Miller Similarity and Scaling of Capillary Properties

1
Miller Similarity and Scaling of Capillary
Properties

• One of many SCALING tools

Williams, 2002
http//www.its.uidaho.edu/AgE558 Modified after
Selker, 2000
http//bioe.orst.edu/vzp
2
The assumptions...

• How can we take information about the hydraulic
  properties of one media to make quantitative
  predictions of the properties of another media?
• 1956 Miller and Miller presented a comprehensive
  methodology.
• Bounds on our expectations
• cant expect to make measurements in a sand and
  hope to learn about the behavior of clays
• fundamentally differing in their chemical
  properties and pore-scale geometric
  configuration.
• To extrapolate from one media to another, the two
  systems must be similar in the geometric sense
  (akin to similar triangles).

3
Characterizing each medium

• We need characteristic microscopic length
  scales of each media with a consistent
  definition.
• Need ?s in some dimension which can be
  identified readily and reflects the typical
  dimensions at the grain scale.
• In practice people often use the d50 as their ?
  as it is easy to measure and therefore widely
  reported.
• Any other measure would be fine so long as you
  are consistent in using the same measure for each
  of the media.

4
Enough generalities, lets see how this works!

• Assumptions Need For Similarity
• Media Uniform with regard to position,
  orientation and time (homogeneous, isotropic and
  permanent).
• Liquid Uniform, constant surface tension,
  contact angle, viscosity and density. Contact
  angle must be the same in the two systems,
  although surface tension and viscosity may
  differ.
• Gas Move freely in comparison to the liquid
  phase, and is assumed to be at a uniform
  pressure.
• Connectivity Funicular (continuous) states in
  both the air and water no isolated bubbles or
  droplets.
• What about hysteresis? No problem, usual
  bookkeeping.

5
Rigorous definition of geometric similarity

• Necessary and sufficient conditions
• Media 1 and 2 are similar if, and only if, there
  exists a constant ? ?1/?2 such that if all
  length dimensions in medium 1 are multiplied by ?
  , the probability of any given geometric shape to
  be seen in the scaled media1 and medium 2 are
  identical.
• Most convenient to define a scaled medium which
  can then be compared to any other similar medium.
  
• Scaled quantity noted with dot suffix K is
  scaled conductivity.

6
Getting to some math..

• Consider moisture content of two media with
  geometrically similar emplacement of water.
• Volumetric moisture content will be the same in
  the two systems.
• For any particular gas/liquid interface Laplaces
  equation gives the pressure in terms of the
  reduced radius Rwhere ? is the contact
  angle, ? is the surface tension, and p is the
  difference in pressure between the gas and
  liquid.

7

• Multiplying both sides by ???
• Stuff on left side is the same for any similar
  media,
• THUS
• Stuff on the right side must be constant as well.
  
• We have obtained a method for scaling the
  pressure!

8
Example Application

• Lets calculate the pressure in media 1 at some
  moisture content ? given that we know the
  pressure in media 2 at ?. From above, we note
  that the pressures are related simply asso we
  may obtain the pressure of the second media as

9
Some data from Selkers lab

• Scaling of the characteristic curves for four
  similar sands. Sizes indicated by mesh bounds.

10
Scaling hydraulic conductivity

• Need to go back to the underlying physical
  equations to derive the correct expression for
  scaling.
• Identify the terms which make up K in Darcys law
  in the Navier-Stokes equation for creeping flow
• Compared to Darcys law
• which can be written

11

• Solving for K we findNow looking at this for a
  one-dimensional case for the moment we see that
  where l is a microscopic unit of length,
  andso equation 2.131 may be rewritten
  asPutting the unscaled variables on the left
  we see that

12
From last slide

• The right-hand side of 2.135 is only dependent
  on the properties of the scaled media, implying
  that the left-hand side must be as well
  Careful p is not the scaled pressure! Should
  writeThe scaling relationship for
  permeability!

13
Example

• Two similar media at moisture content ? Scaled
  conductivities will be identical or, solving
  for K2 in terms of K1 we find which can also
  be written in terms of pressure

14
Last scaling parameter required time

• By looking to the macroscopic properties of the
  system, we can obtain the scaling relationship
  for time
• Consider Darcys law and the conservation of
  mass.
• In the absence of gravity Darcys law
  states
• Multiplying both sides by ????, we find

15
Scaling time...

• From a macroscopic viewpoint, both v and ? are
  functions of the macroscopic length scale, say L.
  The product L? is the reduced form of the
  gradient operator. So we can multiply both sides
  by L to put the right side of this equation in
  the reduced formSince right side is, then
  left side is in reduced form, thus the reduced
  macroscopic velocity is given by

16
Finishing up t scaling

• Would like to obtain the scaling parameter for
  time, say ?, such that ? t t. Using the
  definition of velocity we can write and using
  the fact that xx/L and t?t, v can be
  rewritten
• Now solving for ? we find
• and solving for t
• JOB DONE!!

17

• Squared
• scaling of K
• w.r.t.
• particle size

18

• Data from Warrick
• et al. demonstrating
• scaling of
• saturation -
• permeability
• relationship

19

• Warrick et al.
• demonstration
• of scaled
• pressure -
• saturation
• relationship