Surfactants Adsorption onto Algerian Rock Reservoir for Enhanced Oil Recovery Applications: Prediction and Optimization Using Design of Experiments, Artificial Neural Networks, and Genetic Algorithm (GA)

Abstract

This study investigates the adsorption of surfactants on Algerian reservoir rock from Hassi Messaoud. A new data generation method based on a design of experiments (DOE) approach has been developed to improve the accuracy of adsorption modeling using artificial neural networks (ANNs). Unlike traditional data acquisition methods, this approach enables a methodical and structured exploration of adsorption behavior while reducing the number of required experiments, leading to improved prediction accuracy, optimization, and cost-effectiveness. The modeling is based on three key parameters: surfactant type (SDS and EOR ASP 5100), concentration, and temperature. The dataset required for ANN training was generated from a polynomial model derived from a full factorial design (DOE) established in a previous study. Before training, 32 different ANN configurations were evaluated by varying learning algorithms, adaptation functions, and transfer functions. The best-performing model was a cascade-type network employing the Levenberg–Marquardt learning function, learngdm adaptation, tansig activation function for the hidden layer, and purelin for the output layer, achieving an R² of 0.99 and an MSE of 6.84028 × 10⁻⁹. Compared to DOE-based models, ANN exhibited superior predictive accuracy, with a performance factor (PF/3) of 0.00157 and the same MSE. While DOE showed a slight advantage in relative error (9.10 × 10⁻⁵% vs. 1.88 × 10⁻⁴% for ANN), ANN proved more effective overall. Three optimization approaches—ANN-GA, DOE-GA, and DOE-DF (desirability function)—were compared, all converging to the same optimal conditions (SDS at 200 ppm and 25 °C). This similarity between the various optimization techniques confirms the strength of genetic algorithms for optimization in the field of EOR and that they can be reliably applied in practical field operations. However, ANN-GA exhibited slightly better convergence, achieving a fitness value of 2.3247.

1. Introduction

Energy is a key driver of economic development, and oil remains a crucial resource to meet global demand. However, many mature reservoirs have reached a stage where conventional extraction techniques are no longer sufficient. Enhanced oil recovery (EOR) offers an advanced approach to extracting additional oil when traditional methods, such as natural reservoir pressure or water and gas injection, become ineffective [1,2]. Enhanced oil recovery (EOR) has emerged as an advanced solution to maximize oil production when traditional techniques, such as relying on natural reservoir pressure or injecting water and gas, become insufficient [3,4,5,6]. Despite these advances, a significant amount of oil remains trapped in the pores of reservoir rock due to capillary forces, which hold the oil in microscopic spaces, and its high viscosity, which impedes flow. Using gases, chemicals, and thermal energy, EOR has evolved into a diverse set of techniques designed to optimize oil displacement and improve extraction efficiency [3,7,8].

To overcome the challenge of residual oil trapped in reservoirs, enhanced oil recovery (EOR) techniques are designed to alter the physical and chemical properties of the reservoir, improving oil mobility and facilitating its flow toward production wells. These techniques include thermal, chemical, and gas-based processes, such as CO₂ injection or hydrocarbon gas injection, each tailored to the specific reservoir conditions. Among them, surfactant injection stands out as a particularly promising approach. Due to their amphiphilic nature, surfactants effectively reduce the interfacial tension between water and oil [9,10,11,12,13,14]. This reduction enhances oil dispersion in water, thereby improving mobilization and boosting extraction efficiency. This method is especially beneficial in heterogeneous or complex reservoirs, where oil is often confined in hard-to-reach zones [15,16,17,18,19,20,21].

While surfactants show great promise in enhancing oil recovery, their practical application is not without challenges. The effective use of surfactants in EOR is hindered by challenges related to their adsorption onto rock surfaces. Excessive surfactant adsorption reduces their concentration in the aqueous phase, diminishing their ability to modify interfacial properties and ultimately lowering their effectiveness in the recovery process [22,23]. Surfactant adsorption mechanisms are complex and influenced by multiple factors, including the chemical composition of the surfactant, the mineralogical properties of the rock, temperature, pH, and the ionic strength of the surrounding medium. Therefore, a deep understanding of these mechanisms and optimization of operational conditions becomes obligatory for the enhancement of the efficiency of the whole EOR process [24,25].

To overcome these challenges, modeling and optimization techniques are employed to maximize the effectiveness of surfactants in EOR applications. ANN techniques have proven particularly effective in capturing the complex, nonlinear relationships between input variables and surfactant adsorption [26]. Since these models rely directly on experimental data, they have demonstrated a strong ability to predict adsorption behavior with high accuracy [27]. Although ANN provides accurate predictions of surfactant adsorption, it does not inherently determine the optimal input conditions. The integration of GA ensures an optimal combination of parameters to maximize adsorption efficiency under given constraints. Genetic algorithms (GAs), inspired by natural selection, complement ANNs by efficiently exploring vast solution spaces to optimize surfactant performance [28]. Through key mechanisms such as selection, which identifies the best-performing solutions; crossover, which combines features of parent solutions to generate new ones; and mutation, which introduces random variations, GAs effectively navigate complex and nonlinear solution spaces [26]. When integrated with ANNs, GA-based systems form a powerful framework for both modeling and optimizing surfactant adsorption in EOR processes [29,30].

