Porosity – Permeability...
Transcript of Porosity – Permeability...
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Porosity – Permeability Relationships
Permeability and porosity trends for various rock types
[CoreLab,1983]
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Porosity – Permeability Relationships
Influence of grain size on the relationship
between porosity and permeability
[Tiab & Donaldson, 1996]
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Porosity – Permeability Relationships
• 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
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Porosity – Permeability Relationships
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
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Porosity – Permeability Relationships
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
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Carmen – Kozeny Equation
Where
Kz, Kozeny constant-shape factor to account for variability in shape and length
Porosity – Permeability Relationships
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
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Carmen – Kozeny Equation
Tortuosity,
ko is a shape factor
= 2 for circular
= 1.78 for square
Porosity – Permeability Relationships
2
L
aL
2
pvS
zk
k
Carmen – Kozeny Equation
Where
Kz, Kozeny constant-shape factor to account for variability in shape and length
8
2rk
Spv = 2/r *oz kk
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Porosity – Permeability Relationships
Example: spherical particles with diameter, dp
2172
23
pdk
2
pvS
zk
k
??
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Distribution of Rock Properties
Porosity Distribution
Expected porosity histogram [Amyx,et at., 1960]
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Distribution of Rock Properties
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
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Distribution of Rock Properties
Permeability Distribution
Expected Skewed normal and log normal histograms for permeability [Craig,1971]
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Distribution of Rock Properties
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
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Distribution of Rock Properties
Permeability Variation
Dykstra-Parsons Coefficient
Characterization of reservoir heterogeneity by permeability variation
[Willhite, 1986]
50k
1.84k
50k
V
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Distribution of Rock Properties
Permeability Variation
Example of log normal permeability distribution [Willhite, 1986]
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Distribution of Rock Properties
Permeability Variation
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
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Distribution of Rock Properties
Flow capacity vs storage capacity distribution [Craig, 1971]
Permeability Variation
Lorenz Coefficient
ADCAArea
ABCAArea
kL
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Distribution of Rock Properties
Permeability Variation
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
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Distribution of Rock Properties
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
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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.
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Distribution of Rock Properties
• 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
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Distribution of Rock Properties
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
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Distribution of Rock Properties
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