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```									                   Agron 677 Term Project

Comparison of Finite Difference Method, Philip’s Method
and Green-Ampt Model in Infiltration Simulation

Zhiming Qi

Department of Ag Engineering
Outline

   Introduction
   Materials and Methods
   Results and Discussion
   Conclusion
   Suggestion
Introduction
   Infiltration is a key component to hydrological process.

   Mathematical Methods have been developed for computing and
simulating infiltration:
Green-Ampt Model (1911)
Philip’s method for Richards equation (1957)
Finite difference method for Richards equation

Boundary Conditions:
First class boundary: constant WC upper boundary WC= c;
Second class boundary: constant water influx q = c;
2nd boundary in most natural infiltration process
Third class boundary: constant change of water influx dq/dt = c;

   Computer Models have been developed to simulate the
infiltration procedure.
DRAINMON (Green-Ampt)
MIKE-SHE and HYDRUS (Richards, Finite difference method)
Introduction
       Mathematical Methods for infiltration Modeling:
1. Green-Ampt Model (1911)
Green-Ampt model calculates cumulative
infiltration by assuming water flow into a
vertical soil profile like a piston flow
(first boundary condition)

 
ft  K (           1)
Ft
Ft  t  
Ft  t     Ft  Kt   ln[               ]
Ft  
by Chow et al. (1988)
Parameters required:
K      
Introduction
   Mathematical Methods for infiltration Modeling:
2. Finite difference method for Richards equation
Explicit scheme
Implicit scheme
Crank-Nicolson scheme
(any boundary conditions)

         h
 ( K (h)  1)
t z       z
http://hydram.epfl.chf
Parameters required:
 (h) K (h)
Introduction
   Mathematical Methods for infiltration Modeling:
Philip’s method for Richards equation (1957)
(only for first boundary condition)

                              K
 ( D( ) ) 
t z                    z       z
h
D( )  K ( )

X ( , t )   ( )t 1 / 2   ( )t 2 / 2   ( )t 3 / 2   ( )t 4 / 2    f m ( )t m / 2

Parameters required:
 (h) K (h) D ( )                    ( )  ( )         …….
Materials and Methods
from Haverkamp et al. (1977).

   Soil
 ( s   r )    K s =34 cm/h, A=1.175×106,  K =4.74
            T
h
A
K  Ks               k    s =0.287,  r =0.075;  =1.611×106,  T =3.96
A h

   Boundary Condition (2nd)
t <0   0<z<70 cm,             n =0.10 cm-3/cm-3 or hn=-61.5 cm
t >= 0 z = 0                   q = 13.69 cm/h
t >= 0 z >= 70 cm                = n = 0.10 cm-3/cm-3
Materials and Methods
   Parameters for Green-Ampt Model
Parameters required:   K    
Initial estimation:

K = K s =34 cm/h.
 = hn=-61.5 cm
   s  n    = 0.287-0.10 = 0.187 cm3/cm3
Materials and Methods
   Mathematical Method for Green-Ampt Model
Ft  t  
Ft  t    Ft  Kt   ln[               ]
Ft  

Ft  t  
Y ( Ft  t )  Ft  Kt   ln[               ]  Ft  t
Ft  
Ft  t
Y ( Ft  t )                   0
Ft  t  
Newton’s Iteration for the
equation after ponding:

Y ( Ft  t ) i
( Ft  t ) i 1    ( Ft  t ) i 
   http://www.krellinst.org/
Y ( Ft  t ) i
Materials and Methods
   Parameters for Implicit Finite Difference (Richards)
Parameters required:
 (h) K (h)                       are known

    Parameters for Philip Method (Richards)
Parameters required:
 (h) K (h) D ( )                                     ( )  ( )
h                    A                          ( s   r )  ( s   r )
D( )  K ( )       Ks                                                      [                 ](1 /    T   1)

            ( s   r )
 K / T
 T (   r )  T (   r )
A                
 r

                  d                                      d    d ( ) dK
 ( )               )
n  ( )d  2D d ( )                                 d
(P
d       d
Materials and Methods
   Evaluation Criteria
Cumulative Infiltration (Mass Balance):
Percentage Error

Vm  Vin
PE            100 %
Vin
Model Efficiency:
Program running time

Wetting Front:
Wetting front plotting
Results and Discussion
   Implicit Finite Method
Implicit Method 0.1~0.8h