Several studies have been conducted to model enhanced oil recovery (EOR) processes using ANNs, with a particular focus on various EOR techniques, such as surfactant and polymer injection, which are designed to enhance the efficiency of these recovery methods. For instance, Ahmed F. Belhaj et al. conducted a comprehensive study on the adsorption behavior of two surfactants, alkyl polyglucoside (APG) and alkyl ether carboxylate (AEC), on rock surfaces under harsh reservoir conditions. They employed ANNs to model and predict these complex phenomena, analyzing experimental data on both individual and binary surfactant adsorption while simulating the effects of key parameters such as temperature. The ANNs showed a strong correlation with the experimental data, with an R² of 0.9915 and 0.9926 for APG and AEC, respectively, confirming their accuracy in predicting adsorption behaviors and significantly reducing the time and costs associated with experimentation [31]. Similarly, SeyedehRahaMoosavi et al. also used ANNs to forecast the final oil recovery, RF/, and injection quality (Q) during a CO₂ injection combined with surfactants. The following were the anionic Bio-Terge AS-40, the non-ionic Triton X-100, the cationic Stephantex VT-90, and the alpha-olefin sulfonate AOS C14/16 that were tested against the brine. The ANNs considered MLP and RBF were trained using 214 experimental records with six normalized input parameters, including porosity, permeability, and injected volume. These models demonstrated excellent performance, achieving an R² above 0.997 with remarkably low RMSE [32]. In addition, the work of Qian Sun et al. focused on developing ANNs to predict and optimize polymer flooding projects in the context of EOR. Two models were designed: one to predict the temporal responses of projects, such as final oil recovery (RF), and one to predict injection quality (Q). ANNs were trained with synthetic data generated from high-fidelity numerical simulations, including rock and fluid properties and injection configurations. The RBF-based model was able to produce an R² of 0.9991 and a root mean square error of 0.0092 against the injection quality (Q), while similar results were obtained for the final recovery, RF, with an R² value of 0.9991 and an RMSE of 0.0197. Both of these indicate the efficiency and swiftness with which ANNs are able to provide reliable estimates [33]. Lastly, Si Le Van et al. proposed an innovative approach to using ANNs to model and evaluate the performance of alkali–surfactant–polymer (ASP) flooding in viscous oil reservoirs. Utilizing a dataset generated from representative numerical simulations of typical projects, they optimized the ANN through a rigorous training, validation, and testing process with a 50%–20%–30% data split. The optimized network exhibited high predictive accuracy, achieving an R² of 0.986 and an RMSE of 1.63% in forecasting the performance of the studied scenarios [34].

These studies highlight the power of artificial neural networks (ANNs) in accurately predicting complex behaviors, such as surfactant adsorption, and the efficiency of enhanced recovery techniques based on experimental data.

The main objective of this study is to predict and optimize surfactant adsorption using artificial neural networks (ANNs) based on three key parameters: surfactant type (SDS and EOR ASP 5100), surfactant concentration, and temperature, with the goal of determining the equilibrium adsorption capacity (Qe). To achieve this, experimental data were generated from a polynomial model derived from a full factorial design (DOE) established in a previous study, serving as the foundation for training the ANN. To enhance the network’s performance, 32 mathematical models were evaluated, exploring various neural network architectures and function combinations, including transfer and activation functions. The optimization of operational parameters was then carried out by coupling genetic algorithms with both ANN (ANN-GA) and a full factorial design (DOE-GA). Finally, a comprehensive comparative analysis was performed between the two predictive models (ANN and DOE) and the optimization methods (ANN-GA and DOE-GA) while also incorporating optimization based on the desirability factor (DOE-DF). This analysis aimed to identify the most effective approach for predicting and optimizing surfactant adsorption in enhanced oil recovery (EOR) processes.

2. Materials and Methods

2.1. Surfactants

Sodium dodecyl sulfate (SDS), an anionic surfactant with 98% purity and a molecular weight of 288 g/mol, was supplied by Merck (Darmstadt, Germany). In parallel, EOR ASP 5100, an industrial surfactant specifically designed for enhanced oil recovery, was provided by Solvay (Brussels, Belgium). The surfactant solutions were prepared with distilled water at concentrations ranging from 200 to 800 ppm for both SDS and EOR ASP 5100. To prepare these solutions, the appropriate amounts of surfactants were dissolved in 100 mL of distilled water using a magnetic stirrer (Thermo Fisher Scientific, Waltham, MA, USA) to ensure complete homogenization. These two surfactants were used in the experiments [35].

2.2. Sample Preparation and Batch Adsorption

The rock samples from the sandstone reservoir of Hassi Messaoud, located in southern Algeria, were crushed, washed with distilled water, and dried, then sieved into 4 μm fractions to obtain a uniform size. After grinding, the samples were washed again, decanted, and dried before being used for static adsorption experiments. Batch adsorption tests were then carried out in the laboratory, where a determined volume of surfactant solution was brought into contact with the rock under agitation at 130 rpm for several hours to reach equilibrium. Based on the equilibrium concentration and the initial concentration of the surfactant, the amount of surfactant adsorbed was calculated, providing precise results with a margin of error of 4% for all experiments. To minimize experimental errors, special attention was given to preventing any contamination of the solutions and ensuring thorough homogenization of the mixtures. Knowing the equilibrium concentration and the initial concentration of the surfactant, the amount of surfactant adsorbed can be calculated as follows:

$$\mathrm{A}\mathrm{d}\mathrm{s}\mathrm{o}\mathrm{r}\mathrm{b}\mathrm{e}\mathrm{d}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}(\mathrm{Q}\mathrm{e})={\displaystyle \frac{{\mathrm{m}}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}\ast \left({\mathrm{C}}_0-\mathrm{C}\right)}{{\mathrm{m}}_{\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{e}}}}\ast {10}^{-3}$$

(1)

where Qe represents the adsorption of the surfactant on the rock surface at equilibrium (mg/g of rock), m_solution denotes the total mass of the solution in the initial volume (g), C₀ corresponds to the initial concentration of the surfactant in the solution before equilibrium (mg/L), C indicates the concentration of the surfactant in the aqueous solution after equilibrium (mg/L), and m_sandstone is the total mass of the crushed sandstone [32].

