Improved mathematical model for OGIP prediction

Journal of Unconventional Oil and Gas Resources 14 (2016) 56–61

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Improved mathematical model for OGIP prediction Haohan Liu ⇑ Southwest Petroleum University, Chengdu 615000, China Sichuan College of Architectural Technology, Deyang 618000, China

a r t i c l e

i n f o

Article history: Received 6 March 2015 Revised 31 December 2015 Accepted 3 January 2016 Available online 27 February 2016 Keywords: OGIP Linear relationship Material balance equation 2-Order Taylor expansion

a b s t r a c t Many petroleum engineers apply the conventional formation pressure method to certain gas reservoir, especially the abnormally pressured reservoirs which can lead to errors of up to 100% in the original gas in place (OGIP) extrapolation, and there is still lack of 2-order pressure related equation to estimate the OGIP. Through introducing the transformations and Taylor expansion to the material balance equation, a new 2-order pressure related equation to calculate the OGIP is established; this new method gets advantages of conventional formation pressure method and Law of water cutting volume factor. Meanwhile, it modifies the old linear relationship for conventional P/Z’s in normally pressured reservoirs, and it allows direct extrapolation of OGIP in gas reservoirs concerning rock and water compressibility; after applying this new method to two different abnormally pressured reservoirs, comparisons of predicted results with different methods have been given, and the adaptability and affectivity of the new method have been proved. Ó 2016 Elsevier Ltd. All rights reserved.

Introduction Nowadays, OGIP calculation method, in general, is divided into two parts: static method and dynamic method. The common used static method is volumetric method (Shilun, 2003), and it is useful after oil and gas being proved. The latter one contains material balance method, production decline law (Yuanqian, 1984; Sorrell et al., 2012; Engelder, 2012; Clarkson, 2013) and water drive characteristic curve method (Bear, 2013; Song et al., 2013; Zhaojie et al., 2013; Yan et al., 2013), etc. Most of them are empirical analysis methods, and they are used after the oil and gas field being developed to a certain stage. Material balance equation is the basis for determining the driving mechanism and forecasting the future developing dynamics. Its effective application depends on the feasible geological data, developing dynamics data and reservoir physics data, etc. Based on it, there are three common methods in calculating original gas in place: the conventional formation pressure method (P/Z formation pressure method) (Yuanqian, 2000), the Law of water cutting volume factor (Yuanqian, 1990) and Havlena–Odeh method (Dake, 2001). These methods are widely used in many branches of petroleum engineering (Kim et al., 2011; Jinglin, 2006; Gang, 2013). While, many petroleum engineers apply the conventional formation pressure method to abnormally pressured reservoirs which can lead to errors of up to 100% in the OGIP extrapolation. Here, ⇑ Address: Southwest Petroleum University, Chengdu 615000, China. http://dx.doi.org/10.1016/j.juogr.2016.01.005 2213-3976/Ó 2016 Elsevier Ltd. All rights reserved.

a new method will be proposed to effectively predict the OGIP and reflect the detailed changing trend of parameters like cumulative gas production, water cut and effective compressibility.

Comparisons of OGIP calculation methods for gas reservoir after development Conventional formation pressure method When ignoring edge water or bottom water invasion in gas reservoir gives:


 pi Gp p 1
¼ z zi G

ð1Þ

Advantage: this method can be used to analyze the closed gas reservoir with a constant volume-the curve of pz and Gp is a linear relationship in different development phase. Disadvantage: this method ignores the bond water expansion and pore volume reduction caused by the decline of formation pressure, so the calculation accuracy is low; for abnormally pressured gas reservoir, the effective compressibility of rock can reach to 40  10
4 MPa
1, although there is no water invasion, this conventional formation pressure method is infeasible; this method can only be available when extracting 30–40% of the initial geologic reserves.

