12
Hillel, pp. 190 - 192
Predictive Models for
Saturated Hydraulic Conductivity
CE/ENVE 320 – Vadose Zone Hydrology/Soil Physics
Spring 2004
Copyright © Markus Tuller and Dani Or 2002-2004
The Kozeny-Carman Predictive Model for Ksat
● Efforts to predict hydraulic conductivity based on soil structural
attributes such as porosity, particle size, etc. date back to early 1900.
● The idea is extremely attractive because of the obvious connection
between water flow and the distribution of solids and pore spaces in
the soil medium.
● In 1927 Kozeny introduced a model relating soil permeability to
porosity.
Permeability

g

Fluidity
 g
K s  k

liquid density [M L ]
  
-3
acceleration of gravity [L t-2]
viscosity coefficient of the liquid in [Pa s] [M L-1 t-1]
Fluidity at 20oC is: 9.81x106 [m s]-1.
Note that permeability has dimensions of L2.
Copyright© Markus Tuller and Dani Or2002-2004
The Kozeny-Carman Predictive Model for Ksat
 Kozeny’s model was based on application of Poiseuille’s law to flow
through granular porous media lacking structure or consolidation.
 He idealizes the soil porous system as a bundle of cylindrical
capillaries with a single representative radius.
 Rearranging Poiseuille’s law for laminar flow in tubes, considering
the fraction of pores given by porosity n, yields Kozeny’s model in
its simplest form:
nR
k  
2 2 
2
2
[m ]
A R2  h
RH  
P 2 R h
RH= Hydraulic Radius (ratio pore volume to wetted surface area)
n
Porosity
Copyright© Markus Tuller and Dani Or2002-2004
The Kozeny-Carman Predictive Model for Ksat
Kozeny’s approach was extended by Carman to take the form:
3
n
k 2 2
c Av (1  n ) 2
[m 2 ]
c Empirical coefficient representing particle size and shape
(2-3 for non-cubic particles)
 Tortuosity of the flow path (21/2 for main flow direction 45o
from the vertical axis)
Av Specific surface per unit volume.
Above equation can be simplified by introducing a hydraulic radius RH:
RH  Av (1  n)
n3
k 2 2
c RH
Copyright© Markus Tuller and Dani Or2002-2004
The Kozeny-Carman Predictive Model for Ksat
Tortuosity:
Ratio of the average roundabout path of a fluid molecule to the
apparent or straight flow path.
Tortuosity is a dimensionless geometric parameter of porous
materials, which, though difficult to measure precisely, is always
greater than 1 and may exceed 2.
Copyright© Markus Tuller and Dani Or2002-2004
The Kozeny-Carman Predictive Model for Ksat
k
n3
c 2 R h 2
The product c2 is commonly about 5.
The pore shapes, surfaces, and tortuosity are
interdependent, thus the product c2 Rh2 is
often lumped into a single “formation factor”,
F2.
n3
k 2
F
Ks is estimated as:
g n g
  2 