2.3. XRD and BET Analysis

X-ray diffraction (XRD) analysis was performed using a PANalytical diffractometer (PANalytical, Almelo, The Netherlands), covering a wide range of Bragg angles (2θ ranging from 2° to 70°). The collected data were processed using High Score Plus software (Version 3.0.5) and a CuKα source (λ = 1.54186 Å) to determine the semi-quantitative composition of the samples. Additionally, the specific surface area of the rock samples was determined using the Brunauer–Emmett–Teller (BET) method with an Autosorb-3b analyzer (Quantachrome, Boynton Beach, FL, USA). Nitrogen adsorption was used to measure the surface area after the sandstone powder had been pre-dried at 120 °C to remove any traces of water and adsorbed substances [35].

2.4. Modeling Methods

2.4.1. Full Factorial Design (DOE) Modeling

In this study, a full factorial design 2³ was employed, leading to a total of eight experiments to assess the effects of three factors on the equilibrium adsorption capacity (Qe) of the surfactants SDS and EOR ASP 5100 on Hassi Messaoud rock. The selected factors were surfactant type (X₁), surfactant concentration (X₂), and temperature (X₃), as these parameters play a crucial role in surfactant retention within porous media [36,37,38]. The response variable (Y) corresponds to the equilibrium adsorption capacity Qe (mg/g) (see

Table 1. Factors and levels used in the factorial design 2³ [35].

Factors	Low level (−1)	Highlevel (+1)
X1 (Surfactant choice)	SDS	EOR ASP 5100
X2 (Concentration of surfactant)	200	800
X3 (Temperature in °C)	25	80

).

To model the effects of these three factors and their interactions, the following mathematical equation was applied [39,40]:

$$y=a_0+a_1{X}_1+a_2{X}_2+a_3{X}_3+a_{12}{X}_1{X}_2+a_{13}{X}_1{X}_3+a_{23}{X}_2{X}_3$$

(2)

where y represents the response (equilibrium adsorption capacity Qe); a₀ is the overall mean; a₁, a_2, and a₃ are the coefficients for the individual effects of each factor; and a₁₂, a₁₃, and a₂₃ capture the interaction effects between the factors [40].

In a previous study, this design was developed and extensively detailed to optimize the experimental parameters [35]. In the present research, we leveraged this design to train an artificial neural network (ANN), allowing for the modeling of complex relationships between the experimental variables and the adsorption response.

2.4.2. ANN Modeling Development

Artificial neural networks (ANNs) are powerful tools for analyzing experimental datasets and modeling complex nonlinear relationships between input and output variables [41,42]. Given the intricate relationships observed in surfactant adsorption studies, ANNs were employed in this research to capture and model these complexities effectively. Its implementation follows several key steps, as illustrated in Figure 1, ranging from data preparation to the optimization of network parameters to produce accurate and reliable models [43].

Inspired by the human brain, an ANN processes information through interconnected artificial neurons. Each neuron receives data via weighted connections, where each connection has a specific weight. In this study, the input variables (X₁, X₂, and X₃) were processed by the input layer, which applied an activation function before transmitting the data to the subsequent hidden layers. These layers allow the ANN to establish intricate relationships between input variables and the adsorption response (Qe) [44,45]. The ANN layers capture nonlinear relationships by applying hierarchical transformations to the data using transfer and nonlinear activation functions. These components enable the neural network to effectively learn and model complex interactions between input and output variables.

The neurons in the hidden layers first perform a calculation, which consists of summing the weighted inputs (Equation (3)). Next, an activation function is applied to this sum. Finally, the results are sent to the next layer. This iterative process allows the network to learn complex nonlinear relationships, making it particularly well suited for modeling complex systems [46].

$$S_{ij}=W_0+\left[{\sum }_n^i{w}_{ij}\ast X_i\right]$$

(3)

where n is the number of input elements, X_i is the value of the neuron output of the previous layer, W_ij is the weight value between neuron i and neuron j, and W₀ is the bias.

To predict the system outputs, the production equation of a three-layer network (input, hidden, and output) is used. This equation combines the weights and activation functions of the neurons (Equation (4)) [46].

$${\mathit{Y}}_{\mathit{k}}={\mathit{f}}_{\mathit{k}}\left({\sum }_{\mathit{j}=1}^{\mathit{m}}{\mathit{w}}_{\mathit{j}\mathit{k}}\times {\mathit{f}}_{\mathit{i}}\left({\sum }_{\mathit{i}=1}^{\mathit{n}}{\mathit{x}}_{\mathit{i}}\times {\mathit{w}}_{\mathit{i}\mathit{j}}\right)\right)+{\mathit{w}}_0$$

(4)

where Y_k is one of the outputs of the system, f_k and f_i are successively the functions activation of the neuron k of the output layer and the neuron j of the hidden layer, W_jk and W_ij are successively the weights between the same neuron and the k-th exit neuron and the weight between the i-th neuron and the j-th neuron, W₀ is the bias. Weights and biases are iteratively adjusted during the training process to minimize the error between predicted and actual outputs [47]. The general structure of the ANN, consisting of input, hidden, and output layers, is shown in Figure 2. Each layer’s neurons establish relationships between the input variables and the output response.

