Applicability of the Elastic Water Column Method to Pressurized Pipeline Emptying: Dimensionless Pressure Analysis Under Different Air Pocket Configurations

Abstract

Pressurized pipelines are critical components in hydraulic engineering systems, including urban water supply networks and hydroelectric power plants. These systems are susceptible to air entrapment during operations such as filling and emptying, which can reduce the effective flow area and trigger critical pressure surges or sub-atmospheric conditions. One-dimensional approaches, namely the Rigid Water Column (RWC) and Elastic Water Column (EWC) models, are the most widely used due to their balance between physical accuracy and computational practicality. EWC models have been widely used to analyze transient phenomena in pipe filling and water hammer processes; however, their application to emptying operations is limited. For this reason, this study develops an EWC-based formulation for emptying operations and assesses pressure behavior through a dimensionless analysis for different air pocket configurations. The developed model couples the Method of Characteristics (MOC) with a polytropic air pocket model, enabling the representation of wave propagation effects that RWC-based models cannot capture. The formulation is verified against 24 experimental cases, yielding a mean absolute error of 0.35% in minimum pressure prediction. The results show that dimensionless air pocket ratios $x_0/L_T$ between 0.17 and 0.83 produce minimum pressures between 0.309 and 0.877 $p_{atm}^{*}$, confirming that smaller initial air pocket volumes generate the most severe depressurization conditions. The inclusion of an air valve in the most critical scenario effectively prevents sub-atmospheric pressure development, underscoring the protective role of air admission devices. These findings provide a dimensionless framework for characterizing transient pressure risk during pipeline emptying across different operational conditions.

1. Introduction

Piping systems are fundamental in many commercial and industrial infrastructures, such as hydroelectric power plants, pumping stations, thermal and nuclear power plants, and water and rainwater drainage systems [1]. These infrastructures depend on pipelines to transport fluids under controlled conditions, which implies that they are a central element in ensuring the safe operation of these facilities.

Transient phenomena are highly complex fluid mechanics processes that may occur during the start-up or abrupt interruption of facilities such as hydroelectric plants or drinking water distribution networks [2]. These phenomena may result in pressure surges, cavitation, system instability, and material fatigue [3,4,5].

A particular case is pipes that travel long distances, since they usually present challenges in their operation, especially during maintenance and repair processes in which it is necessary to fill or empty the system, which can induce multiphase flows (water-air) and the entry of air pockets in the pipe [6,7].

Another recurrent phenomenon is the formation of air pockets in systems with undulating elevation profile pipes, in which there may be a separation of the water column at the points, forming air pockets that, if not released before the opening of the valves, generate pressure peaks due to the sudden compression of the air. This type of behavior is complex to model due to the nonlinear behavior of the pressure generated by the air, which depends on multiple factors such as the system’s geometry, the amount of trapped air, and the initial flow conditions [8,9,10].

Depending on the level of detail, one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) models can be used. The 1D models are based on differential equations and iterative procedures, resulting in low computational costs. Meanwhile, 2D and 3D models resolve the flow field across multiple spatial dimensions, capturing complex geometries and localized phenomena at significantly higher computational cost [11,12].

In the case of 1D models, there are two main approaches: the Rigid Column Model (RWC), which neglects the elasticity of water and the pipe, and the Elastic Column Model (EWC), which accounts for these effects [6,13]. EWC models are typically solved using the Method of Characteristics (MOC), while rigid models employ numerical solutions to ordinary differential equations; however, solutions such as characteristic-circuit methods [14] and higher-order MOC formulations [15] have also been implemented, aiming to improve computational efficiency and accuracy in complex pipe networks [16].

During filling operations, the compression of the air pockets can produce significant pressure surges, and transient phenomena occur due to the difference in speed between the water phase and the air phase, which can cause pipeline damage [17,18,19,20]. On the other hand, the emptying processes are characterized by the expansion of the air pockets trapped in the pipe, causing sub-atmospheric pressures that increase the risk of structural collapse of the pipe [21,22].

Although there are multiple investigations associated with transient phenomena in pipelines, most focus on the phenomenon of filling [8,18,23,24,25], while emptying has received less attention despite its relevance in maintenance and emergency operations [11,26,27,28,29].

For this reason, the present article aims to (i) develop an elastic water column formulation for pipeline emptying operations and validate it against experimental data, (ii) assess transient pressure behavior through dimensionless analysis under different initial air pocket configurations, and (iii) evaluate the protective role of air admission devices in the most hydraulically severe emptying scenario.

2. Modeling Approaches for Hydraulic Transients

Existing studies on transient phenomena in pipelines can be broadly classified into four main model types: one-dimensional (1D) models, two-dimensional (2D) and three-dimensional (3D) models, coupled models, and Machine Learning (ML) or AI-based models (Figure 1).

One-dimensional models include the Rigid Water Column Model (RWC) and the Elastic Water Column Model (EWC). Both models are solved using various numerical methods, such as the Method of Characteristics (MOC), Finite Difference Method (FDM) and Finite Volume Method (FVM), and solved iteratively using techniques such as Newton-Raphson.

In contrast, 2D and 3D models utilize Computational Fluid Dynamics (CFD) and methods like FVM, Finite Element Method (FEM), and FDM to capture complex spatial details. Coupled models can apply 1D formulations to long pipe sections and 3D CFD to localized regions where complex flow patterns occurs. Additionally, machine learning-based models, particularly artificial neural networks, have been applied to predict pressure and flow rate changes during transient events, reducing the computational cost associated with high-fidelity simulations.

In hydraulic engineering, 1D models remain popular due to their effective balance between accuracy and computational efficiency. Meanwhile, CFD and AI-based models are increasingly used as advanced tools to handle more complex simulations (Figure 2).

In the field of 1D models, notable works include that of Liu et al. [8], who used the MOC to simulate the rapid filling process based on the RWC theory and validated the results with experimental data, concluding that the air comprehensive coefficient significantly affects the Maximum Air Pressure (MAP) of the entrapped air pockets.

