FLOW THROUGH POROUS MEDIA

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```					FLOW THROUGH POROUS
MEDIA
z

Dz
x
Dy
Dx

y
Porous Media Flow       1
DERIVATION OF RICHARD’S
EQUATION IN RECTANGULAR
COORDINATES
The general continuity equation is:
q=av
where
q is the flow rate, volume/time (L3/T)
a is the cross-section area perpendicular
to the flow, (L2)
v is the flow velocity, length/time (L/T)

Porous Media Flow       2
Flow in the x-direction

qxin  v x DyDz
 vx
qxout    (v x       Dx )DyDz
x

Porous Media Flow   3
Flow in the y-direction

qyin  v y DxDz
 vy
qyout  (v y                   Dy)DxDz
y

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Flow in the z-direction

qin  v z DxDy
 vz
qzout    (v z       D z ) Dx D y
z

Porous Media Flow      5
From Continuity of mass


q   in     qout       
t
DxDyDz

Where  is the volumetric water content
and t is time.

Porous Media Flow   6
From Continuity of mass
v x DyDz  v y DxDz  v z DxDy
 vx
 (v x       Dx )DyDz
 x
 vy
 (v y       Dy )DxDz
 y
 vz
 (v z       Dz )DxDy
 z
 
        DxDyDz
t
Porous Media Flow    7
By canceling out terms
 v x  v y  v z           
 x   y   z DxDyDz   t DxDyDz 
                  
                  

  v x  vy  vz                    

 x                              
 t
         y    z                  
Porous Media Flow          8
Applying the Darcy Law to each
velocity term:
 h
v x  K x
 x
 h
v y  K y
y
h
v z  K z
z
Porous Media Flow     9
FLOW THROUGH POROUS
MEDIA

   h    h    h  
  x    y  Ky  y    z  Kz  z    t
Kx                                 
 x                                   

Porous Media Flow        10
FLOW THROUGH POROUS
MEDIA
In unsaturated soil the total potential can
be estimated as the sum of the
matric potential and the gravity potential:

h z
Since the gravity potential only acts in the
vertical, or z-direction, the total potential, h,
can be replaced by the matric potential, ,
in all terms except the one involving z:
Porous Media Flow          11
FLOW THROUGH POROUS
MEDIA
  h                    
Kz     K z z   z 
 z  z  z                
    z  
    K z  z  z 
z              
    K z
    Kz       
z       z  z
Porous Media Flow    12
FLOW THROUGH POROUS
MEDIA

           
Kx     Ky        Kz   
 x   x y
      y   z
       z

 Kz  
     
z   t
This equation is known as the Richard's Equation.
When only the terms involving z on the left are used,
this equation can be used to simulate the vertical
infiltration of water into the soil profile.
Porous Media Flow         13
FLOW THROUGH POROUS
MEDIA

   h    h    h  
Kx
  x    y  Ky  y    z  Kz  z    t
 x                                   
                              
For saturated flow h will be the total head,
F and there can be no change in moisture
content with time.

Porous Media Flow        14
FLOW THROUGH POROUS
MEDIA
For a homogeneous, isotropic soil
K x = Ky = K z = K
   F    F    F
K       K       K    
 x  x  y   y   z  z 
                       
 F  F  F
2           2           2
     0    K 2 
 x        2
     y 2
z 

Porous Media Flow       15
FLOW THROUGH POROUS
MEDIA
 F  F  F
2              2         2
0  K 2 
 x           2
        y 2
z 
Since K is not 0, the term inside
the bracket must be 0
 F  F  F
2        2                   2
 2 
 x        2 0

     y 2
z 
Porous Media Flow           16
FLOW THROUGH POROUS
MEDIA

 F  F  F
2         2                   2
 2 
 x        2 0

     y 2
z 
This is called the LaPlace Equation
for Saturated Flow in a homogeneous
soil.
Porous Media Flow       17
Porous Media Transport of
Chemicals
• DIFFUSION: transport from points or
higher concentration to points of
lower concentration.
• Molecular Diffusion is due to the
random movement of molecules
• Turbulent Diffusion is due to the
random movement of the fluid
carrier. Also called dispersion.
Porous Media Flow    18
Molecular Diffusion

