Correct understanding and application of waterflooding characteristic curves

PETROLEUM EXPLORATION AND DEVELOPMENT Volume 46, Issue 4, August 2019 http://www.sciencedirect.com/journal/petroleum-exploration-and-development Cite this article as: PETROL. EXPLOR. DEVELOP., 2019, 46(4): 796–803.

RESEARCH PAPER

Correct understanding and application of waterflooding characteristic curves DOU Hongen1,*, ZHANG Hujun1, SHEN Sibo2 1. Research Institute of Petroleum Exploration & Development, PetroChina, Beijng 100083, China; 2. School of Energy Resources, China University of Geosciences, Beijing 100083, China

Abstract: Through reviewing the generation process and essential characteristics of waterflooding curves, the essence and characteristics of Zhang Jinqing waterflooding curve and Yu Qitai waterflooding curve recommended in Chinese Petroleum Industry Standard “Calculation methods for Recoverable Oil Reserves (SY/T5367—1998)” were discussed, and some technical issues related to the curves were examined in-depth. We found that: (1) All the waterflooding curves are based on empirical formulas derived from oilfield production experience and statistics methods, and can characterize oil displacement features by water quite well. (2) A new waterflooding curve can be derived by combining waterflooding parameters and using different mathematical calculations as long as the parameter combinations and mathematical operation meet a linear relationship, so proposing new waterflooding curves by changing the combination mode has no practical significance anymore. (3) The upwarp of waterflooding curve in the extremely high water cut stage is because the mobility ratio curve has an inflection point with the rapid rise of water cut after reaching a certain value, and the later rapid rise of mobility ratio changes the original two-phase flow dynamics. (4) After entering into water cut stage, all the waterflooding curves with linear relationship can be used to make prediction, even curves with inflection points, as long as they have a straight section above the inflection point. (5) Actual data of waterflooding oilfields has proved that Type A, Zhang Jinqing and Yu Qitai waterflooding curves all can predict accurately oil recoverable reserves in extremely high water cut stage and can be promoted. Key words: extremely high water cut stage; waterflooding characteristic curve; intrinsic essence; reason of upwarp; adaptability

Introduction Researchers of former Soviet Union proposed type A, type B, type C and type D water drive characteristic curves based on statistics and development laws of water drive field from 1950s to 1980s [17]. These four types of characteristic curves have been widely used since introduced into China. Up to now, several new types of water drive characteristic curves have been developed through further study by Chinese researchers[818]. The two most representative of them are the new practical water drive characteristic curve[8] and the new generalized water drive curve[9] published on Volume 3 in 1998 and Volume 5 in 1998 of Petroleum Exploration and Development respectively. The methods proposed by these two papers are simple and pragmatic. The recoverable reserves predicted by these methods agree well with the actual production data in water drive oil fields, and the methods have been therefore widely used, and were included in the Chinese petroleum industry standard SY/T5367—1998 in 1998, "Methods for calculating oil recoverable reserves"[10]. The

water drive characteristic curves proposed in references [8] and [9] are named as Zhang Jinqing water drive characteristic curve and Yu Qitai water drive characteristic curve respectively. In 2010, these two water drive characteristic curves were deleted from the petroleum industry standard SY/T5367—2010 "The estimation methods of oil recoverable reserves". In this work, by looking into the nature and essential characteristics of water drive curves, we discuss some technical issues associated with water drive characteristic curves in Chinese petroleum industry standard, in the hope of helping the correct understanding and application of water drive characteristic curves.

1. Historical review of water drive characteristic curves The water drive characteristic curve is a curve reflecting changes in flow rate of various fluids, which characterizes the macroscopic features during water drive development.

Received date: 11 Oct. 2018; Revised date: 04 Jan. 2019. * Corresponding author. E-mail: [email protected] Foundation item: Supported by China National Science and Technology Major Project (2016ZX05016-006). https://doi.org/10.1016/S1876-3804(19)60237-5 Copyright © 2019, Research Institute of Petroleum Exploration & Development, PetroChina. Publishing Services provided by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

