A novel hybrid CCD-ML approach for predicting permeability alterations in carbonate reservoir rocks during waterflooding under scale inhibitor treatment

Abstract

Scale formation during waterflooding presents a persistent challenge to reservoir productivity, frequently resulting in significant permeability reduction. This study introduces an innovative hybrid modeling approach combining response-surface-methodology (RSM) and machine-learning (ML) techniques to predict and optimize permeability loss in carbonate reservoirs subjected to operational variables in the presence of the scale inhibitor DTPMP. A series of 45 coreflood experiments, designed using Central-Composite-Design of RSM (CCDRSM), were performed on carbonate core samples to evaluate the effects of key variables—inhibitor dosage, temperature, sulfate concentration, pore volume, and injection rate—on the permeability ratio (K_d/K_i). The synthetic formation water was rich in calcium and chloride ions, while the injection water contained variable sulfate concentrations (1000–5000 ppm) to induce controlled calcium sulfate scaling. Experimental data were used to develop an RSM-based statistical model and six ML-based models, including Linear-Regression (LR), Support-Vector-Regression (SVR), Gaussian-Process-Regression (GPR), Regression-Tree (RT), Random-Forest (RF), and LSBoost. Among the developed CCD-RSM and CCD-ML models, the CCD-MLGPR exhibited superior accuracy (R² = 0.9991; RMSE = 0.0056), positioning it as the most robust and reliable predictive tool. Sensitivity analysis further revealed that pore volume and inhibitor dosage were the most influential variables affecting K_d/K_i, with pore volume exerting a negative impact and inhibitor dosage a positive one. Furthermore, integration of the MLGPR model with Particle Swarm Optimization (PSO) enabled the identification of optimal operational strategies that effectively minimized formation damage, maintaining K_d/K_i above 0.90 even under worst-case conditions. These findings demonstrate the efficacy of the proposed CCD-ML hybrid methodology as a predictive and optimization tool, offering practical insights for proactive scale control and improved reservoir performance in waterflooding applications.

Introduction

The main reason for the precipitation of inorganic salts in near-wellbore region and well is the supersaturation of formation water with poorly soluble salts of alkaline earth metals, in particular, calcium carbonate and sulfate, magnesium, barium and strontium salts¹. These salts can appear during water flooding into the reservoir in order to maintain the reservoir pressure. As a result of supersaturation, salt solutions lose stability and release poorly soluble carbonates and sulfates of alkaline earth metals, which form into a polydisperse sediment that settles in the bottomhole zone and production equipment². Over time, precipitation turns into solid deposits, which lead to a decrease in rock permeability, production rate and damage of production equipment^3–6.

The most important method of salt control in the petroleum industry is the injection of scale inhibitor into the reservoir and well at the required concentration and volume^7–9. The phosphorus-containing compounds are the wide-used reagents for scale inhibition. They have various mechanisms for prevention of salt formation and growth^10,11. The scale inhibitors must be fully compatible with the formation water without the formation of sediment. The protection period of scale inhibitor is mainly dependent of the reservoir condition, pH, the ionic content of the water and adsorption capability of the reagent^12–14. Diethylene triamine pentamethylene phosphoric acid (DTPMP) is usually used for inhibition of calcium carbonate, calcium sulfate and barium sulfate in the oilfield. In the work¹⁵, the injection of DTPMP to the core sample for analysis of calcium carbonate prevention was examined. The authors observed that at the optimum concentration of DRPMP the reagent adsorption occurs at the maximum level. DTPMP had a high inhibition efficiency for calcium carbonate prevention under static and dynamic conditions.

The effectiveness of technologies to combat salt deposition during oil production depends on the accuracy of the forecast of salt deposition, and this requires an understanding of the physicochemical processes and reasons that cause the formation and deposition of salts in various conditions^16–18. Solving this problem, in turn, is impossible without establishing the exact chemical composition of the formation and injection waters, the chemical composition of the reservoir rocks and the reservoir pressure and temperature. All this data is used to predict possible scaling using various calculation methods that allow assessing the stability of solutions under reservoir conditions^19–22.

The formation damage and reduction in rock permeability is one of the main disadvantages of the water flooding process^23–25. Formation damage refers to the impairment of reservoir productivity or injectivity caused by alterations in the porous medium due to physical, chemical, or mechanical processes occurring during operations such as water flooding²⁶. One of the primary manifestations of formation damage is the reduction in rock permeability, which occurs when pore throats are partially or completely blocked by scale precipitation, fines migration, clay swelling, or other plugging mechanisms^27–29. The reduction in rock permeability is dependent on the type of precipitated salt, injection conditions, pressure, temperature, ionic content of mixture of injection and formation water and initial values of the rock properties such as porosity and permeability. The solubility and amount of calcium sulfate scale deposition in the near wellbore region is strongly dependent on the changes in the pressure and temperature^21,30,31. A sharp drop in pressure, as well as an increase in temperature above 80 °C or a decrease below 30 °C can cause supersaturation of solutions leading to calcium sulfate formation. An increase in temperature above 50 °C promotes the deposition of anhydrite, which occurs in the bottomhole zone of production wells^32–34.

The prediction of the amount of scale precipitation in the core samples under dynamic conditions through analysis of rock permeability can help to better manage the salt formation in the water flooding way. Various models and methods are developed to predict the rock permeability in different processes^35–38. In recent years, artificial intelligence and Machine Learning (ML) methods have been actively used to solve practical problems^39,40. From the very beginning to the present day, the petroleum industry has been at the forefront of developing and implementing these methods to optimize processes. Machine Learning (ML) approaches, after successful training, represent an alternative way to obtain reliable and more accurate results for the required parameter⁴¹.

Okon et al. used ML for model development in order to predict the rock properties such porosity and permeability⁴². By comparing the predicted and real field data, they concluded that ML is a robust way for prediction of reservoir properties. They observed a 0.98 coefficient of determination between real and predicted value of rock permeability. The model developed by ML has lower mean square error than existing models in the literature for permeability prediction. In the work⁴³, an ML model was developed for determination of oil recovery value in water flooding of oil reservoirs. They analyzed the accuracy of the developed model by error metrics and model runtime. The authors mentioned that the suggested model can predict the oil recovery of water flood processes very fast and can be used prior to real operation in the oil field. Abdel Azim and Aljehani tried to predict the rock permeability using well log data through ML⁴⁴. The accuracy of ML model for the 500 real data of rock permeability was about 98%. The authors observed that the mean square error of the developed model for permeability prediction was less than the already developed correlation of “dual water”. Tian et al. applied the combination of ML and genetic algorithm to predict the rock permeability⁴⁵. They observed that the hybrid of ML and genetic algorithms can increase the performance of rock permeability prediction. In this case, R-squared between predicted and calculated values of rock permeability was about one (0.998). They emphasized that this hybrid idea can be used for the accurate evaluation of the reservoir rock properties. Erofeev et al. evaluated various ML models for predicting rock properties including permeability³⁶. They obtained accurate models that made it possible to determine the most influential parameters on the change in porosity and permeability. The authors emphasized that different algorithms should be used for different conditions to select the most suitable one, as a result of which the number of experimental runs and time can be reduced while maintaining high accuracy of the conditions. Tran et al. used ML to predict the carbonate formation permeability from mercury injection capillary pressure tests⁴⁶. The authors mentioned that the performance of ML is strongly dependent amount of used data and their range. The used model could with a high accuracy predict the permeability, in which R² between predicted and real data was 0.96. Mahdaviara et al. analyzed rock permeability values (0.1 to 10 mD) from different oil fields in EOR operations using ML for low permeability oil reservoirs⁴⁷. They obtained models for predicting permeability, using which they were able to determine the most suitable methods for enhancing oil recovery. The proposed models have proven themselves well and have shown appropriate sensitivity to real aspects. Dargi et al. used different machine learning models used to predict permeability for acidizing in the petroleum industry⁴⁸. They concluded that ML models provide a robust basis for predicting permeability changes for acidizing treatment of oil reservoirs. The results highlighted the potential of machine learning algorithms to accurately predict post-acidification permeability, thereby aiding in acid design, evaluating its effectiveness, and ultimately improving productivity.

