# Prediction of barite scale formation and inhibition in hydrocarbon reservoirs using AI modeling: Focus on different optimization algorithms

Ouafa Belkacem, Ahmed Rezrazi, Kamel Aizi, Lokmane Abdelouahed, Maamar Laidi, Abdelhafid Touil, Leila Cherifi, Salah Hanini

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## To cite this version:

Ouafa Belkacem, Ahmed Rezrazi, Kamel Aizi, Lokmane Abdelouahed, Maamar Laidi, et al. Prediction of barite scale formation and inhibition in hydrocarbon reservoirs using AI modeling: Focus on different optimization algorithms. Results in engineering, 2025, 26, 10.1016/j.rineng.2025.105222. hal-05313834

HAL Id: hal-05313834

https://hal.science/hal-05313834v1

Submitted on 14 Oct 2025

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Research paper

# Prediction of barite scale formation and inhibition in hydrocarbon reservoirs using AI modeling: Focus on different optimization algorithms

Ouafa Belkacem<sup>a,b</sup>, Ahmed Rezrazi<sup>a</sup>, Kamel Aizi<sup>c</sup>, Lokmane Abdelouahed<sup>d, *, e</sup>, Maamar Laidi<sup>a</sup>, Abdelhafid Touil<sup>b</sup>, Leila Cherifi<sup>a</sup>, Salah Hanini<sup>a</sup>

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## ARTICLE INFO

### Keywords:

- Barite

- Inhibition

- Formation

- Intelligent models

- Water flooding

- Data augmentation

## ABSTRACT

Barite scale formation is a persistent and detrimental challenge in the oil industry during water-flooding operations, primarily due to the incompatibility between injected water and formation water. This study introduces a novel and highly effective approach for predicting barite scale formation and inhibition by leveraging advanced artificial intelligence (AI) techniques. Innovative intelligent models, including Random Forest (RF), k-nearest Neighbors (KNN), Extreme Learning Machine (ELM), Support Vector Regression (SVR), Decision Trees (DT), and Multilayer Perceptron (MLP), were developed and optimized for this purpose. Notably, the MLP and SVR models were enhanced through training with the particle swarm optimization algorithm (PSO) and data augmentation techniques. The performance of the models was rigorously evaluated using statistical criteria and graphical tools, demonstrating exceptional accuracy with minimal errors. The ELM model emerged as the most precise (RMSE = 0.0005, R² = 0.999), closely followed by the SVR-PSO model (RMSE = 0.0008, R² = 0.999). Additionally, the ELM model successfully estimated over 95 % of the data points, with an applicability domain of 97.82 %. Moreover, we conducted an additional validation using a new independent dataset, which showed an excellent correlation coefficient (0.99) and a very low RMSE (0.01), further demonstrating the robustness and generalizability of the proposed models. This work shows the transformative potential of AI-driven modeling in addressing complex challenges such as barite scale formation. The findings underscore the practical implications of AI techniques in optimizing oilfield operations, enhancing efficiency, and reducing costs, thereby offering a significant advancement in the field.

## 1. Introduction

Oil and gas remain vital but costly forms of energy for many

industries. However, their efficient exploitation is often limited by various operational and reservoir challenges, including high crude oil viscosity, decreasing reservoir pressure, formation damage, and mineral

**Abbreviations:** AI, Artificial Intelligence; ANN, Artificial Neural Network; ANNs, Artificial Neural Networks; AARE, Average Absolute Relative Error; ARE, Absolute Relative Error; BFGS, Broyden-Fletcher-Goldfarb-Shanno; BR, Bayesian Regularization; C<sub>Ba</sub>, Barium concentration; C<sub>SO4</sub>, Sulfate concentration; C<sub>inb</sub>, Concentration of barite inhibitor; Cp, Solubility of barite scale inhibitors; C, SVR specific parameter; CFBPN, Cascade-Forward Back-Propagation Network; DETPMP, Diethylenetriaminepenta (methylene phosphonic acid); DT, Decision Tree; EOR, Enhanced Oil Recovery; ELM, Extreme Learning Machine; GEP, Gene Expression Programming; GBDT, Decision Trees Optimized by Gradient Boosting; GRNN, Generalized Regression Neural Network; GWO, Grey Wolf Optimizer; K<sub>i</sub>, Initial permeability; K<sub>d</sub>, The detected permeability; K<sub>d</sub><sub>exp</sub>, Experimental permeability damage; K<sub>d</sub><sub>pred</sub>, Predicted permeability damage; Knn, The k-nearest neighbors; LSSVM, Least Squares Support Vector Machine; MLP, Multi-Layer Perceptron; MSE, Mean Square Error; MAE, Mean Absolute Error; Q, Flow rate; PPCA, Poly-Phosphono Carboxylic Acid; PSO, Particle Swarm Optimization; RELU, Rectified Linear Unit; RBF, Radial Basis Function; RMSE, Root Mean Square Error; R<sup>2</sup>, Correlation Coefficient; RF, Random Forest; SCG, Scaled Conjugate Gradient; SVR, Support Vector Regression; SVM, Support Vector Machine; SD, Standard Deviation; SHAP, Shapley additive explanations; T, Temperature; t, Injection time; V<sub>inj</sub>, Volume of water injection; ΔP, Differential pressure; ε, Epsilon, SVR specific parameter.

<footnote><sup>a</sup> Laboratory of Biomaterials and Transport Phenomena (LBMTP), Yahia Fares University, Medea, Algeria</footnote>

<footnote><sup>b</sup> The Central Directorate of Research and Development (DCRD) SONATRACH, Boumerdes, Algeria</footnote>

<footnote><sup>c</sup> Laboratoire des sciences et technologies de l'information, de la communication et de la connaissance (Lab-STICC), University of Bretagne Sud, France</footnote>

<footnote><sup>d</sup> LSPC—Laboratoire de Securité des Procédes Chimiques, INSA Rouen Normandie, UNIROUEN, Normandie University, 76000 Rouen, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: Lokmane.abdelouahed@insa-rouen.fr (L. Abdelouahed).</footnote>
scale deposition, as highlighted in previous studies [1,2]. Among these, mineral scale deposition, particularly sulfate scales as Barium sulfate, calcium sulfate, strontium sulfate, and calcium carbonates, represents a critical obstacle in secondary oil recovery operations [2]. The efficiency of these resources is typically hampered by impacts like reservoir depletion, formation damage, and scale deposition, which significantly reduce the oil industry's productivity [3,4].

Water flooding, the most commonly used enhanced oil recovery (EOR) technique, involves injecting water into oil reservoirs to maintain pressure [5]. However, the presence of formation water and treatment processes practiced in exploration and production operations, such as water flooding and crude oil desalting, creates a conducive environment for the development of mineral scales. Scale deposition creates extreme threats to field production in terms of reduced permeability, clogged pipelines, and equipment damage, leading to massive economic losses and production downtime [2]. Previous studies have quantified these issues, showing that scale formation can reduce permeability by up to 80 %, increase operational costs due to frequent cleaning and maintenance, and cause severe reductions in well infectivity and productivity [2]. The principal cause of mineral scale buildup in secondary oil production is incompatibility between formation water and injection water. The water in this sector is most often prohibitively expensive to apply, so supplementary water sources like seawater that contains sulfate anions are applied. Formation water contains cations such as strontium, barium, and calcium, though. The two brines react with one another to generate sulfate scales such as barite (barium sulfate), calcium sulfate, and strontium sulfate [4,6]. These scales reduce oil production, permeability, and porosity, as well as enhance the issue of the oil and gas industry [7].

