Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 1369639, 12 pages
https://doi.org/10.1155/2018/1369639

Research Article

A New Production Prediction Model Based on Taylor Expansion Formula

Xianfeng Ding,¹ Dan Qu,¹ and Haiyan Qiu²

¹School of Science, Southwest Petroleum University, Cheng Du, Sichuan 610500, China
²College of Chemistry and Chemical Engineering, Southwest Petroleum University, Cheng Du, Sichuan 610500, China

Correspondence should be addressed to Xianfeng Ding; fxxd@163.com

Received 28 May 2018; Revised 2 November 2018; Accepted 11 November 2018; Published 3 December 2018

Academic Editor: Mohamed Shaat

Copyright © 2018 Xianfeng Ding et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

On the basis of the analysis of the cumulative production growth curve model, the model variables are adjusted, and the Taylor formula is expanded on the adjusted model. Then the appropriate expansion order n is selected, and the new model for the prediction of cumulative production is established. Furthermore, the error of the new model is discussed, and the model can theoretically achieve any given precision. The model can forecast oil and gas production, cumulative production, and recoverable reserves. Finally, the example analyses show that the greater the order number (n) is, the smaller the error between the prediction data and the actual data and the greater the correlation coefficient become. Compared with other models, the results show that the model has higher prediction accuracy and wider application range and can be used to forecast the production of oil and gas field.

1. Introduction

For an engineer of oil and gas reservoir, oil and gas production, and recoverable reserves, prediction is an essential work, which is an important basis for oil and gas field planning and development, production allocation optimization, and daily dynamic analysis. The method of decline analysis is widely used which is based on system theory, fuzzy mathematics, information theory, grey theory, and cybernetics. Many studies had been focused on the production decline analysis. In 1945, Arps [1] proposed 4 types of declines: exponential, hyperbolic, harmonic, and ratio decline basing on decline trend observed in the field. Then Arps [2] (1956) simplified the proposed decline curves. In 1972, Gentry [3] studied the characteristics of each type of decline and then obtained two dimensionless equations and discussed the decline index in detail. Fetkovich [4] (1971 and 1980) constructed type curves combining the transient rate and the pseudo-steady-state decline curves and derived single-phase flow from material balance and Darcy law. Experts in China have also put forward some decline curves. In this respect, Yu and Chen are the representatives, and these are also empirical statistical formulas. In 1996, based on the distribution of in statistics, Chen [5] deduced the original Weng's prediction model systematically, obtained the generalized model for predicting oil and gas field production, maximum annual production, recoverable reserves, and their occurrence time, and proposed the linear trial and error method for solving the model. In 1997, Hu [6] first proposed the inverse tangent differential distribution model according to the theory of production decline of Arps. The production decline theory of Arps is further enriched. In 1999, Yu [7] presented two new types of decline curve and and then demonstrated the validity of these two curves. On the basis of Weng's cycle model and Weibull's model, Song et al. [8] (2000) introduce a new model which can not only predict the oil and gas field production in the future, but also fix the time that the maximum annual production occurs in the recoverable cycle and put forward a geometric progression method to calculate parameters. In 2003, Li [9] improved Hu's antitangent differential distribution method, which made it more extensive. In 2004, Ding et al. [10] proposed a new model for predicting oil and gas field performance based on Weng's model, logistic model, and Chen-Hu prediction model. The model can predict recoverable reserves, cumulative production, and production. By changing the parameters of the model, the model can be simplified into logistic model and Chen-Hu prediction model. In 2005, Zhu et al. [11] put forward a generalized production decline equation , which can be transformed into Arps equation, tangent differential distribution production decline equation, and improved Hu decline equation by changing the parameters. In 2007, Yao et al. [12] analyzed oil and gas production in Tarim by the grey correlation degree method, and the influencing elements with higher relational degree were selected to establish nonlinear prediction model using grey theory GM(1,N) to solve the model. The model is utilized to predict the oil production and achieved good results. There are many other kinds of oil and gas field production prediction methods, such as experience trend method, historical matching method [13](Zhou, 2012), mathematical model method [14–16](Saraiva et al., 2014; Li et al., 2013; Keven and Roland, 2007), the unit proven reserve ratio method, analogy method [17](Charpentier and Cook, 2010), and intelligent prediction method [18](Weiss et al., 2002).

