Pdf (1)

University of Regina

Petroleum Systems Engineering

Production Engineering Chapter 2

Dr. F. Torabi, P. Eng. Petroleum Systems Engineering Faculty of Engineering and Applied Science

Petroleum Production Operations, Dr. Farshid Torabi

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Well Inflow Performance 2-1 Introduction

The process of flow from the reservoir and into the well sandface can be expressed by a simple expression of Darcy’s law (1856) in radial coordinates:

q

kA dp  dr

(2-1)

Wells drilled in oil reservoirs drain a porous medium of porosity ϕ, net thickness h, and permeability k. Where A is a radial area at a distance r and is given by 𝐴 = 2𝜋𝑟. (This expression assumes a single-phase fluid flowing and saturating the reservoir.)

dp  q  dr   q 

k , A,

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Well Inflow Performance 2-2 Steady-State Well Performance

In a well, within a reservoir, the area of flow at any distance, r, substituting 2𝜋𝑟ℎ in Eq. 2-1:

2krh dp  dr q  cte p q r dr dp  pwf 2kh rw r Finally , q r p  pwf  ln 2kh rw q

(2-2)

(2-3)

(2-4)

Much of the pressure drop occurs in the near-wellbore region, so it is extremely important in well production. Petroleum Production Operations, Dr. Farshid Torabi

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Well Inflow Performance Van Everdingen and Hurts (1949) quantified the condition of the near-wellbore region with the introduction of the concept of the skin effect. The skin effect results in an additional steady-state pressure drop:

ps 

q s 2kh

(2-5)

Which can be added to the pressure drop in the reservoir p  pwf 

 q  r  ln  s  2kh  rw 

(2-6)

 q  re  ln  s  2kh  rw 

(2-7)

At constant-pressure outer boundary 𝑝𝑒 (at 𝑟𝑒 ), the well operates at steady-state condition pe  pwf 

In oilfield units: pe  pwf 

 141.2qB  re  ln  s  kh  rw 

Petroleum Production Operations, Dr. Farshid Torabi

(2-8) 4

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Well Inflow Performance kh p  p  q e

wf

(2-9)

 r  141.2qB  ln e  s   rw 

Two important concept applied to all other types of flow 

The effective wellbore radius, 𝑟 ′ 𝑤 , can be derived:

pe  pwf 

 141.2qB  re  ln  ln e s  kh  rw 

(2-10)

r  141.2qB   ln es  kh  rwe 

(2-11)

Thus

pe  pwf 

Effective wellbore radius, 𝑟 ′ 𝑤 is defined by:

rw  rwe  s Petroleum Production Operations, Dr. Farshid Torabi

(2-12) 5

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Well Inflow Performance  The productivity index, J, of a well is simply the production rate divided by the pressure difference. Thus:

J

q kh  pe  pwf 141.2 B ln re rw   s 

(2-13)

One of the main purposes of “production engineering” and job tasks for production engineer is to maximize the productivity index in a cost-effective manner,  Increasing the flow rate in for a given driving force(drawdown)  Minimizing the drawdown for a give rate • Decreasing the skin effect (through matrix stimulation and removal of nearwellbore damage • Superposition of a negative skin effect from an induced hydraulic fracture

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Well Inflow Performance Example 2.1 Assume that a well in the reservoir described in Appendix A has a drainage area equal to 640 acres (𝑟𝑒 = 2980 𝑓𝑡) and is producing at steady-state with an outer boundary (constant) pressure equal to 5651 psi. calculate the steady-state production rate if the flowing bottomhole pressure is equal to 4500 psi. use a skin effect equal to +10. Describe two mechanisms to increase the flow rate by 50%. Show calculations. k H  8.2md kV  0.9md h  53 ft pi  5651 psi pb  1323 psi co  1.4  10 5 psi 1 cw  3  10  6 psi 1 c f  2.8  10  6 psi 1

  1.03cP B  1.2resbbl / STB Rs  150 SCF / STB

  0.19 S w  0.34 API   28 rw  0.328 ft 7 7 8well 

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Well Inflow Performance Solution: Eq. (2-9) and rearrangement.

