D, version 3.0+ (includes major ion
XModels
Introduction - BioGeoChemical Models
generally affected by a large number of
nonlinear and often interactive physical,
chemical and biological processes.
X Simulating these processes requires a coupled
reactive transport code that integrates the
physical processes of water flow and
advective-dispersive transport with a range of
biogeochemical processes.
X Contaminant transport in the subsurface is
Introduction
1
174
cw volumetric concentrations of CO2 in the dissolved phase
ca volumetric concentrations of CO2 in the gas phase [L3L -3]
Dija effective soil matrix diffusion coefficient of CO2 in the gas
phase [L2T-1]
Dijw effective soil matrix dispersion coefficient of CO2 in the
dissolved phase [L2T -1]
qi soil water flux [LT -1]
qa volumetric air content [L3L -3]
P
CO2 production rate [L3L -3T -1]
Scw dissolved CO2 removed from the soil by root water uptake
[L3L-3]
∂ ( ca θ a + c w θ )
∂
∂ ca
∂
∂
∂
=
+
θ Dijw c w q cw - S cw + P
θ a Daij
∂t
∂ xi
∂ x j ∂ xi
∂ x j ∂ xi i
Šimůnek and Suarez [1993]:
Carbon Dioxide Transport
HYDRUS-1D
HYDRUS-1D (Šimůnek et al., 1998)
γ0
s, p
f(x, z)
f(T)
f(h), f(hφ)
f(cCO2)
f(t)
i
γ p( x,z ) = L pγ p 0 ∏ f
pi
i
CO 2
refers to the soil microorganism and plant roots
reduction coefficient dependence on the depth [L-2]
reduction coefficient dependence on the temperature [-]
reduction coef. dependence on the pressure and osmotic head [-]
reduction coefficient dependence on the concentration of CO2 [-]
reduction coefficient dependence on time [-]
optimal production of soil microorganisms or plant roots at
20oC under optimal moisture, nutrient and O2 conc. conditions
[L3L-2 T-1]
i
∏ f =f ( x, z) f (h) f (T ) f (c ) f (hφ ) f (t )
i
γ s ( x,z ) = L sγ s 0 ∏ f si
P(x, z) = γ s ( x, z ) + γ p ( x, z )
Production of Carbon Dioxide
- variably saturated water flow
- heat transport
- root water uptake
- solute transport
UNSATCHEM (Šimůnek et al., 1996)
- carbon dioxide transport
- major ion chemistry
- cation exchange
- precipitation-dissolution (instantaneous and kinetic)
- complexation

2
175
4
7
3
4 Sorbed species
(exchangeable)
5 CO2-H2O species
6 Silica species
H4SiO4, H3SiO4-, H2SiO42-
PCO2, H2CO3*, CO32-, HCO3-, H+, OH-,
H2O
Ca, Mg, Na, K
CaCO3, CaSO4⋅ 2H2O, MgCO3⋅ 3H2O,
Mg5(CO3)4(OH)2⋅ 4H2O,
Mg2Si3O7.5(OH) ⋅ 3H2O, CaMg(CO3)2
- valence of species i, j
K ij =
y+
( ci )1/ y
x+
x+
( c j )1/ x
y+
cT = ∑ ci
cj
ci
cT - cation exchange capacity [McM-1] (constant and independent of pH)
Kij - selectivity constant [-]
y, x
Gapon equation:
Cation Exchange Selectivity
Kinetic reactions: calcite precipitation/dissolution, dolomite dissolution
Activity coefficients: extended Debye-Hückel equations, Pitzer expressions
6
3 Precipitated
species
10 CaCO3o, CaHCO3+, CaSO4o, MgCO3o,
MgHCO3+, MgSO4o, NaCO3-, NaHCO3o,
NaSO4-, KSO4-
2 Complexed
species
Ca2+, Mg2+, Na+, K+, SO42-, Cl-, NO3-
7
1 Aqueous
components
- constants depending on the dielectric constant, density, and temperature
- ionic charge
- adjustable parameters
- ionic strength
A z2 I
+ bI
1 + Ba I
j
j
*
k
K a2 =
+
2HCO3 ⇔ H + CO3
*
(CO2-3 )( H + )
( HCO-3 )
( H + )( HCO-3 )
( H 2 CO*3 )
( H 2 CO3 )
P CO2 ( H 2 O)
KW
- dissociation constant for water
PCO2
- partial pressure of CO2(g)
KCO2
- Henry's Law constant
H2CO3* - both aqueous CO2 and H2CO3
Ka1, Ka2 - 1st and 2nd dissociation constant of carbonic acid
K a1 =
K CO2 =
( H + )(OH - )
KW =
( H 2 O)
*
+
H 2 CO3 ⇔ H + HCO3
H 2 CO3 ⇔ CO2(g) + H 2 O
+
H 2 O ⇔ H + OH
CO2 - H2O System
- molality
γiDH - modified Debye-Hückel activity coefficient
Bij, Cijk - specific coefficients for each interaction
mi
DH
ln γ i = ln γ i + ∑ B ij ( I ) m j + ∑ ∑ C ijk m j m k + ...
Pitzer expressions [Pitzer, 1979]:
A, B
z
a, b
I
ln γ = -
Extended version of the DebyeDebye-Hückel equation [Truesdell and Jones, 1974]:
Activity Coefficients
3
176
( H + )( H 3 SiO-4 )
( H 4 SiO4 )
K 12 =
( H + )2 ( H 2 SiO2-4 )
( H 4 SiO4 )
( Na + )( HCO-3 )
( NaHCOo3 )
2+
(Ca 2+ )( HCO-3 )
(CaHCO3+ )
(Mg )( HCO-3 )
+
(MgHCO3 )
K9 =
K6 =
K3 =
*
3
k 4 = k1′ +
2+
3
( Ca )( HCO )
1
*
k 2 ( H2 CO3) + k 3 ( H2 O)
( HS+ )
K SP
C
K a2
k1, k2, k3 - first order rate constants representing the
forward reactions
k4 - function representing the back reactions
where
C
R = k1 ( H ) + k 2 ( H2 CO ) + k 3 ( H2 O) - k 4
+
Reaction rates are calculated with the rate equation of
Plummer et al. [1978]:
Kinetic Model of Calcite Precipitation-Dissolution
Ki - equilibrium constants of the ith complexed species [-]
K 11 =
( Na + )(CO2-3 )
( NaCO-3 )
( K + )(SO2-4 )
( KSO-4 )
K8 =
( Na + )(SO24 )
( NaSO-4 )
K7 =
K 10 =
K5 =
2+
( Ca 2+ )(CO2-3 )
(CaCOo3 )
(Mg )(CO2-3 )
o
(MgCO3 )
K2 =
(Mg )(SO2-4 )
o
(MgSO4 )
2+
(Ca 2+ )(SO2-4 )
(CaSOo4 )
K4 =
K1 =
Complexation Equations
4
4
6
6
4.5
( O)
2+
(Mg )2 ( HCO-3 )4= K SSP K CO2 K4a1 P CO2 H 23
K w ( H 4 SiO4 )
4
6
2+ 5
10
H K CO 2 K a1 P CO 2
(Mg ) ( HCO-3 ) = K SP
4
2
K a2 K w
2+
P CO2
N K CO 2 K a1
(Mg )( HCO-3 )2= K SP
2
K a2 ( H 2 O )
k1, k2, k3 - first order rate constants representing the
forward reactions
k4 - function representing the back reactions
0.5
D
+ 0.5
* 0.5
R = k1 ( H ) + k 2 ( H2 CO3) + k 3 ( H2 O) - k 4 ( HCO3)
Reaction rates are calculated with the rate equation
of Busenberg and Plummer [1982]
Kinetic Model of Dolomite Dissolution
Sepiolite:
2
G
2+
2K SP = (Ca )(SO4 )( H 2 O )
( H O)
(Ca 2+ )( HCO-3 )2= K CSP K CO2 K a1 P CO2 2
K a2
Hydromagnesite:
Nesquehonite:
Calcite:
Gypsum:
Precipitation-Dissolution
4
0
20
A)
237.9y
234.2
20
200
50
100
40
SAR
10
60
0
80
100
80
60
40
20
0
B)
200d
0
150
100
60
20
20
40
SAR
10
60
0
80
100
80
60
40
20
0
C)
0
120d
80
20
60
40
20
40
SAR
0
60
80
A. Irrigation with high quality water and no amendments
B. Irrigation with gypsum-saturated
C. Irrigation with high quality water and no gypsum incorporated in top 20 cm
100
80
60
40
0
Reclamation Examples [Šimůnek and Suarez, 1997]
We compare reclamation by:
1) incorporation of gypsum into the top 30 cm of the soil profile
2) addition of gypsum to the irrigation water
3) dissolution of the calcite present in the soil (assuming fixed CO2,
and alternatively, utilizing the model predicted CO2)
4) acidified irrigation water, dissolution of the calcite present in the
soil
D)
Cl
1.0
5.0
92.5
pH
7.93
7.74
1.09
100d
80
20
60
40
40
20
SAR
0
60
80
100
80
60
40
20
0
E)
60
70d
0
50
40
30
20
20
10
40
SAR
0
60
80
100
80
60
40
20
0
F)
0
16d
14
12
20
10
6
2
40
SAR
0
60
80
D. Irrigation with high quality water and calcite throughout the soil profile
E. Irrigation with acid water at pH 2.05 and calcite throughout the soil profile
F. Irrigation with acid water at pH 1.09 and calcite throughout the soil profile
100
80
60
40
20
0
0
Suarez, 1997]
Reclamation Examples [Šimůnek and
Composition of the Applied Water (mmolcL-1)
Ca
Na
HCO3 SO4
Dilute water
1.5
2.0
0.5
2.0
Gypsum saturated water 32.6
4.8
0.4
32.0
Acidified water
1.5
2.0
-100.
