A New Growth Curve for Predicting Production Performance of Water-Flooding Oilfields

Oil production and water cut prediction is one of the most important research contents of reservoir production performance analysis. The growth curve method has the advantages of the general water drive curve method and the combined solution model method with fewer parameters and simple and fast calculation process and so it has been widely used in well production prediction. Based on the analysis of 4W and 4Y4 model growth curves, a new generalized growth curve of the well production performance is proposed. The new model can forecast cumulative oil production, annual oil production, and water cut at different oilfield development periods. A MATLAB program was developed to derive the parameters in the new model. The built model was applied to the production data of the Samattalol oilfield and Daqing oilfield. The predicted cumulative oil production, annual oil production, and water cut are all close to the actual production data, and satisfactory results are obtained, which demonstrates the practicability and reliability of the new model.


Introduction
Water flooding is the most common oilfield development technology around the world. With the oilfields gradually entering the stage of medium or high water cut, accurate prediction of reservoir production performance is very important for reservoir engineers to formulate reasonable measures to stabilize oil production and control water production and to adjust oilfield development plans [1][2]. Oil production and water cut are two major contents of reservoir production performance analysis. Generally, the main methods to predict the annual oil production and water cut of the reservoir include numerical reservoir simulation, water drive curve, and analytical model. e numerical reservoir simulation requires a complex geological model, and it is time-consuming and has low efficiency [3][4][5][6][7][8]. Since the water drive curve method was proposed in the 1980s, there have been about 50 kinds of water drive curves, but many water drive curves have certain limitations in calculating different types of reservoirs, and the calculation accuracy is not high and the calculation results sometimes differ greatly [9][10][11][12][13][14]. In contrast, the analytical method has better practical conditions and is suitable for the whole process of oilfield production and water cut changes [15,16]. Ling et al. proposed an oil cut power function fitting method for the decline rate of the type A water drive curve. However, the method is only suitable for predicting the production of reservoirs with water cut greater than 70% which conforms to the characteristics of type A water drive [17]. Liu et al. established a new dynamic calculation method of displacement efficiency and volume sweep coefficient of water drive reservoirs. e new method overcomes the dependence of most methods on the relative permeability curves of oil and water phases [18]. Based on the correlation between the oil-water relative permeability ratio and water saturation, a new water-flooding type curve was created by combination with the conventional reservoir engineering methods [19][20][21].
Based on the previous studies about the growth curve model, combined with the 4W and 4Y4 types of growth curve, the new prediction model of annual oil production, cumulative oil production, and water cut are established. e reliability of these built models is verified by the actual production data of the Samottorol oilfield and Daqing oilfield.

Establishment of the New Model
Weibull, Gompertz, and logistic growth curves are used to predict the oilfield development performance. Yu [22] studied the growth curves extensively and summarized the following laws: when time t tends to 0, cumulative oil production N P tends to 0; when t tends to infinity, N P tends to N Rmax . e study of growth curve includes the characteristics, application, recoverable reserves, and development index of the curve. It can be seen that the growth curve plays an important role in the analysis and prediction of reservoir production performance. Due to the complexity of production history, it is very difficult to describe the whole process of oilfield development quantitatively. e water cut variations of many oilfields do not follow the single 4W or 4Y4 water drive type curve but a type in-between these two water drive type curves. A simple and practical comprehensive water drive type curve is established. e new model expands the application range of the water drive curve, including both the 4W water drive type curve and the 4Y4 water drive type curve.
Yu [22] proposed the 4W growth curve for predicting oilfield development performance. e curve form is simple, and only has three coefficients α, β, and c. e growth curve of 4W type is given by Suppose that a � 1/α and b � β, the 4W growth curve equation can be rewritten as e growth curve of the 4Y4 type is shown as Equations (2) and (3) have similar mathematical expressions. According to the relationship between cumulative oil production and development time in equations (2) and (3), the new growth curve for predicting oilfield development performance is established: Equation (4) is the new cumulative oil production prediction model.
When parameter d takes different values, a series of different cumulative oil production curves can be obtained. When d � 0, equation (4) is transformed into equation (2), and when d � 1, equation (4) is transformed into equation (3). erefore, both equations (2) and (3) are special cases of equation (4).
Taking the derivation with respect to the time t on both sides of equation (4), the corresponding new prediction model of annual oil production can be obtained as follows: According to the law of growth curve, when t tends to 0, N P tends to 0 and then water cut f w tends to 0; when t tends to infinity, N P tends to N Rmax and then f w tends to 1 (taking the limit water cut as 1). erefore, equation (4) can be rewritten as follows: Equation (6) is the new established water cut prediction model. e parameter d in equations (4)-(6) is unknown, which overcomes the limitation of the original model and expands the application range of the growth curve in reservoir production performance analysis.
When d � 0, the corresponding water cut prediction model of the 4W growth curve is When d � 1, the corresponding water cut prediction model of the 4Y4 growth curve is given as follows:

