Porosity – Permeability Relationships PDF

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Porosity – Permeability Relationships Permeability and porosity trends for various rock types [CoreLab,1983] Porosity – Permeability Relationships Influence of grain size on the relationship between porosity and permeability [Tiab & Donaldson, 1996] Porosity – Permeability Relationships • Darcy’...

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Porosity – Permeability Relationships

Permeability and porosity trends for various rock types [CoreLab,1983]

Porosity – Permeability Relationships

Influence of grain size on the relationship between porosity and permeability [Tiab & Donaldson, 1996]

Porosity – Permeability Relationships • Darcy’s Law 18 permeability

– empirical observations of flow to obtain

• Slichter (1899) – theoretical analysis of fluid flow in packed uniform spheres • Kozeny (1927),Carmen (1939) – capillary tube model

Porosity – Permeability Relationships Capillary Tube Model   ntr 2  Define porosity Where r is radius of the capillary tube, nt is number of tubes/ unit area

Define permeability

k

n t r

Porosity-permeability relationship

4

8 

 r2 k 8

Porosity – Permeability Relationships Example

For cubic packing shown, find  and k. Number of tubes per unit area: 4 tubes/(4r)2 Porosity

Tortuosity

Permeability



2  1 * r  2 4 4r

 La   L 

2   1  

 r 2  r 2 r 2  *  k 8 4 8(1) 32

r

Porosity – Permeability Relationships Carmen – Kozeny Equation

Define specific surface area Spv – specific surface area per unit pore volume Spv = 2/r (for cylindrical pore shape)

k

Sbv- …u it bulk volu e Sgv- …u it grai volu e S bv   * S pv

    S S gv   pv  1   

 r2 k 8

L  a  L 

   

2

Spv = 2/r

 2 k z S pv

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

Porosity – Permeability Relationships Carmen – Kozeny Equation k

r 8

Spv = 2/r

2

k

 2 k z S pv

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

Carmen – Kozeny Equation k z  ko *

Tortuosity, 

 La   L 

   

2

ko is a shape factor = 2 for circular = 1.78 for square

Porosity – Permeability Relationships

Example: spherical particles with diameter, dp k



2 k z S pv

?? k

 3d 2p

721   2

Distribution of Rock Properties Porosity Distribution

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

Distribution of Rock Properties Porosity Distribution 1.2

10 9

Frequency

7

0.8

6 5

0.6

4 0.4

3 2

0.2

1 0

0.0 4

6

8

10

12

14

16

18

20

22

24

26

28

Porosity , %

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

Cum ulative Frequency

1.0

8

Distribution of Rock Properties

Permeability Distribution

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

Distribution of Rock Properties

Permeability Distribution 25

frequency

20

15

10

5

0 0.01

0.10

1.00

10.00

100.00

1,000.00

Permeability, md

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

Distribution of Rock Properties

Permeability Variation Dykstra-Parsons Coefficient k 50  k 84.1 V k 50

Characterization of reservoir heterogeneity by permeability variation [Willhite, 1986]

Distribution of Rock Properties Permeability Variation

Example of log normal permeability distribution [Willhite, 1986]

Distribution of Rock Properties Permeability Variation 10000.000 1000.000 Flow units

k,md

100.000 10.000

y = 578.37e-4.647x R² = 0.9917

1.000 0.100 0.010 0.001 0.0

0.2

0.4

0.6

0.8

probability of samples with permeability >

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

1.0

Distribution of Rock Properties Permeability Variation Lorenz Coefficient

Lk 

Area ABCA Area ADCA

Flow capacity vs storage capacity distribution [Craig, 1971]

Distribution of Rock Properties Permeability Variation

Flow Capacity Distribution 1

 0.643

0.9 Fraction of total Flow Capacity

Lorenz Coefficient Area ABCA Lk  Area ADCA

0.8 0.7 0.6

0.5 0.4 0.3 y = -3.8012x4 + 10.572x3 - 11.01x2 + 5.2476x - 0.0146 R² = 0.9991

0.2 0.1 0 0

0.2

0.4

0.6

0.8

Fraction of total Volume

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

1

Distribution of Rock Properties Drawback of statistical approaches • Sequential ordering of data depth

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

un-arranged

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

Distribution of Rock Properties 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.

Distribution of Rock Properties • Start with CK equation

     1        1     k  S   o gv  k

• Take the log

log( RQI )  log(  r )  log( FZI )

where the Reservoir quality index (RQI) is given by, RQI {m}  0.0314



k{md}

the Flow Zone Indicator (FZI) is, FZI 

1 S gv k z

and the pore-to-grain volume ratio is expressed as r 



1

Plot of RQI vs r for East Texas Well [Amaefule, et al.,1993]

Distribution of Rock Properties 10.000

10.000

FZI 4.0 2.6 1.8

1.000

1.000

RQI

RQI

0.5

0.100

0.100

0.010 0.010

0.100

1.000

0.010 0.010

Porosity Ratio

HFU [NBU42W-29, North Burbank Field]

0.100 Porosity Ratio

1.000

Distribution of Rock Properties 10

10000.000

9 1000.000

8

Flow units

7

10.000

Frequency

k,md

100.000 y = 578.37e-4.647x R² = 0.9917

1.000 0.100 0.010

6

FZI4

5

FZI3

4 3

FZI2

2

FZI1

1 0

0.001 0.0

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

0.2 0.4 0.6 0.8 1.0 probability of samples with permeability >

Porosity, %

10.000

1E+04

FZI

1E+03

4.0 2.6

1E+02 1.8 0.5

RQI

permeability

1.000 1E+01 1E+00 1E-01 1E-02 1E-03 0.00

0.100

k = 6E+066.9644 R2 = 0.9014 0.10

0.20 porosity

0.30

0.40

0.010 0.010

0.100 Porosity Ratio

1.000...