MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2018/1369639 1369639 Research Article A New Production Prediction Model Based on Taylor Expansion Formula http://orcid.org/0000-0002-0275-1292 Ding Xianfeng 1 http://orcid.org/0000-0001-5541-4620 Qu Dan 1 Qiu Haiyan 2 Shaat Mohamed 1 School of Science Southwest Petroleum University Cheng Du Sichuan 610500 China swpu.edu.cn 2 College of Chemistry and Chemical Engineering Southwest Petroleum University Cheng Du Sichuan 610500 China swpu.edu.cn 2018 3122018 2018 28 05 2018 02 11 2018 11 11 2018 3122018 2018 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.

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  proposed 4 types of declines: exponential, hyperbolic, harmonic, and ratio decline basing on decline trend observed in the field. Then Arps  (1956) simplified the proposed decline curves. In 1972, Gentry  studied the characteristics of each type of decline and then obtained two dimensionless equations and discussed the decline index n in detail. Fetkovich  (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 χ2 in statistics, Chen  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  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  presented two new types of decline curve qt=qi(a+1)/(bt+a) and qt=a/(tp+b)q and then demonstrated the validity of these two curves. On the basis of Weng's cycle model and Weibull's model, Song et al.  (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  improved Hu's antitangent differential distribution method, which made it more extensive. In 2004, Ding et al.  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.  put forward a generalized production decline equation qt=qi/(at2+bt+1)1/m, 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.  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 (Zhou, 2012), mathematical model method (Saraiva et al., 2014; Li et al., 2013; Keven and Roland, 2007), the unit proven reserve ratio method, analogy method (Charpentier and Cook, 2010), and intelligent prediction method (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 . 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.

Prediction model parameters
N p ( t ) = N R / 1 + c e - α t (Chen, et al., 1996)  N p : cumulative production of oil and gas fields, 104t (oil),108m3 (gas)NR: Recoverable reserves of oil and gas fields, 104t (oil),108m3 (gas)t: development years, a;c: model undetermined constant

N p ( t ) = N R / 1 + c e - α t (Hu, et al., 1997)  N p : cumulative production of oil and gas fields, 104t (oil),108m3 (gas)NR: Recoverable reserves of oil and gas fields, 104t (oil),108m3 (gas)t: development years, a;c,α: model undetermined constant

N p ( t ) = N R t b / t b + a Np(t)=NR(bt-1)/bt+a (Yu, 2000)  N p : cumulative production of oil and gas fields, 104t (oil),108m3 (gas)NR: Recoverable reserves of oil and gas fields, 104t (oil),108m3 (gas)t: development years, a;a,b: model undetermined constant

N p ( t ) = N R / 1 + c e - α t t - β (Ding et al. 2004)  N p : cumulative production of oil and gas fields, 104t (oil),108m3 (gas)NR: Recoverable reserves of oil and gas fields, 104t (oil),108m3 (gas)t: development years, a;c,α,β: 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 t tends to infinity (t), tn cannot be infinitely small. In order to facilitate the expansion of Taylor, we choose 1/t as the variable; the above models turn to be the forms as follows:(1)Np1t=NR1+ctβ,Np1t=NRt-bt-b+a,Np1t=NRb1/t-1b1/t+a,Np1t=NR1+ce-α/tandNp1t=NR1+ce-α/ttβ,t=1,2,.The Taylor expansion formula of the function f(x) at the point x0 is as follows:(2)fx=fx0+fx0x-x0+fx02!x-x02++fnx0n!x-x0n+Rnxwhere Rn(x)=f(n+1)(ξ)/(n+1)!(x-x0)n+1 and ξ is a value between x0 and x.

When t tends to infinity, obviously 1/t tends to zero. The cumulative production model with adjusted variables is expanded by the Taylor formula:(3)Np1t=Np0+Np01t+Np02!1t2+Np03!1t3++Npn0n!1tn+Rn1twhere Rn1/t=Np(n+1)(θ/t)/(n+1)!1/tn+1, 0<θ<1, t=1,2,, and Np(0) is the recoverable reserve (NR) 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 Np1/t to calculate the cumulative production approximately, the error is(4)Rn1t=Npn+1θ/tn+1!1tn+1

For a fixed n, when t(0,+), if Np(n+1)(1/t)M, the error estimation equation is(5)Rn1t=Npn+1θ/tn+1!1tn+1Mn+1!1tn+1

Given any precision ε>0 and choosing the appropriate time t, as long as M/(n+1)!1/tn+1ε, the order n can be achieved. This implies that, for any precision, a suitable expansion order n can be found to ensure that the new prediction model is strictly true.

