The artificial intelligence (AI) method has attracted special attention from researchers in the oil industry due to its outstanding performance in handling highly complex problems^[1,2,3,4]. Traditional artificial neural network method (ANN) has been widely used in the field of petroleum engineering. Li et al.^[5] used implicit method of ANN to predict the log data of unknown years, and then proposed a method for time series processing by combination of implicit curve and neural network^[6]. Asadisaghandi et al.^[7] used ANN to predict the pressure-volume-temperature (PVT) of oil. Enab et al.^[8] developed forward and inverse ANN to predict gas recovery profile, etc. Singh et al.^[9] estimated the porosity with ANN technique. Memon et al.^[10] constructed the dynamic well Surrogate Reservoir Model (SRM) by radial basis neural network to predict the bottom hole flowing pressure. Kim et al.^[11] proposed an approach based on ANN to select completion mode for shale gas reservoirs. Recently, Choubineh et al.^[12] estimated the minimum miscible pressure of gas-crude oil by using neural network models.

Since the 1990s, the traditional ANNs have been applied in well test interpretation. Athichanagorn et al.^[13] used ANN to identify the features of the derivative plot. Deng et al.^[14] used peak value and horizontal position of radial flow in derivative plot as the input of three-layer feedforward neural network to predict well test parameters. Jeirani et al.^[15] inputted Horner plot into neural network to estimate reservoir pressure, permeability and skin factor. Adibifard et al.^[16] took the coefficients of interpolation Chebyshev polynomials of pressure derivative data as the input of ANN to estimate reservoir parameters. Ghaffarian et al.^[17] identified the gas condensate reservoir model by using the pseudo-pressure derivative data as the input of single and coupled multi-layer perceptron network. However, the well test interpretation method based on traditional ANN has the following problems. First, taking a part of characteristics of pressure derivative curve as the input of ANN makes it difficult to realize true automatic well test interpretation. For example, only coefficients of Chebyshev polynomial are taken as input of network^[16]. Secondly, the well test curves are complex and changeable, which require a large amount of data to train the network. However, traditional 3-layer or 4-layer networks often fail in training the big data volume, limiting the improvement of its automatic interpretation.

Deep learning (DL) is a new field in machine learning. The essence of deep learning is to build network model with multiple hidden layers, and to obtain more representative features by learning large amount of data, thereby to improve the accuracy of prediction and classification. In recent years, deep learning has also been applied to the oil industry. Tian et al.^[18] used recursive neural network to learn permanent downhole pressure gauge (PDG) data for reservoir model identification and production prediction. Sudakov et al.^[19] applied deep learning to permeability prediction. ZHA et al.^[20] used deep learning to reconstruct porous media. Zhang et al.^[21] studied the generation and repair of logging curves by recurrent neural network. Convolutional neural network (CNN) is an important part of deep learning algorithm. It differs from traditional ANNs in the following aspects: (1) CNNs break the limitations of traditional ANNs on the number of layers, increase the number of network layers and forming deep network; (2) CNNs adopt the method of feature learning and extract feature layer by layer to make the prediction or classification more easy to realize. Research on CNNs began in the 1980s. Time-delay network and LeNet-5 network were the earliest CNN algorithms^[22,23]. In the 21st century, with the introduction of deep learning theory, the increase of data volume, and the improvement of computing equipment, CNNs have developed rapidly. The AlexNet network proposed by Krizhevsky et al.^[24] has been widely used, and later CNNs have been applied in the oil industry^[25].

Based on CNN, an automatic well test interpretation approach for radial composite reservoir has been proposed in this work. By this method, the formation parameters can be interpreted by inputting pressure change and derivative into the network, without manual parameter adjustment and fitting, realizing automatic interpretation. The effectiveness and accuracy of this method were verified by the field measurement data of Daqing Oilfield.

The radial composite reservoir model is composed of two regions with different parameter attributes, namely, a circular inner zone with the well-centered and an infinite outer zone. The radial composite reservoir model can describe pollution or stimulation of the zone around well, change of radial lithology and fluid property in the zone far from the well. The basic assumptions of the radial composite reservoir model are as follows: (1) The formation is horizontal, equal in thickness, homogeneous and isotropic. (2) The fluids in both the inner and outer zones are slightly compressible single-phase fluids and their flow obeys the Darcy's law. (3) The pressure is equal everywhere in the formation before the well is opened. (4) The effect of wellbore storage and wellbore pollution are considered, but gravity is ignored.

