Artificial Neural Networks (ANNs) are computational models designed to simulate the structure and function of neurons in the human brain. Similar to the way the human brain works, artificial neural networks can automatically process multiple input variables in parallel, passing information from the input layer to the output layer through the learning process [36,37]. In addition, ANNs also have the ability to reveal potential linear or nonlinear relationships between input and output data. A complete neural network consists of an input layer, an output layer, and at least one hidden layer. The basic structural unit of a neural network is the neuron, which can receive, process, and output signals. It is a nonlinear element with multiple inputs and a single output. A large number of neurons are interconnected via connection weights, bias values, and transfer functions between neurons in each layer, forming a neural network with robust information processing capabilities. A back propagation neural network (BPNN) is a type of feedforward neural network that incorporates a backpropagation learning algorithm. Due to its adjustable architecture, capability for forward signal propagation, and mechanism of error backpropagation, the BPNN has emerged as the most widely implemented neural network paradigm. The schematic architecture of a standard three-layer BPNN is depicted in Fig. 2. However, ANN models are prone to challenges, including overfitting, suboptimal parameter selection, and computational inefficiency during training. To mitigate these limitations, a hybrid framework integrating a Genetic Algorithm (GA) and BPNN is employed to optimize network weights and biases, thereby enhancing prediction accuracy [38].

The GA, first proposed by American scholar John Holland, it a global optimization algorithm that simulates the process of biological evolution.GA mimics principles of natural selection, crossover, and mutation to enhance individual fitness within a population. Its workflow comprises four stages: parameter encoding, population initialization, fitness evaluation, and iterative genetic operations [39,40]. Genetic operations include selection, crossover, and mutation. The selection operation can inherit excellent chromosomes to the next generation or generate new individuals through crossover operation and pass them on to the next generation. By adopting an optimal preservation method, the fitness values of individuals within the same population are sorted in descending order, with priority given to selecting individuals with higher fitness values for inheritance to the next generation. The crossover operation is the process of selecting chromosomes from a population and recombining them into new chromosomes through crossover operations. This operation can effectively inherit excellent genes from the population. The mutation operation involves using other genes in an individual to replace specific genes, which can enhance local search ability and effectively preserve population diversity. In general, selection operations ensure that the population evolves toward higher fitness, crossover operations accelerate the speed of genetic algorithm searching for optimal solutions, and mutation operations help prevent the algorithm from falling into local optima too early. Together, the three genetic operations of selection, crossover, and mutation enable GA to efficiently search the solution space and gradually approach the optimal solution.

Optimizing the BPNN model through GA and developing the GA-BPNN hybrid model. The specific steps are as follows:

(1) Initialization: Define the BPNN architecture and initialize GA parameters and population.

(2) Fitness Evaluation: Train the BPNN on the dataset and calculate individual fitness based on prediction accuracy.

(3) GA Execution: Perform selection, crossover, and mutation operations within the GA.

(4) Population Update: Replace old individuals with newly generated individuals to form a new population.

(5) Termination Check: Determine whether the termination condition is satisfied. If the termination condition is not satisfied, return to step 2 to continue the iteration.

(6) Model Finalization: Deploy the BPNN with optimized weights and biases from the highest-fitness individual.

RFR is an ensemble algorithm composed of multiple decision trees based on the bootstrap aggregating method; it was initially designed for classification tasks. As research progressed, the RFR model was further applied to predict CO₂-oil MMP. For instance, Al Khafaji et al. [34] used the RFR model to predict MMP. The basic principles of RFR can be summarized as four steps: sample selection, feature selection, decision tree construction, and prediction. The number of trees in a random forest is represented as B, and the predicted result of the b-th tree is represented as F_b(x). Therefore, the prediction formula for RFR can be expressed as: (1)y^=1B∑b=1Bfb(x)

The steps for establishing a random forest regression model are as follows:

(1) Data Standardization: Standardize all data to ensure that their values are between –1 and 1, in order to obtain accurate prediction results and accelerate training speed.

(2) Import Data and Select Parameters: Import all data and output the dataset to the model for model training and testing.