K s  k
   F   
3
Copyright© Markus Tuller and Dani Or2002-2004
The Kozeny-Carman Predictive Model for Ksat
The Kozeny-Carman approach
provides unsatisfactory estimates
of Ks in many soils, due to the
assumption of uniform pore radii.
Works well for sands and other
materials with uniform pore size
distribution.
Copyright© Markus Tuller and Dani Or2002-2004
Revil & Cathles Predictive Model for Ksat
Revil and Cathles [1999] developed a predictive model for
permeability using concepts from electrical conductance in porous
media.
They use the concept of a formation factor F that is a dimensionless,
scale invariant parameter that characterizes the pore space
topology.
F is related to the porosity n and pore geometry by the empirical
Archie relationship:
Fn
m
m is the so-called “cementation exponent” and varies with pore
geometry in a range of 1 to 4. (ratio of pure solution electrical
conductivity to the bulk electrical conductivity of the saturated
porous medium.
Copyright© Markus Tuller and Dani Or2002-2004
Revil & Cathles Predictive Model for Ksat
Cementation Exponent m
For media with perfectly spherical grains m=1.5
For sands with porosities between 0.03 and 0.3 m=1.5 to 2.0
For media where the pore space consists entirely of open
interconnected fractures and cracks m=1.1 to 1.3 (most of the
porosity conducts electrical current)
If large pores are connected to narrow throats m>2.5
The derivations of Revil and Cathles [1999] yielded a simple
expression:
2
3m
d n
k
24
d is the mean grain diameter (from particle size analysis) of
granular materials.
Copyright© Markus Tuller and Dani Or2002-2004
Revil & Cathles Predictive Model for Ksat
The approach yields remarkable match for granular materials, and was
later extended to predict hydraulic conductivities of sand-clay mixtures.
1 mD (milidarcy) = 1x10-15 m2 ~10-8 m/s (water 200 C)
Copyright© Markus Tuller and Dani Or2002-2004
Characterization of Flow Conditions
Flow regimes in a porous medium are characterized according to:
1) Changes in flow attributes in time and space
(steady vs. non steady flows)
2) Saturation state of the system
(saturated vs. unsaturated)
3) Geometry of flow
(one-dimensional or multi-dimensional).
Constant Head Method for Ksat: Steady-state,
saturated, and one-dimensional flow
conditions.
(water flux and other flow attributes such as hydraulic
potentials and water content do not change over time at
any given location).
Falling Head Method for Ksat:
Non steady, saturated, and one-dimensional
flow conditions.
(Flux density and hydraulic potentials vary with time.)
Copyright© Markus Tuller and Dani Or2002-2004
Unsaturated Flow
CE/ENVE 320 – Vadose Zone Hydrology/Soil Physics
Spring 2004
Copyright © Markus Tuller and Dani Or 2002-2004
Flow of Water in Unsaturated Soils
 Most flow processes in the field occur under unsaturated conditions, or
in the presence of an air-phase.
 This drastically modifies water flow pathways, with decreasing water
content flow through water filled pores become constrained to narrower
and tortuous channels.
 Darcy's law, originally conceived for saturated flow only, was extended by
Edgar Buckingham (1907) to unsaturated water flow. Buckingham's main
assumption was that unsaturated hydraulic conductivity is a function of
water content or matric potential, i.e. the degree of wetness.
 H 
 (h  z) 
 h 
Jw  K(h)   K(h)
 K(h)
 1

 z 
 z 
 z 
 dh 
Jw  K(h)   1
 dz 
Unsaturated Hydraulic Conductivity expressed as a function of
Matric Potential h
Copyright© Markus Tuller and Dani Or2002-2004
Saturated versus Unsaturated Flow
The driving force for flow in a saturated soil under constant gravitational
potential is the gradient of positive pressures, whereas under unsaturated
conditions the gradient is of negative matric potentials.
When the soil is completely water-saturated, the pores conduct at a
maximal rate Ks. When the soil desaturates, some of the water-filled pores
become partially or completely air-filled, and the cross-sectional area
available for mass flow decreases K(h).
Equations for saturated flow
have a constant saturated
hydraulic conductivity (Ks)
Unsaturated flow equations
have an unsaturated
hydraulic conductivity
function, K(h).
Copyright© Markus Tuller and Dani Or2002-2004
Unsaturated Hydraulic Conductivity Function
crossover
● The unsaturated hydraulic conductivity
is a highly nonlinear function of water
content, K(), or matric potential, K(h)often spanning more than 8 orders of
magnitude!
● At saturation a sandy soil has higher
conductivity than clay due to larger
water-filled pore spaces.
● At lower suctions large pores in sandy
soil drain resulting in a steep decrease
in conductivity.
● Loamy and clayey soils, having smaller
(on average) pores, retain greater
number of water-filled pores resulting in
higher hydraulic conductivity than sandy
soils at low potentials.
Copyright© Markus Tuller and Dani Or2002-2004
Unsaturated Hydraulic Conductivity Function
Estimates for unsaturated hydraulic conductivity for various soil textural
classes can be obtained from the UNSODA and NRCS Soil Survey
databases.
4
4
2
1
a
2
2
3
7
12
11
8
0
1
b
2
9
0
10
8
6
5
9
12
-2
-2
3 4
11
-4
lo g K (c m /d )
lo g K (cm /d)
-4
-6
-8
-6
-8
UNSODA
-10
-12
6
-14
7
-16
0.0
1 Sand
7 Sandy clay loam
2 Loamy sand 8 Clay loam
3 Sandy loam 9 Silty clay loam
4 Loam
11 Silty clay
5 Silt
12 Clay
6 Silt loam
4
0.1
5
0.2
0.3

0.4
0.5
Soil Survey
-10
1 Sand
7 Sandy clay loam
2 Loamy sand 8 Clay loam
3 Sandy loam 9 Silty clay loam
4 Loam
10 Sandy clay
5 Silt
11 Silty clay
6 Silt loam 12 Clay
-12
-14
-16
0.0
0.1
0.2
0.3
0.4
0.5

