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RESEARCH PAPER

PETROLEUM EXPLORATION AND DEVELOPMENT Volume 40, Issue 2, April 2013 Online English edition of the Chinese language journal

Cite this article as: PETROL. EXPLOR. DEVELOP., 2013, 40(2): 216–223.

Received date: 28 Dec. 2011; Revised date: 26 Oct. 2012. * Corresponding author. E-mail: [email protected] Foundation item: Supported by the National Natural Science Foundation of China (51174178); National Science and Technology Major Project of China (2011ZX05016-006). Copyright © 2013, Research Institute of Petroleum Exploration and Development, PetroChina. Published by Elsevier BV. All rights reserved.

Derivation of water flooding characteristic curve for high water-cut oilfields

SONG Zhaojie1,*, LI Zhiping1, LAI Fengpeng1, LIU Gang2, GAN Huohua3 1. School of Energy Resources, China University of Geosciences, Beijing 100083, China; 2. Research Institute of Yanchang Petroleum (Group) Co. Ltd, Xi’an 710054, China; 3. China National Oil-Gas Exploration & Development Corporation, Beijing 100034, China

Abstract: The linear relationship between relative permeability ratio (Kro/Krw) and water saturation (Sw) on the semi-log coordinate in the stage of middle water-cut is the theoretical basis for the derivation of traditional water flooding characteristic curve. However, the re-lationship of Kro/Krw versus Sw deviates from the straight line in the high water-cut stage, which results in the upwarping of water flooding characteristic curve. In order to accurately predict the production performance and recoverable reserves in the late development stage, the relative permeability curves of actual cores were averaged. Furthermore, using the core data in the reference, a new expression of Kro/Krw

versus Sw was obtained by regression for the high water-cut stage. On the basis of the frontal-drive equation and the average water satura-tion equation proposed respectively by Buckley-Leverett and Welge, a new water flooding characteristic curve was derived which is more applicable for high water-cut oilfields. The calculation results indicate that the recoverable reserve calculated by the new approach is al-most equal to the result of the production decline method, proving it is a practical tool in the prediction of production indexes in the late development stage of oilfields.

Key words: water flooding characteristic curve; high water-cut; relative permeability; water saturation; production decline; recoverable reserves

Introduction

Water flooding characteristic curves have a widespread ap-plication in China because of its ability to predict recoverable reserves based on the cumulative oil production, cumulative water production, cumulative liquid production and water-oil ratio of target oilfields. Type-A, type-B, type-C, and type-D curves, among tens of water flooding characteristic curves, are the most traditional methods that have been derived theoreti-cally by Chen [1] and Yu [2], respectively. Besides, a new type of water flooding characteristic curve proposed by Zhang [3] is applicable for a wider scope. In actual application, traditional water flooding characteristic curves are always upwarping in the late development stage of oilfields. In order to solve this problem, based on the principle of oil and water flow in po-rous media, both type-A and type-B water flooding character-istic curves were attested to exhibit a second straight line of different slope in the late development stage of oilfields, and the second equation was derived for both type-A and type-B water flooding characteristic curves [4]. The recoverable re-serve results predicted by various methods are much different

because of their distinct theoretical foundations. Generally, the prediction results of type-A and type-C water flooding characteristic curves are more reliable while that of type-D water flooding characteristic curve is much bigger [5]. The same method should be selected and utilized in different de-velopment stages of a target oilfield to predict the recoverable reserves for comparison analysis [6]. Additionally, the optimal prediction result should be in accordance with the production performance rather than the average value of different meth-ods.

This study focuses on the applicability of traditional water flooding characteristic curves for the purpose of deducing a new water flooding characteristic curve for high water-cut oilfield. By analyzing the non-linear plot of relative perme-ability ratio Kro/Krw versus water saturation Sw on semi-log coordinate, a new correlation of Kro/Krw versus Sw was ob-tained by regression in high water-cut stage, and further, a new water flooding characteristic curve was derived that is more applicable in high water-cut oilfield. The calculation results have proved the convenience and applicability of the

SONG Zhaojie et al. / Petroleum Exploration and Development, 2013, 40(2): 216–223

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Table 1 Equations of traditional water flooding characteristic curves and relevant formulas to predict recoverable reserves

