American Oil and Gas Reporter - January 2015 - 168

FIGURE 1
Example Application of Parallel-Flow Model
A

B
Normalized Oil Rate, qDData (fraction)

1.0

Normalized Oil Rate,qD
(fraction)

1.0
qD-Model
qD-Data

R3 = 0.94

0.5

0.5

0.0

0.0
0

500
1,000
Time, t (days)

0.0

1,500

C

0.5
1.0
Normalized Model Oil Rate, qD-Model
(fraction)

D
1.0

0.30
Normalized Oil Rate Error,
e (fraction)

Normalized Oil Rate,qD
(fraction)

qD-Model

0.15

qD-Data

0.5

0.00
0

500

1,000

1,500

-0.15

0.0
0

-0.30

50,000
100,000
150,000
Cumulative Oil, NP (STB)

Time, t (days)

FIGURE 2
Example Application of Logistic-Growth Model
A

B
1.0
qD-Data
qD-Model

Normalized Oil Rate, qDData (fraction)

Normalized Oil Rate,qD
(fraction)

1.0

R2 = 0.83

0.5

0.5

0.0
0.0

0.0
0

500
1,000
Time, t (days)

1,500

C

0.5
Normalized Model Oil Rate, qD-Model
(fraction)

D

1.0

0.8
qD-Data
qD-Model

0.5

0.0
0

Normalized Oil Rate Error,
e (fraction)

Normalized Oil Rate,qD
(fraction)

1.0

50,000
100,000
150,000
Cumulative Oil, NP (STB)

0.4
0.0
0

500

1,000

1,500

-0.4
-0.8

Time, t (days)

TABLE 1
Statistical Summary of Model-Fitting Parameters
(Logistic-Growth Model)
N (bbl)

n

a

Mean

133,135.35

0.62

5,897.30

Median

111,940.75

0.58

75.96

Standard Deviation

69,011.60

0.23

52,876.12

Kurtosis

3.12

63.29

84.00

Skew

1.32

7.47

9.16

Range

408,248.15

2.13

484,738.60

Minimum

20,000.00

0.46

4.52

Maximum

428,248.15

2.59

484,743.12

168 THE AMERICAN OIL & GAS REPORTER

four. The panels in Figure 1 show an example application of the parallel-flow
model to well UT-ID4. The model fits
rate and cumulative production data quite
well. The A panel shows the normalized
rate/time plot for the data and model history match, B is the cross plot of normalized rate data and normalized model rate
prediction, C is the normalized cumulative
rate plot for the data and model history
match, and D is a plot of the error in normalized rate prediction versus time. Error
(defined as the difference between model
predictions and data) was larger at early
time because there was more scatter in
the data than at late time production. The
coefficient of determination was large at
0.94.
After applying the parallel-flow model
to all 87 wells in the dataset, 28 wells required two exponential terms, 58 wells
required three, and one well required
four. The distributions of initial production
rates were log-normal; that is, initial production rates close to the mean were
more frequent than initial production
rates with large values.
The distributions of the time constants
for the initial production rates also appeared to follow a log-normal distribution.
Physically, the time constants can be interpreted as how fast the fluids in a reservoir will drain. Small values indicate
fluids will drain very fast and large values
imply that it will take longer for the
fluids to drain. There were at least two
time scales in the dataset, with one time
scale accounting for the high-transmissibility, "low-storativity" fractures, and another accounting for the low-transmissibility, high-storativity reservoir matrix.
The panels in Figure 2 show an example application of the logistic-growth
model to the same well (UT-ID4). As in
Figure 1, the A panel is the normalized
rate/time plot for the data and model history match, B is the cross plot of normalized rate data and normalized model rate
prediction, C is the normalized cumulative
rate plot for the data and model history
match, and D is the error in normalized
rate prediction versus time.
The coefficient of determination was
found to be 0.83, which is not as large as
that obtained for the parallel-flow model
(0.94). There, again, was more error at
early-time because of data scatter. The
carrying capacity obtained for the well
was 1.1×105 STB. Therefore, the EUR
from the well based on the logistic-growth
model also was 1.1×105 STB. All of the
model parameters appeared to be log
normally distributed.
Table 1 shows the statistical summary
of the logistic-growth model parameters
for wells in the dataset. This summary
American Oil and Gas Reporter - January 2015

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