CO₂ Corrosion Rate Prediction for Submarine Multiphase Flow Pipelines Based on Multi-Layer Perceptron

Abstract

The implementation of corrosion detection in submarine pipelines is difficult, and a combined PCA-MLP prediction model is proposed to improve the accuracy of corrosion prediction in submarine pipelines. Firstly, the corrosion rate of a submarine multiphase flow pipeline in the South China Sea is simulated by the De Waard 95 model in the multiphase flow transient simulation software OLGA and compared with the actual corrosion rate; then, according to the corrosion data simulated by OLGA, principal component analysis (PCA) is used to reduce the dimensionality of the corrosion factors in the pipeline, and the multiple linear regression model (MLR), multi-layer perceptron neural network (MLPNN), and radial basis function neural network (RBFNN) were optimized. The PCA-MLPNN model has an average relative error of 3.318%, an average absolute error of 0.0034, a root mean square error of 0.0082, a residual sum of squares of 0.0020, and a coefficient of determination of 0.8609. Compared with five models, including MLR, MLPNN, RBFNN, PCA-MLR, PCA-MLPNN, and PCA-RBFNN, PCA-MLPNN has higher prediction accuracy and better prediction performance. The above results indicate that the combined PCA-MLPNN model has a more reliable application capability in CO₂ corrosion prediction of submarine pipelines.

1. Introduction

According to statistics, the total length of China’s submarine pipelines has exceeded 9000 km, and submarine pipelines account for about 5% of the total length of the country’s long-distance oil and gas pipelines. This is a relatively low percentage, but its transmission volume reaches almost 25% of the total national oil and gas transmission volume [1]. Compared to the corrosion detection methods and environment of onshore pipelines, the implementation of corrosion detection in submarine pipelines is currently more difficult and expensive [2,3]. Therefore, to avoid pipeline leakage accidents caused by corrosion cracking or perforation [4], the internal corrosion condition of pipelines can be understood by studying corrosion prediction models to predict the corrosion rate of pipelines promptly [5], so that reasonable measures and suggestions can be made for safe pipeline operation.

In recent years, multi-factor corrosion prediction models have been mostly established by regression analysis, support vector machines, artificial neural networks, etc. Kimiya Zakikhani et al. [6] combined the geological environmental parameters of pipelines and performed multiple regression analyses on the collected historical data of natural gas pipelines to establish a corresponding failure prediction model, which can predict the critical time of external corrosion failure of pipelines. Shuai Zhao et al. [7] used an autoclave to study the effects of CO₂ partial pressure, velocity, temperature, and Cl^- concentration on the corrosion rate of pipelines; they proposed a correction model for CO₂ corrosion rate prediction based on the obtained experimental data with a multiple regression method, which has a prediction error of 6.29%. This is much better than the Norsok M506 model. Carlos A. da Silva et al. [8] used multiple regression analysis and the Box-Cox transformation method to establish a prediction model for multiphase flow-induced corrosion of API X80 steel in a CO₂/H₂S environment. Wang Chen et al. [9] proposed a combined IA-SVM (support vector machine) model based on the influencing factors of submarine pipeline corrosion and using the immune algorithm (IA), which has a small mean absolute error and root mean square error and is suitable for corrosion rate prediction of submarine pipelines. Ya-jun Lv et al. [10] used a particle swarm optimization support vector machine and grid search support vector machine to predict the predicted corrosion rate of steel, and the results showed that the PSO-SVM (particle swarm optimization) model predicts more accurately. Martins Obaseki et al. [11] established an artificial neural network model and successfully predicted the corrosion rate between 0.02 mm/a and 0.17 mm/a with relative errors of 0.013–0.047%, and found that the corrosion rate increases with increasing temperature, flow rate, sediment deposition, and pipe age. Yong Gu et al. [12] proposed a Gaussian-distribution-based conditional expansion method to study the distribution law of corrosion characteristic parameters under each working condition and established a three-layer back-propagation neural network model to predict some flow-accelerated corrosion parameters with good prediction performance. María Jesús Jiménez-Come et al. [13] proposed a three-step model based on an artificial neural network to simulate the corrosion behavior of EN 1.4404 stainless steel in the pitting behavior of EN 1.4404 stainless steel in the marine environment, the model steps are: (1) estimation of the breakdown potential values, (2) dimensionality reduction using principal component analysis (PCA) or linear discriminant analysis (LDA), and (3) prediction of the pitting state of EN 1.4404 stainless steel. The prediction accuracy of the LDA-combined model in this study was as high as 99. 2%. Xiaoxu Chen et al. [14] proposed a dynamic fuzzy neural network model based on principal component analysis to predict the corrosion rate in natural gas pipelines; the model had an RMSE of 0.4232, an MAPE of 5.91%, and a TIC of 0.2352 on the test data set, and the prediction performance was better than that of the other three models.

