Rock Physics Laboratory - Gary Mavko

Fluid Flow

215

Fluid Flow and Permeability

Rock Physics Laboratory - Gary Mavko

Fluid Flow

216

Viscosity describes the shear stresses thatdevelop in a flowing fluid.

Shear stress in the fluid is proportional to the fluidvelocity gradient.

V

Stationary

z

x

Fluid VelocityProfile

where η is the viscosity. Or in terms of the strainrate:

Units:

σxz =η∂Vx

∂z

σxz = 2η∂εxz∂t

∂εxz∂t

=12∂Vx

∂z

1Poise =1dyne − seccm2 = 0.1newton − sec

m2

η ≈ .01Poise ≈1centiPoiseWater at 20oC

Rock Physics Laboratory - Gary Mavko

Fluid Flow

217

Darcy’s Law:

where

volumetric flow rate

permeability of the medium

viscosity of the fluid

cross sectional area

Differential form:

where is the filtration velocity

PU•

∆ l

P + ∆P

Darcy found experimentally that fluid diffusesthrough a porous medium according to the relation

Q = −κηA ∆P

∆lQ =

κ =

η =

A =

V = −κηgrad P( )

V

Rock Physics Laboratory - Gary Mavko

Fluid Flow

218

Units

Darcy’s law:

Permeability κ has dimensions of area, or m2 in SIunits. But the more convenient and traditional unitis the Darcy.

In a water saturated rock with permeability of 1Darcy, a pressure gradient of 1 bar/cm gives aflow velocity of 1 cm/sec.

Q = −κηA ∆P

∆l

1Darcy ≅10−12m2

Rock Physics Laboratory - Gary Mavko

Fluid Flow

219

Kozeny-Carman RelationThe most common permeabilitymodel is to assume that rocks havenice round pipes for pore fluids toflow.

Compare this with general Darcy’s law:

Combining the two gives the permeability of a circularpipe:

We can rewrite this permeability in terms of familiar rockparameters, giving the Kozeny-Carman equation:

where: φ is the porosity S is the specific pore surface area τ is the tortuosity d is a typical grain diameter B is a geometric factor

The classical solution for laminar flow through acircular pipe gives:

strong scale dependence!

2R

Q = −κηA ∆P

∆l

Q = −πR 4

8η∆P∆l

κ =πR4

8A =πR 2

A

R2

8

κ =Bφ3

τ 2S2 κ =Bφ3d2

τ

Rock Physics Laboratory - Gary Mavko

Fluid Flow

220

Schematic porosity/permeability relationship in rocks from Bourbié,Coussy, Zinszner, 1987, Acoustics of Porous Media, Gulf Publishing Co.

H.1

Rock Physics Laboratory - Gary Mavko

Fluid Flow

221

Demonstration of Kozeny-Carman relation in sintered glass, from Bourbié, Coussy, and Zinszner, 1987,

Acoustics of Porous Media, Gulf Publishing Co.

1

1 0

1 0 0

1000

0 1 0 2 0 3 0 4 0 5 0

280 µm spheres50 µm spheres

κ/d

2

(x10

e-6)

Porosity (%)

Sintered Glass

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.

Rock Physics Laboratory - Gary Mavko

Fluid Flow

222

Porosity/permeability relationship in Fontainebleau sandstone, from Bourbié, Coussy, and Zinszner, 1987,

Acoustics of Porous Media, Gulf Publishing Co.

H.3

A particularly systematic variation of permeability withporosity 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.

Rock Physics Laboratory - Gary Mavko

Fluid Flow

223

Kozeny-Carman Relation with Percolation

Hot-pressed Calcite (Bernabe et al, 1982),showing a good fit to the data using the Kozeny-Carman relation modified by a percolationporosity.

As porosity decreases from cementation and compaction, it is common to encounter a percolationthreshold where the remaining porosity is isolated ordisconnected. This porosity obviously does not contribute to permeability. Therefore, we suggest,purely heuristically, replacing giving

H.4

φ→ φ −φP

κ = B φ −φP

3d2

Rock Physics Laboratory - Gary Mavko

Fluid Flow

224

Fused Glass Beads (Winkler, 1993)

H.5

Rock Physics Laboratory - Gary Mavko

Fluid Flow

225

Fontainebleau Sandstone (Bourbié et al, 1987)

H.6

Here we show the same Fontainebleau sandstonedata as before with the Kozeny-Carman relationmodified by a percolation porosity of 2.5%. Thisaccounts for the increased slope at low porosities,while retaining the exponent of 3.

Rock Physics Laboratory - Gary Mavko

Fluid Flow

226

Diffusion

The stress-strain law for a fluid (Hooke’s law) is

which can be written as

combining with Darcy’s law:

gives the classical diffusion equation:

where D is the diffusivity

εαα =1K P

∇•V =1K∂P∂t

V =−κη

∇P

∇2P =−ηκK

∂P∂t

∇2P =−1D∂P∂t

Rock Physics Laboratory - Gary Mavko

Fluid Flow

227

Examples of Diffusion Behavior

1-D diffusion from an initial pressure pulse

Standard result:

P = P0δ x

P x,t = P0

4πDte

x2

–4Dt = P04πDt

e τ–t

Characteristic time scale

τ = x2

4D

Rock Physics Laboratory - Gary Mavko

Fluid Flow

228

Examples of Diffusion Behavior

Sinusoidal pressure disturbance

Disturbance decays approximately as

τd = λ2

4D

λ