Overall process of prediction

Considering that the short-term time series data of daily oil production of a single well are comprehensively affected by many factors, such as porosity, permeability, oil-water saturation, oil-water viscosity ratio of crude oil and inter-well connectivity, showing greater randomness and volatility, and the internal mechanism of the data is relatively complex, it is difficult to accurately predict using a single model. In this regard, combined with data decomposition and a multi-agent combination model, this paper proposes a short-term prediction method for oil production of a single well in the oilfield, and constructs the short-term prediction process of oil production of a single well in the oilfield as shown in S1 Fig, including data decomposition, dataset generation, component aggregation, model allocation, and prediction fusion.

Step 1. Data decomposition.

Raw data was decomposed and reconstructed by Complete Ensemble Empirical Mode Decomposition with Adaptive Noise Analysis (CEEMDAN) to form Intrinsic Mode Function (IMF)s and the residual. After calculating the Fuzzy Entropy (FE value and autocorrelation coefficient (AC) value of each IMFs without the residual. the two values form a vector as the corresponding IMF feature vector.

Step 2. Dataset generation.

The raw data of daily well production and the IMFs obtained after decomposition are one-dimensional sequential data, which cannot be directly adopted for model training and learning.

Step 3. Data distribution.

IMFs are obtained after decomposing the training data are kept, while the residual components are discarded. Firstly, FE value and autocorrelation of each IMF are calculated, which are then normalized to the range of 0,1 in order to form the data vector space. Feature vectors in the vectors space above are clustered into two groups, after which the group including the vector of minimal vector Euclidean Distance (ED) [28] value is signed as "ML-group" while the other is signed as "ELM-group". By the procedure described above, the IMFs are reasonably distributed.

Step 4. Data prediction.

The data output in step (3) is grouped and predicted. Two sub-prediction models. machine learning and deep learning, are used to form a multi-agent combination model. The machine learning model is an Online Sequential Extreme Learning Machine (OSELM) model based on the Whale Optimization Algorithm (WOA)The deep learning model is a convolutional bidirectional gated cyclic network model with self-attention mechanism (ED-CNN-BiGRU-Att) based on Encoder-Decoder architecture.

Step 5. Fusion of predicted results.

The two values predicted in the above step are again added together to obtain the final well predicted yield.

Data decomposition algorithm

CEEMDAN is used to decompose the training set, to reconstruct and form multiple groups of IMFs and residuals, obtain data components under different frequency characteristics, and reduce the complexity of data influencing factors, which is used to decompose the training set as data decomposition approach in this paper, because the result decomposed by CEEMDAN can ameliorate the modal aliasing phenomenon [29], and the completeness of the data reconstruction is better via adding adaptive noise as well. The basic principles of the CEEMDAN algorithm are briefly described as follows.

Step 1: Signal T(t) = T₀(t)+a₀nⁱ(t) is constructed from the original signal T₀(t) by adding noise, where a₀ refers signal-to-noise ratio control coefficient, and nⁱ(t) (i = 1,2, …, N) is the white Gaussian noise. In the first phase, the newly generated data signal needs to be initially decomposed by EMD to obtain via (1). after which the first-order residual data signal is given by T₁(t) = T(t)-IMF₁(t).

(1)

Step 2: A new signal T₂(t) = T₁(t)+a₁E₁[nⁱ(t)] is constructed by adding new noise signal. Here, E_k[*] indicates the function that generates the first phase. And then, repeating calculation by Step 1, IMF₂(t) and T₃(t) are both achieved.

Step 3: Following steps 1 and 2, multiple mode components, as well as residual signals, can be obtained with the expression as follows: (2)

Dataset generation algorithm

As shown in S2 Fig, the sliding window segmentation method is used to divide the dataset into several sample sets for data generation in this paper. This method takes W data as the input sequence of the model from the Nth position of the data set, and then takes 1 data value as the model’s output to form the data set of one-step prediction. N counts from 1 until all the training sets are segmented, and W represents the input length of a group of data [30]. For example, starting from the dataset, a continuous sequence of data with a length of W from the first data point are selected as the first input for the training set, denoted as X₁ = (x₁, x₂, …, x_w). At this point, the output of the training set is Y₁ = x_w+1. Subsequently, the input to the second training set is denoted as X₂ = (x₂, x₃,…, x_w+1) and its corresponding output is denoted as Y₂ = x_w+2. This process continues until all data has been allocated.