In addition, response surface methodology (RSM) is another effective technique for model development in order to predict the required parameters. RSM allows to obtain a visual representation of the relationship between the input and output data of the model through the construction of response surface and analysis of the uncertainty of the input parameters. In RSM optimization, the input variables are defined as independent variables and the effect of these variables on the output (dependent) variables is studied. The main advantage of RSM is reducing the number of repetitions of experiments to evaluate multiple parameters and their interactions.

Mosallanezhad and Kalantariasl used RSM for the determination of performance of engineered water injection processes through model development and its optimization⁴⁹. The authors concluded that the proposed model can be effectively used to examine the optimum value of ion composition of injection water in order to have the best production performance. The comparison of model and experimental data of residual oil saturation during water flooding showed a good agreement between them. Among the evaluated parameters, ion (sodium) concentration and initial rock wettability were the most affecting parameters on the efficiency of the engineered water injection process into the reservoir. Liang et al. could optimize the values of polymer viscosity and residual resistance factor in the production of heavy oil using RSM⁵⁰. The authors could analyze the recovery factor for the polymer flooding process using binary quadratic equations in RSM. The deviation of recovery factor predicted by RSM with the values of recovery factor in the literature was not high, which indicated the importance of process modeling by RSM. Brantson et al. used both RSM and ANN techniques for prediction of water saturation in oil reservoirs⁵¹. The obtained results proved that both models had high accuracy for determination of water saturation. The quadratic model of RSM was developed using ANOVA, which is statistically significant for application under reservoir conditions. The coefficient of determination and mean squared error for ANN model was better than RSM.

Despite extensive studies on scale formation during waterflooding, a critical gap remains in the ability to accurately and efficiently predict permeability damage under varying reservoir conditions. Most existing models either overlook the dynamic nature of scale formation or rely heavily on time-consuming experimental workflows. This study aims to bridge this gap by developing a novel hybrid modeling approach that couples CCDRSM (Central-Composite-Design of Response-Surface-Methodology) with advanced ML techniques, to predict permeability loss during waterflooding in the presence of DTPMP scale inhibitors. By incorporating real coreflood experimental data and optimizing using Particle Swarm Optimization (PSO), this work uniquely enables precise permeability forecasting and inhibitor dosage optimization across a wide range of operational scenarios. The proposed approach provides a more robust and generalizable framework than traditional methods, supporting proactive scale management and enhanced reservoir performance.

Materials and methods

The properties of the used water and scale inhibitor

The mixture of synthetic formation and injection waters was prone to the calcium sulfate precipitation under dynamic conditions in the core samples. To prepare the synthetic formation and injection brines, the required salts were determined and then added to the distillated water at room conditions. The source and purity of all chemicals used in this study are as follows: Diethylenetriamine penta(methylene phosphonic acid) (DTPMP, 50 wt% aqueous solution, industrial grade, Sigma-Aldrich) was used as the scale inhibitor; calcium chloride dihydrate (CaCl₂·2H₂O, > 99% purity, Merck) and sodium sulfate (Na₂SO₄, > 99% purity, Merck) were used as sources of calcium and sulfate ions, respectively. All chemicals were of analytical grade and used without further purification, and distilled water with an electrical conductivity below 2 µS/cm was used for preparing all synthetic brines.

The formation water was characterized by high concentrations of Na⁺ (33542 ppm), Ca²⁺ (13647 ppm), K⁺ (401 ppm), Mg²⁺ (625 ppm), Cl⁻ (70487 ppm), and SO₄²⁻ (427 ppm), with a total dissolved solids (TDS) content of approximately 119,129 ppm. The injection water contained Na⁺ (12487 ppm), Ca²⁺ (301 ppm), K⁺ (54 ppm), Mg²⁺ (1458 ppm), Cl⁻ (29005 ppm), and variable sulfate concentrations (1000–5000 ppm), resulting in a TDS range of 44,305–48,305 ppm. The elevated sulfate concentration in the injection brine was intentionally varied to promote controlled calcium sulfate precipitation under dynamic coreflood conditions. In addition, some coreflood tests were conducted in the presence of DTPMP as scale inhibitor. DTPMP was added to the formation water at various concentrations to determine the its optimum dosage for effective prevention of formation damage.

Coreflood tests

Coreflood experiments were conducted to quantitatively evaluate formation damage induced by calcium sulfate precipitation and to assess the performance of the scale inhibitor under dynamic flow conditions representative of near-wellbore environments. Cylindrical carbonate core plugs with an average diameter of 2.5 cm and length of 4.0 cm were used in all experiments. The Iranian carbonate core samples used in this study consisted predominantly of limestone (calcite) with subordinate dolomite, representing a typical mixed carbonate mineralogy commonly observed in Iranian reservoir formations. The average porosity of the samples was 19%, determined using the gravimetric method, and the pore volume (PV) of each core was calculated using PV = 3.14 * r² * L * porosity, resulting in an approximate PV of 3.7 mL. Prior to testing, the cores were cleaned by sequential solvent extraction using toluene and methanol to remove residual hydrocarbons and impurities, followed by oven drying at 60 °C until constant weight was achieved. The dried cores were then vacuum-saturated with synthetic formation water for at least 6 h to eliminate trapped air and ensure complete saturation, simulating in-situ reservoir conditions. The coreflood apparatus consisted of a stainless-steel core holder equipped with temperature control, a high-precision constant-rate injection pump, separate accumulators for formation and injection waters, and differential pressure transducers connected to a data acquisition system for real-time monitoring of pressure drop across the core (Fig. 1).

Fig. 1.

Schematic diagram of the coreflood experimental setup used for evaluating permeability reduction due to calcium sulfate precipitation under controlled temperature and flow conditions⁵².