Copyright© Markus Tuller and Dani Or2002-2004
Parametric Models for K()
Some researchers have extended the concepts discussed in the
previous section and derived closed-form analytical predictions of
the unsaturated hydraulic conductivity. The basic step is replacing
the summations with integrals of SWC analytical (parametric)
expressions
The van Genuchten and Brooks and Corey SWC Models are
amenable to such integration. Van Genuchten (1980) derived an
expression for the unsaturated hydraulic conductivity based on the
same parameters he used for the SWC curve and on a modification
of Jackson’s model proposed by Mualem (1976).
The resulting water content-dependent unsaturated hydraulic
conductivity K() is given by:
 
1 
K ( )
  1  1   m 
 

Ks

 
1
2
m 2




  r
s   r
Copyright© Markus Tuller and Dani Or2002-2004
Parametric Models for K(h)
van Genuchten-Mualem (VGM) model:
The unsaturated hydraulic conductivity can also be expressed in
terms of matric head (h):
K (h )

Ks

 1  (h ) n 1 1  (h )

1 

n m


2

m
(h ) n 2
Copyright© Markus Tuller and Dani Or2002-2004
Parametric Models for K(h)
Brooks and Corey (BC) model:
Similarly expressions can be derived for the Brook's and Corey
model.
K(h )  h 
  
Ks
 hb 
23
where hb is the matric head at air-entry value or "bubbling pressure",
and  is the pore distribution index parameter.
Expressed as K():
K ( )

Ks
3
2

  r

s   r
Copyright© Markus Tuller and Dani Or2002-2004
Parametric Models for K() & K(h)
As input information for the VG and BC models for unsaturated
hydraulic conductivity we need to measure a few h- pairs (SWC) and
saturated hydraulic conductivity Ks:
(1) We fit the parametric expressions for the SWC to measured
data to obtain free model parameters , n, r (VG) or hb,  (BC)
(2) We use measured Ks and the free model parameters from the
SWC to predict unsaturated hydraulic conductivity
Copyright© Markus Tuller and Dani Or2002-2004
Parametric Models for K(h)
Another widely used parametric model for unsaturated hydraulic
conductivity was proposed by Gardner (1958):
K(h )  K s e
bh
Where b (dimensions of L-1) is a parameter related to typical pore
size of the medium.
The primary advantage of Gardner’s model is that it allows
analytical solutions to certain flow problems.
Why do we need parametric models?
•
For obtaining analytical solutions to flow and transport
problems
•
For comparison between different types of soils or other
porous media.
Copyright© Markus Tuller and Dani Or2002-2004
Steady State Unsaturated Flow in Soils
CE/ENVE 320 – Vadose Zone Hydrology/Soil Physics
Spring 2004
Copyright © Markus Tuller and Dani Or 2002-2004
Steady State Unsaturated Flow
Steady-state unsaturated flow may develop under a variety of
conditions.
(1) When a constant flux at the soil surface, less than the soil's
Ks is maintained for extended periods of time by use of a
sprinkler or other means.
(2) The evaporation rate from a shallow water table may also
attain (approximately) a steady state.
For vertical steady-state unsaturated flow we may write the
Buckingham-Darcy equation:
Jw
 dh 
 K (h )   1
 dz 
dh
 dz
Jw
1
K (h )
This ordinary differential equation may be integrated to yield the
integral form of the Buckingham-Darcy equation (next slide).
Copyright© Markus Tuller and Dani Or2002-2004
Steady State Unsaturated Flow
Integral form of the Buckingham-Darcy equation:
dh
 dz
J
1 w
K (h )
h2
z2
dh
 J w    dz  z1  z 2
h1 1 
z1
K (h )
E.g., Gardner’s Model
K(h )  K s e b h
When K(h) has a known functional form, the integral on the left side
may be evaluated exactly to produce an analytical solution for the
water flux or for the distribution of the matric potential in the soil
profile, h(z).
Copyright© Markus Tuller and Dani Or2002-2004
Steady State Unsaturated Flow
Analytical Solution for Unsaturated Steady State Flow to a Water
Table:
Given a constant flux (Jw < Ks) from the soil surface to a water table
at a depth L. We assume that the unsaturated hydraulic conductivity,
K(h), is given by Gardner's model K(h)=Ks ebh
We want to find the distribution of soil matric potential in the soil
profile from the soil surface to the water table.
(1) The first step is to introduce the parametric expression of
K(h) into the integral form of the Buckingham-Darcy law
h2
z
2
dh
 J w    dz  z1  z 2
h1 1 
z1
K (h )
h ( L)