In this study, corrosion prediction of submarine pipelines was first performed using Schlumberger OLGA 2020.1.0, a multiphase flow transient simulation software, to obtain data on corrosion factors and corrosion rates [15]. Multiple linear regression (MLR), multi-layer perceptron neural network (MLPNN), radial basis function neural network (RBFNN), and other models were improved and optimized based on principal component analysis (PCA), and then compared to select an optimal model to provide scientific guidance for corrosion prediction in submarine pipelines.

2. Methods

2.1. Data

The pipeline data were derived from [16]. Inlet boundary conditions: set the fluid inlet temperature to 32 °C, when the transmission volume is the actual flow rate, the mass flow rate is 1720 kg/h, gas–oil ratio is 560 m³/m³. Outlet boundary conditions: set the fluid outlet pressure to 0.9 MPa. Wall boundary conditions, the ambient temperature is 21.6 °C, and the total heat transfer coefficient of the pipe is 0.045 W/(m²·°C). A natural gas submarine pipeline in the South China Sea, with a total length of 15.965 km, was used in this study. The submarine pipeline transport medium components are shown in

Table 1. Transport medium components.

Component	Molar Fraction/%	Component	Molar Fraction/%	Component	Molar Fraction/%
CO2	1.51	C3H8	3.77	n-C5H12	0.16
N2	0.65	i-C4H10	0.54	C6+	0.536
CH4	83.5	n-C4H10	0.64	O2	0.01
C2H6	8.47	i-C5H12	0.19	H2O	0.024

, the pipeline dimensions and material parameters are shown in

Table 2. Pipe size and material parameters.

Sections	Pipeline Length/m	Pipeline Outside Diameter/mm	Wall Thickness/mm	Pipeline Materials	Anticorrosive Coating
Standpipe section 1	47.7	114.3	8.6	X52	Polyethylene
Bend section 1	43.4	114.3	8.6	X52	Polyethylene
Flat pipe section 1	3000	114.3	8.6	X52	Polyethylene
Flat pipe section 2	12,766	114.3	12.7	X52	Polyethylene
Bend section 1	57.5	114.3	8.6	X52	Polyethylene
Standpipe section 2	50.4	114.3	8.6	X52	Polyethylene

, and the pipeline elevation map is shown in Figure 1.

2.2. Algorithm and Model Introduction

2.2.1. Principal Component Analysis

Principal component analysis (PCA) is a class of analytical statistical algorithms capable of simplifying and making explicit the hidden correlations, influences, and complex mathematical connections among high-latitude factors [17]. This method allows the original hidden functional relationships between multiple factors to be downscaled to principal components while retaining the relationships conveyed by the original data sample. Each principal component reflects a large amount of information about the original factors and is a linear combination of the original individual factors [18]. The main steps of principal component analysis are as follows:

Suppose X_mn is a sample matrix with m samples and n factor data, i.e., X_mn = (X₁, X₂, ..., X_j, ..., X_j, ..., X_n), and X_j = (x_1j, x_2j, ..., x_xj, ..., x_mj)^T.

(1)

Calculate the standardized data sample matrix X^′_mn according to Equation (1):

$$x_{xj}^{\prime }=\frac{x_{xj}-{\overline{x}}_j}{s_j}$$

(1)

where $i=1,2,...m;j=1,2,...n;{\overline{x}}_j=\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_{ij}};s_j=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{\left(x_{xj}-{\overline{x}}_j\right)}^2}};$

(2)

Calculate the standardized data sample covariance matrix $S_{nn}$ according to Equation (2):

$$S_{nn}=\left|\begin{array}{l}\mathrm{cov}\left(X_1^{\prime },X_1^{\prime }\right)\hspace{1em}\mathrm{cov}\left(X_1^{\prime },X_2^{\prime }\right) \cdot \cdot \cdot \mathrm{cov}\left(X_1^{\prime },X_n^{\prime }\right)\\ \mathrm{cov}\left(X_2^{\prime },X_1^{\prime }\right)\hspace{1em}\mathrm{cov}\left(X_2^{\prime },X_2^{\prime }\right) \cdot \cdot \cdot \mathrm{cov}\left(X_2^{\prime },X_n^{\prime }\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\cdot \cdot \cdot \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\cdot \cdot \cdot \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\cdot \cdot \cdot \\ \mathrm{cov}\left(X_n^{\prime },X_1^{\prime }\right)\hspace{1em}\mathrm{cov}\left(X_n^{\prime },X_2^{\prime }\right) \cdot \cdot \cdot \mathrm{cov}\left(X_n^{\prime },X_n^{\prime }\right)\end{array}\right|$$