Type of water flooding characteristic curves Equation of traditional water flooding

characteristic curves Formula to predict recoverable reserves

Type-A water flooding characteristic curve [1] p 1 1 plgW a b N= + w maxpmax 1

1 1 w max

1 lg2.303 (1 )

fN ab b f⎡ ⎤

= −⎢ ⎥−⎣ ⎦

Type-B water flooding characteristic curve [1] 2 2 plgWOR a b N= + pmax max 2

2

1 [lg ]N WOR ab

= −

Type-C water flooding characteristic curve [2] p

3 3 pp

La b L

N= +

pmax 3 w max

3

1 1 (1 )N a fb⎡ ⎤= − −⎣ ⎦

Type-D water flooding characteristic curve [2] p4 4 p

p

La b W

N= +

4 w max

pmax4 w max

( 1)(1 )1 1 a fNb f⎡ ⎤− −

= −⎢ ⎥⎣ ⎦

ZHANG Jin-qing water flooding characteristic curve [3] p p5 5 2

p p

W Wa b

N N= − +

( )( )

5 wmaxpmax 5 5

wmax 5 wmax

11

a fN b b

f a f−

= −+ −

p 1 1 plgW A B N= + w maxpmax 1

1 1 w max

1 lg2.303 (1 )

fN AB B f⎡ ⎤

= −⎢ ⎥−⎣ ⎦

CHEN Yuan-qian two straight lines water flooding characteristic curve [4]

2 2 plgWOR A B N= + [ ]pmax max 22

1 lgN WOR AB

= −

new approach that is beneficial to the evaluation method of water flooding in a high water-cut oilfield.

1 Evaluation of traditional water flooding characteristic curves

Table 1 summarizes the equations of traditional water flooding characteristic curves and the relevant formulas for recoverable reserves prediction. Water flooding characteristic curves are utilized to match the straight line to predict recov-erable reserves, therefore, it is important to determine the initial point of the straight line precisely. It is generally con-sidered that all the water flooding characteristic curves would exhibit a straight line when water cut exceeds 50% [6]. For the application standardization of water flooding characteristic curves, the initial point of the straight line should be the same one when this method is applied to predict recoverable re-serves in different development stages of a target oilfield so as to avoid the possible phenomenon that the prediction results change from large to small [6].

The hypothesis of traditional water flooding characteristic curves is the linear relationship of Kro/Krw versus Sw on semi-log coordinate in the intermediate water saturation stage [7]. The formula is

wro

rw

e mSK nK

−= (1)

The traditional water flooding characteristic curves are generally upwarping in the late development stage of oilfields. As Fig. 1 depicts, a steam tube method was proposed to cal-culate and plot the water flooding characteristic curve [8]. It makes clear that the plot is upwarping in the later stage. When the economic limit of water-oil ratio is assumed 49, the re-covery factor predicted by type-B water flooding characteris-tic curves is clearly higher than the actual result. Therefore, traditional curves are not applicable for the high water-cut oilfield.

Fig. 1 Type-B water flooding characteristic curve in the late development stage of a bulky sandstone reservoir in Shengli Oil-field

2 New correlation of Kro/Krw versus Sw in high water saturation stage

From Equation (1), the linear relationship of relative per-meability ratio Kro/Krw versus water saturation Sw on semi-log coordinate is very typical in the intermediate water saturation stage. However, the plot of Kro/Krw versus Sw deviates from the straight line in high water saturation stage [4, 9], which can not be correctly described by Equation (1). Applying the standardized method [10], the relative permeability data of eight cores in a block of Beier Oilfield and five cores in Dong 14 Block of the Yushulin Oilfield were normalized as shown in Table 2. Besides, referring to the core data in Zhuang 19 Block of Xifeng Oilfield [11], the semi-log plots of Kro/Krw versus Sw were drawn in Fig. 2.

Fig.2 shows that the plots of Kro/Krw versus Sw using the core data from different oilfields all deviate from the straight line in high water saturation stage. In order to deduce the equation of water flooding characteristic curve for high wa-ter-cut oilfield, a new regression formula of Kro/Krw versus Sw

was proposed to match the deviated core data in high water saturation stage. Fig. 3 and Table 3 show the regression re-sults.