In this study, we explored various neural network architectures and tested multiple function combinations to optimize model performance. Specifically, Feed Forward and Cascade Forward networks were evaluated alongside different learning algorithms and transfer functions to determine the most effective configurations. A summary of the tested architectures, algorithms, and functions is provided in

Table 2. Neural network types, algorithms, and functions for modeling and optimization.

Network Types	Feed Forward Backpropagation
	Cascade Forward Backpropagation
Training function	Trainscg
	TrainLM
Adaptation learning function	learngd
	learngdm
Transfer function for Hiden	tansig
	logsig
Transfer function for output layer	tansig
	purelin
Number of neurons in the hidden layer	12

[48].

The performance of Feed Forward Backpropagation and Cascade Forward Backpropagation was evaluated using the mean squared error (MSE) (Equation (5)) and the correlation coefficient (R) (Equation (6)) [49,50,51].

$$\mathrm{M}\mathrm{S}\mathrm{E}={\displaystyle \frac{1}{\mathrm{Q}}}{{\sum }_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{Q}}({\mathrm{y}}_{\mathrm{i},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}-{\mathrm{y}}_{\mathrm{i},\mathrm{e}\mathrm{x}\mathrm{p}})}^2$$

(5)

$$R^2=1-{\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{Q}}{\left({\mathrm{y}}_{\mathrm{p},\mathrm{i}}-{\mathrm{y}}_{\mathrm{e}\mathrm{x}\mathrm{p},\mathrm{i}}\right)}^2}{{\sum }_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{Q}}{\left({\mathrm{y}}_{\mathrm{p},\mathrm{i}}-{\mathrm{y}}_{\mathrm{m}}\right)}^2}}$$

(6)

where Q is the number of data points, y_p represents the network prediction, y_exp is the experimental response, y_m is the mean of the experimental values, and i is the data index.

2.5. Data Generation for Network Training

To develop an effective artificial neural network (ANN) model, it is essential to identify the input and output variables, define the experimental space, and collect sufficient data to enable reliable learning and avoid errors related to overfitting.

However, practical constraints such as time and budget often limit the number of experiments that can be conducted. To address these challenges and maximize data acquisition within these constraints, we employed a design of experiments (DOE) approach to develop a polynomial model. This model enabled the generation of a sufficient dataset for training the ANN. The methodology follows a systematic structure, as outlined in Figure 3, ensuring robust data preparation for the ANN training process.

This approach aims to minimize laboratory-scale manipulations while accurately modeling experimental responses. It begins with the development of an experimental design to establish a polynomial model that predicts responses based on predefined factors. These predicted responses must closely align with experimental results, achieving a coefficient of determination (R²) of at least 99%. Once validated, the model generates input variables based on defined factor levels, which are then processed through the polynomial model to obtain corresponding output variables. The compiled results are subsequently used to train the artificial neural network (ANN) via the nntool in MATLAB (version: R2015a)

In this study, we applied this approach using a full factorial design (DOE); however, any experimental design capable of generating a polynomial model could be employed to implement this method.

2.6. GA Optimization

The genetic algorithm (GA) model is based on Darwinian evolutionary theory, where a population of solutions evolves to reach an optimal solution. In this study, two distinct approaches are used to optimize surfactant adsorption at equilibrium [52,53,54,55]: ANN-GA and DOE-GA. The ANN-GA approach combines an artificial neural network with the genetic algorithm, where the trained ANN model serves as the fitness function to guide the optimization. The DOE-GA approach, on the other hand, employs a full factorial design, using the genetic algorithm to optimize the results predicted by the polynomial model derived from the experimental design [56].

The evolutionary process in the genetic algorithm (GA), utilized in both the ANN-GA and DOE-GA approaches, is driven by three key mechanisms: selection, which identifies the fittest individuals; crossover, which generates new solutions by combining parent characteristics; and mutation, which introduces diversity to prevent premature convergence. These mechanisms play a crucial role in optimizing surfactant adsorption. The experimental data establish constraints for the input variables by defining their minimum and maximum values. While there is no universal method for configuring GA parameters, key factors such as population size, mutation rate, selection method, and crossover type were fine-tuned based on the collected data [57,58]. This adjustment ensures efficient convergence and minimizes error in both optimization approaches. The hybrid optimization, combining the design of experiments (DOE), artificial neural networks (ANNs), and genetic algorithms, is illustrated in Figure 4.

The computer codes for optimizing genetic algorithms were developed using MATLAB. Specific constraints were imposed during the optimization process: in both the ANN-GA and DOE-GA approaches, X₁ was fixed at −1 (representing the surfactant SDS) or +1 (representing EOR ASP 5100), while X₂ and X₃ were allowed to vary between −1 and +1. The parameter values used are listed in

Table 3. Parameter settings for ANN-GA and DOE-GA.

Population size	50
Number of generation	200
Crossover fraction	0.8
Mutation rate	0.01

.

3. Results and Discussion

3.1. Characterization Analysis

The XRD pattern shown in Figure 5 presents the X-ray diffraction profile for the Hassi Messaoud rock, with a dominant peak around 26° (2θ), which indicates a high concentration of quartz (SiO₂), consistent with a quartz content of 84.64% in the chemical analysis. This suggests a predominantly siliceous surface, which is expected to influence the adsorption behavior of surfactants due to the electrostatic interactions between the surfactant molecules and the mineral surface. The presence of secondary minerals such as kaolinite and illite may also contribute to adsorption through additional surface charges and specific interactions. Other lower-intensity peaks correspond to the presence of secondary minerals such as kaolinite, illite, siderite, and halite. Regarding the BET analysis, the specific surface area of the rock, measured under a nitrogen environment, is 5.71 m²/g, which provides insight into the available adsorption sites. A larger surface area generally favors higher adsorption capacities, as it increases the contact between the surfactant molecules and the rock surface. These factors collectively impact the efficiency of surfactant adsorption, which is crucial for enhanced oil recovery applications. These analyses were published in a previous work [35].

