injection well ratio in waterflooding oilfields

PETROLEUM EXPLORATION AND DEVELOPMENT Volume 38, Issue 2, April 2011 Online English edition of the Chinese language journal Cite this article as: PETROL. EXPLOR. DEVELOP., 2011, 38(2): 211–215.

RESEARCH PAPER

Calculation on a reasonable production/injection well ratio in waterflooding oilfields Zou Cunyou1,*, Chang Yuwen1, Wang Guohui1, Lan Lifeng2 1. PetroChina Research Institute of Petroleum Exploration & Development, Beijing 100083, China; 2. Exploration and Development Research Institute of PetroChina Daqing Oilfield Company Ltd., Daqing 163712, China

Abstract: According to the accurate definition of the reasonable production/injection well ratio and injection/production ratio for waterflooding oilfields, the formula of the reasonable production/injection well ratio under the same or different formation pressures between oil well and injection well is derived based on the principle of extreme value. The formula can be used in the balanced or unbalanced injection/production system such as reservoirs in high water cut stage or low/permeability reservoirs. All the previous formulas have flaws: the method of “injectivity, productivity indexes” is based on balanced injection/production system; the method of “injectivity, productivity indexes and injection/production ratio” ignores the influences of crude oil properties; and there is no theoretical and practical significance in the method of “injectivity index to liquid productivity index ratio and injection/production pressure difference”. The method presented in this paper is applied and verified in the Chaoyanggou Oilfield of Daqing. The daily fluid rate of the oilfield can reach the maximum value of 5 155.8 t at the reasonable production/injection well ratio of 3.28. Key words: waterflooding; injection/production ratio; reasonable production/injection well ratio; injection/production equilibrium; low-permeability reservoir

Introduction With most of the oilfields in China entering into a high water-cut stage and more and more low-permeability reservoirs being put into development, it has been found in field practices that traditional injection/production balance development can not meet the need to keep reasonable formation pressure[1-2]. Some abnormal phenomena[3] had been found, such as that formation pressure is still less than initial reservoir pressure when the accumulative injection to production ratio is over 1.0; formation pressure can only keep stable or even decline when the annual injection to production ratio is over 1.0. In the early 1980s, Tong Xianzhang[4] obtained a set of formula and chart by use of water injectivity index, oil productivity index and material balance principle, to find the relationship between maximized production and production/injection well ratio. In the 1990s, Fang Lingyun et al. [5] derived the calculation formula of reasonable production/injection well ratio for the injection/production balance oilfield. These studies provide positive guidance for well pattern deployment and injection/production system adjustment for oilfield development in China. However, because existing calculation formulas on reasonable production/injection well ratio are mainly based on the balanced injection/production

system, there is no guidance significance or even wrong results would appear if they are applied in the unbalanced system. Although literature [6] takes the influence factor of injection/production ratio into account in the formula, the accurate definition of injection/production ratio is neglected, so the influence of crude oil properties (oil density and volume factor) is omitted, thus accurate formula for reasonable produciton/injeciton well ratio is still not obtained. It should be pointed out, although there are many methods to calculate the reasonable production/injection well ratio, the concrete derivation process is seldom seen, their correctness cannot be verified. Moreover, it has also been found by field practices that the formation pressures for oil and injection well are not identical; especially in low-permeability reservoirs, there are great pressure difference between oil well and injection well because of the influence of start-up pressure gradient[7-9]. Therefore, how to calculate the reasonable production-injection well ratio when there is pressure difference between oil well and injection well should be considered. In view of the above reasons, according to the accurate definition of the injection-production ratio for waterflooding oilfields, the formula of the reasonable production-injection

Received date: 17 Aug. 2009; Revised date: 16 Jan. 2011. * Corresponding author. E-mail: [email protected] Foundation item: Supported by the "11th Five-Year Plan" Research Projects of Science and Technology Development of PetroChina (06-06D-01-06-04) Copyright © 2011, Research Institute of Petroleum Exploration and Development, PetroChina. Published by Elsevier BV. All rights reserved.