When t tends to infinity, Rn1/t=Np(n+1)(θ/t)/(n+1)!1/tn+1 is a higher order infinitesimal of 1/tn. Therefore, the remainder can be written as(6)Rn1t=o1tn

When n=2, Np1/t=Np(0)+Np(0)1/t+Np(0)/2!1/t2+o1/t2, ignoring the infinite small terms, the above equation can be written as(7)Np1t=a+b1t+c1t2where a,b,c are undetermined parameters. Equation (7) can be solved by regression analysis, and then it can be used to predict the cumulative production.

When(8)n=3,Np1t=Np0+Np01t+Np02!1t2+Np03!1t3+o1t3,ignoring the infinite small items, the above equation can be written as (9)Np1t=a+b1t+c1t2+d1t3where a,b,c,d are undetermined parameters. They can be solved by regression analysis, and then it can predict the cumulative production.

Let(10)Y-=Np1t,X1=1t,X2=1t2,X3=1t3,Y=Npt,t=1,,m,then (9) can be written as (11)Y-=a+bX1+cX2+dX3

When different values are taken by t, the error sum of square between predicted data and the actual data can be written as(12)E=t=1mYt-Y-t2=t=1mYt-a-bX1t-cX2t-dX3t2

For obtaining the optimal coefficient a, b, c, d, we must make a minimum error E; that is,(13)minE=t=1mYt-Y-t2=t=1mYt-a-bX1t-cX2t-dX3t2

Let the partial derivatives of (13) with respect to a, b, c, d be zero; then we can get the linear equation set (14). (14)ma+bt=1mX1t+ct=1mX2t+dt=1mX3t=t=1mYtat=1mX1t+bt=1mX12t+ct=1mX1tX2t+dt=1mX1tX3t=t=1mX1tYtat=1mX2t+bt=1mX1tX2t+ct=1mX22t+dt=1mX2tX3t=t=1mX2tYtat=1mX3t+bt=1mX1tX3t+ct=1mX2tX3t+dt=1mX32t=t=1mX3tYt

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.(15)Np1t=a0+a11t+a21t2+a31t3++an1tnwhere a0,a1,,an are undetermined parameters.

For the calculation of production, the method in literature (Ding et al., 2004) can be adopted. That is,(16)qt=Npt-Npt-1If 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 n, 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 .

2.3.1. Logistic Production Prediction Model

Logistic production prediction model is Np(t)=NR/1+ce-αt. We choose 1/t as the variable, Np1/t=NR/1+ce-α(1/t)-1=NR/1+ce-αt (the cumulative production of t year); Taylor expansion formula is (17)Np1t=Np0+Np01t+Np02!1t2+Np03!1t3++Npn0n!1tn+Rn1tWe can ignore the infinite item: (18)Np1t=a0+a11t+a21t2+a31t3++an1tn.

For t, a0=NR, the recoverable reserves are equal to the cumulative production at this time.

The model parameters are obtained by self-regression in , and the results are α=-lnβ, NR=1-β/α, c=NR/NP-1eαt, c=1/mj=1mNR/Npj-1eαtj (considering the influences on the parameter c, 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: α=-ln0.8027=0.2198, NR=1-0.8027/3.0479×10-5=6473(104t), c=22.2072.

We choose 1/t as the variable; the Logistic production prediction model turns to be the forms (19)Np1t=64731+22.2072e-0.2198/t.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 , 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 Np(t)=A2eB2e-bt (the cumulative production of t year). We choose 1/t as the variable: (20)Np1t=A2eB2e-b1/t-1=A2eB2e-btCumulativeproductionoftheyeart(21)Np1t=Np0+Np01t+Np02!1t2+Np03!1t3++Npn0n!1tn+Rn1tWe can ignore the infinite item:(22)Np1t=a0+a11t+a21t2+a31t3++an1tnFor t, a0=NR. The model parameters are obtained by linear regression in , and the results are A2=6621, B2=-7.1279, b=0.1647.

We choose 1/t as the variable; the G&H production prediction model turns to be the forms(23)Np1t=6621e-7.1279e-0.1647tWhen 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.

The actual data and the prediction data of the three models.

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 x and y be prediction value and real value, respectively. The error is defined by (x-y)/y. 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:(24)n=2:A=1.00000.99330.99331.0000,B=1.00000.99510.99511.0000n=7:C=1.00000.99450.99451.0000,D=1.00000.99910.99911.0000

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, 1/t 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.

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

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

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

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.

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

The actual data and the prediction data of the cumulative production of Bavly  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

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

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

The actual data and the prediction data of the cumulative production of Bavly  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

The actual data and the prediction data of the cumulative production of Bavly  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

The actual data and the prediction data of the cumulative production of Bavly  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

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

The comparison of actual and predicted data of cumulative production.

The comparison of actual and predicted data of cumulative production.

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.