(3) Bootstrap Sampling: The bootstrap resampling method is employed to randomly sample the original dataset with replacement, generating multiple subsets. Each subset trains a decision tree, while the unsampled instances (termed out-of-bag (OOB) data) are reserved for validation.

(4) Tree Growth: Each decision tree randomly selects several features from the input variable x for the current node’s split subset. The optimal feature for splitting is determined by maximizing variance reduction, which enables maximal tree growth.

(5) OOB Validation: The OOB data, which are excluded from the training process, serve to evaluate model performance and accuracy. The optimal number of decision trees, T, is determined by iteratively refining the model based on OOB prediction errors until convergence.

(6) Prediction: For the input variable x_i (i = 1, 2, … , k) that needs to be predicted. Each decision tree will output a corresponding predicted value y_i (i = 1, 2, … , k).The predicted value of RFR is the average of the predicted values of all decision trees.

SVR is a regression analysis method derived from Support Vector Machines (SVM). While SVR shares conceptual similarities with traditional SVM classification algorithms, its objective diverges by focusing on identifying an optimal fitting function for continuous prediction problems, as opposed to delineating classification boundaries [41]. The basic idea of the SVR model is to find a function that can predict the output value as accurately as possible while keeping the complexity of the model low to avoid overfitting. To achieve this goal, SVR uses a loss function called “ε-insensitive loss function”, which means that the loss is only calculated when the prediction error exceeds a certain threshold, ε [42]. Its basic definition is as follows: (2)L(y,f(x))={0if|y−f(x)|≤ε|y−f(x)|−εotherwise

In the nonlinear case, by introducing relaxation variables εi and εi∗, as well as a penalty coefficient C, the penalty coefficient is used to control the trade-off between model complexity and data fitting. This transforms the optimization problem of SVR into solving the following optimization problem: (3)12‖w‖2+C∑i=1mεi+εi∗

The model is subject to the following constraints: (4){yi−(w⋅x+b)≤ε+εi(w⋅x+b)−yi≤ε+εi∗ε≥0,εi∗≥0

The solution to this problem can be achieved by introducing Lagrange functions L(w,b,εi,εi), transforming the original problem into a dual problem, and using KKT conditions to find the optimal solution. The specific process is as follows: (5)L(w,b,εi,εi)=12‖w‖2+C∑i=1mεi+εi∗−∑i=1mai(ε+εi−yi+w×xi+b)−∑i=1mai∗(ε+εi∗+yi−w×xi−b)−∑i=1m(ηiεi+ηi∗εi∗)where, ai,ai∗,ηi,ηi∗ are Lagrange operators, partial derivatives are obtained for w,b,εi,εi, respectively, to obtain saddle points. Then, the results are substituted into the original optimization problem, which can be transformed into the following dual problem: (6)maxW(a)=−12∑i,j=1m(ai−ai∗)(aj−aj∗)K(xi,xj)−ε∑i=1m(ai+ai∗)+∑i=1myi(ai−ai∗)

The constraints imposed on the SVR model are as follows: (7){0≤ai,ai∗≤C∑i=1m(ai−ai∗)=0where, K(xi,xj) is a kernel function used to calculate the inner product of data points, xi and xj in high-dimensional space. The commonly used kernel functions include linear kernel, polynomial kernel, Gaussian kernel, etc. Among them, Gaussian kernel has the best applicability and is also known as radial basis function kernel (RBF).

Based on the above analysis, obtain the optimal solution and derive the final regression function: (8)f(x)=∑i=1m(ai−ai∗)K(xi,xj)+b¯where, b¯ is the bias term determined through training data.

The steps for establishing a support vector machine regression model are as follows:

(1) Read and preprocess the dataset.

(2) Build a model and define a hyperparameter search space.

(3) Use grid search cross-validation to find the optimal hyperparameters.

(4) Train the model using the optimal hyperparameters and conduct model evaluation.

(5) Use the trained model to predict new input data and output the prediction results.

GPR is a data-driven method proposed by Williams and Rasmussen. Its essence lies in learning the kernel function with probabilistic significance [43]. The optimal hyperparameters are obtained by analyzing historical sample data, after which a prediction model is developed to forecast subsequent samples. The prediction generated by the GPR method carry probabilistic significance and can describe associated with these predictions. Compared with ANN, SVR and other methods, the hyperparameters required for prediction are easier to obtain and are suitable for high-dimensional nonlinear systems. Fig. 3 illustrates the specific process of GPR. The GPR algorithm is described below.