q

kh pe  pwf 

 8.2535651  4500 q  150STB / d 141.21.21.03ln 2980 / 0.328  10

 r  141.2qB  ln e  s   rw 

To increase the production rate by 50%,  First possibility is to increase the drawdown, 𝑝𝑒 − 𝑝𝑤𝑓 , by 50%. Therefore

5651  p 

wf 2

 1.55651  4500  pwf  3925 psi

 Second possibility is to reduce the skin effect. In this case  2980   2980   s2    ln  10  1.5  s  3.6  ln   0.328   0.328 Petroleum Production Operations, Dr. Farshid Torabi

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Well Inflow Performance Example 2.2 Demonstrate the effect of drainage area on oilwell production rate by calculating the ratios of production rates from 80-, 160-, and 640-acre drainage areas to that obtained from a 40-acre drainage area. The well radius is 0.328 ft. Solution: Assuming that the skin effect is zero (this would result in the most pronounced difference in the production rate), the ratios of the production rates :

ln re rw 40 q  ln re rw  q40 For calculating the drainage radius, it will be assume that the well is in the center of a circular drainage area. Thus

re 

 A43,560 

According to the results shown in Table 2-1, these ratios indicate that the drainage area assigned to a well has a small impact on the production rate. Petroleum Production Operations, Dr. Farshid Torabi

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Well Inflow Performance 2-3 Transient Flow of Undersaturated Oil

The diffusivity equation describes the pressure profile in an infinite-acting, radial reservoir, with a slightly compressible and constant viscosity fluid (undersaturated oil or water).

 2 p 1 p  ct p   r 2 r r k t

(2-20)

Its generalized solution is

pr ,t  pi 

q Ei ( x) 4kh

(2-21)

Where 𝐸𝑖 𝑥 is the exponential integral and x is given by

x

 ct r 2

(2-22)

4kt

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Well Inflow Performance For 𝑥 < 0.01 (i.e., for large values of time or for small distances, such as at the wellbore) the exponential integral 𝐸𝑖 (𝑥) can be approximated by −ln(𝛾𝑥), where γ is Euler’s constant and is equal to 1.78. Therefore, Eq. (2-21), at the wellbore and shortly after production, can be approximated by 𝑝𝑟𝑤,𝑡 ≡ 𝑝𝑤𝑓

pwf  pi 

q 4kt ln 4kh  ct rw2

(2-23)

 162.6qB  k  log t  log   3 . 23 2 kh  ct rw  

(2-24)

Finally, pwf  pi 

Above equation is in oilfield units. This expression is often known as the pressure drawdown equation describing the declining flowing bottomhole pressure, 𝑝𝑤𝑓 , while the well is flowing at a constant rate q. Petroleum Production Operations, Dr. Farshid Torabi

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Well Inflow Performance The more common constant-bottomhole-pressure situation results in a similar expression:

kh pi  pwf    k  log t  log  q  3 . 23 162.6 B   ct rw2 

1

(2-25)

Where the time, t, must be in hours. Including skin factor will be:

kh pi  pwf    k  q log t  log  3.23  0.87 s  2  162.6 B   ct rw 

Petroleum Production Operations, Dr. Farshid Torabi

1

(2-26)

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Well Inflow Performance Example 2-3 Using the well and reservoir variables in Appendix A (data given below), develop production rate profile for 1 year assuming that no boundary effects emerge. Do this in increments of 2 months and use a flowing bottomhole pressure equal to 3500 psi. k H  8.2md kV  0.9md h  53 ft pi  5651 psi pb  1697 psi T  220  F co  1.4  10 5 psi 1 cw  3  10  6 psi 1 c f  2.8  10  6 psi 1 ct  1.29  10 5 psi 1 Rs  250 SCF / STB

  0.19 S w  0.34

 o  28 API  g  0.71

rw  0.328 ft 7 7 8wellbore  p sep  100 psi

Tsep  100  F

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Well Inflow Performance Solution: From Eq. (2-25) and substitution of the appropriate variables in Appendix A, the well production rate is given by

 8.2535651  3500 log t  log 8.2 q  3 . 23  162.61.21.03  0.191.031.29 105 0.3282 

1



4651 log t  4.25 (2-27)

For 𝑡 = 2 months, from Eq. (2-27), the production rate 𝑞 = 627 𝑆𝑇𝐵/𝑑. Figure 2-2 is a rate-decline curve for this oil well for the first year assuming infinite acting behavior. The rate decline is from 627 STB/d (after 2 months) to 568 (STB/d) (after 1 year).

