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AMERICAN Journal of Engineering,  
Mechanics and Architecture 

Volume 3, Issue 3, 2025 ISSN (E): 2993-2637 

 
  

Development of Predictive Model for Corrosion Management of Oil 

and Gas Pipelines, Validatiion of the Model Using Artificial Neural 

Network (Ann) and Optimization Using Genetic Algorithm in the 

Niger Delta Area of Nigeria 
 

Igbesi, F. C., Dolor G. A. 

Department of Mechanical Engineering, Delta State Polytechnic, Otefe-Oghara 

 

Abstract: The potential rupture of pipelines poses significant threats to the environment, human 

safety and infrastructure integrity. To mitigate these risks, pipelines require constant monitoring 

and maintenance to detect and rectify defects such as corrosion before they lead to failure. 

However, the regular monitoring and non-destructive testing of pipelines incur substantial costs. 

Consequently, there is a growing interest in research focused on predictive corrosion monitoring 

of pipelines based on easily measurable operational parameters. This study was aimed to develop 

predictive model for corrosion management in oil and gas pipelines in the Niger Delta area of 

Nigeria. 

Secondary data on mean corrosion rates, mean pH levels, mean temperatures, mean pressures, 

and mean aqueous CO2 partial pressures were obtained from an oil and gas multinational 

company spanning the years 2007 to 2011. Polynomial regression and Artificial Neural Network 

(ANN) methodologies were chosen as suitable methods for data analysis. Polynomial regression 

and ANN models were developed and subsequently optimized using a genetic algorithm. The 

models' validity was assessed using Goodness of Fit Indices (GFI). 

For the full second-degree quadratic polynomial model yielded the following results for both 

training and testing data sets: Coefficient of Determination (R2) was 0.9869/0.9361, Root Mean 

Square Error (RMSE) was 0.0007/0.0012, Mean Biased Error (MBE) was 0.0000/0.0002, Mean 

Absolute Biased Error (MABE) was 0.0004/0.0008, Mean Percentage Error (MPE) was 

0.0006/0.0636, and correlation coefficient (r) was 0.9934/0.9689. The goodness of fit for the 

reduced second-degree model for both training and testing datasets provided the following 

results: R2 was 0.9859/0.9341, RMSE was 0.0007/0.0012, MBE was 0.0000/0.0001, MABE was 

0.0004/0.0008, MPE was 0.0006/0.0390, and r was 0.9929/0.9676. Hypothesis testing of the full 

second-degree polynomial model for significance revealed that all model parameters were 

significant at the 95% confidence interval, except for the coefficients related to the interaction of 

mean pressure and mean aqueous CO2 partial pressure (x_(2) x_3) and the square of mean 

aqueous CO2 partial pressure (x_4^2). Comparatively, the ANN model exhibited slightly higher 

accuracy than the polynomial model, reinforcing the validity of the polynomial modelling and 

interaction analyses. Furthermore, the goodness of fit indices for the ANN model during both 

training and testing phases were as follows: R2 was 0.9965/0.9158, RMSE was 0.0004/0.0014, 

MBE was 0.0000/-0.0003, MABE was 0.0003/0.0011, MPE was 0.0003/-0.0905, and r was 

0.9982/0.9608. Through optimization with a genetic algorithm, it was determined that the 

minimal corrosion rate occurred at mean pH of 8.446, mean temperature was 23.692°C, mean 

pressure was 15.725 bar, and mean aqueous CO2 partial pressure was 2.022 bar. This research 

contributes valuable insights into the technological applications and policy implications of 



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predictive corrosion monitoring for pipelines. It also highlights avenues for further research in 

this critical field. 

Keywords: Oil and gas; pipelines; corrosion; model; hazard; validation; Optimization. 

 

 

1. Introduction 

Pipelines represent safety and pinnacle of efficiency in transporting oil and gas, owing to their 

inherent low health risk profile in the absence of structural failures (Adegboye et al., 2019; 

Parlak, and Yavasoglu, 2023). However, the aging and deterioration of pipeline infrastructure 

emerge as formidable challenges within the pipeline industry. In comparison to alternative 

modes of fuel transportation such as highways and railways, pipelines exhibit significantly 

diminished safety and environmental risks. This assertion is supported by Popescu, and Gabor 

(2021) who substantiated this claim through a quantitative analysis, revealing a rate of about 

0.03 fatalities per billion ton–miles for pipelines in contrast to approximately 1.20 fatalities for 

railway transportation and close to 9.22 fatalities for highway transportation. 

It is notable that the proliferation of pipeline networks, surpassing 1,243 miles in length, has 

occurred in more than 60 countries. This expansion followed the construction of the inaugural oil 

pipeline, which measured 109 miles in length and had a diameter of 6 inches, in Pennsylvania, 

USA, in 1879 (El-Abbasy et al., 2015; Borden, 2022). The progressive deterioration of pipelines 

over time stems from their exposure to natural forces in their surroundings and the dynamic 

pressures exerted by the transported fluids. If left unaddressed, this degenerative process 

culminates in failure, accompanied by grave consequences, encompassing environmental 

contamination, health hazards, and substantial economic losses (El-Abbasy et al., 2015). 

The repercussions of pipeline failures extend to encompass severe economic, social, and 

environmental devastations, resulting in substantial repair expenses, human injuries, extensive 

environmental pollution, and widespread disruptions to daily life. Nigeria, a prominent oil-

producing nation, relies significantly on the income generated from oil and gas sales. According 

to a report by the World Economic Outlook, between August 2022 and February 2023, primary 

commodity prices fell by 28.2%, mainly due to a substantial 46.4% drop in energy commodity 

prices. European natural gas prices notably plunged by 76.1%, driven by reduced consumption 

and high storage levels. In contrast, base and precious metal prices rebounded, increasing by 

19.7% and 3.3%, respectively. Food prices also saw a modest rise of 1.9% during this period. 

This Special Feature analyzes how these declines, particularly in fossil fuel and mineral 

extraction, have impacted the macroeconomic activity of commodity exporters (WEO, 2023). 

Notably, data from 2016, reveals that petroleum exports by Nigeria in 2016 amounted to 27,788 

million USD, representing a remarkable 80.1% contribution to the total export value of 34,704 

million USD (Heim, 2019). 

The extensive network of pipelines that crisscrosses Nigeria, as highlighted by Umar et al. 

(2021) serves as the primary conduit for transporting the majority of the petroleum products of 

the nation, covering a vast expanse of approximately 3,106 miles. Consequently, the issue of 

pipeline failures, stemming from factors including material defects, corrosion, third-party 

activities, mechanical loss, and environmental factors, has been extensively documented within 

the Nigerian context. Furthermore, when compared to developed nations, Nigeria has 

experienced an unacceptably high rate of pipeline failures in recent years (Umar et al., 2021). 