Variations or combinations of solution methods have also been used to analyze emptying scenarios under RWC conditions, as carried out by Bonilla-Correa et al. [24], who used iterative methods to investigate the air-water interface location in a pipe during the filling process and proposed a mathematical approach for directly solving air-water interaction in water-emptying processes with entrapped air, based on a one-dimensional formulation of the rigid water column model, piston flow, and the polytropic law, obtaining satisfactory results when validated with experimental data and 3D CFD simulations. Another example is the work of Bucur et al. [30], who proposed a two-equation model based on the experimental analysis of transient flow in a horizontal pipe containing an air pocket; as the model does not explicitly account for air compression and entrapped air effects, it presents some limitations in evaluating maximum pressure for relatively small orifices; however, a large amount of experimental data was used to achieve a good fit of the results.

In the work of Malekpour et al. [31], a global energy-auditing framework is applied to analyze the transient response of confined pipe systems with entrapped air under sudden pressurization, revealing that the maximum pressure is directly related to the energy absorbed by the air pocket; the study demonstrates that neglecting fluid and pipe elasticity overestimates pressure by allowing the air pocket to receive energy that would otherwise be stored as strain energy in the system.

The study by Chaiko and Brinckman [32] evaluates three 1D models to analyze water hammer in pipelines with entrapped air. They show that for air volumes below 5%, a simplified model with uniform gas compression and fixed liquid domain is sufficient; however, for larger air pockets, neglecting interface movement leads to significant errors in peak pressure prediction.

Similarly, other researchers have employed one-dimensional (1D) models to analyze the water–air interaction dynamics either during pipeline filling processes or during the transient events associated with system start-up, particularly when the pipeline remains water-filled following an operational interruption [33,34,35,36].

On the other hand, in the field of 1D models based on EWC, an important referent is the work of Zhou et al. [1], which developed an improved elastic model using general water hammer equations and MOC, considering three phases—water, air, and the water-air interface—while incorporating valve opening time and elevation changes, achieving the same accuracy as a complete EWC model while avoiding interpolation complexity. Wang et al. [37] developed a numerical model based on the EWC theory to investigate the transient behavior during rapid filling in bypass pipelines, finding that entrapped air cushions the impact of colliding waterfronts and concluding that smaller air volumes lead to higher pressure peaks.

The EWC model has also been applied to the analysis of water distribution networks, as demonstrated by Zeng et al. [16], who proposed a graph-theoretic formulation and validated on networks of 6 and 51 pipes, successfully capturing the hydraulic behavior, including valve effects.

The hybrid model proposed by Liu et al. [14], combines characteristic lines and equivalent circuits for precise and efficient simulation of pipe networks. The model was validated by comparing the proposed method with commercial software results, finding significantly enhanced computational speed at comparable accuracy.

The Discrete Air Model (DAM) proposed by Feng et al. [38] integrates the air continuity and momentum equations with the water control equation and the interface continuity equation; validation through comparison with results obtained from the Uniform Air Model (UAM) as well as experimental data from Zhou et al. [15] shows that this model effectively represents pressures in large-scale water pipelines during filling operations. The study of Zhou et al. [15] combines a second-order Godunov-type scheme (GTS) with the MOC. When comparing this approach to both MOC results and laboratory experiments, it was found that the proposed scheme, which incorporates unsteady friction, more accurately reproduces experimental pressure oscillations and demonstrates greater robustness and efficiency than the MOC.

In the work of Do Nascimento Rocha et al. [39], a numerical model that accounts for two-phase flow and fluid–structure interaction is proposed. The validation of the results is conducted through comparisons with experimental data from Chaudhry et al. [40] as well as analytical data. Recently, the study by Aghaei et al. [41] applies the MOC to solve water hammer equations and integrates the Discrete Gas Cavity Model to capture column separation effects. Its results were validated using two experimental setups and comparisons with existing numerical models, highlighting the need for a balanced design approach that mitigates transient risks while maintaining operational efficiency in large-scale pipeline systems.

In this way, it is evident that most of the literature focuses on the analysis of rapid pipeline filling. The characteristics of the systems and validation methods used in some of the 1D model applications identified for rapid filling are shown in

Table 1. Main applications of 1D models for transient phenomena identified.

Modeling Approach	System Characteristics	Validation	Reference
RWC	Based on Martins et al. [42]. The system consists of a pressurized tank connected to a dead-end pipe through a valve, with an internal diameter of 0.0536 m and a total length of 3.62 m.	Validated through experimental verification.	[8]
RWC	Single pipe of 600 m; internal diameter of 0.30 m; pipe slope of 0.02 rad, polytropic coefficient of 1.2; initial air pocket of 500 m; a constant friction factor of 0.018, resistance coefficient of 0.11 s2/m5; initial pressure supplied by a tank of 202,650 Pa.	Validated through experimental verification.	[24]
RWC	The experimental setup consisted of a $L = 14.1 \mathrm{m}$; $D = 0.094 \mathrm{c}\mathrm{m}$ acrylic pipeline laid on a horizontal slope and upstream reservoir with a cross-sectional area of $2.96 {\mathrm{m}}^2$. Three piezoresistive pressure transducers, Two MicroADV (Acoustic Doppler Velocimeter) and Digital camcorder (30 frames per sec)	Validated through experimental verification.	[33]
RWC	A domestic water supply pressure tank in the upstream. A 10 m long pipe was made of galvanized steel and had an inside diameter of 0.035 m. Three quarter-turn ball valves. Three high-frequency-response strain-gauge pressure transducers.	Validated through experimental verification.	[34]
EWC	The system consists of an upstream reservoir with $5 {\mathrm{m}}^2$ cross section, a gate valve, a quarter-turn ball valve, a water vent, and an approximately 4.44 m long pipeline with a 0.09 m diameter. The measuring system consists of one pressure gauge and five pressure transducers.	Comparison with complete elastic model and experimental results	[1]
EWC	Four equal diameter (100 mm) pipes of length 6, 3, 9, and 10 m constitute a simple bypass pipeline.	Comparison with experimental data for a single pipeline with dead-end presented by Zhou [43]	[37]
EWC	Experimental apparatus from Zhou et al. [15] and simulations using a 2400 m pipeline with an intermediate valve and a diameter of 0.30 m to investigate air behavior in a large-scale water pipe.	Comparison with UAM and experimental results from Zhou et al. [15].	[38]
EWC	Upstream Pressure Tank, 8.862 m pipe with 0.4 cm diameter, a flow meter, a pressure gauge, six pressure transducers, flow range from 0–25 m3/h.	Validated through experimental verification.	[15]
EWC	Upstream Pressure Tank with a pressure regulator, 30.6 m pipe with 0.026 m diameter, high-frequency-response pressure transducers at three locations ($x = 8.0 \mathrm{m},$ $21.1 \mathrm{m}$ and $30.6 \mathrm{m}$, respectively, from the upstream end).	Comparison with experimental results from Chaudhry et al. [40]	[39]

.

