richards equation,green ampt model

1. A presentation on
Subsurface water
K.Bhargav
Kumar
154104063

2. Subsurface Water
 unit volume of subsurface consists of soil/rock,
and pores which may be filled with water and/or
air
 total porosity= volume voids/total volume
 water content=volume water/total volume
 saturation=volume water/volume voids
 degree of saturation delineates various zones of
subsurface water

3. Definitions
 soil water - Ground surface to bottom of root zone
depth depends on soil type and vegetation. May
become saturated during periods of rainfall otherwise
unsaturated (soil pores partially filled with air). Plants
extract water from this zone. Evaporation occurs from
this zone.
 intermediate vadose zone - Between soil water zone
and capillary fringe. Unsaturated except during
extreme precipitation events. Depth of zone may
range from centimeters to 100s of meters.

4. Definitions Continued
 capillary zone - Above saturated zone. Water rises
into this zone as a result of capillary force. Depth of
this zone is a function of the soil type. Fractions of a
meter for sands (mm) to meters for fine clays. All
pores filled with H2O, p < 0. Effect seen if place
bottom of dry porous media (soil or sponge) into
water. Water will be drawn up into media to a height
above water where soil suction and gravity forces are
equal.
 saturated zone - All pores filled with water, p > 0.
Formations in this zone with ability to transmit water
are called aquifers.

5. Unsaturated Zone
 Water can exist in all its phases in the
unsaturated zone.
 Liquid water occurs as:
 hygroscopic water - adsorbed from air by molecular
interaction (H-bonds)
 capillary water - held by surface tension due to
viscosity of liquid
 gravitational water-water in unsaturated zone in
excess of field capacity which percolates downward
due to gravity ultimately reaching saturated zone as
recharge.

6. Unsaturated Zone
 Hygroscopic and capillary waters are held by
molecular electrostatic forces (between polar bonds
and particles -- surface tension) in thin films around
soil particles  drier soil, smaller pores  hygroscopic
and capillary forces
 Hygroscopic water - held at -31 to -10,000 bars.
Water is unavailable to plants or for recharge to
groundwater.
 Capillary water - Held at -0.33 to -31 bars. More
water filling pores but discontinuous except in
capillary fringe. This water can be used by plants.

7. Definitions
 Permanent wilting point: tension (suction,
negative pressure) below which plant root
system cannot extract water. Depends on soil
and type of vegetation. Typically -15 bars (-
15x105 Pa, -15000cm
 Field capacity: tension (suction, negative
pressure) below which water cannot be drained
by gravity (due to capillary and hygroscopic
forces) Depends on soil type. Typically about -
0.33 bars

8. Typical Moisture Profiles
 rain after a long dry period
direction of
moisture
movement
moisture content
depth
root zone
hygroscopic
wilting
point field
capacity
saturation

9. Typical Moisture Profiles
 Drying process
moisture
depth
field
capacity
saturation
1 - Drying in upper layers by ET.
2 - Bottom part of wetting front continues
Upper part continues to dry.
3 - At some point  and  movement resu
in no moisture gradient
4 - Dry front established. Lower zones ar
being depleted to satisfy PET at surfa
Drying continues until capillary forces
are unable to move water to surface.

10. Dacry-Buckingham law
 Flow in unsaturated porous media governed by a modified
Darcy’s law called Darcy-Buckingham law :
  - suction head (capillary head) or negative pressure
head. Energy possessed by the fluid due to soil suction
forces. Suction head varies with moisture content,   n,
 0,  < n ,  is negative.
 K() - hydraulic conductivity is a function of water content 
, K() because more continuously connected pores,
more space available for water to travel through, until at
 = n, K(n) = Ksat
 
zh
z
h
Kqz






11. Measuring Soil Suction
 Soil Suction () head measured with
tensiometers, an airtight ceramic cup and tube
containing water.
 Soil tension measured as vacuum in tubes
created when water drawn out of tube into
soil. Comes to equilibrium at soil tension
value.
 Tensiometers often used to schedule
irrigation.

12. Tensiometer

13. Why different flow equations?
Steady-state Transient
Saturated
Unsaturated
Darcy’s law
Darcy’s law
(with K(q))
N/A
Richards’
equation
Darcy’s law:
L
AKq




q
changes
with time
No K(q)
No Dq
No q(y)

14. Equation of Continuity
(Conservation of Mass)
Steady-state Transient
Saturated
Unsaturated
Darcy’s law
Darcy’s law
(with K(q))
Richards’
equation
Input – Output = Change in Storage
x
q


 =
t

tx
q








15. Richards’ equation
L
Kq




Given Darcy’s law:











x
K
xx
q Let things change
from place to place
(say, in the x-direction)
tx
q






We also want
conservation of
mass











x
K
xt
So we substitute it in
to the left-hand side

16. Richards’ equation
  










x
K
xt



Remember that the
potential gradient,
,
combines elevation, osmotic, pressure, and
matric components (among others).
x

Sometimes it’s
convenient to
separate out
the elevation
part:
  



















1
x
K
xt



Vertical
  



















0
x
K
xt



Horizontal
Just remember that this y doesn’t include
elevation!

