8. Fluid Flow

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179Stanford Rock Physics Laboratory - Gary Mavko

Fluid Flow and Permeability

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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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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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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

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