1. Hydrodynamics in Porous Media
We will cover:
1. How fluids respond to local potential gradients (Darcy’s Law) 2. Add the conservation of mass to obtain Richard’s equation
Williams,
Modified after Selker,

2. Darcy’s Law for saturated media
In 1856 Darcy hired to size sand filters for the town’s central water supply.
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 rg).
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. Darcy’s Law
Darcy then observed that the flow of water in a vertical column was well-described by the equation
Darcy’s 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
ÑH is the gradient in hydraulic head (dimensionless)

4. The Dell Operator The “dell” operator: short hand for 3-d derivative
The result of “operating” on a scalar function (like potential) with Ñ is the slope of the function
ÑF points directly towards the steepest direction of uphill with a length proportional to the slope of the hill.

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... Notes on K:
Is the right side of Darcy’s law indeed a vector?
h is a scalar, but ÑH is a vector
Since K is a tensor, KÑH 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., Kab=0 where a¹b). 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. Darcy’s Law is Linear
Consider the intuitive aspects of Darcy’s 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 completely linear; all properties scale linearly with changes in system forces and dimensions.

9. Why is Darcy Linear?
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 Darcy’s 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 Darcy’s Law? No! Shows that the creeping flow assumption is sufficient to get from N-S equation to Darcy’s Law.

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).
Let’s derive the equation for steady flow through a capillary of radius ro
Consider forces on cylindrical control volume shown
S F = [2.75]

12. Force Balance on Control Volume
end pressures:
at S = 0 F1 = Ppr2
at S = DS F2 = (P + DS dP/dS) pr2
shear force: Fs = 2pDSt
where t is the local shear stress
Putting these in the force balance gives
Ppr2 - (P + DS dP/dS) pr2 - 2pDSt = 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
Ppr2 - (P + DS dP/dS) pr2 - 2pDSt = 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 dq 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/pr2
(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 k.
g/m is a function of the fluid alone
NOTICE:
Recovered Darcy’s law!
See why by pulling g/m 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= k g/m . 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# > 1 Why does Darcy's law break down so soon?
Often noted that Darcy's Law breaks down at Re# > 1.
Laminar flow holds capillaries for Re < 2000; Hagen-Poiseuille law still valid
Why does Darcy's law break down so soon?
Laminar ends for natural media at Re#>100 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 Re#>1
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 Darcy’s law
(a) The effect of inertial terms becoming significant at Re>1.
(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.