Predicting Seismic-Induced Settlement of Pipelines Buried in Sandy Soil Reinforced with Concrete and FRP Micropiles: A Genetic Programming Approach

Abstract

Unstable sandy soils pose significant challenges for buried pipelines due to soil–infrastructure interaction, leading to settlement that increases the risk of displacement and stress-induced fractures. In earthquake-prone regions, seismic-induced ground deformation further threatens underground infrastructure. Fiber-reinforced polymer (FRP) composites have emerged as a sustainable alternative to conventional piling materials, addressing durability issues in deep foundations. This paper introduces novel explicit models for predicting the maximum settlement of oil pipelines supported by concrete or polymer micropiles under seismic loading. Using genetic programming (GP), this study develops closed-form expressions based on simplified input parameters—micropile dimensions, pile spacing, soil properties, and peak ground acceleration—improving the models’ practicality for engineering applications. The models were evaluated using a dataset of 610 data points and demonstrated good accuracy across different conditions, achieving coefficients of determination (R²) as high as 0.92, among good values for other evaluation metrics. These findings contribute to a robust, practical tool for mitigating seismic risks in pipeline design, highlighting the potential of FRP micropiles for enhancing infrastructure resilience under challenging geotechnical scenarios.

1. Introduction

Preventing earthquake-induced damage to utility pipes is important for sustaining urban life. Buried pipeline engineering is a crucial infrastructure project worldwide. However, the geological environment along pipelines is complex and variable, leading to frequent collapse incidents that cause significant losses to human life and property. Earthquakes are natural disasters capable of causing extensive damage to essential infrastructure, including water supply systems, sewage systems, and oil and gas pipelines [1]. Since earthquake-related settlement of pipelines used for drinking water, stormwater, wastewater, and oil can severely impact daily life, it is believed that certain design measures can effectively mitigate this risk [2]. Several researchers have investigated the behavior of underground pipes during earthquakes [3,4,5,6,7]. Understanding how buried pipes respond to ground movement is crucial for designing and assessing risks under challenging conditions, such as when earthquakes cause ground shifts, as highlighted by [8,9,10,11,12,13,14]. The increasing frequency and impact of geohazards, particularly earthquakes, have emphasized the need for resilient infrastructure systems and accurate predictive models of soil–structure interaction. Recent studies have examined the dynamic behavior of geotechnical systems under seismic loading, including the influence of soil heterogeneity and wave propagation effects on foundations and buried structures [15,16,17]. These works highlight the complexity of seismic responses and the importance of developing robust yet practical modeling tools. In this context, the present study aims to introduce explicit settlement prediction models for pipelines supported by micropiles, using genetic programming based on validated seismic response data.

Micropiles have traditionally been made from materials such as wood, steel, or concrete, each facing durability issues such as wood decay, steel corrosion, and concrete degradation, which limit their lifespan and increase maintenance costs [18]. For example, steel piles exhibit significant corrosion over time, and concrete degrades in aggressive environments, leading to costly repairs, up to USD 1 billion annually in the U.S. for marine waterfront communities [19]. Recently, compared with traditional materials, polymer materials have emerged as promising alternatives for geotechnical applications, offering superior fluidity, expansibility, rapid setting, high strength, and long-lasting durability [20,21]. The use of fiber-reinforced polymers (FRPs) in engineering represents a promising alternative to traditional pile materials, which often deteriorate rapidly under harsh conditions [7]. There has been increasing interest in FRP composite materials for deep foundations because of their beneficial properties, such as being lightweight, having high specified strength, their exceptional durability, and their ease of application [22]. Beyond their mechanical advantages, recent research has further demonstrated the durability of FRP materials under harsh environmental conditions, including moisture, temperature cycles, and chemical exposure. For instance, long-term studies have shown that FRP-reinforced concrete components maintain structural performance over time despite environmental degradation mechanisms [23,24]. Research on the structural behavior of FRP piles has focused predominantly on their performance under lateral loading, particularly their use as fender piles [25,26,27]. Accurately predicting the horizontal behavior of micropiles is essential for providing strong support for various structures, including pipelines [28].

Prediction techniques help simplify engineering analysis and reduce the time needed for product design. In construction, traditional methods for assessing underground pile settlement, such as static and dynamic load tests, are reliable but are often criticized for their time-consuming and costly nature [29,30,31]. To overcome these challenges, some researchers suggest the use of semiempirical formulas derived from in situ test results [32,33], whereas others turn to finite element simulations with tools such as MIDAS GTS (version 2022R1) [31]. Recognizing these limitations, recent studies have explored artificial intelligence (AI) applications. The recent growth of AI has introduced unique possibilities in various research domains, especially geotechnical engineering [34,35]. Recent research efforts have resulted in the creation of innovative solutions, including artificial neural networks (ANNs) and advanced machine learning techniques [29,36,37,38]. ANNs, inspired by the structure and function of the human brain, have recently shown great potential in geotechnical engineering. Although the concept of artificial neurons originated in 1943, major advancements occurred in 1986 when Rumelhart et al. introduced the backpropagation algorithm for training feedforward ANNs [39].