h (z)
dh
J w b h
1
e
Ks
L
   dz  L  z
z
Copyright© Markus Tuller and Dani Or2002-2004
Steady State Unsaturated Flow
(2) We set z=0 at the soil surface, and z=-L at the water table
(where h=0). From a table of integrals we find the following
solution:


dx
x 1
ax


ln
p

qe
C
 p  qe ax p ap
(3) Applying this solution to our problem with the appropriate
boundary conditions yields:
 J w b h (z)  1  J w 
z  L  h (z)  ln 1 
e
  ln 1 

b  Ks
 b  Ks 
1
(4) Rearranging and log-transforming yields the desired form in
terms of h(z):
h ( z) 
 J

J
ln  w  e b ( L z )  w e b ( L z ) 
b  Ks
Ks

1
One can verify that at z=-L, h(z=-L)=0. The term in the square brackets is
limited to values between 0 and 1. This is important for analyses of maximal
upward flow from a water table.
Copyright© Markus Tuller and Dani Or2002-2004
Steady State Unsaturated Flow
Graph showing matric potential distributions within the soil
profile for various potentials (previous equation).
Copyright© Markus Tuller and Dani Or2002-2004
Steady State Unsaturated Flow - Example
Problem Statement:
Compute and plot the vertical distributions of the soil matric
potential under steady flow into a water table at a depth of 3 m
from the surface for the flow rates of Jw/Ks= -0.001, -0.01, -0.1,
and -0.5. Note that the sign of the downward flux is negative. The
unsaturated hydraulic conductivity of the soil is based on
Gardner's (1958) model, and is characterized by b=5 m-1.
Estimate the hydraulic gradient (dH/dz) from 0 to 1m below the
soil surface.
Solution:
1. The depth dependent matric potential may be calculated
according to the following relationship:
 Jw
J w  b ( L z ) 
 b ( L z )
h (z)  ln 
e

e

b  Ks
Ks

1
Results are listed on the next slide.
Copyright© Markus Tuller and Dani Or2002-2004
Steady State Unsaturated Flow - Example
Tabulated Results:
 Jw
J w  b ( L z ) 
 b ( L z )
h (z)  ln 
e

e

b  Ks
Ks

1
Jw/Ks
Z [m]
0.00
-0.25
-0.50
-0.75
-1.00
-1.25
-1.50
-1.75
-2.00
-2.25
-2.50
-2.75
-3.00
-0.001
h(z) [m]
-1.3815
-1.3813
-1.3808
-1.3790
-1.3727
-1.3522
-1.2936
-1.1667
-0.9725
-0.7419
-0.4978
-0.2495
0.0000
-0.01
h(z) [m]
-0.9210
-0.9210
-0.9210
-0.9208
-0.9201
-0.9179
-0.9104
-0.8861
-0.8188
-0.6805
-0.4788
-0.2451
0.0000
-0.1
h(z) [m]
-0.4605
-0.4605
-0.4605
-0.4605
-0.4604
-0.4602
-0.4595
-0.4571
-0.4487
-0.4221
-0.3499
-0.2055
0.0000
-0.5
h(z) [m]
-0.1386
-0.1386
-0.1386
-0.1386
-0.1386
-0.1386
-0.1385
-0.1382
-0.1373
-0.1340
-0.1229
-0.0882
0.0000
Copyright© Markus Tuller and Dani Or2002-2004
Steady State Unsaturated Flow - Example
Solution – Graph:
S o il M a t r ic P o t e n t ia l ( m )
- 1 .6
- 1 .4
- 1 .2
-1
- 0 .8
- 0 .6
- 0 .4
- 0 .2
0
0 .0 0
0 .5 0
S o il D e p t h ( m )
1 .0 0
1 .5 0
2 .0 0
2 .5 0
3 .0 0
3 .5 0
Copyright© Markus Tuller and Dani Or2002-2004
Steady State Unsaturated Flow - Example
Solution - Continued:
2. To estimate the hydraulic gradient we have to solve:
dH d (h  z ) dh