Two-dimensional (2D) and three-dimensional (3D) CFD models have been widely used to analyze transient events. In Warda et al. [19], a 2D model based on Reynolds-Averaged Navier–Stokes (RANS) equations and the Volume of Fluid (VOF) method demonstrated that pressure peaks and oscillations depend on the orifice size: smaller orifices caused intermitted flow choked by water column while larger ones led to water hammer dominance, findings that were experimentally validated at the Alexandria University Hydraulics Laboratory. Similarly, in Zhu et al. [44] the 3D-CLSVOF (Coupled Level-Set and Volume of Fluid) model and URANS (Unsteady Reynolds-Averaged Navier–Stokes) equations with energy equation were implemented to study the hydraulic transient and thermodynamic characteristics of water flows impacting an air pocket at the vertical end of an elbow pipe; the results showed that the CLSVOF model outperformed the original VOF model in capturing pressure peaks and water–air interface fragmentation.

The approach of Martins et al. [45] focused on developing a 3D model in ANSYS Fluent using FVM with the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm. The model was validated against experimental data collected under unsteady conditions from a pipe-rig facility. Two types of valve maneuvers were examined: one instantaneous, and another represented by a hyperbolic function fitted to experimental data. The results demonstrated high accuracy in reproducing the pressure waveforms, particularly when the valve maneuver was represented by a hyperbolic fit.

The study of Cao et al. [46] proposes a 3D model to evaluate in greater detail the dynamic characteristics of water hammer induced by the closure of a globe valve. This 3D CFD model enables a more accurate reproduction of the experimentally observed pressure oscillations, outperforming the results obtained from 1D models.

Regarding coupled models, the study presented in Wang et al. [47] evaluates a 1D–3D coupling strategy within a CFD environment, in which MOC code is implemented as a boundary condition in OpenFOAM. The performance of this model is assessed by simulating a hydraulic system where the flow is driven by the elevation difference between two free surfaces and modulated by the actuation of a valve. The results demonstrate satisfactory agreement with experimental data. Similarly, in Zhang et al. [48], the same software is used to implement a 1D–3D model capable of capturing water compressibility and gas–liquid interactions. Additionally, Zhang & Cheng [49] apply the MOC approach to sections of the pipeline where the flow predominantly varies along the main streamline, while 3D CFD is employed in regions with complex flow patterns over short distances, aiming to evaluate transient phenomena in hydroelectric systems.

In addition to CFD-based coupled models, recent advances in Machine Learning have also been applied to transient analysis in pipelines. Particularly, models based on Physics-Informed Neural Networks (PINNs) have shown great potential for predicting flow and pressure variations in pipelines, especially within networked systems where implicit or incomplete information is present in the measurements. For instance, Ye et al. [50] propose a PINN-based approach that integrates both physical laws and experimental data into the training process. This enables the model to infer hidden dynamics from the data and achieve accurate predictions of transient behavior, yielding highly satisfactory results. Similarly, machine learning techniques have also been successfully applied to the identification of topological features in pipeline networks, such as intersections. In Bohorquez et al. [51], Artificial Neural Networks (ANNs) are employed to predict the location of pipe junctions with an error of less than 2.32 m in 95% of the test cases, while simultaneously achieving near-perfect accuracy in estimating the diameters of the connected branches. Furthermore, the same approach demonstrates high precision in leak detection, locating leaks with an error below 3.0 mm in 95% of the cases, and achieving an average absolute error of just 0.03 mm in estimating leak size using only transient pressure traces as input. Along the same lines, Ayati et al. [52] present a methodology combining machine learning and transient hydraulic modeling to detect leaks in water distribution systems, also reporting satisfactory performance.

As shown, different approaches can be used to study transient phenomena,

Table 2. Summary of modeling approaches suitability for the analysis of hydraulic transients.

Model	Strengths	Limitations	Suitability
1D Models	This method is efficient and widely validated. It includes both RWC and EWC components, making it well suited for engineering design and real-time applications.	May overlook detailed spatial phenomena; accuracy is limited for rapid transients involving air interaction unless EWC is utilized.	Designing pressurized systems involves analyzing water hammer effects, utilizing surge tanks, and managing the processes of filling and emptying pipelines.
2D/3D Models	High spatial and temporal resolution; captures detailed transient interactions including turbulence and multiphase flow.	Very high computational demand; difficult to apply to large-scale systems; requires detailed geometry and calibration.	Research settings, validation studies, multiphase flow analysis, and localized transient investigations.
Coupled Models	Combine accuracy of 3D with efficiency of 1D; adaptable to complex systems; enhance representativity while reducing cost.	Interface complexity; requires expertise in both domains; sensitive to model coupling and boundary definition.	Urban networks, smart water systems, integrated hydraulic and structural analyses.
AI based Models	Fast prediction once trained; can model nonlinear and complex behavior; useful for system monitoring and control.	Requires large and high-quality datasets; lacks physical interpretability; generalization can be limited.	Real-time monitoring, leak detection, failure prediction, and performance optimization.

presents a summary of their strengths and limitations.

3. Modeling of Transient Flow During Emptying of Pressurized Pipelines

Most published studies have focused on the rapid filling process. However, several investigations have also focused on pipeline emptying and the associated transient phenomena, which are the focus of this section. Among these, Laanearu et al. [28] and Collins et al. [21] provide experimental investigations that reveal the transient phenomena occurring during pipeline emptying events. The investigation of Laanearu et al. [28] employs a simple Control Volume (CV) model to interpret results from the emptying of a horizontal PVC pipeline initially filled with water and subjected to different upstream compressed air pressures and outflow restrictions. In contrast, Collins et al. [21] emphasizes the destructive potential of depressurization events, based on experimental observations of sudden valve openings in static, pressurized pipelines, where negative pressures, rapid pressure surges, and strong acoustic effects were recorded—highlighting the importance of addressing these transients with appropriate caution due to the risk of structural failure.