11.0
Soil Water Composition
Ca = 0.2 mmolcL-1
Na = 4.8
HCO3 = 0.4
Cl = 4.6
Properties
Soil Texture: Loam
KS = 60.072 cm d-1
θS = 0.48
CEC = 100 mmolc kg -1
Calcite = 0.5 mol kg -1
We demonstrate the use of UNSATCHEM to evaluate the reclamation of
a sodic soil using different amendments and reclamation strategies
(Šimůnek and Suarez, 1997). Among the important considerations for
reclamation are
1) quantity of water needed,
2) cost of chemical amendments,
3) quantity of amendment to be used,
4) time required for reclamation to be completed.
D e p th [c m ]
Initial Cond.
θi = 0.24
Temp = 20oC
ESP = 95 %
CO2 = 1%
Soil Characterization and Initial Conditions
Reclamation Example
D e p th [c m ]
177
5
178
HYDRUS (2D/3D)
Gonçalves et al. (2006)
Lysimeter Study
[Langergraber and Šimůnek, 2005]

CWs are used successfully with different quality of the
influent water and under various climatic conditions
CWs are effective in treating organic matter, nitrogen,
phosphorus, and additionally for decreasing the
concentrations of heavy metals, organic chemicals, and
pathogens
Constructed wetlands (CWs) or wetland treatment systems
CWs are wetlands designed to improve water quality
CWs use the same processes that occur in natural
wetlands but have the flexibility of being constructed
Constructed Wetlands
soil: 2. Parameter selection, sensitivity analysis and comparison of model predictions to
field data, Water Resour. Res., 29(2), 499-513, 1993.
Šimůnek, J., and D. L. Suarez, Two-dimensional transport model for variably saturated
porous media with major ion chemistry, Water Resour. Res., 30(4), 1115-1133, 1994.
Šimůnek, J., and D. L. Suarez, Sodic soil reclamation using multicomponent transport
modeling, ASCE J. Irrig. and Drain. Engineering, 123(5), 367-376, 1997.
Suarez, D. L., and J. Šimůnek, UNSATCHEM: Unsaturated water and solute transport
model with equilibrium and kinetic chemistry, Soil Sci. Soc. Am. J., 61, 1633-1646, 1997.
Gonçalves, M. C., J. Šimůnek, T. B. Ramos, J. C. Martins, M. J. Neves, and F. P. Pires,
Multicomponent solute transport in soil lysimeters irrigated with waters of different
quality, Water Resour. Res., 42, W08401, doi:10.1029/2006WR004802, 17 pp., 2006.
Corwin, D. L., J. D. Rhoades, J. Šimůnek, Leaching requirement for soil salinity control:
Steady-state vs. transient-state models, Agric. Water Management, 90(3), 165-180, 2007.
Herbst, M., H. J. Hellebrand, J. Bauer, J. A. Huisman, J. Šimůnek, L. Weihermüller, A.
Graf, J. Vanderborght, and H. Vereecken, Multiyear heterotrophic soil respiration:
evaluation of a coupled CO2 transport and carbon turnover model, Ecological Modelling,
214, 271-283, 2008.
Buchner, J. S., J. Šimůnek, J. Lee, D. E. Rolston, J. W. Hopmans, A. P. King, and J. Six,
Evaluation of CO2 fluxes from an agricultural field using a process-based numerical
model, J. Hydrology, doi: 10.1016/j.jhydrol.2008.07.035, 361(1-2), 131– 143, 2008.
Suarez, D. L., and J. Šimůnek, Modeling of carbon dioxide transport and production in
soil: 1. Model development, Water Resour. Res., 29(2), 487-497, 1993.
Šimůnek, J., and D. L. Suarez, Modeling of carbon dioxide transport and production in
HYDRUS-UNSATCHEM Publications
6
179
lysis (for heterotrophic and
autotrophic MO)
aerobic growth of Nitrosomonas
and Nitrobacter (nitrification)
anoxic growth of heterotrophic
MO using nitrate and nitrite
(denitrification)
aerobic growth of heterotrophic
MO (degradation of OM)
hydrolysis (slowly Î readily
biodegradable)
Heterotrophic Organisms
Hydrolysis
Aerobic growth of heterotrophs on readily biodegradable OM
NO3-growth of heterotrophs on readily biodegradable OM
NO2-growth of heterotrophs on readily biodegradable OM
Lysis
Nitrosomonas
Aerobic growth of N.somonas on NH4
Lysis of N-somonas
Nitrobacter
Aerobic growth of N.bacter on NO2
Lysis of N.bacter
9 Processes:
Dissolved oxygen O2
Organic matter: readily biodegradable CR, slowly biodegradable CS , inert CI
Nitrogen: NH4+, NO2-, NO3-, N2
Inorganic phosphorus IP
Heterotrophic micro-organisms: XH
Autotropic micro-organisms: Nitrosomonas & Nitrobacter, XAN
12 Components:
cCS c XH
c XH
K X + cCS c XH
cO 2
cCR
f N ,Het c XH
K Het ,O 2 + cO 2 K Het ,CR + cCR
rc3 = μ DN
K DN ,O 2
K DN , NO 2
cNO 3
cCR
f N ,DN c XH
K DN ,O 2 + cO 2 K DN ,NO 3 + cNO 3 K DN ,NO 2 + cNO 2 K DN ,CR + cCR
(denitrification) - consumes nitrate (NO3-) and readily biodegradable organic matter (CR).
Nitrate is reduced to dinitrogen (N2). Again, ammonium (NH4+) and inorganic phosphorus (IP)
are incorporated in the biomass
3. NO3-based growth of heterotrophs on readily biodegradable COD
rc2 = μH
biodegradable organic matter (CR), while ammonium (NH4+) and inorganic phosphorus (IP)
2. Aerobic growth of heterotrophic bacteria - consumes oxygen (O2) and readily
rc1 = K h
biodegradable organic matter CR, with a small fraction being converted into inert organic
matter CI. Ammonium (NH4+) and inorganic phosphorus (IP) are released.
1. Hydrolysis - conversion of slowly biodegradable organic matter CS into readily

Processes
(Langergraber and Šimůnek , 2005)
dissolved oxygen [mg O2/l],
readily and slowly biodegradable,
and inert OM [mg COD/l],
ammonia, nitrite, nitrate, and N2
[mg N/l]
inorganic phosphorus [mg P/l],
heterotrophic MO [mg COD/l],
autotrophic MO (Nitrosomonas
and Nitrobacter) [mg COD/l],
Organic N and P are modelled as
N and P content of the COD.
Constructed Wetlands Module

Components
(Langergraber and Simunek, 2005)
Constructed Wetland Module – CW2D
Constructed Wetland Module – CW2D
Subsurface vertical flow constructed wetlands:
Subsurface horizontal flow constructed wetlands:
Constructed Wetlands
7
cO 2
cNH 4
cIP
c XANs
K ANs ,O 2 + cO 2 K ANs , NH 4 + cNH 4 K ANs ,IP + cIP
sand - gravel 0/4 mm
CHAMBER A
DOWNFLOW
CHAMBER B
UPFLOW
drainage layer - gravel 16/32 m m
sand - gravel 0/4 mm
top layer - gravel 4/8 mm
INLET
Δ 10
geotextile
OUTLET
Two-stage subsurface vertical flow constructed wetland
rc6 = μ ANs
oxygen (O2), and produces nitrite (NO2-). Inorganic phosphorus (IP) and a small portion of
ammonium are incorporated in the biomass
6. Aerobic growth of Nitrosomonas on NH4N - consumes ammonia (NH4+) and
rc5 = bH c XH
ammonium (NH4+), and inorganic phosphorus (IP)
5. Lysis of heterotrophic bacteria - Lysis produces organic matter (CS, CR, and CI),
rc4 = μ DN
K DN ,O 2
cNO 2
cCR
f N ,DN c XH
K DN ,O 2 + cO 2 K DN ,NO 2 + cNO 2 K DN ,CR + cCR
Nitrite is reduced to dinitrogen (N2). Again, ammonium (NH4+) and inorganic phosphorus (IP)
4. NO2-based growth of heterotrophs on readily biodegradable COD
(denitrification) - consumes nitrate (NO2-) and readily biodegradable organic matter (CR).