The well test curve used in this study is Gringarten-Bourdet composite curve, which is composed of Gringarten pressure curve and Bourdet pressure derivative curve^[26,27]. Considering the universality, the method proposed adopts dimensionless parameters. Therefore, the well test interpretation parameters of the radial composite reservoir model include: mobility ratio M, storage ratio F, dimensionless composite radius R_fD and dimensionless group C_De^2S. The parameters are transformed into logarithmic forms of lg(M), lg(F), lg(R_fD) and lg(C_De^2S). The dimensionless group characterizes well storage and skin effect. The dimensionless composite radius is defined as:

Equations (2) and (3) define the mobility ratio and storage ratio of the inner and outer zones, respectively:

Convolutional neural network is a kind of deep feed-forward neural network. It is mainly composed of input layer, convolutional layer, pooling layer, fully connected layer and output layer. There are usually several convolutional layers. The pooling layer is located behind the convolutional layer. The fully connected layer is usually at the end of the network. The activation function follows the convolutional and fully connected layer to add non-linearity to the network. The convolution operation is performed between the input data and convolution kernel in the convolution layer, and is usually indicated by the sign “*”. Let f(x) and g(x) be two integrable functions in the real number field, then their convolution results are:

The convolution result of number sequence x(n) and h(n) is:

The input of convolution layer and convolutional kernel are usually multidimensional array data. Convolutional operation is actually the sliding process of the kernel over the input of convolutional layer. Multiply the neurons value on the kernel by the corresponding value on input data and the multiplied values are added up as an inactive neuron of next layer. For example, given a 2D input x_ij and a 2D kernel f_uv, where 1≤i≤N₁, 1≤j≤N₂, 1≤u≤n₁, 1≤v≤n₂. Generally, n₁≤N₁, n₂≤N₂. In this case, the convolution result is:

Fig. 1 shows an example of a 2D array convolution^[28].

Different from the traditional neural network, convolution operation improves the network performance through two important characteristics: local connection and weight sharing. The traditional ANN usually adopts matrix multiplication to construct the connection between input and output, this kind of connection is full connection, which means that each output neuron is connected with each input neuron. But CNNs have the feature of local connection, that is, each output is connected to a part of the input, which greatly reduces the number of parameters and makes the network easy to train. The local connection makes the neuron perceive only the local area, and the lower level local area information is integrated to obtain global information at the higher level, which greatly enhances the network feature extraction capability. Each connection has a different weight in full connection. However, in the convolution layer, rather than having different weights, connections in a set can share the same network weight. Local connection and weight sharing make CNN far superior to traditional ANN in statistical efficiency and storage requirements. There are 18 convolution layers in the network in this paper with matrix of 100×100 as input. The network only has 590 688 parameters. If the full connection mode is adopted with 200 neurons in each hidden layer, 2.621 44×10⁴⁵ parameters would be generated from the input to the 18th hidden layer, far more than the number of parameters using convolution operation.

Pooling, also known as down sampling, extracts the overall characteristic of the adjacent region of a location as the output, and thus reduces the amount of data, characteristic dimensions and network parameters. The common pooling methods are max-pooling and average-pooling. Max-pooling takes the maximum characteristic value of the adjacent areas, while average-pooling takes the average characteristic value of the points in adjacent areas. Fig. 2 shows an example of max- pooling. The network input in this paper is a matrix of 100× 100, which becomes the matrix of 50×50 after the first layer of pooling layer, greatly reducing the feature dimension and making the network easier to train.

It can be seen that compared with the traditional ANNs, CNNs have fewer parameters, easier training, more efficient feature extraction, and much better learning ability and performance.

The detailed steps of the automatic well test interpretation method based on CNN are as follows.

(1) Data collection and preprocessing. Artificial intelligence is a kind of data science, CNN is no exception. In this study, a total of 2×10⁵ log-log plots of radial composite reservoir and their corresponding reservoir parameter combinations lg(M), lg(F), lg(R_fD), and lg(C_De^2S) were obtained, of which 199 820 sets were generated by the analytical method, the rest were field measured data. When generating the simulation data, the value ranges of the parameters are shown in Table 1.

The pressure change and derivative curve in each simulated log-log plot consist of 200 data points. However, the actual field plots usually have 100 to 110 data points in each curve. In order to maintain the consistency of the input dimensions, 100 consecutive data points were selected in each iterative training.

In practical applications, CNN is usually used for object recognition and its input is the pixel values of the image, which is a 2D or 3D matrix. Therefore, before training, the log-log plot data needs to be converted into matrix: for each log-log plot, the 100 pressure data and 100 pressure derivative data points were copied and connected together to form a matrix of 100×100.