Further research has focused on physically based modeling of the emptying process. In Coronado-Hernández et al. [53], a RWC formulation is used for the liquid phase, while the air–water interface is resolved and compared with experimental measurements of flow oscillations and gauge pressures, showing good agreement. The validation was carried out using a PVC–steel pipeline 271.6 m long and 232 mm in internal diameter, equipped with a high-pressure air tank at the upstream end and a manual butterfly valve downstream. Similarly, Coronado-Hernández et al. [54] presents an experimental setup to measure pressure and velocity during emptying for different air-pocket sizes and valve opening times. The phenomenon was then simulated using a one-dimensional rigid model with constant friction and diameter, achieving satisfactory agreement with the observed draining process. In Fuertes-Miquel et al. [27], a rigid-column computational model is proposed to couple the water motion, air–water interface, and air-pocket equations in two cases—(i) a single pipe with the upstream end closed, and (ii) a single pipe with an upstream air valve—accurately reproducing experimental results from a methacrylate pipeline (internal diameter 42 mm, length 4.36 m).

At the network scale, Coronado-Hernández et al. [55] presents a comprehensive framework for simulating the emptying process, combining a RWC model for the water phase, a piston-flow assumption for the interface, a polytropic equation for the air phase, the continuity equation for the air pocket, and an air-valve characterization to quantify the admitted air during the transient. Along similar lines, Coronado-Hernández et al. [22] proposes a model designed for planning emptying operations, which simulates water-phase propagation using a rigid formulation and models the air-pocket behavior through thermodynamic equations, validated with experimental data. The influence of operational conditions is examined in which evaluates the effects of air-pocket size and valve opening percentage and timing on pressure variation, demonstrating—after validation—the capability of a CFD model to reproduce the transient event. More recently, Sun et al. [56] proposes a mathematical model under non-constant flow conditions for the emptying process, examining the influence of drain valve opening patterns, initial air-pocket size, and air-valve configuration on the resulting transient pressure behavior, extending the analysis of operational parameters beyond the configurations previously reported.

In a comparative analysis, Besharat et al. [57] simulates the emptying process using both 2D CFD and 1D models, finding generally consistent predictions of pressure and velocity. A further development is reported in Bonilla-Correa et al. [58], where a new mathematical model is introduced that provides more direct results than previous formulations and can estimate the minimum sub-atmospheric air-pocket pressure head with a root-mean-square error of 1.13%, slightly higher than that obtained with the 1D model (0.33%) due to the consideration of instantaneous valve opening. Validation was achieved through comparison with experimental data and 3D CFD simulations performed in OpenFOAM.

Regarding experimental contributions from independent research groups, Chen et al. [59] reports a comprehensive dataset covering both filling and emptying processes in a large-scale pipeline, documenting the evolution and breakup of the water–air interface under pressure-driven conditions, broadening the empirical basis available for model validation beyond the configurations previously reported in the literature. In addition, Romero et al. [29] validates the model developed by Coronado-Hernández et al. [26] using data from a real large-scale installation (nominal diameter 400 mm, total length 1020 m), representing a significant step forward in the study of transient behavior during actual emptying operations and providing the basis for further investigation at real scale.

Most available studies rely on one-dimensional RWC models to simulate the emptying of pressurized pipelines, while a smaller but growing body of work employs CFD techniques that offer greater spatial detail, albeit at a significantly higher computational cost. Figure 3 shows that RWC models account for 55.6% of the identified contributions, CFD approaches 33.3%, and purely experimental studies for the remaining 11.1%. Notably, no study was found to apply an EWC formulation to the emptying process—a gap that has been consistently highlighted in the literature [11] and that motivates the theoretical developments presented in Section 5.

4. Challenges and Perspectives in Emptying Modeling

Despite the progress achieved in understanding emptying hydraulics, research in this area remains concentrated within a small number of specialized groups. The co-authorship analysis conducted using VOSviewer (version 1.6.20) identified two of the main and largely independent clusters of collaboration within the research on pipeline emptying processes (Figure 4). The network was constructed using co-authorship as the linking criterion, with a minimum threshold of two documents per author; node size reflects publication count and link strength represents the frequency of co-authored works. Although additional contributions from other researchers and groups exist outside these clusters, the analysis reveals two dominant collaborative networks. The cluster on the right, led by Janek Laanearu and Arris S. Tijsseling, includes contributors such as Qingzhi Hou, Ivar Annus, and Anton Bergant. This group has primarily focused on the experimental and theoretical characterization of large-scale filling and emptying phenomena, emphasizing air–water interactions, stratified flow behavior, and the development of improved one-dimensional transient models validated through experimental campaigns in facilities such as HYDRALAB III.

The cluster on the left, which represents a significant portion of the recent literature on pipeline emptying, is led by Vicente S. Fuertes-Miquel and Óscar E. Coronado-Hernández, with recurrent collaborations involving Pedro L. Iglesias-Rey, Helena M. Ramos, María T. Viseu, Modesto Pérez-Sánchez, and Alfonso Arrieta-Pastrana. This research network, linking the Universitat Politècnica de València (Spain) and Instituto Superior Técnico (Portugal), has contributed significantly to the development and validation of rigid-column formulations, CFD-based analyses, and hybrid air–water interaction models. Their collective work provides valuable insight into transient pressure behavior, sub-atmospheric conditions, and the dynamics of air-valve operation during emptying events.

The absence of links between these two clusters in the VOSviewer network indicates limited collaboration between research groups with different methodological approaches. As a result, a significant portion of the active contributions in this field is concentrated within a limited number of specialized groups. Most of the available studies rely on controlled laboratory experiments or small-scale test facilities, while applications in real systems and complex network configurations are still rarely addressed. Only a limited number of studies such as Romero et al. [29] and Coronado-Hernández et al. [55] have addressed operational-scale pipelines or urban water supply systems. This suggests the need for stronger collaboration and for multi-scale modeling frameworks capable of capturing the coupled hydraulic, thermodynamic, and structural processes involved. The concentration of research within two dominant clusters points to the need for diversification in both experimental methods and theoretical frameworks—in particular, combining the large-scale experimental expertise of the right cluster with the modeling and CFD capabilities of the left one could accelerate the development of elastic-based formulations for emptying, precisely the gap this work aims to address.

5. Considerations for Extending the Elastic Method to Emptying Operations

The study of pipeline emptying processes necessarily involves the analysis of two-phase flow, due to the continuous entry and displacement of air within the system. For this reason, most investigations addressing this type of phenomenon rely on boundary condition assumptions that simplify the governing equations while still allowing a reasonable estimation of the effects produced by entrapped or ingressed air pockets.

