P324 03A Hwk 07 BorPBU - Texas A&M University · Bourdet, D.P.,Ayoub, J.A., and Pirard, Y.M.: "Use of Pressure Derivative in Well Test Interpreta -tion," SPEFE (June 1989) 293-302

Petroleum Engineering 324 — Well Performance Homework No. 7 — Analysis of Pressure Buildup Data (Bourdet (Radial Flow) Example)

19 March 2003 — Due: Wednesday 26 March 2003

Homework 7: Analysis of Pressure Buildup Data (Bourdet (Radial Flow) Example)

This problem set considers the "classic" Bourdet example for a pressure buildup test analyzed using derivative type curve analysis. For completeness, the Bourdet, et al. paper is also attached — however, you must provide your own analysis — you are to only use the Bourdet analysis as a guide, there are con-siderable differences of opinion as to what the "right" answers should be.

Be sure to work the problem using BOTH shut-in time (∆t) and effective shut-in time (∆te). Recall that ∆te is the time function for the equivalent pressure drawdown case. A Horner plot is provided, as well as the semilog ∆t and effective shut-in time ∆te plots — you are to use all of these plots.

Finally, for some reason, Bourdet, et al. chose not to work in terms of pressure, pws, but instead to work in terms of the pressure drop, ∆p. This is not a limitation, but it does require that you use ∆p1hr rather than pws,1hr-pwf(∆t=0) in the semilog skin factor relation.

For your convenience, the governing relations for type curve analysis using the "Bourdet -Gringarten" type curves are:

Formation Permeability:

[ ][ ]

or

or 2.141

MP'

MP'wDwD

pp

pp

hqB

k∆∆

=µ

Dimensionless Wellbore Storage Coefficient:

[ ][ ]

/

or 0002637.0

2MPDD

MPe

wtD Ct

tt

rc

kC

∆∆=

φµ

Skin Factor:

[ ] ln

21

2

=

D

MPs

DC

eCs

Notes:

a. The Bourdet-Gringarten type curves for radial flow behavior, including wellbore storage and skin effects, are provided in a 1 inch-by-1 inch format in this handout.

b. You have also been provided with log-log "pressure and pressure integral RATIO function" plots and type curves. Use of these materials is at your discretion, BUT you should note that these functions may significantly improve your ability to assess the transition and end of wellbore storage effects. You are strongly encouraged to use these resources.

You are NOT required to use these pressure and pressure integral RATIO functions, they are provided to assist your analysis—and are not intended to confuse you.

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19 March 2003 — Due: Wednesday 26 March 2003 Bourdet-Gringarten Type Curve — Dimensionless Pressure and Pressure Derivative Functions

Bourdet-Gringarten Type Curve: Dimensionless Pressure and Pressure Derivative Functions

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19 March 2003 — Due: Wednesday 26 March 2003 Bourdet-Gringarten Type Curve — Dimensionless Pressure Integral and Pressure Integral-Derivative Functions

Bourdet-Gringarten Type Curve: Dimensionless Pressure Integral and Pressure Integral-Derivative Functions

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19 March 2003 — Due: Wednesday 26 March 2003 Problem Definition and Requirements

Given:

These data are taken from the Bourdet, et al. reference and are to be considered accurate enough for engineering analysis. Assume that wellbore storage and skin effects are present.

Reservoir properties: φ=0.25 rw=0.29 ft ct=4.2x10-6 psia-1 h=107 ft

Oil properties: Bo=1.06 RB/STB µo=2.5 cp

Production parameters: pwf(∆t=0) = ? psia qo=174 STB/D (constant) tp=15.33 hr

References:

1. Bourdet, D.P., Ayoub, J.A., and Pirard, Y.M.: "Use of Pressure Derivative in Well Test Interpreta-tion," SPEFE (June 1989) 293-302.

Required:

1. For this problem, you are to perform the following analyses:

l "Preliminary" log-log analysis. l Cartesian analysis of "early" time (wellbore storage distorted) data. l Semilog analysis of "middle" time (radial flow) data. l Log-log type curve analysis. l Cartesian analysis of "late" time (boundary-dominated) data (i.e., the "Muskat Plot").

You are to complete the table on the next page provided for you to tabulate your results.