15
55
15
45
180
cO 2
cNO 2
f N , ANb c XANb
K ANb,O 2 + cO 2 K ANb, NO 2 + cNO 2
50
100
150
200
250
Simulated steady-state distribution of heterotrophic organisms XH
0
rc9 = bHANb c XANb
(NH4+), and inorganic phosphorus (IP)
9. Lysis of Nitrobacter - produces organic matter (CS, CR and CI), ammonium
rc8 = μ ANb
and oxygen (O2), and produces nitrate (NO3-). Ammonium (NH4+) and inorganic
phosphorus (IP) are incorporated in the biomass
8. Aerobic growth of Nitrobacter on NO2N - consumes nitrite (NO2-)
rc7 = bHANs c XANs
ammonia (NH4+), and inorganic phosphorus (IP)
7. Lysis of Nitrosomonas - produces organic matter (CS, CR and CI),
8
181
10
20
30
40
50
60
70
After loading, NH4N increased whereas NO3N decreased. Further on
NH4N is nitrified resulting in an increasing NO3N concentration. At 15 cm
depth changes in concentration occurred mainly due to advection since
oxygen was already a limiting factor.
nitrate nitrogen NO3N (right). (downflow chamber)
Simulated time series of ammonia nitrogen NH4N (left) and
Simulated steady-state distribution of Nitrosomonas XANs
0
HP1 – HYDRUS-PHREEQC
(downflow chamber).
Simulated time series of dissolved oxygen DO
After loading, the DO concentration
decreased at the surface and at the
5 cm depth. At 15 cm depth,
however, the DO concentration
increased due to advective DO
transport with the infiltrating
water. Still, oxygen decreased
quickly (within 15 minutes after
loading) due to the consumption of
oxygen. No oxygen was found at
15 cm depth (i.e. in the saturated
zone) during the remainder of the
simulation. In the unsaturated zone
(5 cm depth) the DO concentration
increased again due to re-aeration,
and reaching oxygen saturation
after about 1 hour after loading.
9
182
Variably Saturated Water Flow
Solute Transport
Heat transport
Root water uptake

transport for element master/primary species (inert
transport) (HYDRUS)
¼Calculate speciations, equilibrium reactions, kinetic
reactions, … for each cell (PHREEQC)
¼Solve convection-dispersion equation for solute
¼Solve heat transport equation (HYDRUS)
¼Solve water flow equation (HYDRUS)
Coupling method: non-iterative sequential
approach (weak coupling)
Within a single time step:
HP1 - Coupling Procedure
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
Available chemical reactions:
PHREEQC [Parkhurst and Appelo, 1999]:

HYDRUSimůnek et al., 1998]:
HYDRUS-1D [Šimů
HP1 - Coupled HYDRUS-1D and PHREEQC

1D FE water flow in variably-saturated media
1D FE transport of multiple solutes by CDE
1D heat transport
Mixed equilibrium / kinetic biogeochemical reactions
¼Aqueous speciation (reactions in pore-water)
¼Cation exchange (on clay, organic matter, …)
¼Surface complexation (e.g. iron oxyhydroxides)
¼Mineral dissolution / precipitation
¼Any kinetic reactions (oxidation/reduction,
(bio)degradation, dissolution/precipitation)
HP1 – Model Features
Biogeochemical model
PHREEQC-2.4
A Coupled Numerical Code for
Variably Saturated Water Flow,
Solute Transport and
Biogeochemistry
in Soil Systems
Flow and transport model
HYDRUS-1D 4.0
Simulating water flow, transport and biogeochemical reactions in environmental
soil quality problems
Simulation Tool – HP1
10
0
Al
Na
3
6
9
Time (days)
PHREEQC
Hydrus1D-PHREEQC
0
0.002
0.004
0.006
0.008
0.01
12
Ca
Cl
15
0E+000
2E-004
4E-004
6E-004
8E-004
0
3
9
Time (days)
6
Cd
Pb
Zn
Transport and Cation Exchange
X
X
X
X
X
Concentration (m ol/l)
12
15
q=2 cm/d, λ=0.2 cm, CEC=11 mmol/cell.
Al=0.5, Br=11.9, K=2, Na=6, Mg=0.75, Cd=0.09,
Pb=0.1,
Cd
Pb
Zn=0.25
mmol/L.
Zn
Al= 0.1, Br=3.7, Cl=10, Ca=5,
Mg=1 mmol/L.
Ca
Al3+, Al(OH)2+, Al(OH)2+, Al(OH)3, Al(OH)4-, Br-, Cl-, Ca2+,
Ca(OH)+ , Cd2+, Cd(OH)+, Cd(OH)2, Cd(OH)3-, Cd(OH)42-,
CdCl+, CdCl2, CdCl3-, K+, KOH, Na+, NaOH, Mg2+,
Mg(OH)+, Pb2+, Pb(OH)+, Pb(OH)2, Pb(OH)3-, Pb(OH)42-,
PbCl+, PbCl2, PbCl3-, PbCl42-, Zn2+, Zn(OH)+, Zn(OH)2,
Zn(OH)3-, Zn(OH)42-, ZnCl+, ZnCl2, ZnCl3-, ZnCl42
AlX3, AlOHX2, CaX2, CdX2, KX, NaX, MgX2, PbX2, ZnX2
2ADNT
4ADNT
TAT
TNT -> 66% is transformed in 2ADNT and
34% is to 4ADNT
Transformation constants [1/hour]
TNT
0.01
2ADNT
0.006
4ADNT
0.04
Degradation
(Šimůnek et al., 2006)
Adsorption Coefficients Kd
[L/kg]:
TNT
3
2ADNT
5
4ADNT
6
Sorption (instantaneous)
Soil profile: 100 cm, loam, Ks=1 cm/h, 10 days
TNT in top 5 cm of soil: 1 mg/kg (6.61e-6 mol)
TNT dissolution: rate = 4.1 mg/cm2/hour (1.8e-5 mol/cm2/hour)
Solid 2ADNTT at equilibrium with the solution; Solubility = 2,800 mg/L
TNT
Transport of TNT and its Daughter Products
Exchange Species:
Boundary concentration:
Species and Complexes:
Parameters:
Initial concentrations:
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.
(cations - Ca, Mg, Na, K, Cd, Pb, Zn; anions – Cl, Br, Al)
Transport and Cation Exchange (major ions and heavy metals):
X Transport of heavy metals (Zn2+, Pb2+, and Cd2+) subject
to multiple cation exchange
Transport with mineral dissolution of amorphous SiO2 and
gibbsite (Al(OH)3)
Heavy metal transport in a medium with a pH-dependent
cation exchange complex
Infiltration of a hyperalkaline solution in a clay sample
(this example considers kinetic precipitation-dissolution of
kaolinite, illite, quartz, calcite, dolomite, gypsum,
hydrotalcite, and sepiolite)
Long-term transient flow and transport of major cations
(Na+, K+, Ca2+, and Mg2+) and heavy metals (Cd2+, Zn2+,
and Pb2+) in a soil profile.