Arps J. J. Analysis of decline curies SPE-G 1945 Arps J. J. Estimation of primary oil reserves SPE 627-G 1956 Gentry R. W. Decline-curve analysis SPE 3365-PA 1972 Fetkovich M. J. Decline curve analysis using type curves SPE 4629-PA 1980 Chen Y. C. The derivation and application of Weng's prediction model Research Institute of Petroleum Exploration and Development of CNPC 1996 16 2 22 26 Hu Y. D. Exponential distribution and inverse tangential distribution of production decline Applied Mathematics & Information Sciences 1997 24 6 76 81 Yu Q. T. A new decline curve Petroleum Exploration and Development 1999 26 3 72 75 Song C. Z. Ma X. J. Generalized Weibull, Wengcycle model for predicting oil and gas production Mineral Rock 2000 20 2 37 38 Li C. L. Han S. G. Cheng L. S. Research on a new production decline curve Journal of Southwest Petroleum Institute 2003 25 5 31 33 Ding X. F. Liu Z. B. A new model for oil and gas production prediction Petroleum Exploration and Development 2004 31 3 104 106 Zhu Y. J. Gao W. J. Establishment and research of a new generalized production decline equation Fault Block Oil and Gas Fields 2005 12 6 47 49 Yao J. M. Bing-Song Y. U. Che C. B. Application of modified GM model in prediction of oil production in Tarim Basin Petroleum Geology and development in Daqing 2007 26 1 92 96 Zhou F. History matching and production prediction of a horizontal coalbed methane well Journal of Petroleum Science and Engineering 2012 10 96 97 10.1016/j.petrol.2012.08.013 2-s2.0-84866547624 Li K. Horne R. N. Comparison and verification of production prediction models Journal of Petroleum Science and Engineering 2007 55 3-4 213 220 2-s2.0-33846307086 10.1016/j.petrol.2006.08.015 Li X. Chan C. W. Nguyen H. H. Application of the Neural Decision Tree approach for prediction of petroleum production Journal of Petroleum Science and Engineering 2013 104 11 16 2-s2.0-84876740209 10.1016/j.petrol.2013.03.018 Saraiva T. A. Szklo A. Lucena A. F. P. Chavez-Rodriguez M. F. Forecasting Brazil's crude oil production using a multi-Hubbert model variant Fuel 2014 115 24 31 2-s2.0-84880862080 10.1016/j.fuel.2013.07.006 Charpentier R. R. Cook T. A. Improved USGS methodology for assessing continuous petroleum resources using analogs US Geological Survey Open-File Report 2010 1309 27 Weiss W. W. Balch R. S. Stubbs B. A. How artificial intelligence methods can forecast oil production SPE-75143-MS 2002 10.2118/75143-MS Insperger T. On the approximation of delayed systems by Taylor series expansion Journal of Computational and Nonlinear Dynamics 2015 10 2 024503 Min H. Jia W. Zhao Y. Zuo W. Ling H. Luo Y. LATE: a level-set method based on local approximation of Taylor expansion for segmenting intensity inhomogeneous images IEEE Transactions on Image Processing 2018 27 10 5016 5031 10.1109/TIP.2018.2848471 MR3829447 Yonthanthum W. Rattana A. Razzaghi M. An approximate method for solving fractional optimal control problems by the hybrid of block-pulse functions and Taylor polynomials Optimal Control Applications and Methods 2018 39 3 873 887 10.1002/oca.2383 MR3796971 Nadh A. Samuel J. Sharma A. Aniruddhan S. Ganti R. K. A Taylor Series Approximation of Self-Interference Channel in Full-Duplex Radios IEEE Transactions on Wireless Communications 2017 16 7 4304 4316 2-s2.0-85028994578 10.1109/TWC.2017.2696938 Lee S.-J. Kang M.-C. Uhm K.-H. Ko S.-J. An edge-guided image interpolation method using Taylor series approximation IEEE Transactions on Consumer Electronics 2016 62 2 159 165 2-s2.0-84979603376 10.1109/TCE.2016.7514715 Chen Y. Q. Hu J. G. Zhang D. J. Derivation of logistic model and its self-regression method Xinjiang Petroleum Geology 1996 17 2 150 155 Hu J. G. Zhang D. J. Chen Y. Q. A model investigation on forecasting the production of oil and gas field Natural Gas Industry 1997 17 5 31 34 Yu Q. T. A current formula of generalized increase and decline curves for predicting oilfield development indexes Petroleum Exploration and Development 2000 27 1 50 53 Hu J. G. Chen Y. Q. A simple model for predicting oil and gas field production China Offshore Oil and Gas 1995 9 1 53 59 Ji T. Lu S. Tang M. Wang M. Wang W. Liang H. Min C. Application of BP neural network model in fracturing productivity prediction of fuyu tight oil reservoir in jilin oilfield Acta Geologica Sinica 2015 89 s1 154 155 10.1111/1755-6724.12303_2