(2) Training and optimization of CNN. The matrix in step (1) and corresponding parameter combinations lg(M), lg(F), lg(R_fD), and lg(C_De^2S) were used as the input and output of CNN respectively. 90%, 5% and 5% of the data were used for training, verification and testing respectively. The loss function was mean square error (MSE). The MSE is calculated as:

The smaller the MSE of the training data, the better the network fitting result is. The network was optimized through repeated experiments, and optimization process includes changing network depth, width, super parameters, etc, and adding the regularization method or other methods. The optimal network configuration is obtained through optimization at last. The optimal CNN obtained in this study consists of 18 convolution layers, 3 pooling layers and 3 fully connected layers. The outputs of each convolution layer and full connection layer were activated by ReLU (linear rectification function). The output value of the ReLU function is the maximum value between the independent variable and zero, which makes the output of some neurons zero, increasing the network sparsity and reducing the training time. The network adopted the "dropout" method^[24], which set the output of each hidden neuron to zero at a certain probability in forward propagation to avoid overfitting. The initial learning rate was 0.0001. In order to make the model more stable in the later stages of training, this study used exponential decay method to gradually reduce learning rate during each iterative training.

(3) The trained optimal network was used for automatic initial fitting of well test parameter interpretation. The measured pressure data was transformed into log-log plot, then rearranged into a matrix, input into the trained optimal CNN, the output were lg(M), lg(F), lg(R_fD), and lg(C_De^2S), and then M, F, R_fD and C_De^2S were obtained, realizing the automatic initial fitting of well test interpretation parameters.

(4) Move curves and explain the wellbore and formation parameters. The typical curve was obtained by inputting M, F, R_fD and C_De^2S into the commercial software, and the typical curve was moved to make it coincide with the measured log-log plot. Take any point from the measured curve, record the pressure change value Δp and time value t of the point, and find out the dimensionless pressure change value p_D and dimensionless time value t_D on typical curve at this point. With values of Δp, t, p_D and t_D of this point and parameters M, F, R_fD and C_De^2S, wellbore storage coefficient, skin coefficient, permeability of inner and outer zones and other wellbore and reservoir parameters can be calculated according to a series of formulas^[29].

In the method presented in this paper, the pressure change and its derivative data are directly taken as the input of the network. In the previous methods based on artificial neural networks, the inputs are some characteristics obtained from the pressure change and its derivative data. For example, Deng et al.^[14] used the hump value of derivative plot and the horizontal position of radial flow as the inputs of neural network. Adibifard et al.^[16] took the coefficient of interpolation Chebyshev polynomial as the network input.

The validation set and test set data were used to verify the interpretation accuracy of the optimal trained CNN. Table 2 presents the mean absolute errors (MAEs) of the interpretation values for the validation and test sets. It can be seen all the four reservoir parameters have small average absolute errors, indicating that the network is well trained and has good generalization ability.

In order to exhibit the interpretation effect of the method presented in this paper more clearly, three samples were extracted from the verification set and test set respectively to further demonstrate the performance of the trained CNN. Samples Val1, Val2, and Val3 were taken from the validation set, samples Test1, Test2, and Test3 were taken from the test set. The interpretation results are given in Table 3.

It can be seen from Table 3 that for the six samples, the errors between the interpreted parameters and the real parameters are small on the whole. However, the interpretation errors of lg(F) for sample Val2, lg(M) and lg(C_De^2S) for Val3, lg(M), lg(F), and lg(C_De^2S) for Test2 are relatively large. The large errors in interpretation values of lg(M) and lg(F) are mainly caused by data intercepting errors, as not all features were intercepted, the interpretation results are poor. The sample Test2 is taken as an example to explain the reason in details. The original and intercepted curves of the Test2 are shown in Fig. 3. It can be seen that the curve intercepted during the test doesn’t include the upwarping part of late stage in the original curve, and this part of curve is the key to characterize the mobility ratio and storage ratio. This is the reason for the poor interpretation effect of the mobility ratio and storage ratio.

Although the absolute error of the parameter lg(C_De^2S) of the sample Test2 is relatively large, its relative error is only 0.167 5. Fig. 4 shows that the early stage of the two log-log plots obtained from the interpreted values and the real values almost coincide, which indicates the interpretation error is small.

The validity of the method presented in this paper was further verified by six field cases from Daqing Oilfield. Table 4 presents the basic parameters of the 6 field cases.