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Table 2 Relative permeability data of different blocks

A block in Beier Oilfield Dong 14 Block of Yushulin Oilfield Zhuang 19 Block of Xifeng Oilfield

Relative permeability/f Relative permeability/f Relative permeability/f Water saturation/f

Oil phase Water phase Water saturation/f

Oil phase Water phaseWater saturation/f

Oil phase Water phase

0.622 0 1.000 0 0 0.477 9 1.000 0 0 0.367 0 1.000 0 0

0.632 8 0.871 7 0.019 2 0.488 6 0.862 4 0.005 9 0.383 1 0.883 2 0.053 7

0.643 6 0.754 2 0.027 5 0.499 4 0.737 8 0.010 9 0.399 1 0.774 8 0.072 4

0.654 3 0.647 2 0.034 0 0.510 1 0.625 6 0.015 8 0.415 2 0.674 6 0.086 3

0.665 1 0.550 2 0.039 4 0.520 8 0.525 2 0.020 5 0.431 2 0.582 5 0.097 7

0.675 9 0.462 9 0.044 3 0.531 6 0.435 9 0.025 0 0.447 3 0.498 2 0.107 6

0.686 7 0.384 8 0.048 7 0.542 3 0.357 2 0.029 5 0.463 3 0.421 6 0.116 5

0.697 4 0.315 5 0.052 8 0.553 0 0.288 4 0.033 9 0.479 4 0.352 3 0.124 5

0.708 2 0.254 7 0.056 6 0.563 7 0.228 9 0.038 3 0.495 4 0.290 2 0.131 9

0.719 0 0.201 7 0.060 2 0.574 5 0.178 1 0.042 6 0.511 5 0.235 1 0.138 8

0.729 8 0.156 3 0.063 6 0.585 2 0.135 3 0.046 8 0.527 5 0.186 6 0.145 2

0.740 5 0.117 9 0.066 8 0.595 9 0.099 8 0.051 0 0.543 6 0.144 6 0.151 4

0.751 3 0.086 0 0.069 9 0.606 7 0.071 0 0.055 2 0.559 6 0.108 7 0.157 2

0.762 1 0.060 1 0.072 9 0.617 4 0.048 3 0.059 3 0.575 7 0.078 7 0.162 7

0.772 9 0.039 8 0.075 8 0.628 1 0.031 0 0.063 4 0.591 7 0.054 2 0.168 0

0.783 6 0.024 4 0.078 5 0.638 9 0.018 3 0.067 5 0.607 8 0.034 8 0.173 1

0.794 4 0.013 4 0.081 2 0.649 6 0.009 6 0.071 5 0.623 8 0.020 3 0.178 0

0.805 2 0.006 2 0.083 8 0.660 3 0.004 2 0.075 6 0.639 9 0.010 1 0.182 7

0.816 0 0.002 1 0.086 4 0.671 0 0.001 3 0.079 6 0.655 9 0.003 8 0.187 3

0.826 7 0.000 3 0.088 8 0.681 8 0.000 2 0.083 5 0.672 0 0.000 7 0.191 7

0.837 5 0 0.091 3 0.692 5 0 0.087 5 0.688 0 0 0.196 0

Fig. 2 Plot of Kro/Krw versus Sw in different blocks

Fig. 3 Matching plot of deviated data in high water saturation stage

Table 3 Regression results of the deviated data of plots of Kro/Krw versus Sw

Oilfield Block Regression formula Correlation coefficient

A Block in Beier Oilfield ln(Kro/Krw)=－876.47Sw2+1 311.5Sw－490.58 0.991 2

Zhuang 19 Block of Xifeng Oilfield ln(Kro/Krw)=－548.1Sw2+639.25Sw－187.56 0.995 0

Dong 14 Block of Yushulin Oilfield ln(Kro/Krw)=－1 160.1Sw2+1 419.7Sw－434.69 0.993 4

As shown in Fig. 3 and Table 3, applying binomial expres-

sion to match the data of ln(Kro/Krw) versus Sw enabled us to obtain a good agreement in the high water saturation stage, which indicates that the relationship of Kro/Krw versus Sw can

be represented exactly by the binomial equation in high water saturation stage. The binomial equation is given as

2row w

rw

ln K fS gS hK

= + + (2)


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Transforming Equation (2) gives

2

w w( )ro

rw

e aS bSK dK

− += (3)

where, d=eh, a=−f, b=−g. It can be seen that the new correla-tion of Kro/Krw versus Sw in high water saturation stage (Equa-tion (3)) is different from that in the intermediate water satu-ration stage (Equation (1)).