2. You are to provide a critical and detailed review (at least 1 page) for the following paper(s):

a. Bourdet, D.P., Ayoub, J.A., and Pirard, Y.M.: "Use of Pressure Derivative in Well Test Interpretation," SPEFE (June 1989) 293-302.

l For each paper you are to address the following questions: (Type or write neatly)

n Problem: — What is/are the problem(s) solved? — What are the underlying physical principles used in the solution(s)?

n Assumptions and Limitations: — What are the assumptions and limitations of the solutions/results? — How serious are these assumptions and limitations?

n Practical Applications: — What are the practical applications of the solutions/results? — If there are no obvious "practical" applications, then how could the solutions/results be used in

practice?

n Discussion: — Discuss the author(s)'s view of the solutions/results. — Discuss your own view of the solutions/results.

n Recommendations/Extensions: — How could the solutions/results be extended or improved? — Are there applications other than those given by the author(s) where the solution(s) or the concepts

used in the solution(s) could be applied?

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19 March 2003 — Due: Wednesday 26 March 2003 Required Results

Required: Analysis of Pressure Buildup Data (Bourdet (Radial Flow) Example)

You are to estimate the following:

l "Preliminary" log-log analysis: a. The wellbore storage coefficient, Cs. b. The dimensionless wellbore storage coefficient, CD. c. The formation permeability, k.

l Cartesian analysis of "early" time (wellbore storage distorted) data: a. The pressure drop at the start of the test, ∆pwi(∆t=0) — this should be ˜ 0 psi. b. The wellbore storage coefficient, Cs. c. The dimensionless wellbore storage coefficient, CD.

l Semilog analysis of "middle" time (radial flow) data: (use both Horner and MDH methods) a. The formation permeability, k. b. The near well skin factor, s. c. The radius of investigation, rinv, at the end of radial flow or the end of the test data.

l Log-log type curve analysis: (use both ∆t and ∆te methods) a. The formation permeability, k. b. The near well skin factor, s. c. The wellbore storage coefficient, Cs.

l Cartesian analysis of "late" time (boundary-dominated) data: "Muskat Plot" a. Average pressure DIFFERENCE, )0( =∆−=∆ tppp wf (if applicable).

Results: Pressure Falloff Test Analysis (West Texas Carbonate Reservoir)

Log-log Analysis:

Wellbore storage coefficient, Cs = RB/psi

Dimensionless wellbore storage coefficient, CD =

Formation permeability, k = md

Cartesian Analysis: Early Time Data

Pressure drop at the start of the test, ∆pwi(∆t=0) = psi

Wellbore storage coefficient, Cs = RB/psi

Dimensionless wellbore storage coefficient, CD =

Semilog Analysis: Horner or ∆te ∆t (MDH)

Formation permeability, k = md = md

Near well skin factor, s = =

Radius of investigation, rinv (end of radial flow or end of test) = ft = ft

Log-Log Type Curve Analysis: ∆t ∆te

Formation permeability, k = md = md

Near well skin factor, s = =

Wellbore storage coefficient, Cs = RB/psi = RB/psi

Dimensionless wellbore storage coefficient, CD = =

Cartesian Analysis: Late Time Data ("Muskat Plot" — if applicable)

Average pressure DIFFERENCE, )0( =∆−=∆ tppp wf = psia

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19 March 2003 — Due: Wednesday 26 March 2003 Well Test Data Functions

Well Test Data Functions:

Point ∆t, hr ∆te, hr ∆p, psi ∆p'(∆t), psi ∆p'(∆te), psi 1 8.330E-03 8.325E-03 3.81 6.75 6.76 2 1.250E-02 1.249E-02 6.55 9.87 9.88 3 1.667E-02 1.665E-02 10.03 13.98 13.99 4 2.083E-02 2.080E-02 13.27 17.29 17.32 5 2.500E-02 2.496E-02 16.77 20.08 20.12 6 2.917E-02 2.911E-02 20.01 23.33 23.37 7 3.333E-02 3.326E-02 23.25 21.12 21.17 8 3.750E-02 3.741E-02 26.49 23.48 23.53 9 4.583E-02 4.569E-02 29.48 30.80 30.90