DOU Hongen et al. / Petroleum Exploration and Development, 2019, 46(4): 796–803

Water drive characteristic curves are a kind of empirical models with certain limitations based on statistics which should be selected and used according to practical situation of the reservoir. According to Zhou Weisi et al.[1], the statistics from a parent body is only fit for this parent body itself, and not fit for any subsample not pertaining to this parent body. But we argue that as long as the statistics of the data possess the attributes of the parent body or the statistical data pertains to the parent body, the model from the statistics can be applied. Besides type A, type B, type C and type D water drive characteristic curves recommended by petroleum industry standards SY/T5367-1998 “Methods for calculating oil recoverable reserves” and SY/T5367-2010 “The estimation methods of oil recoverable reserves”, SY/T5367-1998 recommended Zhang Jinqing and Yu Qitai water drive characteristic curves as well. The Zhang Jinqing water drive characteristic curve is based on type C and type D curves, while Yu Qitai water drive characteristic curve has mathematic nature similar with type A curve. After reviewing the setting of type A, type C and type D curves, this paper discusses the technical issues related to Zhang Jinqing and Yu Qitai water drive characteristics curves. 1.1.

Np

 A3  B3 Lp

Wp  a1 e

b1 N p

(1)

By calculating the common logarithm on both sides of the equation, the equation (1) can be expressed as: lgWp  A1  B1 N p (2) Equation (2) was named as type A water drive characteristic curve by late academician Tong Xianzhang of the Chinese Academy of Sciences in 1978, and has been widely applied in the water drive oilfields in China. This curve was included in the petroleum industry standard in 1998[10]. The relationship between cumulative oil production and water cut can be derived from equation (1) as:

Np 

f S  lg ws w  A1  lg  2.303B1  1  f ws  S w  B1

(3)

If the ultimate water cut is fwl, the geologic reserve of the reservoir is N, then the oil recovery factor can be expressed as:

lg ER 

f wl  A1  lg  2.303B1  1  f wl NB1

(4)

Type C and type D water drive characteristic curves

The former Soviet Union scholar Semyon Bachev proposed type C water drive characteristic curve in 1981, the curve is expressed as:

(5)

The former Soviet Union scholar Nazhalohff advanced type D water drive characteristic curve in 1972, the curve is expressed as:

Lp Np

 A4  B4Wp

(6)

The relationship between cumulative oil production and water cut can be derived from equation (5) and (6), and expressed as;

Np 

1  A3 1  f ws  S w   B3

 A4  1

1 Np 

1  f ws  S w  f ws  S w 

B4

(7)

(8)

If the water cut is ultimate water cut fwl, the geologic reserve of the reservoir is N, then oil recovery factor under boundary conditions of type C and type D are respectively:

ER 

1  A3 1  f w l  NB3

Type A water drive characteristic curve

In 1959, based on actual production data of Grozny oilfield, the former Soviet Union scientist Maksimov expressed the relationship between cumulative oil production and cumulative water production through statistical analysis as[1]:

1.2.

Lp

1 ER  1.3.

 A4  1

1  f wl f wl

NB4

(9)

(10)

Zhang Jinqing water drive characteristic curve

A simplified new water drive characteristic curve was proposed by Zhang Jinqing via reviewing various relationships between water cut and recovery percent of reserves. It was included in petroleum industry standard in 1998, and is expressed as:

Wp Np

  A5  B5

Wp NP 2

(11)

The relationship between cumulative oil production and water cut can be derived from equation (11), and is expressed as;

N p  B5  B5

A5 1  f ws  S w  

f ws  S w   A5 1  f ws  S w  

(12)

If the water cut is ultimate water cut fwl, geologic reserve of the reservoir is N, then oil recovery factor at limit condition is expressed as:

B5  B5 ER 

A5 1  f wl  f wl  A5 1  f wl 

(13) N In reference [8], Zhang Jinqing re-wrote equation (5) and (6) respectively as: Np N p  b  c  (14) Lp

 797 

N p  b  c 

Np Lp  N p

(15)

DOU Hongen et al. / Petroleum Exploration and Development, 2019, 46(4): 796–803

Zhang Jinqing did not discriminate the model constants A3, B3, A4 and B4, and used the same coefficients in two different water drive characteristic curves type C and D, both replaced by parameter a and b to unify the characteristic curve expression of type C and type D. In his paper, Zhang Jingjin did not make any transformation between these two equations, and did not confuse their constants either. By adopting the idea mentioned in reference [19], equation (14) and (15) can be derived from equations (5) and (6) respectively. It can be derived from equation (5) that

Np 

Lp  A3 N p B3 Lp

(16)

be a new type of water drive characteristic curve. Therefore, the new constructed functions, for instance equations (14) and (15) are all correct. Zhang Jinqing water drive characteristic curve is a new simple and practical water drive characteristic curve firstly proposed by Chinese scholars, which has been widely applied [13, 17, 1920] . 1.4.