Traditional methods for scale prediction include thermodynamic modeling, empirical equations, and computational fluid dynamics (CFD) simulations. These approaches, while useful, often require extensive experimental data and complex calculations, making them time-consuming and difficult to apply to dynamic reservoir conditions [8]. Experimental techniques as well as thermodynamic models were employed in early studies to predict scale formation. These techniques are lengthy and involve a series of laboratory tests. On the other hand, artificial intelligence (AI) techniques are now powerful methods for scale prediction, simulation, and optimization in petroleum engineering [9]. Artificial intelligence (AI) modeling has been a powerful tool during the last decade for addressing prediction, simulation, and optimization problems in most branches of engineering. Compared to traditional approaches, AI models can handle incomplete datasets, adapt to complex nonlinear relationships, and provide fast and cost-effective predictions [10]. Traditional laboratory-scale water-flooding tests are typically time-consuming, expensive, and of limited scope. On the other hand, AI models offer a cost-effective and reliable option by using data available in the literature to predict and optimize complex processes [4, 9].

For instance, some of the initial applications of AI in petroleum engineering are the works of Kalam et al. (1996), who used Artificial Neural Networks (ANNs) with a backpropagation algorithm to predict relative permeability curves for oil field damage formation [11]. Similarly, Zuluaga et al. (2002) proposed an ANN and Fuzzy model to predict permeability loss caused by external particle invasion in non-consolidated porous media [12]. AI modeling for sulfate scaling prediction has also evolved by Zabihi et al. (2011) [5], who used a multilayer neural network with a backpropagation algorithm to predict permeability damage due to sulfate scaling from thermodynamic properties and reservoir conditions. Foroutan and Moghadasi (2013) [13], created an ANN model for the prediction of calcium sulfate precipitation, while Kamari et al. (2014) [14] suggested the first application of least squares support vector machines (LSSVM) with simulated annealing (CSA) for the prediction of solubility product data for barite in oil field brines. Recently, Rostami et al. (2019) employed a gene expression programming (GEP) model to forecast permeability damage due to

mineral scaling [15], and More recent studies, Larestani et al. (2022) proposed some intelligent modeling methods, including gradient-boosted decision trees (GBDT) and multilayer perceptron (MLP), for sulfate scale formation prediction in oil fields [3].

Despite these advances, previous studies present several limitations in dealing with incomplete data and scarcity of data. To overcome these gaps, the present study introduces an enhanced AI framework for predicting barite scale formation and inhibition, Key contributions include These limitations are overcome within this research through an additional step of data augmentation that amplifies the dataset from 552 to 5501 points via linear interpolation. This approach ensures a more comprehensive representation of real-world reservoir conditions, reducing the bias caused by limited data availability [16,17]. Adding the concentrations of the inhibitors as the input parameters further facilitates predicting the barite inhibition and the inhibition efficiency and presents a clearer explanation of the formation of scale as well as prevention of it. Unlike previous studies that primarily focused on predicting scale formation, this research emphasizes the integration of inhibitor effects into AI models, enhancing their practical applicability in oilfield operations. The primary objective of this study is to introduce a comprehensive review of novel AI techniques, including Extreme Learning Machine (ELM), Support Vector Regression (SVR), k-nearest neighbors (KNN), and Particle Swarm Optimization (PSO), for predicting and inhibiting barite scale formation.

As compared to earlier studies, which often relied on limited datasets and did not account for inhibitor concentrations, our study covers a deficiency by employing data augmentation, including significant input parameters, and advanced optimization techniques to enhance model accuracy and robustness. By improving prediction accuracy and incorporating scale inhibition mechanisms, this study directly contributes to reducing operational costs, preventing unplanned shutdowns, and optimizing oil production efficiency These findings have direct uses in increasing oilfield productivity by optimizing scale inhibition, reducing operations costs, providing efficient alternatives to laboratory-scale tests that are expensive, and offering a reduction in scale deposition downtime.

## 2. Materials and modeling techniques

This section presents the methodology employed to study and solve scale deposition in oilfield pipelines. The study is experimental data and literature-represented conditions under which barite (BaSO₄) and other mineral scales precipitate. As can be seen in Fig. 1, mineral deposition within Algerian oil fields is due to long-term operation of pipelines, leading to gradual scaling. The heterogeneous nature of the deposits shows a complex composition, primarily consisting of iron oxides, sulfates (BaSO₄, CaSO₄, SrSO₄), and carbonates (CaCO₃). Formations of these scales occur primarily due to the following factors:

- Supersaturation of mineral salts in the injected water.

- Incompatible mixing of water sources, leading to precipitation reactions.

- High temperature and pressure conditions, which speed up scale deposition.

- Corrosive conditions, cause iron-based deposits.

Fig. 1 gives a quantitative indication of the degree of pipeline clogging by scaling, pipe (d) is ~30% blocked (moderate scaling), pipe (b) is ~70% blocked (significant scaling), pipe (c) is ~80% blocked (severe scaling), and pipe (d) is ~90-95% blocked (almost complete clogging). The reduction in effective pipe diameter by scaling has a direct impact on fluid dynamics, where the flow rate is a function of the cross-sectional area of the pipe. Thus, more scaling results in: Increased pressure differentials, with increased energy input required for fluid transfer, increased operational costs as a result of higher pumping power demands, and System failure risks that may be occasioned by possible
<figure><img src="image_4.png" /><figcaption>Fig. 1. Problematic of deposit formation in Algerian oil field after a period of operation [18].</figcaption></figure>

clogging, particularly in large flow lines where enormous obstruction may result in complete clogging.

Further, the present section elaborates on the data augmentation techniques employed, describes the modeling approaches adopted to predict and avoid scale formation, and evaluates the effectiveness of the inhibition techniques suggested.

## 2.1. Experimental data gathering and preprocessing

A total of 552 experimental data points were gathered from the published literature [19,20]. The database analysis and statistical description, including standard deviation, maximum, and minimum values of each data vector, could give some information to adapt the optimal IA model. The statistical summary of the data is presented in Table 1. Data preprocessing step is one of the important stages of developing intelligent models [21]. This dataset includes a wide range of thermodynamic properties and reservoir conditions, such as the ionic concentration of barium and sulfate ($C_{Ba2}$, $C_{So4z}$), temperature (T), flow rate (Q), differential pressure ($\Delta P$), initial permeability ($K_i$), volume of injection water ($V_{inj}$), injection time (t), concentration of barite inhibitor ($C_{inb}$), and solubility of barite scale inhibitors ($C_p$) as inputs. The detected permeability ($K_d$) was considered as an output. In contrast to earlier research, we included the concentration of chemical additives and their properties as inputs, where these additives can inhibit or delay the production of barite scale, thereby influencing the mechanism of sedimentation formation. The chemical agents used in this data are Poly-Phosphono Carboxylic Acid (PPCA), Diethylenetriaminepenta (methylene phosphonic acid) (DETPMP) which are commercial anti-scales for mineral scales control, and a locally produced scale inhibitor by Amer Badr Mohamed Bin Merdhah [20] Phosphonate- based. A complex synergy of thermodynamic, kinetic, and hydrodynamic processes regulates Barite ($BaSO_4$) scale formation in oilfield activities. From a thermodynamic perspective, precipitation is governed by the