Data distribution process

Since entropy is a measure of the complexity and chaos of sequence data, the greater its value indicates that, the more frequent the data sequence contains, the greater the chaos of the sequence [31], which can accurately reflect the complexity of daily oil production of a single well. Given a sequence x(i), i = 1 to N, the sequences , i = 1 to N-m+1, is established as given below. (3) where m refers to the length of the sequences, x₀(i) is a base line, and denotes the similarity degree using fuzzy membership function for the vector and . (4) where n and r are width of the exponential function and predefined gradient, . denotes the maximum absolute difference between . and , and the function is defined as follow: (5)

. as the sequences is generated by m = m+1, and φ^m(n,r) is constructed afterwards. Finally, the fuzzy entropy is calculated as given below: (6)

In addition, autocorrelation coefficient, as another quantitative metric, is calculated as follow: (7)

The IMFs feature vectors, constructed as (V_FE,V_acf), are normalized because of the uneven scale. DBSCAN clustering method was used to cluster according to feature vectors, which classified IMFs with similar stationary characteristics into same group. ED is used to solve the distance of all vectors, then the minimum distance value of the calculated vector is found out, and the group with the minimum value was marked as "ML-group", and the other group is marked as "DL-group". The calculation formula of distance is as follow. (8) where V_FE represents FE value and V_acf represents the autocorrelation coefficient.

Subsequently, all of the IMFs were dichotomized by the clustering algorithm, and the Euclidean distance was used to solve the distance of all vectors, then the minimum distance value of the calculated vector was found out, and the group with the minimum value was marked as "ML-group", and the other group was marked as "DL-group".

Prediction model

In this paper, we use a combination of machine learning and deep learning to predict data. The machine learning model used in the paper is adopted as OSELM, which is currently a popular prediction model due to its ability to effectively capture the inline relationships of nonlinear and non-stationary data. In another way, this machine models developed from extreme learning machines can process feed-in input data sets continuously. For better performance of prediction, this machine learning model is optimized by training dataset. In the ED-CNN-BiGRU-Att model as machine learning in this paper, of which layers include convolutional neural network (CNN) layer, pooling layer, bidirectional gate recurrent unit (BiGRU) layer, and self-attention layer, and its overall organizational structure uses encoder-decoder mode for data interaction. In the overall structure, the CNN is used to obtain the one-dimensional data characteristics of oil production. The BiGRU is used to learn the characteristics of oil production time series and strengthen the information of the front data. The self-attention layer retains the probability weight of the hidden state of the time characteristics of the oil production series in the learning process, and redistributes the characteristic parameters of the key information of the data. The Encoder-Decoder framework shares the learning weight. The Encoder pre-trains the dataset, extracts the data information, and forms low-rank vector. Then the Decoder processes and extracts the information and finally outputs it, so as to retain the effective information of the data to the greatest extent.

Deep learning network

In this study, CNN is chosen as the protrusion layer because of its ability to map the data to multiple different scales to extract data features through convolutional operations [34]. However, it is difficult to identify the time series causality of the internal data of single well production time series data. In order to obtain the timeseries association information of the data set, the GRU network layer is added to analyze the time series data association information, and the information is extended backward and retained. GRU is a variant of the recurrent neural network after streamlining the internal structure of LSTM, whose internal structure includes a reset gate and an update gate composed of different activation functions, as shown in S3 Fig. The specific mathematical expression of interna data flow of GRU is as follows: (9) where W_z, W_r and W are weight coefficients; H_t-1 is the implicit information, and H_t represents the intermediate hidden content.

Although the GRU can extract the correlation characteristics of data time scale, it will continue to transfer back calculation with the amount of time series, weakening or even losing the previous correlation information.

Therefore, this paper uses bidirectional Gated Recurrent Unit (BiGRU) to construct deep learning network instead of GRU, which can relearn the features of the previous data [35], as shown in S4 Fig. The network stacks a one-dimensional CNN layer, max-pooling layer, BiGRU and linear layer. Through the unique recursive reverse link mode of BiGRU, it completes forward propagation and then backpropagation, to retrieve the front-end time series data information, ensure the continuity of time series information, and strengthen the front-end time series information. In order to enable CNN-BiGRU to consider the global information fully, the attention mechanism is introduced to allocate the calculation weight of the sequence value to obtain the global information of the data [36]. The output of the CNN-BiGRU network is taken as the three components of Q, K and V, which are transformed by the linear laver and then calculated and output according to the dot product of the following formula. (10) where K^T represents the transpose matrix of the last two dimensions of the K component, d_k represents the dimensions of Q and K, is regarded as a scaling factor, and softmax is a normalized exponential function, which can obtain the weight information vector containing the probability of primary and secondary characteristics of the data. The introduction of attention mechanism calculation can enhance the main characteristics of output data and weaken secondary information.