In general, the reliability of prediction results not only depend on the accuracy of the information and data, the user's quality, and work experience, but also depend on the accuracy of the selected prediction methods and the established mathematical model. Among these factors, the most primary factor is the established mathematical model. Without appropriate forecasting method, it is impossible to have good prediction results. As we all know, Taylor formula is a very important formula in advanced mathematics and has been popular in prediction research and application [19–23]. Based on the analysis of the cumulative production growth curve model, the model variables are adjusted and the adjusted model is expanded by the Taylor formula. Next, a new model for the prediction of cumulative production is established, which can be used to predict oil and gas production, cumulative production rate, and recoverable reserves.

2. Methods

2.1. Problem Description

The prediction of oil and gas production and recoverable reserves is a very important task in reservoir work. It is an important content in compiling oil and gas field development plan, designing oil and gas field development (adjustment) plan, and analyzing oil and gas field development performance. Although there are many production prediction models for oil and gas fields now and these models have been widely used in the prediction of oil or gas production and recoverable reserves, but there are still some limitations. In order to make each model complement each other, this article established a new model for predicting oil and gas field production, cumulative production, recoverable reserves, maximum annual gas production, and its occurrence time.

2.2. The Prediction Model Based on Taylor Expansion Formula

The cumulative growth curve models in the literatures are listed in Table 1.

Table 1

Prediction model
parameters

(Chen, et al., 1996) [24]
: cumulative production of oil and gas fields, (oil), (gas)
: Recoverable reserves of oil and gas fields, (oil), (gas)
: development years, a;
: model undetermined constant

(Hu, et al., 1997) [25]
: cumulative production of oil and gas fields, (oil), (gas)
: Recoverable reserves of oil and gas fields, (oil), (gas)
: development years, a;
: model undetermined constant

(Yu, 2000) [26]
: cumulative production of oil and gas fields, (oil), (gas)
: Recoverable reserves of oil and gas fields, (oil), (gas)
: development years, a;
: model undetermined constant

(Ding et al. 2004) [10]
: cumulative production of oil and gas fields, (oil), (gas)
: Recoverable reserves of oil and gas fields, (oil), (gas)
: development years, a;
: model undetermined constant

These formulas of cumulative growth curve yield prediction model are based on the experience and then the formulas of oil field development are summarized, so the undetermined parameters in the formula must be attained through the actual data when solving the cumulative production and recoverable reserves.

Considering time t is an incremental variable in the cumulative production model, when tends to infinity (), cannot be infinitely small. In order to facilitate the expansion of Taylor, we choose as the variable; the above models turn to be the forms as follows:

The Taylor expansion formula of the function at the point is as follows:

where and is a value between and .

When tends to infinity, obviously tends to zero. The cumulative production model with adjusted variables is expanded by the Taylor formula:

where , , , and is the recoverable reserve () when the original growth curve model as t tends to infinity. Selecting the appropriate expansion order n, (3) could be used to predict the recoverable reserves.

Error Analysis. Using the expansion order n of to calculate the cumulative production approximately, the error is

For a fixed n, when , if , the error estimation equation is

Given any precision and choosing the appropriate time , as long as , the order can be achieved. This implies that, for any precision, a suitable expansion order can be found to ensure that the new prediction model is strictly true.

When t tends to infinity, is a higher order infinitesimal of . Therefore, the remainder can be written as

When , , ignoring the infinite small terms, the above equation can be written as

where are undetermined parameters. Equation (7) can be solved by regression analysis, and then it can be used to predict the cumulative production.

When

ignoring the infinite small items, the above equation can be written as

where are undetermined parameters. They can be solved by regression analysis, and then it can predict the cumulative production.

Let

then (9) can be written as

When different values are taken by , the error sum of square between predicted data and the actual data can be written as

For obtaining the optimal coefficient a, b, c, d, we must make a minimum error E; that is,

Let the partial derivatives of (13) with respect to a, b, c, d be zero; then we can get the linear equation set (14).

The solution of (14) is the value of the coefficient a, b, c, d, and so on, until we find the appropriate order n in (15) to make the predicted results well in accordance with the actual data. Therefore, the following equation can estimate cumulative production and recoverable reserves.

where are undetermined parameters.

For the calculation of production, the method in literature (Ding et al., 2004) can be adopted. That is,

If the actual data is daily production, we can predict cumulative daily production and daily production by (15) and (16).