Solution of the Model
ere are two methods to solve cumulative oil production in equation (4) and water cut model in equation (6). One is to apply the MATLAB programming method to obtain the values of parameters a, b, c, d, and N Rmax through multiple regression. e other method is to rewrite equation (4) into e left and right sides of equation (9) are double logarithmic linear relations.
rough the linear trial and error method, when the linear correlation is the best, the values of a, b, c, d, and N Rmax are obtained and the variation of cumulative oil production with time can be calculated by equation (4). Due to the complexity of equation (5), it is difficult to calculate the relevant parameters by the above two methods. e annual oil production can be calculated by the following formula: e water cut equation parameters a, b, c, and d can also be calculated by the MATLAB programming method or the linear trial and error method, and the change of water cut with time can be calculated by substituting the above parameters into equation (6). Different c and d values are selected and tried to determine whether the two sides of equation (11) are linear. If c and d are appropriate values, the left and right sides of equation (11) show the linear relationship. Generally, this attempt needs to be done many times. However, a MATLAB program is developed in the calculation, and the program can quickly calculate the best c and d values by using the least square method. e MATLAB program gives more accurate results than the linear trial and error method. e water cut double logarithmic linear equation is

Model Validation
Using the new prediction model, the production performances of the Samattalol oilfield and Daqing oilfield, which are relatively advanced in water-flooding development, are predicted. e predicted results are compared with actual production data to verify the accuracy and reliability of the new model.

Samattalol Oilfield.
e Samattalol oilfield was put into operation in 1969 and has been developed since more than 50 years. e production data of the oilfield from 1969 to 1990 are shown in . (12) If the cumulative oil production is substituted into equation (10), the annual oil production of the Samattalol oilfield can be calculated. Based on the data of water cut in Table 1

Daqing Oilfield.
e Daqing oilfield was put into operation in 1965 and has been developed since more than 50 years, too. e production data of the oilfield from 1965 to 1987 are shown in Table 2 [24]. Substituting the cumulative oil production data and time into equation (4)  When the cumulative oil production is substituted into equation (10), the annual oil production of the Daqing oilfield can be calculated, too. Based on the water cut data in Table 2 rough the comparison of the predicted and actual data in Figures 1 and 2 and Tables 1 and 2, it can be seen that the cumulative oil production, annual oil production, and water cut calculated by the new prediction model are all close to the actual values and most of the prediction relative errors are less than 4.0%, which is generally in line with the development trend of the production performance data of the above two oilfields over time.
ere are 3 abnormal parameters in the oil well, and the error between the predicted value of the model and the actual value is large. Because there were only a few oil wells at the beginning of the production, the oil well production and water cut were unstable. e newly added wells can be responsible for the production and water cut, and the oilfield production enters a stable stage with the development of time. e results show that the new model has a good application effect in the Samattalol oilfield and Daqing oilfield, which provides reference for similar   oilfields to apply the new model to predict production performance.

Conclusion
(1) According to the growth curves of 4W and 4Y4, the new generalized growth curve for predicting development performance of water-flooding oilfields is established. e new model overcomes the limitation of the original parameters and has wider applicability and flexibility. e new model can predict the changes of cumulative oil production, annual oil production, and water cut with time.
(2) rough the development of a MATLAB program, the actual values of parameters in the new model can be easily and quickly solved and the performance prediction equation in line with the actual oilfield production can be obtained. e cumulative oil production can be obtained by the growth curve, and the expression of annual oil production can be calculated by the cumulative oil production. e water cut prediction model is suitable for the whole stage of oilfield development. Cumulative oil production, 10 4 t Q: Annual oil production, 10 4 t t: Time, year f w : Water cut, f.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Mathematical Problems in Engineering 5