Soil Physics with HYDRUS:
Chapter 3
Equations, Tables, and Figures
3.1 – 3.5
 R 4 w g H
Q
8 L
Q
H
 K s
A
L
Jw  Ks
H
L
dH
J w  K s
dz
J w  K s
H
z
3.6 – 3.9
H 2  H1
bL
J w  Ks
 Ks
L
L
J w  Ks
H 2  H1
hz
  Ks
L
z
h z
bL
Ks
 Ks
z
L
b
h z
L
Ks 
k l g

3.10 – 3.13
N
K eff 
L
j 1
N
Lj
K
j 1
J w   K eff
j
j
H
L
N
K eff 
L K
j 1
N
j
L
j 1
J x  Ks
j
dz
dx
j
3.14 – 3.17
zmax 
J w   K ( h)
Rs 2
 z02
Ks
H
 (h  z )
 h z 
  K (h)
  K (h)   
z
z
 z z 
 h 
  K ( h)   1 
 z 
K (h)  Ks e h
 
K ( )  K s  
 s 
m
3.18 – 3.20
K ( h) 
Ks
h
1  
a
N
Se
K ( Se )  K s S el

dSe
h 2 ( Se )
0
1

dSe
h 2 ( Se )
0
2
 dSe 
  h ( Se ) 

0

l 
K ( Se )  K s Se
2
1
 dSe 
  h ( Se ) 
0

Se
3.21 – 3.25
K (Se )  Ks Se1l 2/ 
K ( Se )  K s Sel 1  (1  Se1/ m )m 
m  1  1/ n,
K ( Se )  K s

( w1Se1  w2 Se2 ) w11 1  (1  S
l
n 1
)   w2 2 1  (1  S
( w11  w2 2 )2
1/ m1 m1
e1
h0
c 
2
 K (h)dh
hi
K (h0 )  K (hi )

) 
1/ m2 m2
e2
2
3.26 – 3.30
c 
1

x
a y( x)dx  3  y1  4 y2  2 y3  4 y4  2 y5  ...  2 yn2  4 yn1  yn 
b
K (h)  K s
h  ha
2  3
 ha 
K (h)  K s  
h
J w  K  h 
h2
dh
 z1  z2
h
Jw
1 1
K ( h)
h  ha
3.31 – 3.34

Ks
ha
 1  tanh 
i


is   J w
1  i / Ks  i / Ks
a
z 0
 4c 
 K (h0 ) 1 
 r 

 2 H c H 2 
Qs  K s   r 


G
G 


H c
H2 
is  K s 1 

2
2 
G

r
G

r



( L  z)