High Concentration

Low Concentration
• Many molecules move from High to Low
• Few molecules move from Low to High
• Result is decrease in high concentration
and increase in low concentration
Porous Media Flow      19
HYDRODYNAMIC DISPERSION

DUE TO UNEVEN
FLOW VELOCITY
WITHIN A PORE

Porous Media Flow   20
HYDRODYNAMIC DISPERSION

DUE TO UNEVEN
VELOCITIES
BETWEEN PORES

Porous Media Flow   21
HYDRODYNAMIC DISPERSION

DUE TO VARYING
FLOW PATHS

Porous Media Flow   22
FICKS LAW OF DIFFUSION
• D = Diffusion Coefficient, L2 / T
• C = Chemical Concentration , M / L3
• qx = Rate of mass transport in the x-
direction. M/ L2T

C
qx   D
x
Porous Media Flow         23
FICKS LAW OF DIFFUSION
COMBINED WITH THE CONTINUITY

Aq x              Dx

A

       q x    
A q x       Dx 
       x      
Porous Media Flow                      24
FICKS LAW OF DIFFUSION
• Considering one-dimensional flow in
the x-direction, for continuity:
q x        C
x          t
q x C
or         
x     t
Porous Media Flow    25
FICKS LAW OF DIFFUSION
The Combined Diffusion Equation

C    qx       C 
       D    
t     x    x  x 
C  2
D 2
x

Porous Media Flow   26
• Chemical Transport due to bulk
movement of the fluid.
• The fastest form of chemical
transport in porous media.
• Concentration decreases in the
direction of fluid movement.
C      C
 V
t      x
Porous Media Flow   27
Dispersion Equation

C    C   c 2
D 2 V    s
t   x    x
where s represents all the source and
sink terms that occur in the real
environment.
Porous Media Flow     28
Dispersion Equation
Assumptions

•   one dimensional flow
•   uniform flow velocity in the column
•   constant moisture content
•   linear, instantaneous, reversible

Porous Media Flow        29
Chemical Breakthrough
Co
Vt  AL
A
Vv   s AL     s                L
C/Co
= relative
concentration
C
Porous Media Flow                  30
Plug Flow
Q

Porous Media Flow   31
Breakthrough Curve for Plug Flow

1

0.9
RELATIVE CONCENTRATION

0.8

0.7

0.6                        Plug Flow
0.5

0.4

0.3

0.2

0.1

0
0    0.5       1              1.5   2    2.5
RELATIVE TIME FOR ONE PORE VOLUME
Porous Media Flow               32
Q

Porous Media Flow   33

1

0.9
RELATIVE CONCENTRATION

0.8

0.7

0.6

0.4

0.3

0.2

0.1

0
0    0.5       1               1.5     2   2.5
RELATIVE TIME FOR ONE PORE VOLUME
Porous Media Flow             34

1

0.9
RELATIVE CONCENTRATION

0.8

0.7

0.6

0.5

0.3

0.2

0.1

0
0    0.5       1               1.5            2     2.5
RELATIVE TIME FOR ONE PORE VOLUME
Porous Media Flow                      35
Sorption Retardation
Coefficient

BdKd
R  1

R = number of pore volumes
@ C/C0=0.5

Porous Media Flow   36
Dispersion Equation

C   C    C   2

R    D 2 V
t   x    x

Porous Media Flow   37
Breakthrough Curve for Preferential Flow

1

0.9
RELATIVE CONCENTRATION

0.8

0.7

0.6
Preferential
flow
0.5

0.4

0.3

0.2

0.1

0
0     0.5        1               1.5     2     2.5
RELATIVE TIME FOR ONE PORE VOLUME
Porous Media Flow               38
TYPICAL BREAKTHROUGH CURVE FOR CHEMICAL TRANSPORT THROUGH POROUS

1

0.9
Plug Flow
RELATIVE CONCENTRATION

0.8

0.7
0.6        Preferential
0.5            flow
0.4

0.3

0.2

0.1

0
0           0.5            1                 1.5            2     2.5
RELATIVE TIME
Porous Media Flow                      39
Typical Chemical Breakthrough Curves
1
0.9
0.8
0.7
0.6
C/Co