Although AI has demonstrated significant advantages and promising features in geotechnical engineering, limitations remain. Most current models rely heavily on conventional machine learning (ML) algorithms, with advanced and efficient ML algorithms being infrequently utilized [40,41]. ML offers significant potential in geotechnical engineering by utilizing data-driven approaches to improve understanding, decision-making, and efficiency in various applications [42,43]. This particular study examines the effectiveness of genetic programming (GP) models in predicting pipeline settlement under seismic conditions on the basis of both experimental and extensive numerical analyses. GP, introduced by Koza in the early 1990s [44], is an evolutionary computing method that creates a clear and structured representation of the provided data. Recent advancements in machine learning have significantly enhanced predictive modeling in geotechnical engineering. For instance, Zhao et al. (2023) [45] developed an XGBoost model optimized with Bayesian techniques to predict tunnel uplift due to overlying excavation, demonstrating the efficacy of ensemble learning in capturing complex geotechnical behaviors. Similarly, Hu et al. (2023) [46] employed various machine learning algorithms to predict the triaxial compressive strength of rocks subjected to high-temperature treatments, highlighting the potential of AI-driven approaches in material property estimation. These studies underscore the growing role of machine learning in addressing complex geotechnical challenges.

According to Al-Jeznawi et al. in 2024 [7], earlier studies in pipeline engineering have focused mainly on the factors influencing pipeline stress, strain, and the interaction between pipes and soil. However, there is a notable lack of research on the intelligent prediction of nonuniform pipeline settlement. Some scholars have started applying neural network technology to predict ground subsidence caused by engineering activities. For example, Huang et al., in 2022 [47], used neural networks to develop a prediction model for ground subsidence due to tunnel excavation. The advantage of ANNs lies in their ability to identify nonlinear relationships between instrumentation parameters and the predicted outcomes. Compared with other settlement prediction methods, ANN models can handle nonlinear problems with numerous parameters efficiently and are not time-consuming [48,49,50,51,52].

Importantly, studies on this important topic are scarce, and traditional methods like manual calculations and analytical approaches often struggle to accurately account for the distinctive characteristics of ground soil. Since research and databases on the use of FRP composite materials for piling foundations are currently very limited [7], this study incorporates data that consider the impact of soil conditions along with the various materials used in the micropiles. Consequently, there has been increasing emphasis on developing new explicit models to accurately predict pipeline settlement under seismic loading. This study aims to fill the existing research gap by creating comprehensive explicit predictive models for vertical displacement in pipelines subjected to seismic loads.

The primary objective of this study is to develop and validate novel explicit models for predicting the maximum settlement of oil pipelines buried in sandy soils and supported by concrete or polymer micropiles under seismic loading. These models are derived using GP and are designed to balance accuracy with practical applicability by simplifying input requirements. This paper is structured as follows: the Section 2 details the algorithms employed and discusses the dataset, including preprocessing steps. The Section 4 presents the proposed models and analyzes the effects of various soil–micropile parameters. Finally, the Section 5 summarize the key contributions, limitations of the models, and potential future research directions.

2. Research Methodology and Data Collection

Figure 1 shows a visual representation of the current research methodology. As shown in Figure 1a, the process begins with comprehensive data collection based on validated finite element (FE) simulations conducted by Al-Jeznawi et al. [53], where various micropile geometries, soil states (dry/saturated), and seismic records were modeled. This was followed by the preparation of a structured dataset comprising key input variables such as pile spacing, diameter, length, and PGA. Figure 1b highlights the current work, which involves the use of GP to develop closed-form predictive models. These models were trained, validated, and tested using a stratified data-splitting strategy. The final step involves performance evaluation using R², MAE (mean absolute error), and RMSE (root mean squared error) to assess accuracy and generalizability. This structured workflow enables a transparent and replicable approach to model development for seismic-induced pipeline settlement. To achieve this, a comprehensive dataset from the literature is utilized to accomplish the following objectives:

• Use an evolutionary machine learning approach to develop closed-form predictive equations for pipeline settlement with both concrete and polymer micropiles during seismic activity;
• Perform a parametric analysis to evaluate how model parameters influence pipeline settlement, thus providing a deeper understanding of the impact of soil and pile design parameters.