Kinetic biodegradation of NTA (biomass, cobalt)
Verification of HYDRUS-PHREEQC
HP1 Examples
Concentration (m ol/l)
183
11
0
50
100
150
200
S1
1e-016
1e-014
1e-012
1e-010
1e-008
1e-006
1e-016
1e-014
0
50
150
200
250
S4
S3
S2
S1
0
50
150
Time [hours ]
100
200
Surface complexation reactions
Specific binding to charged surfaces (≡FeOH)
Related to amount of Fe-oxides
Multi-site cation exchange reactions
Related to amount of organic matter
Increases with increasing pH
UO22+ adsorbs
Aqueous speciation reactions
Chemical components: C, Ca, Cl, F, H, K,
Mg, N(5), Na, O(0), O(-2), P, S(6), U(6)
250
S4
S3
S2
S1
Breakthrough Curves
S1 – TNT
S2 – 2ADNT
S3 – 4ADNT
S4 – TAT
1e- 016
1e- 014
1e- 012
U-transport in agricultural field soils
Time [hours ]
100
Time [hours ]
250
S4
S3
S2
1e- 010
1e-012
1e- 008
1e- 006
1e-010
50 cm
1e-008
1e-006
100 cm
10 cm
0
0
-100
0.000000000
-80
-60
-40
-20
0
-100
-80
-60
-40
-20
C onc [mol/L]
0.000000002
C onc [mol/L]
2.5e-007
T5
T4
T3
0.000000004
5e-007
T10
T9
T8
T7
T1
T2
T6
T0
TAT
2ADNT
0
-100
-80
-60
-40
-20
0
0
-100
0.00000000
-80
-60
-40
-20
C onc [mol/L]
5e-009
C onc [mol/L]
0.00000001
1e-008
0.00000002
0
20
40
60
80
100
2
3
Total
SC
5
Increased U-sorption
Increased deprotonation
pH
4
CEC
6
by other cations
U-species replaced
Changing processes in U adsorption with increasing pH
4ADNT
TNT
% U(VI) adsorbed
184
12

No U initially present in soil profile (<> few 10 Bq/kg)

151
152
153
154 155 156
Time (year)
Steady-state
157
158
159
160
5 cm depth
• pH variations => variations in sorption potential (low pH => low sorption)
• Water content variations induce pH variations (dry soil => low pH)
150
3.4
3.6
3.8
4
4.2
Atmospheric
Applied each year on May 1 (1 g P/m2)
1.6×10-1 mol Ca(H2PO4)2 /m² in 1 cm of rain
=>3.8×10-6 mol U /m2 in 1 cm of rain (~105
Bq/ha)
200-year time series of synthetic meteorological data to
calculate precipitation and potential evaporation
Composition rain water from measurements
P-fertilizer (Ca(H2PO4)2): ~3000 Bq 238U/kg
Boundary condition
Transient flow conditions =>
transient geochemical conditions

Initial condition
0
8.0x10
-4
1.6x10
-3
(b)
0
0
-3
-3
(d)
-3
0
2.0x10
-9
4.0x10
-9
(f)
Transient
100 year
150 year
200 year
U (mol / 1000 cm³ soil)
0.0x10
100
75
50
25
0
• U moved faster under transient than under steady-state
• U-breakthrough after 100 y
• Ca, P, U accumulation in Bh-horizon (rich in o.m. & Fe-ox.)
P (mol / 1000 cm³ soil)
0.0x10 1.0x10 2.0x10 3.0x10
50
40
30
20
10
Steady-state
Ca (mol / 1000 cm³ soil)
0.0x10
50
40
30
20
10
0
3.4
Atmospheric
Steady-state
3.6
3.8
pH
4
4.2
5 cm depth
25 cm depth
•At least one order of magnitude variation in K
1x101
1x102
1x103
1x104
∆pH results in time variations of U-mobility
Depth (cm)
Depth profiles of Ca, P, and U after 200 years
of P-fertilization
Depth (cm)

pH
Depth (cm)
Initial and Boundary Conditions
K = adsorbed U (mol/l) / aqueous U (mol / l)
185
13
186
0
50
100
1x10
200
0
: steady-state
50
100
/ : transient
Long-term U flux = U application rate:~105 Bq/ha/y
E-horizon
150
100 cm
200
Saturated Flow and Transport Model HP1, Description, Verification and
Examples, Version 1.0, SCK•CEN-BLG-998, Waste and Disposal,
SCK•CEN, Mol, Belgium, 79 pp., 2005.
Jacques, D., J. Šimůnek, D. Mallants, and M. Th. van Genuchten, Operatorsplitting errors in coupled reactive transport codes for transient variably
saturated flow and contaminant transport in layered soil profiles, J.
Contam. Hydrology, 88, 197-218, 2006.
Šimůnek, J., D. Jacques, M. Th. van Genuchten, and D. Mallants,
Multicomponent geochemical transport modeling using the HYDRUS
computer software packages, J. Am. Water Resour. Assoc., 42(6), 1537-1547,
2006.
Jacques, D., J. Šimůnek, D. Mallants, and M. Th. van Genuchten, Modeling
coupled hydrological and chemical processes in the vadose zone: A case
study on long term uranium migration following mineral phosphorus
fertilization, Vadose Zone Journal, Special Issue “Vadose Zone Modeling”,
7(2), 698-711, 2008.
Jacques, D., J. Šimůnek, D. Mallants and M. Th. van Genuchten, Modelling
coupled water flow, solute transport and geochemical reactions affection
heavy metal migration in a Podzol soil, Geoderma, 145, 449-461, 2008.
Jacques, D., and J. Šimůnek, User Manual of the Multicomponent Variably-
HP1 Publications
1x10
150
0
-6
1x10
3
6
-6
19 cm
1x10
1x10
1x10-3
0
3
6
1x10-3
1x10
1x10
1x10
U-fluxes: steady-state vs. transient
U flux (Bq year-1 ha-1)
conditions (∆ pH =>∆ sorption)
Atmospheric boundary conditions important
when assessing U-flux to groundwater
Due to changing flow and geochemical
New biogeochemical transport code HP1 provides
useful insight into complex U-migration processes
U migration under atmospheric boundary
conditions faster than under steady-state flow
conditions

HYDRUS-1D coupled with UNSATCHEM
to simulate transport of major ions
HYDRUS-2D coupled with CW2D to
simulate processes in constructed wetlands
HYDRUS-1D coupled with PHREEQC to
simulate …..
Three new HYDRUS-based programs:
SUMMARY

Conclusions
14
Computer Session 10
HP1 Tutorials
Example 1: Transport and Cation Exchange (single pulse)
This example is adapted from Example 11 of the PHREEQC manual [Parkhurst and Appelo,
1999]. We will simulate the chemical composition of the effluent from an 8-cm column
containing a cation exchanger. The column initially contains a Na-K-NO3 solution in equilibrium
with the cation exchanger. The column is flushed with three pore volumes of a CaCl2 solution.
Ca, K and Na are at all times in equilibrium with the exchanger. The simulation is run for one
day; the fluid flux density is equal to 24 cm/d (0.00027777 cm/s).
The column is discretized into 40 finite elements (i.e., 41 nodes). The example assumes that the
same solution is initially associated with each node. Also, we use the same exchanger
composition for all nodes.
The initial solution is Na-K-NO3 solution is made by using 1 x 10-3 M NaNO3 and 2 x 10-4 M
KNO3 M. The inflowing CaCl2 solution has a concentration of 6 x 10-4 M. Both solutions were
prepared under oxidizing conditions (in equilibrium with the partial pressure of oxygen in the
atmosphere). The amount of exchange sites (X) is 1.1 meq/1000 cm³ soil. The log K constants
for the exchange reactions are defined in the PHREEQC.dat database and do not have to be
therefore specified at the input.
In this example, only the outflow concentrations of Cl, Ca, Na, and K are of interest.
Input
Project Manager
Button "New"
Name: CEC-1
Description: Transport and Cation Exchange, a single pulse
Button "OK"
Main Processes
Heading: Transport and Cation Exchange, a single pulse
Uncheck "Water Flow" (steady-state water flow)
Check "Solute Transport"
Select “HP1 (PHREEQC)”
Button "Next"
Depth of the soil profile: 8 (cm)
Button "Next"
Time Information
Time Units:
Seconds (Note that you can also just put it in days, this would also
be OK)
187
Computer Session 10
Final Time:
86400 (s)
Initial Time Step:
180
Minimum Time Step: 180
Maximum Time Step: 180 (Note: constant time step to have the same conditions as in
the original comparable PHREEQC calculations).
Button "Next"
Print Information
Button "Next"
Print Times
Button: "Default"
Button: "OK"
HP1 – Print and Punch Controls
Button: "Next
Lower Time Step Multiplication Factor: 1
Button "Next"
Button "Next"
Catalog of Soil Hydraulic Properties: Loam
Qs: 1 (Note: to have the same conditions as in the original comparable PHREEQC
calculations)
Ks: 0.00027777 (cm/s)
Button "Next"
Upper Boundary Condition: Constant Pressure Head
Lower Boundary Condition: Constant Pressure Head
Button "Next"
Number of Solutes: 7
Button "Next"
Solute Transport – HP1 Components and Database Pathway
Add seven components: Total_O, Total_H, Na, K, Ca, Cl, N(5)
Check: "Create PHREEQC.IN file Using HYDRUS GUI"
Button: "Next"
188
Computer Session 10
Solute Transport – HP1 Definitions
Definitions of Solution Composition
Define the initial condition 1001:
• K-Na-N(5) solution
• use pH to charge balance the solution
• Adapt the concentration of O(0) to be in equilibrium with the partial
pressure of oxygen in the atmosphere
Define the boundary condition 3001:
• Ca-Cl solution
• Use pH to charge balance the solution
• Adapt the concentration of O(0) to be in equilibrium with the partial
pressure of oxygen in the atmosphere
Solution 1001 Initial condition
-units mmol/kgw
pH 7 charge
Na 1
K 0.2
N(5) 1.2
O(0) 1 O2(g) -0.68
Solution 3001 Boundary solution
-units mmol/kgw
pH 7 charge
Ca 0.6
Cl 1.2
O(0) 1 O2(g) -0.68
Geochemical Model
Define for each node (41 nodes) the geochemical model, i.e., the cation exchange
assemblage X (0.0011 moles / 1000 cm³) and equilibrate it with the initial
solution (solution 1001).
