Hydrodynamics in Porous Media

1
Hydrodynamics in Porous Media

• We will cover
• How fluids respond to local potential gradients
  (Darcys Law)
• Add the conservation of mass to obtain Richards
  equation

2
Darcys Law for saturated media

• In 1856 Darcy hired to figure out the water
  supply to the towns central fountain.
• Experimentally found that flux of water porous
  media could be expressed as the product of the
  resistance to flow which characterized the media,
  and forces acting to push the fluid through the
  media.
• Q - The rate of flow (L3/T) as the volume of
  water passed through a column per unit time.
• hi - The fluid potential in the media at
  position i, measured in standing head equivalent.
  Under saturated conditions this is composed of
  gravitational potential (elevation), and static
  pressure potential (L force per unit area
  divided by ?g).
• K - The hydraulic conductivity of the media.
  The proportionality between specific flux and
  imposed gradient for a given medium (L/T).
• L - The length of media through which flow
  passes (L).
• A - The cross-sectional area of the column (L2).

3
Darcys Law

• Darcy then observed that the flow of water in a
  vertical column was well described by the
  equation
• Darcys expression is written in a general form
  for isotropic media as
• q is the specific flux vector (L/T volume of
  water per unit area per unit time),
• K is the saturated hydraulic conductivity tensor
  (second rank) of the media (L/T), and
• ?H is the gradient in hydraulic head
  (dimensionless)

4
Aside on calculus ...What is this up-side-down
triangle all about?

• The dell operator short hand for 3-d
  derivative
• The result of operating on a scalar function
  (like potential) with ? is the slope of the
  function
• ?F points directly towards the steepest direction
  of up hill with a length proportional to the
  slope of the hill.
• Later well use ?F. The dot just tells us to
  take the dell and calculate the dot product of
  that and the function F (which needs to be a
  vector for this to make sense).
• dell-dot-F is the divergence of F.
• If F were local flux (with magnitude and
  direction), ?F would be the amount of water
  leaving the point x,y,z. This is a scalar result!
  
• ?F takes a scalar function F and gives a vector
  slope
• ?F uses a vector function F and gives a scalar
  result.

5
Now, about those parameters...

• Gradient in head is dimensionless, being length
  per length
• Q Aq Q has units volume per unit time
• Specific flux, q, has units of length per time,
  or velocity.
• For vertical flow speed at which the height of
  a pond of fluid would drop
• CAREFUL q is not the velocity of particles of
  water
• The specific flux is a vector (magnitude and
  direction).
• Potential expressed as the height of a column of
  water, has units of length.

6
About those vectors...

• Is the right side of Darcys law indeed a vector?
  
• h is a scalar, but ?H is a vector
• Since K is a tensor (yikes), K?H is a vector
• So all is well on the right hand side
• Notes on K
• we could also obtain a vector on the right hand
  side by selecting K to be a scalar, which is
  often done (i.e., assuming that conductivity is
  independent of direction).

7
A few words about the K tensor
flux in x-direction
flux in y-direction
flux in z-direction

• Kab relates gradients in potential in the
  b-direction to flux that results in the
  a-direction.
• In anisotropic media, gradients not aligned with
  bedding give flux not parallel with potential
  gradients. If the coordinate system is aligned
  with directions of anisotropy the "off diagonal
  terms will be zero (i.e., Kab0 where a?b). If,
  in addition, these are all equal, then the tensor
  collapses to a scalar.
• The reason to use the tensor form is to capture
  the effects of anisotropy.

8
Looking holistically

• Check out the intuitively aspects of Darcys
  result. The rate of flow is
• Directly related to the area of flow (e.g., put
  two columns in parallel and you get twice the
  flow)
• Inversely related to the length of flow (e.g.,
  flow through twice the length with the same
  potential drop gives half the flux)
• Directly related to the potential energy drop
  across the system (e.g., double the energy
  expended to obtain twice the flow).
• The expression is patently linear all
  properties scale linearly with changes in system
  forces and dimensions.

9
Why is Darcy Linear?

• Because non-turbulent?
• No.
• Far before turbulence, there will be large local
  accelerations it is the lack of local
  acceleration which makes the relationship linear.
  
• Consider the Navier Stokes Equation for fluid
  flow. The x-component of flow in a velocity
  field with velocities u, v, and w in the x, y,
  and z (vertical) directions, may be written

10
Creeping flow

• Now impose the conditions needed for which
  Darcys Law
• Creeping flow acceleration (du/dx) terms small
  compared to the viscous and gravitational terms
  Similarly changes in velocity with time are
  small so N-S is
• Linear in gradient of hydraulic potential on
  left, proportional to velocity and viscosity on
  right (same as Darcy).
• Proof of Darcys Law? No! Shows that the
  creeping flow assumption is sufficient to obtain
  correct form.