A total of 612 data points for each type of micropile material were collected from a study conducted by Al-Jeznawi et al. in 2024 [7], who numerically developed a 3D multipipe grouting micro anti-slide pile model to enhance the seismic performance of buried pipelines. The prediction models developed in this study are based on seven primary input variables: micropile dimensions (length L and diameter D), pile spacing S, soil friction angle (ϕ), soil unit weight (γ), and peak ground acceleration (PGA). The performance of these models was assessed using three metrics: mean absolute error (MAE), mean squared error (MSE), and the coefficient of determination (R²). Al-Jeznawi et al. in 2024 [7] utilized concrete and polyurethane polymer slurries as grouting materials and examined several parameters, including the pile spacing, diameter, and length. Additionally, the effects of soil wetting and different earthquake intensities were assessed through simulations of 50 real earthquakes. The research also assessed the effect of surface loads from a fully loaded truck on a pipeline.

It is important to note that the GP models were developed as surrogate predictive tools trained on output data generated from detailed finite element simulations. The FE framework was employed first to generate settlement data under varied conditions (e.g., micropile dimensions, spacing, and PGA). These results were then used to train the GP algorithm. The GP models do not feed outputs back into the FE simulations, nor were they used to calibrate any numerical model parameters. Instead, they serve as practical, closed-form approximations of the complex relationships embedded in the FE outputs, allowing for rapid settlement prediction in engineering practice without re-running full-scale simulations.

The choice of PGA as the seismic intensity parameter was based on its wide availability, interpretability, and established relevance in seismic design. Although energy- and duration-based measures can provide a more comprehensive view of ground motion, they were not included in this study due to data limitations and the aim to maintain model simplicity. Future studies may explore the integration of such parameters. Additionally, while the GP model structure inherently reflects variable influence, future work could include a formal feature importance analysis to quantify the contribution of each input parameter to model predictions.

3. Analysis Model for Assessing the Seismic Response of Pipelines

The methodology is divided into five subsections, each addressing the objective of this paper of presenting a method to evaluate the seismic resilience of pipelines buried in sandy soils reinforced with concrete and FRP micropiles.

3.1. Finite Element Model for Building the Database

Al-Jeznawi et al. in 2024 [7] developed and validated computational models on the basis of the research conducted by Radwan in 2022 [51]. The objective of their study was to develop a numerical model for a 3D multipipe grouting micro anti-slide pile designed to improve the seismic response of oil pipelines and compare the results with those of concrete micropiles. They used a polyurethane polymer slurry as the grouting material, following the approach of Wang et al. (2021) [52]. In 2024, Al-Jeznawi et al. [7] examined factors like pile spacing, diameter, and length, as well as the impact of soil moisture and different earthquake magnitudes (simulating 50 real earthquakes). The effect of surface loads from a fully loaded truck on the pipeline was also included in the analysis. The model features micropiles with a diameter of 30 cm and includes a high-polymer soil ring pile, 2.5 cm thick, around each pile for analysis. The proposed numerical model, validated in Section 3.2, accurately simulates the cyclic and nonlinear behavior of soil by closely matching the real stress–strain response under cyclic loading. This method outperforms equivalent linear approaches by accounting for nonlinear effects and incorporating strain-dependent soil modulus and damping coefficients. The simulation models both dry and saturated soil conditions using the modified Mohr–Coulomb (MMC) and modified UBCSAND models, respectively. The finite element equations were solved using the modified Newton–Raphson technique.

The MMC model has been demonstrated to effectively represent the dynamic behavior of dry cohesionless soil, as observed by Al-Jeznawi in 2022 [47,53]. Moreover, the modified UBCSAND model is used to simulate the increase in pore water pressure, thereby capturing the potential liquefaction of saturated cohesionless soil, as discussed by Al-Jeznawi et al. in 2022 [47,53]. Rayleigh damping with a damping ratio of 5% was applied. The soil, pavement surface, and piles were modeled using four-node tetrahedral elements. Figure 2 (adapted from [7]) illustrates the use of a fine mesh to guarantee accurate modeling.

To achieve an accurate simulation, a series of nonlinear static–dynamic time–history coupled analyses were performed across various design phases. Interactions at the soil–pile interface were managed using a strength reduction method and an interface reduction coefficient, accounting for the undrained effective stress response of the soil. Following the recommendations from Al-Jeznawi et al. in 2024 [54] and Sharifi et al. in 2020 [55], the finite element analysis covered several stages to capture the transition from elastic to plastic ground behavior. The static phases involved several load increments, ranging from 25 to 40. Furthermore, a time step of 0.01 s was utilized to achieve numerical convergence and accurately simulate pore water pressure generation.