3.2. Prediction with Factorial Design of Experiments

A full factorial experimental design was used to collect the data necessary for training the artificial neural network (ANN). This approach was based on the work developed by Lebouachera et al., who initially focused on analyzing the effects of factors on responses. In our study, we adapted their polynomial model to generate the data required for ANN training. Once optimized, this approach aims to improve both the accuracy and efficiency of the ANN training process [35]. We applied the polynomial model developed by Lebouachera et al. to express the relationship between the input variables and the measured response, as shown in the following:

$$\begin{gathered} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{Q}}_{\mathrm{e}}\left({\displaystyle \frac{\mathrm{m}\mathrm{g}}{\mathrm{g}}}\right)=4.8647625+(0.8340125\ast {\mathrm{X}}_1)+(1.7596875\ast {\mathrm{X}}_2)+(0.7846875\ast {\mathrm{X}}_3)+ \\ (0.6303375\ast {\mathrm{X}}_1\ast {\mathrm{X}}_2)+(0.0066375\ast {\mathrm{X}}_1\ast {\mathrm{X}}_3)+(0.2054625\ast {\mathrm{X}}_2\ast {\mathrm{X}}_3) \end{gathered}$$

(7)

According to Lebouachera et al., the model exhibits a very high coefficient of determination (R²). The high R² value of 99.58% and the adjusted R² of 99.28% indicate that the model effectively correlated the response with the studied parameters [35]. While the high R² (99.58%) and adjusted R² (99.28%) values indicate a strong correlation between the model and the experimental data, such a high level of fit can sometimes suggest overfitting, where the model captures not only the true relationship but also noise or minor fluctuations in the data. To mitigate this risk, we relied on an adjusted R² value, which accounts for the number of predictors and provides a more reliable measure of the model’s generalization capability. Additionally, cross-validation techniques or external validation with independent data can be considered to further assess the model’s robustness. In our case, the ANN training process, which follows the polynomial model, helps in refining the predictive accuracy while reducing overfitting risks.

Subsequently, it was essential to generate values within the defined study range. To achieve this, all input parameter values were carefully selected within the appropriate intervals, as shown in Table 1. One of the main challenges encountered was that the polynomial model used to predict and describe the relationships between factors and responses was expressed in coded values. This introduced complexity in both manipulation and interpretation. To generate a usable dataset and extract modeled responses from the polynomial equation, we developed a specific conversion formula (Equation (8)) to transform uncoded values into coded values:

$${\mathrm{X}}_{\mathrm{c}}={\mathrm{x}}_{\mathrm{m}\mathrm{i}\mathrm{n}}+\left[\left({\mathrm{x}}_{\mathrm{n}}-{\mathrm{x}}_{\mathrm{i}}\right)\times {\displaystyle \frac{{\mathrm{I}}_{\mathrm{c}}}{{\mathrm{I}}_{\mathrm{n}}}}\right]$$

(8)

where X_c is the value in coded units, X_min is the minimum value in coded units, X_n is the chosen value in uncoded units, Xi is the initial value in uncoded units, I_c is the interval between coded units, and I_n is the interval between uncoded units.

By converting the values, we generated datasets that accurately reflected the experimental conditions, ensuring consistency with the polynomial model. This approach enabled us to obtain reliable and precise responses based on the experimental design. The final results are presented in

Table 4. Data collection: generated input variables and polynomial-predicted responses.

N	Input						Output
1	Surfactant		Concentration of surfactant		Temperature		Qe
2	uncoded value	coded value	uncoded value	coded value	uncoded value	coded value
3	SDS	−1	200	−1	25	−1	2.3288125
4	EOR ASP 5100	1	200	−1	25	−1	2.7228875
5	SDS	−1	800	1	25	−1	4.1765875
6	EOR ASP 5100	1	800	1	25	−1	7.0920125
7	SDS	−1	200	−1	80	1	3.4739875
8	EOR ASP 5100	1	200	−1	80	1	3.8946125
9	SDS	−1	800	1	80	1	6.1436125
10	EOR ASP 5100	1	800	1	80	1	9.0855875
11	SDS	−1	220	−0.93333333	26	−0.96363636	2.41172445
12	EOR ASP 5100	1	220	−0.93333333	26	−0.96363636	2.89032718
13	SDS	−1	240	−0.86666667	28	−0.89090909	2.51745014
14	EOR ASP 5100	1	240	−0.86666667	28	−0.89090909	3.08106332
15	SDS	−1	260	−0.8	30	−0.81818182	2.62516818
16	EOR ASP 5100	1	260	−0.8	30	−0.81818182	3.27379182
17	SDS	−1	280	−0.73333333	32	−0.74545455	2.73487859
18	EOR ASP 5100	1	280	−0.73333333	32	−0.74545455	3.46851268
19	SDS	−1	300	−0.66666667	34	−0.67272727	2.84658136
20	EOR ASP 5100	1	300	−0.66666667	34	−0.67272727	3.66522591
21	SDS	−1	320	−0.6	36	−0.6	2.9602765
22	EOR ASP 5100	1	320	−0.6	36	−0.6	3.8639315
23	SDS	−1	340	−0.53333333	38	−0.52727273	3.075964
24	EOR ASP 5100	1	340	−0.53333333	38	−0.52727273	4.06462945
25	SDS	−1	360	−0.46666667	40	−0.45454546	3.19364386
26	EOR ASP 5100	1	400	−0.33333333	44	−0.30909091	4.6786775
27	SDS	−1	420	−0.26666667	46	−0.23636364	3.55863764
38	EOR ASP 5100	1	420	−0.26666667	46	−0.23636364	4.88734491
69	EOR ASP 5100	1	760	0.86666667	67	0.52727273	8.28126423
70	SDS	−1	770	0.9	69	0.6	5.62494475
71	EOR ASP 5100	1	770	0.9	69	0.6	8.43554225
72	SDS	−1	780	0.93333333	71	0.67272727	5.737231
73	EOR ASP 5100	1	780	0.93333333	71	0.67272727	8.59081645
74	SDS	−1	785	0.95	73	0.74545455	5.82913822
75	EOR ASP 5100	1	785	0.95	73	0.74545455	8.70470038
76	SDS	−1	790	0.96666667	75	0.81818182	5.92154352