0.5
0.4
Preferential
0.3                                              Plug
0.1
0
0   0.5    1     1.5          2      2.5       3      3.5
Relative Pore Volume of water passed
Porous Media Flow                          40
Sinks

dC   dC2
dC
R    D 2 V     KC  s
dt   dx    dx

Porous Media Flow   41
Travel Time of a Chemical
Through Porous Media
• Continuity Equation: Q = AV
• Q is the flow rate (Volume / Time)
• A is the total media flow cross-
section area.
• V is the average flow velocity
through the media.
•  is the water fraction by volume in
the media.
Porous Media Flow        42
Travel Time of a Chemical
Through Porous Media
Q=AV
V                           V=Q/A
A

Porous Media Flow           43
Travel Time of a Chemical
Through Porous Media
Q=AaVa
Va                          Va=Q/Aa
Aa=A
Aa               Va = V/

Porous Media Flow              44
Velocity Through Porous
Media
Flow Rate
= 1 CMS                          Total
Area
V=Q/A=1/2= 0.5MPS                   = 2 M2
 = 1/2 = 0.5
Open
Va=V/ = 0.5/0.5 = 1 MPS          Area
= 1 M2
Porous Media Flow       45
Travel Time of a Chemical
Through Porous Media
• Va is the actual flow velocity in the
pores of the media.
• Va = V / 
• Mw = Cw = Mass of chemical in the
flowing water
• MT = Cw(+KdBd) = The total mass of
chemical in the media.

Porous Media Flow         46
Travel Time of a Chemical
Through Porous Media
• Z = the distance over which travel
occurs
• T = Travel time of the chemical
over distance Z.
• V=Z/T
• Therefore T = Z / V
• Actual Ta = Z / Va = Z / V
Porous Media Flow       47
Travel Time of a Chemical
Through Porous Media
• When adsorption of the chemical
occurs within the media, the travel
time must account for this by:

• Tr = Z (  + Kd Bd ) / V

• V may be the infiltration rate
Porous Media Flow      48
Factors Influencing Chemical
Leaching (Rate of Water
Movement)
• Soil Properties
– Infiltration Rate
– Porosity
– Soil Moisture
• Physical Properties
– Surface Roughness
– Slope

Porous Media Flow   49
Factors Influencing Chemical
Leaching (Rate of Water
Movement)
• Physical Properties
– Rainfall amount and intensity
– crop species and stage of growth
– weather
• temperature
• wind velocity

Porous Media Flow   50
Factors Influencing Chemical
Leaching (Soil-Chemical Interactions)
• Solute Properties
– solubility
– vapor pressure
– disassociation and ionization properties
– chemical reactions

Porous Media Flow         51
Factors Influencing Chemical
Leaching (Soil-Chemical Interactions)
• Soil Properties
– organic matter content
– soil texture
– pH

Porous Media Flow   52
Factors Influencing Chemical
Leaching (Management Factors)
• Application Rate
• Chemical placement
– incorporation
– banding
– formulation
• Timing of application
– split application
– time of application

Porous Media Flow   53
Chemical Leaching
Screening Model Example
•   Silty Clay Soil
•   Topsoil 1 meter thick
•   Moisture Content = 0.4 L/L
•   Topsoil Bulk Density = 1.3 kg/L
•   Atrazine Applied at a rate of 1.1 kg/ha
•   Kd = 2 L/kg
•   Half-Life = 60 days
Porous Media Flow       54
Chemical Leaching
Screening Model Example
• Assume average Iowa conditions
with annual infiltration = 1 meter
• 1 m/yr = 1/365 = 0.0027 m/day = V
• T = Z( + KdBd ) / V
• = 1 ( 0.4 + 2 * 1.3 ) / 0.0027
• = 3 / 0.0027 = 1095 days
• = 1095 / 60 = 18.25 half-lives
Porous Media Flow      55
Chemical Leaching
Screening Model Example
• 1.1 kg/ha-yr atrazine / 1m/yr water =
0.11 mg/L initial chemical
concentration.
• 0.11 mg / L divided by 218
• = 0.00000041962 mg/L at the bottom
of the root zone.

Porous Media Flow         56

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