All experiments were conducted at a temperature range of 50–100 °C and an injection rate range of 1–5 mL/min, corresponding to near-wellbore flow conditions. Initial permeability (K_i) was determined by injecting synthetic formation water at steady-state conditions and calculating permeability using Darcy’s law based on measured differential pressure⁵³. The average initial rock permeability before scale precipitation was 85 mD. After establishing baseline permeability, a mixture of injection water and formation water was injected to induce calcium sulfate precipitation within the porous medium. In inhibitor-assisted runs, DTPMP was added to the formation water at a concentration range of 0–50 ppm. Following the scaling stage, damaged permeability (K_d) was recalculated under identical flow conditions, and formation damage was quantified using the permeability ratio (K_d/K_i), where a smaller reduction in permeability indicated more effective scale inhibition and improved formation protection. These experimental results were then used to develop predictive models. After each experiment, the system was flushed thoroughly with deionized water to remove residual brine and inhibitor prior to the subsequent test, ensuring repeatability and experimental consistency.

Design of experiments (DOE)

Figure 2 illustrates the overall framework adopted in this study, encompassing the experimental design, modeling procedures, and process optimization strategy. The experimental design methodology is detailed in the current subsection, whereas the subsequent sections.  (2–4 and 2–5) cover the approaches used for modeling and optimization.

Fig. 2.

Overall workflow of the study integrating experimental design, modeling, and optimization for predicting and minimizing permeability damage.

The Design of Experiments (DOE) is a statistical technique employed to explore and quantify the relationship between input and output variables⁵⁴. This approach involves the deliberate and structured variation of independent variables to examine their influence on a response variable. The primary objective is to determine which factors have the most significant impact on the outcome, enabling targeted optimization of a process. Among various DOE methodologies, the Central Composite Design (CCD) is widely recognized for its effectiveness⁵⁵. A key benefit of CCD lies in its capacity to capture and model non-linear interactions among variables. This allows researchers to detect intricate dependencies and synergies between factors that other experimental designs may overlook. Moreover, CCD enhances experimental efficiency by minimizing the total number of tests required, as it enables simultaneous evaluation of multiple parameters within a systematic framework. This makes it a valuable tool for achieving process improvements while conserving time and experimental resources.

This study systematically evaluated the impact of five primary operational parameters on permeability ratio (K_d/K_i). The investigated variables included inhibitor dosage, temperature, sulfate concentration, pore volume, and injection rate. To enable comprehensive analysis, each parameter was assigned three levels—low (− 1), medium (0), and high (+ 1)—within the experimental design, as detailed in Table 1. The experimental matrix developed using CCD, comprising a total of 45 experimental runs along with the corresponding laboratory measurements of K_d/K_i ratios, is presented in Table 2.

Table 1.

Independent operational parameters with coded and actual values used in the experiments.

Parameter	Symbol	Unit	Experimental levels (coded and actual)
			− 1	0	+ 1
Inhibitor dosage	A	ppm	0	25	50
Temperature	B	°C	50	75	100
Sulfate concentration	C	ppm	1000	3000	5000
Pore volume (PV)	D	–	0	5	10
Injection rate	E	mL/min	1	3	5

Table 2.

DOE-CCD experimental matrix and corresponding results utilized for RSM and ML analyses.

Run	Parameter					Response
	A	B	C	D	E	Kd/Ki	Ki	Kd
1	− 1.000	− 1.000	1.000	− 1.000	− 1.000	1	85.00	85.00
2	− 1.000	− 1.000	− 1.000	− 1.000	− 1.000	1	85.00	85.00
3	− 1.000	1.000	1.000	− 1.000	− 1.000	1	85.00	85.00
4	− 1.000	1.000	− 1.000	− 1.000	− 1.000	1	85.00	85.00
5	− 1.000	− 1.000	1.000	1.000	− 1.000	0.43	85.00	36.55
6	− 1.000	− 1.000	− 1.000	1.000	− 1.000	0.57	85.00	48.45
7	− 1.000	1.000	1.000	1.000	− 1.000	0.39	85.00	33.15
8	− 1.000	1.000	− 1.000	1.000	− 1.000	0.51	85.00	43.35
9	− 1.000	0.000	0.000	0.000	0.000	0.65	85.00	55.25
10	− 1.000	− 1.000	1.000	− 1.000	1.000	1	85.00	85.00
11	− 1.000	− 1.000	− 1.000	− 1.000	1.000	1	85.00	85.00
12	− 1.000	1.000	− 1.000	− 1.000	1.000	1	85.00	85.00
13	− 1.000	1.000	1.000	− 1.000	1.000	1	85.00	85.00
14	− 1.000	− 1.000	1.000	1.000	1.000	0.47	85.00	39.95
15	− 1.000	− 1.000	− 1.000	1.000	1.000	0.59	85.00	50.15
16	− 1.000	1.000	1.000	1.000	1.000	0.4	85.00	34.00
17	− 1.000	1.000	− 1.000	1.000	1.000	0.55	85.00	46.75
18	0.000	0.000	0.000	0.000	− 1.000	0.76	85.00	64.60
19	0.000	0.000	0.000	− 1.000	0.000	1	85.00	85.00
20	0.000	− 1.000	0.000	0.000	0.000	0.79	85.00	67.15
21	0.000	0.000	0.000	0.000	0.000	0.77	85.00	65.45
22	0.000	0.000	0.000	0.000	0.000	0.77	85.00	65.45
23	0.000	0.000	− 1.000	0.000	0.000	0.81	85.00	68.85
24	0.000	0.000	0.000	0.000	0.000	0.78	85.00	66.30
25	0.000	0.000	1.000	0.000	0.000	0.74	85.00	62.90
26	0.000	1.000	0.000	0.000	0.000	0.75	85.00	63.75
27	0.000	0.000	0.000	1.000	0.000	0.75	85.00	63.75
28	0.000	0.000	0.000	0.000	1.000	0.78	85.00	66.30
29	1.000	− 1.000	1.000	− 1.000	− 1.000	1	85.00	85.00
30	1.000	− 1.000	− 1.000	− 1.000	− 1.000	1	85.00	85.00
31	1.000	1.000	1.000	− 1.000	− 1.000	1	85.00	85.00
32	1.000	1.000	− 1.000	− 1.000	− 1.000	1	85.00	85.00
33	1.000	− 1.000	− 1.000	1.000	− 1.000	0.96	85.00	81.60
34	1.000	− 1.000	1.000	1.000	− 1.000	0.93	85.00	79.05
35	1.000	1.000	1.000	1.000	− 1.000	0.92	85.00	78.20
36	1.000	1.000	− 1.000	1.000	− 1.000	0.94	85.00	79.90
37	1.000	0.000	0.000	0.000	0.000	0.95	85.00	80.75
38	1.000	− 1.000	− 1.000	− 1.000	1.000	1	85.00	85.00
39	1.000	− 1.000	1.000	− 1.000	1.000	1	85.00	85.00
40	1.000	1.000	− 1.000	− 1.000	1.000	1	85.00	85.00
41	1.000	1.000	1.000	− 1.000	1.000	1	85.00	85.00
42	1.000	− 1.000	1.000	1.000	1.000	0.94	85.00	79.90
43	1.000	− 1.000	− 1.000	1.000	1.000	0.96	85.00	81.60
44	1.000	1.000	1.000	1.000	1.000	0.92	85.00	78.20
45	1.000	1.000	− 1.000	1.000	1.000	0.95	85.00	80.75

RSM methodology

RSM is a widely adopted statistical and mathematical tool for modeling, analyzing, and optimizing complex engineering processes, particularly those influenced by multiple interdependent variables. In petroleum and chemical engineering applications, RSM facilitates the identification of optimal operating conditions by capturing the underlying relationship between process variables and performance indicators. In this study, a second-order (quadratic) polynomial model was employed to quantify the effects of various input parameters on the system response. The general form of this model is defined as follows:

1

where Y denotes the response variable of interest, Xi corresponds to the ith independent factor, β₀ is the constant intercept, βi and βii represent the coefficients associated with the linear and quadratic effects respectively, and βij accounts for the interaction between the ith and jth input independent variables.