Given the training set D:{Xi,ti}i=1N, N is the number of training samples, X_i is the vector defining time t_i, and the joint probability distribution function of the random variables in the Gaussian process is: (9)P(t|C(Xm,Xn;Θ),{Xn})=1Zexp⁡(−12(t−μ)T⋅C(Xm,Xn;Θ)−1(t−μ))where, X_m and X_n represent the m-th and n-th vectors, C(Xm,Xn;Θ) is parameter covariance function, μ represents mean vector.

The most widely used covariance function is: (10)C(Xm,Xn;Θ)=θ1exp⁡(−12∑i=1L(Xm(l)−Xn(l))δ2)where, L represents the number of elements in vector X_i, δ is variance, θ1 and θ2 are parameters.

For a new input vector X_N+1, use a predictive model to estimate its t_N+1 value rate distribution, expectation, and variance are used for prediction, and the model function is represented as: (11)tn=y(Xn)+vnwhere, y(X_n) represents modeling functions, νn is the prediction error.

So the model probability is: (12)P(TN|{Xn},A,B)=∫P(TN|{Xn},y,v)P(y|A)⋅P(v|B)dydvwhere, P(y|A) is the pre probability distribution of y(x), A is a set of hyperparameters of P(y|A), P(ν|B) is the pre probability distribution of prediction error, B is a parameter representing the error ν.

Let T_N = (t₁, t₂, … , t_n), T_N+1 = (t₁, t₂, … , t_n, t_n+1), therefore the conditional distribution of t_N+1 can be represented by Eq. (13), and it can be used to predict t_N+1: (13)P(tN+1|D,A,B,XN+1)=P(TN+1|{Xn},A,B,XN+1)P(TN|{Xn},A,B)

According to Bayesian method, it can be inferred that the distribution function of t_N+1 is: (14)P(tN+1|D,C(Xn,Xm;Θ),XN+1,Θ)=P(TN+1|C(Xn,Xm;Θ),Θ,Xn+1,{Xn})P(TN|C(Xn,Xm;Θ),Θ,{Xn})

By substituting the Gaussian distribution formula, the preliminary calculation can be simplified as: (15)P(tN+1|D,C(Xn,Xm;Θ),XN+1,Θ)=ZNZN+1exp⁡[−12(tN+1TCN+1−1tN−tNTCN−1tN)]where, Z_N and Z_N+1 are two normalization constants.

After further simplification, the following equation can be obtained: (16)P(tN+1|D,C(Xn,Xm;Θ),XN+1,Θ)=1zexp⁡(−(tN+1−t^N+1)22σt^N+12)where, t^N+1 represented expected value. Therefore, predictions can be made based on their expectations and variances.

The predictive reliability and accuracy of machine learning models are critically contingent upon the representativeness and robustness of the training dataset. In the frameworks of ANN, RFR, GPR, and SVR, input data are conventionally partitioned into training and testing subsets. The former serves to calibrate model parameters and optimize predictive performance, whereas the latter is employed to assess the accuracy and stability of the models.

The experimental dataset employed in this investigation comprises 151 data entries, partitioned into two categories: 89 records of pure CO₂ injection and 62 records of impure CO₂ injection. The dataset was divided into training (90%, n = 136) and testing (10%, n = 15) subsets to ensure unbiased model validation. These data were collected from various mature experimental works in the literature [13,15,44,45]. Data sets include experimental values for reservoir temperature, composition of drive gas (y_CO2, y_H2S, y_N2, and y_C2–C4), molecular weight of the C₇₊ fraction in crude oil (MW_C7+), the ratio of volatile (X_vol) to intermediate (X_int) components in crude oil, pure and impure CO₂-oil MMP. Table 3 presents a statistical description of the dataset used. A detailed analysis of the experimental data highlights that the reservoir temperature ranges from 31.00°C to 148.90°C, the experimental values for MMP range from 6.55 to 40.22 MPa.