Dawatola (2012) reported an alarming increase in pipeline ruptures in Nigeria, with more than 

twenty cases in 2007 and over thirty cases in 2008. More recent data from the Nigerian National 

Petroleum Corporation (NNPC, 2016) indicated a concerning trend, recording forty-nine cases of 

pipeline rupture in 2015 and an even higher fifty-five cases in 2016. These statistics underscore 

the urgent need for Nigeria, more so than any other nation, to address the pervasive issue of 

pipeline failures and reduce them to levels comparable to those in developed economies. Such 



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efforts are essential to harness the wasted resources and mitigate the severe consequences 

associated with these failures, redirecting them towards more productive endeavors. 

This study represents a significant step in this direction, focusing on model-based maintenance 

strategies to minimize the risks of pipeline failure and its associated catastrophic consequences. 

It is crucial to highlight that pipeline failures not only lead to economic losses but also pose 

challenges to human lives and the environment. To put the gravity of pipeline failures into 

perspective, one can recall the tragic incident in Lagos, Nigeria, in 2006, where a pipeline 

explosion claimed the lives of over two hundred people (Dawotola, 2012). Such harrowing 

incidents have left indelible marks on the histories of many other nations as well. Addressing this 

issue is paramount to safeguard lives, protect the environment, and promote sustainable 

development. The development of model-based approaches has emerged as a cornerstone in 

modernizing pipeline transportation, offering a systematic and data-driven methodology to 

optimize operations, enhance safety, and ensure the integrity of pipeline infrastructure (Aldoseri, 

Al-Khalifa, and Hamouda, 2023). Model-based approaches leverage mathematical models, 

simulation techniques, and advanced analytics to predict pipeline behavior, assess risks, and 

inform decision-making across various aspects of pipeline operations. 

Historically, pipeline operations relied on manual inspections, reactive maintenance strategies, 

and periodic maintenance schedules. However, as technological advancements have accelerated, 

there has been a paradigm shift towards model-based approach that harness the power of data 

and computational modeling to drive improvements in efficiency, reliability, and safety (Molęda 

et al., 2023). Corrosion involves the gradual loss of metallic material due to electrochemical and 

mechanical processes. It poses a common threat to the structural integrity of aging oil and gas 

pipelines. Over time, corrosion weakens the pipes, eventually leading to their failure 

(Mahmoodian and Li, 2016), making it the primary cause of pipe failures in many countries. 

During corrosion, metal atoms detach from the bulk material, forming compounds with oxygen 

and water. Therefore, the proper management of pipeline integrity requires vigilant corrosion 

monitoring through corrosion growth rate models, ensuring that corrosion does not exceed a 

certain threshold to minimize the risk of failure. 

In the corrosion process of pipelines, the main cathode reactants are oxygen, carbonic acid, free-

state H2S, and organic acids (Kahyarian et al., 2019). The rate of corrosion in a pipeline is 

influenced by both external and internal factors. External factors encompass the working 

environment of the pipe, such as soil chemistry and moisture for buried pipes, water chemistry 

for submerged pipes, and air content for above-ground pipes. Internal factors include the 

thermodynamic conditions (temperature and pressure), flow rate, pH level, and relative 

electromotive force of the various materials within the pipeline system. 

Dawotola et al. (2011) and Huang et al. (2023) recognized an extensively and robust validated 

approach for the maintenance of oil and gas pipelines, focusing on those susceptible to long-term 

corrosion-induced failure. Their primary objective was to optimize the inspection intervals for 

these petroleum pipelines. The assessment criteria considered included both the frequency of 

failure, determined by fitting a homogeneous Poisson process and power law, and the 

consequences of failure. The latter aspect was evaluated in terms of economic losses, 

environmental impact, and human safety, encompassing scenarios involving small leaks, large 

leaks, and pipeline ruptures. The dataset employed in their analysis covered a range of corrosion 

mechanisms, including uniform corrosion, pitting corrosion, and stress corrosion cracking. The 

central focus of their innovative risk-based integrity maintenance optimization approach 

revolved around determining the economic losses associated with pipeline failure while 

considering human risk and maintenance budget constraints. To enhance the assessment's 

accuracy, Dawotola and their team incorporated structured expert judgment to provide frequency 

of failure assessments for identified failure mechanisms within the studied pipeline. They also 

employed the Analytic Hierarchy Process (AHP) to obtain relative likelihood values for 

attributes of failure mechanisms with very low probability of occurrence, thus contributing to a 



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comprehensive risk management assessment for crude pipelines susceptible to rupture. Their risk 

assessment methodology considered various failure mechanisms, calculating both the frequency 

of failure and the potential consequences, typically measured in terms of historical costs 

associated with failures, for different segments of crude oil pipelines. To demonstrate the 

application of their methodology, they employed historical data from a real-world crude oil 

pipeline owned by the Nigerian Petroleum Development Company (NPDC). This pipeline, 

commissioned in 1989 to supply crude oil to the southwestern region of Nigeria, served as the 

basis for combining frequency of failure and consequences of failure (Huang et al., 2023). 

Dai et al. (2017) also delved into estimating the failure rate of cross-country crude product 

pipelines using historical failure data. They conducted a comparative study involving two 

hypotheses: one assuming minimal repair models along with a Homogeneous Poisson Process 

(HPP), and the other assuming the same repair models but employing a non-Homogeneous 

Poisson Process (nHPP). Their analysis led to the acceptance of the null hypothesis, indicating 

that the number of corrosions occurring in onshore crude product pipelines follows the HPP 

rather than the nHPP. This finding suggested that pipelines installed at the same time would 

exhibit similar mean times between failures and failure intensities, all else being equal. This 

conclusion was drawn from a comparative analysis of data from three different API 5L X42 

pipelines owned by the Nigerian National Petroleum Company (NNPC). 

Dawotola et al. (2012) and Marhavilas, and Koulouriotis (2021) employed probabilistic methods 

to develop an innovative methodology capable of assessing the acceptability of risk levels based 

on cost-benefit analysis. They derived these acceptability criteria by considering historical trends 

in non-voluntary deaths and overall national fatalities. Through this approach, they established 

acceptable levels of individual and societal risk and showcased the methodology's applicability 

to critical infrastructure, particularly petroleum pipelines. Within the scope of their study, 

Dawotola et al. (2012) pursued an optimal maintenance strategy for a petroleum pipeline. This 

strategy involved a structured expert judgment process that considered various failure 

mechanisms. This judgment process led to the calculation of failure frequencies and the 

determination of optimal maintenance intervals for petroleum pipelines based on these failure 

frequencies. The optimization of maintenance intervals was carried out through two alternative 

approaches: Use-based minimization, with the objective function being the expected total cost 

associated with a petroleum pipeline. Maximizing the benefit/cost ratio of the pipeline while 

simultaneously minimizing operational and failure costs. The latter approach was deemed less 

data-intensive, making it a practical and efficient choice for optimizing maintenance strategies 

(Marhavilas, and Koulouriotis 2021).  