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Well Inflow Performance 2-4 Pseudo-Steady-State Flow

The steady-sate condition implied a constant-pressure outer boundary. For no-flow boundaries, drainage areas can either be described by natural limits such as faults, pinchouts, etc., or can be artificially induced by production of adjoining wells. This condition is often referred to as “pseudo-steady state.” the pressure at the outer boundary is no longer constant but instead declines at a constant rate with time, that is, 𝜕𝑝𝑒 𝜕𝑡 = 𝑐𝑜𝑛𝑠𝑡. The pressure p at any point r in a reservoir of radius 𝑟𝑒 according to the radial diffusivity equation (Dake, 1978)

141.2qB  r r2   ln  2  p  pwf  kh  rw 2re 

(2-28)

At 𝑟 = 𝑟𝑒

141.2qB  re 1   ln   pe  pwf  kh  rw 2  Petroleum Production Operations, Dr. Farshid Torabi

(2-29) 15

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Well Inflow Performance 2-4 Pseudo-Steady-State Flow Since 𝑝𝑒 is not known at any given time, Eq. (2-28) is not useful under pseudosteadystate condition. Therefore, it is suggested to use the average reservoir pressure, 𝑝, for the pseudo-steady-state equation. This is defined as volumetrically weighted pressure as follow:

p



re

rw 2 e

pdV

 r  rw2 h

 

re

rw

pdV

re2 h

(2-30)

Since 𝑑𝑉 = 2𝜋𝑟ℎ∅𝑑𝑟

2 p 2 re



re

rw

prdr

(2-31)

The expression for pressure at any point r, can be substituted from Eq. (2-28) to determine 𝑝:

p  pwf

2 141.2qB re  r r2   2 rw  ln rw  2re2 rdr re kh

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(2-32) 16

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Well Inflow Performance 2-4 Pseudo-Steady-State Flow

The result of integration is

p  pwf 

141.2qB  re 3   ln   kh  rw 4 

(2-33)

Introducing the skin effect and incorporating the term 3 4 into the logarithmic expression leads to the inflow relationship for a no-flow boundary oil reservoir:

p  pwf 

 141.2qB  0.472re  ln  s  kh rw  

(2-34)

Equation (2-35) provides the relationship between the average reservoir pressure, 𝑝, and the rate q for pseudosteady-state.

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Well Inflow Performance 2-4 Pseudo-Steady-State Flow

We must be very careful to not get confused because of similarity of equations: For steady-state:

For Pseudosteady-state

JD 

JD 

1

1

 re   ln   s  rw 

 0.472re   ln   s r w  

Petroleum Production Operations, Dr. Farshid Torabi

(2-36)

(2-37)

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Well Inflow Performance Example 2-4 What would be the average reservoir pressure if the outer boundary pressure is 6000 psi, the flowing bottomhole pressure is 3000 psi, the drainage area is 640 acres, and the well radius is 0.328 ft? what would be the ratio of the flow rates before 𝑞1 and after (𝑞2 ) the average reservoir pressure drops by 1000 psi? Assume that s=0. Solution: A ratio of Eqs. (2-8) and (2-32) results in

pe  pwf p  pwf



ln re rw   1 2 ln re rw   3 4

The drainage area 𝐴 = 640 and therefore 𝑟𝑒 = 2980 𝑓𝑡. Therefore, by substituting variable in Eq. (2-31)

p

6000  30008.36  3000  5913 psi 8.61

q2 4913  3000   0.66 q1 5913  3000 Petroleum Production Operations, Dr. Farshid Torabi