2.3. Model Validation

When using our proposed formula (15) to predict accumulative production and recoverable reserves, we need to solve the unknown parameters in the formula, and the regression analysis method is used here. In order to determine the appropriate , regression analysis was carried out to obtain the parameters according to the actual data, which obtained the only certain Taylor expansion. To further validate the applicability of this model, we use the Taylor expansion formula of the production prediction model proposed in the paper [27].

2.3.1. Logistic Production Prediction Model

Logistic production prediction model is . We choose as the variable, (the cumulative production of year); Taylor expansion formula is

We can ignore the infinite item:

For , , the recoverable reserves are equal to the cumulative production at this time.

The model parameters are obtained by self-regression in [19], and the results are , , , (considering the influences on the parameter , taking the average); the linear regression between the cumulative yield and time is used to obtain the slope of the straight line. Bring in actual data: , , .

We choose as the variable; the Logistic production prediction model turns to be the forms

When n=2, n=3, n=4, n=5, and n=7, Taylor expansion formula is used to predict the cumulative production in different years, and the results are shown in Table 7.

2.3.2. G&H Production Prediction Model

To further verify the effectiveness of this model, we use the proposed model in [28], which can be simplified as the famous Gompertz model and the Herbert model. So the model is referred to as the G&H yield prediction model. G&H production prediction model is (the cumulative production of t year). We choose as the variable:

We can ignore the infinite item:

For , . The model parameters are obtained by linear regression in [28], and the results are , , .

We choose as the variable; the G&H production prediction model turns to be the forms

When n=2, n=3, n=4, n=5, and n=7, Taylor expansion formula is used to predict the cumulative production in different years, and the results are presented at Table 8.

2.3.3. Comparison

Table 9 is the prediction results of the proposed Taylor expansion model (T) and the prediction results of Taylor expansion of logistic model (L) and G&H model (G&H). We compare a smaller order of n=2 and larger order of n=7. The comparison of actual data and predicted data of cumulative production are shown in Figures 1 and 2.

Figure 1: The actual data and the prediction data of the three models.

Figure 2: The actual data and the prediction data of the three models.

Figure 1 or Figure 2 can explain that the Taylor model predicts more accurately when compared with the logistic model and Gompertz model and the Herbert model. Comparing Figure 1 and Figure 2, we can conclude that the obtained results by Taylor prediction model are closer to the real value as the increase of Taylor expansion order. Table 9 is the error of prediction value and the real value. Next we give the definition of error. Let and be prediction value and real value, respectively. The error is defined by . Tables 9 and 10 show that the effect of prediction reduced over time when Taylor's order is certain and that the effect of prediction increased over time when the time is certain. Using the proposed Taylor expansion model (T) and the Taylor expansion of logistic model (L) and G&H model (G&H) to predict recoverable reserves, the error is around 5% in the first 10 years. This forecast error is reasonable, because different models exit minor difference. Although the effect of the latter is not particularly good, on the whole, we can draw the conclusion that the proposed prediction model based on Taylor expansion is feasible.

2.3.4. Correlation Analysis

The results of the three kinds of Taylor expansion of different models are compared when n=2 and n=7, respectively, and the results are represented by a matrix. The elements of the matrix represent the correlation coefficient of sequence X and sequence Y, the result is a 22 matrix, the diagonal elements represent X and Y autocorrelation, and nondiagonal elements represent the correlation coefficient of X and Y and Y and X, two of which are equal. The results are as follows:

The correlation coefficient matrices A, C present the prediction results of Taylor expansion prediction model and the Taylor expansion of logistic model, respectively. The correlation coefficient matrices B, D present the prediction results of Taylor expansion prediction model and the Taylor expansion of G&H model, respectively. According to the correlation coefficient matrix, the values become closer to 1. The cumulative production predicted by this method is consistent with the other two Taylor expansion models. It shows that the prediction model established in this paper is practical and reliable.

2.4. The Novelty of Model

Compared with other models in the literatures, the Taylor model has two distinct advantages.

(1) It can be applied to forecast the production of oil and gas fields. We can predict cumulative annual production and daily production by (15) and (16). And as time goes on, of (15) closes to 0 and results predicted by the model are more accurate. As the expansion order n increases, the prediction error becomes smaller and smaller, and the correlation coefficient between the predicted value and the actual value becomes larger and larger.

