Tarek Ahmed. Reservoir engineering handbook

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Table 9-3

Results of Example 9-14

Total Production Rate is 1500 STB/day

Time Oil Rate Water Rate Water Cut Cum. Oil Oil Rec.

days STB/day STB/day Fraction MSTB %

80. 742.1 757.9 0.505 67.98 1.56

260. 734.6 765.4 0.510 160.34 3.69

275. 727.1 772.9 0.515 171.31 3.94

290. 719.6 780.4 0.520 182.16 4.19

770. 541.8 958.2 0.639 481.00 11.07

785. 537.6 962.4 0.642 489.10 11.26

800. 533.5 966.5 0.644 497.13 11.44

1295. 423.5 1076.5 0.718 732.08 16.85

1310. 420.8 1079.2 0.719 738.42 17.00

1325. 418.1 1081.9 0.721 744.71 17.14

2060. 315.7 1184.3 0.790 1011.42 23.28

2075. 314.0 1186.0 0.791 1016.14 23.39

2090. 312.4 1187.6 0.792 1020.84 23.50

2105. 310.8 1189.2 0.793 1025.51 23.60

2120. 309.2 1190.8 0.794 1030.16 23.71

2135. 307.6 1192.4 0.795 1034.79 23.82

2705. 255.5 1244.5 0.830 1194.54 27.50

3545. 200.1 1299.9 0.867 1384.48 31.87

3650. 194.4 1305.6 0.870 1405.19 32.34

where

= distance between the WOC and the horizontal well

= distance between the GOC and the horizontal well

= critical oil rate, STB/day

r=density, lb/ft

= horizontal permeability, md

h = oil column thickness, ft

= half distance between two lines of horizontal wells

(half drainage length perpendicular to the horizontal well)

L = length of the horizontal well

= dimensionless function

¢¢

(9 - 43)

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Joshi (1991) correlated the dimensionless function F with the parame-

ter a≤:

Example 9-15

A 1,640-ft.-long horizontal well is drilled in the top elevation of the

pay zone in a water-drive reservoir. The following data are available:

h = 50 ft k

= 60 md k

= 15 md B

= 1.1 bbL/STB

= 0.73 cp r

= 0.3 ft D

= 50 ft r

= 47.5 lb/ft

= 63.76 lb/ft

= 1320 ft

Using the Chaperson method, calculate:

a. The oil critical flow rate for the horizontal well.

b. Repeat the calculation assuming a vertical well with h

= 15¢ and

= 1489 ft.

Solution

Critical rate for a horizontal well:

Step 1. Solve for a≤ by applying Equation 9-43.

Step 2. Solve for the dimensionless function q

by applying Equation 9-44.

= 4.6821

Step 3. Calculate the critical rate from Equation 9-41.

= 0.0783

1640 4.6821

1320

(63.76 47.5)

.73 1.1

138.4 STB/day

¢¢

a =

1320

= 13.20

.. .()=+

¢¢

3 9624955 0 0616438 0 000504

aa (9 - 44)

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Critical rate for a vertical well:

Step 1. Solve for a≤ by using Equation 9-19.

a≤ = 14.89

Step 2. Solve for q

by applying Equation 9-18.

= 0.8616

Step 3. Calculate the critical rate for the vertical well from Equation 9-17.

The ratio of the two critical oil rates is

This rate ratio clearly shows the critical rate improvement in the case

of the horizontal well over that of the vertical well.

Efros’ Method

Efros (1963) proposed a critical flow rate correlation that is based on

the assumption that the critical rate is nearly independent of drainage

radius. The correlation does not account for the effect of the vertical per-

meability. Efros developed the following two relationships that are

designed to calculate the critical rate in oil-water and gas-oil systems:

Water coning

= 0.0783

()h(hD)L

+(h )

---

[]

(9 - 45)

Rate ratio =

138.4

14@

= 0.0783

60 (50 15

)

(0.73) (1.1)

(63.76 47.5) (0.8616)

= 10 STB/day

¥-

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Gas coning

where L = length of the horizontal well, ft

= half distance between two lines of horizontal wells

r=density, lb/ft

h = net pay thickness

k = permeability, md

Example 9-16

Using the horizontal well data given in Example 9-15, solve for the

horizontal well critical flow rate by using Efros’ correlation.

Solution

Step 1. Calculate the critical oil flow rate by applying Equation 9-45 to give

Karcher’s Method

Karcher (1986) proposed a correlation that produces a critical oil flow

rate value similar to that of Efros’ equation. Again, the correlation does

not account for the vertical permeability.

Water coning

= 0.0783

()

(h B)

)

(1 / 24)

¥-

(9 - 47)

= 0.0783

60 (63.76 47.5) (1640)

(1.1) (0.73) 1320 +

15 STB/day

1320

= 0.0783

( ) [h

(h D )]

+y+(h/3)

[]

(9 - 46)

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where B = h - D

= distance between WOC and horizontal well, ft

Gas coning

where T = h - D

= distance between GOC and horizontal well, ft

Example 9-17

Resolve example by using Karcher’s method.

Solution

Joshi’s Method

Joshi (1988) suggests the following relationships for determining the

critical oil flow rate in horizontal wells by defining the following para-

meters:

• Horizontal well drainage radius r

where A is the horizontal well drainage area in acres.