.

A total of 76 experiments were generated. Of these, 70% were used for training, 15% for validation, and 15% for testing. All input and output data were then integrated into the “nntool” toolbox of MATLAB, which is designed for developing and analyzing artificial neural networks.

3.3. ANN Modeling

After generating a dataset based on the predictions of our polynomial model, we used 76 data points to train an artificial neural network. Two ANN architectures were explored: Feed Forward Backpropagation and Cascade Forward Backpropagation [42]. To optimize training, we tested two training functions (trainlm and trainscg) alongside two learning adaptation functions (learngd and learngdm). The hidden layer utilized tansig and logsig transfer functions, while the output layer incorporated purelin and logsig, as detailed in Table 2. This approach resulted in 32 distinct models, each with a unique combination of algorithms and functions (Table 2). To ensure consistency, the number of neurons in the hidden layer was fixed at 12 across all models [46,59].

According to

Table 5. ANN algorithm results.

Network No.	Network Type	Training Function	Adaptation Learning Function	Transfer Function		Output (Qe)
				Hidden Layer	Output Layer	MSE	R2
1	Feed Forward Backpropagation	trainlm	learngd	tansig	purelin	0.00192256	0.999718645
2					tansig	0.049768699	0.992988724
3				logsig	purelin	0.000654851	0.996762602
4					tansig	0.000104353	0.999984644
5			learngdm	tansig	purelin	0.013302013	0.998028686
6					tansig	0.001047364	0.999845396
7				logsig	purelin	0.020678007	0.997024576
8					tansig	0.01179353	0.998244096
9		trainscg	learngd	tansig	purelin	0.010965378	0.998402231
10					tansig	0.02787814	0.995591963
11				logsig	purelin	0.059602925	0.991226048
12					tansig	0.031101368	0.995437096
13			learngdm	tansig	purelin	0.001270703	0.999812559
14					tansig	0.014380639	0.997867883
15				logsig	purelin	0.014368615	0.997889598
16					tansig	0.083067505	0.987482642
17	Cascade Forward Back propagation	trainlm	learngd	tansig	purelin	0.000309774	0.999954487
18					tansig	0.032258779	0.995362984
19				logsig	purelin	0.004887721	0.99927919
20					tansig	9.10106 × 10−6	0.999998662
21			learngdm	tansig	purelin	8.90312×10−7	0.999999869
22					tansig	0.000254115	0.9999626
23				logsig	purelin	0.001304255	0.999809106
24					tansig	0.000634489	0.99990587
25		trainscg	learngd	tansig	purelin	0.004647879	0.999318984
26					tansig	0.025080224	0.996315409
27				logsig	purelin	0.004459436	0.99934599
28					tansig	0.015544225	0.997674503
29			learngdm	tansig	purelin	0.191194234	0.970380954
30					tansig	0.00494906	0.999252877
31				logsig	purelin	0.042068385	0.993766037
32					tansig	0.009195451	0.998631633

, which summarizes the different algorithms used for ANN modeling of equilibrium adsorption capacity along with the corresponding MSE and R² values, the cascade-type neural network emerged as the most effective configuration. This model employed the Levenberg–Marquardt learning function with learngdm adaptation, tansig as the hidden layer transfer function, and purelin for the output layer. It achieved an impressive MSE of 8.90312 × 10⁻⁷ and an R² of 0.99, demonstrating exceptional accuracy. These results highlight the robustness of the Levenberg–Marquardt algorithm in minimizing error and the adaptability of the learngdm function, making this approach highly suitable for precise predictions in adsorption studies.

The next step in optimizing the functions and network architecture involves determining the optimal number of hidden neurons [60]. To strike a balance between computational efficiency and model complexity, we evaluated a range from 1 to 12 neurons.

Table 6. Effect of the number of hidden neurons on the ANN model performance.

Number of Neurons	MSE	R2
1	0.009853974956	0.998539413722
2	0.005177182735	0.999239763890
3	0.000036998429	0.999994569068
4	0.000000006840	0.999999998995
5	0.000000008452	0.999999999611
6	0.000000087957	0.999999987078
7	0.000003739225	0.999999450701
8	0.000000668454	0.999999901797
9	0.000000171151	0.999999974857
10	0.000000890312	0.999999869212
11	0.283142838389	0.958860619811
12	0.061986921414	0.990618200261

presents the impact of the number of hidden neurons on the ANN model’s performance. Identifying the optimal configuration is expected to enhance predictive accuracy while minimizing the risk of overfitting.