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H. Liu / Journal of Unconventional Oil and Gas Resources 14 (2016) 56–61

Nomenclature Gp G pi p Zi Z Bg Bgi Vgi Cw Cf Swi

cumulative gas production, 108 m3 original gas in place, 108 m3 initial formation pressure, MPa present formation pressure, MPa initial gas compressibility factor present gas compressibility factor volume factor under pressure p free gas volume factor under primitive conditions initial gas contributing pore volume, m3 compressibility coefficient of bound water, 1/MPa compressibility coefficient of formation rock, 1/MPa bond water saturation in original state, %

Wp We Bw psc T Tsc / /i

q qi

RD

w

Law of water cutting volume factor

Introducing Eqs. (9) and (10) to Eq. (8), gives

When ignoring the connate water expansion and the decreased pore volume caused by formation pressure decline gives:

2 3
 p pi 4 Gp 1 5 ¼ 1
z zi G 1
W e
W p Bw

ð2Þ

It is known to us all that

W e
W p Bw V gi pzi w¼ ; pi z Gp RD ¼ ; G 1
RD : 1
x

ð3Þ ð4Þ ð5Þ

ð6Þ

ð7Þ

Eq. (7) shows the degree of gas reserve recovery RD is a p4 declining line to the relative pressure w for a closed gas reservoir with a constant volume. Advantage: this method can be used to analyze water invasion gas reservoir. Disadvantage: the same as the conventional formation pressure method. Havlena–Odeh method The Havlena–Odeh method shows us:

F We ¼Gþ : Eg Eg

ð8Þ

where,

F ¼ Gp Bg þ W p Bw ; Eg ¼ Bg
Bgi :

W p Bw GðBg
Bgi Þ

and dividing

ð12Þ Bg Bgi

on both sides of Eq. (12),

gives

Gp Bgi W e
W p Bw þ : ¼1
G Bg GBg

ð13Þ

By simplifying Eq. (13), gives


 Bgi W e
W p Bw Gp ¼1
: 1
Bg GBgi G

ð14Þ

Because

W e
W p Bw W e
W p Bw ¼ ; V gi GBgi Bgi pzi ¼ ¼ w; Bg pi z

x¼

For gas reservoir without water invasion (x = 0) gives:

w ¼ 1
RD :

ð11Þ

Gp Bg þ W p Bw We ¼1þ : GðBg
Bgi Þ GðBg
Bgi Þ Subtracting

Combing Eqs. (2)–(5), gives

w¼

Gp Bg þ W p Bw We ¼Gþ : Bg
Bgi Bg
Bgi Dividing G on both sides of Eq. (11), gives

V gi

x¼

water cut under pressure p, % water cut under pressure pi, % volume factor of formation water under pressure p atmospheric pressure in standard state, MPa absolute temperature, °C gas temperature in standard state, °C fluid density under formation pressure p rock porosity under formation pressure pi fluid density under formation pressure p, g/mL fluid density under formation pressure pi, g/mL degree of reserve recovery, % apparent pressure coefficient

ð15Þ ð16Þ

Combing Eqs. (14)–(16), gives Eq. (2). Generally speaking, the Havlena–Odeh method is just a transformation of ‘Law of water cutting volume factor’. New method for the OGIP calculation in different types of gas reservoir For a gas reservoir with natural water drive, the elastic expansion of natural gas, bound water of formation and rock will be caused by the decline of gas reservoir pressure; meanwhile, the former gas section will be invaded by the edge water and bottom water. The general material balance equation (Dake, 2001) is shown in formula:

Gp Bg þ W p Bw ¼ GðBg
Bgi Þ þ W e þ GBgi

C w Swi þ C f Dp: 1
Swi

ð17Þ

By simplifying Eq. (17), gives

ð9Þ ð10Þ

When EFg is linear to WEge , the ordinate intersections is the OGIP. However, The Havlena–Odeh method makes no difference with the Law of water cutting volume factor, please see the following deduction.

Gp Bgi W e
W p Bw Bgi C w Swi þ C f þ þ Dp: ¼1
G Bg GBg Bg 1
Swi

ð18Þ

It is known that:

G¼

V gi ; Bgi

ð19Þ

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H. Liu / Journal of Unconventional Oil and Gas Resources 14 (2016) 56–61

Gp ; G

RD ¼

ð20Þ

RD ¼ 1 þ ðg
1Þw þ ðg2
gÞw2 ;

Psc Z i T ; Bgi ¼ Pi T sc Bg ¼

ð21Þ

Psc ZT : PT sc

Because

ð22Þ C w Swi þC f 1
Swi

Dp is linear to

Gp G

¼ RD , state parameter g can be

used to satisfy Eq. (23):

C w Swi þ C f Dp ¼ gRD : 1
Swi

ð23Þ

By substituting Eqs. (19)–(23) to Eq. (18), gives

RD ¼

1 þ xw
w g þ x
1 1 x
1 ¼
; 1
wg 1
wg g g

ð24Þ

By introducing 2-order Taylor polynomial to expression of wg gives

1 ¼ 1 þ wg þ w2 g2 þ oðw2 g2 Þ; 1
wg

ð25Þ

Combining Eq. (24) with Eq. (25) gives

RD ¼ 1 þ ðg þ x
1Þw þ gðg þ x
1Þw2 ;