<table><caption>**Table 1**
The descriptive statistics of experimental data.</caption><thead><tr><th>Category</th><th>Parameters</th><th>Unit</th><th>Min</th><th>Max</th><th>Std</th></tr></thead><tbody><tr><td>Input</td><td>$C_{Ba}$</td><td>Ppm</td><td>250</td><td>2200</td><td>878.818</td></tr><tr><td>Input</td><td>$C_{So4}$</td><td>Ppm</td><td>2750</td><td>2960</td><td>94.641</td></tr><tr><td>Input</td><td>T</td><td>°C</td><td>50</td><td>95</td><td>16.322</td></tr><tr><td>Input</td><td>Q</td><td>cc/min</td><td>2.25</td><td>28.34</td><td>7.120</td></tr><tr><td>Input</td><td>ΔP</td><td>Psig</td><td>75</td><td>200</td><td>44.192</td></tr><tr><td>Input</td><td>K<sub>i</sub></td><td>Md</td><td>11.64</td><td>141.130</td><td>42.701</td></tr><tr><td>Input</td><td>V<sub>inj</sub></td><td>Pv</td><td>1.89</td><td>268.760</td><td>56.262</td></tr><tr><td>Input</td><td>t<sub>inj</sub></td><td>Min</td><td>10</td><td>120</td><td>34.551</td></tr><tr><td>Input</td><td>$C_{inb}$</td><td>Ppm</td><td>0</td><td>1000</td><td>302.521</td></tr><tr><td>Input</td><td>C<sub>p</sub></td><td>Cp</td><td>0.862</td><td>1.381</td><td>0.192</td></tr><tr><td>Output</td><td>K<sub>d</sub></td><td>Md</td><td>8.720</td><td>139.630</td><td>33.272</td></tr></tbody></table>

saturation index (SI), while temperature, pressure, and salinity reduce the solubility of $BaSO_4$ [22], and ionic strength and activity coefficients influence ion interaction. Kinetically, supersaturation degrees control nucleation and crystal growth, and the rates are further determined by ion diffusivity, nucleation sites [23,24], and the presence of inhibitors that alter crystal morphology. Hydrodynamic conditions, such as flow velocity and shear stress, determine whether crystals deposit or remain suspended, with surface roughness and wettability exerting additional influences. Reservoir-specific conditions, including pH, dissolution of $CO_2$, and chemical treatments [25] further control scale development via their impact on ion speciation and solubility. Scale deposition reduces permeability and porosity, impairs injectivity, and alters wettability, leading to serious operational issues [26,27]. An understanding of these interconnected factors is essential for the formulation of predictive models and effective scale mitigation strategies in oilfield operations.

Machine learning and deep learning techniques depend on large datasets to achieve optimal performance, making data augmentation a crucial strategy for improving dataset preparation and addressing the limitations of small dataset sizes [16,17,28]. Our analysis of the collected data revealed a significantly restricted range of input variables, with noticeable gaps due to limited diversity in the input values. To mitigate these issues, this study incorporated data augmentation to boost the number of data points and fill in the missing values effectively. Using Python code, we applied linear interpolation to generate synthetic data between each pair of consecutive original data points [17], filling the missing with additional data points. Following this step, the novel database grew to a total of 5501 points. We used 5001 points to develop the IA models and reserved the remaining data for the validation of the optimal model.

## 2.2. Modeling techniques

### 2.2.1. Artificial neural network (ANN)

Artificial Neural Networks are the base of deep learning and a sub-field of machine learning. Its structure was inspired by the human central nervous system (brain neural) [29,30]. The literature recommends using the ANN for a range of applications, such as regression problems, classification problems, and decision-making. Similar to natural neural networks, ANN model has a significant number of neurons arranged into several layers [31]. The input layer in the first is responsible for receiving information, while the output layer makes predictions about the ultimate output. Between them, we find the hidden layers which are the key components and are responsible for the majority of our network calculations. In all ANN types, the process of treating the dataset is ensured by the reception of the information, and each information parameter is fed as input to each neuron of the first layer, which is connected to the neurons of the next layer through
channels. Each of these channels is assigned a value called weight [32]. The inputs are multiplied by the corresponding weights, and their sum is sent as input to the neurons in the hidden layer. Each of these neurons is associated with a numerical value minus the bias, which is then added to the input sum. This value is then passed through a threshold function called the activation function, as described by Sigmoid and RELU [31]. The ANN models have a high ability to study and identify the linear and nonlinear relationships of complex systems [32,33]. There are many classes of artificial neural networks, and this study includes the multi-layer neural network and the extreme learning machine.

### Multi-layer neural network (MLP)

The multi-layer perceptron is one of the ANN types developed in the 1980s [3], based on a mathematical model of data processing. MLP any type of ANN model is composed of elements noned neurons that are interconnected in structured layers with weights and biases between the layers [34]. In our current study, the MLP model has a single hidden layer where RELU is the activation function and Broyden-Fletcher-Goldfarb-Shannon (BFGS) are the optimization algorithms used to adjust the network weights and biases. The predicted output is then computed as follows:

$$ \text{Predicted value} = \text{RELU}(w_2 \times (\text{BFGS} \times x + b_1) + b_2) \quad (1) $$

Where w and b are weights and biases, respectively, and x are the input variables.

### Extreme Learning Machine (ELM)

The Extreme Learning Machine (ELM) is a recent feedforward network algorithm proposed by Huang et al. [21,35]. With a single hidden layer, an ELM is structured, whose input weights and hidden-layer biases are generated randomly [36]. The ELM structure is composed of three parts: the input layer, the hidden layer, and the output layer. These layers are connected by neurons or nodes. The transmission of information occurs through the link's weight [36]. This study examines the ELM neural network, which is trained using a sigmoid function as an activation function, where the model is configured with a specific number of neurons in the hidden layer. The equation representing our ELM model is as follows:

$$ \begin{aligned} H &= \sigma(X.W_{\text{input}}) \\ Y &= (H.W_{\text{output}}) \end{aligned} \quad (2) $$

Where: H is the hidden layer output, X is the input data matrix, Y is the predicted output, σ is the sigmoid activation function, Winput is the matrix of output weights, and Woutput is the matrix of input weights.

### 2.2.2. Support vector regression (SVR)

SVR is one of the excellent machine learning models and a variant of support vector machine (SVM) adapted for regression analysis, as proposed by Vapnik in the 1990s [30]. It offers a global optimal solution, while other neural network models offer a local optimal solution [37]. SVR aims to determine a hyperplane that best fits the data while taking into account a predetermined margin. C and epsilon (ε) are two crucial parameters that determine the margin in SVR. They are very useful in mathematical relationships where the inputs to the process match the unknown outputs, since it's hard to get these relationships because of the many separate factors that can change the outcome [38]. In our study, we examined support vector regression (SVR), which is established with its specific parameters. The equation representing our SVR model is as follows:

$$ Y(x) = \sum_{i=1}^{N} a_i k(x, x_i) + b \quad (3) $$

Where: Y is the predicted values for the inputs x, N is the number of

support vectors, a$_i$ is the lagrange multipliers, k(x, xi) is the kernel function, representing the inner product of the input vectors in a higher dimensional space, and b is the bias term.

### 2.2.3. Decision tree (DT)

A Decision Tree is a machine learning approach utilized for treating regression and classification problems [39]. The algorithm uses a multilevel model like a tree to make predictions by analyzing the interconnections between the data's features [39]. It operates by dividing the data inputs into smaller groups and then determining the input values for each group. In our study, we examined the capacity of the DT algorithm for this problem.