Consequently, employing these model-based methods presents the most viable avenue for 

implementing condition monitoring and failure control across the extensive oil and gas pipeline 

network in Nigeria. The study aims to further improve the predictive precision of state-of-the-art 

methodologies and adapt the emerging new techniques to the Nigerian context, thereby 

establishing a robust framework for addressing the critical issue of pipeline failures in the Niger 

Delta Area of Nigeria. 

2. Methods 

a. Data collection 

The secondary corrosion data obtained are: mean pH, mean temperature, mean pressure, mean 

aqueous CO2 and mean corrosion rate were obtained from an oil and gas multinational company 

in Nigeria. However, for data privacy and confidentiality reason, the company name is not 

provided. The data set were collected from 2007 to 2011. 

Data analysis 

The polynomial regression analysis was used to analyze the obtained data. The first and second 

polynomial regression were used, models were developed in both case. Polynomial regression 



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was chosen for the analysis because it gave high performance indices using the Goodness of Fit 

Indices (GFI). Models were developed. 

b. Model Development 

The much more advanced generalized approach which cover general-order polynomial analyses 

is effectively handled on the framework of vector/matrix algebra. For such analysis the response 

is generally related to the predictors as follows (Levin, 1998); 

𝑜𝑦(𝒙) = [𝒂(𝒙)]𝑇{∑ 𝒂(𝒙𝑖)[𝒂(𝒙𝑖)]
𝑇𝑛

𝑖=1 }−1∑ 𝒂(𝒙𝑖)𝑦𝑖
𝑛
𝑖=1 + 𝑒   (1) 

Where: 𝒂(𝒙) = {𝑎1(𝑥)𝑎2(𝑥)…𝑎𝑟(𝑥)}
𝑇is the polynomial of basis vectors  

𝑎𝑗(𝑥) represent any of all the possible interactions of the independent variables from zero to 

order 𝑝.  

𝑛 is the number of experimental/sampling runs  

𝑑 is the number of independent variables.  

The length 𝑟 of 𝒂(𝑥)is generally given as 𝑟 =
(𝑑+𝑝)!

𝑑!𝑝!
. In the polynomial model, the numerical 

term  

𝒃 = {∑ 𝒂(𝒙𝑖)[𝒂(𝒙𝑖)]
𝑇𝑛

𝑖=1 }−1∑ 𝒂(𝒙𝑖)𝑦𝑖
𝑛
𝑖=1      (2) 

represent the vector of the model coefficients in terms of the data sets of the predictors and the 

targets. A good understanding of how to handle equation (2) is all that is need in polynomial 

regression. For illustration with some of the simplest multiple linear regression cases, consider 

two predictors, 𝑥1 and 𝑥2.  

For the first degree (1stPRA) case: 

𝒂(𝒙) = {
1
𝑥1
𝑥2
}         (3) 

and 

𝒃 = [∑ {
1
𝑥1𝑖
𝑥2𝑖
} {1 𝑥1𝑖 𝑥2𝑖  }

𝑛
𝑖=1 ]

−1

∑ {
1
𝑥1𝑖
𝑥2𝑖
} 𝑦𝑖

𝑛
𝑖=1       (4) 

where the summation signs in equation (3.4) apply element-by-element to the matrix 

{
1
𝑥1𝑖
𝑥2𝑖
} {1 𝑥1𝑖 𝑥2𝑖  } and the vector {

1
𝑥1𝑖
𝑥2𝑖
} 𝑦𝑖. Therefore 

𝒃 = [

∑ 1𝑛
𝑖=1 ∑ 𝑥1𝑖

𝑛
𝑖=1 ∑ 𝑥2𝑖

𝑛
𝑖=1

∑ 𝑥1𝑖
𝑛
𝑖=1 ∑ 𝑥1𝑖

2𝑛
𝑖=1 ∑ 𝑥1𝑖𝑥2𝑖

𝑛
𝑖=1

∑ 𝑥2𝑖
𝑛
𝑖=1 ∑ 𝑥1𝑖𝑥2𝑖

𝑛
𝑖=1 ∑ 𝑥2𝑖

2𝑛
𝑖=1

]

−1

{

∑ 𝑦𝑖
𝑛
𝑖=1

∑ 𝑥1𝑖𝑦𝑖
𝑛
𝑖=1

∑ 𝑥2𝑖𝑦𝑖
𝑛
𝑖=1

}    (5) 

For the quadratic case 

𝒂(𝒙) =

{
 
 

 
 

1
𝑥1
𝑥2
𝑥1
2

𝑥1𝑥2
𝑥2
2 }
 
 

 
 

         (6) 

and 



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𝒃 =

[
 
 
 
 
 

∑

{
 
 

 
 

1
𝑥1𝑖
𝑥2𝑖
𝑥1𝑖
2

𝑥1𝑖𝑥2𝑖
𝑥2𝑖
2 }
 
 

 
 

{1 𝑥1𝑖 𝑥2𝑖  𝑥1𝑖
2  𝑥1𝑖𝑥2𝑖 𝑥2𝑖

2 }𝑛
𝑖=1

]
 
 
 
 
 
−1

∑

{
 
 

 
 

1
𝑥1𝑖
𝑥2𝑖
𝑥1𝑖
2

𝑥1𝑖𝑥2𝑖
𝑥2𝑖
2 }
 
 

 
 

𝑦𝑖
𝑛
𝑖=1    (7a) 

This becomes 

𝒃 =

[
 
 
 
 
 