(2) Taylor model is a power function in nature. Compared with other models which are containing exponential function, it is faster in calculation speed and occupies less CPU, so it is more suitable for oil and gas production prediction.

3. Result and Discussion

3.1. Application of Prediction Model

To demonstrate the scalability of our model, the experiments were conducted on oilfield block of Hudson oilfield in Tarim China and Bavly oilfield in this part. The Bavly oilfield was discovered in 1946 and developed in 1950, which is one of the outer edge water-flooding sandstone oilfields in the Soviet Union, developed with keeping the formation pressure. The oil bearing area of the oilfield is 118 km². The block was formally put into development in August 2000. The block's month cumulative production from January 2004 to July 2013 was calculated by our model and comparison and analysis were made between the prediction data by models in literatures and the actual data.

3.2. Numerical Results

3.2.1. Bavly Oilfield

The practical data and the prediction data of the cumulative production and production are shown in Figures 3 and 4 and Tables 5 and 6, respectively. Recoverable reserves are shown in Table 2 and the correlation coefficient of the prediction data and the practical data is shown in Table 3.

Table 2: Cumulative recoverable reserves of Bavly oilfield (million tons).

expansion order
n=4
n=5
n=7
n=20
n=50

recoverable reserves
1252.071953
10282.45991
66452.93302
66452.93302
66452.93301

Table 3: The correlation coefficient of the prediction data and the actual data of Bavly oilfield.

expansion order
n=2
n=3
n=4
n=5
n=7

correlation coefficient
0.9979
0.9998
0.999917099
0.999944399
0.999970846

Figure 3: The actual data and the prediction data of the cumulative production of Bavly oilfield.

Figure 4: The actual data and the prediction data of the annual production of Bavly oilfield.

It follows from Tables 2 and 3 that the prediction data is very close to the practical data of the field development. With the increase of the expansion order n, there are no major changes for the prediction data of recoverable reserves, and the correlation coefficient is gradually approximating close to 1. Furthermore, these data are also quietly close to the prediction data in the literatures.

3.2.2. An Oilfield Block of Hudson Oilfield in Tarim China

The comparison of actual data and predicted data of month cumulative production and production are shown in Figures 5, 6, and 7 and the comparison of recoverable reserves is shown in Table 4.

Table 4: The correlation coefficient and recoverable reserves of a block of Hudson oilfield.

n=5
n=6
in ref. Chen et al. 1996

in ref. Ding et al. 2004

correlation coefficient
0.999951
0.999985
0.999865
0.999898

recoverable reserves (million tons)
4001.015
3111.262
3000
3000

Table 5: The actual data and the prediction data of the cumulative production of Bavly [10] oilfield.

year
t(a)
cumulative production (million tons)

practical data
prediction data n=2
prediction data n=3
prediction data n=4
prediction data n=5
prediction data n=7

1959
12
2450
2318.789
2478.5996
2454.5865
2446.3177
2450.3951

1960
13
2830
2836.9906
2793.9021
2822.8706
2843.0266
2827.0884

1961
14
3200
3281.8253
3171.0607
3194.672
3195.8839
3206.4152

1962
15
3565
3667.8429
3556.2227
3561.4828
3549.8524
3562.595

1963
16
3905
4005.9848
3923.4511
3913.4051
3900.8663
3902.4385

1964
17
4235
4304.637
4261.2805
4243.0933
4236.546
4228.1552

1965
18
4535
4570.3352
4565.7695
4546.1807
4547.2328
4535.8497

1966
19
4805
4808.2484
4836.8296
4820.7178
4827.7119
4819.6946

1967
20
5065
5022.5182
5076.2599
5066.4659
5076.3844
5074.9303

1968
21
5305
5216.5021
5286.6897
5284.2989
5294.047
5299.0852

1969
22
5515
5392.9507
5471.0127
5475.7569
5482.8775
5492.0571

1970
23
5665
5554.1387
5632.0934
5642.7353
5645.7241
5655.6689

1971
24
5785
5701.9639
5772.6209
5787.2774
5785.6499
5793.0743

1972
25
5895
5838.0216
5895.0469
5911.4414
5905.6629
5908.201

1973
26
5995
5963.6633
6001.5667
6017.2188
6008.568
6005.3019

1974
27
6085
6080.0409
6094.1253
6106.4878
6096.8977
6088.6276

1975
28
6170
6188.1426
6174.4338
6180.9887
6172.8895
6162.2078

1976
29
6240
6288.8208
6243.9918
6242.3153
6238.4913
6229.7177

1977
30
6300
6382.8148
6304.1099
6291.9142
6295.3804
6294.4093

1978
31
6350
6470.7689
6355.933
6331.0909
6344.9897
6359.0873

Table 6: The actual data and the prediction data of the annual production of Bavly oilfield [10].

