7 HW Solution - PEGN 423 – Petroleum Reservoir Engineering I – Fall 2016 PDF

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PEGN 423 – Petroleum Reservoir Engineering I – Fall 2016 HOMEWORK #7 Decline Curve Analysis for Unconventional Gas Reservoirs Assignment: Thursday, November 12, 2016 Due date: Thursday, November 22, 2016 NAME: Solut Solution ion Problem #1: Follow these steps to modify your Excel spreadsheet built in Homework #6, Problem #5. The modification of your spreadsheet consists of the implementation of the Modified Hyperbolic Decline method for low permeability unconventional reservoirs. (50 points) 1. Set up a cell for the predetermined minimum nominal decline rate, Dmin. 2. Set up a cell to calculate the time to reach the minimum nominal decline rate, t*, using, Di 1 t* Dmin bDi this calculation should use the values of b and Di obtained from the nonlinear regression to the production data using solver. 3. Set up a cell to calculate the production rate at the time of the minimum nominal decline rate, t*, using the hyperbolic decline curve equation, qi qt* 1 1 bDit * b this value should be used as the initial flow rate, qi, when switching to the exponential decline period. 4. In the column that calculates the production rate using the hyperbolic decline curve equation, include the following IF statement,

if elseif

t t* t

t*

then then

qt '

qi 1 bDi t

qt ' qt*e

1

(hyperbolic decline) b

Dmin t t*

(exponential decline)

5. Initialize the solver to optimize the contents of the cells that contain b, Di and qi by maximizing the cell that contains R2. 1

Note: The deliverables for problem #1 is your developed Excel spreadsheet using the data from problem #2. Table 7.1 Important Parameters for Decline Analysis Hyperbolic -Decline Exponent b (unitless)

Initial Nominal Decline Rate Di (1/month)

Initial Gas Production Rate qi (Mscf/Day)

Minimum Nominal Decline Rate Dmin (1/Month)

1.6196

0.3732

1915.16

0.0050

Average Gas Production Rate at Time to Reach the Minimum Nominal Production Rate time of Minimum Nominal Decline Rate, Decline Rate q avg t* (months) qt* (Mscf/Day) (Mscf/D) 121.83

432.54

133.58

Table 7.2 Production da ta for problem #6

Year

Time, t (months)

Time, t (months)

Gas production rate, q g (Mscf/day)

Estima ted Gas Production Rate, Type of Decline qt' (Mscf/Day)