 
∑ 1𝑛
𝑖=1 ∑ 𝑥1𝑖

𝑛
𝑖=1 ∑ 𝑥2𝑖

𝑛
𝑖=1 ∑ 𝑥1𝑖

2𝑛
𝑖=1 ∑ 𝑥1𝑖𝑥2𝑖

𝑛
𝑖=1 ∑ 𝑥2𝑖

2𝑛
𝑖=1

∑ 𝑥1𝑖
𝑛
𝑖=1 ∑ 𝑥1𝑖

2𝑛
𝑖=1 ∑ 𝑥1𝑖𝑥2𝑖

𝑛
𝑖=1 ∑ 𝑥1𝑖

3𝑛
𝑖=1 ∑ 𝑥1𝑖

2 𝑥2𝑖
𝑛
𝑖=1 ∑ 𝑥1𝑖𝑥2𝑖

2𝑛
𝑖=1

∑ 𝑥2𝑖
𝑛
𝑖=1 ∑ 𝑥1𝑖𝑥2𝑖

𝑛
𝑖=1 ∑ 𝑥2𝑖

2𝑛
𝑖=1 ∑ 𝑥1𝑖

2 𝑥2𝑖
𝑛
𝑖=1 ∑ 𝑥1𝑖𝑥2𝑖

2𝑛
𝑖=1 ∑ 𝑥2𝑖

3𝑛
𝑖=1

∑ 𝑥1𝑖
2𝑛

𝑖=1 ∑ 𝑥1𝑖
3𝑛

𝑖=1 ∑ 𝑥1𝑖
2 𝑥2𝑖

𝑛
𝑖=1 ∑ 𝑥1𝑖

4𝑛
𝑖=1 ∑ 𝑥1𝑖

3 𝑥2𝑖
𝑛
𝑖=1 ∑ 𝑥2𝑖

2 𝑥2𝑖
2𝑛

𝑖=1

∑ 𝑥1𝑖𝑥2𝑖
𝑛
𝑖=1 ∑ 𝑥1𝑖

2 𝑥2𝑖
𝑛
𝑖=1 ∑ 𝑥1𝑖𝑥2𝑖

2𝑛
𝑖=1 ∑ 𝑥1𝑖

3 𝑥2𝑖
𝑛
𝑖=1 ∑ 𝑥1𝑖

2 𝑥2𝑖
2𝑛

𝑖=1 ∑ 𝑥2𝑖𝑥2𝑖
3𝑛

𝑖=1

∑ 𝑥2𝑖
2𝑛

𝑖=1 ∑ 𝑥1𝑖𝑥2𝑖
2𝑛

𝑖=1 ∑ 𝑥2𝑖
3𝑛

𝑖=1 ∑ 𝑥2𝑖
2 𝑥2𝑖

2𝑛
𝑖=1 ∑ 𝑥2𝑖𝑥2𝑖

3𝑛
𝑖=1 ∑ 𝑥2𝑖

4𝑛
𝑖=1 ]

 
 
 
 
 
 
−1

{
  
 

  
 

∑ 𝑦𝑖
𝑛
𝑖=1

∑ 𝑥1𝑖𝑦𝑖
𝑛
𝑖=1

∑ 𝑥2𝑖𝑦𝑖
𝑛
𝑖=1

∑ 𝑥1𝑖
2 𝑦𝑖

𝑛
𝑖=1

∑ 𝑥1𝑖𝑥2𝑖𝑦𝑖
𝑛
𝑖=1

∑ 𝑥2𝑖
2 𝑦𝑖

𝑛
𝑖=1 }

  
 

  
 

  

(7b) 

Many regression models have more than two predictor variables. For such systems, it can be 

inferred from the foregoing that the response in linear regression is simply represented in the 

form of simple summation as follows; 

𝑦 = 𝑎0 + ∑ 𝑎𝑖𝑥𝑖
𝑑
𝑖=1 + 𝑒     (8) 

Where: 𝑎𝑖 are the coefficients, 

𝑥𝑖 is the ith of 𝑑 predictors  

𝑒 is the error of the analysis which is assumed to be normally distributed.  

Excluding the error and substituting 𝑛 measured/sampled data points in equation (8) becomes  

𝑦(1) = 𝑎0 + ∑ a𝑖𝑥𝑖
(1)𝑑

𝑖=1

𝑦(2) = 𝑎0 + ∑ a𝑖𝑥𝑖
(2)𝑑

𝑖=1

⋮
⋮

𝑦(𝑛) = 𝑎0 + ∑ a𝑖𝑥𝑖
(𝑛)𝑑

𝑖=1

       (9a) 

On minimizing the sum of square errors between the left- and right-hand-side of equation (9b), 

the normal equations  

∑ 𝑦(𝑗)𝑛
𝑗=1  =  𝑛𝑎0 + ∑ 𝑎𝑖 ∑ 𝑥𝑖

(𝑗)𝑛
𝑗=1

𝑑
𝑖=1       (10a) 

∑ 𝑥1
(𝑗)
𝑦(𝑗)𝑛

𝑗=1  =  𝑎0∑ 𝑥1
(𝑗)𝑛

𝑗=1 + ∑ 𝑎𝑖 ∑ 𝑥1
(𝑗)
𝑥𝑖
(𝑗)𝑛

𝑗=1
𝑑
𝑖=1     (10b) 

∑ 𝑥2
(𝑗)
𝑦(𝑗)𝑛

𝑗=1  =  𝑎0∑ 𝑥2
(𝑗)𝑛

𝑗=1 + ∑ 𝑎𝑖 ∑ 𝑥2
(𝑗)
𝑥𝑖
(𝑗)𝑛

𝑗=1
𝑑
𝑖=1     (10c) 

⋮ 

∑ 𝑥𝑑
(𝑗)
𝑦(𝑗)𝑛

𝑗=1  =  𝑎0∑ 𝑥𝑑
(𝑗)𝑛

𝑗=1 + ∑ 𝑎𝑖 ∑ 𝑥𝑑
(𝑗)
𝑥𝑖
(𝑗)𝑛

𝑗=1
𝑑
𝑖=1      (10d) 

are generated. These normal equations are solved simultaneously to give the coefficients 𝑎0 and 

𝑎𝑖. 

The general quadratic model for the problem deriving from equation (1) reads  

𝑦 = 𝑎0 + ∑ 𝑎𝑖𝑥𝑖
𝑑
𝑖=1 + ∑ 𝑎𝑖,𝑖𝑥𝑖

22𝑑
𝑖=𝑑+1 + ∑ 𝑎𝑖,𝑘𝑥𝑖𝑥𝑘𝑖≠𝑘    (11) 



23   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

Substituting 𝑛 measured/sampled data points in equation (8) becomes 

𝑦(1) = 𝑎0 + ∑ 𝑎𝑖𝑥𝑖
(1)𝑑

𝑖=1 + ∑ 𝑎𝑖,𝑖𝑥𝑖
(1)22𝑑

𝑖=𝑑+1 + ∑ 𝑎𝑖,𝑘𝑥𝑖
(1)𝑥𝑘

(1)
𝑖≠𝑘

𝑦(2) = 𝑎0 + ∑ 𝑎𝑖𝑥𝑖
(2)𝑑

𝑖=1 + ∑ 𝑎𝑖,𝑖𝑥𝑖
(2)22𝑑

𝑖=𝑑+1 + ∑ 𝑎𝑖,𝑘𝑥𝑖
(2)𝑥𝑘

(2)
𝑖≠𝑘

⋮
⋮

𝑦(𝑛) = 𝑎0 + ∑ 𝑎𝑖𝑥𝑖
(𝑛)𝑑

𝑖=1 + ∑ 𝑎𝑖,𝑖𝑥𝑖
(𝑛)22𝑑

𝑖=𝑑+1 + ∑ 𝑎𝑖,𝑘𝑥𝑖
(𝑛)𝑥𝑘

(𝑛)
𝑖≠𝑘

   (12) 