year
t(a)
annual production (million tons)

actual data
prediction data n=2
prediction data n=3
prediction data n=4
prediction data n=5
prediction data n=7

1959
12
390
158.789
418.5996
394.5865
386.3177
390.3951

1960
13
380
518.2016
315.3025
368.2841
396.7089
376.6933

1961
14
370
444.8347
377.1586
371.8014
352.8573
379.3268

1962
15
365
386.0176
385.162
366.8108
353.9685
356.1798

1963
16
340
338.1419
367.2284
351.9223
351.0139
339.8435

1964
17
330
298.6522
337.8294
329.6882
335.6797
325.7167

1965
18
300
265.6982
304.489
303.0874
310.6868
307.6945

1966
19
270
237.9132
271.0601
274.5371
280.4791
283.8449

1967
20
260
214.2698
239.4303
245.7481
248.6725
255.2357

1968
21
240
193.9839
210.4298
217.833
217.6626
224.1549

1969
22
210
176.4486
184.323
191.458
188.8305
192.9719

1970
23
150
161.188
161.0807
166.9784
162.8466
163.6118

1971
24
120
147.8252
140.5275
144.5421
139.9258
137.4054

1972
25
110
136.0577
122.426
124.164
120.013
115.1267

1973
26
100
125.6417
106.5198
105.7774
102.9051
97.1009

1974
27
90
116.3776
92.5586
89.269
88.3297
83.3257

1975
28
85
108.1017
80.3085
74.5009
75.9918
73.5802

1976
29
70
100.6782
69.558
61.3266
65.6018
67.5099

1977
30
60
93.994
60.1181
49.5989
56.8891
64.6916

1978
31
50
87.9541
51.8231
39.1767
49.6093
64.678

Table 7: The actual data and the prediction data of the cumulative production of Bavly [10] oilfield.

year
t(a)
cumulative production (million tons)

practical data
prediction data n=2
prediction data n=3
prediction data n=4
prediction data n=5
prediction data n=7

1959
12
2450
2500.6
2500.6
2500.6
2500.6
2500.6

1960
13
2830
2843.1
2845.2
2845.2
2845.2
2845.2

1961
14
3200
3186.2
3199.3
3199.5
3199.1
3198.9

1962
15
3565
3520.6
3556.6
3557.3
3555.0
3553.6

1963
16
3905
3841.4
3911.6
3913.3
3906.3
3900.9

1964
17
4235
4146.3
4260.5
4263.9
4248.1
4233.2

1965
18
4535
4434.4
4600.8
4606.3
4576.8
4544.1

1966
19
4805
4706.0
4930.7
4938.8
4890.3
4828.6

1967
20
5065
4961.5
5249.0
5260.3
5187.1
5083.2

1968
21
5305
5201.7
5555.3
5570.3
5466.8
5305.6

1969
22
5515
5427.5
5849.4
5868.3
5729.5
5494.9

1970
23
5665
5639.9
6131.3
6154.5
5975.5
5650.8

1971
24
5785
5839.8
6401.3
6429.0
6205.4
5773.9

1972
25
5895
6028.1
6659.7
6692.1
6420.1
5865.2

1973
26
5995
6205.6
6907.0
6944.2
6620.3
5926.1

1974
27
6085
6373.2
71435
7185.6
6807.0
5958.4

1975
28
6170
6531.6
7369.8
7417.0
6981.1
5963.7

1976
29
6240
6681.4
7586.3
7638.6
7143.4
5944.1

1977
30
6300
6823.3
7793.6
7851.0
7294.7
5901.4

1978
31
6350
6957.9
7992.2
8054.5
7435.9
5837.5

Table 8: The actual data and the prediction data of the cumulative production of Bavly [10] oilfield.

year
t(a)
cumulative production (million tons)

practical data
prediction data n=2
prediction data n=3
prediction data n=4
prediction data n=5
prediction data n=7