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Well Inflow Performance 2-4-1 Transition to Pseudo-Steady State from Infinite Acting Behavior Earlougher (1977) has shown that the time,𝑡𝑝𝑠𝑠 , at which pseudo-steady state begins is

t pss 

 ct A 0.000264k

t DApss

(2-41)

A: drainage area 𝑡𝐷𝐴𝑝𝑠𝑠 : characteristic value that depends on the drainage shape • Circle/square: 𝑡𝐷𝐴 =0.1 • Well in a 1×2 rectangle: 𝑡𝐷𝐴𝑝𝑠𝑠 =0.3 • Well in a 1×4 rectangle: 𝑡𝐷𝐴𝑝𝑠𝑠 =0.8 • Off-centered wells in irregular patterns have even larger values of 𝑡𝐷𝐴 , implying that the well will “feel” the farther-off boundaries after a significantly longer time. The drainage area approximated by a circle with an equivalent drainage radius, 𝑟𝑒 , Earlougher equation (with 𝑡𝐷𝐴 =0.1); 𝑡𝑝𝑠𝑠 is in hours and other variables in oilfield units

t pss  1200

 ct re2 k

Petroleum Production Operations, Dr. Farshid Torabi

(2-42) 20

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Well Inflow Performance 2-5 Well Draining Irregular Patterns In practice, rarely wells’ drainage area are being considered regular-shaped. Even if they are assigned regular geographic drainage areas, these are distorted after production commence because of: • The presence of natural boundaries, • Lopsided production rates in adjoining wells To account for irregular drainage shapes or assymetrical positioning of a well within its drainage area, a series of “shape” factors was developed by Dietz (1965). For a well at the center of a circle, Eq. (2-33) was applied. The logarithmic expression can be modified into

re 3 1 4re2 ln   ln rw 4 2 4e3 2 rw2

(2-43)

𝜋𝑟𝑒2 : Drainage area of a circle of radius 𝑟𝑒 4𝜋𝑒 3 2 = 56.32 = 1.78 × 31.6 𝛾 = 1.78 is Euler’s constant 𝐶𝐴 = 31.6 is a shape factor for a circle with a well at the center Petroleum Production Operations, Dr. Farshid Torabi

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Well Inflow Performance 2-5 Well Draining Irregular Patterns Dietz (1965) has shown that all well/reservoir configurations that depends on drainage shape and well position have a characteristic shape factor. Therefore, Eq. (2-32) can be generalized for any shape into:

p  pwf 

 141.2qB  1 4A  ln   s 2 kh 2  C r A w   (2-44)

Figure 2-3 (Earlougher, 1977) contains some commonly encountered drainage shape and well positions. Petroleum Production Operations, Dr. Farshid Torabi

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Well Inflow Performance Example 2-5 Assume that two wells in the reservoir, described in Appendix A, each drain 640 acres. Furthermore, assume that 𝑝 = 5651psi (same as 𝑝𝑖 ) and that 𝑠 = 0. the following bottomhole pressure in both is 3500 psi. However, well A is placed at the centre of a square, whereas well B is at the center of the upper right quadrant of a square drainage shape. Calculate the production rates from the two wells at the onset of pseudo-steady state. (This calculation is valid only at very early time. At late time, either drainage shapes will change, if they are artificially induced, or the average reservoir pressure will not decline uniformly within the drainage areas because of different production rates and resulting different rates of depletion. Solution: For well A, from Fig. 2-3 shape factor, 𝐶𝐴 = 30.9 and from Eq. (2-44)

q

8.2532151  640 STB d 141.21.21.030.5ln 464043560 1.7830.90.3282 

For well B, since it is located at the center of the upper right quadrant 𝐶𝐴 = 4.5

q

8.2532151  574 STB d  10%reduction 2 141.21.21.030.5ln 464043560 1.784.50.328 

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Well Inflow Performance Example 2-6 The following data were obtained on a three-well fault block. A map with well locations is shown in Fig. 2-4. (Use properties as for the reservoir described in Appendix A.) k H  8.2md kV  0.9md h  53 ft pi  5651 psi pb  1323 psi co  1.4  10 5 psi 1 cw  3  10  6 psi 1 c f  2.8  10  6 psi 1

  1.03cP B  1.2resbbl / STB Rs  150 SCF / STB

  0.19 S w  0.34 API   28 rw  0.328 ft 7 7 8well 

Each well produced for 200 days since the previous shut-in. At the end of 200 days, the following rates and skin effects were obtained from each well: Petroleum Production Operations, Dr. Farshid Torabi

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Well Inflow Performance

If the bottomhole pressure is 2000 psi for each well, calculate the average reservoir pressure within each drainage area.