Yu Qitai water drive characteristic curve

According to water drive development pattern of an oilfield, Yu Qitai proposed the statistical relationship between cumulative liquid production, cumulative water production and cumulative oil production [9]:

 Lp Np  a   Wp 

where

Lp  Wp  N p

(17)

Substituting equation (17) into (16), we have:

1 A3 N p Np   B3 B3 Lp

(18)

lg N p  a2  b lg

(19)

Substituting equation (19) into equation (18), the same form as equation (14) can be obtained. In a similar way, by alternating the form of equation (6), we obtain:

Np 

1 A4  1 N p  B4 B4 Wp

(20)

Np Wp  nN p

Lp

(24)

Wp

b In equation (1) b  e 1 , x=Np; in equation (23) x  ln

(21)

Substituting equation (21) into equation (20) to obtain the expression as equation (15). The general equation for both equation (14) and equation (15) can be written as:

N p  a0  b0

(23)

Equation (24) is called Yu Qitai generalized water drive characteristic curve, which was included in petroleum industry standard in 1998[10]. Afterwards some researchers [21-24] carried out many studies with Yu Qitai water drive characteristics curve, but this curve was deleted from petroleum industry standard in 2010[11]. By comparing statistical relationships of equation (1) proposed by Maksimov and equation (23) of Yu Qitai, a general expression of them can be expressed as: y  ab x (25)

Let

1  a0  B  4  b   A4  1  0 B4

b

A water drive characteristic curve can be derived from expression (23):

Let

 1 1 a0  B  3  b1   A3  0 B3

  

.

From the above analysis, the Yu Qitai curve equation (23) and Maksimov curve of equation (1) have the same form. In addition, the Yu Qitai water drive characteristic curve is proved to be simple, practical and accurate in water drive development performance analysis and recoverable reserves estimation [22-23]. In order to broaden the application of Yu Qitai water drive characteristic curve, a water drive recovery factor formula is derived and expressed as:

(22)

When n=1, equation (22) is type C water drive characteristic curve, i.e. equation (14); when n=0, the equation (22) is type D water drive characteristic curve, i.e. equation (15). Thus it can be seen that equations (14) and (15) are obtained from equations (5) and (6) through mathematical transform, and are both correct. Considering from another perspective, if we cannot derive equations (14) and (15) from equations (5) and (6), then we can construct a new function based on equations (5) and (6) as well. This new constructed function only needs to meet linear relation through combination of 2 or 3 parameters, then it will

Lp Wp

ER 

10a2 N

  b 1  f wl   1  f wl    2bf wl   b

2 1  f wl  b  1   4bf wl 1  f wl    2bf wl  

(26)

2. Correct application of water drive characteristic curves The water drive characteristic curve was the relationship firstly proposed by former Soviet Union scientists through statistics on actual production data and laboratory data able to

