Pruebas de Presion 4

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3.4.7 Type Curve Matching

A type curve is a graph of dimensionless pressure together with the

corresponding welltest derivative function versus dimensionless time, often on

a log-log scale. Such a graph is a convenient way to present the solution tothe diffusivity equation obtained by numerical computations. Figure 3.17 is

an example of type curves obtained by solving the diffusivity equation in the

presence of wellbore storage and skin. From the definitions of P wD , t D and C D ,

BOURDET TYPE CURVES

0.1

1

10

100

0.1 1 10 100 1000 10000

tD/CD

P w

D -

P w

D '

C D e2S

10 60

10 50

10 40

10 30

10 20

10 15

10 10

10 8

10 6

10 4

10 3

10 2

Figure 3.17. Bourdet et al type curves for a homogeneous reservoir withwellbore storage and skin for C De 2S from 10 2 to 10 60 .


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log log log141.2wD

khP P

q B

= +

(3.91)

42.951 10log log log

D

D

t x kht C C

= + (3.92)

Eqs.(3.91) and (3.92) show that the graph of log wDP versus log D

D

t C

is the

same as the graph of log P versus log t with the axes shifted by the constants

log141.2

khq B

and42.951 10

log x kh

C

. Thus, the P versus t obtained from a

transient pressure test can be matched graphically with the appropriate typecurve to estimate the reservoir and well parameters.

The procedure for using the Bourdet et al.'s type curves to analyze a

transient pressure test is as follows:

1. Prepare a log-log plot of P versus t on tracing paper using the same

scale as the type curves. Use only major grid lines on this plot. This is

the field plot.

2. Slide the field plot over the type curves both vertically and horizontally

to obtain the best match of the P D function and the corresponding

welltest derivative function. Trace the matched type curve on the field

plot to obtain a visual record of the best match.

3. Choose a convenient match point and record DM P , D D M

t C

, M t and

( )2 S D M C e .

4. Calculate the formation permeability from the pressure match as

( )( )( )( )

141.2 DM

M

q B Pk

h P

= (3.93)


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5. Calculate the wellbore storage coefficient from the time match as

( )0.0002951 M

D

D M

kht

C t C

=

(3.94)

6. Calculate the dimensionless wellbore storage coefficient.

7. Calculate the skin factor as

( )21ln

2

S D M

D

C eS

C

=

(3.95)

Figure 3.18 shows qualitatively a type match using the Bourdet et al.'s type

curves.

Figure 3.18. Example type curve match.


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3.4.8 Radius of Investigation of a WellTest

An approximate equation for estimating the radius investigated by a

welltest is given by

0.03248invt

kt r

c = (3.96)

where the variables are in oilfield units.

3.4.9 Field Example of a Well Test Analysis

A pressure buildup test was conducted on a new well in a new reservoir with

the results shown in Table 3.5.

Table 3.5: Pressure Buildup Data for Field Example

t t ( )wsP t (hrs) (hrs) (psia)15.33000 0.00000 3086.3315.33417 0.00417 3090.5715.33833 0.00833 3093.8115.34250 0.01250 3096.5515.34667 0.01667 3100.0315.35083 0.02083 3103.27

15.35500 0.02500 3106.7715.35917 0.02917 3110.0115.36333 0.03333 3113.2515.36750 0.03750 3116.4915.37583 0.04583 3119.4815.38000 0.05000 3122.4815.38833 0.05833 3128.9615.39667 0.06667 3135.9215.40500 0.07500 3141.1715.41333 0.08333 3147.6415.42583 0.09583 3161.9515.43833 0.10833 3170.68

15.45083 0.12083 3178.3915.46333 0.13333 3187.1215.47583 0.14583 3194.2415.49250 0.16250 3205.9615.50917 0.17917 3216.6815.52583 0.19583 3227.8915.54250 0.21250 3238.3715.55917 0.22917 3249.0715.58000 0.25000 3261.79


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15.62167 0.29167 3287.2115.66333 0.33333 3310.1515.70500 0.37500 3334.3415.74667 0.41667 3356.2715.78833 0.45833 3374.98

15.83000 0.50000 3394.4415.87167 0.54167 3413.9015.91333 0.58333 3433.8315.95500 0.62500 3448.0515.99667 0.66667 3466.2616.03833 0.70833 3481.9716.08000 0.75000 3493.6916.14250 0.81250 3518.6316.20500 0.87500 3537.3416.26750 0.93750 3553.5516.33000 1.00000 3571.7516.39250 1.06250 3586.23

16.45500 1.12500 3602.9516.51750 1.18750 3617.4116.58000 1.25000 3631.1516.64250 1.31250 3640.8616.70500 1.37500 3652.8516.76750 1.43750 3664.3216.83000 1.50000 3673.8116.95500 1.62500 3692.2717.08000 1.75000 3705.5217.20500 1.87500 3719.2617.33000 2.00000 3732.2317.58000 2.25000 3749.71

