8. Fluid Flow
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8/12/2019 8. Fluid Flow
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179Stanford Rock Physics Laboratory - Gary Mavko
Fluid Flow and Permeability
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180Stanford Rock Physics Laboratory - Gary Mavko
Viscosity describes the shear stresses that develop in aflowing fluid.
Shear stress in the fluid is proportional to the fluid velocity gradient.
Where is the viscosity. Or in terms of the strain rate:
Units: Water at 20oC:
V
Stationary
z
x
FluidVelocityProfile
x z Vx
z
x z 2x z
t
x z
t1
2
Vx
z
1Poise 1dyne sec
cm2 0.1
newton sec
m2
.01Poise 1centiPoise
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181Stanford Rock Physics Laboratory - Gary Mavko
Darcys Law:
where volumetric flow rate
permeability of the medium
viscosity of the fluid
cross sectional area
Differential form:
where is the filtration velocity
Darcy found experimentally that fluid diffuses through a porous
medium according to the relation
Q
A
Pl
Q
A
V
grad P
V
P
l
Darcys Law
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Units
Darcys law:
Permeability has dimensions of area, or m2in SI units. But themore convenient and traditional unit is the Darcy.
In a water saturated rock with permeability of 1 Darcy, a pressure
gradient of 1 bar/cm gives a flow velocity of 1 cm/sec.
Q
A
Pl
1Darcy 1012 m2
Darcys Law
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Kozeny-Carman Relation
The most common permeability model is to assume
that rocks have nice round pipes for pore fluids to flow.
Compare this with general Darcys law:
Combining the two gives the permeabilityof a circular pipe:
We can rewrite this permeability in terms of familiar rock parameters, giving
the Kozeny-Carman equation:
where: is the porosity
S is the specific pore surface area
is the tortuosity
d is a typical grain diameter
B is a geometric factor
The classical solution for laminar flow
through a circular pipe gives:strong scale
dependence!Q
AP
l
Q R
4
8
Pl
R
4
8A R
2
A
R2
8
B3
2S2 B
3d2
2R
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184Stanford Rock Physics Laboratory - Gary Mavko
Schematic porosity/permeability relationship in rocks fromBourbi, Coussy, Zinszner, 1987,Acoustics of Porous
Media,Gulf Publishing Co.
H.1
10-9
10 -7
10-5
10-3
10-1
101
1 10
Permeability(Darcy)
Porosity (%)
Claysand
shales
Silts
Micritic
sandstones
Shalysandstones
Granularlimestones
Crystallinerocks
Tightsediments
Clean coarse-grained sandstones
Strong dependence ofpermeability on grain
and pore size
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Demonstration of Kozeny-Carman relation in sintered glass, from Bourbi, Coussy, and Zinszner,1987, Acoustics of Porous Media, Gulf Publishing Co.
H.2
Here we compare the permeability for two synthetic porous materials having very
different grain sizes. When normalized by grain-size squared, the data fall on top
of each other -- confirming the scale dependence.
1
10
100
1000
0 10 20 30 40 50
280 m sphere s50 m spheres
/d2
(x10e-6)
Porosity (%)
Sintered Glass
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H.3
A particularly systematic variation of permeability with porosity for
Fontainebleau sandstone. Note that the slope increases at small
porosity, indicating an exponent on porosity larger than the power of 3
predicted by the Kozeny-Carman relation.
1
10
100
1000
10000
2 4 6 8 10
Permeability
(mD)
Porosity (%)
2 30
n = 8
n = 3
= a n
Demonstration of Kozeny-Carman relation in sintered glass, from Bourbi, Coussy, and Zinszner,1987, Acoustics of Porous Media, Gulf Publishing Co.
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Kozeny-Carman Relation with Percolation
Hot-pressed Calcite (Bernabe et al, 1982), showing a good fit to the datausing the Kozeny-Carman relation modified by a percolation porosity.
As porosity decreases from cementation and compaction, it is commonto encounter a percolation threshold where the remaining porosity is
isolated or disconnected. This porosity obviously does not contribute topermeability. Therefore, we suggest, purely heuristically, replacinggiving
H.4
P
B P
3
d2
B .045 3d20.00001
0.0001
0.001
0.01
0.1
Permeability(mD)
.05
Porosity
.20.10
p 0.045
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Fused Glass Beads (Winkler, 1993)
H.5
B .035 3d20.0001
0.001
0.01
0.1
1
10
100
Permeability/D
2 200 micron
Porosity
100 micron
50 micron
.05 .50.10
p 0.035
Permeability(mD)
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Fontainebleau Sandstone (Bourbi et al, 1987)
H.6
Here we show the same Fontainebleau sandstone data as before with
the Kozeny-Carman relation modified by a percolation porosity of 2.5%.
This accounts for the increased slope at low porosities, while retaining
the exponent of 3.
B .025
3
d2
1
10
100
1000
10000
Permeability
(mD)
Porosity
.02 .30.05 .10
p 0.025
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Sandstone Data
H.7
d.5mm; p 0
d.18mm; p 0
d.06mm; p 0.03
d.02mm; p 0.04
Data from Tiab and Donaldson, 1996
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Diffusion
The stress-strain law for a fluid (Hookes law) is
which can be written as
combining with Darcys law:
gives the classical diffusion equation:
where D is the diffusivity
1
KP
V 1
K
Pt
V
P
2P K
Pt
2P1D
Pt
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Examples of Diffusion Behavior
1-D diffusion from an initial pressure pulse
Standard result:
Characteristic time scale =x24D
P x,t =
P04Dt e
x2
4Dt = P04Dt e
t