The study by Al-Jeznawi et al. in 2024 [7] utilized polymer micropiles with different spacing (25, 50, 75, and 100 cm), diameters (10, 15, and 20 cm), and lengths (2, 3, and 4 m) and incorporated 50 different earthquake records. These micropiles were installed in cohesionless soil with a relative density (D_r) of 25%. The UBCSAND model, originally developed by Beaty and Byrne [49], requires parameters such as the relative density (D_r = 25%), the initial void ratio (e₀ ≈ 0.80), the shear modulus (G₀, derived from (N₁)₆₀ values), and empirical factors like α, β, and Kσ to capture shear strength degradation under cyclic loading. The soil properties were derived from Radwan in 2022 [51] and followed the methodology outlined by Beaty and Byrne in 2011 [56]. The calibration process relies on the equivalent SPT (Standard Penetration Test) blow count for clean sand, (N₁)₆₀, as determined by ASTM D 1586-99 [57], forming the basis for the primary calibration equations proposed by Beaty and Byrne in 2011 [56]. The simulation included components to model the interaction between the soil and the piles. The key input parameters for the ground materials are detailed in Al-Jeznawi et al. in (2024) [7]. The micropile models were tested using a standard fully loaded truck, specifically a heavy concrete truck with dual wheels. A fully loaded concrete truck exerts a total force of 66,000 pounds (293.5 kN) on the pavement, with 28,000 pounds (124.5 kN) distributed on each of its rear axles. Before proceeding further, it is essential to validate the numerical model by comparing it with previous studies, as detailed in Al-Jeznawi et al. in 2024 [7].

To avoid boundary effects, the lateral boundaries were placed at a minimum distance of five times the pipeline diameter from the pipeline axis, based on established recommendations for dynamic analyses. Absorbing boundary conditions were applied along the lateral and bottom edges using non-reflective elements and viscous dashpots, ensuring that outgoing seismic waves were not reflected back into the domain. This configuration allowed for accurate simulation of free-field conditions and was verified through preliminary sensitivity analyses.

Additional information about the used numerical models and their validation can be found in [7].

3.2. Genetic Programing

Genetic programming (GP) is a subset of evolutionary algorithms (EAs) that addresses problems without the need for the user to define the form or structure of the solution beforehand. In this study, the GP algorithm was selected due to its unique ability to generate closed-form equations that provide interpretable models. Unlike black-box machine learning approaches, such as ANNs or support vector machines (SVMs), GP explicitly represents relationships between variables in a mathematical format, making it particularly suited for engineering applications where interpretability is crucial. In addition, GP is a flexible algorithm that does not require predefined model structures, allowing it to explore a wide range of potential solutions during the optimization process. Also, GP autonomously finds nonlinear relationships and interactions among input variables (automatic feature interaction), which is essential for accurately modeling complex geotechnical phenomena. Finally, the resulting closed-form expressions from GP can be directly applied in practice without the need for specialized computational tools, unlike other algorithms that often require runtime execution of trained models.

In GP, a population of potential solutions is initially generated at random, and each solution’s fitness is assessed via a designated fitness function. The most effective solutions are carried forward through generations, whereas crossover and mutation operations are applied to fewer fit solutions to produce new ones for further evaluation [50,58].

Several key terms, as outlined by Willis et al. in 1997 [59], are essential in understanding GP. The term “population” refers to the set of solutions generated in each generation, with the initial set termed the parent solutions and subsequent sets as child solutions or programs. “Crossover” involves creating a new child program by combining segments from two parent solutions. “Mutation” denotes the creation of a new child program by randomly altering a segment of a parent solution. The “fitness function” evaluates the performance of each GP solution, with metrics such as the coefficient of determination (R²) or error measures such as the mean squared error (MSE) and mean absolute error (MAE) commonly used in regression problems. The general structure of the GP algorithm, adapted from Poli et al. in 2008 [60], is summarized in

Table 1. General outline of the GP algorithm.

1	Generate an initial population of random solutions
2	Repeat
3	Evaluate the fitness of each solution in the population.
4	Select one or more solutions for genetic operations, with selection probability determined by fitness.
5	Apply genetic operations (e.g., crossover and mutation) with predefined probabilities to create new solutions.
6	Until a satisfactory solution is identified, or a termination criterion is met (e.g., reaching a maximum number of generations)
7	Return the best-performing solution

.

In this study, the GP algorithm was implemented using a symbolic regression approach optimized for geotechnical engineering applications. To ensure reproducibility, the following parameters were used throughout the model development process: a population size of 500 individuals per generation, a maximum of 1000 generations, and tournament selection with a tournament size of 5. The genetic operations included a crossover rate of 0.85, a mutation rate of 0.10, and a reproduction rate of 0.05. The termination criterion was defined as either reaching the maximum number of generations or observing no improvement in the best fitness value over 100 consecutive generations. These parameter values were selected based on preliminary tuning and established practices in the literature to ensure an optimal balance between convergence speed and model performance.

3.3. Data Description

The data used in this research were sourced from a comprehensive numerical database specifically designed to evaluate the seismic response of oil pipelines supported by micropiles. This database includes 612 data points for each type of micropile material (concrete or polyurethane polymer). It incorporates key parameters, including micropile dimensions (L and D), S, ϕ, γ, and the PGA, offering a comprehensive view of the factors influencing pile behavior.

Table 2. Statistical description of the dataset.