Also, for the quadratic (2ndPRA) case, on minimizing the sum of square errors between the left- 

and right-hand-side of equation (12), the normal equations  

∑ 𝑦(𝑗)𝑛
𝑗=1  =  𝑛𝑎0 + ∑ 𝑎𝑖 ∑ 𝑥𝑖

(𝑗)𝑛
𝑗=1

𝑑
𝑖=1 + ∑ 𝑎𝑖,𝑖 ∑ 𝑥𝑖

(𝑗)2𝑛
𝑗=1

2𝑑
𝑖=𝑑+1 + ∑ 𝑎𝑖,𝑘 ∑ 𝑥𝑖

(𝑗)
𝑥𝑘
(𝑗)𝑛

𝑗=1𝑖≠𝑘   

    (13a) 

∑𝑥1
(𝑗)
𝑦(𝑗)

𝑛

𝑗=1

 =  𝑎0∑𝑥1
(𝑗)

𝑛

𝑗=1

+∑ 𝑎𝑖∑𝑥1
(𝑗)
𝑥𝑖
(𝑗)

𝑛

𝑗=1

𝑑

𝑖=1
+∑ 𝑎𝑖,𝑖∑𝑥1

(𝑗)
𝑥𝑖
(𝑗)2

𝑛

𝑗=1

2𝑑

𝑖=𝑑+1

+∑ 𝑎𝑖,𝑘∑𝑥1
(𝑗)
𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)

𝑛

𝑗=1
𝑖≠𝑘

 

(13b) 

∑𝑥2
(𝑗)
𝑦(𝑗)

𝑛

𝑗=1

 =  𝑎0∑𝑥2
(𝑗)

𝑛

𝑗=1

+∑ 𝑎𝑖∑𝑥2
(𝑗)
𝑥𝑖
(𝑗)

𝑛

𝑗=1

𝑑

𝑖=1
+∑ 𝑎𝑖,𝑖∑𝑥2

(𝑗)
𝑥𝑖
(𝑗)2

𝑛

𝑗=1

2𝑑

𝑖=𝑑+1

+∑ 𝑎𝑖,𝑘∑𝑥2
(𝑗)
𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)

𝑛

𝑗=1
𝑖≠𝑘

 

(13c) 

⋮ 

∑𝑥𝑑
(𝑗)
𝑦(𝑗)

𝑛

𝑗=1

 =  𝑎0∑𝑥𝑑
(𝑗)

𝑛

𝑗=1

+∑ 𝑎𝑖∑𝑥𝑑
(𝑗)
𝑥𝑖
(𝑗)

𝑛

𝑗=1

𝑑

𝑖=1
+∑ 𝑎𝑖,𝑖∑𝑥𝑑

(𝑗)
𝑥𝑖
(𝑗)2

𝑛

𝑗=1

2𝑑

𝑖=𝑑+1

+∑ 𝑎𝑖,𝑘∑𝑥𝑑
(𝑗)
𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)

𝑛

𝑗=1
𝑖≠𝑘

 

(13d) 

are generated and solved simultaneously to give the coefficients 𝑎0, 𝑎𝑖, 𝑎𝑖,𝑖 and 𝑎𝑖,𝑘. 

For four-predictor system, which specially apply to this study, this is expanded to read 

𝑦 = 𝑎0 + 𝑎1𝑥1 + 𝑎2𝑥2 + 𝑎3𝑥3 + 𝑎4𝑥4 + 𝑎1,2𝑥1𝑥2 + 𝑎1,3𝑥1𝑥3 + 𝑎1,4𝑥1𝑥4 + 𝑎2,3𝑥2𝑥3 + 𝑎2,4𝑥2𝑥4
+ 𝑎3,4𝑥3𝑥4 + 𝑎1,1𝑥1

2 + 𝑎2,2𝑥2
2 + 𝑎3,3𝑥3

2 + 𝑎4,4𝑥4
2 

(14) 

Then, the arising normal equations are 



24   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

∑𝑦(𝑗)
𝑛

𝑗=1

 =  a0𝑛 + a1∑𝑥1
(𝑗)

𝑛

𝑗=1

+ a2∑𝑥2
(𝑗)

𝑛

𝑗=1

+ a3∑𝑥3
(𝑗)

𝑛

𝑗=1

+ a4∑𝑥4
(𝑗)

𝑛

𝑗=1

+ a1,2∑𝑥1
(𝑗)
𝑥2
(𝑗)

𝑛

𝑗=1

+ a1,3∑𝑥1
(𝑗)
𝑥3
(𝑗)

𝑛

𝑗=1

+ a1,4∑𝑥1
(𝑗)
𝑥4
(𝑗)

𝑛

𝑗=1

+ a2,3∑𝑥2
(𝑗)
𝑥3
(𝑗)

𝑛

𝑗=1

+ a2,4∑𝑥2
(𝑗)
𝑥4
(𝑗)

𝑛

𝑗=1

+ 𝑎3,4∑𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥3
(𝑗)
𝑥4
(𝑗)

𝑛

𝑗=1

+ a1,1∑𝑥1
(𝑗)2

𝑛

𝑗=1

+ a2,2∑𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥2
(𝑗)2

𝑛

𝑗=1

+ a3,3∑𝑥3
(𝑗)2

𝑛

𝑗=1

+ a4,4∑𝑥4
(𝑗)2

𝑛

𝑗=1

 

∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑦(𝑗)𝑛

𝑗=1  =  a0∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)𝑛

𝑗=1 + a1∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥1
(𝑗)𝑛

𝑗=1 + a2∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥2
(𝑗)𝑛

𝑗=1 +

a3∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥3
(𝑗)𝑛

𝑗=1 + a4∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥4
(𝑗)𝑛

𝑗=1 + a1,2∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥1
(𝑗)
𝑥2
(𝑗)𝑛

𝑗=1 +

a1,3∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥1
(𝑗)
𝑥3
(𝑗)𝑛

𝑗=1 + a1,4∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥1
(𝑗)
𝑥4
(𝑗)𝑛

𝑗=1 + a2,3∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥2
(𝑗)
𝑥3
(𝑗)𝑛

𝑗=1 +

a2,4∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥2
(𝑗)
𝑥4
(𝑗)𝑛

𝑗=1 + 𝑎3,4∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥3
(𝑗)
𝑥4
(𝑗)𝑛

𝑗=1 + a1,1∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥1
(𝑗)2𝑛

𝑗=1 +

 a2,2∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥2
(𝑗)2𝑛

𝑗=1 + a3,3∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥3
(𝑗)2𝑛

𝑗=1 + a4,4∑ 𝑥𝑖
(𝑗)
𝑥𝑘
(𝑗)
𝑥4
(𝑗)2𝑛

𝑗=1   

 (15) 

where i=1, 2,…, d and k=1, 2,..., d. The 15 linear simultaneous equations in 15 unknown 

coefficients can then be solved for the quadratic model. 