17.70500 2.37500 3757.1917.83000 2.50000 3763.4418.08000 2.75000 3774.6518.33000 3.00000 3785.1118.58000 3.25000 3794.0618.83000 3.50000 3799.8019.08000 3.75000 3809.5019.33000 4.00000 3815.9719.58000 4.25000 3820.2019.83000 4.50000 3821.9520.08000 4.75000 3823.7020.33000 5.00000 3826.45

20.58000 5.25000 3829.6920.83000 5.50000 3832.6421.08000 5.75000 3834.7021.33000 6.00000 3837.1921.58000 6.25000 3838.9422.08000 6.75000 3838.0222.58000 7.25000 3840.7823.08000 7.75000 3843.0123.58000 8.25000 3844.52


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24.08000 8.75000 3846.2724.58000 9.25000 3847.5125.08000 9.75000 3848.5225.58000 10.25000 3850.0126.08000 10.75000 3850.75

26.58000 11.25000 3851.7627.08000 11.75000 3852.5027.58000 12.25000 3853.5128.08000 12.75000 3854.2528.58000 13.25000 3855.0729.08000 13.75000 3855.5029.83000 14.50000 3856.5030.58000 15.25000 3857.2531.33000 16.00000 3857.9932.08000 16.75000 3858.7432.83000 17.50000 3859.4833.58000 18.25000 3859.99

34.33000 19.00000 3860.7335.08000 19.75000 3860.9935.83000 20.50000 3861.4936.58000 21.25000 3862.2437.58000 22.25000 3862.7438.58000 23.25000 3863.2239.58000 24.25000 3863.4840.58000 25.25000 3863.9941.58000 26.25000 3864.4942.58000 27.25000 3864.7343.83000 28.50000 3865.2345.33000 30.00000 3865.74

Other pertinent data are as follows:

Producing time before the test = 15.33 hours

Production rate before the test = 174 STB/D

Wellbore radius = 0.29 ft

Formation thickness = 107 ft

Porosity = 25 %

Total compressibility = 4.2x10 -6 psi -1

Formation volume factor = 1.06 RB/STB

Oil viscosity = 2.5 cp


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15.39667 0.06667 0.06638 230.94 3135.92 49.59 49.03 48.82

15.40500 0.07500 0.07463 205.40 3141.17 54.84 53.03 52.76

15.41333 0.08333 0.08288 184.97 3147.64 61.31 77.43 76.99

15.42583 0.09583 0.09523 160.97 3161.95 75.62 88.83 88.32

15.43833 0.10833 0.10757 142.51 3170.68 84.35 71.73 71.24

15.45083 0.12083 0.11989 127.87 3178.39 92.06 80.09 79.46

15.46333 0.13333 0.13218 115.98 3187.12 100.79 85.25 84.53

15.47583 0.14583 0.14446 106.12 3194.24 107.91 92.29 91.40

15.49250 0.16250 0.16080 95.34 3205.96 119.63 110.52 109.37

15.50917 0.17917 0.17710 86.56 3216.68 130.35 119.27 117.89

15.52583 0.19583 0.19336 79.28 3227.89 141.56 129.06 127.44

15.54250 0.21250 0.20959 73.14 3238.37 152.04 136.87 135.00

15.55917 0.22917 0.22579 67.89 3249.07 162.74 146.06 143.92

15.58000 0.25000 0.24599 62.32 3261.79 175.46 155.10 152.61

15.62167 0.29167 0.28622 53.56 3287.21 200.88 172.44 169.27

15.66333 0.33333 0.32624 46.99 3310.15 223.82 192.65 188.52

15.70500 0.37500 0.36605 41.88 3334.34 248.01 212.55 207.52

15.74667 0.41667 0.40564 37.79 3356.27 269.94 208.64 203.21

15.78833 0.45833 0.44502 34.45 3374.98 288.65 216.24 209.94

15.83000 0.50000 0.48421 31.66 3394.44 308.11 241.12 233.50

15.87167 0.54167 0.52318 29.30 3413.90 327.57 265.11 256.0515.91333 0.58333 0.56195 27.28 3433.83 347.50 247.95 239.07