The optimal architecture for the artificial neural network (ANN) model was found to have four neurons in the hidden layer. This configuration achieved an MSE of 6.84028 × 10⁻⁹ and an R² of 0.99, demonstrating excellent predictive accuracy. The selection of 4 neurons was based on a performance evaluation of architectures ranging from 1 to 12 neurons. Beyond four neurons, the model could show signs of overfitting, while fewer neurons could lead to underfitting. This configuration provided the best balance between accuracy and generalization. Figure 6 presents the schematics of the optimal neural networks used in this study, generated using MATLAB.

Figure 7 presents the correlation between the model’s predicted outputs and the target values for the training, validation, and test phases, as well as for the complete dataset. The graph shows that the correlation coefficient (R) remains consistently close to 1 across all phases of training, validation, and testing, as well as for the overall dataset. This high correlation signifies an excellent alignment between the model’s predictions and the actual values, indicating that the model successfully captures the underlying relationships between the input variables and the target values, thereby ensuring reliable and accurate predictions [61,62].

Figure 8, which illustrates the performance graph, depicts the progression of the mean squared error (MSE) as a function of the number of training epochs. The error gradually decreases for the training, validation, and test sets, indicating continuous model improvement. The stabilization of the green curve (validation) after epoch 386 suggests that the model reached its optimal validation point at this stage, with an MSE of 1.9825 × 10⁻⁸. After 386 epochs, the ANN model reaches its stabilization point, indicating that the network has correctly learned the relationships between the inputs and the output. This point ensures an optimal balance between learning and the model’s ability to predict the data accurately without overfitting [62,63].

The error histogram in Figure 9 illustrates the distribution of the differences between the predicted and actual values for the artificial neural network model across the training, validation, and test datasets. Most errors are concentrated around zero, indicating strong model performance and predictions that closely match the actual targets. The maximum error is approximately 0.001957, while the minimum error is around 0.08264, demonstrating that even the largest deviations remain relatively low. These results suggest that the model is well-fitted and generalizes effectively to unseen data, as evidenced by the low errors observed in both the validation and test sets [63].

• Comparison of the developed DOE and ANN models

A comparative analysis was conducted between the two developed models of artificial neural networks (ANNs) and the full factorial design of experiments (DOE) to model the adsorption of surfactants on Algerian rock. Figure 10 illustrates the performance comparison for predicting the equilibrium adsorption capacity (Qe) based on both predicted and experimental responses. The results indicate a minimal difference between the two methods, highlighting their reliability in accurately modeling surfactant adsorption.

To enhance this comparative analysis, several statistical indicators were utilized, including the performance factor (PF/3) (Equation (10)), relative error (RE) (Equation (9)), coefficient of determination (R²) (Equation (6)), and mean squared error (MSE) (Equation (5)). These metrics provide a precise evaluation of each model’s reliability and consistency in predicting adsorption behavior relative to the experimental data [64].

$$RE={\displaystyle \frac{1}{n}}\sum {\displaystyle \frac{{\left|E_i-P_i\right|}_i}{E_i}}\times 100$$

(9)

$${\displaystyle \raisebox{1ex}{$PF$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}={\displaystyle \frac{100\left[\left(\gamma -1\right)+V_{AB}+\frac{CV}{100}\right]}{3}}$$

(10)

where n represents the number of data sets used, E represents the actual experimental data, and P represents the data predicted by the model. The PF/3 is a composite metric that integrates three different statistical measures, as follows:

Coefficient of Variation (CV): Measures the relative variability of the data with respect to the mean.

$$CV={\displaystyle \frac{\sqrt{\frac{1}{n}\sum (P_i-E_i(\frac{\sum P_i{E}_i}{\sum P_i^2}){)}^2}}{\overline{P}}}\times 100$$

(11)

Gamma Factor (γ): An indicator of the model’s fit quality, which can vary depending on the context.

$${log}_{10}\left(\gamma \right)=\sqrt{{\displaystyle \frac{1}{n}}\sum {\left({log}_{10}\left({\displaystyle \frac{P_i}{E_i}}\right)-\overline{{log}_{10}\left({\displaystyle \frac{P_i}{E_i}}\right)}\right)}^2}$$

(12)

Variance of Absolute Bias (V_AB): Assesses the dispersion of absolute biases between the experimental data and the model’s predictions.

$$V_{AB}=\sqrt{{\displaystyle \frac{1}{n}}\sum \left({\displaystyle \frac{{\left(P_i-E_i\sqrt{\frac{\sum P_i/E_i}{\sum E_i/P_i}}\right)}^2}{E_i{P}_i\sqrt{\frac{\sum P_i/E_i}{\sum E_i/P_i}}}}\right)}$$

(13)

Table 7. Comparison of the statistical indicators for DOE and ANN.

	DOE	ANN
$\gamma$	1.00014	1.000016
CV	0.010022	0.00165
VAB	0.000132	0.000015
PF/3	0.01212	0.00157
Coefficient of determination (R2)	0.99	0.99
Relative error (%)	9.10927 × 10−5	1.87966 × 10−4
MSE	2.37656 × 10−7	6.84028 × 10−9

presents a comparative analysis of the statistical indicators for the full factorial design of experiments (DOE) and artificial neural networks (ANNs). Both methods achieve a high coefficient of determination (R² = 0.99), indicating a strong fit to the experimental data. However, while the DOE exhibits slightly better accuracy in terms of relative error, the ANN outperforms in key metrics such as the performance factor (PF/3 = 0.001570579) and mean squared error (MSE = 6.84028 × 10⁻⁹). These results highlight ANN’s superior ability to model complex relationships with higher precision, making it a more reliable choice for predicting surfactant adsorption behavior.

