Introduction

The reliable and economically efficient operation of gas transmission pipeline networks remains one of the primary objectives of the energy sector. Despite significant technological advances in pipeline monitoring, control, and maintenance, completely eliminating gas losses from transmission systems has not yet been achieved. Consequently, international research has shifted its focus from the concept of zero losses toward establishing acceptable loss thresholds under normal operating conditions. Current practice indicates that acceptable losses generally range from 0.5–1.0 % in most European countries, whereas values between 1.5–2.5 % are considered reasonable for developing economies [1]. Azerbaijan's main gas transmission network extends to approximately 9,800 km, making it one of the largest gas pipeline systems within the CIS region.

Gas losses can generally be classified into three major categories according to their origin. The first category comprises hydraulic (frictional) losses, which arise from viscous interaction between the flowing gas and the internal pipe surface. Although these losses do not directly reduce the transported gas volume, they increase pressure losses along the pipeline, thereby raising compressor station energy demand and fuel consumption. The second category consists of mass losses caused by leakage, including seal failures in valve assemblies, corrosion-induced microcracks, and controlled gas releases during maintenance or depressurization operations. The third category includes technological losses, representing natural gas consumed as compressor fuel, gas used during equipment testing, and other operational requirements associated with transmission facilities [2, 3].

A comprehensive evaluation of these three loss mechanisms forms the basis of the gas balance concept. In principle, the amount of gas entering a transmission system must equal the sum of the delivered gas volume, the change in line-pack inventory, and all forms of gas loss. In practice, however, establishing an accurate gas balance remains challenging because each component is subject to measurement uncertainty. Various metering technologies—including ultrasonic flow meters, turbine meters, and differential-pressure measurement systems—operate with different accuracy levels and calibration characteristics, introducing unavoidable uncertainties into balance calculations.

Existing national regulations, including AZS 004–2018 «Technical Operation Rules for Main Pipelines», primarily recommend the conventional measurement-based gas balance method for estimating overall losses. However, they do not provide a unified methodology for separately quantifying hydraulic, leakage, and technological losses. This methodological gap complicates operational planning and investment decision-making, as transmission system operators often lack sufficient information to identify which loss mechanism should be prioritized for mitigation.

Several alternative approaches have been proposed internationally to overcome these limitations. In the United States, the Pipeline Research Council International (PRCI) developed an inline loss assessment methodology based on distributed pressure measurements collected along pipeline routes [4]. Within Europe, the European Gas Research Group (GERG) has introduced thermodynamic simulation techniques for estimating gas losses under varying operating conditions [5]. While both methodologies have demonstrated satisfactory performance, their practical implementation often requires either expensive instrumentation or calibration procedures that are not readily transferable to local geological and operational environments.

The principal scientific contribution of the present study lies in the development of an integrated methodology specifically designed for high-pressure gas transmission pipelines operating within the 40–75 bar pressure range. The proposed approach combines the Colebrook–White iterative friction model with the Papay compressibility correction to improve hydraulic loss prediction. Furthermore, an empirical degradation model describing the age-related decline in compressor polytropic efficiency has been incorporated, together with a leakage model that accounts for compressible gas flow behavior. The developed methodology has been validated using field measurements collected from a 300 km pipeline section operating under actual transmission conditions.

Theoretical Background and Computational Methodology

For steady single-phase natural gas flow in transmission pipelines, friction-induced pressure loss is evaluated using the Darcy–Weisbach equation, which remains one of the most widely accepted hydraulic models for high-pressure pipeline analysis:

where:

— λ — Darcy friction factor (dimensionless);

— L — pipeline section length (m);

— D — internal pipe diameter (m);

— ρ — average gas density (kg/m³);

— v — average gas flow velocity (m/s).

Since natural gas is a compressible fluid, its density varies with pressure and temperature. Therefore, the gas density under operating conditions is determined using the real-gas equation of state:

where:

— P — absolute operating pressure (Pa);

— M — molar mass of natural gas (approximately 0.0175 kg/mol);

— Z — gas compressibility factor;

— R — universal gas constant (8.314 J·mol⁻¹·K⁻¹);

— T — absolute gas temperature (K).

Unlike incompressible liquids, the density of natural gas continuously changes along the pipeline because of pressure reduction caused by friction. Consequently, incorporating the compressibility factor Z significantly improves the accuracy of hydraulic calculations, particularly for transmission pipelines operating above 40 bar.