Solution: The drainage volume formed by the production rate of each well are related by

VA q A  VB qB VA q A  VC qC Petroleum Production Operations, Dr. Farshid Torabi

(2-46)

(2-47) 25

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Well Inflow Performance If the reservoir thickness were the same throughout, then the ratios of the areas would have sufficed. However, since h varies the volume can be replaced by 𝑉𝑖 = ℎ𝑖 𝐴𝑖 , where i refers to each well. Finally:

AA  AB  AC  Atotal  480acres

(2-48)

From Eqs. (2-39), (2-40) and (2-41); (each square on the fault block map represents 40,000 𝑓𝑡 2 (200 × 200 𝑓𝑡)

   243 ac1.06 10 ft   265 squares  108 ac4.7 10 ft   118 squares

AA  129 ac 5.6 106 ft 2  140 squares AB AC

7

6

2

2

By sketching these areas on the fault block map, the approximate drainage area shown in Fig. 2-4, can be drawn. These describe shape and therefore approximate shape factors. From Fig. (2-3) Well A is shape no. 12 (𝐶𝐴 = 10.8) Well B is shape no. 7 (𝐶𝐴 = 30.9) Well C is shape no. 10 (𝐶𝐴 = 3.3) Petroleum Production Operations, Dr. Farshid Torabi

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Well Inflow Performance From Eq. (2-44) For well A:

p  pwf 

 141.2qB  1 4A  ln   s 2 kh  2 C A rw 

   141.21001.21.03  45.6 106    p  2000  0 . 5 ln  2    2565 psi 2 8.238     1 . 78 10 . 8 0 . 328   For well B:

   141.22001.21.03  41.06 107    p  2000  0 . 5 ln  0    2838 psi 2 8.240     1 . 78 30 . 9 0 . 328   For well C:

   141.2801.21.03  44.7 106    p  2000  0 . 5 ln  5    2643 psi 2 8.236     1 . 78 3 . 3 0 . 328  

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University of Regina

Petroleum Systems Engineering

Well Inflow Performance 2-6 Well Inflow Performance Relationship The bottomhole pressure is a function of the wellhead pressure, which, in turn, depends on production engineering decisions, separator or pipeline pressure, etc. Therefore, what a well will actually produce must be combination of what the reservoir can deliver and what the imposed wellbore hydraulics would allow. It is then useful to present the well production rate as a function of the bottomhole pressure. This type of presentation is known as an “inflow performance relationship” (IPR) curve. Usually, the bottomhole pressure, 𝑝𝑤𝑓 , is graphed on the ordinate and the production rate, q, is graphed on the abscissa. Equations (2-9), (2-26) and (2-44) can be used as IPR curves.

kh pi  pwf    k  log t  log  q  3 . 23 162.6 B   ct rw2 

For transient

 141.2qB  re  ln  s  kh  rw 

For steady-state

 141.2qB  1 4A  ln   s 2 kh  2 C A rw 

For pseudo-steady-state

pe  pwf 

p  pwf 

1

Petroleum Production Operations, Dr. Farshid Torabi

28

University of Regina

Petroleum Systems Engineering

Well Inflow Performance Example 2-7 Using the well and reservoir data in appendix A, construct transient IPR curves for 1, 6, and 24 months. Assume zero skin. Solution: Eq. (2-24): pwf  pi 

q

 162.6qB  k  log t  log   3 . 23 2 kh  c r t w  

2.165651  pwf  log t  4.25

The relationship between q and 𝑝𝑤𝑓 will depends on time, t (in hours). Figure 2-5 is a graph of the transient IPR curves for the three different times. Petroleum Production Operations, Dr. Farshid Torabi

Figure 2-5 Transient IPR curves for Example 2-7.