1
dz
dz
dz
J w K s  0.001
dH  1.38149  1.37268

 1  0.99119
dz
1
J w K s  0.01  dH dz  0.99911
J w K s  0.1  dH dz  0.99992
J w K s  0.5  dH dz  0.99999
Copyright© Markus Tuller and Dani Or2002-2004
Prediction of Unsaturated Hydraulic Conductivity
The relationships between the hydraulic conductivity and the degree
of saturation or matric potential arouse interest in methods for
predicting the soil hydraulic conductivity based on SWC.
Most models are based on simplifying assumptions:
1)
The complicated soil pore geometry is represented by a
bundle of continuous cylindrical capillaries.
2)
The flow is assumed to occur only in these capillaries
(no bypass flow)
3)
The length of the bundle of capillaries (Lc), is assumed to
be larger than the length (L) of the soil column it
represents; the ratio Lc/L is the tortuosity factor ().
Soil Pores
Copyright© Markus Tuller and Dani Or2002-2004
Predictive Models for Soil Hydraulic Conductivity
Poiseuille's law for laminar flow in cylindrical tubes is applied to the
bundle of capillaries with a gradient H across its length. This
enables computation of the total volume of flow through the column
by:
Soil Pores
Poiseuille's Law
 R 4 P
Q
8 L
  w g H M
4
  N jQ j 
N
R

j j
8 L c j1
j1
M
Q tot
Nj
is the number of capillaries of radius Rj in the bundle
M
is the number of different (arbitrarily assigned) capillary size
classes
Copyright© Markus Tuller and Dani Or2002-2004
Predictive Models for Soil Hydraulic Conductivity
The flux (Jw) through the column is:
Jw
nj
Q tot   w g H M
4


n jR j

A
8 L c j1
is the number of tubes with radius Rj per unit area (nj =Nj/A)
The next step is to convert the soil water characteristic curve into a
soil water content vs. equivalent pore radii curve. This is done by
using the capillary-rise equation to convert a matric potential value to
an equivalent capillary radius (see next slide).
Copyright© Markus Tuller and Dani Or2002-2004
Predictive Models for Soil Hydraulic Conductivity
1) The total saturation
water contend s is
divided into a certain
number M of equal
increments v where
the matric potential hj
corresponds to a water
content of s- jxv
2) We assume that all
tubes with radii greater
than Rj have drained at
matric potential hj
 2
rRj 
w g h j
Copyright© Markus Tuller and Dani Or2002-2004
Predictive Models for Soil Hydraulic Conductivity
SWC-Curve
1.E+04
1.E+02
1.E+01
1.E+00
1.E-01
1.E-02
0
0.1
0.2
0.3
0.4
Water Content- Equivalent
Capillary Radius Curve
0.5
Volumetric Water Content [m3/m3]
1.E-02
1.E-03
Capillary Rise Equation
 2
Rj 
w g h j
Equivalent Radius [m]
Matric Potential [-m]
1.E+03
Empty
1.E-04
1.E-05
1.E-06
1.E-07
Full
1.E-08
1.E-09
0
0.1
0.2
0.3
0.4
0.5
Volumetric Water Content [m3/m3]
Copyright© Markus Tuller and Dani Or2002-2004
Predictive Models for Soil Hydraulic Conductivity
3) The number of capillaries
having a radius Rj per
unit area in each water
content interval is:
nj=v/Rj2 (where v is
interpreted as the
fraction of the water-filled
cross sectional area that
is reduced when all
capillaries having a
radius of Rj drain).
Copyright© Markus Tuller and Dani Or2002-2004
Predictive Models for Soil Hydraulic Conductivity
4) Now we can plug in the number of capillaries nj into Poiseuille’s
law and introduced a multiplication by -L/z=1 (where z=0-L) to
obtain a form similar to Darcy's law :
2

  w g H M  v

 v M 1  L H
4

Jw 
R j  


2
2
8 L c j1  R j
 2  w g j1 h j  L c z
Saturated Hydraulic
Conductivity
 Tortuosity Factor
  2  v M 1
Ks 

2 w g j1 h j 2
Copyright© Markus Tuller and Dani Or2002-2004
Predictive Models for Soil Hydraulic Conductivity
As water content decreases from s to s- i x v the tubes larger
than Ri drain and do not contribute to water flow. The effect on the
unsaturated hydraulic conductivity, which is a function of the water
content, is given by
  2  v M 1
Ks 