Machine-learning (ML) methodology

For permeability damage estimation, the data were randomly split into two groups: 80% for training and 20% for testing. The machine-learning methods were selected to cover different modeling categories including linear, tree-based, ensemble-based, and kernel-based approaches while considering the limited size of the experimental dataset (45 CCD-based coreflood runs). This selection enabled a systematic comparison of diverse algorithms and avoided potential overfitting that could arise with more complex neural network models.

Subsequently, six machine learning methods, including Linear Regression (LR), Support Vector Regression (SVR), Gaussian Process Regression (GPR), Regression Tree (RT), Random Forest (RF), and LSBoost, were employed. A brief description of each method will be provided in the following sections.

Linear regression

LR is a statistical method used to examine the relationship between a dependent variable and one or more independent variables. The primary goal is to predict the dependent variable based on the independent variables. In linear regression, it is assumed that the relationship between the variables can be estimated by a linear equation. Although the data may exhibit fluctuations, their overall relationship can be represented by a simple linear line^56,57. To construct a linear regression model, a set of observed data is used. Then, using the least squares method, the parameters of the regression line are adjusted to minimize the sum of the squared distances between the data points and the predicted line. The main advantage of linear regression is its simplicity and interpretability. Since the lines that fit the data are straightforward to understand, examining the effects of independent variables on the dependent variable becomes easier^58,59. The general form of linear regression is expressed as follows⁶⁰:

2

In the above equation, the output represents the estimated parameter, C₀ is the intercept of the linear regression, C₁−C_n are the coefficients representing the influence of the independent variables X₁ to X_n on the output and ε denotes the discrepancy between the actual values and the predicted values⁶⁰.

Regression tree

RT is a forecasting model that predicts a continuous outcome using one or more input variables^61,62. This approach uses machine learning techniques to split the feature space into multiple uniform regions, where the dependent variable is predicted based on the mean or median of its values within each region^62,63. Building a regression tree involves repeatedly dividing the dataset into smaller subsets based on a chosen criterion (e.g., minimizing the sum of squared errors)⁶⁴. This process continues until each subset has a minimum number of data points or no further gains in prediction accuracy can be achieved⁶⁵. Regression trees are valued for their simplicity and interpretability, their ability to model both linear and nonlinear relationships, and their capacity to handle missing data. However, they are prone to overfitting, where the model fits the training data too closely and performs poorly on new data^66,67. To address this, methods like tree pruning or ensemble approaches such as Random Forests are used to improve model robustness and accuracy^68,69.

Random forest

The theoretical foundation of the RF algorithm is based on the assumption that combining the predictions of multiple decision trees trained on different subsets of data and features can improve prediction accuracy⁷⁰ and reduce the problem of overfitting, which is commonly seen in single decision tree models⁷¹. This approach makes the Random Forest more robust against variations in the training data and enhances its generalization capability. Additionally, Random Forest can calculate feature importance, which aids in understanding the data and interpreting the model^72,73. Equation (3) is used for RF analyses related to the estimation of predictions for data points⁷⁴:

3

In the RF algorithm, Eq. (3) represents the calculation of the final prediction for an input data point x. This prediction is obtained by averaging the predictions of all M trees in the forest. Each Fi(x) denotes the prediction function of the i-th tree for the data point x, and by summing and averaging these predictions, the final prediction , which represents the RF estimate for the output x, is computed. This approach helps to improve prediction accuracy and reduce the risk of overfitting.

Support Vector Regression

Support Vector Regression (SVR) is an advanced machine learning technique widely used for accurate prediction of continuous variables^75,76. Derived from the Support Vector Machine (SVM) model originally developed for classification tasks, SVR adapts a margin-based approach to regression problems⁷⁷. The primary objective is to construct a function that predicts target values while allowing a specified margin of tolerance (ϵ, epsilon), minimizing errors that fall outside this margin⁷⁸. To handle nonlinear relationships, SVR maps input data into a higher-dimensional feature space using kernel functions, enabling linear regression in this transformed space without explicitly computing the high-dimensional mappings⁷⁹. This kernel-based mechanism makes SVR highly effective for modeling complex, nonlinear systems^80–82. The performance of SVR depends on key hyperparameters, including the penalty parameter C, which controls the trade-off between model complexity and training error, and ϵ, which defines the width of the margin of tolerance. A larger C emphasizes accuracy on training data, while a larger ϵ improves generalization by reducing sensitivity to noise.

Gaussian process regression

This method is based on probability theory and Gaussian processes, which use a Gaussian distribution to define the probabilistic distribution of functions. A Gaussian process is defined by a mean function and a covariance function (or kernel), which provide information about the structure of the data and the correlation between data points. These features help the model to make more accurate predictions and to model uncertainty in the predictions⁸³. In GPR, it is assumed that the data follow a Gaussian distribution. The model is expressed as follows:

4

In Eq. (4), m(x) represents the mean function, and k(x, xʹ) is the covariance function, which determines the degree of correlation between different data points. To train the model, the training data is used to estimate the parameters of the mean and covariance functions. The model can then be used to predict new values and also to estimate the uncertainty of these predictions. These features make Gaussian Process Regression well-suited for applications such as time series prediction, modeling natural phenomena, and optimizing complex functions⁸⁴. The main advantage GPR over other regression methods is its ability to model uncertainty and provide higher accuracy predictions. This method not only offers predicted values but also provides confidence intervals for these predictions. This is particularly useful in applications where risk assessment and decision-making under uncertainty are required⁸⁵.

LSBoost

LSBoost stands for “Least Squares Boosting,” and it combines simpler models in an incremental manner to reduce overall error. This method begins with a base model and then iteratively adds new models that predict the residual errors of the previous models. This process is mathematically optimized to enhance the final model’s performance. The primary advantage of LS Boost is that by adding new models to the base model, the prediction ability is improved and the overall error is reduced⁸⁶. The main formula used in LSBoost is as follows:

5

where represents the final estimate of the model, M is the number of simple models added to the final model, h_m(x) is the simple model at the mmm-th iteration, and β_m is the weight coefficient for the model h_m(x), chosen to minimize prediction error. The selection of β_m is based on minimizing the least squares error, aiming to reduce the sum of squared prediction errors. In other words, at each stage, a new model is added that attempts to minimize the residual error from previous models⁸⁷.