 798 

DOU Hongen et al. / Petroleum Exploration and Development, 2019, 46(4): 796–803

characterize water drive development performance. Later on, the curve was expanded by researchers and several expressions were derived from it and named after the researchers, Academician Tong Xianzhang named Maksimov water drive characteristic curve as type A water drive characteristic curve in 1978, hereafter Chinese researchers have named water drive characteristic curves as type B, type C and type D curves sequentially. Even though water drive characteristic curves given by scholars of former Soviet Union are fundamentally obtained by statistical regression of data instead of theoretical derivation, these characteristic curves all can be expressed as the relationship of water cut and recovery percent, which make the recoverable reserves prediction easier. It is found by analyzing the shape of the curves that the characteristic curves present different features of concave and convex due to difference in ratio of oil viscosity to water viscosity in the reservoir. In recent years, more than 100 new water drive characteristic curves have been put forward, and the sequential naming method is deficient to name these curves. The water drive characteristic curve is an empirical method for analyzing performance of reservoir development, the precondition of its establishment is that the reservoir is developed by water drive and water is already produced. All the water drive characteristic curves characterize the variation pattern of oil production, water production and liquid production after water free period, and reflect the production performance of water drive reservoirs. Any water drive characteristic curve cannot be used to calculate the parameters such as cumulative oil production and recovery percent of reserves during water free production period. The Yu Qitai water drive characteristic curve has one more parameter than Maksimov’s 2-parameter model, so it is called generalized water drive characteristic curve, which reflects explicit relation of 3 parameters, cumulative oil production, cumulative liquid production and cumulative water production during water drive. According to statistical analysis of various parameters of water drive (or other drives) reservoir, as long as a function mathematically satisfying linear relation of 2 or 3 drive characteristic parameters (i.e. cumulative oil production, cumulative water production, cumulative liquid production, water-oil ratio, liquid-oil ratio and water cut) is constructed, a new type of water drive characteristic curve can be formed. Totally 35 combinations can be created by randomly combining 2 or 3 parameters. Moreover, a number of mathematical operations can satisfy linear relationships, including multiplication and division, square and extraction of square root, exponent or power, logarithm or double logarithmic, and some kind of mixed computation. If derived by 20 different mathematical operations and the 35 parameters combinations, 700 types of water drive characteristic curves can be created. Therefore, one thing must be underlined here that the water drive characteristic curve can be only used to calculate pa-

rameters such as cumulative oil production and recovery percent of reserves after water breakthrough, but the calculated parameters such as recoverable reserves from different water drive characteristic curves cannot be compared to verify the reliability of such results. But these calculated parameters can be compared with the estimated results from Arps decline analysis and numerical simulation.

3. Upwarp of water drive characteristic curve at high water-cut period 3.1. Intrinsic cause of upwarp of water drive characteristic curve at high water cut stage In recent years, the upwarp of water drive characteristic curve has been intensively studied, and a number of characteristic curves characterizing development performance at high water cut stage have been proposed [2532]. It is universally accepted that the water drive characteristic curve upwards because the ratio of oil relative permeability to water relative permeability and water saturation don’t follow exponential relationship anymore at the stage of high water cut.

K ro  S w 

K rw  S w 

 m e  nSw

(27)

In the reference [25, 29], the relationship between

K ro K rw

and Sw at high water cut stage is expressed as:

K ro  S w 

K rw  S w 

 me



 nS w2  cS w

Does the deviation of relationship of



(28)

K ro and Sw from K rw

semi-log straight line really cause the upwarp of the water drive characteristic curve at the high water cut stage? This question will be answered by the following derivation with the assumption of radial fluid flow of oil in the reservoir. The water cut in the formation can be expressed as:

f w  Sw   where

Qw  Bw Qw s 

Qo  Bo Qo s 

Qw Qw  Qo

2 π KK rw  S w  h p r w ln e rw

2 π KK ro  S w  h p r o ln e rw

(29)

(30)

(31)

The surface water cut can be expressed as:

f ws  S w  

Qws Qws  Qo s

(32)

According to equations (30) and (31), the formation oil/water ratio can be expressed as:

 799 

WOR  WOR s From the equation (32), we have:

Bw Bo

(33)

DOU Hongen et al. / Petroleum Exploration and Development, 2019, 46(4): 796–803

WOR s 

1 1 1 f ws ( S w )

(34)

By substituting (34) into (33), the formation water/oil ratio can be further expressed as:

WOR 

Bw 1 1  1 Bo f ws ( S w )

(35)

The mobility ratio can be expressed as:

M 

o K rw  S w   w K ro  S w 

(36)

Substituting equations (30), (31) and (36) into (32) and after reduction, we obtain:

M 

1 1 f ws  S w 

Bw  1 Bo

(37)

By neglecting the effect of capillary pressure and gravity, the relations of water cut with mobility ratio and water-oil ratio can be expressed respectively as:

f w  Sw  

f w  Sw  

1

 K S  1  w ro w o K rw  S w 



1 1 1 M

Bw Qws 1  1 Bw Qw s  BoQos 1  WOR

(38)

(39)

Substituting equations (33) and (34) into (39), the relationship between formation water cut and surface water cut can be obtained as:

f w  Sw  

1  B  1 1 o   1 Bw  f ws ( S w ) 

(40)