2 w g j1 h j 2
  2  v
K ( s  i v ) 
2 w g
M
1
ji 1
j
h2
This equation, originally Developed by Childs and Collis-George
(1950), serves as the basis for many predictive models of unsaturated
hydraulic conductivity, most of which are based on SWC information
and assume a capillary bundle geometry.
Copyright© Markus Tuller and Dani Or2002-2004
Predictive Models for Soil Hydraulic Conductivity
Even with a detailed SWC, there is still an unknown, i.e. the
tortuosity (). One way to circumvent this problem is to measure the
saturated hydraulic conductivity (which is relatively easy to
measure) and use it to solve for , assuming it is constant. One such
formulation was proposed by Jackson (1972):


j 
2
K(i )  K s 
 s








c
M

2 j  1  2i
j i
M

j 1
h j2
2j  1
h j2
where j=(s-j x v), c is a constant usually taken as c=1, and
the water content is divided to M equal increments (of v)
Copyright© Markus Tuller and Dani Or2002-2004
Jackson Model - Example
The matric potential-soil water content function (SWC) for Sarpy
loam soil is given by the VG model expressed for the matric potential
(h) as:

1
n
1


1
h   m  1



  r
s   r
s  0.4 m3 m3  r  0.03 m3 m3
  2.79 m 1
n  1.6
m  1  1 n  0.375
K s  50 mm hr
c 1
M  10
Copyright© Markus Tuller and Dani Or2002-2004
Jackson Model - Example
First we split the SWC into 10 equal increments and calculate water
content  and corresponding matric potential h for the midpoint of
each increment.
1
n
1


1  2 j 1 m 
h j  1 
 1

 
2M 


1.E+04
Matric Potential [-m]
1.E+03
1.E+02
1.E+01
1.E+00
1.E-01
1.E-02
0
0.1
0.2
0.3
Volumetric Water Content [m3/m3]
0.4
0.5
 j  s  s   r 
(2 j  1)
2M
Copyright© Markus Tuller and Dani Or2002-2004
Jackson Model - Example
Now we can apply Jackson’s equation and calculate K() for the
midpoints of each increment. If we consider for example the
calculation of K(0.289):
K(0.289)  50
*
0.289 13.0457
 2.2038
0.4 213.844
mm hr 
j, i
hj (m)
1/hj2
j+/2
(2j+1-2i)hj2
K(j)
1
0.1079
85.83
0.400
213.844
50.000
2
0.2446
16.72
0.363
70.8035
15.024
3
0.3919
6.51
0.326
30.3092
5.7757
4
0.5790
2.98
0.289
13.0457
2.2038
5
0.8424
1.41
0.252
5.27549
0.7771
6
1.253
0.637
0.215
1.89699
0.2384
7
1.982
0.254
0.178
0.56445
0.0587
8
3.556
0.079
0.141
0.12307
0.0101
9
8.430
0.0141
0.104
0.01515
9.21E-4
10
52.806
3.59E-4
0.067
0.00036
1.40E-5
j is a pointer whereas i is a counter
** /2=0.37/(10x2)=0.0185; note that hj corresponds to j, which is less than the water content indicated in
the table by the amount of /2.
Copyright© Markus Tuller and Dani Or2002-2004
Jackson Model - Example
When we compare our results with the analytically derived van
Genuchten-Mualem (VGM) K() model (see next slides) we see good
agreement.
Copyright© Markus Tuller and Dani Or2002-2004
15
Hillel, pp. 210 - 219
Unsteady and Unsaturated Flow –
The Richards Equation
CE/ENVE 320 – Vadose Zone Hydrology/Soil Physics
Spring 2004
Copyright © Markus Tuller and Dani Or 2002-2004
Unsaturated Nonsteady Flow
Unsaturated nonsteady flow is the most common flow regime
under field situations, where water content , the matric potential
h, and the flux Jw vary in time and space.
The application of Buckingham-Darcy's law to describe the flow
is no longer sufficient as the flux varies from one location to
another, a condition called flux divergence.
The difference between the flux entering a volume element of soil
and that exiting the same volume element is the change in water
content, or storage (CONSERVATION OF MASS = CONTINUITY
PRINCIPLE).
Copyright© Markus Tuller and Dani Or2002-2004
The Richards Equation
The principal of continuity may
be formalized mathematically for
one-dimensional vertical flow as:
 v J w

t
z
Note that the partial derivatives indicate possible variations of flux
and water content in space and in time.
The three-dimensional form of the continuity equation is given as:
 J wz J wy J wx 