Zou Cunyou et al. / Petroleum Exploration and Development, 2011, 38(2): 211–215

well ratio when injection rate and producing rate is unbancled and also formation pressures between oil well and injection well are different is derived in detail based on the principle of extreme value in this paper. Also, the flaws in existing formulas are analyzed in this paper.

1 Theoretical derivation of the formula of reasonable production/injection well ratio 1.1 Reasonable production/injection well ratio under the same formation pressure between production well and injection well To get the accurate formula of reasonable production/injection well ratio, the accurate definition of the reasonable production/injection well ratio and injection-production ratio should be understood firstly. The reasonable production/injection well ratio is defined as the ratio of production well number to injection well number at highest fluid rate under the conditions of constant bottom flowing pressure between injection wells and production wells and the constant number of development wells[1013]. Injection/production ratio refers to the underground volume ratio of injected fluid to produced fluid. When the average water cut of oilfield is fw, the underground volume of average daily fluid rate for single well can be written as: qL J L ( pR  pwf ) > (1  f w ) Bo / Uo  f w Bw / U w @ (1)

If the production well number is no, the underground volume of daily fluid rate in the oilfield is: QL no qL no J L ( pR  pwf )

>(1  f w ) Bo / Uo  f w Bw / U w @

(2)

If the injection well number is nw, the underground volume of daily water injection rate in the oilfield is: Qinj nw I w ( pinj  pR ) Bw / U w (3)

By the definition of injection/production ratio, RIP Qinj / QL

(4)

(5)

ª º BU RIP J L «(1  f w ) o w  f w » Bw Uo ¬ ¼ c Iw

b

Then Eq. (10) can be expressed simply as: aR QL (1  R )(bR  c)

(11)

By Eq. (11), the derivative of QL to R can be calculated as following equation: dQL dR

a (c  bR 2 )

>(1  R)(bR  c)@2

(12)

Let dQL/dR = 0, we can obtain: c  bR 2

0

(13)

Therefore c b

R

Iw ª º BU RIP J L «(1  f w ) o w  f w » Bw Uo ¬ ¼

(14)

Generally, we take water density ȡw=1.0 g/cm3 and water volume factor Bw=1.0. Putting them into Eq. (14), the reasonable production-injection well ratio can be expressed in the end as follows: R

Putting Eq. (2) and Eq. (3) into Eq. (4) and arranging, we obtain nw I w ( pinj  pR ) Bw / U w RIP no J L ( pR  pwf ) u

>(1  f w ) Bo / Uo  f w Bw / U w @

Assuming the total well number in oilfield is nt, that is nt=no+nw, combined with Eq. (6), we can obtain Rnt no (9) 1 R Putting Eq. (9) into Eq. (8), R nt J L > (1  f w ) Bo / Uo  f w Bw / U w @ I w ( pinj  pwf ) QL 1 R ª º BU RIP J L «(1  f w ) o w  f w » R  I w B U w o ¬ ¼ (10) Let a nt J L > (1  f w ) Bo / Uo  f w Bw / U w @ I w ( pinj  pwf )

Iw RIP J L > (1  f w ) Bo / Uo  f w @

(15)

By principle of extreme value[14], we know that R calculated by Eq. (14) and Eq. (15) is just the reasonable R which enables the fluid rate to reach the maximum.

(6)

1.2 Reasonable production/injection well ratio under different formation pressure between production well and injection well

Divided by nw in both sides of Eq. (5), the formation pressure PR can be obtained: ª º BU I w pinj  RIP RJ L «(1  f w ) o w  f w » pwf Bw Uo ¬ ¼ pR (7) ª º Bo U w RIP RJ L «(1  f w )  fw »  Iw Bw Uo ¬ ¼ Putting Eq. (7) into Eq. (2) and arranging, the daily fluid rate of the oilfield can be expressed by the following equation: >(1  f w ) Bo / Uo  f w Bw / U w @ I w ( pinj  pwf ) QL no J L (8) ª º Bo U w RIP RJ L « (1  f w )  fw »  Iw Bw Uo ¬ ¼