As previously discussed, the presence of air within pressurized pipelines plays a central role not only during emptying operations but also in the general behavior of hydraulic systems, since its influence can trigger severe transient events that may cause significant structural damage to the conduits [60]. For instance, the selection of the polytropic coefficient ($m$) is a key factor because it controls the thermodynamic stiffness of the air phase during transient events. Particularly in emptying operations where heat transfer and mass exchange between air and water are substantial, an inadecuate selection of m could lead to an overestimation of sub-atmospheric pressures or to unrealistic oscillation frequencies.

The work of Fuertes-Miquel et al. [11] highlights that values of $m \approx 1.20$ generally provide good agreement between experimental data and results from RWC models. However, in formulations based on the EWC approach, $m$ does not only represent the thermodynamic behavior of the air but also the coupled stiffness of the air–water system.

Experimental studies (e.g., [61,62]) clearly show that during the initial stage of the transient, when air compression occurs within less than one second, the process behaves adiabatically ($m \approx 1.4$). As the event progresses, heat transfer between the air, the surrounding water, and the pipe wall causes the behavior to evolve toward quasi-isothermal conditions ($m \approx 1.0-1.2$). This observation is consistent with the results of Zhou et al. [62], who demonstrated that the adiabatic assumption accurately reproduces the first pressure peak but tends to overpredict subsequent cycles once thermal exchange becomes significant. This means that assuming a constant $m$ throughout the entire transient event can lead to unrealistic predictions of pressure attenuation and wave celerity.

Similarly Zhou et al. [63], analyzed rapid air expulsion through an orifice in a vertical pipe, adopting a constant $m = 1.4$ to represent adiabatic behavior during fast transients. Although the model accurately reproduced the first pressure peak, it overestimated subsequent oscillations. The authors attributed this discrepancy to the oversimplified assumption of a constant $m$, noting that real air–water interactions involve significant heat exchange after the first compression.

Beyond its thermodynamic behavior, extending the elastic method to pipeline emptying also requires considering the dynamic influence of trapped air on the overall hydraulic response. The elasticity of air pockets acts as a temporary energy reservoir that absorbs pressure surges during the initial depressurization and releases part of that energy as the system evolves. This elastic feedback modifies both the local wave celerity and the global momentum of the water phase, often producing oscillatory patterns comparable to those observed during filling operations [11], though typically with asymmetric amplitudes due to the continuous air ingestion that characterizes emptying events [27,28]. An accurate elastic-based description of the process should account for the time-dependent stiffness of the air–water interface and its effect on the water column inertia [55,57].

The experimental validation of elastic-based emptying models remains challenging. Laboratory facilities capable of simultaneously capturing transient pressures, air pocket dynamics, and interface deformation under rapid drainage are scarce [27,64]. In addition, reproducing realistic boundary conditions—particularly at the upstream end, where compressed air expands or enters the system—is complex because small deviations in air inflow or leakage can alter the transient response [28]. Scaling effects further complicate the design: while the elastic behavior of water and pipe walls can be reproduced at laboratory scale, the thermodynamic response of the air phase and its heat transfer characteristics do not always scale linearly [11]. Existing experimental data describing elastic effects during emptying are limited, making direct validation of numerical models difficult.

In addition to the difficulties associated with experimental validation, the EWC framework introduces additional modeling challenges that have not yet been fully resolved. While unsteady friction formulations have been incorporated into elastic water hammer models for single- phase flow [11], their extension to emptying operations—where a moving air pocket interacts with a decelerating water column—introduces velocity gradients that evolve differently from those observed in pressurization events. Consequently, the associated energy dissipation mechanisms may not be adequately represented by conventional steady-friction assumptions.

Another unresolved aspect is the representation of air-valve boundary conditions in elastic formulations. Air valves are installed at high points to admit air when internal pressure drops below atmospheric. Without them, sub-atmospheric conditions can develop and cause pipe collapse [60]. Their behavior is commonly modeled through an isentropic nozzle-flow analogy, as described by Wylie et al. [65], which distinguishes two operating regimes depending on the ratio between internal pipe pressure $p_t^{*}$ and atmospheric pressure $p_{atm}^{*}$. When $p_t^{*}>0.528·p_{atm}^{*},$ the flow through the valve orifice is subsonic and the admitted mass flow rate increases as the internal pressure decreases. Once $p_t^{*} \le 0.528·p_{atm}^{*}$, sonic conditions are reached and the volumetric flow rate becomes constant regardless of further pressure reduction—a condition known as sonic blocking [66]. However, within an EWC framework the air valve performs a dual function: it regulates the admitted air mass flow while simultaneously acting as a wave-reflection boundary that influences the local wave celerity and the upstream pressure response. A consistent formulation capturing this coupled behavior has not yet been fully addressed in the literature. As illustrated in Figure 5, the characteristic curves provided by manufacturers can exhibit substantial differences even for valves of the same nominal diameter. For DN 250 air valves, Manufacturer B demonstrates greater air admission capacity in the pressure range approaching the sonic threshold. Nevertheless, the two curves intersect at approximately 10,000 Nm³/h, underscoring the importance of carefully selecting air valves according to the pressure range at which the system predominantly operates. These discrepancies, which have also been reported in laboratory testing [26], introduce additional uncertainty into the characterization of air valve boundary conditions and directly affect the accuracy of any model incorporating such devices—a limitation that becomes particularly consequential within an EWC framework, where small variations in admitted airflow may produce measurable changes in wave speed.

This coupling is further complicated by uncertainties associated with discharge coefficients and dynamic valve closure, since small variations in admitted airflow may produce measurable changes in wave speed when pipe-wall elasticity is considered [11]. In topographically irregular systems, multiple air pockets may develop simultaneously at different high points during drainage further complicating the modeling task [55]. In elastic models, each air–water interface can behave as a partial wave reflector, potentially generating interference patterns between successive pressure fronts. Although this phenomenon has been documented during filling operations, its implications for elastic formulations of pipeline emptying remain largely unexplored and may lead to underestimation of local pressure extremes [67].

