porosity peameability realation

Porosity – Permeability Relationships

Permeability and porosity trends for various rock types

[CoreLab,1983]


Influence of grain size on the relationship

between porosity and permeability

[Tiab & Donaldson, 1996]


• Darcy’s Law (1856) – empirical observations of flow to obtain permeability

• Slichter (1899) – theoretical analysis of fluid flow in packed uniform spheres

• Kozeny (1927),Carmen (1939) – capillary tube model


Capillary Tube Model Define porosity

Where r is radius of the capillary tube,

nt is number of tubes/ unit area

Define permeability

Porosity-permeability relationship

2rnt

8

4r

tn

k

8

2rk


Example

For cubic packing shown, find and k.

Number of tubes per unit area: 4 tubes/(4r)2

Porosity

Tortuosity

Permeability

r

4

2*

24

1 r

r

1

2

L

aL

32)1(8*

48

222 rrrk

Carmen – Kozeny Equation

Where

Kz, Kozeny constant-shape factor to account for variability in shape and length


Define specific surface area

Spv – specific surface area per unit pore volume

Spv = 2/r (for cylindrical pore shape)

Sbv- …unit bulk volume

Sgv- …unit grain volume

pvS

gvS

pvS

bvS

1

*

2

pvS

zk

k

2

L

aL

8

2rk

Spv = 2/r


Tortuosity,

ko is a shape factor

= 2 for circular

= 1.78 for square


2

L

aL

2

pvS

zk

k


Where

Kz, Kozeny constant-shape factor to account for variability in shape and length

8

2rk

Spv = 2/r *oz kk


Example: spherical particles with diameter, dp

2172

23

pdk

2

pvS

zk

k

??

Distribution of Rock Properties

Porosity Distribution

Expected porosity histogram [Amyx,et at., 1960]


Porosity Distribution

Actual porosity histogram [NBU42W-29, North Burbank Field]

0

1

2

3

4

5

6

7

8

9

10

4 6 8 10 12 14 16 18 20 22 24 26 28

Porosity , %

Fre

qu

en

cy

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Cu

mu

lati

ve

Fre

qu

en

cy


Permeability Distribution

Expected Skewed normal and log normal histograms for permeability [Craig,1971]


Permeability Distribution

Actual permeability histogram [NBU42W-29, North Burbank Field]

0

5

10

15

20

25

0.01 0.10 1.00 10.00 100.00 1,000.00

fre

qu

en

cy

Permeability, md


Permeability Variation

Dykstra-Parsons Coefficient

Characterization of reservoir heterogeneity by permeability variation

[Willhite, 1986]

50k

1.84k

50k

V



Example of log normal permeability distribution [Willhite, 1986]



Actual Dykstra-Parsons probability plot [NBU42W-29, North Burbank Field]

y = 578.37e-4.647x R² = 0.9917

0.001

0.010

0.100

1.000

10.000

100.000

1000.000

10000.000

0.0 0.2 0.4 0.6 0.8 1.0

k,m

d

probability of samples with permeability >

Flow units


Flow capacity vs storage capacity distribution [Craig, 1971]


Lorenz Coefficient

ADCAArea

ABCAArea

kL



Lorenz Coefficient

643.0

ADCAArea

ABCAAreaLk

Actual Lorenz plot [NBU42W-29, North Burbank Field]

y = -3.8012x4 + 10.572x3 - 11.01x2 + 5.2476x - 0.0146 R² = 0.9991

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Fra

cti

on

of

tota

l F

low

Cap

acit

y

Fraction of total Volume

Flow Capacity Distribution


dep

th

arranged un-arranged

Schematic of statistical approach of arranging data in comparison to true reservoir data, which is not ordered.

Drawback of statistical approaches

• Sequential ordering of data

• reliance only on permeability variations for estimating flow in layers. Does not account for: – phase mobility, pressure gradient, Swirr and the k/ ratio


Hydraulic Flow Unit • unique units with similar

petrophysical properties that affect flow. – Hydraulic quality of a rock is

controlled by pore geometry

– It is the distinction of rock units with similar pore attributes, which leads to the separation of units into similar hydraulic units.

– not equivalent to a geologic unit. The definition of geologic units or facies are not necessarily the same as the definition of a flow unit.

HFU1 HFU2

HFU3

HFU4

Schematic illustrating the concept of flow units.


• Start with CK equation

• Take the log

where the Reservoir quality index (RQI) is given by,

the Flow Zone Indicator (FZI) is,

and the pore-to-grain volume ratio is expressed as Plot of RQI vs r for East Texas Well

[Amaefule, et al.,1993]

)log()log()log( FZIr

RQI

}{0314.0}{

mdkmRQI

zk

gvS

FZI1

1r

gvS

ok

k

1

1


HFU [NBU42W-29, North Burbank Field]

0.010

0.100

1.000

10.000

0.010 0.100 1.000

RQ

I

Porosity Ratio

0.010

0.100

1.000

10.000

0.010 0.100 1.000

RQ

I

Porosity Ratio

4.0

2.6

1.8

0.5

FZI


y = 578.37e-4.647x R² = 0.9917

0.001

0.010

0.100

1.000

10.000

100.000

1000.000

10000.000

0.0 0.2 0.4 0.6 0.8 1.0

k,m

d

probability of samples with permeability >

Flow units

0

1

2

3

4

5

6

7

8

9

10

4 6 8 10 12 14 16 18 20 22 24 26 28

Fre

qu

en

cy

Porosity, %

FZI4

FZI3

FZI2

FZI1

k = 6E+066.9644 R2 = 0.9014

1E-03

1E-02

1E-01

1E+00

1E+01

1E+02

1E+03

1E+04

0.00 0.10 0.20 0.30 0.40

pe

rme

ab

ilit

y

porosity

0.010

0.100

1.000

10.000

0.010 0.100 1.000

RQ

I

Porosity Ratio

4.0

2.6

1.8

0.5

FZI

porosity peameability realation

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