Error metrics

After constructing and optimizing the models, to evaluate their performance on both training and test data, several error metrics were computed, including Mean Absolute Error (MAE), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Coefficient of Determination (R²). The formulas for these metrics are provided below⁸⁸.

6

7

8

9

In the formulas above, , , and N represent the actual values of the output parameter, the predicted values of the output parameter, the mean of the actual output values, and the number of samples, respectively.

Results and discussion

Data pre-processing

The collected data includes inhibitor concentration, temperature, sulfate concentration, pore volume, injection rate, and permeability ratio (K_d/K_i). To build the models, scatter plots for each pair of parameters were first created to visualize the data distribution. According to Fig. 3, the data exhibit a suitable level of dispersion with no apparent bias, which can contribute to the development of more comprehensive models.

Fig. 3.

Scatter and KDE plots of input and output parameters.

The Pearson correlation coefficient was employed to evaluate the strength and direction of relationships between the input variables, with the results presented in Fig. 4.

Fig. 4.

Pearson correlation coefficient values for the studied variables.

As illustrated, pore volume exhibits the strongest correlation with K_d/K_i, followed by inhibitor concentration, both indicating a significant influence on the response. In contrast, temperature, sulfate concentration, and injection rate show negligible correlations with K_d/K_i, suggesting a limited effect under the studied conditions. Moreover, inhibitor concentration and injection rate demonstrate positive correlations with permeability damage, whereas pore volume, temperature, and sulfate concentration are negatively correlated with it. In addition, it should be noted that the apparent zero or very low points in Fig. 4 result from the natural dispersion of the experimental data. Pearson correlation primarily reflects the dataset distribution and does not necessarily indicate a direct influence between variables, reflecting the variability inherent in the CCD-based experimental design.

RSM-based modeling

Based on the experimental data presented in Table 2, a reduced quadratic model (in coded form) was developed using RSM to effectively predict variations in permeability loss. A summary of the model—encompassing its statistical significance, predictive equation, and accuracy metrics—is provided in Table 3.

Table 3.

Summary of the RSM-model including significance, formulation, and accuracy metrics.

Model and terms significance			Model formula, Kd/Ki (coded)		Model accuracy
Source	F-value	p-value	Factor	Coefficient	Fit-statistics
Model	377.00	< 0.0001	Intercept	0.7773	R²	0.9898
A-Inhibitor dosage	935.54	< 0.0001	A	0.1150	Adjusted	0.9872
B-Temperature	5.88	0.0206	B	− 0.0091	R²	0.9830
C-Sulfate concentration	29.99	< 0.0001	C	− 0.0206	Predicted	58.7184
D-Pore volume	1421.69	< 0.0001	D	− 0.1418	R²
E-Injection rate	1.38	0.2486	E	0.0044	AP
AC	12.02	0.0014	AC	0.0134
AD	847.33	< 0.0001	AD	0.1128	Error metrics
CD	25.81	< 0.0001	CD	− 0.0197	MSE	0.00037
D²	113.35	< 0.0001	D²	0.0810	RMSE	0.01933

The fit-statistics for the developed RSM-model are summarized in Table 3. Key indicators—including the coefficient of determination (R²), adjusted R² (adj-R²), predicted R² (pred-R²), and adequate precision (AP)—were employed to assess how well the model represents the experimental data. Generally, models with an R² value of 0.80 or higher are considered statistically significant⁸⁹. In this case, the model achieved an R² of 0.9898, indicating that 98.98% of the variation in the response variable can be explained by the model. Furthermore, the adj-R² and pred-R² values were found to be 98.72% and 98.30%, respectively. The close agreement between these values (difference < 0.2) confirms both the robustness and the predictive reliability of the model⁹⁰. Additionally, adequate precision, which evaluates the signal-to-noise ratio, yielded a value of 58.718—well above the acceptable threshold of 4—demonstrating a strong signal. Therefore, the model is deemed highly reliable for exploring and optimizing the design space⁵⁵.

Figure 5 presents the plot of predicted versus experimental (actual) values for K_d/K_i. As depicted, the data points are closely distributed around the diagonal line, indicating a strong agreement between the experimental observations and the values predicted by the developed RSM model⁹¹. This close alignment confirms the model’s high accuracy and predictive capability in estimating permeability ratios.

Fig. 5.

Predicted versus experimental (actual) values for permeability ratio (K_d/K_i). The diagonal line represents perfect agreement between experimental data and model predictions.

The F-statistics and p-values presented in Table 3 serve as key indicators for evaluating the statistical significance of the developed RSM model. The F-value is a test statistic used to assess whether a given model term has a meaningful relationship with the response variable, while the corresponding p-value determines the likelihood that the observed effect occurred by chance. A model term is considered statistically significant when its p-value is less than the selected significance level⁹². In this study, model significance is confirmed when the p-value is ≤ 0.05 and the F-value is relatively high. The model’s F-value of 377.00 indicates a highly significant model. According to the results, model terms A, B, C, D, AC, AD, CD, and D² are statistically significant. Among these, pore volume (D) and inhibitor dosage (A) exerted the most substantial effects on permeability variation, as reflected by their high F-values of 1421.69 and 935.54, respectively. Conversely, injection rate (E) had the least impact. The relative importance of parameters affecting the K_d/K_i response in the developed RSM-model follows the descending order: D > A > AD > D² > C > CD > AC > B > E. It should be noted that although the injection rate (E) exhibited a p-value of 0.2486, exceeding the conventional 0.05 significance threshold, it was retained in the final quadratic RSM model. Despite its limited statistical contribution within the investigated range (1–5 mL/min), injection rate may affect hydrodynamic behavior, ion transport mechanisms, and fluid residence time within the porous medium, thereby enhancing the model’s robustness, interpretability, and applicability under broader operational scenarios.

Additionally, the influence of each individual parameter on K_d/K_i was evaluated through a sensitivity analysis, as illustrated in Fig. 6. In this plot, each curve represents the effect of a single input variable varied across its entire range, while all other variables were held constant at a predefined reference point within the design space. This reference point corresponds to the coded level “0” for all parameters and includes: inhibitor dosage (25 ppm), temperature (75 °C), sulfate concentration (3000 ppm), pore volume (5), and injection rate (3 mL/min). The sensitivity curves clearly demonstrate that pore volume and inhibitor dosage have the most pronounced influence on K_d/K_i, with pore volume exerting a negative effect and inhibitor dosage a positive one. In contrast, injection rate was found to have the least impact on the response variable under the given conditions.

Fig. 6.

Sensitivity analysis of operational parameters affecting permeability ratio (K_d/K_i) based on the developed RSM model. Each curve represents the variation of K_d/K_i with one parameter while other variables are fixed at their central coded level (0). Curve labels correspond to coded variables defined in Table 1: A—Inhibitor dosage, B—Temperature, C—Sulfate concentration, D—Pore volume (PV), and E—Injection rate.