1959
12
2450
2465.9
2465.9
2465.9
2465.9
2465.9

1960
13
2830
2864.3
2865.0
2864.9
2864.9
2864.9

1961
14
3200
3250.6
3255.3
3254.0
3253.6
3253.5

1962
15
3565
3618.8
3631.6
3626.7
3624.7
3624.2

1963
16
3905
3966.5
3911.5
3979.6
3973.2
3971.4

1964
17
4235
4293.0
4333.8
4310.8
4296.6
4291.7

1965
18
4535
4598.8
4658.1
4620.3
4593.6
4583.2

1966
19
4805
4884.4
4964.9
4908.5
4864.4
4845.1

1967
20
5065
5152.1
52454.7
5176.3
5109.9
5077.8

1968
21
5305
5402.2
5528.3
5425.1
5331.3
5282.3

1969
22
5515
5636.3
5786.8
5656.1
5530.2
5459.8

1970
23
5665
5855.7
6030.9
5870.8
5708.4
5612.0

1971
24
5785
6061.5
6261.7
6070.4
5867.6
5740.6

1972
25
5895
6254.7
6480.0
6256.2
6009.5
5847.6

1973
26
5995
6436.5
6686.6
6429.3
6135.6
5934.6

1974
27
6085
6607.7
6882.4
6590.8
6247.4
6003.4

1975
28
6170
6769.2
7068.1
6741.8
6346.4
6055.7

1976
29
6240
6921.7
7244.4
6883.0
6443.7
6093.1

1977
30
6300
7065.9
7411.9
7015.2
6510.6
6117.0

1978
31
6350
7202.4
7571.2
7139.3
6571.8
6128.7

Table 9: The actual data and the prediction data of the cumulative production of Bavly [10] oilfield.

year
t(a)
cumulative production (million tons)

T
L
G&H
T
L
G&H

practical data
prediction data n=2

prediction data n=7

1959
12
2450
2318.789
2500.6
2465.9
2446.3177
2500.6
2465.9

1960
13
2830
2836.9906
2843.1
2864.3
2843.0266
2845.2
2864.9

1961
14
3200
3281.8253
3186.2
3250.6
3195.8839
3198.9
3253.5

1962
15
3565
3667.8429
3520.6
3618.8
3549.8524
3553.6
3624.2

1963
16
3905
4005.9848
3841.4
3966.5
3900.8663
3900.9
3971.4

1964
17
4235
4304.637
4146.3
4293.0
4236.546
4233.2
4291.7

1965
18
4535
4570.3352
4434.4
4598.8
4547.2328
4544.1
4583.2

1966
19
4805
4808.2484
4706.0
4884.4
4827.7119
4828.6
4845.1

1967
20
5065
5022.5182
4961.5
5152.1
5076.3844
5083.2
5077.8

1968
21
5305
5216.5021
5201.7
5402.2
5294.047
5305.6
5282.3

1969
22
5515
5392.9507
5427.5
5636.3
5482.8775
5494.9
5459.8

1970
23
5665
5554.1387
5639.9
5855.7
5645.7241
5650.8
5612.0

1971
24
5785
5701.9639
5839.8
6061.5
5785.6499
5773.9
5740.6

1972
25
5895
5838.0216
6028.1
6254.7
5905.6629
5865.2
5847.6

1973
26
5995
5963.6633
6205.6
6436.5
6008.568
5926.1
5934.6

1974
27
6085
6080.0409
6373.2
6607.7
6096.8977
5958.4
6003.4

1975
28
6170
6188.1426
6531.6
6769.2
6172.8895
5963.7
6055.7

1976
29
6240
6288.8208
6681.4
6921.7
6238.4913
5944.1
6093.1

1977
30
6300
6382.8148
6823.3
7065.9
6295.3804
5901.4
6117.0

1978
31
6350
6470.7689
6957.9
7202.4
6344.9897
5837.5
6128.7

Table 10: Errors.