### 2.2.4. Random forest (RF)

Random Forest regression is a machine learning model developed by Breiman in 2001 [40]. The algorithmic technique is employed for addressing regression and classification problems [41]. The algorithm works by training multiple decision trees on random subsets of the data and then combining their predictions to generate a final prediction [40, 42]. This study investigates the RF regression model and examines its capacity for this problem. The equation that represents our algorithm model is as follows:

$$ Y = \frac{1}{N} \sum_{i=1}^{N} y_i \quad (4) $$

Where: Y is the predicted values, N is the number of decision trees in the forest, and y$_i$ is the prediction made by the i-th decision tree.

### 2.2.5. K nearest neighbors (Knn)

The K-nearest neighbor technique is an important approach used for classification and regression analysis. It operates under the assumption that the class of each instance in the training data has been labeled. During the classification process, a new instance is categorized based on the class of its k closest neighboring training instances. Therefore, the Knn technique lacks a distinct learning mechanism. It partitions the feature space based on the class of the training data and employs this procedure as the classification model [42,43]. The variance between the test data and training data, the selection of the k value, and the classification statement all have pivotal roles in Knn models. Given that the feature space in Knn is an n-dimensional real vector space, there are two viable approaches for computing the distance between two instance points: Euclidean distance and distance. The Knn analysis in this paper utilizes the Euclidean distance. Moreover, the selection of k will significantly affect the categorization output of the model [42,43]. In our study, we examined the Knn technique for this problem. The equation that represents our Knn model is as follows:

$$ Y = \frac{1}{n} \sum_{i=1}^{n} y_i \quad (5) $$

Where: Y is the predicted values, n is the number of neighbors, and y$_i$ is the prediction made by the Knn model.

### 2.2.6. Optimization algorithms

Optimization algorithms are utilized as techniques to identify the most optimal solution for various issues, and they hold significant significance in different domains. Particle swarm optimization PSO is one of the algorithms inspired by nature and is a significant algorithm within the realm of swarm intelligence. The algorithm was developed by Kennedy and Eberhart in 1995 using the simulation of birds' foraging behavior to locate a food source [44,45]. The grey wolf optimizer GWO, in another kind of swarm intelligence optimization algorithm, when it is a method inspired by grey wolves' [46]. There are many studies about the using of the optimization algorithms in the training of IA models and how they improve their efficacy. In this study, the PSO algorithm was coupled with MLP neural network and SVR technique to optimize the
best weights and bias for the MLP, and get the ideals parameters for the SVR model, as illustrated in the schematic diagram in Fig. 2. The graphical representations of the developed models are mentioned on the Annex/Appendix part in Fig. A.1.

AI models such as ANNs and SVMs are superior to traditional methods as they are capable of handling vast, complex datasets and learning to improve their predictive capacities continuously. These models have the ability to integrate a wide range of variables such as socio-economic, demographic, and environmental variables, allowing for a more integrated analysis across disciplines [9]. Their nonlinearity and capability to detect nonlinear patterns make them particularly effective in environments where variable interactions are dynamic and intricate. In contrast, standard models are grounded primarily on linear processes and limited predictors, which restricts their capacity to detect intricate interactions in data. These models are likely to possess lower predictive accuracy due to their linearity and stationarity assumptions, thus performing worse in dynamic environments. Additionally, they are also more likely to experience issues like lag in data, overfitting when the data is small, and complexity in processing noisy or missing data [10, 26]. As seeing in Fig. 2,

Even though traditional models can have interpretability and computational efficiency, they might lack the power and flexibility of AI-based techniques. Moreover, AI models have improved generalization capability, especially where they are trained on diverse and augmented data sets. Unlike traditional models, which may require manual selection and feature preprocessing, AI techniques take

<figure><img src="image_5.png" /><figcaption>Fig. 2. A schematic figure of the PSO algorithm.</figcaption></figure>

* $\bar{y}$: New position of the particle.

* $\bar{x}$: Current or previous position of the particle.

* $\bar{v}$: Current velocity of the particle.

* $\vec{p} - \bar{x}$: Difference between the personal best position found by the particle ($\vec{p}$) and a reference average position ($\bar{x}$).

* $\vec{g} - \bar{x}$: Difference between the global best position (found by the whole swarm, denoted by $\vec{g}$) and the average position.

* $\vec{p}_g - \bar{x}$: Another form of global best position (possibly a weighted or time-updated version), depending on the PSO variant.

* $\varphi, \varphi_g, \varphi_t$: Coefficients that control the influence of each component in the update rule.

advantage of automated feature extraction and optimization, reducing the role of human intervention and increasing scalability. This advantage is particularly vital in fields like engineering, medicine, and finance, where accurate prediction is required for decision-making and risk assessment. In general, AI-based techniques confer a monumental superiority over traditional models by providing more precise predictions, greater ease of handling nonlinear relations, and higher flexibility in adjusting to the real-world complexity [10]. These make them invaluable resources for modern predictive modeling and decision support systems.

## 2.3. Model's development

This part of work describes the preprocessing and modeling steps undertaken to develop the considered intelligent models, as illustrated in Fig. 3.

Before the modeling of our AI models namely MLP, MLP-PSO, SVR, SVR-PSO, ELM, KNN, DT, and RF an essential preprocessing step was applied. Linear interpolation augmentation data technique was employed to address missing data points and to smooth inconsistencies in the dataset, thereby ensuring a more reliable and continuous representation of the input features. Thereafter, the complete dataset was normalized to a range of 0 to 1 utilizing Eq. (6), to standardize variable scales and enhance convergence conditions during training [46]. We utilized 5001 data points for the development of the AI models. The residual data was allocated for validating the optimal model to evaluate its generalization performance. The preprocessing steps were essential for ensuring the robustness and accuracy of the developed models.

$$X_{Nor} = \frac{X - X_{min}}{X_{max} - X_{min}} \qquad (6)$$

Where Xmax and Xmin represent the maximum and minimum values of a given variable X, respectively, X is the measured value, and $X_{Nor}$ points out the normalized value. The data were randomly split into two subsets: 80% for training and 20% for testing to assess the performance of our categorical models. The modeling of our IA models was affected using the Python language in Jupyter notebook. To adapt the best specific parameters for each IA model, a trial-and-error technique was applied, and Table 2 presents guidelines to identify the best parameters for our models. The model's efficiency and reliability were evaluated using various statistical and graphical criteria, including Root Mean Square Error (RMSE), Correlation Coefficient (R²), Standard Deviation (SD), Mean Square Error (MSE), Mean Absolute Error (MAE), and Average Absolute Relative Error (AARE), for the statistical one, which are listed on annex and appendix section [3,47]. In terms of the graphical criteria, several plots are employed to evaluate and compare our developed models, such as a cross plot to compare the experimental values with the predicted values, cumulative frequency plot, Taylor plot, and error bars to visualize the results of our models and calcify them. The best model has the highest correlation coefficient and the smallest errors. After selecting the ideal model for prediction, it must be tested in its domain of applicability. The leverage mathematical technique was used as a means to identify the outlier outside the specified range [48]. For more validate the efficacy of our model, we tested its predictions using a new dataset of 500 points.