It should be also noted that although injected pore volume (PV) and injection rate seem to be operationally related, they represent fundamentally different physical effects in the porous medium. Pore volume corresponds to the cumulative amount of brine that has passed through the core, directly controlling total ion exposure, supersaturation buildup, and the extent of scale accumulation⁹³. In contrast, injection rate primarily affects flow velocity and residence time at a given moment⁹⁴. Within the investigated range (1–5 mL/min), variations in injection rate were insufficient to significantly alter nucleation and crystal growth mechanisms, resulting in a comparatively minor statistical effect. Therefore, PV was treated as a cumulative damage-governing parameter, while injection rate was considered a kinetic modifier with secondary influence under the studied conditions. This distinction explains why PV exhibited the strongest impact on permeability reduction, whereas injection rate showed minimal effect on inhibition performance in RSM analyses.

The higher extent of permeability reduction observed at lower injected pore volumes can be attributed to the rapid nucleation and growth of calcium sulfate crystals immediately after incompatible brine mixing⁹⁵. At the early stage of injection, the contact between formation and injection waters generates maximum local supersaturation, which promotes intense nucleation within pore throats and on mineral surfaces. This initial precipitation preferentially occurs in smaller pore channels where flow velocities and concentration gradients are higher, leading to significant permeability reduction during the first injected pore volumes⁹⁶. As scale deposition progresses, several factors contribute to a decline in precipitation rate^3,6: (i) partial blockage of pore throats reduces available reactive surface area, (ii) ion concentrations decrease due to ongoing precipitation, thereby lowering supersaturation, and (iii) flow redistribution toward less obstructed pathways limits further deposition in already damaged regions. Consequently, the majority of permeability impairment occurs during the early stage of coreflooding, while subsequent injected pore volumes produce comparatively smaller incremental damage.

ML-based modeling

To estimate reduction-in-permeability, the dataset was randomly divided into training and testing subsets, with 80% of the data allocated for model training and the remaining 20% reserved for validation. Six machine learning methods—LR, SVR, GPR, RT, RF, and LSBoost—were implemented to develop predictive models. The performance of these models was evaluated through a combination of error metrics and cross plots. In the cross plots, predicted values are plotted against the corresponding experimental measurements, where data points that lie closer to the X = Y line indicate greater predictive accuracy and model reliability⁹⁷. Figure 7 illustrates the cross plots for all models, enabling a clear visual comparison of their predictive capabilities. According to these plots, LR shows suboptimal performance, with the data not aligning well along the line. Although the data in LR do not generally align with the line, they are somewhat concentrated around it. The performance of the RT and RF methods appears similar, with relatively good performance in medium to high K_d/K_i ranges. The GPR and LSBoost methods show the best alignment with the line, with the data points closely following the line across all ranges. The performance of these two methods is very similar on the training dataset, but it appears that GPR has shown better performance in estimating values for the test dataset.

Fig. 7.

Cross plots comparing experimentally measured and predicted permeability ratios (K_d/K_i) for the six developed machine learning models (LR, SVR, GPR, RT, RF, and LSBoost.

To assess the performance of the models more precisely, error metrics were used. The values of various error metrics for each model are presented in Table 4. As expected from the cross plots, the linear regression model demonstrates the weakest performance among all models in both the training and test datasets, with R² = 0.6956 and RMSE = 0.1056. Following linear regression, the RT and RF methods show the next weakest performance. Although their performance is quite similar, the random forest method has a slight advantage with R² = 0.9628 compared to the decision tree’s R² = 0.9553. SVR exhibits excellent performance with R² = 0.9912 and RMSE = 0.0180, providing predictions with an average error of less than 0.02. Both LSBoost and GPR demonstrated outstanding performance in predicting the training data, accurately estimating permeability-changes values. However, LSBoost performed slightly weaker than GPR in predicting the test data, indicating the higher generalizability of the developed GPR model.

Table 4.

Error metrics of developed ML-models.

Taking into account both quantitative error metrics and graphical methods, the GPR emerged as the most reliable and robust ML-model for predicting permeability loss. The optimized hyperparameters of the developed MLGPR-model are summarized in Table 5. A low noise level (Sigma = 8.2952e−04) indicates high data precision and minimal measurement uncertainty. The use of a linear basis function suggests an assumed linear relationship between inputs and outputs. The ARD Squared Exponential kernel enhances model flexibility by assessing the relevance of each input dimension, supporting feature selection in high-dimensional spaces. Moreover, the choice to omit input standardization implies that the dataset features are already on comparable scales.

Table 5.

Optimized hyperparameters of the MLGPR-model.

Hyperparameter	Optimized value
Sigma	8.2952e−04
Basis function	Linear
Kernel function	ARDSquaredExponential
Standardize	False

Compression of ML and RSM-based models

In this study, both RSM and ML approaches were applied to predict permeability ratio (K_d/K_i). The RSM model, developed using a reduced quadratic equation, exhibited strong predictive power, with an R² of 0.9898 and an adequate precision value of 58.718—well above the generally acceptable threshold. It also yielded an RMSE of 0.0193, indicating relatively low prediction error and good model reliability. On the other hand, six machine learning models were evaluated, with the GPR model outperforming all others. It achieved an exceptional overall R² value of 0.9991 and a notably low RMSE of 0.0056, reflecting excellent predictive capability and minimal error. In addition, the GPR model exhibited strong generalization performance on test data, confirming its robustness and practical applicability.

A comparative overview is presented in Table 6. Although the RSM model provides valuable interpretability and facilitates a clear assessment of individual parameter contributions, the GPR model exhibits superior predictive accuracy and a robust capacity to capture complex, nonlinear relationships. As a result, the GPR model is identified as the most appropriate and robust framework for in-depth analysis of the factors influencing K_d/K_i, as well as for process optimization in the subsequent sections.

Table 6.

Comparison of best RSM and ML models.

Metric	RSM-model	GPR-model
R² (total)	0.9898	0.9991
RMSE (total)	0.0193	0.0056
MSE (total)	0.0004	0.0000
Generalization	Good	Excellent
Flexibility	Limited (quadratic)	High
Parametric insight	High	Moderate

To place the results of the study in a broader scientific context and highlight their contribution, a critical comparison with previously published studies is necessary. Several researchers have reported strong predictive capabilities of ML techniques for permeability estimation. For example, Okon et al.⁴² achieved an R² value of approximately 0.98 for permeability prediction, while Tian et al.⁴⁵ reported R² values close to 0.998 using a hybrid ML-genetic algorithm approach. Similarly, Tran et al.⁴⁶. obtained an R² of 0.96 for carbonate permeability prediction. The GPR model developed in this study achieved an overall R² of 0.9991, which is comparable to or slightly higher than values reported in the literature, indicating excellent predictive accuracy. In addition, in terms of RSM applications, previous studies such as Mosallanezhad and Kalantariasl⁴⁹ and Liang et al.⁵⁰ demonstrated that quadratic RSM models can effectively capture parameter interactions and optimize injection processes with acceptable deviations from experimental data. The RSM model developed in the present study (R² = 0.9898) shows similar reliability while specifically addressing permeability damage due to scale precipitation under dynamic coreflood conditions. Moreover, consistent with prior investigations^36,48, the present results confirm that hybrid or advanced ML approaches outperform traditional statistical models in capturing complex nonlinear interactions. However, the current study uniquely integrates CCD-based experimental design, ML modeling, and PSO optimization within a unified framework for scale-induced permeability damage prediction. This integrated methodology extends beyond previous permeability prediction studies by simultaneously providing predictive accuracy, parameter sensitivity insight, and operational optimization capability.