3 Theoretical derivation of the new water flooding characteristic curve

Based on the new regression formula of Kro/Krw versus Sw in high water saturation stage discussed previously, the equation of water flooding characteristic curve in high water-cut oil-field is derived theoretically in this section.

On the basis of the regression of relative permeability data in high water saturation stage, the relationship of relative permeability ratio at the exit-end of the core Kroe/Krwe versus the exit-end water saturation Swe is given as

2

we we( )roe

rwe

e aS bSK dK

− += (4)

Ignoring gravity and capillary force, in the steady state flow condition of oil and water, the relationship of relative perme-ability ratio Kroe/Krwe versus the transient production rate of oil and water at the exit-end of the core is expressed as [12]

roe o o o o

rwe w w w w

K Q BK Q B

μ γμ γ

= (5)

Substituting Equation (4) into Equation (5) gives the corre-lation of water-oil ratio WOR as:

2

we wew o o o

o w w w

eaS bSQ BWORQ d B

μ γμ γ

+= = (6)

Applying the frontal-drive equation and the average water saturation equation proposed by Buckley-Leverett [13] and Welge [14], respectively, and combining the experimental the-ory of Ilflos, the relationship of the exit-end water saturation versus recovery degree was presented in Reference [1] as

( ) ( ) ( )p oiwe wi or oi wi or

3 1 3 11 12 2 2 2

N SS S S S R S S

N⎛ ⎞

= + − − = + − −⎜ ⎟⎝ ⎠

(7) By logarithmic transformation of Equation (6), and substi-

tuting Equation (7) into it, we get 2

oi oio o o

w w w

3 32 2lg lg =

2.303

a S R E b S R EBWOR

d Bμ γμ γ

⎛ ⎞ ⎛ ⎞+ + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠= +

22oi oi9 (6 3 )

9.212 4.606aS aE b SR R+

+ +

2

o o o

w w w

lg2.303

B aE bEd Bμ γμ γ

++ (8)

where, wi or1 (3 1)2

E S S= + − . The new equation of water

flooding characteristic curve in high water-cut oilfield is de-rived as 2lgWOR AR BR C= + + (9)

where, 2

oi99.212aSA = , oi(6 3 )

4.606aE b SB +

= ,

2o o o

w w w

lg2.303

B aE bECd Bμ γμ γ

+= + .

Substituting p /R N N= into Equation (9) obtains the rela-tionship of WOR versus cumulative oil production Np in high water-cut oilfield as 2

p plgWOR A N B N C′ ′= + + (10)

where, 2

AAN

′ = , BBN

′ = .

When the water-oil production ratio reaches the economic limit WOR max, the water flooding recoverable reserve can be computed as

2

maxpmax

4 ( lg )2

B B A C WORN

A′ ′ ′− + − −

=′

(11)

4 Applications and discussions

4.1 Case of Craig F. F. Jr.

In order to evaluate the applicability and feasibility of the new water flooding characteristic curve in a high water-cut oilfield, the actual production data from References [5] and [16] were selected to conduct calculation and result analysis.

Table 4 summarizes the relationship of relative permeabil-ity of oil and water, and the fractional flow of water, i.e., wa-ter cut, versus water saturation, while the fist four rows of Table 5 show water flooding performance of an example res-ervoir after water breakthrough [15]. The analytic equation of recovery degree in a water flooding reservoir was computed in Reference [1] as

p wa wi wa wi

oi wi1N S S S SRN S S

− −= = =

− (12)

By applying Equation (12), the recovery degree, cumulative oil production and cumulative water production are calculated at different exit-end water saturations. In the calculation, the initial water saturation is 0.1. The exit-end water saturation is 0.469 and the cumulative water production is 0 at the time of water breakthrough.