In this section, we will statistically analyze the values of four statistical error functions, with Table 5 displaying the implementation performance of each model. In this table, the performance of the training and the testing data is listed separately. From Table 5, it can be observed that the RFR model has R² values close to 1 compared to the other three models on both the training and testing sets, specifically are 0.98 and 0.95, respectively. The MAPE and RMSE values on the testing set are also quite low, at 4.31% and 1.00%, respectively. These indicators suggest that the RFR model exhibits the highest prediction accuracy and strong stability. Besides, the GA-BPNN model demonstrates good prediction performance, ranking second to RFR. In comparison, it is evident that GPR has the lowest performance among the four models.

Among the four models evaluated, the RFR operates under the ensemble learning paradigm, which aggregates predictions from multiple decision trees to mitigate overfitting and enhance generalization. This method has obvious advantages over neural networks in processing small and medium-sized data sets (<10,000 samples).The GPR model is distinguished by kernel flexibility. By selecting the appropriate covariance function, GPR can adapt to different data structures (periodic/non-periodic), and provide probability output to quantify the prediction uncertainty, which is a very valuable feature for risk perception decision-making in reservoir engineering. However, GPR 's dependence on computationally intensive hyperparameter optimization often leads to local optimum, especially in high-dimensional parameter space, so advanced global optimization techniques such as Bayesian optimization are needed. Due to its ε-insensitive loss function, SVR is theoretically robust to outliers and noise, and can ignore the error within the tolerance threshold. Its performance depends heavily on the choice of kernel functions (e.g., linear, polynomial) and hyperparameter tuning, but it has interpretability defects and sensitivity to parameter initialization. The predictive performance of the neural network model in this study was not outstanding, and the BPNN algorithm optimized with GA did not demonstrate its outstanding predictive ability. It is speculated that this may be due to the lack of massive data in related research fields, and the big data fitting advantage of the GA-BPNN model has not been fully utilized.

To better illustrate the model performance, Fig. 4 displays the MMP data predicted by RFR, GA-BPNN, SVR, and GPR models. From Fig. 4, it can be seen that the RFR, GA-BPNN, and SVR models have achieved good alignment near the unit slope compared to the GPR model. Among the four models, the error distribution of the RFR and the GA-BPNN models is relatively balanced, with fewer outliers, thus ensuring good stability. The error visualization of the GPR model around the unit slope is quite evident, but it remains within an acceptable range.

To further validate the predictive performance of the model, a statistical error analysis will be conducted based on previous models. The prediction results of the RFR model and GA-BPNN model developed in this paper will be compared with six forecasting MMP methods proposed by Lee [46], Cronquist [47], Yellig and Metcalfc [16], Alston et al. [18], Emera et al. [28], and Yuan et al. [48]. The six comparative models selected in this article simultaneously consider the influence of CO₂ injection gas impurities on the prediction results of CO₂-oil MMP, as well as the influencing factors of MMP prediction under pure CO₂ injection. Therefore, they were used for comparison with RFR and GA-BPNN models.

Fig. 5 visually displays the comparison results between the simulated MMP values and literature methods. The unit slope line in Fig. 5 serves as the evaluation criterion; the closer the data points are to the unit slope line, the higher the consistency between the simulated and experimental values. Table 6 shows the comparison of various indicators of the model. These results indicate that the RFR model and GA-BPNN model have strong predictive abilities.

Analyze the impact of the characteristic parameters of the four models used in this article on predicting MMP in order to deepen our understanding of the CO₂-oil interaction mechanism and its intrinsic correlation. In this study, the relevance factor (R) was utilized to assess the extent of influence that each parameter has on MMP. For input variables, the R is a value ranging from −1 to 1. When R is negative, it represents that the input variable has a negative effect on the output variable, and when R is positive, it indicates that the input variable has a positive effect on the output variable. The absolute value of R represents the impact of the input variable on the output variable, and the larger the absolute value of R, the greater the impact of the input variable on the output variable. The calculation formula for R is as follows: (22)R(Xj,MMP)=∑i=1n(Xj,i−XJ¯)∑i=1N(MMPi−MMP¯)∑i=1n(Xj,i−XJ¯)2∑i=1N(MMPi−MMP¯)2where, Xj,i and Xj¯ is the i-th and average number of input j, MMP_i indicates the i-th, MMP¯ represents the average value of the MMP output.