EXCHANGE 1-41 @Layer 1@
X 0.0011
-equilibrate with solution 1001
Button: "OK"
Additional Output
Since output is required only for the total concentrations and such output is
available in the automatically generated file obs_node.out, there is not need to
define additional output.
Button: "Next"
189
Computer Session 10
Bulk Density: 1.5 (g/cm3)
Disp.:
0.2 (cm)
Button "Next"
Upper Boundary Condition:
Concentration Flux
Add the solution composition number (i.e., 3001) for the upper boundary condition
Lower Boundary Condition:
Zero Gradient
Button "Next"
Menu: Conditions->Profile Discretization
or Toolbar: Ladder
Number (from sidebar):
41
Button "Edit condition", select with Mouse the entire profile and specify 0 cm pressure
head.
Button "Insert", Insert a node at the bottom
Menu: File->Exit
Button "Next"
Close Project
Run project
Note: This exercise will produce following warnings: "Master species N(3) is present in solution
n but is not transported.". The same warning occurs for N(0). N(3) and N(0) are two secondary
master species from the primary master species N. Only the secondary master species N(5) was
defined as a component to be transported (Solute Transport – HP1 Components). HP1, however,
checks if all components, which are present during the geochemical calculations, are defined in
the transport model. If not, a warning message is generated. In our example, the concentrations
of the components N(0) and N(3) are very low under the prevailing oxidizing conditions.
Therefore, they can be neglected in the transport problem. If you want to avoid these warnings,
you have to either include N(0) and N(3) as components to be transported or define an alternative
using
primary
master
species
representing
nitrate
(such
as
Nit-)
SOLUTION_MASTER_SPECIES and SOLUTION_SPECIES.
190
Computer Session 10
OUTPUT
Display results for “Observation Points” or “Profile Information”. Alternatively, the graph below
can be created using information in the output file obs_nod.out.
0.0014
Concentration [mol/kg]
0.0012
0.001
Cl
0.0008
Ca
Na
0.0006
K
0.0004
0.0002
0
0
14400
28800
43200
57600
72000
86400
Time [s]
Outflow concentrations of Cl, Ca, Na and K for the single-pulse
cation exchange example.
Results for this example are shown in the figure above, in which concentrations for node 41 (the
last node) are plotted against time. Chloride is a conservative solute and arrives in the effluent at
about one pore volume. The sodium initially present in the column exchanges with the incoming
calcium and is eluted as long as the exchanger contains sodium. The midpoint of the
breakthrough curve for sodium occurs at about 1.5 pore volumes. Because potassium exchanges
more strongly than sodium (larger log K in the exchange reaction; note that log K for individual
pairs of cations are defined in the database and therefore did not have to be specified), potassium
is released after sodium. Finally, when all of the potassium has been released, the concentration
of calcium increases to a steady-state value equal to the concentration of the applied solution.
191
Computer Session 10
Example 2: Transport and Cation Exchange (multiple pulses)
This example is the same as the one described in the previous example, except that time variable
concentrations are applied at the soil surface.
Following sequence of pulses are applied at the top boundary:
0 – 8 hr: 6 x 10-4 M CaCl2
8 – 18 hr: 5 x 10-6 M CaCl2, 1 x 10-3 M NaNO3, and 2 x 10-4 M KNO3
18 – 38 hr: 6 x 10-4 M CaCl2
38 – 60 hr: 5 x 10-6 M CaCl2, 1 x 10-3 M NaNO3, and 8 x 10-4 M KNO3
INPUT
Project Manager
Click on CEC-1
Button "Copy"
New Name: CEC-2
Description: Transport and Cation Exchange, multiple pulses
Button "OK", "Open"
Main Processes
Heading:
Transport and Cation Exchange, multiple pulses
Button "Next"
Button "Next"
Time Information
Time Units:
hours
Final Time:
60 (h)
Initial Time Step:
0.1
Maximum Time Step:
0.1
Number of Time-Variable Boundary Records:
Button "Next"
Print Information
Default
Button "OK"
Button "Next"
192
4
Computer Session 10
Solute Transport – HP1 Definitions
Definitions of Solution Composition
Add additional boundary solution compositions with numbers 3002 and 3003.
Define a bottom boundary solution: Solution 4001 – pure water
-units mmol/kgw
ph 7 charge
Na 1
K 0.2
N(5) 1.2
Ca 5E-3
Cl 1E-2
O(0) 1 O2(g) -0.68
-units mmol/kgw
ph 7 charge
Na 1
K 0.8
N(5) 1.8
Ca 5E-3
Cl 1E-2
O(0) 1 O2(g) -0.68
solution 4001 bottom boundary solution
#pure water
Button: "OK"
Button: "Next"
Time-Variable Boundary Conditions
Fill in the time, and the solution composition number for the top boundary
Time
8
18
38
60
cTop
3001
3002
3001
3003
cBot
4001
4001
4001
4001
Button "Insert", Insert node at 2, 4, 6, and 8 cm
Menu: File->Exit
Button "Next"
Calculations - Execute HP1
193
Computer Session 10
OUTPUT
After the program finishes, explore the output files.
Total concentration of K (mol/kg water)
Figures below give the K concentration at different depths in the profile and show the outflow
concentrations. The first pulse is identical to the single pulse project. Then additional solute
pulses of different solution compositions will restart the cation exchange process depending on
the incoming solution composition.
0.0012
0.001
0.0008
0.0006
0.0004
0.0002
0
0
10
20
30
Time (hours)
40
50
60
-6.0 cm
-8.0 cm
-2.0 cm
-4.0 cm
Time series of K concentrations at four depths for the multiplepulse cation exchange example.
Concentrations (mol/kg water)
0.0012
0.001
0.0008
0.0006
0.0004
0.0002
0
0
10
20
30
Time (hours)
Na
Cl
40
K
50
60
Ca
Outflow concentrations for the multiple-pulse cation exchange
example.
194
n
S
Q = α hm
- unit storage of water (or mean depth) [L]
- discharge per unit width [L2T-1],
- rate of local input, or lateral inflows (precipitation - infiltration)
[LT-1]
S
n
and
m = 5/3
- Manning’s roughness coefficient for overland flow
- slope
α = 1.49
1/ 2
Manning hydraulic resistance law:
h
Q
q(x,t)
∂h ∂Q
+
= q ( x, t )
∂t ∂x
Kinematic wave equation:
Overland Flow
Riverside, CA
and Miroslav Šejna
Present and Future Plans in
HYDRUS Development
(Overview)
0.24
0.26
0.28
0.30
0.32
0.34
0.36
100 m
0.38
0.40
0.42
0.44
0
10000
0.5
1
1.5
2
2m
8000
1m
4m
4000
Length [cm]
6000
10 m
Analytical solution
1 minutes
2 minutes
4 minutes
10 minutes
15 minutes
2000
High intensity rainfall of 0.00666 cm/s (i.e., 24 cm/hour) of 10 minutes duration
Loamy soils with Ks= 25 cm/d, and Ks= 25 m/d in the middle of the transect
Soil transect is 100 m long, with a slope of 0.01
Roughness coefficient n = 0.01
0.5
Overland flow
Infiltration
2.5
ParSWMS – Parallelized Version of HYDRUS
HYDRUS Package For MODFLOW
Selected HYDRUS Applications
New HYDRUS Website
New HYDRUS Book
Overland Flow
X
X
X
X
X
- filtration theory
- colloid facilitated transport
- two-site kinetic sorption
- air-water interface
- site-limited sorption
Facilitated Solute Transport
X Colloid, Virus, and Bacteria Transport, ColloidColloid-
- kinematic wave approach
- diffusion wave approximation
X Overland Flow Module
Present and Future Plans
Depth [cm]
195
15 m
0
1
196
Aaw
qc
Rc
ρ
Dc
Γc
θw
Cc
Sc
⎞ ∂qc Cc
⎟ − ∂x + Rc
⎠
colloid concentration in the aqueous phase [nL-3]
colloid concentrations adsorbed to the solid phase [nM-1]
colloid concentrations adsorbed to the air–water interface [nL-2]
volumetric water content accessible to colloids [L3L-3] (due to ion
or size exclusion, θw may be smaller than the total volumetric
water content θ)
dispersion coefficient for colloids [L2T-1]
bulk density [ML-3]
air-water interfacial area per unit volume [L2L-3]
volumetric water flux density for colloids [LT-1]
various chemical and biological reactions [nL-3T-1]
∂θ w Cc
∂S
∂A Γ
∂C
∂ ⎛
+ ρ c + aw c = ⎜ θ w Dc c
∂t
∂t
∂t
∂x ⎝
∂x
Colloid Transport
immobile in the subsurface since under normal
conditions they are strongly sorbed to soil
X They can also sorb to colloids which often move
at rates similar or faster as non-sorbing tracers
X Experimental evidence exists that many
contaminants are transported not only in a
dissolved state by water, but also sorbed to
moving colloids.