(2)

where $\mathrm{cov}\left(X_i^{\prime },X_j^{\prime }\right)=\frac{{\displaystyle \sum _{i=1}^n\left(X_i^{\prime }-\overline{{\mathrm{X}}_i^{\prime }}\right)}\left(X_j^{\prime }-\overline{{\mathrm{X}}_j^{\prime }}\right)}{n-1}$

(3)

Calculate the eigenvalues and corresponding eigenvectors of the covariance matrix $S_{nn}$, and calculate the variance contribution rate and the cumulative contribution rate of the principal components according to Equations (3) and (4); additionally, calculate the reduced dimensional principal components according to Equation (5):

$${\varphi }_i=\frac{{\beta }_i}{{\displaystyle \sum _{i=1}^n{\beta }_i}}\times 100\%$$

(3)

$${\delta }_k={\displaystyle \sum _{i=1}^k{\varphi }_i}$$

(4)

$$\left\{\begin{array}{c}Z1=a_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n\\ Z2=a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n\\ \cdots \\ Zk=a_{k1}{x}_1+a_{k2}{x}_2+\cdots +a_{kn}{x}_n\end{array}\right.$$

(5)

where $k=1,2,...,n;$ $x_n$ is the normalized data; $Zk$ is the principal component; $a_{kn}$ is the principal component coefficient.

2.2.2. De Waard 95

The De Waard 95 model focuses on two processes, the kinetic process of the corrosion reaction, independent of the flow rate, and the mass transfer process related to the flow rate. At temperatures less than 80°C, the model predicts approximately the same results as the experimental findings of [19]. The De Waard 95 [20] model is calculated as follows:

$$\frac{1}{V_{corr}}=\frac{1}{V_r}+\frac{1}{V_m}$$

(6)

$$V_m=0.245\frac{U^{0.8}}{D^{0.2}}{P}_{C{O}_2}$$

(7)

$$\log V_r=4.93-\frac{1119}{t+273}-0.58\log P_{C{O}_2}-0.34\left(p{H}_{act}-p{H}_{C{O}_2}\right)$$

(8)

$$p{H}_{C{O}_2}=3.71+0.00417t+0.5\log P_{C{O}_2}$$

(9)

where $V_{corr}$ is the corrosion rate, mm/a; $V_r$ is the reaction rate, mm/a; $V_m$ is the mass transfer rate, mm/a; $t$ is the medium temperature, °C; $P_{C{O}_2}$ is the partial pressure of CO₂, MPa; $p{H}_{act}$ is the actual pH; $p{H}_{C{O}_2}$ is the pH of CO₂ saturated solvent; $U$ is the liquid-phase flow rate, m/s; $D$ is the pipe diameter, m.

2.2.3. Multiple Linear Regression

Multiple linear regression models [21] are mathematical models that study the linear relationships between multiple independent variables and a dependent variable with the aim of explanation and prediction.

Assume a multiple linear regression prediction model of the random variable Y with independent variables x₁, x₂,...,x_k [22], whose general expression is:

$$Y={\beta }_0+{\beta }_1{x}_1+\cdots {\beta }_i{x}_i+{\beta }_k{x}_k+\mu$$

(10)

where $k$ is the number of independent variables; ${\beta }_0$ is the regression constant; ${\beta }_k$ is the regression coefficient; $\mu$ is the random error.

2.2.4. Multi-Layer Perceptron Neural Network

An artificial neural network (ANN) is a machine learning algorithm inspired by biological neural networks [23]. MLPNN, as a typical representative of neural networks, consists of an input layer, one or more hidden layers, and an output layer [24], which do not constitute connections between the same layers. The layers are fully connected to each other by neurons containing nonlinear activation functions. Each connection is assigned a different weight, so that the input of each layer is a weighted sum of the output values of the neurons in the previous layer, i.e., as follows [25]:

$$y_i^l=f\left({\displaystyle \sum _{j=1}^n{y}_j^{l-1}}{w}_{ji}^{l-1}+b_i^l\right)$$

(11)

where $y_i^l$ is the output of the first neuron in the first layer; $f$ is the activation function of the first layer; $n$ is the number of neurons in the first layer; $y_i^{l-1}$ is the output of the first neuron of the first layer; $w_{ji}^{l-1}$ is the connection weight of the first neuron and the first neuron of the first layer; $b_i^l$ is the deviation of the first neuron of the first layer.