3.4. GA Optimization

Three distinct approaches were employed to optimize the adsorption process: ANN coupled with a genetic algorithm (ANN-GA), DOE combined with a genetic algorithm (DOE-GA), and DOE integrated with a numerical method based on the desirability function (DOE-DF). For optimization using ANN-GA, the final trained ANN model was used as the fitness function. In the case of DOE-GA optimization, the fitness function was formulated as a polynomial equation, where the response Qe is calculated as a function of the variables X₁, X₂, and X₃.

Figure 11 and Figure 12 illustrate the evolution of fitness values over generations for the DOE-GA and ANN-GA models. In both cases, significant improvements in fitness were observed during the initial generations, followed by rapid stabilization. Both models reached convergence after approximately 20–30 generations, indicating the efficiency of the genetic algorithm. The small difference between the best fitness value and the average fitness further confirms the reliability of both methods.

For both optimization methods, the optimal adsorption conditions were obtained using sodium dodecyl sulfate (SDS) at a concentration of 200 ppm and a temperature of 25 °C (uncoded values). Additionally, a third optimization approach, developed by Lebouachera et al. [35], utilized the desirability function of Derringer and Suich (1980), a widely applied method for multi-response optimization in industrial applications [65]. This method focused on minimizing the adsorption capacity (Qe). According to the results obtained from the prediction profiler function, the optimal conditions were found to be a surfactant concentration of 200 ppm and a temperature of 25 °C, yielding a predicted Qe of 2.3289 mg/g with a desirability value of 0.942118.

• Comparison between DOE-GA, ANN-GA, and DOE-DF

The comparison of the three optimization methods—DOE-GA, ANN-GA, and DOE-DF—reveals that they all converge toward the same optimal adsorption conditions: an SDS concentration of 200 ppm and a temperature of 25 °C. As shown in

Table 8. Comparison of optimization methods for surfactant adsorption in EOR.

Optimization Methods	DOE-GA	ANN-GA	DOE-DF
Optimum conditions	SDS, 200 ppm, and 25 °C
Predicted response	2.32889	2.3247	2.3289

, the predicted values for each approach are remarkably close: 2.32889 for DOE-GA, 2.3247 for ANN-GA, and 2.3289 for DOE-DF. This similarity in results confirms the effectiveness of genetic algorithms (GAs) in optimizing input parameters, regardless of the model used.

Given this convergence, our choice fell on ANN-GA, not due to a significant difference in optimization but rather because of its superior predictive capability. Although the response predicted by ANN-GA is slightly lower than those obtained with DOE-GA and DOE-DF, this difference remains minimal and does not diminish the efficiency of the other methods. Thus, all approaches remain viable, and the choice of one over another depends on the specific requirements of the studied problem.

4. Conclusions

This study successfully achieved its primary objective of predicting and optimizing the adsorption of surfactants on an Algerian reservoir rock from Hassi Messaoud. Artificial neural networks (ANNs) were employed to model the adsorption process, focusing on three key parameters: surfactant type (SDS and EOR ASP 5100), surfactant concentration, and temperature. Following the prediction phase, operational parameters were optimized using genetic algorithms (GAs).

The prediction using ANN was conducted in several stages. First, data were generated from a polynomial model established in a previous study. This dataset enabled effective training of the neural network, leading to precise predictions of adsorption responses.

The network’s performance was then evaluated by testing 32 mathematical models. The best-performing model employed the cascade-type neural network, the Levenberg–Marquardt learning function with learngdm adaptation, tansig as the hidden layer transfer function, and purelin for the output layer. With an optimal configuration of four neurons in the hidden layer, this model achieved an exceptional MSE of 6.84028 × 10⁻⁹ and an R² of 0.99 without overfitting, as confirmed by performance analysis on training, validation, and test sets.

It sounds like all three optimization methods, ANN-GA, DOE-GA, and DOE-DF, converged on the same optimal conditions, which reinforces the robustness of the findings. The minimal differences in predicted adsorption values suggest that all approaches are valid, though ANN-GA demonstrated slightly lower predictions. Given that ANN-GA and DOE-GA achieved rapid convergence within 20–30 generations, it highlights the efficiency of the genetic algorithm in optimizing surfactant adsorption.

Artificial neural networks (ANNs) trained using the design of experiments (DOE) approach have proven highly effective in capturing complex, nonlinear relationships with remarkable accuracy. Their adaptability to experimental data makes them a powerful predictive tool for adsorption processes, significantly reducing both time and laboratory costs. In terms of optimization, each method demonstrated distinct advantages. ANN-GA excelled in handling intricate relationships with superior precision and adaptability, while DOE-GA and DOE-DF provided mathematically structured solutions that are more straightforward and easier to implement in practical applications.

The results obtained in this study provide valuable insights for the oil industry, particularly in optimizing surfactant adsorption processes in EOR. This approach not only enhances the accuracy of adsorption predictions but also facilitates the optimization of operating conditions for key influencing parameters. Moreover, it significantly reduces the reliance on costly and time-consuming laboratory experiments by offering a systematic and efficient predictive framework.

However, the main limitation of ANN and ANN-GA models lies in their dependence on specific reservoir conditions. Variations in key factors such as rock type or other reservoir characteristics require retraining the model, rendering previous versions obsolete and limiting their direct applicability to different geological settings. Additionally, while ANN or ANN-GA excels in prediction and identifying optimal conditions, despite techniques designed to improve the interpretability of ANNs, such as SHAP (Shapley Additive Explanations), saliency maps, and LIME (Local Interpretable Model-agnostic Explanations), artificial neural networks (ANNs) are often perceived as “black boxes” that are difficult to exploit due to the complexity of their internal mechanisms, making it difficult to interpret results in a physically meaningful way. This lack of transparency presents a significant challenge when a deeper understanding of the underlying mechanisms is essential.