Errors Squared, SE

Total Errors Squa red, ST

2009 2009

JAN FEB

0 1

1229.07 1420.79

1915.16 1430.31

Hyperbolic Hyperbolic

470726.23 90.64

2009

MAR

2

1185.07

1174.08

Hyperbolic

120.69

566301.99

2009 2009 2009

APR MAY JUN

3 4 5

1009.13 926.52 814.80

1011.20 896.71 810.93

Hyperbolic Hyperbolic Hyperbolic

4.30 888.78 14.97

332456.48 244016.63 146123.01

2009 2009

JUL AUG

6 7

732.97 682.26

743.78 689.47

Hyperbolic Hyperbolic

116.77 52.01

90258.42 62360.27

2009 2009

SEP OCT

8 9

666.46 582.45

644.46 606.41

Hyperbolic Hyperbolic

484.04 574.10

54718.75 22473.13

2009 2009

NOV DEC

10 11

553.63 535.47

573.73 545.30

Hyperbolic Hyperbolic

404.17 96.64

14662.88 10594.67

2010 2010 2010

JAN FEB MAR

12 13 14

528.16 491.21 458.29

520.28 498.07 478.18

Hyperbolic Hyperbolic Hyperbolic

62.02 47.03 395.44

9143.26 3442.21 663.08

2010 2010

APR MAY

15 16

467.80 448.94

460.24 443.96

Hyperbolic Hyperbolic

57.18 24.77

1243.30 268.97

2010 2010

JUN JUL

17 18

425.87 409.90

429.11 415.50

Hyperbolic Hyperbolic

10.52 31.35

44.48 512.55

2010 2010

AUG SEP

19 20

400.39 388.33

402.96 391.37

Hyperbolic Hyperbolic

6.61 9.22

1033.60 1954.49

2010 2010 2010

OCT NOV DEC

21 22 23

370.65 369.60 365.45

380.61 370.59 361.24

Hyperbolic Hyperbolic Hyperbolic

99.17 0.98 17.75

3830.32 3961.39 4501.02

2011 2011

JAN FEB

24 25

345.90 348.46

352.48 344.25

Hyperbolic Hyperbolic

43.25 17.71

7506.42 7069.38

2011 2011

MAR APR

26 27

338.39 328.90

336.51 329.21

Hyperbolic Hyperbolic

3.53 0.10

8864.15 10741.17

2011 2011 2011

MAY JUN JUL

28 29 30

327.10 320.50 312.23

322.31 315.78 309.58

Hyperbolic Hyperbolic Hyperbolic

22.92 22.29 7.01

11117.51 12552.87 14474.40

2011 2011

AUG SEP

31 32

308.39 303.73

303.69 298.09

Hyperbolic Hyperbolic

22.06 31.82

15413.13 16591.92

2011 2011

OCT NOV

33 34

290.07 300.97

292.75 287.65

Hyperbolic Hyperbolic

7.17 177.39

20297.59 17310.56

2011 2012

DEC JAN

35 36

287.61 278.06

282.78 278.12

Hyperbolic Hyperbolic

23.33 0.00

21004.59 23863.95

2012 2012 2012

FEB MAR APR

37 38 39

274.83 274.03 270.70

273.65 269.37 265.26

Hyperbolic Hyperbolic Hyperbolic

1.38 21.69 29.57

24872.32 25125.30 26192.06

2012 2012

MAY JUN

40 41

264.87 260.10

261.31 257.51

Hyperbolic Hyperbolic

12.66 6.70

28113.10 29735.42

2012 2012

JUL AUG

42 43

256.35 250.35

253.85 250.33

Hyperbolic Hyperbolic

6.23 0.00

31042.78 33193.05

2012 2012

SEP OCT

44 45

245.30 242.55

246.93 243.65

Hyperbolic Hyperbolic

2.65 1.20

35058.67 36096.05

2012 2012 2013

NOV DEC JAN

46 47 48

241.33 235.90 235.68

240.48 237.41 234.45

Hyperbolic Hyperbolic Hyperbolic

0.73 2.29 1.51

36561.11 38667.14 38753.71

2013 2013

FEB MAR

49 50

233.43 225.58

231.58 228.80

Hyperbolic Hyperbolic

3.42 10.39

39644.64 42832.28

2013

APR

51

224.07

226.11

Hyperbolic

4.16

43459.58

2

634460.67 976638.84

Problem #2: Table 2.1 presents production data from a tight gas reservoir well located in the Grad Valley field, producing from the Williams Fork formation of the Piceance Basin. Using these production data: 2.1.- Use the spreadsheet that implements the Modified Hyperbolic Decline method with nonlinear regression from problem #1 to determine: (10 points)     

The hyperbolic exponent b Initial nominal decline rate Di Apparent initial oil rate qi Time to reach the minimum decline rate t* Production rate at the time of minimum decline rate qt*

Use as minimum decline rate Dmin = 6%/year. Solution: Hyperbolic Decline Exponent b (unitless)

Initial Nominal Decline Rate Di (1/month)

Initial Gas Production Rate qi (Mscf/Day)

1.6196

0.3732

1915.16

Minimum Nominal Decline Rate Dmin (1/Month)

Time to Reach the Minimum Nominal Decline Rate t* (months)

Average Gas Production Rate qavg (Mscf/D)

Production Rate at time of Minimum Nominal Decline Rate, qt* (Mscf/Day)

0.0050

121.83

432.54

133.58

2.2.- Estimate the expected production rate in the 200th month. (10 points) Solution: at time t=200th month, the production rate falls into an exponential decline pattern, thus, 𝑞𝑖 = 133.584 𝐷=