When regression model is based on one to two predictors, it will be easy to graphically present 

the variation of the response with the predictor/s in a one- or two-dimensional spaces. These are 

illustrated in Figures 1 and 2. In many cases, as is the case in this work, one- or two-dimensional 

spatial graphs are not practicable when the number of variables is more than 3. This case, 

graphical representation of results is usually shown in terms of the variation of the target 

responses with the predicted responses and the goodness of fit measured in terms nearness of 

slope to unity and intercept to nullity, see Figure 3 which is based on the same data as Figure 2. 

The figure 1 and 2 were plotted using 2018 version of Matlab software. 

 

Figure 1: Variation of a response with a predictor. The MatLab data “census” was used to 

generate this plot. 



25   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

 

Figure 2: Variation of a response with two predictors. The MatLab data “franke” was used 

to generate this plot. 

 

Figure 3: Variation of the response target with the response predictions. The MatLab data 

“franke” was used to generate this plot. 

c. Validation of the model using Artificial Neural Network (ANN) 

The artificial neural network was used to validate the polynomial regression analysis. It was also 

used because it gave a higher performance indices using Goodness of Fit Indices (GFI), 

validating the regression analysis. Model was developed. 

 



26   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

d. Optimization of the model 

Genetic Algorithm is used for the optimization of the mode. Because the model that emerges 

(referred to as equation 17) becomes intricate when considering its four independent variables, 

the genetic algorithm is chosen as the means for optimizing it, specifically to minimize 

corrosion. This genetic algorithm is applied to optimize objectives, whether they entail 

constraints or not. The rationale behind opting for a genetic algorithm is its effectiveness in 

tackling challenging optimization problems that do not align well with conventional optimization 

algorithms. These problematic scenarios include those characterized by objective functions that 

are discontinuous, non-differentiable, stochastic, highly nonlinear, and those involving mixed 

integer programming. 

3. Results and Discussions 

3.1. Second degree (quadratic) model. 

To develop the second-degree (quadratic) model, the data are inserted in the normal equations 

(13). The training data subset are randomly sampled 80% of the full data set while the testing 

subset are the remainder. As done earlier for the first-degree case, the inserted response data 𝑦(𝑗) 

is the mean corrosion rate given in Figure 4 and the inserted predictor data 𝑥𝑖
(𝑗)

 are the mean pH, 

mean temperature, mean pressure and mean aqueous CO2 partial pressure.  

The normal equations are then solved simultaneously to give the coefficients 𝑎0 and 𝑎𝑖. The 

results are graphed given in Figure 4 to show that the second-degree polynomial linear 

regression gives predictions that correlate very well with the input data. It can be seen that the 

model captures the responses very well since the testing results validate the training results as 

can be seen from the co-graphing of training and testing results on Figure 4.8. This points to a 

conclusion that the developed model will perform to similar extent as when the model is trained 

when independent field data are introduced.  

The parameters of the model are summarized Table 1. It can be seen from the table that the 

model is specifically given as  

𝑦 = −0.96139 + 0.088173𝑥1 + 0.043456𝑥2 + 0.015383𝑥3 + 0.23259𝑥4 −
0.0030606𝑥1𝑥2 − 0.0015876𝑥1𝑥3 − 0.11871𝑥1𝑥4 − 9.8756 × 10

−5𝑥2𝑥3 + 0.023881𝑥2𝑥4 +
0.0028826𝑥3𝑥4 + 0.012291𝑥1

2 − 0.00096625𝑥2
2 − 0.00011784𝑥3

2 + 0.0030086𝑥4
2  

        (16) 

The second-degree model coefficients are tested for significance. The numerical measures of the 

significance are given in Table 1. It can be seen that all parameters of the model are significant at 

95% confidence interval except the coefficients of the interaction of mean pressure and mean 

aqueous CO2 partial pressure (𝑥2𝑥3) and the square of mean aqueous CO2 partial pressure (𝑥4
2). 

These mean that the two second-degree variables are not relevant to the model and can be 

excluded without significantly affecting the model. 

Table 1: The second-degree model and the hypothesis testing results 

 Coefficient SE TStat pValue 

Intercept -0.96139 0.063413 -15.161 4.2183e-25 

𝑥1 0.088173 0.0095346 9.2477 3.1964e-14 

𝑥2 0.043456 0.0030395 14.297 1.2793e-23 

𝑥3 0.015383 0.0012426 12.38 3.4653e-20 

𝑥4 0.23259 0.023144 10.05 8.8314e-16 



27   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

𝑥1𝑥2 -0.0030606 0.00021012 -14.566 4.3696e-24 

𝑥1𝑥3 -0.0015876 0.00013376 -11.869 3.0635e-19 

𝑥1𝑥4 -0.11871 0.0027118 -43.777 3.6725e-57 

𝑥2𝑥3 -9.8756e-05 4.2885e-05 -2.3028 0.023921 

𝑥2𝑥4 0.023881 0.00086446 27.625 2.3791e-42 

𝑥3𝑥4 0.0028826 0.00053495 5.3886 7.1403e-07 

𝑥1
2 0.012291 0.00057182 21.495 9.5268e-35 

𝑥2
2 -0.00096625 6.1963e-05 -15.594 7.9054e-26 

𝑥3
2 -0.00011784 1.8073e-05 -6.5203 6.0793e-09 

𝑥4
2 0.0030086 0.0033919 0.88698 0.37778 

 

 

Figure 4: The correlation of the predictions with the measured data for the second-degree 

regression model. 

It can be seen that the model captures the responses very well since the testing results validate 

the training results as can be seen from the co-graphing of training and testing results on Figure 

1. This points to a conclusion that the developed model will perform to similar extent as when 

the model is trained when independent field data are introduced.  