3.35 – 3.39
2
3

1
H
H
H 
G
 A1  A2  A3    A4   
2 
r
 r 
 r  
JwL
Ks 
H1  H 2
Jw 
H  H1
db
 Ks 2
dt
L
L  b0  L 
K s  ln 

t1  b1  L 
H2
is  K s
GG r 2
3.40
2
r
H  r 
sinh       1 
H
 r  H
GG 
2
1
TABLE 3.1
Brooks and Corey (1964) soil hydraulic parameters (Ks, θr, θs, ha, and λ) for 11 textural classes (Rawls et al., 1982).
Macroscopic capillary length (λc) calculated using Equation 3.21 and Equation 3.25.
Ks
(cm
d-1)
θr
(cm3 cm-3)
θs
(cm3 cm-3)
Sand
504.0
0.020
0.417
-7.26 0.592
9.88
Loamy sand
146.6
0.035
0.401
-8.69 0.474
12.28
loam
62.16
0.041
0.412
-14.7 0.322
22.18
Loam
31.68
0.027
0.434
-11.1 0.220
17.79
Silt loam
16.32
0.015
0.486
-20.7 0.211
33.38
Sandy clay loam
10.32
0.068
0.330
-28.1 0.250
44.16
Clay loam
5.52
0.075
0.390
-25.9 0.194
42.27
Silty clay loam
3.60
0.040
0.432
-32.6 0.151
55.03
clay
2.88
0.109
0.321
-29.2 0.168
48.61
Silty clay
2.16
0.056
0.423
-34.2 0.127
58.96
Clay
1.44
0.090
0.385
-37.3 0.131
64.07
Textural class
ha
(cm)
λ
(-)
λc
(cm)
TABLE 3.2
Soil properties (texture, sand, silt, clay, structure, and Ks) for the Cecil loamy sand (plot 4
from Bruce et al., 1983).
Horizon
Texture
Clay
%
Silt
%
Sand
%
Structure
Ks
cm h-1
Weak, medium granular
Weak, medium,
subangular blocky
Strong, medium,
subangular blocky
Strong, medium,
subangular blocky
Weak, medium,
subangular blocky
Massive
19.19
7.69
Ap
BA
Loamy sand
Clay loam
7
37
15
20
78
43
Bt1
Clay
50
20
30
Bt2
Clay
41
25
34
BC
Clay loam
36
27
37
C
Sandy clay
loam
24
24
52
10.73
0.206
0.035
0.467
TABLE 3.3
Soil texture/structure categories for estimation of macroscopic capillary length (λc),
adapted from Elrick and Reynolds (1992).
Soil texture/structure category
Coarse and gravelly sands; may also include some highly structured
soils with large cracks and/or macropores
λc
(cm)
2.8
Most structured soils from clays through loams; also includes
unstructured medium and fine sands
8.3
Soils which are both fine textured (clayey) and unstructured
Compacted, structureless, clayey materials such as landfill caps and
liners, lacustrine or marine sediments, etc
25
100
TABLE 3.4
Dimensionless coefficients for the polynomial (Equation 3.35) describing the geometric
factor G, valid for H/r < 10 (Bosch and West, 1998)
Soil texture/structure
Sand
Structured loams and clays
Unstructured clays
A1
0.079
0.083
0.094
A2
0.516
0.514
0.489
A3
-0.048
-0.053
-0.053
A4
0.002
0.002
0.002
Figure 3.1 A vertical column of soil 40.0 cm in length with 5.0 cm of water ponded on the surface and
water allowed to drip from the bottom. The soil saturated hydraulic conductivity is 10.0 cm h-1
Figure 3.2 A vertical soil column of length L and saturated hydraulic conductivity Ks. Water is
ponded to a depth b at the top of the column and water discharges freely from the bottom at
atmospheric pressure.
Figure 3.3 A vertical soil column with an arbitrary point at height z inside the column, instead of the
top of the column.
Saturated Hydraulic Conductivity (cm/d)
100000
10000
1000
100
10
1
0.1
0.01
0.001
0.01
0.1
Clay Fraction (g/g)
Figure 3.4 Saturated hydraulic conductivity as a function of clay fraction for 324
soils from the UNSODA database (Nemes et al., 2001).
1
Figure 3.5 Saturated hydraulic conductivity as a function of depth in a Cecil series soil from North
Carolina Piedmont (Schoneberger and Amoozegar, 1990).
Figure 3.6 A soil column with N layers of thickness L1...LN and saturated hydraulic conductivites
K1...KN on the left side. An equivalent uniform column on the right with a saturated hydraulic
conductivity Keff.
Figure 3.7 A column consisting of a sand layer 30.0-cm thick overlying a loam layer 20.0-cm thick.
The saturated hydraulic conductivities of the sand and loam layers are 21.0 and 1.32 cm h-1,
respectively. Water is ponded at the top of the column to a depth of 5.0 cm and allowed to drip from
the bottom of the column.
Figure 3.8 Geometry considered for defining the drainage equation. Water height in the ditch or
drain is z0 (Warrick, 2003).
Figure 3.9 Examples of unsaturated hydraulic conductivity curves as a function of pressure head
(top) and volumetric water content (bottom) for the sand, loam, and clay soil textural classes based
on parameters in Table 3.1.
Figure 3.10 Boreholes and wetting fronts in a soil with a small λc and a soil with a large λc.
Figure 3.11 Excel spreadsheet with Brooks and Corey parameters for a loamy sand from Table 3.1.
Figure 3.12 The sum of the Simpson terms in column C is calculated in cell C18. The integral of K(h)
is calculated in cell C19 by multiplying the sum by 10/3 (h/3) (see Equation 3.27). In cell C20, λc is
calculated by dividing the integral by the difference between Ks and K(-100) (see Equation 3.25).
0
-10
-20
Depth (cm)
-30
-40
i = 0.5 cm/d
i = 1.5 cm/d
-50
-60
-70
-80
-90
-100
-40
-30
-20
-10
0
Pressure Head (cm)
Figure 3.13 Distribution of h as a function of z for a steady low infiltration rate (0.5 cm d -1) and a
high infiltration rate (1.5 cm d-1) based on Equation 3.31 (adapted from Jury and Horton, 2004).
Figure 3.14 Steady 3-D infiltration of water from a ring of radius r at the soil surface.
Figure 3.15 Steady 3-D infiltration of water from a borehole.
Figure 3.16 New Project dialog window in HYDRUS-1D.
Figure 3.17 Project panel in HYDRUS-1D.
Figure 3.18 Geometry Information dialog window in HYDRUS-1D.
Figure 3.19 Time Information dialog window in HYDRUS-1D.
Figure 3.20 Soil Hydraulic Property Model dialog window in HYDRUS-1D.
Figure 3.21 Water Flow Parameters dialog window in HYDRUS-1D.
Figure 3.22 Rosetta Lite dialog window in HYDRUS-1D.
Figure 3.23 Predicted parameters for a loamy sand textural class in the output panel.
Figure 3.24 Section of capillary tube of radius R and length L filled with water flowing in response to
a pressure difference P2 - P1 (Jury and Horton, 2004).
1
0.8
r/R
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
V(r)/Vmax
Figure 3.25 Parabolic distribution of velocities within a cylinder. The velocity is zero at the wall
where r = R and at a maximum = Vmax at the center of the cylinder where the r = 0 (Jury and Horton,
2004).
Figure 3.26 Excel spreadsheet for calculating pressure head distribution under steady flow in a
profile of the Cecil soil with properties from Table 2.4 and Table 3.2 . Water is ponded to a depth of
0.1 cm at the surface and there are tile drains at a depth of 250 cm (pressure head equals zero).