𝑀𝑠𝑐𝑓 𝐷𝑎𝑦

0.06 0.005 = 12 𝑚𝑜𝑛𝑡ℎ

𝑞 = 𝑞𝑖 𝑒 −𝐷(𝑡−𝑡 ) = 133.584 ∗

𝑴𝒔𝒄𝒇 𝑀𝑠𝑐𝑓 −( 0.005 ∗(200−121.833)𝑚𝑜𝑛𝑡ℎ𝑠) 𝑒 𝑚𝑜𝑛𝑡ℎ = 𝟗𝟎. 𝟑𝟔𝟖 𝑑𝑎𝑦 𝑫𝒂𝒚 3

2.3.- Estimate the expected cumulative production in the 200th month. (10 points) As t=200 month > t* =121.83 month, the cumulative production can be divided into two parts 𝑁𝑝1 and 𝑁𝑝2 .

𝑁𝑝1 represents the cumulative production from in initial time to t*, the hyperbolic equation should be applied. 𝑁𝑝1 = =

𝑞𝑖𝑏 1−𝑏 (𝑞1−𝑏 − 𝑞𝑡∗ ) (1 − 𝑏)𝐷𝑖 𝑖

1915.161.6196 (1915.161−1.6196 − 133.581−1.6196) 1 𝑚𝑜𝑛𝑡ℎ 1 (1 − 1.6196) ∗ 0.3732 ∗ 𝑚𝑜𝑛𝑡ℎ 30.44 𝑑𝑎𝑦

=1,060,454 MSCF 𝑁𝑝2 represents the cumulative production from in t* to 200th months, the exponential equation should be applied. 𝑁𝑝2 =

𝑞200𝑡ℎ−𝑞𝑡∗ 𝐷𝑚𝑖𝑛

=

133.58 𝑀𝑠𝑐𝑓/𝑑𝑎𝑦−90.367𝑀𝑠𝑐𝑓/𝑑𝑎𝑦 1

=263089.3

1 𝑚𝑜𝑛𝑡ℎ

0.005 𝑚𝑜𝑛𝑡ℎ∗ 30.44 𝑑𝑎𝑦𝑠

𝑵𝒑 = 259,293.14 + 1,046,475.78 =1323543.7 𝑴𝒔𝒄𝒇 2.4.- What is the remaining life of the well if the economic limit is 75 Mscf/day? (10 points) As 𝑞=75 Mscf/day < 𝑞𝑡∗ , we need to find the production time for hyperbolic decline (𝑡1 )

and the production time for exponential decline (𝑡2 )

1 𝑞𝑖 𝑏 1 1915.16 ) 𝑡1 = [( ) − 1] = [( 𝑏𝐷𝑖 𝑞𝑡∗ 1.6196 ∗ 0.3732 133.58 𝑡2 =

1

𝐷𝑚𝑖𝑛

ln (

𝑞𝑡∗ )= 𝑞𝑒𝑙

1

1 𝑚𝑜𝑛𝑡ℎ𝑠 1 0.005 𝑚𝑜𝑛𝑡ℎ ∗ 30.44𝑑𝑎𝑦𝑠

1.6196

ln (

− 1] = 121.83 𝑚𝑜𝑛𝑡ℎ𝑠

133.58 ) = 115.45 𝑚𝑜𝑛𝑡ℎ𝑠 75

𝑇𝑜𝑡𝑎𝑙 𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝑙𝑖𝑓𝑒 = 𝑡1 + 𝑡2 − 𝑡𝑝𝑟𝑜𝑑𝑢𝑐𝑖𝑛𝑔 = 121.833 + 115.45 − 51 = 186.3 𝑚𝑜𝑛𝑡ℎ𝑠 2.5.- Make a plot of gas production rate versus time, including the data from Table 2.1., the estimated gas production rate using the modified hyperbolic decline method, an 4

extrapolation of the estimated gas production rate until t = 200 months. These data should be calculated in your spreadsheet from problem #1. (10 points)

Production Rate， Mscf

10000.00 Production data Prediction of Production rate Extroplation

1000.00

100.00

10.00 0

50

100

150

200

250

Time, Month Figure 1: Gas Production Forecasted Using Modified Hyperbolic Decline

5...