Attribute	Count	Mean	SD	Min	Max
S (cm)	612	62.5	31.48	25	100
D (cm)	612	11.25	2.97	10	20
L (m)	612	2.25	0.59	2	4
Ø (°)	612	30.33	0.47	30	31
γ (KN/m3)	612	19	1.41	18	21
PGA (g)	612	0.46	0.27	0.02	1.17
SFRP (mm)	612	7.93	3.54	1.02	18.5
SConcrete (mm)	612	11.50	4.72	2.2	26.5

presents the statistical data for the input and output variables, including the minimum value (Min), maximum value (Max), mean, standard deviation (SD), and range of the collected data. Figure 3 shows the distribution of each feature.

In this study, S_FRP refers to the predicted settlement (in mm) of pipelines supported by fiber-reinforced polymer (FRP) micropiles, while S_Concrete denotes the predicted settlement (in mm) of pipelines supported by concrete micropiles. These are the target output variables for the respective GP models developed under varying soil and seismic conditions.

3.4. Data Preprocessing

In this section, correlation analysis, feature selection, and data splitting are implemented to prepare the data for machine learning model development.

3.4.1. Correlation Analysis

A correlation matrix was generated and is illustrated in Figure 4 to analyze the linear relationships between attributes in the dataset. This matrix presents the Pearson correlation coefficient (R) for each pair of features, with values ranging from −1 to 1, as defined in Equation (1). In the literature, the Pearson correlation coefficient is considered a crucial indicator for assessing the relationship between variables [61,62]. Coefficients close to −1 or 1 imply strong linear correlations, whereas values near 0 indicate weak or negligible correlations [63].

$$\mathrm{R}={\displaystyle \frac{\mathrm{c}\mathrm{o}\mathrm{v}\left(\mathrm{X},\mathrm{Y}\right)}{{\mathsf{\sigma }}_{\mathrm{X}}{\mathsf{\sigma }}_{\mathrm{Y}}}}$$

(1)

where $\mathrm{c}\mathrm{o}\mathrm{v}\left(\mathrm{X},\mathrm{Y}\right)$ denotes the covariance between variables X and Y, and ${\mathsf{\sigma }}_{\mathrm{X}}$ and ${\mathsf{\sigma }}_{\mathrm{Y}}$ refer to the standard deviations of X and Y, respectively.

3.4.2. Feature Selection

The correlation matrix in Figure 4 reveals the presence of multicollinearity between the features Ø and γ. Multicollinearity, which occurs when two or more predictor variables in a regression model are highly correlated, can adversely affect the model’s development and performance [64]. To address multicollinearity, we chose to exclude both variables Ø and γ from the list of input features. This exclusion offers two significant benefits: firstly, it eliminates the issue of multicollinearity, ensuring that the remaining input features contribute independently to the model’s predictions. Secondly, by limiting the input features to only the micropile geometrical specifications and the earthquake peak ground acceleration (PGA), the need for additional soil testing is eliminated, ensuring efficiency and practicality in diverse geotechnical scenarios. This option simplifies the model, enhancing the efficiency and applicability of the model and making it more straightforward to implement. In addition, the referred exclusion did not adversely affect the predictive accuracy of the model, as the retained parameters provided sufficient information for settlement prediction.

The use of PGA in this study was guided by its widespread availability in seismic records and its strong correlation with pipeline settlement in the dataset. While PGD (peak ground displacement) provides a direct measure of ground deformation, its limited availability restricted its inclusion in the current model. Future studies could explore the combined use of PGA and PGD to enhance predictive accuracy.

While PGD and PGA were evaluated in terms of their suitability as input variables, Peak Ground Velocity (PGV) was not included in this study due to inconsistent availability across the earthquake dataset. Despite its recognized relevance in seismic hazard assessment, PGV data were not uniformly available for all 50 real earthquake records used in the simulations. Future studies may benefit from incorporating PGV, particularly when working with ground motion databases that provide complete and high-resolution velocity time histories.

3.4.3. Data Split

In the ML framework, the dataset is divided into training, validation, and testing sets to develop, validate, and assess the model. During the training phase, the model parameters are determined exclusively via the training dataset. The model is subsequently validated on unseen data from the validation set to evaluate its generalizability. This iterative process of training and validation continues until an optimal model with the best set of hyperparameters is achieved. The final evaluation of the model’s robustness is performed by testing its performance on the testing set, which provides insights into the model’s ability to predict unseen or future data.

As highlighted by Tokar and Johnson in 1999 [65], the method of data partitioning significantly impacts the model performance. To ensure optimal training and testing, the training, validation, and testing sets should maintain comparable feature distributions. To achieve this goal, a specialized algorithm was developed via Python 3.8.18 to allocate 70% of the original data for training, 20% for validation, and the remaining 10% for testing. This allocation strategy ensures that the distributions and attribute ranges are consistent across the training, validation, and testing sets. This methodology is in line with the same one used in previous studies [66,67].