Fig. 5 to Fig. 10 show the comparison of calculation results between the method presented in this paper and analytical method. Table 5 gives the corresponding interpretation parameters. It can be seen from Fig. 5a and Fig. 6a that for the measured data without noise or with slight noise, the method presented in this paper can correctly interpret the parameters, which can be seen from the near coincidence of the measured and the calculated curves. As the noise increases, as shown in Fig. 7a and Fig. 8a, the interpretation results are still good. Even when the measured data has clutter noise, the trained CNN can still explain the formation and wellbore parameters quite accurately (Fig. 9a). Even when the measured data oscillate up and down locally, Fig. 10a illustrates that the interpretation result is still good. It is proved that this method has good effectiveness and robustness. Even if there are some “bad samples” in the training samples, the trained CNN can still correctly interpret the shut-in pressure data. This indicates that a small number of "bad samples" would not affect the performance of the CNN, but the corresponding parameters of “bad samples” couldn’t be interpreted. If the “bad samples” are removed, the CNN after training would have better performance. How to tick out the “bad samples” will be a study content in the future.

In addition, through comparison with analytical method, it is found that the differences between the interpretation results of the method presented in this paper and analytical method are small, and both methods can get good interpretation results. But the analytical method requires professional well testing personnel to complete the operation and costs a lot of manual labor and time. In contrast, the method proposed in this paper can realize automatic interpretation by transforming the measured data into matrix and inputting it into the trained CNN. The outputs are the interpretation parameters. A person without professional knowledge can accomplish well test interpretation aided by the method presented in this paper, which improves work efficiency significantly.

It can be seen that the proposed well test interpretation method based on CNN can interpret radial composite reservoir parameters with high accuracy and efficiency, and realize automatic interpretation of well test data.

Two most representative examples (Case 1 and Case 6) were selected to compare the results from the proposed method and the least square method. The Case 1 has little noise, while Case 6 has a lot of noise. It can be seen from Fig. 11, Fig. 12 and Table 6, the results obtained by the least squares method are poorer. Furthermore, the least square method involves the selection of fitting parameters, so full automatic interpretation can’t be realized by this method.

An automatic well test interpretation method based on CNN for radial composite reservoirs is proposed. By taking the pressure change and derivative data as inputs and the corresponding parameter combinations as outputs, the trained CNN can automatically interpret the pressure data and give the parameters of radial composite reservoir. This method can realize automatic fitting of well test parameters and improve the efficiency of well test interpretation. Its effectiveness and robustness have been verified by the measured data of Daqing Oilfield, and this method has been proved superior to analytical method and the least square method by cases of Daqing Oilfield.

This method can’t be simply copied to other types of reservoirs. As different types of reservoirs differ widely in characteristics of well test curves and the number of parameters, different neural network structures and related processing methods are needed.

B—volume factor, m³/m³;

C—wellbore storage coefficient, m³/MPa;

C_D—dimensionless wellbore storage coefficient;

C_t—total compressibility factor, Pa^-1;

d₁(k), d₂(k), d₃(k), d₄(k)—interpreted values of lg(M), lg(F), lg(R_fD) and lg(C_De^2S) in the k-th parameter combination;

f_uv—2D convolution kernel;

f(x), g(x)—function;

F—storage ratio, dimensionless;

h(n), x(n)—number sequence;

H—formation thickness, m;

i, j—row and column numbers of 2D input;

K—permeability, 10^-3 μm²;

M—mobility ratio, dimensionless;

MSE—mean square error;

n, s—length of number sequence;

n₁, n₂—the number of rows and columns of 2D convolution kernel;

N—number of parameter combinations;

N₁, N₂—number of rows and columns of 2D input;

p_D—dimensionless pressure change, p_D=2πKHΔp/QBμ;

p_D'—dimensionless derivative of pressure change, p_D'=t_Ddp_D/dt_D;

Δp—pressure change, MPa;

Δp'—derivative of pressure change, Δp'=ΔtdΔp/dΔt, MPa;

Q—production, m³/s;

r_w—wellbore radius, m;

R_f—composite radius, m;

R_fD—dimensionless composite radius;

S—skin factor, dimensionless;

t—time, h;

Δt—time change, h;

t_D—dimensionless time, t_D=Kt/fμC_tr_w²;

u, v—row and column numbers of convolution kernel;

x, τ—independent variables of function;

x_ij—2D input;

y₁(k), y₂(k), y₃(k), y₄(k)—real values of lg(M), lg(F), lg(R_fD) and lg(C_De^2S) in the k-th parameter combination;

μ—fluid viscosity, mPa·s;

f—porosity of plugging layer, %.

1—inner zone;

2—outer zone.