By comparing equations (35), (37), (38) and (39), it can be seen that the relation of water-oil ratio and surface water cut and that of mobility ratio and surface water cut are identical, and the relation of mobility ratio with formation water cut and that of water-oil ratio with formation water cut are identical too when the effect of capillary pressure and gravity are neglected. When both formation oil volume factor and formation water volume factor are equal to 1, the formation water cut and surface water cut are the same. Usually, the value of formation oil volume factor ranges from 1.0 to 1.3 and formation water volume factor ranges from 1.00 to 1.05. The water cut ranges from 0 to 100%. Considering high coincidence of the relation of mobility ratio with water cut between theoretical value and field data, a mobility ratio or water-oil ratio curve at water cut of 15% to 99% is plotted with equation (37) or (35), which is abbreviated as theoretical curve. The calculation step is set at 15% at water cut from 15% to 90%, and 1% when water cut exceeds 90% (Fig. 1). Equation (37) or (35) is a monotone increasing function. When water cut is below 80%, mobility ratio (or water-oil ratio) increases slowly and is less than 4.0; when water cut is

Fig. 1.

Relationship of water cut and mobility ratio.

over 80%, mobility ratio (or water-oil ratio) increases quickly, and an inflection point occurs at the water cut of around 85%. When water cut is 90%, which is an increase of 10% from 80%, the mobility ratio (or water-oil ratio) reaches 9.0, increasing by 125%. When the water cut increases another 5% from 90% to 95 %, the mobility ratio (or water-oil ratio) increases by 111%, from 9 to 19. When water cut increases by another 4% from 95% to 99%, the mobility ratio (or water-oil ratio) increases by 521% from 19 to 99. Apparently, small changes in water cut at extremely high water cut stage will cause great change in mobility ratio (or water-oil ratio). Especially after water cut exceeds 90%, the increase of mobility ratio speeds up. From the perspective of oil displacement, the larger the mobility ratio is, the severer the channelling, finguring or tonguing will be, and therefore, the sweeping efficiency will decrease, and the oil displacement effect will be poorer accordingly. Upwarping of a water drive characteristic curve at high water cut stage is mainly becasue of the rapid mobility ratio (water-oil ratio) increase after the inflection point occuring at the water cut above 80%. That rapid rise of mobility ratio or water-oil ratio changes the flow dynamics of two-phase flow, this is the intrinsic reason of upwarping. The deviation of ln

K ro ～Sw relation from semi-log line is the K rw

representation of the dynamic behavior change of two-phase flow. Fig. 2 shows the comparison between the theoretical curve of mobility ratio and water cut with actual curve of lower S1 reservoir in Pucheng oilfield according to data provided by reference [25], these two almost coincide, indicating that the understanding from theoretical curve is supported by the actual development data. The occurring of inflection point and upwarping of the water drive characteristic curve at high water cut stage is an inevitable phenomenon. The development results and economic benefit are poor if the reservoir is developed by water flooding alone, therefore, other development modes must be adopted to reduce the detrimental effect from rapid increase of mobility ratio to improve displacement efficiency and exploitation benefit.

 800 

DOU Hongen et al. / Petroleum Exploration and Development, 2019, 46(4): 796–803

Fig. 2. Comparsion between theoretical and actual mobility ratio curves.

3.2.

Fig. 3. Type A water drive characteristic curve of Pucheng oilfield.

Applicability of water drive characteristic curves

In 1959, through matching of experimental data, Craft and K ro Hawkins found that ～Sw relation followed the expoK rw nential decline expressed in equation (27) and the intermediate section of this curve was almost a straight line, based on this finding, the equation (27) has been widely used to derive water drive characteristic curves. The type A, type B and X-plot characteristic curves of Ershghi I et. al. [33] are all based on equation (27). For many years, a lot of researchers thought that upwarping of water drive curve at high water cut stage was because expressing the exponential term of e with linear function Sw in the equation (27) was insufficient to characterize the performance of high water cut stage. Hence, they thought the quadratic function of Sw should be used and the equation was rewritten as equation (28). In fact, no matter what form of function of Sw is adapted in the exponential term of e, the equation is still an exponential function in nature. If the water drive characteristic curves in forms of quadratic polynomial, complex power function etc. are used to analyze the performance and predict recoverable reserves, the trend lines of the curve in the late high water stage will be very difficult to predict accurately since these curves are mostly nonlinear and the tangent line of these curves are not unique. Even though the early matching effect of nonlinear curves is favorable, these curves cannot be used to predict parameters such as recoverable reserves. The type A, Yu Qitai and Zhang Jinqing water drive characteristic curves were further discussed according to the data of lower S1 reservoir of Pucheng oilfield provided in reference [25]. Fig. 3 is the type A water drive characteristic curve, the square of the correlation factor is 0.999, indicating that it is feasible to predict recoverable reserves with type A water drive characteristic curve. The cumulative oil production of 1992 matched by this curve is 631.91×104 m3, which is 0.89% higher than the actual data, satisfying the accuracy requirement. The predicted water drive ultimate recoverable reserves are 705.38×104 m3. Fig. 4 is the water drive production data of this field

