FLOW THROUGH POROUS MEDIA

1
FLOW THROUGH POROUS MEDIA
2
DERIVATION OF RICHARDS EQUATION IN RECTANGULAR
COORDINATES
The general continuity equation is q a
v 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)
3
Flow in the x-direction
4
Flow in the y-direction
5
Flow in the z-direction
6
From Continuity of mass
Where Q is the volumetric water content and t is
time.
7
From Continuity of mass
8
By canceling out terms
9
Applying the Darcy Law to each velocity term
10
FLOW THROUGH POROUS MEDIA
11
FLOW THROUGH POROUS MEDIA
In unsaturated soil the total potential can be
estimated as the sum of the matric potential and
the gravity potential
Since the gravity potential only acts in the
vertical, or z-direction, the total potential,
h, can be replaced by the matric potential, Y,
in all terms except the one involving z
12
FLOW THROUGH POROUS MEDIA
13
FLOW THROUGH POROUS MEDIA
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.
14
FLOW THROUGH POROUS MEDIA
For saturated flow h will be the total head, F
and there can be no change in moisture content
with time.
15
FLOW THROUGH POROUS MEDIA
For a homogeneous, isotropic soil Kx Ky Kz K
16
FLOW THROUGH POROUS MEDIA
Since K is not 0, the term inside the bracket
must be 0
17
FLOW THROUGH POROUS MEDIA
This is called the LaPlace Equation for Saturated
Flow in a homogeneous soil.
18
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.

19
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

20
HYDRODYNAMIC DISPERSION
DUE TO UNEVEN FLOW VELOCITY WITHIN A PORE
21
HYDRODYNAMIC DISPERSION
DUE TO UNEVEN VELOCITIES BETWEEN PORES
22
HYDRODYNAMIC DISPERSION
DUE TO VARYING FLOW PATHS
23
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

24
FICKS LAW OF DIFFUSIONCOMBINED WITH THE
CONTINUITY
Dx
A
Volume ADx
25
FICKS LAW OF DIFFUSION

• Considering one-dimensional flow in the
  x-direction, for continuity

26
FICKS LAW OF DIFFUSIONThe Combined Diffusion
Equation
27
ADVECTION

• 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.

28
The Combined Advection/ Dispersion Equation
where s represents all the source and sink terms
that occur in the real environment.
29
The Combined Advection/ Dispersion
EquationAssumptions

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

30
Chemical Breakthrough
Co
A
Qs
L
C/Co relative concentration
C
31
Plug Flow
Q
32
(No Transcript)
33
Flow with Advection
Q
34
(No Transcript)
35
(No Transcript)
36
Sorption Retardation Coefficient
R number of pore volumes _at_ C/C00.5
37
The Combined Advection/ Dispersion Equation
considering adsorption
38
(No Transcript)
39
(No Transcript)
40
(No Transcript)
41
Combining Adsorption Advection-Dispersion
Degradation and Source-Sinks
42
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.
• Q is the water fraction by volume in the media.

43
Travel Time of a Chemical Through Porous Media
QAV VQ/A
V
A
44
Travel Time of a Chemical Through Porous Media
QAaVa VaQ/Aa AaAQ Va V/Q
Va
Aa
45
Velocity Through Porous Media
Flow Rate 1 CMS
Total Area 2 M2
VQ/A1/2 0.5MPS
Q 1/2 0.5
Open Area 1 M2
VaV/Q 0.5/0.5 1 MPS
46
Travel Time of a Chemical Through Porous Media

• Va is the actual flow velocity in the pores of
  the media.
• Va V / Q
• Mw CwQ Mass of chemical in the flowing water
• MT Cw(QKdBd) The total mass of chemical in
  the media.

47
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 ZQ / V

48
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 ( Q Kd Bd ) / V
• V may be the infiltration rate

49
Factors Influencing Chemical Leaching (Rate of
Water Movement)

• Soil Properties
• Infiltration Rate
• Porosity
• Soil Moisture
• Physical Properties
• Surface Roughness
• Slope

50
Factors Influencing Chemical Leaching (Rate of
Water Movement)

• Physical Properties
• Rainfall amount and intensity
• crop species and stage of growth
• weather
• temperature
• solar radiation
• wind velocity

51
Factors Influencing Chemical Leaching
(Soil-Chemical Interactions)

• Solute Properties
• adsorption coefficient
• solubility
• vapor pressure
• disassociation and ionization properties
• chemical reactions

52
Factors Influencing Chemical Leaching
(Soil-Chemical Interactions)

• Soil Properties
• organic matter content
• soil texture
• pH
• degradation rate

53
Factors Influencing Chemical Leaching (Management
Factors)

• Application Rate
• Chemical placement
• incorporation
• banding
• formulation
• Timing of application
• split application
• time of application

54
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

55
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(Q KdBd ) / V
• 1 ( 0.4 2 1.3 ) / 0.0027
• 3 / 0.0027 1095 days
• 1095 / 60 18.25 half-lives

56
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.