The summary statistics for these three sets are presented in

Table 3. Dry soil state data split.

Training Set
Attribute	Count	Mean	SD	Min	Max
S (cm)	285	61.84	33.00	25	100
D (cm)	285	11.78	3.40	10	20
L (m)	285	2.37	0.70	2	4
PGA (g)	285	0.46	0.27	0.02	1.17
SFRP (mm)	285	6.06	1.93	1.4	11.2
SConcrete (mm)	285	8.71	2.41	2.2	14.2
Validation set
Attribute	count	mean	SD	min	max
S (cm)	82	67.37	33.72	25	100
D (cm)	82	12.25	3.78	10	20
L (m)	82	2.378	0.69	2	4
PGA (g)	82	0.47	0.25	0.02	1.17
SFRP (mm)	82	6.26	1.74	2.8	10.5
SConcrete (mm)	82	8.95	2.43	3	13.5
Testing set
Attribute	count	mean	SD	min	max
S (cm)	41	57.31	32.23	25	100
D (cm)	41	11.70	3.46	10	20
L (m)	41	2.36	0.62	2	4
PGA (g)	41	0.40	0.30	0.02	1.17
SFRP (mm)	41	5.69	1.74	1.02	11.2
SConcrete (mm)	41	8.62	2.29	3	13

and

Table 4. Saturated soil state split.

Training Set
Attribute	Count	Mean	SD	Min	Max
S (cm)	142	63.02	28.20	25	100
D (cm)	142	10	0	10	10
L (m)	142	2	0	2	2
PGA (g)	142	0.45	0.27	0.02	1.17
SFRP (mm)	142	11.63	3.18	4.5	18.5
SConcrete (mm)	142	16.93	3.21	10.4	26.5
Validation set
Attribute	count	mean	SD	min	max
S (cm)	41	61.58	26.86	25	100
D (cm)	41	10	0	10	10
L (m)	41	2	0	2	2
PGA (g)	41	0.44	0.25	0.04	1.17
SFRP (mm)	41	11.57	3.07	5.8	17.9
SConcrete (mm)	41	16.80	2.99	10.5	23
Testing set
Attribute	count	mean	SD	min	max
S (cm)	21	60.71	30.17	25	100
D (cm)	21	10	0	10	10
L (m)	21	2	0	2	2
PGA (g)	21	0.51	0.28	0.02	1.17
SFRP (mm)	21	11.96	3.10	7	17
SConcrete (mm)	21	17.85	3.11	13	24

for dry soil and saturated soil, respectively, demonstrating their comparable distributions and confirming the appropriateness of the partitioning.

3.5. Model Evaluation

To evaluate the predictive accuracy of the developed models, this study utilizes three evaluation metrics: the coefficient of determination (R²), mean absolute error (MAE), and root mean squared error (RMSE), as described in Equations (2) to (4) [62]. These metrics are commonly used to determine the best-performing regression model. The R² value ranges from 0 to 1, with higher values indicating a more accurate model. In contrast, the MAE and RMSE are error measures that start from 0 and have no upper limit; thus, lower values for these metrics signify better performance of the developed models.

$${\mathrm{R}}^2={\displaystyle \frac{\sum {\left({\widehat{\mathrm{y}}}_{\mathrm{i}}-\overline{{\mathrm{y}}_{\mathrm{i}}}\right)}^2}{\sum {\left({\mathrm{y}}_{\mathrm{i}}-\overline{{\mathrm{y}}_{\mathrm{i}}}\right)}^2}}$$

(2)

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left|{\mathrm{y}}_{\mathrm{i}}-{\widehat{\mathrm{y}}}_{\mathrm{i}}\right|$$

(3)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{y}}_{\mathrm{i}}-{\widehat{\mathrm{y}}}_{\mathrm{i}}\right)}^2}$$

(4)

where ${\mathrm{y}}_{\mathrm{i}}$ is the true data value, in this case, the actual settlement, and where ${\widehat{\mathrm{y}}}_{\mathrm{i}}$ is the predicted value, which equals the settlement value that the specific model predicts.

4. Results

This section presents the closed-form models derived from 1000 generations of the GP algorithm. Four models are introduced: two for concrete micropiles and two for FRP micropiles. Each type of micropile is further categorized on the basis of the soil moisture conditions, with separate models developed for dry and saturated soil states. The following paragraphs provide a detailed overview of each model.