43,560 A

= 0.0783

60 (63.76 47.5)

(50)

1640

(1.1) (0.73) (2 1320)

1320

(1/24) 15 STB/day

¥-

= 0.0783

()

(h T)

)

(1/24)

¥-

(9 - 48)

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Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 622

• Half the major axis of drainage ellipse a

• Effective wellbore radius r¢

For oil-water systems:

For oil-gas systems:

where r=density, lb/ft

= horizontal density, md

= distance between the horizontal well and the WOC, ft

= distance between the horizontal well and GOC, ft

= wellbore radius, ft

Example 9-18

Resolve Example 9-17 by applying Joshi’s approach.

Solution

Step 1. Solve for a by applying Equation 9-49

a = (1640 / 2) 0.5 0.25 +

(2 1489 / 1640)

= 1,606 ft

0.5

[]

oeh

= 0.0246

()

)

/r )

[]

(9 - 52)

eh w

= 0.0246

()

)

l(r/r)

[]

(9 - 51)

[]

1 1 L / (2a)

h /(2r )

h/L

[]

(9 - 50)

aL rL

=++

[]

(/) . . ( /)

2 0 5 0 25 2

(9 - 49)

Gas and Water Coning 623

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Step 2. Calculate r¢

from Equation 9-50.

Step 3. Estimate the critical flow rate from Equation 9-51.

HORIZONTAL WELL BREAKTHROUGH TIME

Several authors have proposed mathematical expressions for determin-

ing the time to breakthrough in horizontal wells. The following two

methodologies are presented in the following sections:

• The Ozkan-Raghavan Method

• Papatzacos’ Method

The Ozkan-Raghavan Method

Ozkan and Raghavan (1988) proposed a theoretical correlation for cal-

culating time to breakthrough in a bottom-water-drive reservoir. The

authors introduced the following dimensionless parameters:

= dimensionless vertical distance = D

/h (9 - 54)

where L = well length, ft

= distance between WOC and horizontal well

H = formation thickness, ft

= vertical permeability, md

= horizontal permeability, md

L dimensionless well length = [L/(2h)] k / k

Dvh

= (9 - 53)

= 0.0246

(63.76 47.5) (60)

(50 50)

(0.73) (1.1) l (1489 / 357)

= 52 STB/day

[]

+- ¥

[]

1489

1 1 1640 / (2 1606)

50/(2 0.3)

= 357 ft

50/1640

1640

2 1606()

[]

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Ozkan and Raghavan expressed the water breakthrough time by the

following equation:

with the parameter f

as defined by:

=f(1 - S

- S

) (9 -56)

where t

= time to breakthrough, days

= vertical permeability, md

= horizontal permeability, md

f=porosity, fraction

= connate water saturation, fraction

= residual oil saturation, fraction

= oil flow rate, STB/day

= sweep efficiency, dimensionless

Ozkan and Raghavan graphically correlated the sweep efficiency with

the dimensionless well length L

and dimensionless vertical distance

as shown in Figure 9-24.

Example 9-19

A 1,640-foot-long horizontal well is drilled in a bottom-water-drive

reservoir. The following data are available:

h = 50 ft k

= 60 md k

= 15 md B

= 1.1 bbl/STB

= 0.73 cp r

= 0.3 ft r

= 47.5 lb/ft

= 63.76 lb/ft

= 1 f=15% S

= 0.25 S

= 0.3

The well is producing at 1000 STB/day. Calculate time to break-

through.

Solution

Step 1. Solve for L

by using Equation 9-53.

= 15 / 60 = 8.2

1640

250()

5.615

(

)

(9 - 55)

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Step 2. Calculate the parameter f

from Equation 9-56.

= 0.15 (1 - 0.25 - 0.30) = 0.0675

Step 3. Estimate the sweep efficiency E

from Figure 9-24.

@ 21

Step 4. Solve for time to breakthrough by applying Equation 9-55.

0.0675

(50)

5.615 1000 1.1

(60 / 15) = 114.7 days

¥¥

626 Reservoir Engineering Handbook

Figure 9-24. Sweep efficiency for horizontal and vertical wells. (After Ozkan, E.,

and Raghavan, R., courtesy SPE, 1988.)

Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 626

Papatzacos’ Method

Papatzacos et al. (1989) proposed a methodology that is based on

semianalytical solutions for time development of a gas or water cone and

simultaneous gas and water cones in an anisotropic, infinite reservoir

with a horizontal well placed in the oil column.

Water coning

Step 1. Calculate the dimensionless rate q

from the following expression:

where r=density, lb/ft

= vertical permeability, md

= horizontal permeability, md

h = oil zone thickness, ft

L = length of horizontal well

Step 2. Solve for the dimensionless breakthrough time t

DBT

by applying

the following relationship:

Step 3. Estimate the time to the water breakthrough t

by using the

water and oil densities in the following expression:

where t

= time to water breakthrough as expressed in days

= oil density, lb/ft

= water density, lb/ft

Gas coning

Step 1. Calculate the dimensionless flow rate q

= 20,333.66

/ Lh ( )

[]

(9 - 60)

o DBT

vw o

22 758 528,.

()

(9 - 59)

DBT

=1 (3

ln (9 - 58)

= 20,333.66

/ L h ( )

[]

(9 - 57)

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