   J w  


t
y
x 
 z
del operator
(gradient of J)
Copyright© Markus Tuller and Dani Or2002-2004
Unsaturated Nonsteady Flow – Richards Equation
To obtain a flow equation for unsaturated nonsteady flow we have
to combine Buckingham-Darcy's law for the flux (Jw) with the
continuity equation.
For one-dimensional vertical flow we obtain:
  
 h 
 K (h )  1
t z 
 z 
For three-dimensional flow we obtain:

 
h   
h   
h  K (h )
   K (h ) H   K (h )    K (h )    K (h )  
t
x 
x  y 
y  z 
z 
z
This equation is known as the RICHARDS EQUATION, and is the
most general unsaturated flow equation used in practice.
Note that this equation is based on the assumption of isothermal
conditions (uniform temperature) and flow through a rigid matrix
(no swelling or shrinking of the soil).
Copyright© Markus Tuller and Dani Or2002-2004
Unsaturated & Unsteady Horizontal Flow
Unsaturated and nonsteady horizontal flow in one dimension may
be represented by the simplest form of the Richards equation,
with no gravitational component, as:

 
h 

K (h ) 

t  x 
x 
In this case we have two variables,  and h, which vary in time and
space, but only one equation.
In some cases it is possible to reduce the equation to only one
variable by using the interrelationship h (the SWC curve).
For water infiltration into a homogeneous soil having uniform initial
water content it is advantageous to eliminate the matric potential
(h).
Copyright© Markus Tuller and Dani Or2002-2004
Unsaturated Nonsteady Horizontal Flow
We first apply the chain rule to the matric potential gradient to
obtain:

 
h 
h dh 

K
(
h
)

t  x 
x 
x
d x
Reciprocal of the
SPECIFIC WATER CAPACITY Cw()
d
C w ( ) 
dh
Cw() is the slope of the SWC curve at any particular water content.
Next we express K(h[]) = K(), and define a new hydraulic function,
the Soil Water Diffusivity D() as:
dh
K ( )
D( )  K ( )

d C w ( )
The Soil Water Diffusivity is
the ratio of unsaturated
hydraulic conductivity to the
specific water capacity.
Copyright© Markus Tuller and Dani Or2002-2004
Unsaturated Nonsteady Horizontal Flow
Soil Water Diffusivity D():
dh
K ( )
D( )  K ( )

d C w ( )
Since the K() and Cw() functions are highly nonlinear, the
diffusivity is also nonlinear. However, the range of variation in D()
is much smaller than the range of K() variation because the
nonlinearity in numerator and denominator cancel out to some
degree. The resulting water content or diffusivity form of the
Richards equation for one dimensional horizontal flow is then:

 
 

D( ) 

t  x 
x 
The driving force is the
gradient in water content
“water concentration”
This is not in accordance with the basic physical
principle of flow occurring due to potential gradients !!
Special care is required when applying the diffusivity
form of the Richards equation!!
Copyright© Markus Tuller and Dani Or2002-2004
Unsaturated Nonsteady Horizontal Flow

 
h 

K (h ) 

t  x 
x 

 
 

D( ) 

t  x 
x 
Copyright© Markus Tuller and Dani Or2002-2004
Unsaturated Nonsteady Horizontal Flow
Graphical presentation of the K(), Cw(), and D() functions
CW()
1.E+00
0.30
-1
]
1.E-01
1.E-02
Water Capacity [m
Hydraulic Conductivity [m /hr]
K()
1.E-03
1.E-04
1.E-05
1.E-06
1.E-07
0.25
0.20
0.15
0.10
0.05
0.00
1.E-08
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
3
Volumetric Water Content [m 3/m 3]
0.5
3
Volumetric Water Content [m /m ]
D()
1.E+00
Diffusivity [m2/hr]
1.E-01
D( )  K ( )
1.E-02
1.E-03
dh
K ( )

d C w ( )
1.E-04
1.E-05
1.E-06
0
0.1
0.2
0.3
0.4
3
0.5
3
Volumetric Water Content [m /m ]
Copyright© Markus Tuller and Dani Or2002-2004
Boltzmann transformation for horizontal flow
  xt
1
2
 d  t 1 d


t d t
2 d
 
   
d   d 
d 
D
(

)

D
(

)

D
(

)
 x 
x   x 
d x  d  
d 
Copyright© Markus Tuller and Dani Or2002-2004
ODE for Unsaturated Nonsteady Horizontal Flow
 d
d 
d 


D( ) 