Determination of the Friction Factor Using the Colebrook–White Equation

Under normal transmission conditions, gas flow is fully turbulent (Re > 4000). For this regime, the Darcy friction factor is determined using the implicit Colebrook–White equation:

where:

— ε — absolute internal pipe roughness (m);

— Re — Reynolds number;

— μ — dynamic gas viscosity (Pa·s).

The Reynolds number is calculated as

Typical values of pipe roughness range from 0.046 mm for new steel pipelines to approximately 0.10–0.20 mm for aged pipelines affected by corrosion and surface degradation.

Because Equation (3) is implicit with respect to the friction factor, a direct analytical solution is not possible. In this study, an iterative numerical procedure was adopted. An initial estimate of the friction factor was obtained from the Swamee–Jain correlation:

The estimated value was subsequently substituted into the Colebrook–White equation, and iterations were repeated until the convergence criterion

was satisfied. For the operating conditions investigated in this study, numerical convergence was typically achieved after 4–6 iterations, providing stable and highly accurate friction factor estimates suitable for hydraulic loss calculations.

Compressibility Factor Estimation Using the Papay Correlation

At elevated operating pressures (P > 40 bar), the thermodynamic behavior of natural gas deviates significantly from the ideal gas assumption. To account for these real-gas effects, the present study employs the Papay (1968) empirical correlation for estimating the gas compressibility factor:

where

— Pr — pseudo-reduced pressure ;

— Tr — pseudo-reduced temperature ;

— Ppc — pseudo-critical pressure (Pa);

— Tpc — pseudo-critical temperature (K).

The pseudo-critical properties of the gas mixture are evaluated according to Kay's mixing rule, expressed as

where:

— yi — mole fraction of component i;

— Pc,i — critical pressure of component i;

— Tc,i — critical temperature of component i.

Compared with the ideal gas assumption (Z = 1), the Papay correlation provides considerably improved accuracy for transmission pipelines operating at high pressures while maintaining relatively low computational complexity. For this reason, it has been incorporated into the proposed hydraulic calculation framework.

Polytropic Compression Model for Compressor Stations

The energy required for gas compression was evaluated assuming a polytropic compression process. The specific compression work per unit mass is calculated as

where:

— n — polytropic exponent (typically 1.27–1.32 for natural gas);

— T₁ — gas temperature at compressor suction (K);

— P₁, P₂ — suction and discharge absolute pressures (Pa);

— Zavg — average compressibility factor,

The corresponding polytropic efficiency of the compressor is determined from

where:

— ṁ — gas mass flow rate (kg/s);

— Nshaft — compressor shaft power (W).

Because compressor performance gradually deteriorates during long-term operation, an empirical degradation model was introduced to represent efficiency loss with operating time:

where:

— η₀ — nominal efficiency of a new compressor;

— t — accumulated operating time (×10³ h);

— kd — degradation coefficient (0.0142);

— α — degradation exponent (0.61).

This relationship enables the model to account for the progressive reduction in compressor performance caused by mechanical wear, surface erosion, and equipment ageing, thereby improving long-term energy consumption estimates.

Estimation of Technological Gas Losses at Compressor Stations

Gas losses associated with compressor station operation were evaluated using a volumetric gas balance. The technological loss is defined as

where:

— Qin — gas flow entering the compressor station (Nm³/h);

— Qout — gas flow leaving the station (Nm³/h);

— Qfuel — gas consumed as compressor fuel (Nm³/h);

— ΔQCS — technological gas loss (Nm³/h).

This parameter includes operational losses arising from equipment leakage, venting during maintenance procedures, and gas released during pressure testing or other routine operational activities.

Analytical Model for Gas Leakage

Leakage through defects such as cracks, corrosion pits, or small openings in the pipe wall was modeled using a generalized Torricelli–Bernoulli approach modified for compressible gas flow. The leakage rate is expressed as

where:

— Cd — discharge coefficient (0.62–0.72);

— Adef — equivalent defect area (m²);

— ΔP — pressure difference between the pipeline and the surrounding atmosphere (Pa);

— ρ — gas density (kg/m³);

— φ(Ma) — compressibility correction factor.

The Mach-number correction is calculated as

where:

— γ — ratio of specific heats (1.31 for methane-rich natural gas);

— Ma — Mach number, defined as the ratio of gas velocity to the local speed of sound.