2.2.5. Radial Basis Function Neural Network

The RBF neural network (radial basis function neural network) has good pattern classification and function-fitting ability. It has the same network structure as the MLP neural network, which can also be divided into the input layer, hidden layer, and output layer. Among them, the radial basis function is the transformation function of the neurons in the hidden layer, and the transfer function of each neuron is a Gaussian function [26], which is given as:

$${\phi }_i(x)=\exp \left(-\frac{{‖x-c_i‖}^2}{2{\delta }_i^2}\right)(i=1,2,\cdots n)$$

(12)

where $x$ is the input vector, dimensional; $c_i$ is the center vector of the first basis function, and n-dimensional; ${\delta }_i$ is the width of the basis function around the centroid.

2.3. Model Evaluation Metrics

To further evaluate the prediction performance of the model, four indicators were selected: mean absolute error (MAE), root mean square error (RMSE), squared sum error (SSE), and coefficient of determination (R²). MAE reflects the true error of the predicted value, RMSE reflects the deviation between the predicted and true values, SSE reflects the degree of error between the predicted and true values, and R² reflects the degree of fit of the model [10]. In general, the closer the MAE, RMSE, and SSE values are to 0, the better the model prediction is, and R² takes values between 0 and 1. The closer it is to 1, the better the fit is represented [22].

$$\mathrm{MAE} =\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|y_{ai}-y_{pi}\right|}$$

(13)

$$\mathrm{RMSE} =\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{\left(y_{ai}-y_{pi}\right)}^2}}$$

(14)

$$SSE={\displaystyle \sum _{i=1}^n{\left(y_{ai}-y_{pi}\right)}^2}$$

(15)

$$R^2=1-\frac{{\displaystyle \sum _{i=1}^n{\left(y_{ai}-y_{pi}\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(y_{ai}-{\overline{y}}_a\right)}^2}}$$

(16)

where $y_{pi}$ denotes the predicted value; $y_{ai}$ denotes the actual value; ${\overline{y}}_a$ denotes the average of the actual values; n and n denote the number of data sets.

3. Results and Discussion

3.1. OLGA Predictive Analysis

The corrosion rate and the variation curve of each corrosion factor with the pipeline mileage were obtained by OLGA numerical simulation. As shown in Figure 2, the variation of corrosion rate is the result of the combined effect of temperature, pressure, wall shear stress, fluid flow velocity, CO₂ partial pressure, pH, liquid holdup (HOL), and other factors [27], so it is important to establish a multi-factor corrosion prediction model for the prediction of corrosion rate in submarine pipelines. The error analysis between the predicted values of the De Waard 95 model (OLGA) and the actual values of the submarine pipeline is shown in

Table 3. The relative error between predicted and actual value.

	Number	Act/(mm/a)	PR/(mm/a)	RE/%	ARE/%
Mean value	1	0.0637	0.0715	12.24	12.42
Maximum value	2	0.1868	0.1634	12.53
Minimum value	3	0.0064	0.0072	12.50

, and the comparison graph is shown in Figure 3. Compared with the actual corrosion rate, OLGA-simulated data are relatively smooth: the average predicted corrosion rate is higher than the actual average, the maximum value of the predicted rate is less than the actual maximum value, and the minimum value is greater than the actual minimum value. These factors indicate that OLGA simulation of complex working conditions of corrosion rate shows high performance in general. In Table 3, Act indicates the actual value, PR indicates the predicted value, RE indicates the relative error, and ARE indicates the average relative error.

3.2. PCA Algorithm Processing Result

The data in this part of the study were obtained from the simulation results of OLGA, and eight factors were selected that have a significant effect on the CO₂ corrosion rate [27]: wall shear (Pa), X1; gas phase flow rate (m/s), X2; liquid-phase flow rate (m/s), X3; liquid holdup, X4; partial pressure of CO₂ (Pa), X5; pH, X6; temperature (°C), X7; pressure (bar), X8; corrosion rate (mm/a), Y. A total of 150 sets of data were selected, and the contents are shown in

Table 4. Corrosion data sheet.