It has been proved by oilfield development theory and practice that, the formation pressure between oil wells and injection wells is not the same, especially in low-permeability reservoirs. Due to the influences of high flow resistance of low-permeability reservoirs, start-up pressure gradient and pressure sensitivity, high pressure can be easily formed between production wells and injection wells[2]. So there is a pressure difference between production wells and injection wells, which is too great to be neglected. Therefore, this pressure difference must be considered in calculation of reasonable production/injection well ratio. Now we make theoretical

The production/injection well ratio is defined as: R no / nw

Zou Cunyou et al. / Petroleum Exploration and Development, 2011, 38(2): 211–215

derivation for the reasonable production/injection well ratio under this condition. Assuming the average formation pressure of production well is pows, the pressure difference between injection well and production well is ǻp, then the formation pressure of injection well can be expressed as: piws pows  'p (16) So the daily liquid rate and daily water injection rate of oilfield can be written by the following equations: QL no J L ( pows  pwf ) > (1  f w ) Bo / Uo  f w Bw / U w @ (17) Qinj

nw I w ( pinj  piws ) Bw / U w

Putting Eq. (16) into Eq. (18), Qinj nw I w ( pinj  pows  'p ) Bw / U w

(18) (19)

Then putting Eqs. (17), (19) and (6) into Eq. (4), the formation pressure of production well can be expressed as: ª º BU I w ( pinj  'p )  RIP RJ L «(1  f w ) o w  f w » pwf B U w o ¬ ¼ pows (20) ª º Bo U w RIP RJ L « (1  f w )  fw »  Iw Bw Uo ¬ ¼ Putting Eqs. (20) and (9) into Eq. (17) and arranging, the daily liquid rate in the oilfield can be obtained as follows: R nt J L > (1  f w ) Bo / Uo  f w Bw / Uw @ I w ( pinj  'p  pwf ) QL 1 R ª º BU RIP J L «(1  f w ) o w  f w » R  I w B U w o ¬ ¼ (21) Let d nt J L > (1  f w ) Bo / Uo  f w Bw / U w @ I w ( pinj  'p  pwf ) Then Eq. (21) can be expressed simply as: dR QL (1  R )(bR  c)

(22)

By Eq. (22), the derivative of QL to R can be calculated as: dQL dR

d (c  bR 2 )

(23)

2

Existing methods and their problems

2.1 Water injectivity index and liquid productivity index method

The formula is deduced in the condition of balanced injection-production system. No doubt it plays a positive role in the injection-production system adjustment in medium and high permeability sandstone reservoirs which was developed early in China. The formula is[5,15] : R

Iw / JL

(25)

2.2 Water injectivity index, liquid productivity index and injection-production ratio method

Jia Zili et al.[6] considered the influence of injection-production ratio in the derivation, and got a formula about reasonable injection-production well ratio for oilfield developed by water injection. The original formula form is: M2

RIP J L I w

(26)

With the concept of reasonable production-injection well ratio in this paper, Eq. (26) can be written as follows:

R

Iw

 RIP J L

(27)

It must be pointed out that the accurate definition of injection/production ratio is neglected during the derivation, which wrongly understands the definition as the ground volume ratio between injected fluid and produced fluid. Therefore, there is no surface oil density ȡo and volume factor Bo in the formula. Obviously, the formula is not accurate. For ease of distinction, this method presented in reference [6] is called as “Water injectivity index, liquid productivity index and injection/production ratio method”. 2.3 Water injectivity index to liquid productivity index ratio and injection-production pressure difference method

(24)

This method is widely adopted to calculate reasonable R by many literatures[13,16], oilfields and research institutes, so we also pay attention to this method. Putting QL in Eq. (17) and Qinj in Eq. (18) into Eq. (4) in this paper, we can easily obtain following expression: nw I w ( pinj  piws ) Bw / U w RIP (28) no J L ( pows  pwf ) > (1  f w ) Bo / Uo  f w Bw / U w @

It is easy to see that whether the pressure between production well and injection well is the same or not, the calculation formula for the reasonable R is not different. Then whether it indicates the pressure difference has no influences upon the calculation of reasonable R? The answer is certainly negative. The differences between them lie in the calculation of water injectivity index Iw and liquid productivity index JL. For high-permeability reservoirs, it is able to approximately think that the pressures are equal. But for low-permeability reservoirs, the obvious pressure difference can not be neglected; otherwise there would be great error to the final calculation result of reasonable R.

