Download Effective and Relative Permeabilities

Statistical Analysis
of
Reservoir Data
Statistical Models
• Statistical Models are used to describe real
world observations
– provide a quantitative model
• prediction
• interpolation
Normal Distribution
• Example
– porosity from cores
or logs
• Two parameters:
– mean
– standard deviation
• Characteristics
– symmetric
– mean, median and
mode occur at same
value
Probability Paper
• Any two parameter
model can be plotted as
a straight line
– cumulative frequency
for normal
distributions plot as
straight line
• standard deviation
from slope
Log Normal Distribution
• Example
– permeability values from
cores or logs
• Two parameters:
– mean (of log(x))
– standard deviation (of log(x))
• Characteristics
– log(x) values have normal
distribution
– assymetric
• large “tail” toward large
values
• mean, median and mode
do not occur at same value
Log Probability Paper
• Cumulative frequency
for log normal
distributions plot as
straight line
• standard deviation
from slope
-2s
-1s
0
+1s
+2s
Reservoir heterogeniety
Usually permeabilities are “lognormally” distributed.
That is, the logarithm of their values
form a normal (bell-shaped) probability
curve.
This can be demonstrated by plotting
permeabilities, arranged in order from
smallest to largest, on a “logprobability” scale.
Dykstra-Parsons permeability
variation =
From Craig
k50  k84.1
V 
k50
Reservoir heterogeniety
• Dykstra-Parsons Perm. Variation, VDP:
• step1--arrange perms in increasing order
• step2--assign percentiles to each perm
number
• step3--plot on log-probability scale
• step4--compute VDP
k 50  k 84.1

k 50
Reservoir heterogeniety
• Dykstra-Parsons Perm. Variation, VDP:
• step1--transform permeability data
[Ln(k)]
• step2--calculate s, the sample standard
deviation, of the transformed data
• step3--compute VDP  1  exp( s)
Example—Calculation of VDP
Permeability Data (Table 6.1 of Craig)
2.9
7.4
30.4
3.8
8.6
11.3
1.7
17.6
24.6
5.5
2.1
21.2
4.4
2.4
5.0
167.0
1.2
2.6
22.0
11.7
3.6
920.0
37.0
10.4
16.5
19.5
26.6
7.8
32.0
10.7
6.9
3.2
13.1
41.8
9.4
50.4
35.2
0.8
18.4
20.1
16.0
71.5
1.8
14.0
84.0
23.5
13.5
1.5
17.0
9.8
Mean =
28.9
Std Dev =
Ln(k) ~ N(m ,s 2)
1.0647
2.0015
3.4144
1.3350
2.1518
2.4248
0.5306
2.8679
3.2027
1.7047
0.7419
3.0540
1.4816
0.8755
1.6094
5.1180
0.1823
0.9555
3.0910
2.4596
1.2809
6.8244
3.6109
2.3418
2.8034
2.9704
3.2809
2.0541
3.4657
2.3702
1.9315
1.1632
2.5726
3.7329
2.2407
3.9200
3.5610
-0.2231
2.9124
3.0007
2.7726
4.2697
0.5878
2.6391
4.4308
3.1570
2.6027
0.4055
2.8332
2.2824
Mean =
2.3744
Std Dev =
Note:
CV = s/k
14.5
5.3
1.0
6.7
11.0
10.0
12.9
27.8
15.0
8.1
93.8
39.9
4.8
3.9
74.0
120.0
19.0
55.2
22.7
6.0
15.4
2.3
3.0
8.4
25.5
4.1
12.4
2.0
47.4
6.3
4.6
12.0
0.6
8.9
1.5
3.5
3.3
5.2
4.3
44.5
9.1
29.0
99.0
7.6
5.9
33.5
6.5
2.7
66.0
5.7
60.0
2.6741
1.6677
0.0000
1.9021
2.3979
2.3026
2.5572
3.3250
2.7081
2.0919
1.2618
3.6864
1.5686
1.3610
4.3041
4.7875
2.9444
4.0110
3.1224
1.7918
2.7344
0.8329
1.0986
2.1282
3.2387
1.4110
2.5177
0.6931
3.8586
1.8405
1.5261
CV =
V DP =
2.4849
-0.5108
2.1861
0.4055
1.2528
1.1939
1.6487
1.4586
3.7955
2.2083
0.5314
0.7169
3.3673
4.5951
2.0281
1.7750
3.5115
1.8718
0.9933
4.1897
1.7405
4.0943
V DP = 1 - exp(-s)