oil wells performance · OIL WELLS PERFORMANCE The productivity index is generally measured during a production test on the well, where the well is shut -in until the static reser

OIL WELLS PERFORMANCE

The productivity index is generally measured during a production

test on the well, where the well is shut-in until the static reservoir

pressure is reached. The well is then allowed to flow at a constant

flow rate of and a stabilized bottom-hole flowing pressure of. It is

important to note that the productivity index is a valid measure only

if the well is flowing at pseudo-steady state conditions. Therefore,

in order to accurately measure the productivity index of a well, it is

essential that the well is allowed to flow at a constant rate for a

sufficient amount of time to reach the pseudo-steady state

condition.

Hassan S. Naji,

Professor,

[email protected]

www.hbsnumerics.com

1

1. Productivity Index of a Well

The productivity index of a well, denoted by 𝐽, is a measure of the ability of the well

to produce. It is given by:

𝐽 =𝑞

𝑝𝑟 − 𝑝𝑤 (1)

Where:

𝐽 Wellbore productivity index, STB/day/psig

𝑞 Wellbore stabilized oil flow rate, STB/day

𝑝𝑟 Average (static) reservoir pressure, psig

𝑝𝑤 Wellbore stabilized bottom-hole flowing pressure, psig

The productivity index is generally measured during a production test on the well,

where the well is shut-in until the static reservoir pressure 𝑝𝑟 is reached. The well is

then allowed to flow at a constant flow rate of 𝑞 and a stabilized bottom-hole flowing

pressure of 𝑝𝑤. It is important to note that the productivity index is a valid measure

only if the well is flowing at pseudo-steady state conditions. Therefore, in order to

accurately measure the productivity index of a well, it is essential that the well is

allowed to flow at a constant rate for a sufficient amount of time to reach the pseudo-

steady state condition.

A plot of the bottom hole pressure, 𝑝𝑤, versus the oil flow rate, 𝑞, of an oil well is

called the Inflow Performance Relationship and is referred to as IPR. The IPR

curve is constructed either for the present reservoir pressure or for the future reservoir

pressure.

The present IPR curve of a well can be generated via many methods. The methods

are grouped into two main categories: The first category, that uses reservoir

parameters, includes the integral and Fetkovitch methods. The second category

includes a set of Vogel-type empirical correlations. To construct the IPR curve of a

well using well performance, perform the following steps:

Use the static reservoir pressure 𝑝𝑟 and the stabilized wellbore rate and pressure

(𝑞 & 𝑝𝑟) to calculate 𝐽 as follows:

𝐽 =𝑞

𝑝𝑟 − 𝑝𝑤 (2)

Construct the IPR curve by assuming various values of 𝑝𝑤 and calculating the

corresponding 𝑞 as follows:

𝑞 = 𝐽(𝑝𝑟 − 𝑝𝑤) (3)

2

2. The Integral Form

Darcy’s equation is expressed mathematically as:

𝑞 = (𝑘ℎ

141.22 [ln (𝑟𝑟𝑤) − 0.75 + 𝑠]

)(∫𝑘𝑟𝑜𝜇𝑜𝐵𝑜

𝜕𝑝𝑝𝑟

𝑝𝑤

) (4)

The IPR curve is constructed by assuming various values of 𝑝𝑤 and calculating the

corresponding 𝑞. To construct the IPR curve of a well using well performance along

with the integral method, perform the following steps:

Use the static reservoir pressure 𝑝𝑟 and the stabilized wellbore rate and pressure

(𝑞 & 𝑝𝑤) to calculate 𝐽 as follows:

𝐽 =𝑞

∫1

𝜇𝑜𝐵𝑜𝜕𝑝

𝑝𝑟

𝑝𝑤

(5)



𝑞 = 𝐽∫1

𝜇𝑜𝐵𝑜𝜕𝑝

𝑝𝑟

𝑝𝑤

(6)

3. Fetkovitch Method

Fetkovitch (1973) started with Darcy’s equation:

𝑞 = (𝑘ℎ

141.22 [ln (𝑟𝑟𝑤) − 0.75 + 𝑠]

)(∫𝑘𝑟𝑜𝜇𝑜𝐵𝑜

𝜕𝑝𝑝𝑟

𝑝𝑤

) (7)

He considered two cases:

Saturated Oil Reservoirs 𝑝𝑟 ≤ 𝑝𝑏

When the reservoir pressure 𝑝𝑟 and the bottom-hole flowing pressure 𝑝𝑤 are both

below the bubble-point pressure 𝑝𝑏, the oil flow rate can be written as:

3

𝑞 = (𝑘ℎ

141.22(𝜇𝑜𝐵𝑜)𝑝𝑏 [ln (𝑟𝑟𝑤) − 0.75 + 𝑠]

)(1

2𝑝𝑏) (𝑝𝑟

2 − 𝑝𝑤2 )

= (𝐽

2𝑝𝑏) (𝑝𝑟

2 − 𝑝𝑤2 ) = 𝐶(𝑝𝑟

2 − 𝑝𝑤2 )

(8)

The constant 𝐶 is referred to as the performance coefficient. The IPR curve is

constructed by assuming various values of 𝑝𝑤 and calculating the corresponding 𝑞

using the above equation.

