The Characteristic Curves

1
The Characteristic Curves

• Now for the real excitement putting the solid
  and fluid together!

2
A simple thought experiment

• (1) Find a Bukner funnel with pores much smaller
  than the pores in the soil sample.
• (2) Attach a long water-filled tube which
  connects the funnel to a graduated cylinder half
  full of water.
• (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.

3
A word about porous media...

• Must be careful not to exceed the air entry
  pressure of the porous plate

4
Back to our experiment...

• We will continue through three stages
• (1) First (main) wetting,
• (2) First (main) drying, and
• (3) re-wetting (primary wetting).

5
The first step get the soil wet

• (1) MAIN WETTING. Incrementally elevate beaker
  until water level is at soil 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 2.54

6
Now dry it, then re-wet...

• (2) MAIN DRAINING. Lower the end of the tube,
  and apply a suction to the water in the soil
  while measuring outflow.
• (3) PRIMARY WETTING. repeat (1).
• This experiment illustrates most of the physics
  which control the retention and movement of
  fluids through porous media

(2)
(3)
7
Simplified System

• Illustration of Haines jumps
• Filling
• 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
• first the outer pores will drain
• When the head becomes less than -2?/r2 all but
  isolated pores drain as air can finally enter the
  necks

8
So lets go through this step by step

• Main wetting curve labeled (1) (2) (3)
• 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)

Pressure
Water Content
9
Following the draining process

• Main draining curve labeled (3) (4) (5)
• Why doesnt this follow the wetting curve?
• Haines jumps and other sources of hysteresis
• Degree of saturation is a function of pressure
  and the history of wetting of the pore
• For this reason, the wetting and drying curves
  for soil are referred to as hysteretic. More on
  this as we proceed...

Pressure
Water Content
10
Particle size to Characteristic Curves

• (a) Particles distributed between dmin and dmax
• (b) Pore size distribution similar
• The ordinate goes from mass of particles, to
  volume of pores.
• (c) Laplaces eq. relate pore size filling
  pressure of each pore.
• Plot becomes filling pressure vs. volume of
  pores.
• (d) Finally note volume of pores degree of
  saturation.

11
Identifying break points

• hw is as the pressure at which the largest group
  of pore bodies fill.
• rmax 2?/hw
• ha is diameter of the typical pore
  throats rthroat 2?/ha
• ?r Why doesnt 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)

Pressure
Water Content
12
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.

13
A few more scanning curves

• 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)
• These are examples of primary, secondary, and
  tertiary scanning curves

Pressure
Water Content
14
Hysteresis

• Sources
• Haines Jumps
• Contact Angle
• How to deal with it
• Independent Domain Models
• General Model
• Similarity Models

15
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
• Along the horizontal plane (right negative, left
  positive)
• ?F 0 Fsl - Fsg Flg cos? 2.55

16
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 sik, so may put in terms
  of the relative surface tensions
• ?sg ?sl ?lg cos? 2.56

17
Interpretation of Youngs

• Solving 2.56 for the contact angle we find
• Physical limits on possible values of ?
• The contact angle is bounded by 0o lt ? lt 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) gt ?lg for water, the
  contact angle for water going into geologic
  material is often taken to be 0o.

18
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.
• 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.

19
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.