## 3. Results and discussions

This part of work presents the results of the methodologies applied to the dataset following the procedures elucidated in the model's development section. The application of data augmentation technique serves to get a new dataset with a big size of 5501 points with boosting the number of data points and fill in the missing values effectively, which give an opportunities of generalization models results. After training the intelligent models, the key parameters associated with each model, capable of providing the best prediction, were identified as ideal
<figure><img src="image_6.png" /><figcaption>Fig. 3. The road map of the developed models.</figcaption></figure>

<table><caption>**Table 2**
The range descriptive of trial-and-error stage.</caption><thead><tr><td>Intelligent model</td><td>Investigated rang</td><td>Number of models</td></tr></thead><tbody><tr><td>MLP</td><td>1 ≤ numbers of hidden neuron ≤ 500</td><td>200</td></tr><tr><td>MLP-PSO</td><td>1 ≤ numbers of hidden neuron ≤ 500</td><td>150</td></tr><tr><td rowspan="2">SVR</td><td>0 ≤ C ≤ 3</td><td rowspan="2">200</td></tr><tr><td>0 ≤ ε ≤ 2</td></tr><tr><td rowspan="2">SVR-PSO</td><td>0 ≤ C ≤ 2</td><td rowspan="2">150</td></tr><tr><td>0 ≤ ε ≤ 2</td></tr><tr><td>ELM</td><td>1 ≤ numbers of hidden neuron ≤ 1000</td><td>250</td></tr><tr><td>KNN</td><td>1 ≤ neighbors ≤ 100</td><td>100</td></tr><tr><td>RF</td><td>1 ≤ decision trees ≤ 200</td><td>100</td></tr><tr><td>DT</td><td>-</td><td>100</td></tr></tbody></table>

parameters for model construction.

### 3.1. Statistical performance evaluation for the intelligent models

The MLP model demonstrated the best predictive performance, exhibiting optimal results with a single hidden layer comprising 100

neurons. RELU was employed as an activation function, and the BFGS algorithm was employed for the adjustment of network weights and biases. ELM neural network, trained with a sigmoid function as an activation function and encompassing 700 neurons, showcased note-worthy predictive capabilities. The SVR model exhibited commendable predictive accuracy with specific parameters set at C=1 and ε=0.011. The RF model, trained with 100 decision trees, and the Knn model trained with a single neighbor, also presented distinctive performances. All these optimized specific parameters for each model give the best predictive results. Furthermore, the optimization of model performance was achieved through the application of the PSO algorithm in conjunction with the MLP neural network and the SVR model. The optimal configuration included 72 neurons for the MLP model, and C=31.34, ε=0.001 for the SVR model, resulting in enhanced predictive accuracy. A comprehensive set of statistical indices was employed to evaluate the performance and the reliability of our models. Table 3 presents the statistical description of the developed models, where ELM model demonstrated superior predictive capabilities, with an RMSE of 0.0005 and a correlation coefficient R² of 0.99999 for all data. Conversely, the MLP model exhibited comparatively lower predictive
<table><caption>**Table 3**
Statistical performance evaluation for the intelligent models.</caption><thead><tr><th></th><th>SVR</th><th>SVR-PSO</th><th>ELM</th><th>MLP</th><th>MLP-PSO</th><th>DT</th><th>RF</th><th>KNN</th></tr></thead><tbody><tr><td>Train data</td><td>4000</td><td>4000</td><td><strong>4000</strong></td><td>4000</td><td>4000</td><td>4000</td><td>4000</td><td>4000</td></tr><tr><td>R2</td><td>0.99900</td><td>0.99998</td><td><strong>0.99999</strong></td><td>0.9956</td><td>0.99688</td><td>0.99979</td><td>0.99979</td><td>1</td></tr><tr><td>MAE</td><td>0.005</td><td>0.0006</td><td><strong>0.0003</strong></td><td>0.008</td><td>0.0069</td><td>0.001</td><td>0.0012</td><td>0</td></tr><tr><td>RMSE</td><td>0.006</td><td>0.0008</td><td><strong>0.0005</strong></td><td>0.013</td><td>0.011</td><td>0.002</td><td>0.002</td><td>0</td></tr><tr><td>MSE</td><td>4.16 × 10<sup>-5</sup></td><td>6.87 × 10<sup>-7</sup></td><td><strong>2.91 × 10<sup>-7</sup></strong></td><td>0.0001</td><td>0.0001</td><td>8.57 × 10<sup>-6</sup></td><td>8.51 × 10<sup>-6</sup></td><td>0</td></tr><tr><td>Test data</td><td>1001</td><td>1001</td><td><strong>1001</strong></td><td>1001</td><td>1001</td><td>1001</td><td>1001</td><td>1001</td></tr><tr><td>R2</td><td>0.99889</td><td>0.99997</td><td><strong>0.99998</strong></td><td>0.9957</td><td>0.99725</td><td>0.99852</td><td>0.99851</td><td>0.9970</td></tr><tr><td>MAE</td><td>0.005</td><td>0.0007</td><td><strong>0.0004</strong></td><td>0.008</td><td>0.0068</td><td>0.003</td><td>0.0032</td><td>0.0049</td></tr><tr><td>RMSE</td><td>0.006</td><td>0.001</td><td><strong>0.0007</strong></td><td>0.013</td><td>0.010</td><td>0.007</td><td>0.007</td><td>0.011</td></tr><tr><td>MSE</td><td>4.69 × 10<sup>-5</sup></td><td>1.12 × 10<sup>-6</sup></td><td><strong>5.09 × 10<sup>-7</sup></strong></td><td>0.0001</td><td>0.0001</td><td>6.29 × 10<sup>-5</sup></td><td>6.31 × 10<sup>-5</sup></td><td>0.0001</td></tr><tr><td>All data</td><td>5001</td><td>5001</td><td><strong>5001</strong></td><td>5001</td><td>5001</td><td>5001</td><td>5001</td><td>5001</td></tr><tr><td>R2</td><td>0.99898</td><td>0.99998</td><td><strong>0.99999</strong></td><td>0.9957</td><td>0.99696</td><td>0.99953</td><td>0.99953</td><td>0.9994</td></tr><tr><td>MAE</td><td>0.005</td><td>0.0006</td><td><strong>0.0003</strong></td><td>0.008</td><td>0.006</td><td>0.001</td><td>0.0016</td><td>0.00098</td></tr><tr><td>RMSE</td><td>0.006</td><td>0.0008</td><td><strong>0.0005</strong></td><td>0.013</td><td>0.011</td><td>0.003</td><td>0.0039</td><td>0.002</td></tr><tr><td>MSE</td><td>4.27 × 10<sup>-5</sup></td><td>7.83 × 10<sup>-7</sup></td><td><strong>3.35 × 10<sup>-7</sup></strong></td><td>0.0001</td><td>0.0001</td><td>1.94 × 10<sup>-5</sup></td><td>1.94 × 10<sup>-5</sup></td><td>2.52 × 10<sup>-5</sup></td></tr></tbody></table>

accuracy, with an RMSE of 0.013 and an R² of 0.9957 for all data, while it still satisfactory. The precision of the models can be ranked as follows: ELM > PSO-SVR > SVR > DT > RF > KNN > PSO-MLP > MLP. In light of the results given in Table 3, it can also be inferred that the performance of the intelligent model doesn't just depend on the type employed, where the integration of optimization algorithms significantly enhances its performance. The collaboration between the PSO algorithm and the SVR model notably reduced the errors and augmented the correlation between experimental and predictive values.