It can be seen that, “Water injectivity index to liquid productivity index ratio and injection/production pressure difference method” is only the simple solution for existing data, and

>(1  R)(bR  c)@2

Let dQL/dR = 0, and considering Bw = 1.0, ȡw = 1.0 g/cm3, based on principle of extreme value, the expression of reasonable R when the formation pressure is not the same between oil and water wells can be obtained: R

Iw RIP J L > (1  f w ) Bo / Uo  f w @

Let Bw=1.0, ȡw=1.0 g/cm3, and considering production-injection well ratio R=no/nw, water injection pressure difference ǻpiw=pinjpiws, producing pressure difference ǻpL  powspwf, then transforming Eq. (28), we can get the calculation formula of “water injection index to liquid productivity index ratio and injection-production pressure difference method”: I w ( f w ) 'piw J L ( f w ) 'pL (29) R >(1  f w ) Bo / Uo  f w @ RIP

Zou Cunyou et al. / Petroleum Exploration and Development, 2011, 38(2): 211–215

the calculated ratio of production wells to injection wells is for current well pattern, not for “maximum liquid production” conditions. Therefore, this method has no any practical significance.

3

Application

The Chaoyanggou Oilfield is a complex fractured extra-low permeability reservoir controlled by many factors such as faults, structure and lithology etc. The average air permeability is 9.3u103 ȝm2, average effective porosity is 16.0% for the reservoir, surface oil density is 0.864 g/cm3, oil volume factor is 1.082. At the end of 2008, there are 2250 opened production wells, daily oil production was 2762 t, and daily water production was 1643 t; and there are 873 opened injection wells, daily water injection 15576 m3. The static formation pressure of production wells and injection wells was 7.56 MPa and 21.04 MPa respectively, the bottom flowing pressure was 1.91 MPa and 22.39 MPa respectively. We can see that there is a obvious pressure difference between injection wells and production wells (piwspows=13.48 MPa) in the Chaoyanggou Oilfield, therefore we cannot approximately substitute average formation pressure of oil wells for that of injection wells. The average value of single wells is adopted for liquid productivity index JL and water injectivity index Iw. The calculation formulas are: Qo  Qw (30) JL no ( pows  pwf ) Iw

Qinj nw ( pinj  piws )

(31)

When using Eq. (24) to calculate the reasonable production/injection well ratio, the application object of the parameters must be paid attention to. Among which, injection/production ratio and water cut are values for the oilfield, and the calculation formulas are as follows: Qinj (32) RIP Qw  Qo Bo Uo fw

Qw (Qo  Qw )

(33)

By Eq. (30)Eq. (33), following indexes for average single well of the Chaoyanggou Oilfield at the end of 2008 can be calculated: JL = 0.35 t/(d·MPa), Iw =13.22 t/(d·MPa), fw =37%, RIP = 3.05. Putting these results and oil density, volume factor into Eq. (24), we can get the reasonable production/injection well ratio in the Chaoyanggou Oilfield, the value is 3.28. By Eq. (22), the daily liquid rate in the Chaoyanggou Oilfield for different production/injection well ratios was calculated (Fig. 1). We can see that the daily liquid rate of the oilfield increases rapidly with the increasing of production-injection well number ratio, when it reaches the highest point, the daily liquid rate decreases slowly. Of course, the production/injection well ratio corresponding to the highest liquid rate is the reasonable production/injection well ratio. The calculation result indicates that the liquid rate can reach the highest value 5 155.79 m3/d in the Chaoyanggou Oilfield