While the elastic method has been primarily developed for filling and pressurization events, its structure provides a foundation for adaptation to emptying processes [61,62]. The Virtual Plug Method (VPM) idealizes the air–water interface as a moving piston between entrapped air pockets, providing a simplified yet effective representation of two-phase interactions during transients [1,67]. This approach could be reformulated to capture the progressive elongation of the air cavity and the reverse movement of the water column during drainage. Similarly, the improved EWC formulation proposed by Zeng et al. [16], which incorporates local interpolation to enhance spatial accuracy and numerical stability, could be adapted to model emptying operations involving moving air–water interfaces. Furthermore, the formulation developed by Zhou et al. [63] for rapid air expulsion through an orifice in a vertical water pipe offers a valuable basis for representing the air release phase typical of emptying operations. Coupling this model with the elastic deformation of both the water phase and the pipe wall could enable a more comprehensive representation of the mutual feedback between air expansion and water compressibility [68].

Overall, the extension of elastic-based formulations to emptying processes demands the incorporation of dynamic coupling terms linking air compressibility, water elasticity, and wall deformation. Such developments would enable a unified modeling framework capable of describing both filling and emptying transients under realistic two-phase flow conditions [11,27,55,61,63].

The key references and their main contributions to the development of elastic-based approaches for pipeline emptying are summarized in

Table 3. Key contributions supporting the adaptation of elastic-based models to emptying phenomena, summarizing their assumptions and validation methods.

Reference	Model or Method	Application or Context	Key Findings/Relevance for Emptying Modeling
Fuertes-Miquel et al. (2019) [11]	Comprehensive literature review on filling and emptying models	Synthesis of transient models and identification of research gaps	Highlights scarcity of studies on elastic behavior during emptying; identifies key gaps in two-phase transient modeling.
Liu et al. (2011) [61]	Rigid-Plug Elastic-Water Model (EWC)	Transient flow in pipes with entrapped air (elastic coupling)	Demonstrates elastic coupling of air-water phases; forms theoretical basis for extending EWC to emptying.
Fuertes-Miquel et al. (2019) [27]	Transient analysis of single-pipe emptying with air-water interaction	Experimental and numerical analysis of air-water interface dynamics	Shows importance of air-pocket dynamics and heat transfer in emptying phenomena.
Laanearu et al. (2012) [28]	Experimental study of large-scale pipeline emptying with pressurized air	Experimental validation of air-pocket motion and transient pressure	Confirms influence of air elasticity on water-column motion; key for model calibration.
Romero et al. (2020) [29]	Large-scale experiments of emptying operations in hydraulic systems	Validation of transient behavior in real-scale drainage systems	Bridges laboratory and field-scale transients; underlines challenges in scaling elastic effects.
Coronado-Hernández et al. (2018) [53]	Rigid Water Column Model for pressurized-air-driven emptying	Simulation of air-pressurized emptying events	Validates RWC formulation; supports EWC development with air compressibility effects.
Coronado-Hernández et al. (2018) [54]	Subatmospheric pressure analysis with air pocket in draining pipeline	Analysis of negative pressure and air-pocket effects during drainage	Quantifies subatmospheric pressure and backflow during draining; essential for validation.
Coronado-Hernández et al. (2017) [55]	Mathematical model for emptying operations in supply networks	Elastic formulation for network-scale emptying operations	Provides conceptual framework for network-scale EWC applications in emptying.
Besharat et al. (2018) [57]	Backflow and air-pocket analysis during emptying (CFD + 1D)	Pressure and backflow prediction in air-pocket-dominated events	Demonstrates coupled air-water transient modeling; useful for calibration of EWC and CFD.
Zhou et al. (2013) [62]	Thermodynamic study of air behavior during rapid filling	Comparison of adiabatic and quasi-isothermal air compression ($m$ = 1.0–1.4)	Identifies time-dependent variation of $m$ (1.4 to 1.0) and its influence on pressure prediction.
Zhou et al. (2019) [63]	Rapid air expulsion model through an orifice in vertical pipe	Simulation of air expulsion and transient pressure oscillations	Validates adiabatic assumption for first pressure peak; shows limits of constant m assumption.
Zhou et al. (2013) [67]	Virtual Plug Method (VPM) for air–water interface representation	Simulation of transients involving two entrapped air pockets in a pipeline	Introduces a simplified piston-type formulation for air–water interface motion; potentially adaptable to emptying processes.

, which provides a comparative overview of the methods, applications, and findings discussed in this section.

Elastic Modeling of Emptying Process for Different Air Pocket Sizes

To illustrate the theoretical considerations previously discussed, a one-dimensional elastic formulation for pipeline emptying is applied to simulate the transient pressure behavior of an entrapped air pocket for different initial air pocket sizes. The formulation is implemented within the MOC framework and represents an elastic extension of the RWC approaches commonly used to analyze drainage transients, enabling the representation of wave propagation effects that are inherently absent in RWC formulations.

Following the domain transformation approach of Chaiko and Brinckman [32], the spatial coordinate is mapped onto a fixed computational domain through the transformation $y = (x - x_0(t\left)\right)/L_{\mathrm{e}}\left(t\right)$, where $y = 0$ corresponds to the air–water interface, $y = 1$ to the downstream valve, $x_0\left(t\right)$ is the instantaneous air pocket length, and $L_{\mathrm{e}}\left(t\right) = L_T - x_0\left(t\right)$ is the instantaneous water column length. Under this transformation, the compatibility relations of the MOC propagate along characteristic lines:

$$\mathrm{d}\mathrm{y}/\mathrm{d}\mathrm{t}={\mathrm{V}}_{\mathrm{w}}(1-\mathrm{y})/{\mathrm{L}}_{\mathrm{e}}\pm \mathrm{a}/{\mathrm{L}}_{\mathrm{e}}$$

(1)

where $V_w = d{x}_0/dt$ is the interface velocity and a is the pressure wave speed. The liquid domain is discretized into $M = 50$ uniformly spaced nodes in the transformed coordinate y, and the time step is determined adaptively at each iteration:

$$\Delta t = CFL · L_{\mathrm{e}} \Delta y/(a + |V_w\left|\right), CFL = 0.80$$

(2)

A value of $CFL = 0.80$ is adopted to ensure numerical stability while maintaining sufficient temporal resolution throughout the transient.