To obtain the sequence feature information in advance and ensure the learning efficiency of the deep network model for data, the above network is optimized and constructed under the encoder-decoder framework to form the ED-CNN-BiGRU-Att network model, as shown in S4 Fig. The network encoder comprises a convolution layer, a pooling layer and a bidirectional gate recurrent unit network layer. The input data is trained and learned in the decoder, and the hidden weight information is output by the last layer. At the same time, after the input data passes through the one-dimensional convolution layer and the max- pooling layer in turn, it combines the implicit weight information returned by the encoder to construct two inputs of the subsequent two-way cyclic network layer. It continues to pass forward into the self-attention layer, and finally the linear layer is the output of the overall network structure.

Data fusion model

After the construction of ML and DL, IMFs are predicted by the two models respectively after redistribution to play their respective advantages.

The series both in “ML-group” and “DL-group”, generated through the step of data distribution process above, are separately added together to build new two series “ML-series” and “DL-series”. Then as illustrated in S5 Fig, “ML-series” and “DL-series”are predicted by WOSELM and ED-CNN-BiGRU-Att, respectively, after which both of the predicted are summed to obtain the final result.

Data preprocessing and analysis

After decomposition by CEEMDAN, seven IMFs and one residual are obtained, as shown in S7 Fig. The residual component cannot continue to be decompose, which is discarded in this study. IMFs1-IMFs3 are high-frequency signals, IMFs4 and IMFs5 are medium-frequency signals, and IMFs6 and IMFs7 are low-frequency signals that reflect the data trends of the actual values.

Entropy is a measure that can describe the complexity and chaos of a time series signal. The larger its value, the more complex the waves of the sequence signal, and the more frequencies it contains accordingly [31]. Moreover, AC can also express correlations before and after data points in a time series dataset to characterize data complexity. FE and AC of IMFs after normalizing are given in S2 Table. It is clear seen in S8 Fig that decomposition can significantly reduce the entropy value of the raw data, resulting in a set of IMFs with reduced complexity. This process can further extract the influence factors of oil production from the data, which is beneficial for subsequent model training and learning. In S2 Table, the label of DL represents the “DL group” after clustering and grouping, while ML represents the “ML group”.

Comparison analysis of prediction performance

Taking dataset 1 as an example, it is divided into a training set and test set according to the ratio of 8:2, and processed according to the short-term prediction method of single well production based on the multi-agent hybrid model proposed in this paper. First, CEEMDAN is used to decompose and calculate the training set data, and seven groups of IMFs and a residual quantity can be obtained as shown in S7 Fig. Among them, the residual quantity is the smallest component of the order that cannot be decomposed after decomposition and reconstruction, which can be considered as the noise quantity after decomposition and can be ignored. It can be seen from S7 Fig that the original data is decomposed into components with less frequency, and the frequency range covers high frequency, intermediate frequency and low frequency. IMFsl-IMFs3 are high-frequency signals, and the data floating range is within the range of [-2.5, 2.5]. The magnitude is smaller than the original data, which can be regarded as random signals; IMFs4 and IMFs5 are intermediate frequency signals. The data floating range is the same as that of high-frequency signals, while their frequency is lower than that of high-frequency signals. They show a certain periodic law, which can be regarded as quasi-periodic signals generated after decomposition; IMFs6 and IMFs7 are low-frequency signals that reflect the actual value’s data trend and can be considered as the trend quantity information. Among them, the amplitude range of IMFs6 is the same as that of high and intermediate frequency, showing a concave curve similar to the data form of the actual value. IMFs7 is the component with the lowest frequency, and the amplitude is the closest to the actual value, showing a gradual decreasing trend, which is consistent with the gradual decreasing trend of the actual value. The above calculation can decompose the data training set into IMFs with different frequencies, and various features implied in the actual value data can be characterized in a data-driven manner.

Since the decomposed and reconstructed IMFs component is one-dimensional data, the sliding window method converts the original one-dimensional data into a data form to generate a data set that the model can identify. Each set of data in the dataset contains 10 predictive training values and 1 tag value. At the same time, to avoid identifying the time correlation characteristics and improve the generalization ability of model prediction, the dataset is randomly mixed.