Table 2: The goodness-of-fit indices of the second-degree model 

GFI Training Testing 

R2 0.9869 0.9361 

RMSE 0.0007 0.0012 

MBE 0.0000 0.0002 



28   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

MABE 0.0004 0.0008 

MPE 0.0006 0.0636 

r 0.9934 0.9689 

 

The goodness-of-fit indices of the second-degree model are summarized in Table 2 to compare 

directly the predicted responses against the measured responses. It can be seen that the R2 is 

almost unity at 0.9869 and the correlation (r) coefficient is almost unity at 0.9934. These indicate 

a very accurate second-degree polynomial regression for corrosion. 

The observation that two second-degree variables (the interaction of mean pressure and mean 

aqueous CO2 partial pressure and of the latter) are not relevant to the model and can be excluded 

without significantly affecting the model can be verified by reworking the regression to seen the 

impact on the model performance indicators. 

 

3.2. The Reduced Second-Degree Model 

The results are presented in Equation 17, Figure 5, Tables 3 and 4 present the veracity of the 

observation in terms similarity of the values with the corresponding indicators for the full model 

in Equation 17, Figure 5, Tables 3 and 4. 

𝑦 = −0.93980 + 0.087521𝑥1 + 0.043510𝑥2 + 0.0136527𝑥3 + 0.2274035𝑥4 −
0.00316755𝑥1𝑥2 − 0.00157643𝑥1𝑥3 − 0.11742780𝑥1𝑥4 + 0.0242314756𝑥2𝑥4 +
0.0026338902𝑥3𝑥4 + 0.012384649𝑥1

2 − 0.000997065𝑥2
2 − 0.0001334789𝑥3

2   (17) 

Table 3: The reduced second-degree model and the hypothesis testing results 

 Coefficient SE tStat pValue 

Intercept -0.93980 0.06432344 -14.6106096 2.04989253e-24 

𝑥1 0.087521 0.0097881 8.94162774 1.0363354e-13 

𝑥2 0.043510 0.00311802 13.9544395 2.9785469e-23 

𝑥3 0.0136527 0.00101308 13.4763858 2.1650894e-22 

𝑥4 0.2274035 0.02198834 10.3420047 1.8162327e-16 

𝑥1𝑥2 -0.00316755 0.000210252 -15.065517 3.3101078e-25 

𝑥1𝑥3 -0.00157643 0.000137267 -11.4843421 1.1233545e-18 

𝑥1𝑥4 -0.11742780 0.002667005 -44.02983497 2.4276929e-58 

𝑥2𝑥4 0.0242314756 0.0007811680 31.0195436 1.0304365e-46 

𝑥3𝑥4 0.0026338902 0.0005068729 5.19635296 1.4876765e-06 

𝑥1
2 0.012384649 0.0005852508 21.1612679 9.5914389e-35 

𝑥2
2 -0.000997065 6.238412378e-05 -15.9826743 9.0802652e-27 

𝑥3
2 -0.0001334789 1.712118786e-05 -7.79612124 1.891749815e-11 

 



29   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

 

Figure 5: The correlation of the predictions with the measured data for the reduced 

second-degree regression model 

 

 

 

Table 4: The goodness-of-fit indices of the reduced second-degree model 

GFI Training Testing 

R2 0.9859 0.9341 

RMSE 0.0007 0.0012 

MBE 0.0000 0.0001 

MABE 0.0004 0.0008 

MPE 0.0007 0.0390 

r 0.9929 0.9676 

 

The goodness-of-fit indices of the reduced second-degree model are summarized in Table 4 to 

compare directly the predicted responses against the measured responses. It can be seen that the 

R2 is almost unity at 0.9859 and the correlation coefficient r is almost unity at 0.9929. These 

results according to Ali et al. (2023) demonstrate that corrosion is not only affected by the 

predictors but also by their self and cross interactions as wide difference in predictive accuracy 

between the first-order and second order polynomial models. From the findings of this study, it 

can be deduced that the first-degree polynomial linear regression gives predictions that fairly 

correlate with the input data as can be seen from then fairly positive trend about the ideal 

(Sarker, 2021). Even though the model captures the responses poorly, the predicted testing 

results validate the predicted training results showing that the developed model will perform to 

similar extend when independent field data are introduced as when the model is trained. It can be 

seen also, that all parameters of the model are significant at 95% confidence interval except the 



30   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

coefficients of the interaction of mean pressure and mean aqueous CO2 partial pressure (𝑥2𝑥3) 

and the square of mean aqueous CO2 partial pressure (𝑥4
2). This suggests that the two second-

degree variables are not relevant to the model and can be excluded without significantly 

affecting the model and can be verified by reworking the regression to see the impact on the 

model performance indicators. 

These results demonstrate that corrosion is not only affected by the predictors but also by their 

self and cross interactions as wide difference in predictive accuracy between the first-order and 

second order polynomial models. 

3.3. The ANN Results 

 

Figure 6: The correlation of the predictions with the measured data for the ANN Model 

MatLab neural network tool is used for modelling the corrosion data. Figure 6 shows modelling 

results which graphically indicates that a highly reliable network has been trained.  

 



31   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

 

Figure 7: A typical setting used for ANN modelling 

Figure 7 summarizes most of the settings used in training the neural network. The architecture is 

simple, having one single hidden layer with ten neurons. The data set was randomly divided into 

training 80% and testing subsets 20% of the data set (the same random subsets used in for the 

above regression analyses).  

As implemented in MatLab, the Levenberg-Marquardt backpropagation algorithm, given as  

𝐱𝑘+1 = 𝐱𝑘 + [𝐉
T𝐉 + 𝜇𝐈]−1𝐉T𝐞       (18) 

Where: 

𝐉 is the Jacobian matrix made of first derivatives of the square errors with respect to the biases 

and weights of the artificial neural networks  

𝐞 is a vector of network square errors, is triggered with the command trainlm.  

Equation (18) is given here to highlight the importance of the parameter 𝜇.  

The algorithm initially employs the Newton's method, which involves approximating the Hessian 

matrix when the scalar µ is set to zero. However, as µ grows larger, the algorithm transforms into 

a gradient descent approach with a small step size. When approaching an error minimum, the 

algorithm prioritizes transitioning to Newton's method, which is faster and more accurate. 

Consequently, the value of µ is decreased after each successful step, and it is only increased 

when doing so would enhance the performance function. 

The Levenberg-Marquardt algorithm that is based on standard back propagation technique 

algorithm is adopted as the training algorithm to iteratively updates weight and bias values. The 

default performance criterion of the tool, the mean square error, is adopted. 

The best training performance of 1.28 × 10−7 occurred at the 171th epoch, see Figure 7. 