Parameters effects on K_d/K_i using CCD-MLGPR model

Figures 8 and 9, and 10 illustrate the predicted effects of different operational variables on permeability ratio using the CCD-MLGPR model. Each figure presents 3D contour plots and trend lines to visually represent the interaction and sensitivity of parameters.

Fig. 8.

The contour plot illustrating the combined effect of pore volume (PV) and inhibitor dosage on permeability ratio (K_d/K_i) predicted by the CCD-MLGPR model (Temperature = 75 °C, Sulfate Concentration = 3000 ppm, Injection Rate = 3 mL/min).

Fig. 9.

The contour plot showing the interaction effect of sulfate concentration and temperature on permeability ratio (Kd/Ki) predicted by the CCD-MLGPR model (Inhibitor Dosage = 25 ppm, Pore Volume = 5, Injection Rate = 3 mL/min).

Fig. 10.

The contour plot demonstrating the combined influence of inhibitor dosage and injection rate on permeability ratio (K_d/K_i) predicted by the CCD-MLGPR model (Sulfate Concentration = 3000 ppm, Pore Volume = 5, Temperature = 75 °C).

Figure 8 shows the combined impact of pore volume (PV) and inhibitor dosage on the permeability ratio (K_d/K_i), highlighting their roles in salt precipitation and formation damage. As PV increases from 1 to 10, K_d/K_i decreases markedly, indicating increased permeability loss due to enhanced ion transport and supersaturation, which drive scale formation⁵³. Without inhibitor, K_d/K_i falls from about 0.88 at PV = 1 to below 0.50 at PV = 10, representing over 50% permeability reduction. This confirms that higher brine volumes raise saturation indices and salt precipitation. However, increasing inhibitor dosage significantly reduces damage. At PV = 5, raising the inhibitor from 0 to 40 ppm improves K_d/K_i from 0.70 to 0.93, limiting damage to under 7%. The inhibitor’s effectiveness is most pronounced at high dosages, nearly preventing scale formation even at PV = 10, where K_d/K_i reaches ~ 0.90 compared to less than 0.55 without inhibitor. These findings confirm that PV is the most impactful parameter promoting scale formation, while inhibitor dosage is the most effective in reducing permeability damage. Their interaction is especially critical: effective inhibition can nearly eliminate permeability loss even under high PV conditions, while insufficient dosage fails to prevent severe formation damage.

Figure 9 illustrates the effect of temperature (50–100 °C) and sulfate concentration (1000–5000 ppm) on the permeability ratio (K_d/K_i). The results show that increasing either parameter leads to greater permeability damage, reflected by a decrease in K_d/K_i. At low sulfate (1000 ppm) and low temperature (50 °C), K_d/K_i is around 0.83, indicating minimal damage (~ 17%). In contrast, at high sulfate (5000 ppm) and high temperature (100 °C), K_d/K_i drops to approximately 0.72, showing a 28% reduction in permeability. The contour pattern reveals a synergistic effect: sulfate increases supersaturation and promotes scale growth, while temperature reduces calcium sulfate solubility and alters crystal structure, accelerating deposition⁹⁸. Significant damage (K_d/K_i < 0.75) occurs when both parameters exceed ~ 4000 ppm and 85 °C. This analysis highlights the importance of managing both sulfate content and reservoir temperature to reduce scale formation and permeability loss during injection operations.

Figure 10 illustrates the combined effect of inhibitor dosage (0–50 ppm) and injection rate (1–5 mL/min) on the permeability ratio (K_d/K_i). The response surface indicates that inhibitor dosage is the dominant factor in mitigating permeability damage, while the effect of injection rate is minimal. At any given inhibitor dosage, increasing the injection rate from 1 to 5 mL/min leads to only a slight improvement of 3–5% in K_d/K_i, indicating a modest reduction in permeability loss. However, increasing inhibitor dosage from 0 to 50 ppm significantly enhances K_d/K_i to above 0.95, corresponding to less than 5% formation damage. This demonstrates that chemical inhibition plays a far more critical role than adjusting flow rate. While a higher injection rate may slightly reduce ion contact time, it is insufficient to prevent nucleation without adequate inhibitor presence. Effective scale prevention therefore depends primarily on optimizing inhibitor concentration.

Based on the conducted analyses, pore volume (PV) is identified as the most influential parameter affecting formation damage. The Pearson correlation analysis (Fig. 4) shows that PV exhibits the strongest negative correlation with Kd/Ki, indicating that increasing injected pore volume significantly intensifies permeability reduction. This finding is further supported by the ANOVA results presented in Table 3, where pore volume (D) is the most influential parameter affecting formation damage, as evidenced by its highest F-value (1421.69), followed by inhibitor dosage (A) with an F-value of 935.54. The sensitivity analysis (Fig. 6) also confirms that pore volume produces the steepest response variation, demonstrating its dominant impact on scale accumulation. Physically, pore volume represents the cumulative exposure of the porous medium to supersaturated brine, directly controlling ion transport, precipitation extent, and scale buildup. In contrast, inhibitor dosage shows a strong positive effect by mitigating crystal growth and reducing permeability damage. The combined influence of these two parameters is clearly illustrated in Fig. 8, where increased pore volume markedly decreases K_d/K_i, while higher inhibitor concentration effectively compensates for this damage. Therefore, pore volume is identified as the primary driver of formation damage, whereas inhibitor dosage is the most effective mitigation parameter.

Process optimization

In the final stage of this study, the Particle Swarm Optimization (PSO) algorithm was employed to accurately identify regions of maximal output response (K_d/K_i). This method offers enhanced insights into the system’s behavior across diverse input parameters, facilitating a more precise and comprehensive understanding of its performance under varying conditions⁵⁵. Optimization was conducted using the PSO-MLGPR hybrid framework to locate the optimal regions that result in the highest values of K_d/K_i. As discussed previously, pore volume (PV) is the most influential parameter contributing to scale formation. The primary objective was to determine the optimal conditions under a high PV scenario (PV = 10) that maximize the permeability ratio. The results for this objective (Goal I) are presented in Table 7. Notably, under the optimal conditions of 50 ppm inhibitor, 50 °C, 1000 ppm sulfate concentration, and an injection rate of 5 mL/min, a maximum K_d/K_i value of 0.97 was achieved—even under the severe condition of PV = 10.

Table 7.

Summary of optimization objectives and corresponding results using PSO-MLGPR.