X Examples: heavy metals, radionuclides,
pesticides, viruses, pharmaceuticals, hormones,
and other contaminants
X Many contaminants should be relatively
Colloid-Facilitated Solute Transport
ψaca
Scstr
ψs
kac
kdc
ψs
kac
kdc
Rsc
ρ
various chemical and biological reactions acting on
kinetically attached colloids to the matrix [nL-3T-1]
first-order colloid attachment coefficient [T-1]
first-order colloid detachment coefficient [T-1]
dimensionless colloid retention function [-]
∂Sc
= θ wψ s kac Cc − ρ kdc Sc − Rsc
∂t
Colloid massmass-transfer between the aqueous and solid phases:
Attached Colloids, Scatt
kdca
Mobile Colloids, Cc
Strained Colloids,
ψsstr
kstr
kaca
Air-Water Interface Colloids, Γc
Colloid Transport
Solid
Water
Air
2
197
solid-phase concentrations of strained colloids [nM-1]
solid-phase concentrations of attached colloids [nM-1]
first-order straining coefficient [T-1]
various chemical and biological reactions acting on
kinetically attached colloids to the matrix [nL-3T-1]
first-order colloid attachment coefficient [T-1]
first-order colloid detachment coefficient [T-1]
Colloid Transport
ψs
kac
kdc
Scstr
Scatt
kstr
Rsc
att
c
∂Sc
∂S
∂S
=ρ
+ρ
= θ wψ sstr k str Cc + (θ wψ s kac Cc − ρ kdc Sc ) − Rsc
∂t
∂t
∂t
str
c
Γc
kdca
kaca
ψaca
Aaw
Rac
colloid concentration adsorbed to the air–water interface [nL-2]
various chemical and biological reactions of attached colloids to
the air-water interface [nL-3T-1]
air–water interfacial area per unit volume [L2L-3]
dimensionless colloid retention function for the air–water interface
(-)
first-order colloid attachment coefficient to the air–water interface
[T-1]
first-order colloid detachment coefficient from the air–water
interface [T-1]
∂Aaw Γ c
= θ wψ aca kaca Cc − Aaw kdca Γ c − Rac
∂t
Colloid massmass-transfer between the aqueous phase and the air–
air–
water interface:
ρ
Colloid massmass-transfer between the aqueous and solid phases:
Colloid Transport
Sc
Scmax
⎛ d 50 + x ⎞
⎟
⎝ d 50 ⎠
ψ sstr = ⎜
ψs = 1−
−β
maximum solid-phase colloid concentration [nM-1]
median grain size of the porous media [L]
fitting parameter [-]
distance from the porous-medium inlet [L]
σ aw
1
aw
∫P
Sw
( S )dS =
σ aw
nρ w g
n
porosity [L3L-3]
Sw water saturation [-]
Paw air-water capillary pressure [ML-1T-2]
σaw surface tension [MT-2]
h
pressure head [L]
ρw density of water [ML-3]
g
gravitational acceleration [LT-2]
Aaw = −
n
Sw
∫ h( S )dS
1
Interfacial area model of Bradford and Leij (1997) is used to
estimate the air–water interfacial area Aaw:
Colloid Transport
Scmax
d50
b
x
ψs
Bradford et al. (2003)
Adamczyk et al. (1994)
Colloid Transport
3
198
Left-hand side sums the mass of contaminant:
- in the liquid phase
- sorbed instantaneously and kinetically to the solid phase
- sorbed to mobile and immobile (attached to solid phase or air–
water interface) colloids
Right-hand side considers various spatial mass fluxes
- dispersion and advective transport of the dissolved contaminant
- dispersion and advective transport of contaminant sorbed to mobile
colloids
∂θ C
∂S
∂S
∂θ C S
∂S S
∂A Γ S
+ ρ e + ρ k + w c mc + ρ c ic + aw c ac
∂t
∂t
∂t
∂t
∂t
∂t
∂ ⎛
∂C ⎞ ∂qC ∂ ⎛
∂Cc ⎞ ∂qcCc Smc
= ⎜θ D
+ ⎜ θ w Smc Dc
+R
⎟−
⎟−
∂x ⎝
∂x ⎠ ∂x
∂x ⎝
∂x ⎠
∂x
Mass Balance of Total Contaminant:
Contaminant
Kd
Dissolved
Contaminant, C
ω
kac
ψi
kdc
volumetric water content [L3L-3] (note that we use the entire water
content for the contaminant)
C
dissolved-contaminant concentration in the aqueous phase [ML-3]
Se, Sk contaminant concentration sorbed instantaneously and kinetically to
the solid phase [MM-1]
Smc, Sic, and Sac
contaminant concentrations sorbed to mobile and
immobile (attached to solid and air–water interface) colloids [Mn-1]
R
various chemical and biological reactions [ML-3T-1]
∂θ C
∂S
∂S
∂θ C S
∂S S
∂A Γ S
+ ρ e + ρ k + w c mc + ρ c ic + aw c ac
∂t
∂t
∂t
∂t
∂t
∂t
∂ ⎛
∂C ⎞ ∂qC ∂ ⎛
∂Cc ⎞ ∂qcCc Smc
= ⎜θ D
+ ⎜ θ w Smc Dc
+R
⎟−
⎟−
∂x ⎝
∂x ⎠ ∂x
∂x ⎝
∂x ⎠
∂x
Mass Balance of Total Contaminant:
Contaminant
θ
kdic
ψm
ψg
Kinetically Sorbed
Contaminant, Sk
kaic
kdmc
kamc
kdac
Contaminant sorbed to immobile colloids, Sic
Instantaneously Sorbed
Contaminant, Se
kstr
kaac
kdca
Solid
Water
Contaminant sorbed to
mobile colloids, Smc
kaca
Contaminant sorbed to colloids at air-water interface, Sac
Air
Applications:
Schijven, J., and J. Šimůnek, Kinetic modeling of virus transport at field scale, J. of Contam. Hydrology, 55(1-2),
113-135, 2002.
Bradford, S. A., S. R. Yates, M. Bettehar, and J. Šimůnek, Physical factors affecting the transport and fate of
colloids in saturated porous media, Water Resour. Res., 38(12), 1327, doi:10.1029/2002WR001340, 63.1-63.12,
2002.
Bradford, S. A., J. Šimůnek, M. Bettehar, M. Th. van Genuchten, and S. R. Yates, Modeling colloid attachment,
straining, and exclusion in saturated porous media, Environ. Sci. & Technology, 37(10), 2242-2250, 2003.
Bradford, S. A., M. Bettehar, J. Šimůnek, and M. Th. van Genuchten, Straining and attachment of colloids in
physically heterogeneous porous media, Vadose Zone Journal, 3(2), 384-394, 2004.
Zhang, P., J. Šimůnek, and R. S. Bowman, Nonideal transport of solute and colloidal tracers through reactive
zeolite/iron pellets, Water Resour. Res., 40, doi:10.1029/2003WR002445, 2004.
Bradford, S. A., J. Šimůnek, M. Bettahar, Yadata Tadassa, M. Th. van Genuchten, and S. R. Yates, Straining of
Colloids at Textural Interfaces, Water Resour. Res., W10404, 17 pp, 2005.
Bradford, S. A., J. Šimůnek, M. Bettahar, M. Th. van Genuchten, and S. R. Yates, Significance of straining in
colloid deposition: evidence and implications, Water Resour. Res., 42, W12S15, doi:10.1029/2005WR004791, 16
pp., 2006.
Gargiulo, G., S. A. Bradford, J. Šimůnek, P. Ustohal, H. Vereecken, and E. Klumpp, Transport and deposition of
metabolically active and stationary phase Deinococcus Radiodurans in unsaturated porous media, Environ. Sci.
and Technol., 41(4), 1265-1271, 2007.