15.95500 0.62500 0.60052 25.53 3448.05 361.72 253.29 243.21

15.99667 0.66667 0.63889 23.99 3466.26 379.93 283.06 271.37

16.03833 0.70833 0.67705 22.64 3481.97 395.64 243.76 233.17

16.08000 0.75000 0.71502 21.44 3493.69 407.36 258.62 246.28

16.14250 0.81250 0.77160 19.87 3518.63 432.30 298.44 283.73

16.20500 0.87500 0.82775 18.52 3537.34 451.01 258.25 244.44

16.26750 0.93750 0.88347 17.35 3553.55 467.22 273.98 258.07

16.33000 1.00000 0.93876 16.33 3571.75 485.42 278.26 261.44

16.39250 1.06250 0.99363 15.43 3586.23 499.90 283.74 265.20

16.45500 1.12500 1.04809 14.63 3602.95 516.62 301.05 280.62

16.51750 1.18750 1.10213 13.91 3617.41 531.08 288.60 267.90

16.58000 1.25000 1.15576 13.26 3631.15 544.82 253.30 234.50

16.64250 1.31250 1.20899 12.68 3640.86 554.53 247.56 227.85


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16.70500 1.37500 1.26182 12.15 3652.85 566.52 281.16 258.06

16.76750 1.43750 1.31426 11.66 3664.32 577.99 263.46 241.04

16.83000 1.50000 1.36631 11.22 3673.81 587.48 247.74 225.68

16.95500 1.62500 1.46926 10.43 3692.27 605.94 227.42 206.12

17.08000 1.75000 1.57070 9.76 3705.52 619.19 210.56 188.93

17.20500 1.87500 1.67066 9.18 3719.26 632.93 224.74 200.33

17.33000 2.00000 1.76919 8.67 3732.23 645.90 208.37 184.96

17.58000 2.25000 1.96203 7.81 3749.71 663.38 162.84 142.20

17.70500 2.37500 2.05641 7.45 3757.19 670.86 150.46 130.44

17.83000 2.50000 2.14947 7.13 3763.44 677.11 140.26 120.70

18.08000 2.75000 2.33172 6.57 3774.65 688.32 140.46 119.19

18.33000 3.00000 2.50900 6.11 3785.11 698.78 138.98 116.46

18.58000 3.25000 2.68151 5.72 3794.06 707.73 115.07 95.49

18.83000 3.50000 2.84944 5.38 3799.80 713.47 133.68 108.08

19.08000 3.75000 3.01297 5.09 3809.50 723.17 150.18 121.27

19.33000 4.00000 3.17227 4.83 3815.97 729.64 107.37 85.60

19.58000 4.25000 3.32750 4.61 3820.20 733.87 64.24 50.83

19.83000 4.50000 3.47882 4.41 3821.95 735.62 40.75 31.50

20.08000 4.75000 3.62637 4.23 3823.70 737.37 56.31 42.75

20.33000 5.00000 3.77029 4.07 3826.45 740.12 79.61 59.90

20.58000 5.25000 3.91071 3.92 3829.69 743.36 87.17 65.0020.83000 5.50000 4.04777 3.79 3832.64 746.31 74.57 55.11

21.08000 5.75000 4.18157 3.67 3834.70 748.37 72.12 52.33

21.33000 6.00000 4.31224 3.56 3837.19 750.86 70.51 50.88

21.58000 6.25000 4.43987 3.45 3838.94 752.61 34.46 25.33

22.08000 6.75000 4.68648 3.27 3838.02 751.69 19.52 12.42

22.58000 7.25000 4.92217 3.11 3840.78 754.45 53.06 36.18

23.08000 7.75000 5.14764 2.98 3843.01 756.68 43.29 28.98

23.58000 8.25000 5.36355 2.86 3844.52 758.19 41.52 26.89

24.08000 8.75000 5.57049 2.75 3846.27 759.94 40.82 26.16

24.58000 9.25000 5.76902 2.66 3847.51 761.18 33.25 20.81

25.08000 9.75000 5.95963 2.57 3848.52 762.19 40.20 24.38

25.58000 10.25000 6.14279 2.50 3850.01 763.68 37.65 22.86

26.08000 10.75000 6.31892 2.43 3850.75 764.42 32.21 18.81

26.58000 11.25000 6.48843 2.36 3851.76 765.43 33.95 19.69


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27.08000 11.75000 6.65168 2.30 3852.50 766.17 36.54 20.56