For the IMFs components with different characteristics in dataset 1, the FE value and autocorrelation coefficient are used to construct the sequence feature vector, which is used as the classification basis for clustering. Due to the different scales of FE value and autocorrelation coefficient, the maximum and minimum normalization method is used to normalize FE value and autocorrelation coefficient, and the results are shown in S2 Table.

Since entropy is a measure of the complexity and chaos of sequence data, the greater its value indicates that, the more frequent the data sequence contains, the greater the chaos of the sequence [30], which can accurately reflect the complexity of daily oil production data of a single well. Therefore, this paper uses the Fuzzy Entropy value and autocorrelation coefficient to quantify the complexity of the data decomposition and reorganization sequence, and constructs the feature vector based on this, combined with the data clustering theory, as the basis for sub-model data grouping input.

The IMFs components were clustered using density-based clustering, and the number of centers was set to 2. The IMFs components were divided into two groups, IMFs1-IMFs4 and IMFs5-IMFs7. Calculating the euclidean distance of the sequence eigenvector, the group of the minimum length is obtained. This data group has low The IMFs components were clustered using density-based clustering, and the number of centers was set to 2. The IMFs components were divided into two groups,IMFs1-IMFs4 and IMFs5-IMFs7. Calculating the euclidean distance of the sequence eigenvector, the group of the minimum length is obtained. This data group has low non-stationary but strong volatility, complex data components, and high pre and post-dependence. Therefore, this group uses the deep learning model for training prediction, while the other group uses the machine learning model. The FE value, autocorrelation coefficient, and grouping obtained through the above clustering process are shown in S2 Table. According to the above clustering grouping, the accumulated values of IMFs1-IMFs4 and IMFs5-IMFs7 components are predicted respectively by ED-CNN-BiGRU-Att deep learning model and OSELM machine learning model based on the WOA algorithm, and then the two groups of predicted values are fused and accumulated to finally obtain the single well production prediction value.

According to the short-term prediction method of oil field single well production based on the multi-agent hybrid model, the selected three-stage data set is predicted. The results of IMFs component FE value obtained from the training set data of the three-segment data set and its decomposition are shown in S8 Fig. It can be seen that the mean value of data set 1 is high, the fluctuation is large, and the FE value of data is large, although the FE value of IMFs component is significantly reduced after decomposition: The FE values of the mean, volatility and decomposition of dataset 2 are lower than those of dataset l; Dataset 3 has the smallest amplitude change, the best stability, and the FE value of the original data and IMFs component is the lowest. Therefore, CEEMDAN decomposition can effectively reduce the complexity of the dataset, and can be quantified by FE value. The FE values of the selected three data sets and their IMFs components all show a downward trend one by one. It can be seen that the FE value of the IMFs component of the single well daily oil production time series data is less than the original data, indicating that the data decomposition and reconstruction method can effectively reduce the degree of data confusion and non-stationarity and can separate the influencing factors of oil production from the, which is conducive to the prediction of subsequent models.

The training sets of the three datasets are further divided into training and validation sets, which are used for tuning hyperparameters of the machine learning and deep learning models. Due to the incorporation of key components such as OSELM, GRU, and linear layers in the proposed predictive model of this study, comparative models including OSELM, BP, LSTM, and CNNBiGRU are included in the predictive research.

Based on the characteristics of oil production data, the main hyper parameters of the ED-CNN-BiGRU-Att network model are given as follows: The out channels value is 32 and the kernel size is 2 in the CNN, respectively. The kernel size is 2 in the pooling layer. In the BiGRU, the hidden size is set to 12, the num layers is set to 3, and bidirectional is set to true value, The number of neurons in the linear layer in the Attention is also set to 12. Also, LSTM model contains 12 intermediate cell layers, amd the model is trained for 2000 iterations with a learning rate of 2E-4. S9 Fig shows the predictive performance of the various models across the three datasets, displayed by the predicted line plots and the predicted Taylor plots. From the prediction line plots, it can be observed that the predicted values of all models are closely aligned with the actual values, indicating that each model exhibits excellent predictive power for well production. Across the three datasets, all models exhibit some level of hysteresis, but the proposed model in this study demonstrates the best tracking performance in terms of its predicted values.