Parameters	Unit	Goals		Optimization results
		I	II	I	II
Pore volume (PV)	–	Highest value (10)	Highest value (10)	10	10
Temperature	°C	in range	highest value (100)	50 °C	100
Injection rate	mL/min	in range	lowest value (1)	5	1
Sulfate concentration	ppm	in range	in range	1000	1000–5000
Inhibitor dosage	ppm	in range	Minimizing	50	36–50
Kd/Ki	–	Maximizing	Maximizing	0.97	0.85–1.00

As stated above, increasing PV and temperature, along with decreasing injection rate, exacerbates permeability impairment, leading to a decline in the K_d/K_i ratio. Therefore, the second objective (Goal II) was to identify the minimum inhibitor dosage range required to maintain K_d/K_i > 0.85, considering various sulfate concentrations under the worst-case operating conditions: PV = 10, temperature = 100 °C, and injection rate = 1 mL/min. The optimization results for this objective are also shown in Table 7 and visualized in Fig. 11. In this figure, the green-shaded regions represent the optimal inhibitor concentrations that effectively minimize permeability reduction and maintain K_d/K_i values above 0.85.

Fig. 11.

Optimization map identifying optimal inhibitor dosage regions (green-shaded areas) that maintain permeability ratio (K_d/K_i) above 0.85 under worst-case operational conditions (PV = 10, temperature = 100 °C, injection rate = 1 mL/min).

Additionally, each point within the green region of Fig. 11 can be interpreted as a viable optimal solution. A selection of these optimal conditions is summarized in Table 8. For instance, at a sulfate concentration of 5000 ppm (the worst-case scenario), applying 40–50 ppm of inhibitor resulted in K_d/K_i values ranging from 0.84 to 0.92, indicating a promising mitigation strategy for permeability loss under harsh operational environments.

Table 8.

Selected optimal conditions derived from PSO results at worst-case scenarios.

Pore volume injected (PV)	Temperature	Injection rate	Sulfate concentration	Inhibitor dosage	Kd/Ki
10	100 °C	1	2000	40	0.87
				45	0.91
				50	0.94
			4000	40	0.85
				45	0.89
				50	0.93
			5000	40	0.84
				45	0.88
				50	0.92

It should be noted that the present study was designed to isolate the mechanism of calcium sulfate scale formation and its impact on permeability under controlled single-phase brine flow conditions. Therefore, oil was not included in the coreflood experiments, which is a common and accepted practice in laboratory investigations focused specifically on inorganic scale precipitation and inhibitor performance evaluation. The exclusion of oil eliminates multiphase flow effects, wettability alteration, and oil–water interfacial interactions, allowing a more accurate quantification of permeability impairment caused solely by mineral deposition within the pore structure. While crude oil composition may influence relative permeability behavior and flow distribution in field conditions, calcium sulfate precipitation is primarily governed by ionic supersaturation, temperature, and mixing ratios of incompatible brines rather than oil chemistry^3,5. Regarding reservoir pressure, the experiments were conducted under constant pressure conditions representative of laboratory near-wellbore simulations. Unlike carbonate scales such as calcium carbonate, whose precipitation strongly depends on pressure-driven CO₂ degassing, calcium sulfate scaling is largely controlled by solubility equilibrium and ion concentration, and is comparatively less sensitive to pressure variations under single-phase aqueous flow⁹⁹. Nevertheless, in field applications, reservoir pressure and multiphase flow conditions may influence flow dynamics and scale deposition patterns. These factors should be considered when extrapolating laboratory results to field-scale systems.

Study limitations and future research directions

While the present study provides valuable insight into calcium sulfate scale formation and inhibition under controlled laboratory conditions, several limitations should be acknowledged. First, the experiments were conducted using synthetic formation and injection brines designed to isolate calcium sulfate precipitation. In real reservoir environments, multiple scale types such as calcium carbonate, barium sulfate, and strontium sulfate may precipitate simultaneously due to complex brine compositions and varying thermodynamic conditions⁵. The coexistence of different scale species can alter nucleation kinetics, crystal morphology, and inhibitor performance, which were not considered in this study. Second, the experiments were performed under single-phase aqueous flow conditions to focus specifically on inorganic scaling mechanisms. In actual reservoir systems, multiphase flow involving oil, gas, and water may influence flow distribution, wettability, and scale deposition patterns. The interaction between crude oil components and scale inhibitors may also affect adsorption behavior and inhibition efficiency¹⁰⁰. Third, the experiments were conducted at constant temperature and pressure representative of near-wellbore laboratory conditions. However, field conditions often involve pressure gradients, temperature variations, and heterogeneous rock properties, which can influence scale precipitation dynamics and inhibitor transport behavior. Additionally, only DTPMP was evaluated as a scale inhibitor in the coreflooding experiments. In field applications, scale control strategies may involve blended inhibitors, synergistic additives, squeeze treatments, or compatibility with other production chemicals such as corrosion inhibitors and demulsifiers^101–103. The interaction between different chemical additives and their combined impact on formation damage require further investigation. Future research should therefore focus on (i) evaluating multi-scale precipitation systems under dynamic conditions, (ii) investigating inhibitor performance under multiphase flow scenarios, (iii) studying adsorption/desorption behavior and long-term retention in carbonate formations, and (iv) integrating geochemical modeling with advanced data-driven approaches to improve predictive capability under field-scale conditions.

Conclusions

This study proposes a hybrid modeling strategy combining RSM and advanced ML techniques to predict and optimize permeability loss in carbonate reservoirs under varying operational conditions, especially in the presence of the scale inhibitor DTPMP. A series of CCDRSM-based coreflood experiments was performed to assess the influence of key factors on the permeability ratio (K_d/K_i), leading to the development of an RSM-based and six ML-based (LR, SVR, GPR, RT, RF, LSBoost) models. While the CCD-RSM model demonstrated reasonable predictive performance, the CCD-MLGPR model showed superior accuracy (R² = 0.9991, RMSE = 0.0056), establishing it as the most reliable predictive tool in the study. The findings indicated that PV was the dominant factor driving scale formation, dosage was the most effective parameter for mitigating permeability damage, and injection rate had the least impact. Without inhibitor treatment, permeability dropped by more than 50% (K_d/K_i < 0.50) at high PV levels. Increasing the DTPMP concentration to 40 ppm improved K_d/K_i to 0.93, limiting damage to below 7%. Additionally, integrating the GPR model with PSO optimization identified ideal operational conditions that preserved K_d/K_i values above 0.85–0.90, even under extreme scenarios.

Abbreviations

RSM

Response surface methodology

ML

Machine learning

GPR

Gaussian Process Regression

SVR

Support Vector Regression

LR

Linear regression

RT

Regression tree

RF

Random forest

LSBoost

Least squares boosting

CCD

Central composite design

DTPMP

Diethylene triamine pentamethylene phosphonic acid

PSO

Particle swarm optimization

K_d/K_i

Permeability ratio

MAE

Mean absolute error

MSE

Mean squared error

RMSE

Root mean squared error

R²

Coefficient of determination

PV

Pore volume

ANOVA

Analysis of variance

AI

Artificial intelligence

AP

Adequate precision

Author contributions

Authors 1 and 2 contributed to the conceptualization, methodology, data collection, and writing of the main manuscript text.Authors 3 and 4 conducted the formal analysis, created visualizations, and interpreted the results.All authors reviewed, revised, and approved the final version of the manuscript.

Funding

No funding was received for conducting this study.

Data availability

No datasets were generated or analysed during the current study.

Declarations

Competing interests

The authors declare no competing interests.