11
Capillary tube model for flow

• Widely used model for flow through porous media
  is a group of cylindrical capillary tubes (e.g.,.
  Green and Ampt, 1911 and many more).
• Lets derive the equation for steady flow through
  a capillary of radius ro
• Consider forces on cylindrical control volume
  shown
• ? F 0 2.75

12
Force Balance on Control Volume

• end pressures
• at S 0 F1 P?r2
• at S ?S F2 (P ?S dP/dS) ?r2
• shear force Fs 2?r?S?
• where ??is the local shear stress
• Putting these in the force balance gives
• P?r2 - (P ?S dP/dS) ?r2 - 2?r?S? 0 2.76
• where we remember that dP/dS is negative in sign
  (pressure drops along the direction of flow)

13
continuing the force balance
P?r2 - (P ?S dP/dS) ?r2 - 2?r?S? 0 2.76

• With some algebra, this simplifies to
• dP/dS is constant shear stress varies
  linearly with radius
• From the definition of viscosity
• Using this 2.77 says
• Multiply both sides
• by dr, and integrate

14
Computing the flux through the pipe...

• Carrying out the integration we find which
  gives the velocity profile in a cylindrical pipe
• To calculate the flux integrate over the area
  in cylindrical coordinates, dA r d? dr, thus

15
Rearranging terms...

• The integral is easy to compute, giving
• (fourth power!!)
• which is the well known Hagen-Poiseuille
  Equation.
• We are interested in the flow per unit area
  (flux), for which we use the symbol q Q/?r2
• (second power)
• We commonly measure pressure in terms of
  hydraulic head, so we may substitute rgh P, to
  obtain

16

• r02/8 is a geometric term function of the
  media.
• referred to as the intrinsic permeability,
  denoted by ?.
• ??? is a function of the fluid alone
• NOTICE
• Recovered Darcys law!
• See why by pulling ??? out of the hydraulic
  conductivity we obtain an intrinsic property of
  the solid which can be applied to a range of
  fluids.
• SO if K is the saturated hydraulic conductivity,
  K ? ??? . This way we can calculate the
  effective conductivity for any fluid. This is
  very useful when dealing with oil spills ...
  boiling water spills ..... etc.

17
Darcy's Law at Re gt 1

• Often noted that Darcy's Law breaks down at Re gt
  1.
• Laminar flow holds capillaries for Re lt 2000
  Hagen-Poiseuille law still valid
• Why does Darcy's law break down so soon?
• Laminar ends for natural media at Regt100 due to
  the tortuosity of the flow paths (see Bear, 1972,
  pg 178).
• Still far above the value required for the break
  down of Darcy's law.
• Real Reason due to forces in acceleration of
  fluids passing particles at the microscopic level
  being as large as viscous forces increased
  resistance to flow, so flux responds less to
  applied pressure gradients.

18
A few more words about Regt1

• Can get a feel for this through a simple
  calculation of the relative magnitudes of the
  viscous and inertial forces.
• FI ? Fv when Re ? 10.
• Since FI go with v2, while Fv goes with v,
• at Re ?1 FI ? Fv/10,
• a reasonable cut-off for creeping flow
  approximation

19
Deviations from Darcys law

• (a) The effect of inertial terms becoming
  significant at Regt1.
• (b) At very low flow there may be a threshold
  gradient required to be overcome before any flow
  occurs at all due to hydrogen bonding of water.

20
How does this apply to Vadose?

• Consider typical water flow where v and d are
  maximized
• Gravity driven flow near saturation in a coarse
  media.
• maximum neck diameter will be about 1 mm,
• vertical flux may be as high as 1 cm/min (14
  meters/day).
• 2.100
• Typically Darcy's OK for vadose zone.
• Can have problems around wells

21
What about Soil Vapor Extraction?

• Does Darcy's law apply?
• Air velocities can exceed 30 m/day (0.035
  cm/sec). The Reynolds number for this air flow
  rate in the coarse soil used in the example
  considered above is
• 2.101
• again, no problem, although flow could be higher
  than the average bulk flow about inlets and
  outlets

22
Summary of Darcy and Poiseuille

• For SATURATED MEDIA
• Flow is linear with permeability and gradient in
  potential (driving force)
• At high flow rates becomes non-linear due to
  local acceleration
• Permeability is due to geometric properties of
  the media (intrinsic permeability) and fluid
  properties (viscosity and specific density)
• Permeability drops with the square of pore size
• Assumed no slip solid-liquid boundary doesn't
  work with gas.