Fig. 6 visually displays the R value of each input variable. From Fig. 6, it can be seen that X_int, X_C5-C6, and y_H2S can reduce the value of MMP. The observed negative correlations between X_int, X_C5-C6, y_H2S, and MMP can be attributed to their thermodynamic roles in reducing CO₂-oil interfacial tension. The ability of y_H2S to reduce MMP is due to the polarity of H₂S molecules, which can effectively reduce the interfacial tension between CO₂ and oil. The C₂-C₆ components in crude oil have similar molecular weights and properties to CO₂, which helps to increase the solubility of CO₂ in crude oil and reduce CO₂-oil MMP.

The increase of the other seven factors (T_R, MW_C7+, X_vol, y_C2-C4, y_CO2, y_C1, y_N2) will lead to an increase in CO₂-oil MMP. The increase of T_R will result in a decrease in the solubility of CO₂ in crude oil, which seriously affects the ability of CO₂ extraction to extract light hydrocarbons, ultimately leading to an increase in CO₂-oil MMP. An increase of MW_C7+ may create a greater difference between crude oil and CO₂ molecules, resulting in an increase in interfacial tension and thus affecting MMP. For the light components (X_vol) in crude oil, due to their easy volatilization, they are prone to entering the CO₂ gas phase from the crude oil during the CO₂ oil multi-stage mixed phase contact process, which results in a decrease in CO₂ purity and an increase in MMP. The increase in CO₂ concentration enhances the frequency of contact between CO₂ and crude oil. As CO₂ continuously extracts lighter components from the crude oil, the proportion of heavier components increases. Since these heavier components exhibit poor miscibility with CO₂, a higher pressure is required to achieve miscibility. Within the parameter curve range, the influence of the ten influencing factors on MMP is ranked in descending order: T_R > X_C5-C6 > MW_C7+ > X_vol > y_H2S > y_C2-C4 > y_CO2 > y_C1, y_N2 > X_int.

The Xinghua Block of Bayan oil field is a typical ultra-deep oil reservoir characterized by a large burial depth (>5000 m), high formation temperature and pressure, complex crude oil composition, and intricate fluid phases [49,50]. To verify the applicability of the established CO₂-oil MMP prediction models for ultra-deep oil reservoirs, experimental measurement data of CO₂ flooding MMP from two wells in Xinghua block of Bayan oil field were selected. The thin tube experiment refers to a simulated displacement test conducted in a thin tube model, and the experimental flowchart is shown in Fig. 7. The criterion for determining whether the thin tube experiment qualifies as a mixed-phase displacement is that the crude oil recovery rate, when injected with 1.2 pore volume(PV) exceeds 90%. Additionally, as the displacement pressure increases, there is no significant rise in displacement efficiency, and mixed-phase fluids can be observed in the observation window. The method for determining MMP is to plot the relationship curve between the recovery degree and displacement pressure when injecting 1.2 PV in each thin tube experiment, while ensuring that the thin tube experiment achieves three cycles of miscible flooding and three cycles of immiscible flooding [51].The pressure corresponding to the intersection of the non-mixed phase and mixed phase curves is the minimum mixed phase pressure (MMP), as shown in Fig. 8. The results of the slim tube test showed that injecting pure CO₂ at formation temperatures of 139°C and 133.9°C resulted in MMP values of 49.79 and 48.48 MPa for the reservoir, respectively.

Calculate the MMP using the development prediction models and compare the calculated value with the experimental value measured by the thin tube experiment, Table 7 shows the comparison results between the MMP values predicted by four models and the experimental values measured by capillary experiments. As summarized in Table 7, both the RFR and GA-BPNN models demonstrate superior predictive accuracy, with RMSE values below 3% and AAPRE less than 0.1%. Furthermore, all four models exhibit minimal deviations between predicted and experimentally measured MMP, with errors consistently confined within engineering-acceptable thresholds (relative error < 5%). These results validate the robustness and applicability of the proposed models for MMP prediction in ultra-deep reservoirs characterized by high-temperature, high-pressure conditions.