## 3.2. Graphical performance evaluation for the intelligent models

Graphical analysis was employed to graph the results. Fig. 4 presents cross plots of experimental permeability damage ($K_{d,exp}$) and the predicted damage in permeability ($K_{d,pred}$) for the eight models. This cross-plot confirms the results of the statistical analysis and reveals that the ELM model is the optimum model, with the predictions being highly near the reference line ($Y = X$).

As evident, the predictions of the proposed models show excellent consistency along the diagonal line ($Y = X$) for all data. Fig. 5 also

supports this conclusion by using cumulative frequency analysis, whereby model accuracy is measured using analysis of the absolute relative error (ARE%) [3]. The ELM model was an excellent prediction model, with an accuracy rate of over 99.36 % and an absolute relative error of under 10 %. These high levels of precision and coverage capability strengthen the credibility of the developed models.

For a comparison of the relative reliability of the developed models, a Taylor diagram Fig. 6 was employed. The Taylor plot effectively illustrates the correlation, standard deviation, and root-mean-square error (RMSE) for all models. The red star represents the reference, with the observation that the ELM model is better than the rest due to its higher correlation and proximity to the reference point. Although all models have comparable standard deviations (between 0.24 and 0.28), there are variations in correlation. Nevertheless, as can be seen from the Taylor diagram, a realistic overlap is present among models like PSO-SVR, SVR, and DT. This overlap indicates that although ELM is the best-performing model, some other models also demonstrate competitive performance in certain situations. The inset section of the Taylor plot further emphasizes these differences, showing how ELM retains greater reliability. The model ranking, based on the Taylor plot analysis, is consistent with the

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** The cross plots for the developed models: DT (1), ELM (2), Knn (3), MLP (4), MLP-PSO (5), RF (6), SVR (7), SVR-PSO (8).</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 5. The cumulative frequency plot for the predictive IA models.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Taylor plot for the predictive IA models.</figcaption></figure>

statistical results: ELM > PSO-SVR > SVR > DT > RF > Knn > PSO-MLP > MLP. This consistency between graphical and statistical results further emphasizes the robustness of the evaluation method.

To ensure a robust evaluation of our models, we applied cross-validation to our large dataset. Cross-validation is a fundamental technique in machine learning and statistical modeling that helps estimate a model's predictive performance while mitigating the risk of overfitting. It involves repeatedly splitting the data into training and test sets, providing a more reliable assessment of model performance [49]. Our results were highly satisfactory, with correlation coefficients reaching approximately 99 % and minimal errors across all tested models, and this negates that our models suffer from overfitting problems. Additionally, these findings confirm the robustness and reliability of our models. Table A.2 in appendix section presents the detailed numerical results of this evaluation. Also, Fig. A.2 presents cross plots of experimental permeability damage ($K_{d,exp}$) and the predicted damage in permeability ($K_{d,pred}$) for the eight models, which are provided in the

annex section.

### 3.3. Sensitivity analysis based on SHAP

In boosting ensemble models, each input variable is evaluated for its significance in contributing to the predictive performance of the models. This process of quantifying feature importance serves as a critical method for enhancing interpretability, as it reveals how individual variables influence the decision-making mechanisms of the ensemble. By analyzing these contributions, practitioners gain deeper insights into the model's behavior, fostering transparency and informed model selection [50].

The SHAP (Shapley additive explanations) analysis showed substantial comprehension of the complex physicochemical mechanisms responsible for permeability reduction ($K_d$) through barite scaling. Among the variables tested, $C_{Ba}^2-$ and $C_{So}^2-$ ion concentrations consistently emerged as the best predictors for all models of machine learning,
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** SHAP interpretability and features importance analysis for the detected permeability (Kd) ELM model: Bar plot left panel and SHAP summary plots right panel.</figcaption></figure>

with the ELM model having the most predictive capability, as evidenced by its highest R² and lowest error metrics. This control is strongly rooted in the inherent stoichiometry of barite (BaSO₄) precipitation, wherein high concentrations of Ba²⁺ and SO₄²⁻ ions act in concert to create supersaturation, favoring pore-blocking crystal growth [51]. The third most important variable was temperature (T), whose effect was more multifaceted. Whereas barite solubility is generally expected to decline with rising temperature in idealized aqueous systems, the interplay of ionic interactions, pressure, and reservoir-specific thermodynamic conditions can function to invert this trend and modulate both nucleation and growth rates of barite [52]. Differential pressure (ΔP) and flow rate (Q) exerted the most powerful nonlinear influences. Specifically, high velocities of flow can facilitate mobilization of the particles and slow down deposition of crystals, whereas reducing hydrodynamic forces facilitates adhesion and scale formation within pore structures. These were also substantiated by the SHAP visualizations in Fig. 7. Systematically increased C<sub>Ba2+</sub> and C<sub>SO42-</sub> concentrations were associated with negative SHAP values, reflecting their unfavorable impact on permeability. In contrast, C<sub>inb</sub> and Cp of the barite inhibitors showed positive contributions, indicating their inhibitory role against permeability loss. Additionally, even secondary operational parameters T, Q, ΔP, and V<sub>inj</sub> were identified to affect permeability behavior, showing their potential influence in field-scale optimization. Compared to other approaches, the ELM model exhibited a better capacity to accommodate the inherent nonlinearity and multivariate nature characterizing the dataset, perhaps because of its universal approximation capability.

While the SVR and RF were equally effective (detailed results of the models can be found in Fig. A.3 in Annex/Appendix section). Their respective sensitivity to feature interactions localized in specific regions of Cp and T revealed vulnerabilities to mitigating reservoir-specific variability. These strengths endow AI-based techniques with a substantial advantage over traditional models, offering enhanced predictive accuracy, seamless management of nonlinear relationships, and indispensable capabilities for contemporary predictive modeling and decision-making. These findings underscore the need for stringent control of Ba²⁺ and SO₄²⁻ levels in water injection operations. Additionally, optimization of operational parameters and development of inhibitors with better solubility profiles (Cp) also appear to be important strategies for long-term permeability management. Apart from its predictive value, the SHAP model proposed here also offers actionable and mechanistic insights and ultimately facilitates the design of tailored scale

mitigation strategies suitable for complicated reservoir conditions.

### 3.4. Selection and valorization of the best model

After comparing all the developed models, the Extreme Learning Machine (ELM) demonstrated superior predictive performance. As a single-layer neural network, ELM offers a balanced trade-off between model simplicity and computational speed, while effectively handling regression tasks. To validate the selected model, Fig. 8 presents a composite visualization summarizing three key analyses related to ELM's reliability and generalization capacity. In Fig. 8a, the Williams plot is used to assess the applicability domain of the ELM model. Results show that 97.82 % of the data points 4892 out of 5001 fall within the acceptable leverage range (−3 ≤ R ≤ 3 and 0 ≤ H ≤ 0.06), with only 109 outliers. This wide domain of applicability suggests the model's reliability for a variety of reservoir scenarios and input combinations. In Fig. 8b, the cross-plot of experimental permeability damage versus predicted values for new, unseen data demonstrates a strong fit to the diagonal (Y = X), indicating excellent prediction accuracy. This is further supported by a high correlation coefficient of 0.99 and a low RMSE of 0.01, confirming the robustness of the ELM model under real-field conditions. To further evaluate the benefit of the data augmentation process, Fig. 8c compares ELM's predictions when trained solely on the original dataset of 552 points (without augmentation). Although the points still align reasonably along the diagonal, a clear reduction in data density is observed. This highlights the limitations of small datasets in capturing complex nonlinearities and diversity in input features. In contrast, the augmented dataset significantly improved model generalization and prediction confidence. The impact of data limitation also explains the practical overlaps observed in the Taylor diagram, as smaller datasets tend to yield converging model performances due to restricted variability. Overall, these results emphasize the ELM model's superiority and confirm the importance of incorporating data augmentation techniques when working with limited experimental data. Such enhancement not only improves the training process but also broadens the model's applicability to real-world reservoir conditions, ensuring better decision-making in field operations.