Gargiulo, G., S. A. Bradford, J. Šimůnek, P. Ustohal, H. Vereecken, and E. Klumpp, Bacteria transport and
deposition under unsaturated conditions: the role of the matrix grain size and the bacteria surface protein, J.
Gargiulo, G., S. A. Bradford, J. Šimůnek, P. Ustohal, H. Vereecken, and E. Klumpp, Bacteria transport and
deposition under unsaturated conditions: the role of bacteria surface hydrophobicity, Vadose Zone Journal, 7(2),
406-419, 2008.
Colloid Transport
4
199
ρ
ψi
kac, kdc
Sc
Sic
Smc
kaic, kdic
Ric
colloid concentration sorbed in the solid phase [nM-1]
contaminant concentration sorbed to immobile colloids [Mn-1]
contaminant concentration sorbed to mobile colloids [Mn-1]
adsorption/desorption rate to/from immobile colloids [T-1]
various reactions for contaminant sorbed to immobile colloids
[ML-3T-1]
first-order colloid attachment/detachment coefficient [T-1]
parameters adjusting the sorption rate to the number of
immobile colloids present [-]
∂S c S ic
= θk aicψ i C − ρk dic S c S ic + θ w (k acψ s + k strψ str )C c S mc − ρk dc S c S ic + Ric
∂t
Rac
kaac, kdac
Aaw
kaca, kdca
Sac
Γc
colloid concentration sorbed to the air-water interface [nL-2]
contaminant concentration sorbed to colloids at the air-water
intergace [Mn-1]
air–water interfacial area per unit volume [L2L-3]
first-order colloid attachment/detachment coefficient to/from
the air–water interface [T-1]
adsorption/desorption rate to/from colloids sorbed at the airwater interface [T-1]
various reactions for contaminant sorbed to colloids at the airwater interface [ML-3T-1]
∂Aaw Γc S ac
= θk aacψ g C − Aaw k dac Γc S ac + θ w k acaψ a C c S mc − Aaw k dca Γc S ac + Rac
∂t
Mass-balance equation for Contaminant Sorbed to Colloids
Attached to the Air–
Air–Water Interface:
Interface
colloid concentration in the aqueous phase [nL-3]
contaminant concentration sorbed to mobile colloids [Mn-1]
adsorption and desorption rates to/from mobile colloids [T-1]
first-order colloid attachment/detachment coefficients to/from
the air–water interface [T-1]
first-order colloid attachment/detachment coefficient [T-1]
various chemical and biological reactions for contaminant
sorbed to mobile colloids [ML-3T-1]
parameters adjusting the sorption rate to the number of mobile
colloids present [-]
Mass-balance equation for
Contaminant Sorbed to Immobile Colloids:
Colloids
ψm
kac, kdc
Rmc
Cc
Smc
kamc, kdmc
kaca, kdca
instantaneous sorption on one fraction of the sites (type-1 sites)
[MM-1]
kinetic sorption on the remaining sites (type-2 sites) [MM-1]
first order rate constant [T-1]
fraction of exchange sites assumed to be in equilibrium with the
solution phase [-]
adsorption isotherm [MM-1] that can be expressed using
Freundlich, Langmuir, or linear adsorption models
various chemical and biological reactions of the kinetically sorbed
contaminant [ML-3T-1]
Rsk
Ψ(C)
f
ω
Sk
Se
∂S
ρ k = ω [ (1 − f )Ψ (C ) − Sk ] + Rsk
∂t
∂θ wCc Smc
∂ ⎛
∂C ⎞ ∂q C S
= ⎜ θ w Smc Dc c ⎟ − c c mc + θ kamcψ mC − θ w kdmcCc Smc −
∂t
∂x ⎝
∂x ⎠
∂x
−θ w ( kacψ s + k strψ str ) Cc Smc + ρ kdc Sc Sic − θ w kacaψ a Cc Smc + Aaw kdca Γ c Sac + Rmc
Contaminant Sorbed to Mobile Colloids:
Colloids
Contaminant Sorbed to the Solid Phase:
Phase
S = Se + Sk
5
200
for message-passing between the different processors. MPI is
free software for LINUX or UNIX operating systems.
X Test
- Water flow and solute transport problem - 492264 finite
element nodes
- Supercomputer with 41 SMP nodes with 32 processors each
(total 1312 processors - Power4+ 1.7 GHz)
X Developed by Forschungszentrum Jülich, Germany.
X SWMS_3D – earlier and simpler version of Hydrus-3D
X MPI (Message-Passing Interface) - a library specification
SWMS_3D (Simunek et al., 1995).
X ParSWMS (Hardelauf et al., 2007) - Parallelized version of
ParSWMS – parallelized version of HYDRUS
System of coupled equations (solved numerically):
a) Five partial differential equations
- total mass of contaminant
- mass of contaminant sorbed kinetically to solid phase
- mass of contaminant sorbed to mobile colloids
- mass of contaminant sorbed to immobile colloids
- mass of contaminant sorbed to colloids at the air-water
interface
b) One algebraic equation
- mass of contaminant sorbed instantaneously to solid
phase (adsorption isotherm)
∂Sc
= θ kac Cc − ρ kdc Sc
∂t
-0.030
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
0
100
200
Bottom Solute Flux
300
Time [min]
400
500
600
Time gain as compared to the one processor run (in log2) as a function of the
number of processors np (in log2) for solute transport scenario with 492264 nodes (open
circles) and water flow with atmospheric upper boundary conditions (diamonds).
ParSWMS – parallelized version of HYDRUS
⎞ ∂qc Cc Smc
+ θ kamcψ m C − θ kdmc Cc S mc − θ kac Cc S mc + ρ kdc Sc Sic
⎟−
∂x
⎠
∂Sc Sic
= θ kaicψ i C − ρ kdic Sc Sic + θ kac Cc S mc − ρ kdc Sc Sic
∂t
∂θ Cc S mc
∂C S
∂ ⎛
= ⎜ θ Dc c mc
∂t
∂x ⎝
∂x
ρ
∂S
∂S
∂θ Cc Smc
∂S S
∂C S ⎞ ∂q C S
∂θ C
∂ ⎛
∂C ⎞ ∂qC ∂ ⎛
+ρ e +ρ k +
+ ρ c ic = ⎜ θ D
+ ⎜ θ Dc c mc ⎟ − c c mc
⎟−
∂t
∂t
∂t
∂t
∂t
∂x ⎝
∂x ⎠ ∂x
∂x ⎝
∂x ⎠
∂x
Solute transport (with colloids):
ρ
∂C c ⎞ ∂q c C c
∂θ C c
∂S
∂ ⎛
+ ρ c = ⎜ θDc
⎟−
∂x ⎝
∂x ⎠
∂x
∂t
∂t
Colloid transport:
6
MODFLOW model domain is grouped in to zones based on similarities in soil
hydraulic characteristics, hydrogeology and meteorology.
A HYDRUS vertical profile is assigned to each of the zones on which the 1D
Richards equation is used.
HYDRUS Package: Zoning
Seo, H. S., J. Šimůnek, and E. P. Poeter, Documentation of the HYDRUS Package
for MODFLOW-2000, the U.S. Geological Survey Modular Ground-Water Model,
GWMI 2007-01, International Ground Water Modeling Center, Colorado School
of Mines, Golden, Colorado, 96 pp., 2007.
Twarakavi, N. K. C., J. Šimůnek, and H. S. Seo, Evaluating interactions between
groundwater and vadose zone using HYDRUS-based flow package for
MODFLOW, Vadose Zone Journal, doi:10.2136/VZJ2007.0082, Special Issue
“Vadose Zone Modeling”, 7(2), 757-768, 2008.