ð26Þ

Combining Eq. (26) with Eq. (20) gives

Gp ¼ a0 þ a1 w þ a2 w2 :

ð27Þ

where, a0 = G; a1 = a0(g + x – 1); a2 = a0(g + x – 1)g. Eq. (27) can also be written as

Gp ¼ a0 þ a01

p p2 þ a02 : z z

ð28Þ

Gonzales et al. (2008) found the water invasion from the recompaction around the mud stone is small during the declining pressure process of abnormally high pressure reservoir, it is, in Eq. (26), when x = 0, gives

RD ¼ 1 þ ðg
1Þw þ gðg
1Þw2 ;

ð29Þ

Gp ðg
1Þzi p gðg
1Þzi p2 þ : ¼1þ z z G pi pi

ð30Þ p , z

In Eq. (28), using the production data of Gp and the coefficients of a0 ; a01 ; a02 can be fitted out with least-square method. When x = 0, the OGIP can be achieved. From Eqs. (28) and (30), gives

w¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a21
4a2 ða0
Gp Þ

; 2a2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 02 0 p
a1 þ a1
4a2 ða0
Gp Þ ¼ : z 2a02

ð31Þ ð32Þ

Advantages: this new method not only gets the advantages of Havlena–Odeh method, but it can also be used to calculate the  e , the water cut x, the relative pressure effective compressibility C coefficient w, the cumulative gas production Gp and the OGIP, meanwhile, it lays a theoretical base for OGIP calculation. For abnormally pressured gas reservoir, fluid and rock deformation cannot be ignored. For q ¼ q0 eC w DP ; / ¼ /0 ð1 þ C f DPÞ;

g > 0.

C w Swi þC f 1
Swi

ð33Þ

(2) When considering the edge and bottom water invasion gives

RD ¼ ð1 þ xÞ þ ðg þ xg
1Þw þ ðg2 þ xg2
gÞw2 :

ð34Þ

Assumed conditions: this new method is fit for abnormally pressured gas reservoir. For atmospheric gas reservoir and condensate gas reservoir, the models have been established, but more data and examples should be selected to test their efficiency and adaptation, further studies should be taken to finish these steps. Disadvantages: (1) This new method cannot be used to calculate state parameter g , and it only proposed a new way to predict OGIP; Efficiency of this method related to parameter g: In general, the effective compressibility of a gas reservoir is near 5  10
4 MPa
1; for an abnormally pressured reservoir, the value is 3–4 times of it, near 1.5  10
3 MPa
1; if the formation pressure is 22.3881 MPa (in Anderson ‘L’ gas reservoir, the RD is probably bigger than 0.5), from Eq. (23), g = 0.0672 . Hence, 2-order Taylor polynomial to expression of wg in Anderson ‘L’ gas reservoir is effective. It is meaningless for us to get a statistical value of g related to the producing data and test the efficiency of the new method. The new method has no problem, it can be used to predict the OGIP and compare the accuracy with other predicting methods. (2) The error of least-square method;

where, a01 ¼ a0 ðg þ x
1Þ pzii ; a02 ¼ a0 ðg þ x
1Þg pzii .


a1 þ

(1) When ignoring the edge and bottom water invasion gives

Dp ¼ gRD > 0 gives

Error analysis:   Given producing data: pz ; Gp i ; i ¼ 1; 2; . . . ; n, where n is the number of producing data. From Eq. (28), error related mathematical model can be established:

min Eða0 ; a01 ; a02 Þ ¼ min


n p2  2 X p ðGp Þi
a0 þ a01 þ a02 : z z i i¼1

ð35Þ a01 ,

a02

After solving this model, the value of a0, can be established, and then confirm the error of each predicting Gp. Models of atmospheric gas reservoir and Condensate gas reservoir Atmospheric gas reservoir For atmospheric gas reservoir, q  qi, /  /i.

q ¼ q0 eCw DP ; / ¼ /0 ð1 þ C f DPÞ:

ð36Þ ð37Þ

When ignoring fluid and rock deformation gives

C w Swi þ C f Dp ¼ gRD ¼ 0: 1
Swi

ð38Þ

Combining Eq. (2) gives

RD ¼ ð1 þ xÞ
w:

ð39Þ

(1) Using Eq. (2), when ignoring the edge and bottom water invasion gives

RD ¼ 1
w;

ð40Þ

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H. Liu / Journal of Unconventional Oil and Gas Resources 14 (2016) 56–61

(2) Using Eq. (2) and Eq. (6), when considering the edge and bottom water invasion gives

RD ¼ ð1 þ xÞ
w;

ð41Þ

Let

B¼

ð1
Sw
S0 Þð1
yw Þ : ð1
Swi Þð1
ywi Þ

ð51Þ

Introducing the same deduction to Eq. (18), gives (3) Using Eqs. (2) and (6), when considering the edge and bottom water invasion and the fluid and rock deformation gives

RD ¼ ð1 þ xÞ þ ðg þ xg
1Þw þ ðg2 þ xg2
gÞw2 :

ð52Þ

ð42Þ Condensate gas reservoir with oil rim For condensate gas reservoir with oil rim, Ralph. J. Schilthuis equation (Schilthuis, 1936) gives

Condensate gas reservoir Condensate gas reservoir without water vapor Assume that the initial pressure is higher than the DPP (dew point pressure). When ignoring the rock compressibility and water vapor gives the material balance equation

ðGt
Gpt ÞBg Gt Bgi ¼ : 1
S0

ð43Þ

Making the corresponding parameter replacements in the above equation gives

Bg Gpt ¼ 1
ð1
S0 Þ i ; Gt Bg

ð44Þ

Ni ¼

RDt ¼ 1
ð1
S0 Þw:

ð1
Sw Þð1
yw ÞBgi ½1
C f ðpi
pÞ ¼ G
Gp : ð1
Swi Þð1
ywi ÞBg

ð46Þ

Let

C f ðpi
pÞ ¼ gRD ;

ð47Þ

ð1
Sw Þð1
yw Þ A¼ : ð1
Swi Þð1
ywi Þ

ð48Þ

Combining Eq. (16) and Eqs. (47) and (48) to Eq. (46), gives

1 þ Aðg
1Þw þ A2 gðg
1Þw2 ¼ RD ; a0 þ a1 w þ a2 w2 ¼ Gp :

ð49Þ

When the initial pressure is lower than the DPP gives:

ð1
Sw
So Þð1
yw ÞBgi G ½1
C f ðpi
pÞ ¼ ðG
Gp Þ: ð1
Swi Þð1
ywi ÞBg

ð50Þ

ð53Þ

Making the corresponding parameter replacement in Eq. (53), gives

Gp ðW e
W p Bw Þ m 1 Bgi þ
¼ ; Gi Bg 1 þ m 1 þ m Bg ð1 þ mÞ Gi

ð54Þ

Making the corresponding parameter replacement in Eq. (54), gives

ð45Þ

Condensate gas reservoir with water vapor When the initial pressure is higher than the DPP gives material balance equation

  Np Bt þ ðRp
Roi ÞBgt
W e
Bw W p :

 ti Bgt
Bgi Bt
Bti þ mB B gi

RD ¼

It is,

G

1 þ Bðg
1Þw þ B2 gðg
1Þw2 ¼ RD :


 1 m 1 1 þ
w mþ1 1þm 1þm m
 m2 1 1 w2 :
þ 2 1þm m ð1 þ mÞ

ð55Þ

Applications and discussions Applications Here, the Anderson ‘L’ gas reservoir and the Louisiana Offshore gas reservoir (two abnormally pressured gas reservoirs of America) (Duggan and Jack, 1972; Ramagost and Farshad, 1981) are selected to test the new method mentioned above. Fundamental data of the two abnormally pressured gas reservoirs are shown in Table 1. By using Matlab (mathematical software) and introducing LSQ method to the new method combined with data in Table 1 gives:

2 Anderson ‘L’ : Gp ¼ 15:5012
0:0006 pz
0:0075 pz

2 Louisana Offshore: Gp ¼ 23:0898 þ 0:0942 pz
0:0103 pz Using conventional formation pressure method combined with data in Table 1 gives: Anderson ‘L’ : Gp ¼ 24:7386
0:5368 pz Louisana Offshore: Gp ¼ 39:8608
0:7500 pz

Table 1 Data of ANDERSON ‘L’ and LOUISIANA Offshore gas reservoir. Anderson ‘L’ gas reservoir