0
0.1hr
-10       0.2hr

-20
0.3hr

0.4hr
PE = 0.964%
0.5hr
-30
0.6hr

-40       0.7hr
Running Time: 8 min
Depth cm

0.8hr

-50

-60

-70

-80

-90

-100
0.00           0.05   0.10             0.15             0.20   0.25

Water Content

Figure 1. Wetting front by finite difference method for Richards equation.
Results and Discussion
                 Philip’s Method            Philip X~Theta

0
0.1hr
-20       0.2hr
0.287
0.3hr
-40
0.4hr

0.5hr
-60
0.6 hr

-80       0.7hr
Distance

-100

-120

-140

-160
PE =160%
-180                                   Running Time:
< 2 sec.
-200
0.00            0.05   0.10        0.15        0.20   0.25

Water Content

Figure 2. Wetting front of Philips method using different WC for the upper boundary condition
Results and Discussion
                 Philip’s Method            Philip X~Theta                                                             Implicit Method 0.1~0.8h

0                                                                           0

0.1hr                                                                       0.1hr
-20       0.2hr
0.287                              -10       0.2hr

0.3hr                                                                       0.3hr
-40                                                                         -20
0.4hr                                                                       0.4hr

0.5hr                                                                       0.5hr
-60                                                                         -30
0.6 hr                                                                      0.6hr

0.7hr                                                             -40       0.7hr
-80

Depth cm
0.8hr
Distance

-50
-100

-60
-120

-70
-140

-80
-160
PE =160%
-90
-180                                   Running Time:
< 2 sec.                             -100
-200                                                                            0.00           0.05   0.10             0.15             0.20   0.25
0.00            0.05   0.10        0.15        0.20   0.25
Water Content
Water Content

Figure 2. Wetting front of Philips method using different WC for the upper boundary condition
Results and Discussion
                 Philip’s Method            Philip X~Theta                                                              Philip X~Theta

0                                                                           0
0.1hr                                                                       0.1hr
-20       0.2hr
0.287                              -10       0.2hr
0.265
0.3hr                                                                       0.3hr
-40                                                                         -20
0.4hr                                                                       0.4hr

0.5hr                                                                       0.5hr
-60                                                                         -30
0.6 hr                                                                      0.6 hr

-80       0.7hr                                                                       0.7hr
-40
Distance

Distance

-100                                                                         -50

-120                                                                         -60

-140                                                                         -70

-160
PE =160%                              -80                               PE =14.5%
-180                                   Running Time:                                                           Running Time:
-90
< 2 sec.                                                                < 2 sec.
-200
-100
0.00            0.05   0.10        0.15        0.20   0.25
0.00            0.05   0.10        0.15        0.20   0.25
Water Content
Water Content
Figure 2. Wetting front of Philips method using different WC for the upper boundary condition
Results and Discussion
                 Philip’s Method            Philip X~Theta                                                              Philip X~Theta

0                                                                           0                                                             20
0.1hr                                                                       0.1hr
-20       0.2hr
0.287                              -10       0.2hr
0.265
15
0.3hr                                                                       0.3hr
-40                                                                         -20
0.4hr                                                                       0.4hr

-60
0.5hr                                                                       0.5hr                                                 10
-30
0.6 hr                                                                      0.6 hr

PE%
-80       0.7hr                                                                       0.7hr
-40
5
Distance

Distance

-100                                                                         -50

0
-120                                                                         -60

-140                                                                         -70                                                              -5
-160
PE =160%                              -80                               PE =14.5%
Running Time:                 -10
-180                                   Running Time:                         -90
< 2 sec.                                                                < 2 sec.                                  Water Content
-200
-100                                                             -15
0.00            0.05   0.10        0.15        0.20   0.25
0.00            0.05   0.10        0.15        0.20   0.25
0.256 0.258   0.26   0.262 0.264
Water Content
Water Content
Figure 2. Wetting front of Philips method using different WC for the upper boundary condition
Results and Discussion
                 Philip’s Method            Philip X~Theta                                                              Philip X~Theta                                                              Philip X~Theta

0                                                                           0                                                                           0

0.1hr                                                                       0.1hr                                                                       0.1hr
-20       0.2hr
0.287                              -10       0.2hr
0.265                                      -10       0.2hr
0.260
0.3hr                                                                       0.3hr                                                                       0.3hr
-40                                                                         -20                                                                         -20
0.4hr                                                                       0.4hr                                                                       0.4hr

0.5hr                                                                       0.5hr                                                                       0.5hr
-60                                                                         -30                                                                         -30
0.6 hr                                                                      0.6 hr                                                                      0.6 hr