The plot of lgWOR versus Np is drawn in Fig. 4 using the calculated data above. The first half of the plot is a straight

Table 4 Relative permeability and fractional flow of water at different water saturations in an example reservoir [15]

Relative permeability /f Water saturation /f Oil phase Water phase

Fractional flow of water /f

0.10 1.000 0 0 0.30 0.373 0.070 0.272 9 0.40 0.210 0.169 0.616 8 0.45 0.148 0.226 0.753 3 0.50 0.100 0.300 0.857 1 0.55 0.061 0.376 0.925 0 0.60 0.033 0.476 0.966 5 0.65 0.012 0.600 0.990 1 0.70 0 0.740 1.000 0


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Table 5 Water flooding performance of an example reservoir after water breakthrough

Exit-end water saturation /f

Fractional flow of water/f

Water-oil ratio

Average water saturation/f

Recovery degree/f

Cumulative oil production/104 m3

Cumulative water production/104 m3

0.469 0.798 3.950 0.563 0.514 2.284 0

0.495 0.848 5.579 0.582 0.536 2.378 0.523

0.520 0.888 7.928 0.600 0.556 2.467 1.227

0.546 0.920 11.500 0.617 0.574 2.550 2.191

0.572 0.946 17.518 0.636 0.596 2.644 3.833

0.597 0.965 27.571 0.652 0.613 2.723 6.009

0.622 0.980 49.000 0.666 0.629 2.792 9.394

0.649 0.990 99.000 0.681 0.646 2.866 16.719

0.674 0.996 249.000 0.694 0.660 2.930 32.688

0.700 1.000 ∞ 0.700 0.667 2.960

Note: Original Oil in Place is 4.440×104 m3.

Fig. 4 Plot of water-oil ratio versus cumulative oil production of an example reservoir in Reference [15]

line, while the latter half exhibits upwarp. Applying type-B and the new water flooding characteristic curve to match the first and latter half of the plot, respectively, regression formu-las and the predicted recoverable reserves are presented in Table 6. Besides, the recoverable reserves predicted by the type-A, type-C, type-D, Zhang Jinqing and Chen Yuanqian two-straight-line water flooding characteristic curves are listed in Table 6 for comparison.

As seen in Fig. 4, good match is obtained by applying the type-B water flooding characteristic curve in the first half of the plot, while the upwarp of plot in the latter stage indicates poor match. Applying the water flooding characteristic curve for the high water-cut oilfield with a pattern of a parabola

enables us to get a better agreement for the upwarping data in the latter half of the plot. In this case, the production data cal-culated by Equation (12) was matched by different water flooding characteristic curves to predict recoverable reserves; therefore, the cumulative oil production calculated by Equa-tion (12) when water cut reaches 0.98, 2.792×104 m3, was considered the standard value of water flooding recoverable reserves to conduct comparison with results of different water flooding characteristic curves. In Table 6, the recoverable reserves predicted by the water flooding characteristic curve in the high water-cut oilfield perfectly matches the value cal-culated by Equation (12), which indicates the new method is better applicable in high water-cut oilfields.

In order to conduct a theoretical analysis of the upwarping of water flooding characteristic curve in the late development stage of oilfields, the plot of relative permeability ratio Kro/Krw versus water saturation Sw in this case is drawn in Fig. 5 to conduct a comparison analysis with the upwarping of water flooding characteristic curve in Fig. 4.

Because the relative permeability data provided by the ref-erence and the cumulative oil production calculated by Equa-tion (12) are scattered points, it is difficult to determine ex-actly from which point the plots deviate from the straight line in Fig. 4 and Fig. 5. However, what can be determined is that the deviating point in Fig. 4 is somewhere between the cumu-lative oil production interval of 2.550×104m3 and 2.644×

Table 6 Water flooding recoverable reserves predicted by different water flooding characteristic curves

Types of water flooding characteristic curves Regression formulas Predicted recoverable reserves/104 m3

Type-A water flooding characteristic curve lgWp=－6.399 2+2.992 5Np 2.734

Type-B water flooding characteristic curve lgWOR=－3.153 5+1.891Np 2.871

Type-C water flooding characteristic curve Lp/Np=0.257 6+0.385 6Lp 2.727

Type-D water flooding characteristic curve Lp/Np=1.042 2+0.412 4Wp 2.665

Zhang Jinqing water flooding characteristic curve Wp/Np=－0.045 3+2.422 7Wp/Np2 2.661