17. 
depth
Wetting Zone
Transmission
Zone
Transition Zone
Saturation Zone
Wetting Front

Infiltration
 General
 Process of water
penetrating from ground
into soil
 Factors affecting
 Condition of soil surface,
vegetative cover, soil
properties, hydraulic
conductivity, antecedent
soil moisture
 Four zones
 Saturated, transmission,
wetting, and wetting front

18. Infiltration
 Infiltration rate, f(t)
 Rate at which water enters the soil at the surface
(in/hr or cm/hr)
 Cumulative infiltration, F(t)
 Accumulated depth of water infiltrating during given
time period

t
dftF
0
)()( 
dt
tdF
tf
)(
)( 
t
f, F F
f

19. Infiltrometers
Single Ring Double Ring
http://en.wikipedia.org/wiki/Infiltrometer

20. Infiltration Methods
 Horton and Phillips
 Infiltration models developed as approximate
solutions of an exact theory (Richard’s Equation)
 Green – Ampt
 Infiltration model developed from an approximate
theory to an exact solution

21. Horton Infiltration Model
• one of earliest infiltration equations developed (1933)
and the most common empirical equation used to
predict infiltration if ponding occurs from above:
• Instantaneous infiltration
• Cumulative infiltration
• fc, minimum infiltration capacity (approximately
saturated hydraulic conductivity)
• fo, maximum infiltration capacity (function of
saturated conductivity and soil tension)
• k constant representing exponential rate of decrease
of infiltration
kt
cc ffftf  exp)()( 0
 

 
t
Ktco
c
K
ff
tfdftF
0
)exp1()()( 

22. Horton’s Infiltration Model
• All are empirical parameters which must be fit to each
soil type using data from a ring infiltrometer
experiment
• Horton’s equations are only valid after ponding.
Therefore all water the soil has potential to infiltrate is
available at soil surface. Ponding will only occur if i >
f(t). Should only be used during very high intensity
precipitation events over small areas
fc
fo rate of decay
governed by k,
increase k,
increase rate of
decay
(analogous to Ksat)
t
F(t)
f(t)
(time after ponding)

23. Green-Ampt Assumptions
Wetted Zone
Wetting Front
Ground Surface
Dry Soil
L

n
i

z
 = increase in
moisture content as
wetting front passes

 = Suction head at
“sharp” wetting front Conductivity, K
L = Wetted depth
K = Conductivity in
wetted zone
Ponded Water
0h
0h
= Depth of water
ponding on
surface (small)

24. Green-Ampt soil water variables
Wetted Zone
Wetting Front
Ground Surface
Dry Soil
L

n
i

z
r e
i = initial moisture content of
dry soil before infiltration
happens
 = increase in moisture
content as wetting front
passes
 = moisture content (volume
of water/total volume of soil)
r = residual water content of
very dry soil
e = effective porosity
n = porosity

25. Green ampt equation:
Infiltration rate:
The cool thing is, though, that what we want (F or f) is a
function of only things we can figure out (porosity, initial
moisture content, soil conductivity, and soil capillary
pressure). The problem is that you can’t easily put F on
one side, and all the other stuff on the other. This
inability to separate the equation means that the
equation is nonlinear.

26. Ponding time
 Elapsed time between the time rainfall begins and
the time water begins to pond on the soil surface
(tp)

27. Ponding Time
 Up to the time of ponding,
all rainfall has infiltrated (i
= rainfall rate)
if 








 1
F
Kf

ptiF *










 1
* pti
Ki

Potential
Infiltration
Actual Infiltration
Rainfall
Accumulated
Rainfall
Infiltration
Time
Time
Infiltrationrate,f
Cumulative
Infiltration,F
i
pt
pp tiF *
)( Kii
K
tp





28. References
 enchartedlearning.com
 tutor.com
 Huggett, J. (2005) Fundamentals of
Geomorphology, Routeledge,
 Horton, Robert E (1933) "The role of infiltration in
the hydrologic cycle" Transactions of the
American Geophysics Union, 14th Annual
Meeting, pp. 446–460.
 Horton, Robert E (1945) "Erosional development
of streams and their drainage basins;
Hydrophysical approach to quantitative
morphology" Geological Society of America
Bulletin, 56 (3): 275–370. doi:10.1130/0016-7606