-80       0.7hr                                                                       0.7hr                                                             -40       0.7hr
-40

Distance
Distance

Distance

-100                                                                         -50                                                                         -50

-120                                                                         -60                                                                         -60

-140                                                                         -70                                                                         -70

-160
PE =160%                              -80                               PE =14.5%                                 -80                       PE =-0.57%
-180                                   Running Time:                                                           Running Time:                             -90
Running Time:
-90
< 2 sec.                                                                < 2 sec.                                                            < 2 sec.
-200                                                                                                                                                    -100
-100
0.00            0.05   0.10        0.15        0.20   0.25                                                                                              0.00            0.05   0.10        0.15        0.20   0.25
0.00            0.05   0.10        0.15        0.20   0.25
Water Content                                                                                                                                           Water Content
Water Content
Figure 2. Wetting front of Philips method using different WC for the upper boundary condition
Results and Discussion
      Philip Method Parameters
h~Theta                                                                              K~Theta
Rumda ~ WatCont
-20                                                                                     10

 ( )
60

h( )                                                                                   K ( )
-25                                                                                      9

-30                                                                                      8                                                      50

Hydraulic Conductivity
7
-35
40
Pressure

6
-40

Rumda
5
-45                                                                                                                                             30
4
-50
3                                                      20
-55                                                                                      2
-60                                                                                      1                                                      10

-65                                                                                      0
0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26                                           0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26            0
0.10   0.12   0.14   0.16        0.18         0.20   0.22   0.24   0.26
Theta                                                                               Theta
Differential of Pressure to Theta                                                                D~Theta                                                                Theta
Gama ~ WatCont
800                                                             2500
60

700
dh
D ( )
2000                                                                             50
600

d
40
Dh_Dtheta

Diffusivity

1500
500

 ( )
Gama
30
400
1000

300                                                                                                                                              20
500
200                                                                                                                                              10

100                                                                                       0
0
0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26                                            0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26
0.10   0.12   0.14   0.16        0.18         0.20   0.22   0.24   0.26
Theta                                                                                Theta
Theta

Figure 3. Parameters of Philip method
Results and Discussion
            Green-Ampt Method        Green-Ampt Model Wetting Front                                                 Green-Ampt Model Wetting Front

0                                                                              0

0.1hr                                                                          0.1hr

-10                                                                            -10       0.2hr
0.2hr
0.3hr
0.3hr
-20
-20                                                                                      0.4hr
0.4hr
0.5hr
0.5hr                                                                -30
-30
0.6hr
0.6hr
-40       0.7hr
-40       0.7hr

Depth cm
0.8hr
Depth cm

0.8hr
-50
-50

-60
-60

-70
-70

-80
0.287                   PE = 0.20%                                   -80
0.265                     PE = 0.19%
-90
Running Time: < 2 sec.                       -90                                 Running Time: < 2 sec.
-100
-100
0.00           0.05   0.10          0.15          0.20   0.25
0.00           0.05   0.10          0.15          0.20   0.25
Theta
Theta

Figure 4. Wetting front from Green-Ampt model
Results and Discussion
   Comparison
Water Content (v/v)                                                         Water Content (v/v)                                                        Water Content (v/v)
0    0.05     0.1     0.15       0.2       0.25    0.3                      0    0.05      0.1    0.15       0.2      0.25     0.3                      0   0.05      0.1      0.15        0.2   0.25   0.3
0                                                                          0                                                                           0

-10                                                                           -10                                                                         -10

-20                                                                           -20                                                                         -20                Finite Diff_ 0.7 hr
Green-Ampt _ 0.7 hr
Depth (cm)

Depth (cm)

Depth (cm)
-30                                                                           -30                                                                         -30                Philip _ 0.7 hr

-40                                                                           -40                                                                         -40

-50                                                                           -50                                                                         -50

-60                                                                           -60                                                                         -60
Finite Diff_ 0.1 hr
(a)                                                                                                  Finite Diff_ 0.4 hr
Green-Ampt _ 0.1 hr
(c)
-70                                                                           -70       (b)                      Green-Ampt _ 0.4 hr                      -70
Philip _ 0.1 hr
Philip _ 0.4 hr
-80                                                                           -80                                                                         -80
0.1 Hour
-90                                                                           -90                                                                         -90