29

University of Regina

Petroleum Systems Engineering

Well Inflow Performance Example 2-8 Assume that the initial reservoir pressure of the well described in Appendix A is also the constant pressure of the outer boundary, 𝑝𝑒 (steady-state). Draw IPR curves for skin effects equal to 0, 5, 10, and 50, respectively. Use a drainage radius of 2980 ft (A=640 acers).

Solution: Equation (2-9) describes a straight-line relationship between q and 𝑝𝑤𝑓 for any skin effect. For example for s=5

pwf  5651 5.66q s (skin effect)

0

10

50

Multipliers of the rate

3.66

7.67

23.7

Petroleum Production Operations, Dr. Farshid Torabi

Figure 2-6 Steady-state and impact of skin effect for Example 2-8.

30

University of Regina

Petroleum Systems Engineering

Well Inflow Performance Example 2-9 Each IPR curve for Pseudo-steady-state IPR reflects a “snapshot” of well performance at a given reservoir pressure. For this exercise, calculate the IPR curves of the well described in Appendix A for zero skin effect but for average reservoir pressure in increments of 500 psi from the “initial” 5651to 3500 psi. Drainage radius of 2980 ft. Solution: For a circular drainage shape, Eq. (2-44) can be used instead of general form Eq. (2-34) for pseudo-steadystate

pwf  5651 3.36q For all average reservoir pressure, the slope will remain the same. Petroleum Production Operations, Dr. Farshid Torabi

Figure 2-7 Pseudosteady-state IPR curves for a range of average reservoir pressures (Example 2-9).

31

University of Regina

Petroleum Systems Engineering

Well Inflow Performance 2-8 Effect of Water Production; Relative Permeability If both oil and free water are flowing, then effective permeabilities must be used. The sum of these permeabilities is invariably less than absolute permeability of the formation (to either fluid). These effective permeabilities are related to the “relative” permeabilities (also rock properties) by (2-53)

ko  kkro

and

k w  kkrw

(2-54)

Relative permeabilities are determined in the laboratory and are characteristic of a given reservoir rock and its saturating fluids. With the relative permeabilities, 𝑘𝑟𝑜 and 𝑘𝑟𝑤 , are being function of 𝑆𝑤 , as shown in Fig. 2-8.

Petroleum Production Operations, Dr. Farshid Torabi

32

University of Regina

Petroleum Systems Engineering

Well Inflow Performance 2-9 Effect of Water Production; relative Permeability Usually, relative permeability curves are presented as functions of the water saturation, 𝑆𝑤 , as shown in Fig. 2-8. when the water saturation, 𝑆𝑤 , is the connate water saturation, 𝑆𝑤𝑐 , no free water would flow and therefore its effective permeability, 𝑘𝑤 , would be equal to zero. Similarly, when the oil saturation becomes the residual oil saturation, 𝑆𝑜𝑟 , then no oil would flow and its effective permeability would be equal to zero. Thus, in an undersaturated oil reservoir, inflow equations must be written for both oil and water. For example, for steady-state production,

qo  and

qw 

kkro h pe  pwf o

141.2 Bo o ln re rw   s 

(2-55)

kkrwh pe  pwf w

141.2 Bw  w ln re rw   s 

(2-56) Figure 2-8 Relative permeability effects, water production.

Petroleum Production Operations, Dr. Farshid Torabi

33

University of Regina

Petroleum Systems Engineering

Well Inflow Performance 2-9 Effect of Water Production; relative Permeability With the relative permeabilities, 𝑘𝑟𝑜 and 𝑘𝑟𝑤 , are being function of 𝑆𝑤 , as shown in Fig. 2-8. Note that the pressure gradients have been labeled with subscripts for oil and water to allow for different pressure within the oil and water phases. The ratio 𝑞𝑤 𝑞𝑜 is referred to as the water-oil ratio. In an almost depleted reservoir it would not be unusual to obtain water-oil ratios of 10 or larger. Such a well is often referred to as a “stripper” with production rates of less than 10 STB/d of oil.

Petroleum Production Operations, Dr. Farshid Torabi

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