Fig. 4. Yu Qitai water drive characteristic curve of Pucheng oilfield.

matched by Yu Qitai water drive characteristic curve, the square of the correlation factor is 0.999, indicating it is feasible to predict recoverable reserves of late high water-cut stage with Yu Qitai water drive characteristic curve. The cumulative oil production of 1992 matched by this curve is 619.40×104 m3, which is 1.10% lower than the actual data, meeting the accuracy requirement too. The predicted ultimate water drive recoverable reserves are 661.46×104 m3. Fig. 5 is the water drive production data of this field matched by Zhang Jinqing water drive characteristic curve, the square of regression correlation factor is 1.000, indicating Zhang Jinqing water drive characteristic curve is also adaptable to predict recoverable reserves of late high water-cut

Fig. 5. Zhang Jinqing water drive characteristic curve of Pucheng oilfield.

 801 

DOU Hongen et al. / Petroleum Exploration and Development, 2019, 46(4): 796–803

stage. The cumulative oil production of 1992 matched by this curve is 614.00×104 m3, which is 2.00% lower than the actual data, meeting the accuracy requirement too. The predicted ultimate recoverable reserves are 648.44×104m3. According to available data[34], the lower S1 reservoir of Pucheng oilfield is a monolithic reservoir with high permeability put into production in 1980, it had geologic reserves of 1 384.15×104 m3 and calibrated recovery factor of 51.30%. Its water cut reached 94.69% in 1992, and increased further to 97.74% in 2013, when the recovery percent of geologic reserves amounted to 50.89%. At present the reservoir is still produced. The water drive recovery factors predicted by type A curve, Yu Qitai water drive characteristic curve and Zhang Jinqing characteristic curve are 50.96%, 47.79%, and 46.84% respectively. Comparing the predicted values from these three water drive characteristic curves with the present actual development result, it is found that the type A curve agrees well with the actual data, while the results predicted by Yu Qitai and Zhang Jinqing water drive characteristic curves are slightly lower than the actual data. The oil recovery factor of lower S1 reservoir of Pucheng oilfield was also predicted by new methods proposed by references [25] and [29], the results are 48.25% and 48.50% respectively, which are also slightly lower than the actual recovery percent. It can be seen that even though the different water drive characteristic curves match well, with square of correlation factors ranging from 0.995 to 1.000, and have high fitting degree of cumulative oil production, the ultimate recovery factors predicted by these curves are not very accurate. The reason is that the predicted results by water drive characteristic curve are related to statistical samples. Factors affecting samples include petroleum physical property of reservoir fluids and physical property of reservoir rock. This illustrates that each water drive characteristic curve has its applicable range, and not every water drive characteristic curve fits all water drive oilfields, so the water drive characteristic curve must be chosen carefully according to the actual situation of the oilfield in concern. In summary, any linear water drive characteristic curve can be used during development in water cut period. Even though with inflection point, reliable results can be obtained by using linear part of water drive characteristic curves before inflection point.

4.

combination and mathematical operation meet linear relation. The number of combinations satisfying linear relationship is over 700, so it has no actual significance to propose new water drive characteristic curves by changing combinations anymore. The water drive characteristic curves cannot be used to calculate the parameters such as cumulative oil production and recovery percent of reserves during water free development period. The major reason of upwarping of water drive characteristic curve at high water cut development period is that an inflection point occurs when mobility ratio increases with water cut after water cut reaches a certain level and the rapid rise of mobility ratio changes the behavior of two- phase flow. Any linear water drive characteristic curve can be used to do prediction of the development during water cut period. Even with the inflection point, reliable results can be obtained with the linear part of water drive characteristic curves. Actual data of water drive oilfields has confirmed that the type A, Zhang Jinqing and Yu Qitai water drive characteristic curves all can accurately predict recoverable reserves at high water cut period, and can be promoted. These curves are recommended to be included in petroleum industry standard.