Chen Yuanqian two straight lines water flooding characteristic curve

lgWOR=－3.816 6+2.175 7Np

(the second straight line) 2.841

Water flooding characteristic curve for high water-cut oilfields

lgWOR=3.112 5Np2－12.292Np+12.98 2.806


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Fig. 5 Plot of Kro/Krw versus Sw in Reference [15]

104m3 and the corresponding water cut interval is between 0.920 and 0.946. The deviating point in Fig. 5 is in the water saturation interval between 0.55 and 0.60 and the correspond-ing water cut interval is between 0.925 0 and 0.966 5. There-fore, the two deviating points in Fig. 4 and Fig. 5 are almost located in the same water cut interval, which indicates that the semi-log plot of Kro/Krw versus Sw deviating from the straight line in high water saturation stage results in the upwarping of traditional water flooding characteristic curves in the late de-velopment stage of oilfields. Meanwhile, the case result proves that the hypothesis of the new water flooding charac-teristic curve is rational.

4.2 Case of Sha 1 Reservoir in the Pucheng Oilfield

The actual production data of Sha 1 Reservoir is given in Reference [16]. This reservoir has an oil volume factor of 1.429 9 m3/m3 and a water volume factor of 1 m3/m3; there-fore, the cumulative oil production, cumulative water produc-tion, cumulative liquid production and water cut on the sur-face can be calculated (Table 7), of which the value of the cumulative water production on the surface equals that inside the reservoir.

Based on the calculated production data above, the plot of lgWOR versus Np is drawn in Fig. 6 where the production data in 1984 was not included because the water cut was just 0.180 4 and the water flooding characteristic curve has not yet ex-hibited the characteristic of a straight line.

Fig. 6 illustrates the first half of the plot corresponds with the traditional relationship of straight line, while the latter half is upwarping clearly. Applying water flooding characteristic curve in high water-cut oilfield to match the latter half of the plot yielded the expression of water-oil ratio versus cumula-tive oil production as 2

p plg 0.000 023 2 0.019 507 4.369WOR N N= − + (13)

Substituting the relevant coefficient into Equation (11) gives the water flooding recoverable reserves of 667.95×104

m3 in Sha 1 Reservoir in the Pucheng Oilfield. There are various evaluation methods for water flooding

performance [17−20] of which the production decline method is the most popular one. In order to verify the result of the new approach, the expanded Arps decline method of generalized decline types is utilized to conduct a comparison analysis. The expanded Arps decline method was proposed by Chen Yuan-qian in 1994 that considers the exponential and harmonic de-clines as two special types of hyperbolic decline, and further, was derived based on hyperbolic decline to predict the recov-erable reserves [21−22].

From the production performance of Sha 1 Reservoir in the Pucheng Oilfield, we can see the production started to decline in 1986 (Fig. 7). However, for precise prediction of the recov-erable reserves, the year of 1988 is determined as the initial point to calculate the maximum cumulative oil production in the selected decline period. Table 8 summarizes the produc-tion data in the selected decline period of Sha 1 Reservoir in the Pucheng Oilfield.

Applying linear regression trail and error method and as-suming decline exponent as 0, we can obtain a straight line with the correlation coefficient of 0.9999 (Fig. 8), which illus-trates the annual oil production corresponds to an exponential decline in the decline period. The correlation of cumulative oil production in the selected decline period versus annual oil production is given by regression as p 141.88 3.069N Q= − (14)

Reference [22] provided the time conversion coefficient of 1.0; further, we can work out the maximum cumulative oil

Table 7 Production data of Sha 1 reservoir in the Pucheng Oilfield

Year Cumulative oil production (inside reservoir)/104 m3

Cumulative water production/104 m3

Cumulative oil production (on the surface)/104 m3

Cumulative liquid production (on the surface)/104 m3

Water cut/f

1983 210.85 10.77 147.46 158.23

1984 296.28 23.92 207.20 231.12 0.180 4

1985 443.66 60.52 310.27 370.79 0.262 0

1986 592.50 149.53 414.36 563.89 0.461 0

1987 692.23 291.27 484.11 775.38 0.670 2

1988 758.31 461.90 530.32 992.22 0.786 9

1989 808.06 657.94 565.12 1 223.06 0.849 3

1990 846.14 904.44 591.75 1 496.19 0.902 5

1991 873.79 1 181.27 611.08 1 792.35 0.934 7

1992 895.56 1 452.87 626.31 2 079.18 0.946 9


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Fig. 6 Plot of water-oil ratio versus cumulative oil production of Sha 1 reservoir in the Pucheng Oilfield

