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Hydrodynamics inPorous Media
We will cover:
How fluids respond to local potential
gradients (Darcys Law)
Add the conservation of mass to
obtain Richards equation
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Darcys Law for saturated mediaIn 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 flowwhich characterized the media, and forces acting to pushthe fluid through the media.
Q - The rate of flow (L3/T) as the volume of water passed through acolumn per unit time.
hi - The fluid potential in the media at position i, measured instanding head equivalent. Under saturated conditions this iscomposed 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).
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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 forisotropic media as
q is the specific flux vector (L/T; volume of water perunit 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)
Q= K )H(HL
A0-1 [2.68
q = -K H [2.69]
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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 alength proportional to the slope of the hill.
Later well use F. The dot just tells us to take the dell andcalculate the dot product of that and the function F (which needs tobe 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 bethe 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.
xi,y
j,z
k
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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, orvelocity.
For vertical flow: speed at which the height of a pondof 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 ofwater, has units of length.
LHH=H 01 [2.70]
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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), KH is a vectorSo all is well on the right hand side
Notes on K:we could also obtain a vector on the right handside by selecting K to be a scalar, which is oftendone (i.e., assuming that conductivity is
independent of direction).
q = -KH [2.70]
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A few words about the K tensor
Kab relates gradients in potential in the b-direction toflux that results in the a-direction.
In anisotropic media, gradients not aligned withbedding give flux not parallel with potential gradients. If
the coordinate system is aligned with directions ofanisotropy the "off diagonal terms will be zero (i.e.,Kab=0 where ab). If, in addition, these are all equal,then the tensor collapses to a scalar.
The reason to use the tensor form is to capture theeffects of anisotropy.
q = -
Kxx Kxy Kxz
Kyx Kyy KyzKzx Kzy Kzz
h
x h
y h
z
= -
Kxxhx
+K xy hy
+K xzhz
;K yxhx
+K yy hy
+K yzhz
;K zxhx
+K zy hy
+K zzhz
flux in x-direction flux in y-direction flux in z-direction
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Looking holisticallyCheck 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., flowthrough twice the length with the same potentialdrop gives half the flux);
Directly related to the potential energy dropacross the system (e.g., double the energyexpended to obtain twice the flow).
The expression is patently linear; all propertiesscale linearly with changes in system forces and
dimensions.
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Why is Darcy Linear?Because non-turbulent?
No.Far before turbulence, there will be large localaccelerations: 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 withvelocities u, v, and w in the x, y, and z (vertical)
directions, may be written
ut
+ u ux
+ vuy
+ wuz
=-1
Px
- gzx
+
u
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Creeping flowNow impose the conditions needed for which Darcys Law
Creeping flow; acceleration (du/dx) terms smallcompared 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, proportionalto velocity and viscosity on right (same as Darcy).
Proof of Darcys Law? No! Shows that the creeping flowassumption is sufficient to obtain correct form.
uxuyuz0 [2.69]
ut0 [2.70]
xPgz
2u [2.71]
u
t+ u
u
x+ v
u
y+ w
u
z=
-1
P
x- g
z
x+
u
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Capillary tube model for flowWidely used model for flow through porous media is a group
of cylindrical capillary tubes (e.g.,. Green and Ampt, 1911and 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]
V
0
s
r
ro
s
Cy lindrica l Control Volum e
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Force Balance on Control Volume
end pressures:
at S = 0 F1 = Pr2at S = S F2 = (P + S dP/dS) r
2
shear force: Fs = 2rS
where is the local shear stress
Putting these in the force balance gives
Pr2 - (P + S dP/dS) r2 - 2rS = 0 [2.76]
where we remember that dP/dS is negative in sign (pressure
drops along the direction of flow)
V
0
s
r
ro
s
Cylindrical Control Volume
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continuing the force balance
With some algebra, this simplifies to
dP/dS is constant: shear stress varieslinearly with radius
From the definition of viscosity
Using this [2.77] says
Multiply both sides
by dr, and integrate
= -dS
dPr
2[2.77
Pr2 - (P + S dP/dS) r2 - 2rS = 0 [2.76]
dr
dv [2.78]
dr
dv= r
2 dPdS
[2.79]
)(
0
rvv
v
dv =
r
rr o
dr
dS
dPr
2
[2.80]
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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
v(r) = ( )r2-r02
4 dPdS [2.81]
Q = Area
vdA [2.82]
Q =
=
r = 0
r = r 0
r 2 - r 02
4
d P
d S r d r d [2 .8 0 ]
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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), forwhich we use the symbol q = Q/r2
(second power)
We commonly measure pressure in terms of hydraulichead, so we may substitute gh = P, to obtain
Q = -
r04
8 dPdS [2.84]
q= -1
r02
8 dP
dS [2.85]
q= -
r02
8
dh
dS [2.86]
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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 solidwhich 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 forany fluid. This is very useful when dealing with oil spills
... boiling water spills ..... etc.
q = -r 02
8
d h
d S[2 .83]
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Darcy's Law at Re# > 1Often 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 fluidspassing particles at the microscopic level being aslarge as viscous forces: increased resistance to flow,so flux responds less to applied pressure gradients.
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A few more words about Re#>1Can get a feel for this
through a simplecalculation of therelative magnitudes ofthe 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 forcreeping flow
approximation
d2
d2
d1
d1
Flow
Isometric View
v1 v2
Cross-Section
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Deviations from Darcys law
(a) The effect of inertialterms becoming significantat Re>1.
(b) At very low flow theremay be a threshold
gradient required to beovercome before any flowoccurs at all due to
hydrogen bonding of water.
q
h
Darcy'sLa
w
0 1 10 100
Re=0
Re=1
Re=10
Re=100
K
1
q
h
Darcy'sLa
w
0
K
1
0
ThresholdPressure
(b)
(a)
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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
R = d 1
v1
= 1gr/cm3
x0.1cmx1cm/min0.01gr/cm-sec
= 0.167
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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 thecoarse 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
R =d 1 v1
=0.001gr/cm 3 x0.1cmx0.035cm/sec
1.8x10 -4 gr/cm-sec
= 0.02
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Summary of Darcy and PoiseuilleFor SATURATED MEDIA
Flow is linear with permeability and gradientin potential (driving force)
At high flow rates becomes non-linear due to
local accelerationPermeability is due to geometric properties of
the media (intrinsic permeability) and fluidproperties (viscosity and specific density)
Permeability drops with the square of poresize
Assumed no slip solid-liquid boundary:
doesn't work with gas.
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