### 3.5. Modeling performance by comparing with previous studies

Compared to previous ML-based studies on scale prediction and
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Evaluation of the ELM model performance and generalization capacity: (a) Williams plot for the ELM model; (b) Prediction with new data for the ELM model; (c) Prediction with 552 data for the ELM model.</figcaption></figure>

<table><caption>**Table 4**

Comparison between our models and published models.</caption><thead><tr><td>Model</td><td>Method</td><td>Data size</td><td>R<sup>2</sup></td><td>Application</td></tr></thead><tbody><tr><td>This study</td><td>ELM</td><td>5001</td><td>0.9999</td><td>Scale prediction / Inhibition</td></tr><tr><td>Larestani et al. [3]</td><td>GBDT</td><td>431</td><td>0.9990</td><td>Scale prediction</td></tr><tr><td>Rostami et al. [15]</td><td>GEP</td><td>431</td><td>0.9800</td><td>Scale prediction</td></tr><tr><td>Foroutan and Moghadasi [13]</td><td>Feedforward Backpropagation ANN.</td><td>550</td><td>0.9700</td><td>Scale prediction</td></tr><tr><td>Zabihi et al. [5]</td><td>MLP ANN</td><td>264</td><td>0.9900</td><td>Scale prediction</td></tr></tbody></table>

inhibition, the models developed in this work exhibit superior predictive performance and enhanced generalization capability. As summarized in Table 4, several relevant studies have been reported in the literature. For instance, Zabihi et al. (2011) [5] applied Artificial Neural Networks (ANNs) to model the effect of barium sulfate scaling on reservoir rock permeability using 264 data points, achieving a correlation coefficient exceeding 0.9900 for predicting damaged permeability ($K_d$). Similarly, Foroutan and Moghadasi (2013) [13] employed an ANN-based model trained on 550 data points to estimate relative permeability during mineral precipitation, also reaching a correlation coefficient higher than 0.9700. In 2019, Rostami et al. [15] proposed a gene expression

programming (GEP) model to predict formation damage induced by mineral scaling, obtaining a correlation coefficient of 0.9800. More recently, Larestani et al. (2022) [3] developed various AI-based models for sulfate scale prediction, achieving a correlation coefficient up to 0.9990 for the GBDT model. Despite the promising results from these studies, our proposed model consistently outperforms all previously published approaches across all available statistical indicators, as evidenced by a higher correlation coefficient and a significantly lower RMSE. This superior performance is largely attributed to the use of an extensive and diverse dataset, which enhances the model's robustness and generalization capacity. Furthermore, unlike previous works, our
model is capable of predicting the effectiveness of scale inhibitors by distinguishing between the permeability observed under injection and non-injection conditions, providing valuable insights for practical scale management.

Note should be taken, however, that while AI-driven approaches present exceptional benefits such as better prediction accuracy, the capability of capturing complex nonlinear relationships and ease of use with large datasets, they have some flaws. Their largest weaknesses are that they demand massive and quality datasets, could overfit if models are not periodically tuned, and some of the AI models are difficult to interpret in terms of physical interpretation. To counter these issues and improve reliability, a worthwhile strategy is to combine AI techniques with traditional mechanistic or empirical models. The resulting hybrid models would use the predictive ability of AI with the interpretability and physical reasonableness of the classic models, particularly for prediction and prevention of barite scale formation in hydrocarbon reservoirs.

## 4. Conclusions

This study has addressed a significant research gap in the prediction and control of barite scale formation in petroleum water-flooding systems by leveraging artificial intelligence (AI). While prior works have often focused on laboratory-scale testing or limited modeling approaches, our contribution lies in the comprehensive construction, optimization, and validation of eight advanced AI models (MLP, MLP-PSO, SVR, SVR-PSO, ELM, KNN, DT, and RF) trained on a substantially augmented and diverse dataset representing real reservoir conditions and inhibitor properties.

Among these models, Extreme Learning Machine (ELM) stood out due to its remarkable performance in accuracy, speed, and generalization, achieving an R² of 0.99999 and RMSE of 0.0005. The success of the ELM model, validated through both internal cross-validation and external testing on an independent dataset, demonstrates the reliability of AI approaches for practical deployment in real oilfield environments.

Beyond statistical performance, this work confirms that AI models can serve as reliable surrogates to experimental screening by accurately predicting inhibitor efficiency, scale inhibition trends, and permeability behavior under varying operational parameters. The integration of inhibitor concentration, thermodynamic variables, and time-based injection factors further allows a nuanced interpretation of scale formation dynamics a major step forward compared to traditional empirical approaches.

The study also highlights the critical role of data augmentation in improving model robustness and reducing bias, especially in cases where complete experimental data are lacking a common issue in reservoir engineering.

However, certain limitations remain. While the current models performed well across simulated and real conditions, their transferability to

other scale types (calcium carbonate or strontium sulfate) or different operational contexts still requires further validation. Moreover, although SHAP-based interpretability was employed, future work could benefit from deeper explainability frameworks to gain more insight into model decisions.

From a broader perspective, the findings demonstrate the potential for AI models to transform oilfield scale management. When integrated into real-time monitoring systems, these models can help operators take preventive action, reducing downtime, chemical usage, and environmental risks. This also aligns with industry needs for cost-effective and sustainable reservoir management practices.

Future research should aim to expand the dataset with field-scale measurements under extreme reservoir heterogeneities, validate these models in diverse geological settings, and test their performance in predicting other mineral scale types. The adoption of hybrid models or coupling AI with mechanistic simulations could further boost accuracy and trustworthiness.

### CRediT authorship contribution statement

**Ouafa Belkacem:** Writing – original draft, Visualization, Validation, Methodology, Investigation, Conceptualization. **Ahmed Rezrazi:** Visualization, Validation, Methodology, Investigation. **Kamel Aizi:** Writing – review & editing, Writing – original draft, Resources, Methodology, Investigation, Conceptualization. **Lokmane Abdelouahed:** Writing – review & editing, Investigation. **Maamar Laidi:** Writing – original draft, Validation, Resources, Methodology, Investigation. **Abdelhafid Touil:** Writing – original draft, Visualization, Validation, Methodology, Investigation. **Leila Cherifi:** Writing – original draft, Visualization, Validation. **Salah Hanini:** Writing – original draft, Visualization, Methodology, Investigation.

### Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Lokmane Abdelouahed reports was provided by INSA Group. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

The Central Directorate of Research and Development - DCRD SONATRACH, as represented by its Central Director, Mr. Mustapha BENAMARA, is acknowledged by the authors for their support of this study. The authors also appreciate Dr. Menad NAIT AMER's advice and assistance.