10 5.000E-02 4.984E-02 32.48 33.76 33.89 11 5.833E-02 5.811E-02 38.96 43.84 44.02 12 6.667E-02 6.638E-02 45.92 57.78 58.06 13 7.500E-02 7.463E-02 51.17 66.91 67.26 14 8.333E-02 8.288E-02 57.64 72.79 73.22 15 9.583E-02 9.523E-02 71.95 77.66 81.64 16 1.083E-01 1.076E-01 80.68 81.64 82.22 17 1.208E-01 1.199E-01 88.39 80.55 89.09 18 1.333E-01 1.322E-01 97.12 88.78 89.60 19 1.488E-01 1.474E-01 104.24 100.16 101.22 20 1.625E-01 1.608E-01 115.90 107.64 108.84 21 1.792E-01 1.771E-01 126.68 117.25 118.69 22 1.958E-01 1.934E-01 137.89 131.18 132.91 23 2.125E-01 2.096E-01 148.37 136.13 138.11 24 2.292E-01 2.258E-01 159.07 144.82 147.06 25 2.500E-01 2.460E-01 171.79 152.26 154.84 26 2.917E-01 2.862E-01 197.12 172.53 175.92 27 3.333E-01 3.262E-01 220.15 155.13 158.47 28 3.750E-01 3.660E-01 244.34 197.70 202.66 29 4.167E-01 4.056E-01 266.27 208.73 214.60 30 4.583E-01 4.450E-01 264.98 246.72 254.94 31 5.000E-01 4.842E-01 304.44 229.26 237.04 32 5.417E-01 5.232E-01 323.90 236.52 245.14 33 5.833E-01 5.619E-01 343.83 283.48 293.97 34 6.250E-01 6.005E-01 358.05 283.25 294.78 35 6.667E-01 6.389E-01 376.25 255.69 267.03 36 7.083E-01 6.770E-01 391.97 254.54 266.47 37 7.500E-01 7.150E-01 403.69 253.64 266.43 38 8.125E-01 7.716E-01 428.63 260.60 274.46 39 8.750E-01 8.278E-01 447.34 261.24 276.33 40 9.375E-01 8.835E-01 463.55 263.38 278.87 41 1.000E+00 9.388E-01 481.75 262.50 279.70 42 1.062E+00 9.936E-01 496.23 255.25 273.22 43 1.125E+00 1.048E+00 512.95 255.65 272.94 44 1.188E+00 1.102E+00 527.41 255.70 270.05 45 1.250E+00 1.156E+00 541.15 250.50 270.79 46 1.312E+00 1.209E+00 550.86 239.87 260.34 47 1.375E+00 1.262E+00 562.85 237.73 259.39 48 1.438E+00 1.314E+00 574.32 229.43 252.16 49 1.500E+00 1.366E+00 583.81 224.24 246.40 50 1.625E+00 1.469E+00 602.27 207.76 229.24

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19 March 2003 — Due: Wednesday 26 March 2003 Well Test Data Functions (continued)

Well Test Data Functions: (continued)

Point ∆t, hr ∆te, hr ∆p, psi ∆p'(∆t), psi ∆p'(∆te), psi 51 1.750E+00 1.571E+00 615.52 197.56 219.81 52 1.875E+00 1.671E+00 629.25 182.27 202.80 53 2.000E+00 1.769E+00 642.23 169.91 191.06 54 2.250E+00 1.962E+00 659.71 151.22 169.07 55 2.375E+00 2.056E+00 667.19 139.88 165.14 56 2.500E+00 2.149E+00 673.44 134.31 153.47 57 2.750E+00 2.332E+00 684.65 116.75 144.48 58 3.000E+00 2.509E+00 695.11 114.03 137.61 59 3.250E+00 2.682E+00 704.06 107.18 122.87 60 3.500E+00 2.849E+00 709.80 96.38 112.89 61 3.750E+00 3.013E+00 719.50 82.69 106.94 62 4.000E+00 3.172E+00 725.97 78.06 100.59 63 4.250E+00 3.328E+00 730.20 72.36 93.66 64 4.500E+00 3.479E+00 731.95 69.85 87.79 65 4.750E+00 3.626E+00 733.70 58.90 78.15 66 5.000E+00 3.770E+00 736.45 48.95 73.95 67 5.250E+00 3.911E+00 739.69 41.46 63.32 68 5.500E+00 4.048E+00 742.64 39.18 55.68 69 5.750E+00 4.182E+00 744.70 38.76 52.89 70 6.000E+00 4.312E+00 747.19 41.04 50.41 71 6.250E+00 4.440E+00 748.94 37.92 51.84 72 6.750E+00 4.686E+00 748.02 32.48 49.57 73 7.250E+00 4.922E+00 750.78 26.91 46.61 74 7.750E+00 5.148E+00 753.01 23.84 40.42 75 8.250E+00 5.364E+00 754.52 21.86 38.54 76 8.750E+00 5.570E+00 756.27 26.74 35.80 77 9.250E+00 5.769E+00 757.51 24.21 33.94 78 9.750E+00 5.960E+00 758.52 23.62 39.45 79 1.025E+01 6.143E+00 760.01 22.26 38.29 80 1.075E+01 6.319E+00 760.75 21.34 36.77 81 1.125E+01 6.488E+00 761.76 20.27 35.63 82 1.175E+01 6.652E+00 762.50 19.60 35.38 83 1.225E+01 6.809E+00 763.51 19.01 34.60 84 1.275E+01 6.961E+00 764.25 18.93 34.14 85 1.325E+01 7.107E+00 765.07 17.59 33.75 86 1.375E+01 7.249E+00 765.50 17.89 33.37 87 1.450E+01 7.452E+00 766.50 17.21 32.52 88 1.525E+01 7.645E+00 767.25 16.33 32.82 89 1.600E+01 7.829E+00 767.99 15.40 31.43 90 1.675E+01 8.004E+00 768.74 15.19 31.90 91 1.750E+01 8.172E+00 769.48 15.04 31.76 92 1.825E+01 8.332E+00 769.99 14.50 31.42 93 1.900E+01 8.484E+00 770.73 13.35 31.89 94 1.975E+01 8.631E+00 770.99 13.31 30.77 95 2.050E+01 8.771E+00 771.49 13.13 30.66 96 2.125E+01 8.905E+00 772.24 12.41 31.19 97 2.225E+01 9.076E+00 772.74 11.84 30.66 98 2.325E+01 9.239E+00 773.22 11.72 30.90 99 2.425E+01 9.392E+00 773.48 10.67 30.07

