Darcy meets theta

1
Darcy meets theta

• The plummeting fortunes of Permeability and other
  distress caused by inadequate fluids

2
Extending Darcy to Unsaturated Media

• 1907 Buckingham saw Darcy could describe
  unsaturated flow
• q - K(q) ÑH 2.102
• K(q) a function of the moisture content.
• Conductivity is not a function of pressure the
  geometry of the water filled pores is all that
  matters, which is dictated by q alone. To
  express K as a function of pressure you must
  employ the hysteretic functional relationship
  between q and H.

3
How does K vary with q?

• Drops like a rock.
• Three factors are responsible for this behavior
• 1. Large pores empty first. These are the pores
  with least resistance to flow, since they have
  the largest diameters (recall the 1/r4 dependence
  of hydraulic resistance in the in the
  Hagen-Poiseuille equation).
• 2. Flow paths increase in length. Instead of
  proceeding straight through a chunk of media, the
  flow must avoid all the empty pores, making the
  path more tortuous.
• 3. There is less cross-section of flow. In any
  given area normal to flow, all the fluid must
  pass through a smaller portion of this area for
  a given aerial flux the pore velocity must be
  higher.

4
Permeability with theta...

• To get a feel for how fast K drops, lets
  consider the effect of tortuosity alone (item 2
  above).
• Must be distinguished from the Darcian flow
  length and the Darcian velocity.

Difference between the true microscopic fluid
flow path length and flow velocity in comparison
with the Darcian values, which are based on a
macroscopic picture of the system.
5
A few illustrative calculations

• So writing the true pressure gradient that acts
  on the fluid we have
• Le/L ratio of the true path length to the
  Darcian length.
• Next write equation for capillary velocity, vf
  (Hagen-Poiseuille)
• Translating into the Darcian velocity, q, we find

6
Conclusions on tortuosity...

• Solving for q we find
• or, after comparing to Darcys law we see that
• where C is some constant.
• K goes down with (tortuosity)2 and (radius)2
• K also hit by big pores emptying first
• Conductivity will drop precipitously as q
  decreases

Characteristics of K(q) 1 at saturation K
Ks. 2-3 K(q) 0 pendular water
7
Adding Conservation of Mass Richards Equation

• Need to add the constraint imposed by the
  conservation of mass (L.A. Richards 1931).
• There are many ways to obtain this result (well
  do a couple).
• Consider arbitrary volume media. Keep track of
  fluid going into and out of this volume.

8

• Volume is independent of time, bring the time
  derivative inside
• Apply the divergence theorem to the right side
• Combining 2.110 and 2.111

9

• Since 2.112 is true for any volume element that
  the integrand must be zero for all points
• Replace q using Darcys law to obtain Richards
  equation
• where H is the total potential. This is also
  referred to as the Fokker-Plank equation

10
In terms of elevation and pressure

• Total potential is sum of gravity potential and
  pressure, H h z
• but
• So we obtain
• noticing that

11
And finally...

• So Richards Equation may be written (drum roll
  please....)
• Before we can get anywhere we need the
  relationships, K(q) and hq.
• First order in time and second order in space
  require
• 1 initial condition and
• 2 boundary conditions (top and bottom)

12
The diffusion form of Rs eq.

• Can put in more familiar form by introducing
• the soil diffusivity. Note that

13
The diffusion form of Rs eq.

• D(q) gives us a diffusion equation in q
• Favorite trick in solving diffusion problems is
  to assume that D is constant over space and pull
  it outside the derivative it will find no place
  here!
• D(q) strongly non-linear function of q and varies
  drastically as a function of space.
• This makes Richards equation quite an interesting
  challenge in terms of finding tidy analytical
  solutions, and even makes numerical modelers
  wince a bit due to the very rapid changes in both
  D and q.

14
A brief aside on how this all might apply to the
movement of gas

• What about that no-slip boundary?

15
Gas Flow in Porous Media

• Might expect movement of gases is a simple
  extension of liquids
• use gas density viscosity with intrinsic
  permeability to get K
• use Darcy's law.
• Well, it ain't quite that simple.
• Recall the "no-slip" boundary condition
• Idea collisions between molecules in liquid so
  frequent that near a fixed surface the closest
  molecules will be constantly loosing all of their
  wall-parallel energy, rendering them motionless
  from the macroscopic perspective.
• Requisite mean free path of travel short
  compared to the aperture through which the liquid
  is moving.

16
about Gases

• Gases
• mean free length of travel on the order of media
  pore size.
• No-slip condition for liquids does not apply
• Taken together, results in "Klinkenberg effect"
  (experimental/theoretical 1941 paper by L.J.
  Klinkenberg).
• Gas permeability, Kg, a function of gas pressure
  (dictates mean free path length)

17

• k intrinsic permeability to liquid flow
• r characteristic radius apertures of the media
• l mean free path length of the gas molecules
• c proportionality factor between the mean free
  path for the free gas compared to the mean free
  path for gas which just collided with the
  capillary wall (c is just slightly less than 1,
  Klinkenberg, 1941).
• Coarse media capillaries larger than the mean
  free path length, the intrinsic permeability for
  liquids is recovered
• Fine media gas permeability exceeds liquid
  permeability.

18
Final notes on gas permeability

• kg/k measure of size of conducting pathways,
• can be used as a diagnostic parameter (Reeve,
  1953).
• As PÞ0 the mean free path length lÞr
• limit for kg 10 k
• Klinkenberg's found max kg 5 - 20 times k
• Practical purposes
• correction for vicinity of 1 bar 20 to 80 over
  the liquid permeability, depending upon the media
  (Klinkenberg, 1941).
• Methods of measurement of gas permeability
• Corey (1986),
• New methods Moore and Attenborough, 1992.
• kg depends on the liquid content, analogous to
  hydraulic conductivity with moisture content (see
  Corey, 1986).

19
Expressions for Conductivity Retention

• To solve Richards equation we need the
  mathematical relationships between pressure,
  moisture content and conductivity.
• Conductivity is a non-hysteretic function of
  moisture content
• Moisture content and pore pressure are related
  through a hysteretic functional (often hysteretic
  ignored gives expressions of conductivity in
  terms of matric potential).
• For analytical solution, must use analytical
  expressions.
• For numerical solutions may use constructed as
  interpolations between successive laboratory data.

20
(No Transcript)
21
(No Transcript)
22
(No Transcript)