Louisiana offshore gas reservoir

p, MPa

z

p/z, MPa

Gp, 108 m3

p, MPa

z

p/z, MPa

Gp, 108 m3

1
x

65.5483 64.0660 61.8459 59.2603 57.4470 55.2200 52.4207 51.0625 48.2770 46.3396 45.0572 39.7413 32.8604 29.6129 25.8562 22.3881

1.440 1.418 1.387 1.344 1.316 1.282 1.239 1.218 1.176 1.147 1.127 1.048 0.977 0.928 0.891 0.854

45.520 45.181 44.590 44.092 43.653 43.073 42.309 41.923 41.052 40.401 39.980 37.921 33.634 31.910 29.0194 26.2156

0 0.11114 0.46502 0.91344 1.20638 1.55842 2.13455 2.47750 2.97590 3.32975 3.62150 4.88820 6.48195 7.96966 9.22187 10.0426

78.924 73.614 69.869 63.814 59.131 54.524 50.897 47.221 44.055 40.186 37.303 34.483 31.034 28.759

1.496 1.438 1.397 1.33 1.28 1.23 1.192 1.154 1.122 1.084 1.057 1.033 1.009 0.988

52.757 51.192 50.014 47.980 46.196 44.328 42.699 40.919 39.265 37.072 35.291 33.381 30.757 29.108

0 0.68398 1.9733 3.6957 5.3553 6.9929 8.2988 9.9984 11.075 12.572 13.633 14.87 16.254 16.955

1 0.9841 0.9661 0.9434 0.9258 0.9086 0.8950 0.8812 0.8693 0.8548 0.8440 0.8334 0.8205 0.8120

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H. Liu / Journal of Unconventional Oil and Gas Resources 14 (2016) 56–61 12 Real data curve New method Conventional formation pressure method

10

G P, 1 0 8 M 3

8

6

4

2

0

-2 26

28

30

32

34

36 p/z,MPa

38

40

42

44

46

Fig. 1. Predicting curves of p/z and Gp of Anderson ‘L’ gas reservoir with different methods.

20 Real data curve New method Law of water cutting volume factor Conventional formation pressure method

15

G p ,1 0 8 m 3

10

5

0

-5 25

30

35

40 p/z,MPa

45

50

55

Fig. 2. Predicting curves of p/z and Gp of Louisiana Offshore gas reservoir with different methods.

Using Law of water cutting volume factor combined with data in Table 1 gives: Louisana Offshore: Gp ¼ 31:7739
0:6127ð1
xÞ pz Finally, OGIP prediction with new method in different oilfields is confirmed: Anderson ‘L’ : G ¼ 15:5012  108 m3 Louisana Offshore: G ¼ 23:0898  108 m3 In Section 2.3, Havlena–Odeh method is just a transformation of Law of water cutting volume factor. Here, comparisons of new method with conventional formation pressure method and Law of water cutting volume factor have been given, Figs. 1 and 2 show the detailed predicting results with them.

Discussions Analysis of variance: (1) Variance of predicting value in the Anderson ‘L’ gas reservoir DA(Gp) Variance with conventional method

P16  DA ðGp Þ ¼

i¼1

2

i 24:7386
0:5368 pz
Gip 16

Variance with new method

¼ 0:0965;

H. Liu / Journal of Unconventional Oil and Gas Resources 14 (2016) 56–61

P16 DA ðGp Þ ¼

i¼1


2   2

i i 15:5012
0:0006 pz
0:0075 pz
Gip 16

¼ 0:0318: (2) Variance of predicting value in the Louisiana Offshore gas reservoir DL(Gp) Variance with conventional method

DA ðGp Þ ¼

2

pi P14  i i¼1 39:8608
0:7500 z
Gp 14

¼ 0:2590;

Variance with Law of water cutting volume factor

DA ðGp Þ ¼

2

 P14  i i p i i¼1 31:7739
0:6127ð1
x Þ z
Gp 14

¼ 0:0570; Variance with new method

DA ðGp Þ ¼


2

pi

pi 2 P14 i i¼1 23:0898 þ 0:0942 z
0:0103ð z Þ
Gp 14

¼ 0:0413:

i where pz , xi and Gip are data sequence values in Table 1. (3) Figs. 1 and 2 show the predicting accuracy with new method is higher than other above-mentioned methods in two different abnormally pressured gas reservoirs. (4) Figs. 1 and 2 show the OGIP is often over-estimated by conventional formation pressure method; meanwhile, the real historical production data point is near to the new method instead of the conventional method and Law of water cutting volume factor. The OGIP is the intersection point with y-axis, it is 15.5012  108 m3 in Anderson ‘L’ gas reservoir, which is smaller than the estimated value with conventional formation pressure method, and the OGIP is 23.0898  108 m3 in Louisiana Offshore gas reservoir, which is bigger than the estimated value with conventional formation pressure method and smaller than the estimated value with Law of water cutting volume factor. Conclusions (1) New OGIP calculation method is established based on the material balance equation, and the intersection point with y axis of the new method curve in the 2-D coordinate plane is the OGIP; (2) In the Anderson ‘L’ gas reservoir, the DA(Gp) is 0.0965 with conventional method, and 0.0318 with new method;

61

(3) In the Louisiana Offshore gas reservoir, the DL(Gp) is 0.259 with conventional method, 0.057 with Law of water cutting volume factor and 0.0413 with new method; (4) The new method changes the old linear relationship between pz and OGIP with parabolic curve, so it gets high and more reasonable calculating accuracy, it is better than the conventional formation pressure method and Law of water cutting volume factor.

Acknowledgments I always acknowledge reviewers and the associate editor. Meanwhile, I offer my regards and blessings to the China Postdoctoral Science Foundation funded project, No: 2014M562509XB, and Scientific Project of Sichuan Provincial Education Department (Nos. 15ZB0447, 15ZB0450). References Bear, J., 2013. Dynamics of Fluids in Porous Media. Courier Dover Publications. Clarkson, C.R., 2013. Production data analysis of unconventional gas wells: review of theory and best practices. Int. J. Coal Geol. 109, 101–146. Dake, L.P., 2001. The Practice of Reservoir Engineering. Elsevier Science. Duggan, Jack, O., 1972. The Anderson L: an abnormally pressured gas reservoir in south Texas. JPT, 132. Engelder, T., 2012. Gas Decline Curves in the Marcellus Shale Play. Pennsylvania State University. Gang, Hu, 2013. A new method for calculating volumetric sweep efficiency in a water-flooding oilfield. Petrol. Explor. Dev. 40 (1), 103–106. Gonzales, F.E., Ilk, D., Blasingame, T.A., 2008. A quadratic cumulative production model for the material balance of an abnormally pressured gas reservoir. SPE, 114044. Jinglin, Feng, 2006. A simple method to calculate single-well dynamic reserves of water-drive gas reservoirs based on Havlena–Odeh method. China Offshore Oil Gas 18 (4). Kim, S.M., Lee, J.D., Lee, H.J., et al., 2011. Gas hydrate formation method to capture the carbon dioxide for pre-combustion process in IGCC plant. Int. J. Hydrogen Energy 36 (1), 1115–1121. Ramagost, Billy Paul, Farshad, Fred F., 1981. P/Z abnormally pressured gas reservoirs. SPE, 10125. Schilthuis, R.J., 1936. Active oil and reservoir energy. Trans. AIME 118 (01), 33–52. Shilun, Li, 2003. Gas Engineering. Petroleum engineering press, Beijing (pp. 187– 189). Song, Z., Li, Z., Lai, F., et al., 2013. Derivation of water flooding characteristic curve for high water-cut oilfields. Petrol. Explor. Dev. 40 (2), 216–223. Sorrell, S., Speirs, J., Bentley, R., et al., 2012. Shaping the global oil peak: a review of the evidence on field sizes, reserve growth, decline rates and depletion rates. Energy 37 (1), 709–724. Yan, Gou, Huaxun, Liu, Shusheng, Gao, et al., 2013. Research and application effect analysis on water-drive characteristic curve for water-drive gas reservoir. Oil Drill. Prod. Technol. 35 (3), 63–65. Yuanqian, Chen, 1984. A method of determining decline characteristics for a constant volume gas reservoir. Petrol. Explor. Dev. 11 (3), 52–59. Yuanqian, Chen, 1990. Petroleum Reservoir Engineering Calculation Methods. Petroleum industry publishing house, Beijing. Yuanqian, Chen, 2000. Calculation methods of recoverable reserves of oil fields. XJPG 21 (2), 130–137. Zhaojie, Song, Zhiping, Li, Fengpeng, Lai, et al., 2013. Derivation of water flooding characteristic curve for high water-cut oilfields. Petrol. Explor. Dev. 40 (2), 201– 208.