27.58000 12.25000 6.80901 2.25 3853.51 767.18 38.36 21.44

28.08000 12.75000 6.96074 2.20 3854.25 767.92 36.51 19.89

28.58000 13.25000 7.10716 2.16 3855.07 768.74 30.55 16.56

29.08000 13.75000 7.24854 2.11 3855.50 769.17 27.64 14.43

29.83000 14.50000 7.45173 2.06 3856.50 770.17 32.70 16.92

30.58000 15.25000 7.64495 2.01 3857.25 770.92 30.23 15.15

31.33000 16.00000 7.82892 1.96 3857.99 771.66 32.51 15.89

32.08000 16.75000 8.00429 1.92 3858.74 772.41 34.83 16.64

32.83000 17.50000 8.17164 1.88 3859.48 773.15 30.98 14.58

33.58000 18.25000 8.33152 1.84 3859.99 773.66 33.60 15.21

34.33000 19.00000 8.48442 1.81 3860.73 774.40 27.78 12.67

35.08000 19.75000 8.63077 1.78 3860.99 774.66 23.22 10.01

35.83000 20.50000 8.77100 1.75 3861.49 775.16 40.28 17.08

36.58000 21.25000 8.90548 1.72 3862.24 775.91 39.27 16.70

37.58000 22.25000 9.07644 1.69 3862.74 776.41 26.72 10.90

38.58000 23.25000 9.23853 1.66 3863.22 776.89 21.33 8.60

39.58000 24.25000 9.39243 1.63 3863.48 777.15 24.52 9.34

40.58000 25.25000 9.53875 1.61 3863.99 777.66 33.76 12.75

41.58000 26.25000 9.67803 1.58 3864.49 778.16 25.91 9.71

42.58000 27.25000 9.81077 1.56 3864.73 778.40 23.88 8.4843.83000 28.50000 9.96817 1.54 3865.23 778.90 30.25 10.62

45.33000 30.00000 10.14560 1.51 3865.74 779.41

The diagnostic plots depicted in Figure 3.19 show the classic response of a

homogeneous reservoir model with wellbore storage and skin. From the

welltest derivative function, it would appear that the effect of wellbore storage

had subsided at about 10 hrs, in which case, the pressure data beyond 10 hrs

might be fitted to the Horner semilog line. Notice the noise in the derivative

function. This is not surprising because numerical differentiation is a noisy

process. The wellbore storage coefficient can be calculated from the unit slope

line as


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( )( )174 1.06 10.0096 RB/psi

24 800C

= =

The dimensionless wellbore storage coefficient is given by

( )( )( )( )( ) ( )

226

5.615 0.00969.09 10

2 0.25 4.20 10 107 0.29 DC x

x = =

Figure 3.19. Diagnostic plots for field example.

Figure 3.20 shows the Horner plot. The Horner straight line is given by

( ) 15.333880.60 78.52logwst

P t t

+ =


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The slope of the Horner line is

78.52 psi/log cyclem =

from which the permeability is calculated as

( )( )( )( )( )( )

162.6 174 2.5 1.068.92 md

78.52 107k = =

Figure 3.20. Horner plot for field example.

From the Horner line,

( )1 3785.35wsP hr = psia

The skin factor can be calculated as


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( )( )( )( )263086.33 3785.35 8.92

1.1513 log 3.2378.52 0.25 2.5 4.20 10 0.29

S x

= +

( )1.1513 8.90 7.61 3.23 5.21S = + =

Thus, the well is damaged. The initial reservoir pressure is given by

* 3880.6iP P= = psia.

The radius of investigation for the drawdown portion of the test is given by

( )( )

( )( )( )6

8.92 15.330.03248 234 ft

0.25 2.5 4.20 10invr

x

= =

The radius of investigation for the buildup portion of the test is given by

( )( )( )( )( )6

8.92 300.03248 330 ft

0.25 2.5 4.20 10invr

x = =

The buildup test cannot investigate what the drawdown did not investigate.

Therefore, the radius of investigation of the buildup test test is obtained from

the drawdown portion of the test as 234 ft. The reservoir bulk volume

investigated by the test is given by

( )( )( )2 2 6 3234 107 18.406 10 ftinvV r h x = = =

The estimated permeability of 8.92 md can be assigned to this volume of the

reservoir. This is in contrast to the core permeability, which is measured on a

minute portion of the reservoir.

The last step in a pressure transient analysis is to simulate the test. A

successful simulation of the test, though non-unique, tends to lend credibility

to the results of the analysis. Using the k, S and C D obtained from the semilog

analysis as preliminary estimates, the homogeneous reservoir model with


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wellbore storage and skin can be used to simulate the test by a trial and error

procedure.

Figures 3.21 through 3.23 show graphical comparisons of the simulated

test and the field test. Figure 3.21 compares the pressure buildup responses;

Figure 3.22 compares the log-diagnostic plots; and Figure 3.23 compares the

Horner plots. Clearly, the agreement between the simulated test and the field

test is good. The best match was obtained with the following parameters:

9.4k = md

6S = 29 10 DC x=

2 81.46 10S DC e x=

( ) 3089.46wf pP t = psia

3880.60iP = psia

Thus, the simulation serves to fine tune the parameters obtained by the

conventional semilog analysis of the test.

Figure 3.21. A comparison of the simulated and the measured pressurebuildup response for field example.


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Figure 3.22. A comparison of the simulated and the measureddiagnostic plots for field example.

Figure 3.23. A comparison of the simulated and the measured Hornerplots for field example.

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