T
L
G&H
T
L
G@H

year
t(a)
prediction data n=2

prediction data n=7

1959
12
0.053556
0.020653
0.00649
0.001503
0.020653
0.00649

1960
13
0.00247
0.004629
0.01212
0.004603
0.005371
0.012332

1961
14
0.02557
0.004313
0.015813
0.001286
0.000344
0.016719

1962
15
0.028848
0.012454
0.015091
0.004249
0.003198
0.016606

1963
16
0.02586
0.016287
0.015749
0.001059
0.00105
0.017004

1964
17
0.016443
0.020945
0.013695
0.000365
0.000425
0.013388

1965
18
0.007792
0.022183
0.014068
0.002697
0.002007
0.010628

1966
19
0.000676
0.020604
0.016524
0.004727
0.004912
0.008345

1967
20
0.008387
0.020434
0.017196
0.002248
0.003593
0.002527

1968
21
0.016682
0.019472
0.018322
0.002065
0.000113
0.004279

1969
22
0.02213
0.015866
0.021995
0.005825
0.003645
0.010009

1970
23
0.01957
0.004431
0.033663
0.003403
0.002507
0.009356

1971
24
0.014354
0.009473
0.047796
0.000112
0.001919
0.007675

1972
25
0.009666
0.022578
0.061018
0.001809
0.005055
0.008041

1973
26
0.005227
0.035129
0.073645
0.002263
0.011493
0.010075

1974
27
0.000815
0.047362
0.0859
0.001955
0.020805
0.01341

1975
28
0.00294
0.058606
0.097115
0.000468
0.033436
0.018525

1976
29
0.007824
0.070737
0.109247
0.000242
0.04742
0.023542

1977
30
0.013145
0.083063
0.121571
0.000733
0.06327
0.029048

1978
31
0.019019
0.095732
0.134236
0.000789
0.080709
0.03485

Figure 5: The comparison of actual and predicted data of cumulative production.

Figure 6: The comparison of actual and predicted data of cumulative production.

Figure 7: The comparison of actual and predicted data of production.

Figures 5, 6, and 7 and Table 4 displayed that the new model has higher prediction accuracy compared with the other models in the literatures. Therefore, it can be used to forecast the actual oil and gas production.

4. Conclusions

(1) Based on the analysis of the cumulative production growth curve model and the Taylor expansion of the model, once the appropriate expansion order n is selected, a new model for the prediction of cumulative production is established. Furthermore, the error of the new model is discussed, and the model can theoretically achieve any given precision.

(2) The analysis of examples of the new model shows that the prediction error gets smaller; the correlation coefficient between predicted and practical data becomes closer to 1 with the increases of the order n. Therefore, in order to meet the requirements of accuracy, an appropriate order n should be selected as far as possible. Because of the machine error of computers, the accuracy of prediction may be not ideal with larger order n.

(3) Compared with other models in the literatures, the results indicate that the model has higher prediction accuracy. It can be applied to forecast the production of oil and gas fields.

Appendix

See Tables 5, 6, 7, 8, 9, and 10.

Abbreviations

G&H model: | Gompertz model and the Herbert model
T model: | Taylor expansion model
L model: | logistic model.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Xianfeng Ding and Dan Qu conceived and designed the study. Xianfeng Ding performed the MATLAB experiments. Haiyan Qiu wrote the paper. Xianfeng Ding and Dan Qu reviewed and edited the manuscript. All authors read and approved the manuscript.

Acknowledgments

At the point of finishing this paper, the authors would like to express their sincere thanks to all those who have lent them hands in the course of my writing this paper. First of all, they would like to take this opportunity to show their sincere gratitude to the supervisor, Mr. Xianfeng Ding, who has given so much useful advices on the writing and has tried his best to improve the paper. Secondly, they would like to express their gratitude to their classmates who offered them references and information on time. Without their help, it would have been much harder to finish this paper.

References

J. J. Arps, “Analysis of decline curies,” SPE-G, 1945.

J. J. Arps, “Estimation of primary oil reserves,” SPE 627-G, 1956.

R. W. Gentry, “Decline-curve analysis,” SPE 3365-PA, 1972.

M. J. Fetkovich, “Decline curve analysis using type curves,” SPE 4629-PA, 1980.

Y. C. Chen, “The derivation and application of Weng's prediction model,” Research Institute of Petroleum Exploration and Development of CNPC, vol. 16, no. 2, pp. 22–26, 1996.

Y. D. Hu, “Exponential distribution and inverse tangential distribution of production decline,” Applied Mathematics & Information Sciences, vol. 24, no. 6, pp. 76–81, 1997.

Q. T. Yu, “A new decline curve,” Petroleum Exploration and Development, vol. 26, no. 3, pp. 72–75, 1999.

C. Z. Song and X. J. Ma, “Generalized Weibull, Wengcycle model for predicting oil and gas production,” Mineral Rock, vol. 20, no. 2, pp. 37-38, 2000.