100 2.525E+01 9.539E+00 773.99 11.14 30.15 101 2.625E+01 9.678E+00 774.49 11.10 30.34 102 2.725E+01 9.811E+00 774.73 10.04 29.98 103 2.850E+01 9.968E+00 775.23 10.06 29.97

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19 March 2003 — Due: Wednesday 26 March 2003 Cartesian Plot — Early-Time Pressure Data

Cartesian Plot: Early-Time Pressure Data

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19 March 2003 — Due: Wednesday 26 March 2003 Semilog Plot — "MDH" Plot — Pressure Data versus Shut-In Time

Semilog Plot: "MDH" Plot — Pressure Data versus Shut-In Time

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19 March 2003 — Due: Wednesday 26 March 2003 Semilog Plot — "Horner" Plot — Pressure Data versus Horner Time

Semilog Plot: "Horner" Plot — Pressure Data versus Horner Time

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19 March 2003 — Due: Wednesday 26 March 2003 Semilog Plot — "Agarwal" Plot — Pressure Data versus Effective Shut-In Time

Semilog Plot: "Agarwal" Plot — Pressure Data versus Effective Shut-In Time

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19 March 2003 — Due: Wednesday 26 March 2003 Log-log Plot — Pressure Drop and Pressure Drop Derivative Data versus Shut-In Time (1 inch x 1 inch)

Log-log Plot: Pressure Drop and Pressure Drop Derivative Data versus Shut-In Time (1 inch x 1 inch)

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19 March 2003 — Due: Wednesday 26 March 2003 Log-log Plot — Pressure Drop and Pressure Drop Derivative Data versus Effective Shut-In Time (1 inch x 1 inch)

Log-log Plot: Pressure Drop and Pressure Drop Derivative Data versus Effective Shut-In Time (1 inch x 1 inch)

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19 March 2003 — Due: Wednesday 26 March 2003 Log-log Plot — Pressure Ratio and Pressure Integral Ratio Data Functions versus Shut-In Time (1 inch x 1 inch)

Log-log Plot: Pressure Ratio and Pressure Integral Ratio Data Functions versus Shut-In Time (1 inch x 1 inch)

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19 March 2003 — Due: Wednesday 26 March 2003 Log-log Plot — Pressure Ratio and Pressure Integral Ratio Data Functions versus Effective Shut-In Time (1 inch x 1 inch)

Log-log Plot: Pressure Ratio and Pressure Integral Ratio Data Functions versus Effective Shut-In Time (1 inch x 1 inch)

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19 March 2003 — Due: Wednesday 26 March 2003 Late-Time Cartesian Plot ("Muskat Plot") — Pressure Buildup Case

Late-Time Cartesian Plot ("Muskat Plot"): Pressure Buildup Case

Documents

P324 03A Hwk 07 BorPBU - Texas A&M University · Bourdet, D.P.,Ayoub, J.A., and Pirard, Y.M.: "Use of Pressure Derivative in Well Test Interpreta -tion," SPEFE (June 1989) 293-302