	Input Variable								Output Variable
NO.	X1	X2	X3	X4	X5	X6	X7	X8	Y
1	4.9586	3.9248	0.6270	0.0187	25,736.8906	4.2342	31.9488	17.0443	0.1065
2	4.9569	3.9205	0.6273	0.0187	25,769.4492	4.2341	32.0150	17.0659	0.1067
3	8.7768	3.8826	1.3514	0.0090	25,770.1992	4.2342	32.0467	17.0664	0.1338
4	8.7762	3.8806	1.3504	0.0090	25,784.0195	4.2342	32.0545	17.0755	0.1634
…	……	……	……	……	……	……	……	……	……
146	2.3312	6.4497	0.6999	0.0200	18,294.5996	4.2735	22.2948	12.1156	0.0686
147	2.3429	6.4760	0.7015	0.0199	18,219.6504	4.2743	22.2638	12.0660	0.0685
148	2.3548	6.5027	0.7032	0.0199	18,144.3809	4.2751	22.2330	12.0161	0.0683
149	2.3668	6.5298	0.7049	0.0198	18,068.7695	4.2759	22.2025	11.9661	0.0682
150	2.3790	6.5572	0.7066	0.0197	17,992.8301	4.2767	22.1723	11.9158	0.0680

.

This section uses IBM SPSS Statistics 26.0 to conduct principal component analysis on corrosion data as follows: to eliminate the influence of different magnitudes of corrosion factors on the results, standardization is performed. Then, the eigenvalues and cumulative contribution rate are calculated, in general, when the cumulative contribution rate of the eigenvalues is greater than 80%, and the corresponding principal component is selected [17]. As shown in Figure 4, the eigenvalues of the first two principal components were greater than 1 and the cumulative contribution rate reached 96.19%, indicating that the two principal components adequately expressed the corrosion data.

3.3. Comparison of Model Predictions

The prediction results of MLR, MLPNN, RBFNN, PCA-MLR, PCA-MLPNN, and PCA-RBFNN were analyzed, as shown in

Table 5. Comparison of model prediction results.

	MLR	PCA-MLR	MLPNN	PCA-MLPNN	RBFNN	PCA-RBFNN
ARE/%	9.564	7.146	4.956	3.318	6.520	4.129

, ARE indicates the average relative error. The predicted values of MLR differed significantly from the actual values, but after PCA eliminated the redundant information in the corrosion influencing factors, the relative error of MLR prediction was reduced by 25.28%, and the prediction accuracy reached 92.854%. MLPNN and RBFNN prediction performance after optimization also greatly improved: the relative error of prediction reduced by more than 30%, and the PCA-MLPNN model prediction achieved an accuracy of up to 96.682%. As shown in Figure 5 and Figure 6, although the fluctuation of the relative error of PCA-MLPNN prediction is small, the average relative error is large and cannot accurately predict the nonlinear output values. In contrast, PCA-MLPNN and PCA-RBFNN predictions are more accurate and perform better overall. Through this section, corrosion prediction under complex working conditions requires consideration of various factors, and global approximation MLPNN, which is applicable to distinguish dispersed data, performs better.

3.4. Model Evaluation

As can be seen from Figure 6 and Figure 7, the predictive stability of MLPNN and RBFNN is comparable, and both models have a good ability to handle the nonlinear relationship between input and output variables, while MLR performs poorly in handling nonlinear problems. After the PCA optimization process, the MAE of MLR was reduced by 24.69%, RMSE by 28.82%, and SSE by 49.34%, and R² improved by 26.33%; the MAE of MLPNN was reduced by 28.71%, RMSE by 7.46%, SSE by 14.37%, and R² improved by 2.79%; the MAE of RBFNN was reduced by 39.34%, RMSE by 23.31%, SSE by 41.18%, and R² was improved by 15.51%. The MAE, RMSE, and SSE of all three models were reduced and R² was increased (closer to 1), indicating that the prediction performance of the models was improved. Among them, PCA-MLPNN (MAE of 0.0034, RMSE of 0.0082, SSE of 0.0020, and R² of 0.8609) has the best prediction performance.

4. Conclusions

(1) Based on PCA, to reduce the dimensionality of influencing factors of corrosion, eliminating the redundant information of factors and weakening the correlation between factors, the relative error of MLR prediction was reduced by 25.28%, and the relative error of prediction of MLPNN and RBFNN were reduced by more than 30%. Compared with five models, including MLR, MLPNN, RBFNN, PCA-MLR, PCA-MLPNN, and PCA-RBFNN, the PCA-MLPNN model had an average relative error of 3.318% and showed the best prediction performance (MAE of 0.0034, RMSE of 0.0082, SSE of 0.0020, and R² of 0.8609). These results show that the PCA-MLPNN model has a more reliable application capability in corrosion prediction. The continuous optimization of the corrosion prediction model is of greater significance to the rapid development of submarine pipeline safety operation monitoring.

(2) For the corrosion prediction of submarine mixed transmission pipelines, higher prediction accuracy can be achieved by OLGA and other authoritative commercial software in the early stage; in the later stage, under the premise of more complete operation and inspection data, PCA-MLPNN and other combined machine learning models can be used for corrosion prediction, to provide scientific guidance for corrosion prediction in submarine pipelines.