2 d d  
d 
Analytical solution with
constant diffusivity D()=D:

x 

 2 Dt 
 ( x, t )   i  ( 0   i ) erfc 
Copyright© Markus Tuller and Dani Or2002-2004
Diffusivity Form of the Richards Equation
Special care is required when applying the diffusivity
form of the Richards equation!!
It should be applied only to homogeneous soils, where there is a
unique relation between h and , neglecting hysteresis, and to
describe horizontal flow where there is no gravitational component.
What happens in layered soils?
We cannot apply
diffusivity form of
Richards equation!!
(the derivative /x
is not defined across
the discontinuity)
Matric potential is
continuous across
layers
Water Content is
discontinuous across
layers
Copyright© Markus Tuller and Dani Or2002-2004
Diffusivity Form of the Richards Equation
The diffusivity form of Richards equation is applicable:
In homogenous soils for conditions where the vertical gravitational
component is very small in comparison with matric influences
(translated into a water content "gradient").
Examples:
1) Early phases of infiltration into relatively dry soil,
2) Evaporation of water from the soil surface.
For special cases the diffusivity may be assumed as
constant over a small range of water content.
This facilitates analytical solutions of the Richards equation.
Copyright© Markus Tuller and Dani Or2002-2004
Parametric Forms of the Diffusivity Function
Diffusivity is defined as:
D( )  K ( )
dh
d
We can express K() with the van Genuchten – Mualem relationship
and substitute with the VG model for :
1 
1 

K ( )


2
m
  1 1 
 

Ks

 
D( ) 
m 2



(1  m) K s
 m ( s   r )

  r 
1



s   r 1   h  n 
1 1

2
 m
m

 
h
1

m

1
n
 1



1 m 
1 m
1   m   1   m   2










Copyright© Markus Tuller and Dani Or2002-2004
Parametric Forms of the Diffusivity Function
Applying the same approach for the Brooks and Corey model yields:
Ks h b
D( ) 

 ( s   r )
2
1

Copyright© Markus Tuller and Dani Or2002-2004
Numerical Solutions of Richards Equation
● The high degree of nonlinearity in the
coefficients of the Richards equation
for unsaturated flow makes it
impractical to obtain analytical
solutions.
● It might be considered as one of the
most challenging equations in
environmental physics (considerably
different than flow of heat or electricity).
● Except for a few special cases,
obtaining solutions to practical flow
problems represented by the Richards
equation is feasible by means of
successive numerical approximations,
generally using computers.
Copyright© Markus Tuller and Dani Or2002-2004
Numerical Solutions of Richards Equation
Numerical Approximations (Finite Difference Method) are based on:
1) Discretization of the flow domain
into a grid: "layers" for one
dimensional flow or a mesh for
two-dimensional flow.
2) Discretization of the continuous
partial differential equation into a
series of approximated difference
equations, one for each of the K
spatial nodes.
i 1 / 2
 i j 1   i j
( hi j  hi j 1  z ) K i 1/ 2 ( hi j 1  hi j  z )

z
z
z
t
3) Application of initial and boundary
conditions.
4) Formulation of the problem into a
computer code which is able to
solve the equations for the entire
flow domain at various time steps.
Copyright© Markus Tuller and Dani Or2002-2004
Numerical Solutions of Richards Equation
Example: A Numerical Approximation to the Richards Equation
Find a numerical approximation to changes in water content with
time under vertical unsaturated and nonsteady flow into a soil with a
known initial water content, and known hydraulic functions (y) and
K(y).
Solution: We first discretize the soil profile (the spatial domain) into
units of z, and the time into units of t. We denote the water content
at the i-th depth increment and at the j-th time step as ij; similarly we
express the corresponding matric head as hij (see graph previous
slide)
The hydraulic conductivity, which is a function of the matric head (or
of the water content), relates transfer between two soil layers and
thus must be averaged to represent both layers as: K([hij + hji+1]/2) =
Kji+1/2. The resulting numerical approximation for BuckinghamDarcy’s law for vertical flow is then given by:
j
j
j
j
j
j
j1
j
  i   i
 
 h  (h i1  h i  z) K i1 2 (h i  h i1  z) K i1 2

 K (h )   1 

2
t
t
z 
z
z 2
 z 
Copyright© Markus Tuller and Dani Or2002-2004
Numerical Solutions of Richards Equation
Example – continued:
Combining terms and rearranging allows us to solve for the only
unknown in this equation, which is ij+1, the water content for the i-th
depth increment at the next (future) time step:
 ij1   ij 

t
j
j
j
j
j
j
(
h

h


z
)
K

(
h

h


z
)
K
i 1
i
i
i 1
i 1 2
i 1 2
2
z

Because this is a discrete approximation of a continuous process,
increments z and t should be kept small. In addition, initial soil
water contents with respect to depth and the conditions at the
boundaries of the flow domain need to be specified.
Copyright© Markus Tuller and Dani Or2002-2004