Fig. 1 Relationship of daily liquid rate vs. production/injection well ratio in the Chaoyanggou Oilfield

when production/injection well ratio is 3.28. The current production/injection well ratio is 2.58 and the daily liquid rate is 5 102.35 m3/d, which has no significant difference with the maximal daily liquid rate under the reasonable production/injection well ratio. This indicates that the current production/injection well ratio in the Chaoyanggou Oilfield is reasonable basically. The reasonable production/injection well ratio calculated by different methods and the corresponding daily liquid rate are shown in Table 1. It is obviously seen that the calculated result by “Water injectivity index to liquid productivity index ratio and injection-production pressure difference method” is just the current production-injection well ratio, which has no any significance. “Water injectivity index and liquid productivity index method” is built on the basis of balanced injection-production system, so the result is obviously different with the reasonable ratio. “Water injectivity index and liquid productivity index and injection/production ratio method” neglects the influence of oil properties and has obvious theoretical flaws, so the calculated result is not accurate. Especially when the oil volume factor Bo and surface oil density ȡo is relatively high, it would lead to great error. Table 1 Comparison of the reasonable production/injection well ratios and daily corresponding liquid rates by different methods Method Water injectivity index to liquid productivity index ratio and injection-production pressure difference method Water injectivity index and liquid productivity index method Water injectivity index and liquid productivity index and injection/production ratio method The method in this paper

4

Reasonable production/injection well ratio/f

Daily liquid ˉ rate/(m3·d 1)

2.58

5 102.35

6.18

4 801.64

3.53

5 150.82

3.28

5 155.79

Conclusions and suggestions

In this paper, the calculation formula of reasonable production/injection well ratio is derived in detail and accurately for the unbalanced injection-production system and under the conditions of different formation pressure between production

Zou Cunyou et al. / Petroleum Exploration and Development, 2011, 38(2): 211–215

well and injection well. We had some new understandings on it, which are very significant for the adjustment of production-injection system in waterflooding oilfields. The following two points need to be noticed when calculating the reasonable production/injection well ratio by the method presented in this paper: (1) For high-permeability reservoirs, when the formation pressures of oil well and injection well have no significant difference, it can be considered that the both are the same. If the difference is too high to be neglected, especially for low-permeability reservoirs, the static formation pressure of oil well and water injection well shall be measured respectively, then liquid productivity index and water injectivity index shall be calculated to obtain the reasonable production/injection well ratio. (2) In consideration of the common phenomena in China that the injection/production ratio is higher than 1.0 for high water-cut and low-permeability reservoirs, so the injection-production ratio should be taken into account during the calculation. In addition, during the development process of oilfield, the value of the reasonable production/injection well ratio is changing with the increasing of the water cut and the changing of the development conditions. Generally, the ratio is decreasing with the increasing of water cut, therefore, water wells should be added continuously to meet the demand of enhancing the liquid rate by turning oil wells into water wells or drilling infill adjustment wells.

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Nomenclature

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qL—average single well daily liquid rate (underground volume), m3/d; JL—liquid productivity index, t/(d·MPa); pR—average formation pressure, MPa; pwf—average bottom flowing pressure of oil wells, MPa; fw—water cut, f; Bo—oil formation volume factor, f; ȡo—surface oil density, g/cm3; Bw—water formation volume factor, f; ȡw—water density, g/cm3; QL—oilfield daily liquid rate (underground volume), m3/d; no—production well number, wells; Qinj—oilfield daily water injection rate (underground volume), m3/d; nw—injection well number; Iw—water injectivity index, t/(d·MPa); pinj—average bottom flowing pressure of injection wells, MPa; RIP—injection/production ratio, f; R—production/injection well ratio, f; nt—total oilfield well number; piws—average formation pressure of injection wells, MPa; pows—average formation pressure of production wells, MPa; ǻp—pressure difference between oil well and injection well, MPa; Qw—oilfield daily water rate, t; Qo—oilfield daily oil rate, t.

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