Undersaturated Oil Reservoirs 𝑝𝑟 > 𝑝𝑏

Two cases are considered:

Case 1: 𝑝𝑤 ≥ 𝑝𝑏

𝑞 = (𝑘ℎ

141.22(𝜇𝑜𝐵𝑜)(𝑝𝑟+𝑝𝑤)2

[ln (𝑟𝑟𝑤) − 0.75 + 𝑠]

) (𝑝𝑟 − 𝑝𝑤) = 𝐽(𝑝𝑟 − 𝑝𝑤) (9)

Case 2: 𝑝𝑤 < 𝑝𝑏

𝑞 = (𝑘ℎ

141.22(𝜇𝑜𝐵𝑜)𝑝𝑏 [ln (𝑟𝑟𝑤) − 0.75 + 𝑠]

) [(𝑝𝑟 − 𝑝𝑏)

+ (1

2𝑝𝑏) (𝑝𝑏

2 − 𝑝𝑤2 )] = 𝐽 [(𝑝𝑟 − 𝑝𝑏) + (

1

2𝑝𝑏) (𝑝𝑏

2 − 𝑝𝑤2 )]

(10)

4. Constant Productivity Index Method

The IPR curve of a well can be generated using the constant productivity index

approach. The productivity index of a well is defined as a measure of the well to

produce. It is given by:

𝐽 =𝑞

𝑝𝑟 − 𝑝𝑤=𝑞

∆𝑝 (11)

Where:

4

𝐽 Wellbore productivity index, STB/day/psig

𝑞 Wellbore stabilized oil flow rate, STB/day

𝑝𝑟 Average (static) reservoir pressure, psig

𝑝𝑤 Wellbore stabilized bottom-hole flowing pressure, psig

The productivity index is generally measured during a production test on the well.

The well is shut-in until the static reservoir pressure is reached. The well is then

allowed to flow at a constant flow rate of 𝑞 and a stabilized bottom-hole flowing

pressure of 𝑝𝑤. It is important to note that the productivity index is a valid measure

only if the well is flowing at pseudo-steady state conditions. Therefore, in order to

accurately measure the productivity index of a well, it is essential that the well is

allowed to flow at a constant rate for a sufficient amount of time to reach the pseudo-

steady state condition. The method is summarized as follows:

Use the stabilized wellbore rate and pressure (𝑞 & 𝑝𝑤) to calculate 𝐽 as follows:

𝐽 =𝑞

𝑝𝑟 − 𝑝𝑤 (12)



𝑞 = 𝐽(𝑝𝑟 − 𝑝𝑤) (13)

5. Vogel's Method

Vogel (1968) used a computer program to generate the IPRs for several hypothetical

saturated oil reservoirs producing under a wide range of conditions. When applying

his method, the only data required are: the average reservoir pressure 𝑝𝑟, the oil

bubble-point pressure 𝑝𝑏, and the stabilized wellbore rate and pressure (𝑞 & 𝑝𝑤).

Vogel’s methodology can be used to predict the IPR curve for the following two types

of reservoirs:


For the saturated oil reservoirs, we follow the following two steps:

Use the stabilized wellbore rate and pressure (𝑞 & 𝑝𝑤) to calculate (𝑞)𝑚𝑎𝑥 as

follows:

(𝑞)𝑚𝑎𝑥 =𝑞

1 − 0.2 (𝑝𝑤𝑝𝑟) − 0.8 (

𝑝𝑤𝑝𝑟)2

(14)

5



𝑞 = (𝑞)𝑚𝑎𝑥 [1 − 0.2 (𝑝𝑤𝑝𝑟) − 0.8 (

𝑝𝑤𝑝𝑟)2

] (15)


For the undersaturated oil reservoirs, we follow the following three steps:

Use the stabilized wellbore rate and pressure (𝑞 & 𝑝𝑤) to calculate the

productivity index 𝐽 as follows:

𝐽 =

{

𝑞

(𝑝𝑟 − 𝑝𝑤) 𝑝𝑤 ≥ 𝑝𝑏

𝑞

(𝑝𝑟 − 𝑝𝑏) +𝑝𝑏1.8

[1 − 0.2 (𝑝𝑤𝑝𝑏) − 0.8 (

𝑝𝑤𝑝𝑏)2

]𝑝𝑤 < 𝑝𝑏

}

(16)

Calculate the oil flow rate at the bubble-point pressure:

𝑞 = 𝐽(𝑝𝑟 − 𝑝𝑏) (17)

Construct the IPR curve by assuming various values of wfP and calculating the


𝑞 = {

𝐽(𝑝𝑟 − 𝑝𝑤) 𝑝𝑤 ≥ 𝑝𝑏

𝑞𝑜𝑏 +𝐽𝑝𝑏1.8

[1 − 0.2 (𝑝𝑤𝑝𝑏) − 0.8 (

𝑝𝑤𝑝𝑏)2

] 𝑝𝑤 < 𝑝𝑏} (18)

6. Wiggins (1993) Method

Wiggins (1993) used four sets of relative permeability and fluid property data as the

basic input for a computer program to generate the IPRs for several hypothetical

saturated oil reservoirs producing under a wide range of conditions. The only data

required are: the average reservoir pressure 𝑝𝑟, the oil bubble-point pressure 𝑝𝑏, and

the stabilized wellbore rate and pressure (𝑞 & 𝑝𝑤). This method is considered for the

following two types of reservoirs:

6




follows:


1 − 0.52 (𝑝𝑤𝑝𝑟) − 0.48 (

𝑝𝑤𝑝𝑟)2

(19)



𝑞 = (𝑞)𝑚𝑎𝑥 [1 − 0.52 (𝑝𝑤𝑝𝑟) − 0.48 (

𝑝𝑤𝑝𝑟)2

] (20)





𝐽 =

{

𝑞


𝑞

(𝑝𝑟 − 𝑝𝑏) +𝑝𝑏1.8 [1 − 0.52 (

𝑝𝑤𝑝𝑏) − 0.48 (

𝑝𝑤𝑝𝑏)2


}

(21)


𝑞 = 𝐽(𝑝𝑟 − 𝑝𝑏) (22)



𝑞 = {



[1 − 0.52 (𝑝𝑤𝑝𝑏) − 0.48 (

𝑝𝑤𝑝𝑏)2

] 𝑝𝑤 < 𝑝𝑏} (23)

7

7. Wiggins (1996) Method

In 1996, Wiggins derived an equation for the prediction of oil well performance.

His equation can be used as follows:




follows:

(𝑞)𝑚𝑎𝑥

=𝑞

1 − 0.0933 (𝑝𝑤𝑝𝑟) − 1.6183 (

𝑝𝑤𝑝𝑟)2

+ 1.5579 (𝑝𝑤𝑝𝑟)3

− 0.8464 (𝑝𝑤𝑝𝑟)4 (24)



𝑞 = (𝑞)𝑚𝑎𝑥 [1 − 0.0933 (𝑝𝑤𝑝𝑟) − 1.6183 (

𝑝𝑤𝑝𝑟)2

+ 1.5579 (𝑝𝑤𝑝𝑟)3

− 0.8464 (𝑝𝑤𝑝𝑟)4

]

(25)





𝐽 =

{

𝑞


𝑞

(𝑝𝑟 − 𝑝𝑏) +𝑝𝑏1.8 [

1 − 0.0933 (𝑝𝑤𝑝𝑏) − 1.6183 (

𝑝𝑤𝑝𝑏)2

+1.5579 (𝑝𝑤𝑝𝑏)3

− 0.8464 (𝑝𝑤𝑝𝑏)4 ]

𝑝𝑤 < 𝑝𝑏

}

(26)


8

𝑞 = 𝐽(𝑝𝑟 − 𝑝𝑏) (27)



𝑞 =

{


𝑞𝑏 +𝐽𝑝𝑏1.8

[ 1 − 0.0933 (

𝑝𝑤𝑝𝑏) − 1.6183 (

𝑝𝑤𝑝𝑏)2

+1.5579 (𝑝𝑤𝑝𝑏)3

− 0.8464 (𝑝𝑤𝑝𝑏)4

]

𝑝𝑤 < 𝑝𝑏

}

(28)

8. Klins-Clark Method

Klins and Clark (1993) proposed an inflow expression similar in form to that of

Vogel’s and can be used to estimate future IPR data. This method is considered for

the following two types of reservoirs:




follows:


[1 − 0.295 (𝑝𝑤𝑝𝑟) − 0.705 (

𝑝𝑤𝑝𝑟)𝑑

]

(29)

Where:

𝑑 = [0.28 + 0.72 (𝑝𝑟𝑝𝑏)] (1.24 + 0.001𝑝𝑏) (30)



𝑞 = (𝑞)𝑚𝑎𝑥 [1 − 0.295 (𝑝𝑤𝑝𝑟) − 0.705 (

𝑝𝑤𝑝𝑟)𝑑

] (31)

9




productivity index𝐽 as follows:

𝐽 =

{

𝑞


𝑞

(𝑝𝑟 − 𝑝𝑏) +𝑝𝑏1.8 [1 − 0.295 (

𝑝𝑤𝑝𝑏) − 0.705 (

𝑝𝑤𝑝𝑏)𝑑


}

(32)


𝑞𝑜𝑏 = 𝐽(𝑝𝑟 − 𝑝𝑏) (33)



𝑞 = {



[1 − 0.295 (𝑝𝑤𝑝𝑏) − 0.705 (

𝑝𝑤𝑝𝑏)𝑑

] 𝑝𝑤 < 𝑝𝑏} (34)

9. FPP Method

The FPP solution method is given by:

𝑦 = 𝑏0 + 𝑏1𝑥𝛾 + 𝑏2𝑥

2𝛾 (35)

Using the least square method, we obtain a system of three equations in three

unknowns 𝑎0, 𝑎1, 𝑎2:

𝑏0𝑁 + 𝑏1∑𝑥𝛾

𝑁

+ 𝑏2∑𝑥2𝛾

𝑁

=∑𝑦

𝑁

𝑏0∑𝑥𝛾

𝑁

+ 𝑏1∑𝑥2𝛾

𝑁

+ 𝑏2∑𝑥3𝛾

𝑁

=∑𝑥𝛾𝑦

𝑁

𝑏0∑𝑥2𝛾

𝑁

+ 𝑏1∑𝑥3𝛾

𝑁

+ 𝑏2∑𝑥4𝛾

𝑁

=∑𝑥2𝛾𝑦

𝑁

(36)

N is the number of data points. In matrix form, this can be written as:

10

[ 𝑁 ∑𝑥𝛾

𝑁

∑𝑥2𝛾

𝑁

∑𝑥𝛾

𝑁

∑𝑥2𝛾

𝑁

∑𝑥3𝛾

𝑁

∑𝑥2𝛾

𝑁

∑𝑥3𝛾

𝑁

∑𝑥4𝛾

𝑁 ]

{

𝑏0𝑏1𝑏2

} =

{

∑𝑦

𝑁

∑𝑥𝛾𝑦

𝑁

∑𝑥2𝛾𝑦

𝑁 }

(37)

The power 𝛾 is solved for to minimize the residual root mean square (RRMS):

𝑅𝑅𝑀𝑆 = √1

𝑁∑(𝑧𝑜𝑏𝑠 − 𝑧𝑐𝑎𝑙)

2

𝑁

𝑖=1

(38)

Once the set of equations has been solved, the following equations are solved:

(𝑞)𝑚𝑎𝑥 = 𝑏0𝑎1 = 𝑏1(𝑞)𝑚𝑎𝑥𝑎2 = 𝑏2(𝑞)𝑚𝑎𝑥

𝑥 =𝑝𝑤𝑝𝑟

𝑦 = 𝑞

(39)

This methodology can be used to predict the IPR curve for the following two types of

reservoirs:


For the saturated oil reservoirs, we follow the following step:



𝑞 = (𝑞)𝑚𝑎𝑥 [1 − 𝑎1 (𝑝𝑤𝑝𝑟)𝛾

− 𝑎2 (𝑝𝑤𝑝𝑟)2𝛾

] (40)



11



𝐽 =

{

𝑞


𝑞

(𝑝𝑟 − 𝑝𝑏) +𝑝𝑏1.8 [1 − 𝑎1 (

𝑝𝑤𝑝𝑏)𝛾

− 𝑎2 (𝑝𝑤𝑝𝑏)2𝛾


}

(41)


𝑞𝑜𝑏 = 𝐽(𝑝𝑟 − 𝑝𝑏) (42)



𝑞 = {



[1 − 𝑎1 (𝑝𝑤𝑝𝑏)𝛾

− 𝑎2 (𝑝𝑤𝑝𝑏)2𝛾

] 𝑝𝑤 < 𝑝𝑏} (43)

10. Prediction Future IPRs

As the average reservoir pressure 𝑝𝑟 declines, the IPR curve is shifted. There are

several methods that are designed to address the problem of how the IPR might shift.