C. L. Li, S. G. Han, and L. S. Cheng, “Research on a new production decline curve,” Journal of Southwest Petroleum Institute, vol. 25, no. 5, pp. 31–33, 2003.

X. F. Ding and Z. B. Liu, “A new model for oil and gas production prediction,” Petroleum Exploration and Development, vol. 31, no. 3, pp. 104–106, 2004.

Y. J. Zhu and W. J. Gao, “Establishment and research of a new generalized production decline equation,” Fault Block Oil and Gas Fields, vol. 12, no. 6, pp. 47–49, 2005.

J. M. Yao, Y. U. Bing-Song, and C. B. Che, “Application of modified GM model in prediction of oil production in Tarim Basin,” Petroleum Geology and development in Daqing, vol. 26, no. 1, pp. 92–96, 2007.

F. Zhou, “History matching and production prediction of a horizontal coalbed methane well,” Journal of Petroleum Science and Engineering, no. 10, pp. 96-97, 2012.

K. Li and R. N. Horne, “Comparison and verification of production prediction models,” Journal of Petroleum Science and Engineering, vol. 55, no. 3-4, pp. 213–220, 2007.

X. Li, C. W. Chan, and H. H. Nguyen, “Application of the Neural Decision Tree approach for prediction of petroleum production,” Journal of Petroleum Science and Engineering, vol. 104, pp. 11–16, 2013.

T. A. Saraiva, A. Szklo, A. F. P. Lucena, and M. F. Chavez-Rodriguez, “Forecasting Brazil's crude oil production using a multi-Hubbert model variant,” Fuel, vol. 115, pp. 24–31, 2014.

R. R. Charpentier and T. A. Cook, “Improved USGS methodology for assessing continuous petroleum resources using analogs,” US Geological Survey Open-File Report, vol. 1309, no. 27, 2010.

W. W. Weiss, R. S. Balch, and B. A. Stubbs, “How artificial intelligence methods can forecast oil production,” SPE-75143-MS, 2002.

T. Insperger, “On the approximation of delayed systems by Taylor series expansion,” Journal of Computational and Nonlinear Dynamics, vol. 10, no. 2, Article ID 024503, 2015.

H. Min, W. Jia, Y. Zhao, W. Zuo, H. Ling, and Y. Luo, “LATE: a level-set method based on local approximation of Taylor expansion for segmenting intensity inhomogeneous images,” IEEE Transactions on Image Processing, vol. 27, no. 10, pp. 5016–5031, 2018.

W. Yonthanthum, A. Rattana, and M. Razzaghi, “An approximate method for solving fractional optimal control problems by the hybrid of block-pulse functions and Taylor polynomials,” Optimal Control Applications and Methods, vol. 39, no. 3, pp. 873–887, 2018.

A. Nadh, J. Samuel, A. Sharma, S. Aniruddhan, and R. K. Ganti, “A Taylor Series Approximation of Self-Interference Channel in Full-Duplex Radios,” IEEE Transactions on Wireless Communications, vol. 16, no. 7, pp. 4304–4316, 2017.

S.-J. Lee, M.-C. Kang, K.-H. Uhm, and S.-J. Ko, “An edge-guided image interpolation method using Taylor series approximation,” IEEE Transactions on Consumer Electronics, vol. 62, no. 2, pp. 159–165, 2016.

Y. Q. Chen, J. G. Hu, and D. J. Zhang, “Derivation of logistic model and its self-regression method,” Xinjiang Petroleum Geology, vol. 17, no. 2, pp. 150–155, 1996.

J. G. Hu, D. J. Zhang, and Y. Q. Chen, “A model investigation on forecasting the production of oil and gas field,” Natural Gas Industry, vol. 17, no. 5, pp. 31–34, 1997.

Q. T. Yu, “A current formula of generalized increase and decline curves for predicting oilfield development indexes,” Petroleum Exploration and Development, vol. 27, no. 1, pp. 50–53, 2000.

J. G. Hu and Y. Q. Chen, “A simple model for predicting oil and gas field production,” China Offshore Oil and Gas, vol. 9, no. 1, pp. 53–59, 1995.

T. Ji, S. Lu, M. Tang et al., “Application of BP neural network model in fracturing productivity prediction of fuyu tight oil reservoir in jilin oilfield,” Acta Geologica Sinica, vol. 89, no. s1, pp. 154-155, 2015.