Acknowledgements I wish to express my sincere thanks to Doctor Qu Debin who provided very valuable suggestions during the writing of this paper.

Nomenclature a, a0, a1, a2, a0 , a10 , A1, A3, A4, A5, b, b1, b, b , b0 , b01 , B1, B3, B4, B5,

c, c , m, n—regression constant of model;

Bo, Bw—volume factor of formation oil and formation water, 3

m /m3; ER—ultimate recovery factor during water drive stage, %; fw(Sw)—formation water cut, %; fwl—ultimate water cut in water drive stage, %; fws(Sw)—surface water cut, %; h—net pay thickness, m; K—reservoir absolute permeability of, m2; Kro(Sw), Krw(Sw)—relative permeability of oil and water phase, dimensionless; Lp—cumulative liquid production at surface condition, 104 m3; M—mobility ratio, dimensionless;

Conclusions

N—original oil geological reserve, 104 m3;

The existent water drive characteristic curves are all derived based on empirical formula obtained from practical experience and statistical method instead of the flow equation of water and oil. What really matters is that the curves can actually characterize the dynamic performance of water drive reservoir, and reflect the relations between cumulative oil production, water production, liquid production and water cut. New type water drive characteristic curves can be obtained by different mathematical operations and combination of the water drive characteristic parameters, as long as the parametric

Np—cumulative oil production at surface condition during water drive stage, 104 m3;

 802 

Qo—oil production under formation condition, m3/s; Qos—oil production under surface condition, m3/s; Qw—water production under formation condition, m3/s; Qws—water production under surface condition, m3/s; re—radius of oil drainage, m; rw—radius of well bore, m; R—correlation coefficient, dimensionless; Sw—water saturation, %;

DOU Hongen et al. / Petroleum Exploration and Development, 2019, 46(4): 796–803

Wp—cumulative water production under surface condition, 104m3; WOR—water-oil volume ratio under reservoir condition, dimensionless; (WOR)s—water oil volume ratio at surface condition, dimensionless; p—production pressure difference, Pa; μo, μw—viscosity of formation oil and formation water respectively, Pas.

References [1]

[2]

[3]

[4]

[5] [6]

[7] [8] [9] [10]

[11]

[12]

[13]

[14]

[15] [16]

[17]

ZHOU Weisi, GAO Canhua, CHEN Yanjin. Application of displacement characteristic curve in oilfield development analysis. Chinese Journal of Theoretical and Applied Mechanics, 1978, 14(1): 72–75. TONG Xianzhang. On exploration of statistical regularity in naturally flooded and artificially water flooded reservoirs. Petroleum Exploration and Development, 1978, 5(6): 40–69, 81. WAN Jiye. A series of displacement characteristic curve and its application on the prediction performance of waterflooded oil reservoirs(Ⅰ). Petroleum Exploration and Development, 1982, 9(6): 65–72. WAN Jiye. A series of displacement characteristic curve and its application on the prediction performance of waterflooded oil reservoirs(Ⅱ). Petroleum Exploration and Development, 1983, 10(1): 44–48. YU Qitai, JIN Hongwei. About the general water drive characteristic curve. Acta Petrolei Sinica, 1995, 16(1): 61–64. ZHANG Hujun. Derivation and application of a new model for predicting recoverable reserves. Well Testing and Production Technology, 1995, 16(1): 38–42. YU Qitai. Study on the water displacement curve: The 1st in series. Xinjiang Petroleum Geology, 1996, 17(4): 364–369. ZHANG Jinqing. A new practical water displacement curve. Petroleum Exploration and Development, 1998, 25(3): 56–57. YU Qitai. A new generalized water drive curve. Petroleum Exploration and Development, 1998, 25(5): 48–50. Standardization Technical Committee for Oil and Gas Field Development. Methods for calculating oil recoverable reserves: SY/T 5367—1998. Beijing: Petroleum Industry Press, 1999. Standardization Technical Committee for Oil and Gas Field Development. The estimated methods of oil recoverable reserves: SY/T 5367—2010. Beijing: Petroleum Industry Press, 2011. YU Qitai. A generalized water displacement characteristic curve: Казаков. Petroleum Geology and Oilfield Development in Daqing, 1997, 16(3): 45–48. YU Qitai. Application of Zhang’s water drive curve and its characteristics of oil-water seepage flow. Xinjiang Petroleum Geology, 1998, 19(6): 507–511. ZHOU Guanghou, ZHANG Yuzhuang. Generalized water flooding characteristic curve. Petroleum Geology and Oilfield Development in Daqing, 2000, 19(2): 21–23. XIANG Tianzhang, WU Yi. A new generalized water drive curve. Fault-Block Oil & Gas Field, 2003, 10(5): 58–60. LIU Shihua, GU Jianwei, YANG Renfeng. New water-flooding characteristic curve at high water-cut stage. Journal of Liaoning Technical University, 2011, 30(S1): 158–163. ZHOU Peng, CHEN Xiaofan, LE Ping, et al. Establishment of a new type of water drive characteristic curve. Petroleum Ge-