The water column is governed by the one-dimensional transient flow equations for a slightly compressible liquid. Within the MOC formulation, these equations reduce to two compatibility relations propagating along the characteristic lines $dx/dt=\pm a$. The entrapped air pocket is represented as a spatially uniform thermodynamic region whose pressure evolves according to a polytropic relation between pressure and volume:

$$p_a^{*}(t)=p_{a,0}^{*}{\left({\displaystyle \frac{V_{a,0}}{V_a\left(t\right)}}\right)}^m$$

(3)

where $p_a^{*}$ is the absolute air pressure, $V_a$ is the instantaneous air pocket volume, $p_{a,0}^{*}$ is the initial air pressure, and $m=1.2$ is the polytropic exponent, representative of intermediate heat transfer conditions during relatively slow emptying transients. Considering that the air pocket occupies a pipe segment of length $x_0,$ the initial air volume is $V_a,0=A{x}_0$, while the instantaneous volume is $Va\left(t\right)=AL\left(t\right)$, where $L\left(t\right)$ denotes the instantaneous position of the air–water interface measured from the upstream end of the pipe.

The motion of the air–water interface is determined from the kinematic relation:

$${\displaystyle \frac{dL}{dt}}={\displaystyle \frac{Q_w}{A}}$$

(4)

which couples the interface position with the water discharge $Q_w$ computed from the characteristic equations of the water column. At each time step, the interface position is obtained by coupling the polytropic air relation with the backward MOC characteristic arriving from the water column interior, resulting in a nonlinear equation that is solved iteratively using a bisection procedure.

At the air–water interface ($y = 0$), the arriving characteristic is $C^{-}.$ Coupling the polytropic relation with the $C^{-}$ compatibility equation yields a nonlinear equation in the interface discharge $Q_{int}$, solved by bisection converging to a tolerance of ${10}^{-8}$ within 60 iterations. The interface position is advanced using a second-order trapezoidal scheme:

$${{\mathrm{x}}_0}^{\mathrm{n}+1}={{\mathrm{x}}_0}^{\mathrm{n}}+(\Delta \mathrm{t}/2)({\mathrm{Q}}_{\mathrm{i}\mathrm{n}\mathrm{t}}^{\mathrm{n}+1}/\mathrm{A}+{{\mathrm{V}}^{\mathrm{n}}}_{\mathrm{w}})$$

(5)

At the downstream valve ($y = 1$), the arriving characteristic is $C^{+}$. The valve head loss $h_v = K_d Q_v^2$ leads to the quadratic expression:

$$K_d{Q}_v^2 + \left({\displaystyle \frac{a}{gA}}\right)Q_v - (C_{+}- H_{atm}) = 0$$

(6)

where $C_{+}$ is the compatibility constant from the $C^{+}$ characteristic and $H_{atm}$ is the total head at atmospheric pressure at the valve location. The numerical model was implemented in Python (version 3.13.9) using in-house scripts.

The Elastic Water Column Model (EWCM) developed was verified against the experimental dataset of Fuertes-Miquel et al. [27], comprising 12 runs across two pipe inclination angles ($\theta = 0.457 \mathrm{r}\mathrm{a}\mathrm{d}$ and $\theta = 0.515 \mathrm{r}\mathrm{a}\mathrm{d}$), three initial air pocket lengths ($x_0 = 0.205 \mathrm{m}$, $0.340 \mathrm{m}$, and $0.450 \mathrm{m}$), and two valve opening schedules, each replicated twice, yielding 24 individual tests.

To evaluate the influence of the polytropic exponent m on the model predictions, three values are analyzed: $m = 1.0, 1.2, \mathrm{a}\mathrm{n}\mathrm{d} 1.4$. Figure 6 presents the simulated air pocket pressure for each value alongside the experimental measurements from Run 1. The results show that $m = 1.4$ produces the closest agreement with the experimental data in terms of both pressure magnitude and oscillation period, particularly at the first pressure minimum. This behavior is consistent with the adiabatic conditions expected during the rapid initial compression phase of the transient. Accordingly, $m = 1.4$ is adopted for the experimental verification presented in the following section, while $m = 1.2$ is retained for the dimensionless analysis in accordance with the reference geometry of Fuertes-Miquel et al. [11].

The EWCM reproduces the minimum air pocket pressure with a mean absolute error of 0.35% and a maximum of 0.66% across all cases. The error in the time to minimum pressure is below 0.10 s for 21 of the 24 comparisons, with a mean absolute value of 0.054 s. The three cases with larger deviations in timing (Runs 11 T1, 11 T2, and 12 T1) are associated with slower valve opening schedules and lower final discharge coefficients, conditions under which small differences in valve characterization have a proportionally greater influence on the simulated pressure minimum timing. Figure 7 illustrates the transient pressure evolution predicted by the EWCM against experimental measurements for Run 1—which yields the closest agreement—and Run 12—which presents the largest deviation in the time to minimum pressure—representing the performance range of the model across the dataset. The complete numerical results for all 12 runs and both repetitions are summarized in

Table 4. EWCM verification against experimental data.

Run	$\mathit{\theta }$ ($\mathbf{r}\mathbf{a}\mathbf{d}$)	${\mathit{x}}_0$ (m)	EWCM ($\mathbf{m}{\mathbf{H}}_2\mathbf{O}$)	T1 Exp. ($\mathbf{m}{\mathbf{H}}_2\mathbf{O}$)	εp T1 (%)	T2 Exp. ($\mathbf{m}{\mathbf{H}}_2\mathbf{O}$)	εp T2 (%)
1	0.457	0.205	8.027	7.995	0.40	8.028	−0.02
2	0.457	0.340	8.232	8.225	0.08	8.212	0.24
3	0.457	0.450	8.348	8.298	0.61	8.304	0.53
4	0.457	0.205	8.211	8.199	0.14	8.199	0.14
5	0.457	0.340	8.346	8.311	0.42	8.304	0.50
6	0.457	0.450	8.465	8.456	0.11	8.449	0.18
7	0.515	0.205	8.055	8.048	0.09	8.015	0.50
8	0.515	0.340	8.097	8.087	0.12	8.068	0.37
9	0.515	0.450	8.243	8.212	0.37	8.206	0.45
10	0.515	0.205	8.189	8.146	0.52	8.140	0.60
11	0.515	0.340	8.279	8.232	0.58	8.225	0.66
12	0.515	0.450	8.341	8.311	0.36	8.304	0.44

.

