Transcript The Characteristic Curves

The Characteristic Curves
Now for the real excitement: putting
the solid and fluid together!
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A simple thought experiment
(1) Find a Bukner funnel with pores
much smaller than the pores in
the soil sample.
Soil
l
Bukner Funnel
(2) Attach a long water-filled tube
which connects the funnel to a
graduated cylinder half full of
water.
H
(3) Place a thin slice of dry soil on
the top of the porous plate.
(4) Keeps track of the amount of
water which enters and exits the
soil sample as you raise and
lower the tube.
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A word about porous media...
Must be careful not to exceed the air entry pressure
of the porous plate
2
Pentry =
rmax
r
[2.53]
Soil
Water
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Back to our experiment...
We will continue through three stages:
(1) First (“main”) wetting,
(2) First (“main”) drying, and
(3) re-wetting (“primary wetting”).
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The first step: get the soil wet
(1)
MAIN WETTING.
Incrementally elevate beaker
until water level is at soil
Bukner Funnel
height. Measure the water
drawn up by the soil as H goes
from Pentry to 0. Each
measurement is taken allowing
the system to come to a steady
state. Measuring elevation, H,
as positive upward, the
pressure applied to the water
in this soil will be given by:
Psoil = wgH
Soil
l
H
[2.54]
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Now dry it, then re-wet...
(2) MAIN DRAINING.
Lower the end of the tube,
and apply a suction toBukner
theFunnel
water in the soil while
measuring outflow.
(3) PRIMARY WETTING.
repeat (1).
Soil
l
(2) (3)
H
This experiment illustrates
most of the physics which
control the retention and
movement of fluids
through porous media
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Simplified System
2r2
r
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Illustration of “Haines jumps”
Filling
(a)
(b)
(c)
(d)
 no water enters until the head
becomes greater than -2/r1
 When this pressure is exceeded,
the pores will suddenly fill
In the draining process
 When the head becomes less than
-2/r2 all but isolated pores drain as
air can finally enter the necks
-2r 2
(d)
(a)
Pressure
 first the outer pores will drain
(d)
-2r 1
(c)
(b)
(b)
0
0  r Mois ture Cont ent 
sa t
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So lets go through this step by step
Main wetting curve
labeled (1) (2) (3)
(1)
Pressure
(6)
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More water is taken up
by the soil as the beaker
comes closer to the
elevation of the soil (i.e.,
as the negative pressure
of the feed water
decreases)
(5)
(8)
ha
(2)
hw
ma i
n
drai
ning
curv
e
(4)
(7)
ma in wett ing
curve
(3)
0
0
r
Water Content
 su  s
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Following the draining process
Main draining curve
labeled (3) (4) (5)
Why doesn’t this follow the
wetting curve?
(1)
(5)
 Haines jumps and other
sources of hysteresis
For this reason, the wetting and
drying curves for soil are
referred to as hysteretic. More
on this as we proceed...
Pressure
ng curve
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 Degree of saturation is a
function of pressure and the
history of wetting of the pore
(6)
(8)
ha
(2)
hw
ma i
n
drai
ning
curv
e
(4)
(7)
ma in wett ing
curve
(3)
0
0
r
Water Content
 su  s
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Particle size to Characteristic Curves
(d) Finally note volume
of pores = degree of
saturation.
Pore Size
rd
r min
dmin
0%
Percent Mass < d
(a)
100%
2
=
h
hmax
0%
r

Percent Volume < r
100%
(b)
hmax
 = nV
hmin
Pressure
(c) Laplace’s eq. relate
pore size filling
pressure of each pore.
Plot becomes filling
pressure vs. volume of
pores.
Particle Size
(b) Pore size
distribution similar:
The ordinate goes from
mass of particles, to
volume of pores.
r max
dmax
Pressure
(a) Particles distributed
between dmin and dmax
hmin
0%
Percent Volume
(c)
100%
0
Moisture Content
(d)
 sat
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Identifying break points
 hw is as the pressure at which
the largest group of pore
bodies fill.
rmax = 2/hw
(5)
Pressure
(6)
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 ha is diameter of the typical
pore throats
rthroat = 2/ha
 r: Why doesn’t the soil drain
completely?
Chemically bound water
Fluid held in the very small
radius regions at particle
contacts.
 su: Some pores don't fill due to
gas trapping ( 10%)
(1)
(8)
ha
(2)
hw
ma i
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drai
ning
curv
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(4)
(7)
ma in wett ing
curve
(3)
0
0
r
Water Content
 su  s
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A bit of Terminology
Pendular: volumes of liquid which are
hydraulically isolated from nearby fluid
Funicular: liquid which is in hydraulic
connection with the bulk fluid.
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A few more “scanning curves”
(5)
(6)
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These are examples of
primary, secondary, and
tertiary scanning curves
(1)
Pressure
So we have gone to and
from the extremes.
Note that we can also
reverse the process in
the middle as shown at
(6) (7) and (8)
(8)
ha
(2)
hw
ma i
n
drai
ning
curv
e
(4)
(7)
ma in wett ing
curve
(3)
0
0
r
Water Content
 su  s
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Hysteresis
Sources
Haines Jumps
Contact Angle
How to deal with it
Independent Domain Models
General Model
Similarity Models
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Contact Angle: Youngs Equation
What should the angle of contact between the solid
and fluid be, and why?
At equilibrium, forces balance at the point of contact.
Considering horizontal components
Flg
Liquid
 lg

Fsl
sl 
sg
Fsg
Solid
Along the horizontal plane (right negative, left positive)
F = 0 = Fsl - Fsg + Flg cos
[2.55]
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Youngs Eq. continued
F = 0 = Fsl - Fsg + Flg cos
[2.55]
Fsg = solid-gas surface force/length;
Fsl = solid-liquid force /length;
Flg = liquid-gas surface force /length.
Per unit length, Fik = ik, so may put in
terms of the relative surface tensions
sg = sl + lg cos
[2.56]
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Interpretation of Youngs
Solving [2.56] for the contact angle we find
sg - sl
 = cos-1

 lg 
[2.57]
Physical limits on possible values of :
The contact angle is bounded by 0o <  <
180o. So if the operand of cos-1 is greater
than 1, then  will be 0o, while if the value is
less than -1, the value will be 180o.
Often true that (sg - sl) > lg for water, the
contact angle for water going into geologic
material is often taken to be 0o.
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Hysteresis: In the Contact Angle
Contact angle differs for advancing and receding cases.
“Rain-drop effect”: Why a drop of water on a flat plate
will not start to move as soon as you tilt the plate:
more energy is required to remove the water from the
trailing edge of the plate than is given up by the sum
of the gravitational potential plus the energy released
wetting the plate.
Di
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The relevance of the rain-drop
effect to capillary hysteresis
is simply an extension of the
observation regarding the plate
and drop: a media will retain
water more vigorously than it
will absorb water.
ect
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rop
Mo
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r
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a
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Contact Angle Hysterisis:
the Rain Drop Effect
What is the physical basis?
At the microscopic level the
Youngs-Laplace equation is
adhered to,
from a macroscopic point of
view, the drip cannot
advance until the apparent
contact angle is quite large.
Upon retreat the
macroscopic contact angle
will be much smaller than the
true microscopic magnitude
Same result from surface
contamination.
Fluid

(a)

Solid


(b)
Direction of Movement
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