 803 

ology and Recovery Efficiency, 2012, 19(4): 99–102. [18] WANG Hua. Formula derivation and application of improved water drive characteristic curve calculation technique for recoverable reserves. Petroleum Geology and Recovery Efficiency, 2012, 19(4): 84–86. [19] LI Feng, ZHANG Weijun, ZHANG Shaowei. Derivation and application of Zhangjinqing water drive characteristic curve. Liaoning Chemical Industry, 2012, 41(6): 611–612. [20] HU Gang. A new method for calculating volumetric sweep efficiency in a water-flooding oilfield. Petroleum Exploration and Development, 2013, 40(1): 103–106. [21] SHI Mingjie, ZHONG Jidong. Comprehensive application of water drive and production decline curve in late period of oilfield development. Petroleum Geology and Oilfield Development in Daqing, 2004, 23(4): 26–27. [22] DENG Chang. Theoretical analysis and prediction of horizontal well production decline in thin bottom water reservoir. Chengdu: Southwest Petroleum University, 2012. [23] SUN Yujin. Evaluation of development effect of water drive in C reservoir and study on measures to exploit potential. Chengdu: Southwest Petroleum University, 2016. [24] ZHOU Yuhui, HU Shuyong, DONG Haijing, et al. Applied condition of type-Y1 water drive characteristic curve in bottom water reservoir. Xinjiang Petroleum Geology, 2010, 31(2): 184–185. [25] SONG Zhaojie, LI Zhiping, LAI Fengpeng, et al. Derivation of waterflooding characteristic curve for high water-cut oilfields. Petroleum Exploration and Development, 2013, 40(2): 201–208. [26] LIU Yujian, WANG Xinhai, DU Zhihua. Theoretical derivation of a new water flooding characteristic curve relationship. China Sciencepaper, 2014, 9(6): 734–738. [27] CUI Chuanzhi, XU Jianpeng, WANG Duanping. A new waterflooding characteristic curve at ultra-high water cut stage. Acta Petrolei Sinica, 2015, 36(10): 1267–1271. [28] LIANG Baohong. New method to determine the time of turning point of water drive characteristic curve in ultra high water cut stage. Petroleum Geology and Recovery Efficiency, 2015, 22(5): 103–106. [29] WANG Xiaolin, YU Lijun, LI Zhiping. New waterflooding characteristic curves at the stage of extra-high water cut. Petroleum Geology and Oilfield Development in Daqing, 2015, 34(6): 54–56. [30] FAN Haijun, ZHU Xueqian. Derivation and application of new waterflooding characteristic curve in high water-cut oilfield. Fault-Block Oil & Gas Field, 2016, 23(1): 105–108. [31] DENG Sen, WANG Nutao, MENG Lingqiang, et al. Establishment and application of the new two-type water-flooding characteristic curves at high water cut stage. Petroleum Geology and Oilfield Development in Daqing, 2017, 36(4): 58–63. [32] WANG Jiqiang, SHI Chengfang, JI Shuhong, et al. New water drive characteristic curves at ultra-high water cut stage. Petroleum Exploration and Development, 2017, 44(6): 955–960. [33] ERSHGHI I, OMORIGIE O. A method for extrapolation of cut vs recovery curves. Journal of Petroleum Technology, 1978, 30(2): 203–204 [34] YANG Changhua, DENG Ruijian, NIU Baolun, et al. CO2 foam sealing channeling system research and application in Pucheng Es1 reservoir. Fault-Block Oil & Gas Field, 2014, 21(1): 118–120, 124.