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Infiltration.ppt

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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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