Hyeyoung Sophia Seo, Navin Twarakavi,
Jirka Šimůnek, and Eileen P. Poeter
The Unsaturated Flow Package
for Modflow-2000
Water table
MODFLOW Grid
Depth to
Ground
Water
Flux (q)
a: Ground Surface
b: Bottom of Soil Column
UNSF Soil Profile
b
a
Layers
3
2
1
Z1
7
6
Rows
ZSURF
5
4
3
2
1
1
K: Hydraulic
Conductivity
K1
K2
K3
K4
K5
K6
K7
K8
K9
K10
K11
K12
Explanation
2
3
4
5
6
7
8
Columns
9
10 11 12 13
K: Hydraulic Conductivity
K3
K2
K1
Zone 2
Zone 1
Explanation
HYDRUS Package for MODFLOW
Zone 1
HYDRUS
Sub-model
Solve for bottom fluxes in
each profile using the
atmospheric data and 1D
Richards Equation
Zone 2
Average water
table depths
Bottom fluxes as recharge at
the water table for the next
MODFLOW time step
MODFLOW
Sub-model
Layer 1
Layer 2
Layer 3
HYDRUS Package for Modflow
Depth
201
7
Layer 1
Layer 2
MODFLOW Grid
Depth to
Ground
Water
b
a: Ground Surface
b: Bottom of Soil Column
UNSF Soil Profile
Flux (q)
Z1
General head boundary
3
1
10
324
314
320
3 18
322
7
6
3
2
3
1
45 2 3
9
14 13 12 11
5
4
5
8
7
6
8
Pumping well
316
4
32
2
33
4
(c) Initial water table depth (m)
310
312
3 20
330 33 2
(a) Land surface elevation (m)
7
9 8
11 12 13
15
Inactive cells
a) Land surface elevation
b) depth to bedrock
c) water table depth at the
beginning of the simulation
Hypothetical regionalscale ground water
flow problem:
5
Uplands (Hydraulic conductivity=11 m/d, specific yield=0.1)
Alluvial aquifer (Hydraulic conductivity=53 m/d, specific yield=0.2)
9
5
2
2
8
3
1
7
Average
water table
depths for
each zone.
Total flux at
the water table
for each zone
(a)
2
W
50
S
4
N
50
6
60
E
334.5
8
144.4
24.3
Legend (in m)
(b)
0
303.8
2
70
50
(b) Aquifer thickness (m)
30
3
4
ROWS
6
5
Total flux at
the water table
for each zone
110
5
4
Average
water table
depths for
each zone.
330
6
3
Total flux at
the water table
for each zone
t=2
HYDRUS - MODFLOW - Case Study
HYDRUS
(vadose zone)
Average water
table depths
for each zone.
MODFLOW
(ground water)
(seasonal)
t=1
40
7
2
start
Meteorological
conditions
(daily)
Time steps
70
8
9
10
11
12
13
14
15
1
10
(Seo et al., 2006)
K: Hydraulic
Conductivity
K1
K2
K3
K4
K5
K6
K7
K8
K9
K10
K11
K12
328
Model domain, spatial distribution of
hydraulic conductivities and specific
yields, wells (red circles) and general
head boundaries.
Hypothetical regionalscale ground water flow
problem:
COLUMNS
Layer 3
Explanation
150
ZSURF
(c)
18.4
0.3
10 Kilometers
50
a
140
Depth
end
HYDRUS-MODFLOW - Interaction in Time
100
80
11
50
70
7
10 11 9 5 6
120
334 336
330 33
2
3 26
4
50
336
90
60
33
130
50
202
60
4
2
8
203
0
S
N
E
1
2
5
8
No Data
Ground water table fluxes
(recharge vs discharge) as
predicted by the HYDRUS
package at the end of stress periods
(a) 3, (b) 6 and (c) 12.
(c)
(a)
4
W
0
4
E
Recharge
Discharge
S
N
(b)
8 Kilometers
Hypothetical regionalscale ground water flow
problem:
3 Kilometers
Precipitation rate factor
3
W
1
2
3
4
5
6
7
8
9
10
Zones
0
Flow and Transport Under the Banana Tree
(Sansoulet et al., 2007)
3
S
E
MODFLOW zones
used to define
HYDRUS soil
profiles
Zonation showing the
spatial distribution of
precipitation
W
Hypothetical regional-scale ground water flow problem:
Hypothetical regional-scale ground water flow problem:
N
11
12
13
14
15
16
17
18
19
20
3 Kilometers
9
204
New HYDRUS web site: Public Libraries
Flow and Transport in the Buddha Statue
New HYDRUS web site: Public Libraries
www.pc-progress.com/en/default.aspx
New HYDRUS web site:
10
205
CRC Press, Taylor & Francis Group
ISBN-10: 142007380X, ISBN-13: 9781420073805
due 4/15/2009
David Radcliffe and Jirka Šimůnek
Introduction to Soil Physics
with HYDRUS:
Modeling and Applications
New HYDRUS Book
New HYDRUS web site: References
New HYDRUS web site: Short Courses
11
206
References:
Abbaspour, K. C., R. Schulin, M. Th. van Genuchten, and E. Schläppi, Procedures for uncertainty analyses applied to a
landfill leachate plume, Ground Water, 36(6), 874-1883, 1997.
Albright, W. H., Application of the Hydrus-2D Model to Landfill Cover Design in the State of Nevada. Water Resources
Center, Desert Research Institute, Publication 41153, 1997.
Bear, J., Dynamics of Fluid in Porous Media. Elsevier, New York, NY, 1972.
Bradford, S. A., S. R. Yates, M. Bettehar, and J. Šimůnek, Physical factors affecting the transport and fate of colloids in
saturated porous media. Water Resour. Res., 38(12), 1327, doi:10.1029/2002WR001340, 63.1-63.12, 2002.
Bradford, S. A., J. Šimůnek, M. Bettehar, M. Th. van Genuchten, and S. R. Yates, Modeling colloid attachment,
straining, and exclusion in saturated porous media, Environ. Sci. Technol., 37(10), 2242-2250, 2003.
Bradford, S. A., M. Bettehar, J. Šimůnek, and M. Th. van Genuchten, Straining and attachment of colloids in physically
heterogeneous porous media, Vadose Zone Journal, 3(2), 384-394, 2004.
Bradford, S. A., S. Torkzaban, F. J. Leij, J. Šimůnek, and M. Th. van Genuchten, Modeling the coupled effects of
pore space geometry and velocity on colloid transport and retention, Wat. Resour. Res., 45, W02414,
doi:10.1029/2008WR007096, 15 pp., 2009.
Benjamin, J. G., H. R. Havis, L. R. Ahuja, and C. V. Alonso, Leaching and water flow patterns in every-furrow and
alternate-furrow irrigation, Soil Sci. Soc. Am. J., 58, 1511-1517, 1994.
Brooks, R. H., and A. T. Corey, Hydraulic properties of porous media, Hydrol. Paper No. 3, Colorado State Univ.,
Fort Collins, CO, 1964.
Buchner, J. S., J. Šimůnek, J. Lee, D. E. Rolston, J. W. Hopmans, A. P. King, and J. Six, Evaluation of CO2 fluxes
from an agricultural field using a process-based numerical model, J. Hydrology, doi:
10.1016/j.jhydrol.2008.07.035, 361(1-2), 131– 143, 2008.
Carsel, R.F., and R. S. Parrish, Developing joint probability distributions of soil water retention characteristics, Water
Resour. Res., 24, 755-769, 1988.
Casey, F. X. M., and J. Šimůnek., Inverse analyses of the transport of chlorinated hydrocarbons subject to sequential
transformation reactions, J. of Environ. Quality, 30(4), 1354-1360, 2001.
Casey, F. X. M., G. L. Larsen, H. Hakk, and J Šimůnek, Fate and transport of 17β-Estradiol in soil-water systems,
Environ. Sci. Technol., 37(11), 2400-2409, 2003.
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.
Das, B. S., J. M. H. Hendrickx, and B. Botchers, Modeling transient water distribution around landmines in bare soils,
Soil Sci., 166(3), 163-173, 2001.
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.
de Vos J. A., D. Hesterberg, P. A. C. Raats, Nitrate leaching in a tile-drained silt loam soil, Soil Sci. Soc. Am. J.,
64(2), 517-527, 2000.
de Vries, D. A., The thermal properties of soils, In Physics of Plant Environment, Editor, R.W., van Wijk, pp. 210235, North Holland, Amsterdam, 1963.
De Wilde, T., J. Mertens, J. Šimůnek, K. Sniegowksi, J. Ryckeboer, P. Jaeken, D. Springael, and P. Spanoghe,
Characterizing pesticide sorption and degradation in micro scale biopurification systems using column
displacement experiments, Environmental Pollution, 157, 463-473, 2009.
Diodato, D. M., Review: HYDRUS-2D, Computer Spotlights in Ground Water, Ground Water, 38(1), 10-11, 2000.
Divine, C. E., STANMOD Software Review, Southwest Hydrology, 3(2), 37, 2003.
Dontsova, K. M., S. L. Yost, J. Šimůnek, J. C. Pennington, C. Williford. 2006. Dissolution and transport of TNT,
RDX, and Composition B in saturated soil columns. J. of Environ. Quality, 35:2043-2054.
Durner, W. 1994. Hydraulic conductivity estimation for soils with heterogeneous pore structure. Water Resour. Res.,
32(9):211-223.
Gargiulo, G., S. A. Bradford, J. Šimůnek, P. Ustohal, H. Vereecken, and E. Klumpp, Transport and deposition of
metabolically active and stationary phase Deinococcus Radiodurans in unsaturated porous media, Environ. Sci.
and Technol., 41(4), 1265-1271, 2007a.
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