For low-speed flow conditions (Ma ≪ 1), the compressibility correction approaches unity , reducing Equation (11) to the classical incompressible Torricelli formulation. Under high-pressure operating conditions, however, incorporating the Mach-number correction substantially improves the prediction of leakage rates and enhances the overall accuracy of the proposed gas loss assessment methodology.

Results and Discussion

Pressure Profile Along the Pipeline

The combined solution of Equations (1)–(6) was applied to a 300 km pipeline section with an internal diameter of 1020 mm and a roughness of 0.06 mm . The inlet pressure was 75 bar , the gas temperature 278 K (5°C) , and the average gas density 58.4 kg/m³ at 65 bar operating pressure. As shown in Figure 1 , the operating pressure decreases from 75 bar to 54.8 bar over the 300 km pipeline, corresponding to a total pressure loss of 20.2 bar .

Fig. 1

Distribution of Gas Loss Sources

Statistical analysis of the field measurement data identified five major categories of gas losses (Figure 2). Considering the indirect effect of friction losses on compressor energy consumption, the combined share of direct mass losses (leakage, valve assemblies, and compressor technological losses) was estimated at 53 % .

Fig. 2

Compressor Efficiency Degradation

The dependence of compressor polytropic efficiency on operating time was modeled using Equation (9) with the calibrated parameters η₀ = 0.882 , kd = 0.0142 , and α = 0.61 . As shown in Figure 3 , the peak efficiency decreases from 0.88 to 0.81 after 5 years of operation, while the optimum operating point shifts toward lower flow rates. This performance degradation results in an additional annual fuel consumption of approximately 1.2 million m³ of natural gas.

Fig. 3

The application of the Colebrook–White iterative method with the Swamee–Jain initial approximation reduced the root mean square error of the friction factor from 1.8 % to 0.4 % compared with the Moody diagram, particularly improving accuracy at high Reynolds numbers ( Re > 10⁷ ). For a 300 km pipeline, a 1 % overestimation of the friction factor results in an outlet pressure underestimation of approximately 0.6 bar and an overestimation of compressor power requirements by about 0.8 % .

Including the Papay compressibility factor in the hydraulic model increased the calculated gas density by 6.4 % at an operating pressure of 65 bar . Without this correction, gas density is underestimated, leading to systematic overprediction of flow velocity and friction losses. The Papay correlation provides an accuracy of approximately ±0.3 % , close to that of the Peng–Robinson equation ( ±0.2 % ), while requiring nearly ten times less computational effort , making it suitable for real-time SCADA applications.

The leakage model exhibited an average deviation of 6.6 % , mainly due to simplified assumptions regarding defect geometry and uncertainties in the discharge coefficient ( Cd ). Future work will focus on integrating CFD-based discharge coefficients to further improve prediction accuracy.

The calibrated compressor degradation coefficient ( kd = 0.0142 ) agrees well with the range reported by Brinkworth (2008) ( 0.012–0.016 ), confirming the physical validity of the proposed degradation model. Reducing the compressor maintenance interval from 18,000 h to 14,000 h can save approximately 1.2 million m³ of natural gas per compressor annually.

Conclusions

1. The integrated model combining the Darcy–Weisbach , Colebrook–White , and Papay equations predicts hydraulic losses in main gas pipelines with an accuracy of 98.7 % , reducing the average friction factor error to 0.4 % .
2. The compressor polytropic efficiency decreases from 0.88 to 0.81 after five years of operation, resulting in approximately 1.2 million m³ of additional annual fuel consumption per compressor unit.
3. The distribution of gas losses was estimated as 41 % friction losses, 23 % leakage, 19 % compressor technological losses, 11 % valve assembly losses, and 6 % other sources. Friction and leakage together account for 64 % of total losses.
4. The implementation of the proposed technical measures during 2022–2023 reduced total gas losses from 1.95 % to 1.44 % , corresponding to an annual saving of approximately 38 million m³ of natural gas.
5. The proposed leakage model achieved an average prediction error of 6.6 % . Future integration of CFD-based corrections is expected to reduce this error to approximately 2–3 % .
6. The developed computational methodology is suitable for integration into real-time SCADA systems. Owing to its low computational cost, the Papay compressibility correlation enables efficient real-time implementation while maintaining high prediction accuracy.

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