A New Production Prediction Model Based on Taylor Expansion Formula

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.


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-steadystate 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  2 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   =   (+1)/(  +) 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  (Hu, et al., 1997) [25]   : cumulative production of oil and gas fields, 10 4  (oil),10 8  3 (gas)   : Recoverable reserves of oil and gas fields, 10 4  (oil),10 8  3 (gas) : development years, a; , : model undetermined constant (Yu, 2000) [26]   : cumulative production of oil and gas fields, 10 4  (oil),10 8  3 (gas)   : Recoverable reserves of oil and gas fields, 10 4  (oil),10 8  3 (gas) : development years, a; , : model undetermined constant (Ding et al. 2004) [10]   : cumulative production of oil and gas fields, 10 4  (oil),10 8  3 (gas)   : Recoverable reserves of oil and gas fields, 10 4  (oil),10 8  3 (gas) : development years, a; , , : model undetermined constant forward a generalized production decline equation   =   /( 2 +  + 1) 1/ , 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][15][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][20][21][22][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.

Methods
. .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.

. . e Prediction Model Based on Taylor Expansion Formula.
The cumulative growth curve models in the literatures are listed in Table 1.
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 1/ as the variable; the above models turn to be the forms as follows: The Taylor expansion formula of the function () at the point  0 is as follows: where   () = ( (+1) ()/( + 1)!)( −  0 ) +1 and  is a value between  0 and .
Error Analysis.Using the expansion order n of  p (1/) to calculate the cumulative production approximately, the error is For a fixed n, when  ∈ (0, +∞), Given any precision  > 0 and choosing the appropriate time , as long as (/( + 1)!)(1/) +1 ≤ , 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,   (1/) = ( p (+1) (/)/( + 1)!)(1/) +1 is a higher order infinitesimal of (1/)  .Therefore, the remainder can be written as 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.
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).
. .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].
The model parameters are obtained by self-regression in [19] We choose 1/ 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.

. . . 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  p () =  2   2  − (the cumulative production of t year).We choose 1/ as the variable: We can ignore the infinite item: For  → ∞,  0 =   .The model parameters are obtained by linear regression in [28], and the results are  2 = 6621,  2 = −7.1279, = 0.1647.
We choose 1/ 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.
. . .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 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.
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.
. .e 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/ 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.our model and comparison and analysis were made between the prediction data by models in literatures and the actual data.

Result and Discussion
. .Numerical Results . . .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.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.

. . . 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.
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.(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.

Conclusions
(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.

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

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

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

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

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

( 1 )Figure 6 :Figure 7 :
Figure 6: The comparison of actual and predicted data of cumulative production.

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

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

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

Table 10 :
Errors.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. the