## Annex/Appendix

### A1. Graphical presentation of the model's architecture
<figure><img src="image_12.png" /><figcaption>Fig. A.1. The graphical presentation of the model’s architecture: DT, Knn, MLP-PSO, SVR-PSO, MLP, ELM, RF, SVR, RF.</figcaption></figure>
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>Fig. A.1. (continued).</figcaption></figure>

## A2. Comparison of various machine learning methods used in this study

<caption>Table A.1</caption>

<table><caption>Table A.1
Comparison of various machine learning methods used in this study.</caption><thead><tr><td>Method</td><td>Type</td><td>Complexity</td><td>Training Speed</td><td>Main Applications</td></tr></thead><tbody><tr><td>MLP</td><td>Neural Network</td><td>High</td><td>Medium to Slow</td><td>Classification, Regression</td></tr><tr><td>ELM</td><td>Simple Neural Network</td><td>Medium</td><td>Very Fast</td><td>Regression, Classification</td></tr><tr><td>DT</td><td>Tree-Based Model</td><td>Medium</td><td>Fast</td><td>Regression, Classification</td></tr><tr><td>SVR</td><td>Kernel-Based Model</td><td>High</td><td>Medium</td><td>Regression</td></tr><tr><td>RF</td><td>Ensemble Model</td><td>Medium to High</td><td>Medium</td><td>Classification, Regression</td></tr><tr><td>Knn</td><td>Instance-Based</td><td>Medium</td><td>No Training</td><td>Classification, Regression</td></tr></tbody></table>
### A3. Mathematical formulation of statistical evaluation

Root mean square error (RMSE).

$$RMSE = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (k_{d \text{ exp}} - k_{d \text{ pred}})^2}$$

Correlation coefficient (R²).

$$R^2 = 1 - \frac{\sum_{i=1}^{N} (k_{d \text{ exp}} - k_{d \text{ pred}})^2}{\sum_{i=1}^{N} (k_{d \text{ exp}} - k_{d})^2}$$

Standard deviation (SD).

$$SD = \sqrt{\frac{1}{N-1} \sum_{i=1}^{N} \left( \frac{k_{d \text{ exp}} - k_{d \text{ pred}}}{k_{d \text{ exp}}} \right)^2}$$

Mean square error (MSE).

$$MES = \frac{\sum_{i=1}^{N} (k_{d \text{ exp}} - k_{d \text{ pred}})^2}{N}$$

Mean absolute error (MAE).

$$MAE = \frac{\sum_{i=1}^{N} |k_{d \text{ exp}} - k_{d \text{ pred}}|}{N}$$

Where $k_{d \text{ exp}}$ and $k_{d \text{ pred}}$ are the experimental and predicted values respectively, N is the number of data set, and $\bar{k}_d$ represent the average value of measured data.

### A4. Statistical performance evaluation for the intelligent models using cross-validation

Table A.2

<table><caption>**Table A.2**
Statistical performance evaluation for the intelligent models using cross-validation.</caption><thead><tr><th></th><th>SVR</th><th>SVR-PSO</th><th>ELM</th><th>MLP</th><th>MLP-PSO</th><th>DT</th><th>RF</th><th>KNN</th></tr></thead><tbody><tr><td>Train data</td><td>4000</td><td>4000</td><td><strong>4000</strong></td><td>4000</td><td>4000</td><td>4000</td><td>4000</td><td>4000</td></tr><tr><td>R2</td><td>0.99800</td><td>0.99989</td><td><strong>0.99998</strong></td><td>0.9966</td><td>0.99788</td><td>0.99879</td><td>0.99879</td><td>1</td></tr><tr><td>MAE</td><td>0.005</td><td>0.0006</td><td><strong>0.0003</strong></td><td>0.008</td><td>0.0069</td><td>0.001</td><td>0.0012</td><td>0</td></tr><tr><td>RMSE</td><td>0.006</td><td>0.0008</td><td><strong>0.0005</strong></td><td>0.013</td><td>0.013</td><td>0.003</td><td>0.003</td><td>0</td></tr><tr><td>MSE</td><td>5 × 10<sup>-5</sup></td><td>6.87 × 10<sup>-7</sup></td><td><strong>2.91 × 10<sup>-7</sup></strong></td><td>0.0001</td><td>0.0001</td><td>8.57 × 10<sup>-6</sup></td><td>8.51 × 10<sup>-6</sup></td><td>0</td></tr><tr><td>Test data</td><td>1001</td><td>1001</td><td><strong>1001</strong></td><td>1001</td><td>1001</td><td>1001</td><td>1001</td><td>1001</td></tr><tr><td>R2</td><td>0.99889</td><td>0.99997</td><td><strong>0.99998</strong></td><td>0.9957</td><td>0.99725</td><td>0.99852</td><td>0.99751</td><td>0.9870</td></tr><tr><td>MAE</td><td>0.005</td><td>0.0007</td><td><strong>0.0004</strong></td><td>0.008</td><td>0.0068</td><td>0.003</td><td>0.0032</td><td>0.0049</td></tr><tr><td>RMSE</td><td>0.006</td><td>0.001</td><td><strong>0.0005</strong></td><td>0.015</td><td>0.010</td><td>0.007</td><td>0.007</td><td>0.011</td></tr><tr><td>MSE</td><td>5 × 10<sup>-5</sup></td><td>2 × 10<sup>-6</sup></td><td><strong>5.1 × 10<sup>-7</sup></strong></td><td>0.0001</td><td>0.0001</td><td>6.29 × 10<sup>-5</sup></td><td>6.31 × 10<sup>-5</sup></td><td>0.0001</td></tr><tr><td>All data</td><td>5001</td><td>5001</td><td><strong>5001</strong></td><td>5001</td><td>5001</td><td>5001</td><td>5001</td><td>5001</td></tr><tr><td>R2</td><td>0.99898</td><td>0.99998</td><td><strong>0.99998</strong></td><td>0.9957</td><td>0.99696</td><td>0.99853</td><td>0.99853</td><td>0.9984</td></tr><tr><td>MAE</td><td>0.006</td><td>0.0006</td><td><strong>0.0003</strong></td><td>0.008</td><td>0.006</td><td>0.001</td><td>0.0016</td><td>0.00098</td></tr><tr><td>RMSE</td><td>0.006</td><td>0.009</td><td><strong>0.0007</strong></td><td>0.013</td><td>0.011</td><td>0.003</td><td>0.0039</td><td>0.002</td></tr><tr><td>MSE</td><td>4 × 10<sup>-5</sup></td><td>8 × 10<sup>-7</sup></td><td><strong>3.4 × 10<sup>-7</sup></strong></td><td>0.0001</td><td>0.0001</td><td>1.94 × 10<sup>-5</sup></td><td>1.94 × 10<sup>-5</sup></td><td>2.52 × 10<sup>-5</sup></td></tr></tbody></table>

### A5. The cross plots for the developed models using cross-validation
<figure><img src="image_15.png" /><figcaption>Fig. A.2. The cross plots for the developed models using cross-validation: ELM (1), SVR (2), SVR-PSO (3), Knn (4), MLP (5), MLP-PSO (6), RF (7), DT (8).</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Fig. A.3.** Global SHAP interpretability and features importance analysis for the detected permeability ($K_d$) models: Bar plot left panel and SHAP summary plots right panel: Knn (1), MLP (2), RF (3), SVR (4), MLP-PSO (5), DT (6), SVR-PSO (7).</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Fig. A.3. (continued).</figcaption></figure>
# Data availability

Data will be made available on request.

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