Figure 6. Wetting fronts from finite difference, Philip methods and Green-Ampt model
Results and Discussion
   Comparison
Water Content (v/v)                                                         Water Content (v/v)                                                        Water Content (v/v)
0    0.05     0.1     0.15       0.2       0.25    0.3                      0    0.05      0.1    0.15       0.2      0.25     0.3                      0   0.05      0.1      0.15        0.2   0.25   0.3
0                                                                          0                                                                           0

-10                                                                           -10                                                                         -10

-20                                                                           -20                                                                         -20                Finite Diff_ 0.7 hr
Green-Ampt _ 0.7 hr
Depth (cm)

Depth (cm)

Depth (cm)
-30                                                                           -30                                                                         -30                Philip _ 0.7 hr

-40                                                                           -40                                                                         -40

-50                                                                           -50                                                                         -50

-60                                                                           -60                                                                         -60
Finite Diff_ 0.1 hr
(a)                                                                                                  Finite Diff_ 0.4 hr
Green-Ampt _ 0.1 hr
(c)
-70                                                                           -70       (b)                      Green-Ampt _ 0.4 hr                      -70
Philip _ 0.1 hr
Philip _ 0.4 hr
-80                                                                           -80                                                                         -80
0.1 Hour                                                                    0.4 Hour
-90                                                                           -90                                                                         -90

Figure 6. Wetting fronts from finite difference, Philip methods and Green-Ampt model
Results and Discussion
   Comparison
Water Content (v/v)                                                         Water Content (v/v)                                                        Water Content (v/v)
0    0.05     0.1     0.15       0.2       0.25    0.3                      0    0.05      0.1    0.15       0.2      0.25     0.3                      0   0.05      0.1      0.15        0.2   0.25   0.3
0                                                                          0                                                                           0

-10                                                                           -10                                                                         -10

-20                                                                           -20                                                                         -20                Finite Diff_ 0.7 hr
Green-Ampt _ 0.7 hr
Depth (cm)

Depth (cm)

Depth (cm)
-30                                                                           -30                                                                         -30                Philip _ 0.7 hr

-40                                                                           -40                                                                         -40

-50                                                                           -50                                                                         -50

-60                                                                           -60                                                                         -60
Finite Diff_ 0.1 hr
(a)                                                                                                  Finite Diff_ 0.4 hr
Green-Ampt _ 0.1 hr
(c)
-70                                                                           -70       (b)                      Green-Ampt _ 0.4 hr                      -70
Philip _ 0.1 hr
Philip _ 0.4 hr
-80                                                                           -80                                                                         -80
0.1 Hour                                                                    0.4 Hour                                                                       0.7 Hour
-90                                                                           -90                                                                         -90

Figure 6. Wetting fronts from finite difference, Philip methods and Green-Ampt model
Results and Discussion
     Comparison

Table 1. Models’ strength and drawback
for infiltration simulation under the 2nd class boundary condition

Methods                           Finite Difference                                 Philip                                      Green-Ampt
PE                                      0.96%                                       -0.57%                                         0.19%
Accuracy                                 High                                        High                                           High
Parameters Acquirement                 Moderate                                  Complicated                                        Easy
Programming                          Complicated                              Very Complicated                                      Easy
Numerical Stability       Not always stable, selecting dt, dz;   Always Stable, but need monitor the iteration                 Always Stable
Runing Time                              8 min                                     < 2 Sec.                                       < 2 Sec.
Boundary Condition (BC)                q = 13.6                                   θ0 = 0.260                                     θ0 = 0.265
Treat with any boudary conditons       Only treat with 1st class boudary condtion      Only treat with 1st class boudary condtion
Still works when changing 2nd to 1st BC        Still works when changing 2nd to 1st BC
But is very sensitive to θ 0 , need adjust                not very sensitive to θ0
Wetting Front                            Good                                        Good                                         Distorted
Conclusion
   All the models could show high accuracies in simulating
cumulative infiltration.

   Finite difference method
best wetting front prediction
treat the second class boundary best among the 3 models
but sometimes numerically unstable;

   Green-Ampt model
required least parameters and the simplest programming
but the wetting front is not precise;

   Philip’s method
solve Richards equation much quicker than finite difference method
high numerical stability
but care should be taken under 2nd boundary condition
programming is much complicated;
Suggestion
   Finite difference method is the best approach for
cumulative infiltration and wetting front prediction ;

   Philip’s method is a good alternative for finite difference
method if there is a problem in numerical stability for
finite difference method or a good computer is not
accessible.

   Green-Ampt model is a simple and good method if only
the cumulative infiltration is concerned.

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