4.1. Dry Soil State

4.1.1. Concrete Micropiles

The developed model is given in Equation (5), while Figure 5 shows the scatter plot and the residual plots for the training, validation, and testing sets.

$$\begin{gathered} {\mathrm{S}}_{\mathrm{C}\mathrm{D}}=-0.25 \mathrm{D}-0.00005 \mathrm{S}+0.007 \mathrm{D}\times \left(43.5 \mathrm{P}\mathrm{G}\mathrm{A}+0.5 \mathrm{S}-2.65 \mathrm{D}\right)+ \\ {\displaystyle \frac{1.5}{1.4\times \left(\left(-3.5 \mathrm{L}+11.5\right)+7.57\right)}}-0.6 \end{gathered}$$

(5)

where ${\mathrm{S}}_{\mathrm{C}\mathrm{D}}$ is the FRP pile settlement (mm), D is the pile’s diameter (cm), S is the pile spacing (cm), L is the length if the pile (m), and PGA is the peak ground acceleration (g).

4.1.2. FRP Micropiles

The developed model is shown in Equation (6), while Figure 6 shows the scatter plot and the residual plots for the training, validation, and testing sets.

$${\mathrm{S}}_{\mathrm{P}\mathrm{D}}=0.01+0.95\times \left(-0.4 \mathrm{D}+0.05 \mathrm{S}-0.25 \mathrm{L}+4 \mathrm{P}\mathrm{G}\mathrm{A}+6.5\right)$$

(6)

where ${\mathrm{S}}_{\mathrm{P}\mathrm{D}}$ is the FRP pile settlement in dry soil (mm), D is the pile’s diameter (cm), S is the pile spacing (cm), L is the length if the pile (m), and PGA is the peak ground acceleration (g).

4.2. Saturated Soil State

For the models developed for the saturated soil state, the spacing of the micropiles and the PGA were sufficient to create highly accurate predictions. However, since these models are data driven, the incorporation of additional data could further refine and improve their accuracy.

4.2.1. Concrete Micropiles

The developed model is shown in Equation (7), while Figure 7 shows the scatter plot and the residual plots for the training, validation, and testing sets.

$${\mathrm{S}}_{\mathrm{C}\mathrm{S}}=8.65+0.68\times \left(12.23 \mathrm{P}\mathrm{G}\mathrm{A}+0.1 \mathrm{S}+0.315\right)$$

(7)

where ${\mathrm{S}}_{\mathrm{C}\mathrm{S}}$ is the concrete pile settlement in saturated soil (mm), S is the pile spacing (cm), and PGA is the peak ground acceleration (g).

4.2.2. FRP Micropiles

The developed model is shown in Equation (8), while Figure 8 shows the scatter plot and the residual plots for the training, validation, and testing sets.

$${\mathrm{S}}_{\mathrm{P}\mathrm{S}}=5.23\times \sqrt[2]{13.42 \mathrm{P}\mathrm{G}\mathrm{A}+\sqrt[2]{57.38+7.6 \mathrm{S}}}-16.3$$

(8)

where ${\mathrm{S}}_{\mathrm{P}\mathrm{S}}$ is the FRP pile settlement in saturated soil (mm), S is the pile spacing (cm), and PGA is the peak ground acceleration (g).

Table 5. Summary of model performance.

Model	Training Set			Validation Set			Testing Set
	R2	RMSE (mm)	MAE (mm)	R2	RMSE (mm)	MAE (mm)	R2	RMSE (mm)	MAE (mm)
${\mathrm{S}}_{\mathrm{C}\mathrm{D}}$	0.84	0.93	0.76	0.80	1.09	0.90	0.83	0.93	0.74
${\mathrm{S}}_{\mathrm{P}\mathrm{D}}$	0.85	0.72	0.57	0.80	0.77	0.61	0.76	0.83	0.66
${\mathrm{S}}_{\mathrm{C}\mathrm{S}}$	0.88	1.10	0.89	0.80	1.31	1.12	0.85	1.18	0.95
${\mathrm{S}}_{\mathrm{P}\mathrm{S}}$	0.92	0.86	0.68	0.92	0.81	0.68	0.92	0.83	0.73

presents a summary of the evaluation metrics for all of the proposed models.

5. Discussion and Conclusions

Addressing the issues caused by unstable sandy soil affecting buried pipelines is essential to reduce the risks of over settlement and fractures due to overstress. The observed reduction in pipeline settlement with FRP micropiles can be attributed to their role in enhancing the overall stiffness of the soil–structure system. As slender, high-strength inclusions, FRP micropiles help distribute seismic-induced loads over a wider area and deeper soil layers, reducing stress concentrations beneath the pipeline. Additionally, the presence of micropiles limits the development of shear strains in the surrounding sandy soil by providing lateral restraint and improving confinement. This increases the composite stiffness and mitigates excessive deformation, ultimately resulting in reduced vertical displacement of the pipeline during seismic excitation. The use of micropiles, especially those constructed from FRPs, offers a practical solution for improving soil stability under pipelines and provides a more sustainable approach to cohesionless soil reinforcement. This study introduces innovative methods to evaluate the seismic resilience of micropiles by predicting the maximum settlement of underground pipelines. It addresses a gap in previous research, which has focused predominantly on static load scenarios and employed complex machine learning models such as ANNs. Instead, this study presents accurate and transparent closed-form models using GP. These models use micropile dimensions (L and D), S, ϕ, γ, and the PGA as inputs. This approach allows for a straightforward application without the need for costly laboratory tests to determine other soil properties.