Four simple approximation methods are presented.

1. Vogel’s Method

Vogel’s method provides a rough approximation of the future (𝑞)𝑚𝑎𝑥,𝑓 at the

specified future 𝑝𝑟,𝑓 as follows:

(𝑞)𝑚𝑎𝑥,𝑓 = (𝑞)𝑚𝑎𝑥,𝑝 (𝑝𝑟,𝑓

𝑝𝑟,𝑝) [0.2 + 0.8 (

𝑝𝑟,𝑓

𝑝𝑟,𝑝)] (44)

Where the subscripts 𝑓 and 𝑝 refer to future and previous respectively. The calculated

(𝑞)𝑚𝑎𝑥,𝑓 can be used to predict the future IPR at 𝑝𝑟,𝑓.

12

2. Fetkovitch

This method, proposed by Fetkovitch (1973), provides a simple approximation of the

future (𝑞)𝑚𝑎𝑥,𝑓 at the specified future 𝑝𝑟,𝑓 as follows:


𝑝𝑟,𝑝)

3

(45)

Where the subscripts f and p refer to future and present respectively. The calculated

(𝑞)𝑚𝑎𝑥,𝑓 can be used to predict the future IPR at 𝑝𝑟,𝑓.

3. Wiggins’ Method

Wiggins (1993) proposed the following relationship:


𝑝𝑟,𝑝) [0.15 + 0.84 (

𝑝𝑟,𝑓

𝑝𝑟,𝑝)] (46)

4. Standing’s Method

Standing (1970) extended the application of Vogel’s to predict future IPR of a

well as a function of reservoir pressure. He noted that Vogel’s equation can be

arranged as:

𝑞

(𝑞)𝑚𝑎𝑥= (1 −

𝑝𝑤,𝑓

𝑝𝑟) [1 + 0.8 (

𝑝𝑤,𝑓

𝑝𝑟)] (47)

Standing introduced the productivity index 𝐽 as follows:

𝐽 =(𝑞)𝑚𝑎𝑥𝑝𝑟

[1 + 0.8 (𝑝𝑤,𝑓

𝑝𝑟)] (48)

He then defined the present (current) zero drawdown productivity index as:

𝐽𝑝∗ = 1.8(

(𝑞)𝑚𝑎𝑥𝑝𝑟

) (49)

Where 𝐽𝑝∗ is Standing’s zero-drawdown productivity index. The 𝐽𝑝

∗ is related to the

productivity index 𝐽 by:

13

𝐽

𝐽𝑝∗=

1

1.8[1 + 0.8 (

𝑝𝑤,𝑓

𝑝𝑟)] (50)

Combining equations (47) and (50) to eliminate (𝑞)𝑚𝑎𝑥 yields:

𝑞 = [𝐽𝑓∗𝑝𝑟,𝑓

1.8] (1 −

𝑝𝑤,𝑓

𝑝𝑟,𝑓) [1 + 0.8 (

𝑝𝑤,𝑓

𝑝𝑟,𝑓)] (51)

Where the subscripts 𝑓 refers to future condition. Standing suggested that 𝐽𝑓∗ is

estimated from the present value of 𝐽𝑝∗ by the following expression:

𝐽𝑓∗ = 𝐽𝑝

∗

(𝑘𝑟𝑜𝜇𝑜𝐵𝑜

)𝑓


)𝑝

(52)

If relative permeability data is not available, 𝐽𝑓∗ can be estimated from:


∗ ((𝑝𝑟)𝑓(𝑝𝑟)𝑝

)

2

(53)

Standing’s methodology for predicting a future IPR is summarized in the following

steps:


follows:


(1 −𝑝𝑤,𝑓𝑝𝑟) [1 + 0.8 (

𝑝𝑤,𝑓𝑝𝑟)]

(54)

Calculate 𝐽𝑝∗ using:

𝐽𝑝∗ = 1.8(

(𝑞)𝑚𝑎𝑥𝑝𝑟

) (55)

Calculate 𝐽𝑓∗ using:


∗


)𝑓


)𝑝

(56)

14

If relative permeability data is not available, 𝐽𝑓∗ can be estimated from:


∗ ((𝑝𝑟)𝑓(𝑝𝑟)𝑝

)

2

(57)



𝑞 = [𝐽𝑓∗𝑝𝑟,𝑓

1.8] (1 −

𝑝𝑤,𝑓

𝑝𝑟,𝑓) [1 + 0.8 (

𝑝𝑤,𝑓

𝑝𝑟,𝑓)] (58)

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