Fig. 7 Production performance of Sha 1 reservoir in the Pucheng Oilfield

Table 8 Production data in the selected decline period of Sha 1 reservoir in the Pucheng Oilfield

Year t /a Annual oil

production/(104 m3⋅a−1) Cumulative oil

production/104 m3 1988 0 46.21 0

1989 1 34.79 34.79

1990 2 26.63 61.42

1991 3 19.34 80.76

1992 4 15.22 95.99

Fig. 8 Linear regression trail and error result of expanded Arps decline method

production of 141.88×104 m3 in the selected decline period. Adding the cumulative oil production of 530.32×104 m3 in the previous period enables us to predict the recoverable reserves of 672.2×104 m3 in Sha 1 Reservoir of Pucheng Oilfield.

By the comparison analysis, the recoverable reserves pre-

dicted by the water flooding characteristic curve, 667.95×104

m3, matches well with the result of the expanded Arps decline method, 672.2×104 m3, which proves the practical applicabil-ity of the new method.

However, for the production decline rules of which the de-cline exponent does not equal 0, 0.5 or 1.0, computer program is needed in the expanded Arps decline method to conduct a linear regression trail and error by setting different values of decline exponents between 0 and 1 according to a fixed step, such as 0.01, which makes the computing complex and time-consuming. Applying the water flooding characteristic curve can avoid a complicated process of determining decline exponent, meanwhile solve the problem of predicting recov-erable reserves caused by the upwarping of traditional water flooding characteristic curve in the late development stage of oilfields.

5 Conclusions

The theoretical hypothesis of traditional water flooding characteristic curves is the linear relationship of relative per-meability ratio Kro/Krw versus water saturation Sw on semi-log coordinate in the intermediate water saturation stage, so the approaches are more applicable to predict the production in-dex for middle water-cut oilfields. However, applying tradi-tional approaches in high water-cut oilfields would make a significant error in the predicted recoverable reserves, which limits the application of the traditional approaches in high water-cut oilfields.

The semi-log plot of Kro/Krw versus Sw deviates from the straight line in the high water saturation stage, which is the main reason of the upwarping of traditional water flooding characteristic curves in the late development stage of oilfields.

The equation of water flooding characteristic curve in high water-cut oilfields is a binomial expression, and the pattern of the curve is a parabola, while the pattern of traditional water flooding characteristic curves is a straight line. Therefore, good agreements can be obtained by applying new approach to match the upwarping data of water flooding characteristic curve in the late development stage of oilfields.

When water flooding characteristic curves exhibit upwarp-ing in the late development stage of oilfields, it is suggested to apply new approach to predict the production index that can solve the problem of predicting recoverable reserves caused by the upwarping of traditional water flooding characteristic curves in the late development stage of oilfields. The calcula-tion results indicate that the new approach has a good practi-cability.

Nomenclature

Kro, Krw—relative permeability of oil and water, f; Sw—water saturation, f; Wp—cumulative water production, 104 m3; Np—cumulative oil production, 104 m3;


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Npmax—water flooding recoverable reserves, 104 m3; fwmax—economic limit of water cut, f; WOR—water-oil ratio, f; WORmax—economic limit of water-oil ratio, f; Lp—cumulative liquid production, 104 m3; a1, a2, a3, a4, a5, b1, b2, b3, b4, b5, A1, A2, B1, B2, a, b, d, f, g, h, m, n,

A, B, A′, B′, C, E—coefficient; Kroe, Krwe—relative permeability of oil and water at the exit-end, f; Swe—exit-end water saturation, f; Qo, Qw—transition production rate of oil and water at the exit-end,

m3/d; μo, μw—viscosity of oil and water in reservoir, mPa•s; Bo, Bw—volume factor of oil and water in reservoir, m3/ m3; γo, γw—gravity of oil and water in reservoir, f; N—original oil in place, 104 m3; Soi—initial oil saturation, f; Swi—initial water saturation, f; Sor—residual oil saturation, f; R—recovery degree, f; Swa—average water saturation, f; t—time, a; Q—annual oil production at time t in the decline period, 104 m3/a.

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