The accuracy of these models was evaluated via the R², MAE, and RMSE metrics. The GP model generally demonstrated good accuracy for both concrete and polymer micropiles. Specifically, for the concrete micropiles under dry conditions, the R², RMSE, and MAE values were 0.83, 0.93, and 0.74, respectively. For the concrete micropiles under saturated conditions, these values were 0.85, 1.18, and 0.95, respectively. For the polymer micropiles under dry conditions, the values were 0.76, 0.83, and 0.66, respectively, and for the polymer micropiles under saturated conditions, the values were 0.92, 0.83, and 0.73, respectively.

The previously referred findings show that the GP model shows a relatively poorer fit for drying conditions. This can be due to the following:

i.

FRP micropiles, due to their inherent material properties, are more sensitive to environmental and drying conditions. This variability could introduce higher variability in the dataset (in fact, the dataset for polymer micropiles under drying conditions had a slightly higher proportion of extreme data points), potentially impacting the models’ ability to capture precise settlements under drying conditions;

ii.

to enhance the models’ applicability and reduce complexity, some secondary parameters, such as moisture loss rate and polymer-specific drying characteristics, were excluded from the input variables. This simplification may have contributed to a reduced fit in this specific condition.

In spite of the aforementioned points, the findings indicate that the developed models are, in general, accurate and also practical and user-friendly for designers. They require only the values of micropile dimensions, spacing, angle of internal friction, soil unit weight, and peak ground acceleration, making them easy to implement in practice without the need for extensive and expensive laboratory tests. Nevertheless, for drying scenarios and for practical applications, it is suggested to complement the GP model with field measurements or incorporate additional polymer-specific parameters, provided that the necessary data are available.

Finally, the intricate interactions among seismic activity, soil properties, and structural designs highlight the need to account for these factors in pipeline design and evaluation to ensure that they can withstand seismic hazards.

Finally, it should be considered that the proposed models in this research focus on micropile dimensions, spacing, and PGA as key inputs, indirectly reflecting the conditions of sandy soil. While specific soil properties, such as cohesion or permeability, were not explicitly included, the models were designed to balance accuracy and practical usability. Future studies could extend the approach to include direct soil property parameters to further enhance model specificity for diverse sandy soil conditions.

Also, in spite of the models’ applicability being reliable within the conditions represented by the current dataset, it could be expanded in future work by including more diverse pipeline scenarios. The training and validation data are sourced from the same database, ensuring consistency in model development. However, this approach limits the generalizability of the results to the conditions within the dataset. Future work should incorporate external datasets representing diverse oil pipeline scenarios to enhance the models’ applicability.

While the proposed GP models provide a practical tool for predicting pipeline settlement under seismic loading, this study does not include a direct performance comparison with other predictive models, such as artificial neural networks (ANNs), random forests, or design code-based frameworks (e.g., ASCE or Eurocode). These models are intended to complement, not replace, standard procedures, especially for preliminary design or in resource-limited settings. Future work should benchmark their performance and cost-effectiveness against established guidelines through case studies and field validation.

While the developed models demonstrated strong predictive capability within the scope of the current dataset, it is important to acknowledge that the numerical database was based on simulations in sandy soil conditions. Although a wide range of pile geometries, soil states (dry and saturated), and real earthquake records were considered, the dataset does not cover other soil types such as clayey or silty soils, nor different pipeline configurations. This represents a limitation in terms of generalizability. Future studies are encouraged to expand the dataset to include diverse geotechnical profiles and pipeline geometries, as well as seismic records with more varied characteristics, to enhance the robustness and applicability of the predictive models in broader engineering contexts.

6. Practical Application and Design Implications

The proposed GP-based predictive models offer practical tools for the preliminary design and assessment of buried pipelines in sandy soils reinforced with micropiles. Engineers can use the closed-form equations to estimate expected pipeline settlement under various seismic loading scenarios, using only a small set of easily obtainable input parameters, micropile dimensions (L and D), spacing (S), and peak ground acceleration (PGA). This enables rapid evaluation of alternative reinforcement layouts during early planning phases.

Table 6. Predicted settlement of FRP micropile-reinforced pipelines at PGA = 0.9 g.

Pile Spacing (cm)	Predicted Settlement (mm, FRP)
100	11.8
50	7.2
25	5.0

provides an example of predicted settlements for FRP micropiles at a fixed seismic intensity (PGA = 0.9 g), showing how closer pile spacing can significantly reduce vertical displacement. Such simplified outputs can serve as quick reference guides for selecting micropile configurations that meet serviceability requirements.

Future work could involve developing simplified design charts or software tools integrating these equations to support more intuitive application in field projects.