The system configuration considered in this analysis is shown in Figure 8. It consists of a single inclined pipeline with total length $L_T=600 \mathrm{m}$, internal diameter $D=0.30 \mathrm{m}$, friction factor $f=0.018$, elevation difference $\Delta z=12 \mathrm{m}$, and pressure wave speed $a=1000 \mathrm{m}/\mathrm{s}$, and polytropic exponent $m = 1.2$. For the geometry analyzed, the acoustic period $T_{ac} = 2L_T/a = 1.2 \mathrm{s}$ is significantly shorter than the characteristic emptying timescale—of the order of hundreds of seconds, as shown in Figure 9—placing the system in the mass-spring regime. Under these conditions, elastic oscillations occur at a frequency much higher than the drainage dynamics, and both RWC and EWC formulations tend to produce similar minimum pressure envelopes. The elastic formulation becomes more critical in shorter pipelines or systems with lower wave speeds, where $T_{ac}$ is comparable to the emptying timescale and acoustic oscillations are resolvable at the drainage scale. No air admission valve is considered; the upstream end remains closed throughout the transient, and the emptying operation is initiated by opening a downstream valve characterized by a resistance coefficient $K_d=0.45 \mathrm{m}/{({\mathrm{m}}^3/\mathrm{s})}^2$. These parameters are adopted from the reference system studied in Fuertes-Miquel et al. [11].

Five values of the initial air pocket length are considered: $x_0\in \left\{100,200,300,400,500\right\} \mathrm{m}$, representing cases in which the entrapped air occupies between approximately 17% and 83% of the total pipe volume. Figure 9 presents the evolution of dimensionless air pocket pressure $p_a^{*}/p_{atm}^{*}$ as a function of time for each scenario.

The results reveal that smaller initial air pockets produce more severe sub-atmospheric pressure troughs, whereas larger pockets lead to more moderate pressure drops. This inverse relationship follows directly from the polytropic law: a smaller initial air volume undergoes a proportionally larger relative expansion for the same interface displacement, resulting in a steeper pressure drop. For $x_0=100 \mathrm{m}$, the minimum pressure reaches approximately $0.309 p_{atm}^{*}$, approaching vacuum conditions and highlighting the structural risk associated with small trapped air volumes in the absence of protective devices. As $x_0$ increases to $500 \mathrm{m}$, the minimum pressure rises to approximately $0.877 p_{atm}^{*}$, reflecting the greater thermodynamic buffering capacity of larger air pockets. This behavior is consistent with trends reported in rigid-column emptying studies, confirming that the present elastic formulation reproduces the qualitative behavior observed in the literature [11,58].

The protective effect of air admission devices is illustrated through results reported by Fuertes-Miquel et al. [11] for the case $x_0 = 100 \mathrm{m}$ operated with an air valve installed at the upstream end, included in Figure 9 for reference. In this configuration, the air valve opens as soon as the pressure drops below atmospheric, admitting air into the system and effectively arresting the depressurization process. As a result, the minimum pressure remains close to $p_{atm}^{*}$, and the system recovers atmospheric conditions once the pipeline is completely drained. The contrast with the corresponding curve without an air valve—which reaches a minimum of ${0.309 p}_{atm}^{*}$—illustrates the critical role of air admission devices in preventing sub-atmospheric pressure development during emptying operations. The case $x_0=100 \mathrm{m}$ as selected because it represents the most critical scenario. In this case the unprotected pressure minimum falls below the sonic blocking threshold ($0.528 p_{atm}^{*}$), meaning that a valve of insufficient capacity could reach its maximum admission rate before fully arresting the pressure drop—a limitation directly linked to the discharge coefficient uncertainties previously discussed.

The results highlight two relevant aspects for the development of elastic-based emptying formulations. First, following the initial pressure drop, the curves stabilize at a minimum value governed by the polytropic relation, with no significant recovery during the simulation period. Second, the divergence between curves increases as $x_0$ decreases, indicating that elastic effects are more pronounced in smaller air pocket configurations—the scenarios that also produce the most severe depressurization conditions.

6. Conclusions

The elastic water column formulation developed in this work, together with the dimensionless analysis, yields the following conclusions:

• The existing literature reveals a significant gap in the study of pipeline emptying: while filling processes have been extensively modeled using elastic formulations, the application of elastic water column models to emptying operations has not been systematically addressed.
• The EWCM developed was validated against laboratory data from Fuertes-Miquel et al. [27], demonstrating mean absolute errors below 0.35% in minimum pressure across 24 experimental cases, confirming the physical consistency of the formulation. The sensitivity analysis conducted on Run 1 shows that $m = 1.4$ yields the closest agreement with experimental data in terms of pressure magnitude and oscillation period, consistent with the adiabatic conditions expected during rapid transients in short pipelines. For the dimensionless analysis, $m = 1.2$ is adopted following the reference geometry of Fuertes-Miquel et al. [11], which corresponds to quasi-isothermal conditions in longer pipelines where heat exchange is more significant. However, the performance of the formulation under different geometric conditions, such as variations in diameter and pipe length, as well as the dynamic variation of m throughout the transient, should be addressed in future work to improve predictions across a wider range of operational conditions.
• The dimensionless analysis confirms that the minimum pressure attained during pipeline emptying is strongly controlled by the relative size of the entrapped air pocket with respect to the total pipe length. Expressing the air-pocket size through the dimensionless parameter $x_0/L_T$ reveals a systematic relationship with the normalized pressure $p_a^{*}/p_{atm}^{*}$, indicating that smaller relative air volumes lead to significantly more severe depressurization conditions. This behavior follows directly from the polytropic pressure–volume relation governing the air pocket and suggests that dimensionless parameters provide a useful framework for generalizing emptying dynamics across different pipeline scales.
• The comparison between protected and unprotected scenarios shows that air admission devices effectively prevent sub-atmospheric pressure development. For the most critical case analyzed ($x_0/L_T = 0.17$), the unprotected minimum pressure of $0.309 p_{atm}^{*}$ falls below the sonic blocking threshold of $0.528 p_{atm}^{*}$, underscoring that both the presence and adequate sizing of air valves are essential conditions for safe emptying operations. For small air pocket configurations, adequate valve sizing is therefore a critical design requirement.

Future work should address the extension of this formulation to irregular pipe profiles, multiple air pocket configurations, and the formal coupling of air valve boundary conditions within the EWC framework. Additionally, a fully dimensionless parametric study incorporating the influence of pipe diameter, friction factor, slope, and wave speed would extend the generality of the present analysis beyond the reference geometry considered here.