32   Journal of Engineering, Mechanics and Architecture                      www. grnjournal.us  

 
 

 

Figure 8: Training and testing performance with epochs 

Maximum number of epochs to train the network is set at the default value of 1000. As shown in 

Figure 8, the performance is tracked during training by computing the mean square error at every 

epoch and comparing with the pre-set performance goal. The default value is zero. If the 

performance goal is reached before the 1000th epoch, the training is stopped. That is why Figure 

8 shows that the training was terminated after 171 epochs.  

 

Figure 9. Error frequency 

Figure 9 shows a histogram of error frequencies. It can be seen that, as typical of a good model, 

the smaller errors are more frequent. 

 

 



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Table 5: The goodness-of-fit indices of the ANN model 

GFI Training Testing 

R2 0.9965 0.9158 

RMSE 0.0004 0.0014 

MBE 0.0000 -0.0003 

MABE 0.0003 0.0011 

MPE 0.0003 -0.0905 

𝑟 0.9982 0.9608 

 

The goodness-of-fit indices of the ANN model are summarized in Table 5 to compare directly 

the predicted responses against the measured responses. It can be seen that the R2 is almost unity 

at 0.9965 and the correlation coefficient is almost unity at 0.9982. These indicate a very accurate 

ANN for corrosion. It can be observed that ANN is somewhat better than the developed 

regression models based on these GOF indices: note that for the quadratic model, R2 is 0.9869 

and the correlation coefficient is 0.9934 

Even though the inclusion of the interaction effects of the predictors improved the prediction 

accuracy of the second-degree polynomial model, ANN approach was still slightly more accurate 

than the linear regression. But, the latter being a black-box model which cannot be used for 

symbolic optimization analysis, it is viewed as a validation for the polynomial model. 

3.4. Optimization of the model 

Given that the model introduced in Equation 17 involves four independent variables, its 

optimization (with the goal of minimizing corrosion) can be quite complex. To tackle this 

challenge, the genetic algorithm is employed. This genetic algorithm is utilized for optimizing 

objectives, whether they involve constraints or not.  

The rationale behind opting for a genetic algorithm lies in its suitability for addressing 

challenging optimization problems that don't align well with conventional optimization 

algorithms. These problematic scenarios encompass cases where the objective function is 

characterized by traits such as discontinuity, non-differentiability, stochasticity, high 

nonlinearity, and mixed-integer programming. 



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Figure 10. The interface of Mat Lab optimization toolbox 

The MatLab optimization toolbox is used in implementing the genetic algorithm in this study, 

see the interface given in Figure 10. In the optimization, each of the variables were constrained 

within its lower and upper bounds and the calculated optimal is 0.191498 mm/year occurring at 

mean pH of 8.446, mean temperature = 23.692°C, mean pressure = 15.725 bar and mean 

aqueous CO2 partial pressure = 2.022 bar. The optimal is attained after 98 iterations.  

 

Figure 11. The progression of the adaptive optimization process 

Figure 11 illustrations the progression of the adaptive optimization process showing how the 

minimum was attained. 

The architecture of artificial neural networks is also optimized sometimes to create the best 

input-output relationship. The architectural parameter of interest here is the number of neurons in 

the hidden layer, called ANN size here, since the model used here is based on a single-layer 

multi-perceptron architecture. Because of the iterative formulation of ANN model, analytical 

optimization is usually not possible.  



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This optimal network size is applied to the data and the results are as shown in Figures 13 to 16 

and Table 6. By looking at the figures and the goodness of fit indices the relative improvement 

can be seen. 

 

Figure 12. Variation of performance of ANN models with size of the hidden layer showing 

the optimal 

By calculating the mean square error for various artificial neural network (ANN) models with 

different quantities of neurons in the hidden layer and visualizing the outcomes as depicted in 

Figure 12, we observe that an ANN with 6 neurons in the hidden layer represents the optimal 

configuration. Furthermore, it's evident that the optimal ANN size falls within the range of 5 to 

13 neurons in the hidden layer. Notably, the choice of 10 neurons in the ANN used falls within 

this favorable range. 

 

Figure 13. The correlation of the predictions with the measured data for the optimal ANN 

model 



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Figure 14. The training settings and indicators for the optimal ANN 

 

 

Figure 15. Training and testing performance with epochs for the optimal ANN model 



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Figure 16. Error frequency for the optimal model 

Figure 16 shows a histogram of error frequencies. It can be seen that, as typical of a good model, 

the smaller errors are more frequent. 

Table 6. The goodness-of-fit indices of the optimal ANN model 

GFI Training Testing 

R2 0.9890 0.9292 

RMSE 0.0006 0.0015 

MBE -0.0000 0.0004 

MABE 0.0004 0.0011 

MPE 0.0005 0.1330 

𝑟 0.9945 0.9737 

 

The goodness-of-fit indices of the optimal ANN model are summarized in Table 6. It can be seen 

that the R2 is almost unity at 0.9890 and the correlation coefficient is almost unity at 0.9945 

4. Conclusions 

The critical issue of corrosion in Nigerian oil pipelines was addressed; this study undertook a 

comprehensive analysis using data on corrosion rate, pH, temperature, pressure, and aqueous 

CO2 partial pressure, obtained from an IOC company in the Niger Delta region. The second-

degree polynomial model was developed, incorporating interaction effects among the predictors. 

This model significantly improved predictive accuracy, achieving near-unity R-squared values 

for both training and testing datasets, highlighting the importance of considering interaction 

effects. To validate the developed model, an ANN was employed. Although the ANN 

demonstrated slightly higher predictive accuracy, its black-box nature limits interpretability. 

Therefore, the second-degree polynomial model, offering greater transparency and 

interpretability, was selected for further optimization. The optimization process, conducted using 

a GA, identified the optimal conditions for minimizing corrosion rates. By adjusting the 



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predictor variables within defined bounds, the algorithm determined the optimal corrosion rate to 

be 0.191498 mm/year, occurring at specific values of pH, temperature, pressure, and aqueous 

CO2 partial pressure. 

This study successfully developed and optimized predictive models for corrosion rates in 

Nigerian oil pipelines. The second-degree polynomial model, enriched with interaction effects, 

proved both accurate and interpretable, while the ANN served as a robust validation tool. The 

findings provide critical insights into the factors driving corrosion and offer a reliable basis for 

implementing preventive maintenance strategies to mitigate pipeline failures and enhance 

operational efficiency. 

It can be recommended that to predict corrosion in oil and gas pipelines, employing polynomial 

regression models with high-degree polynomials is essential to capture interaction effects 

accurately. Through hypothesis testing, the significance of these interactions can be assessed, 

allowing for the identification and removal of unnecessary predictors to streamline models and 

enhance efficiency. 

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