The Taylor diagram is a widely used and provides a comprehensive assessment of multiple model variables by comparing their statistical properties with observations. The diagram consists of a scatterplot that displays the standard deviation and correlation coefficient of each model variable relative to the reference dataset. The distance between each model point and the reference point represents the standard deviation ratio, indicating the spread of the model variable compared to the observed data. A smaller distance suggests a better agreement between the model and observations in terms of variability. From the Taylor diagram, it is evident that the points of the proposed model are noticeably closer to the other models compared to the other models. Secondly, the angle between the model variable vector and the reference vector represents the correlation coefficient. A small angle indicates a strong correlation between the model and observed data, implying a good match in terms of pattern and phase. From this perspective, the proposed model in this study is also more competitive.

To quantify the short-term prediction effect, performance metrics are selected, including Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), R-Square (R²), Mean Absolute Range Normalized Error (MARNE) [37], and Nash-Sutcliff efficiency (NSE) [38], of which the formulas are as follows: (11) (12) (13) (14) (15) (16) where, is the predicted value calculated by the prediction model, y_i is the original data value in the test set, and is the predicted mean value.

In addition, the Diebold-Mariano (DM) test method is a test method to evaluate the prediction performance from a statistical perspective [39, 40]. Eq (17) gives the hypothesis that the prediction performance of the model in this paper is better than that of the comparative model. (17) where H₀ and H₁ represent the original hypothesis and alternative hypothesis respectively. and represent the mean square error between the predicted value and the real value of the model, respectively.

In statistical methodology, the Kruskal-Wallis test (KW test) does not require the assumption of a normal distribution for the data under investigation. Therefore, for the prediction of oil production data, the KW test is employed to evaluate the differences in the predicted values between the different models in this paper. The calculation for KW test is as follows: (18)

In the equation above, k is the number of control factor layers, N_i is the number of samples generated by the ith hierarchy, is the average rank generated by the ith hierarchy, and represents the average rank generated by k levels, whose value is equal to (n+1)/2.

According to the above prediction effect evaluation method, the results of the statistical value of the predicted data deviation of the three-segment dataset model are shown in S3 Table, while DM test and KW test values are given in S4 Table. According to MAE, MAPE, RMSE, R², MARNE and NSE values, the prediction performance of this method has obvious advantages over other models. Taking LSTM as an example for comparison, the MAE value predicted by this model for three data sets is 49.94%, 57.94%, and 16.24% lower than that of the LSTM model, respectively. For the MAPE index, the MAPE value predicted by this model for the three data sets is 52.03%, 58.07%, and 18.93% lower than that of the LSTM model, respectively. For the RMSE index, the RMSE value predicted by this model for the three data sets is 52.53%, 66.89%, and 44.21% lower than that of the LSTM model, respectively. For the R² index, the predicted R² value of this model for the three data sets is 22.90%, 20.01%, and 41.59% higher than that of the LSTM model, respectively. For the MARNE index, the MARNE value predicted by this model for the three data sets is 50.52%, 57.63%, and 16.09% lower than that of the LSTM model, respectively.

In addition, the p-value of the DM test of the proposed model in S4 Table is less than 0.05 on the three data sets, so the original hypothesis can be considered not to be tenable, and because the DM values are negative, it can be considered that the prediction effect of this model is better than that of LSTM model. The difference of the mean sample ranks between the level using the comparative method and proposed method is exhibit in S4 Table via KW test as well. It is indicated that the proposed model does not exhibit significant differences compared to other models in terms of minimizing the deviations from the test values form the KW test results in Dataset 1, which may be attributed to the presence of substantial fluctuations and does not necessarily imply a disadvantage of the proposed model. Nevertheless, in the other datasets, the data behaves differently, which demonstrates that the proposed model has minimal differences between it and the true values compared to the others.

In left side of S10 Fig, the violin plots of raw data with five model predictions are presented to compare their predictions, considering an intuitive statistical representation for the prediction results of models [11]. It can be observed that across the three datasets, the predicted values of each model exhibit remarkable similarity in terms of upper and lower quartiles, median values, and overall graph shape, which indicates that the predicted values of the models are generally aligned with the actual values. From an intuitive statistical point of view, the predicted value of the proposed model is closer to the test value. In right side of S10 Fig, The error boxplots for each model depict the deviation of the predicted values from the actual values across the three datasets. The proposed model overlooks the noisy components in the data, resulting in a smaller overall error and fewer outliers in the boxplots compared to the other models.

Fit plots are commonly used to analyze the predictive results. The distribution of predicted value scatter plots and the angle between the scatter fit line and the actual value centerline are used to determine the prediction performance. In S11 Fig, it is evident that the predicted values of the proposed model are more concentrated compared to the scatter plots of the other compared models across the three datasets. Moreover, the fitted line of the proposed model has the smallest angle with the center line of the actual value.