## Abstract

Finding the most suitable closing law is essential to decrease the shock wave pressure caused by transient flow and minimize the potential damage to equipment. The closure of a valve can occur instantly, rapidly, or gradually, and the appropriate law can be convex, linear, or concave, depending on various factors. These factors include the pipe's characteristics (type, diameter, roughness, and length), the conveyed fluid (nature and temperature), and operating conditions (pressure and flow rate). Other factors that receive less attention, such as the duration of slow closure and the impact of soil load on the pipe, are also considered in this study. The main focus of this article is to investigate how the optimal law evolves based on the time it takes for a valve to gradually close, specifically in the case of a valve located at the end of an underground gravity supply pipe. The findings reveal that when the slow closure time (*t*) exceeds 0.50 times the return period (*t*_{4}), the exponent of the optimal law becomes a damped periodic function. Each closure time corresponds to a unique optimal law, and as the valve closure time increases, the range of optimal laws becomes narrower.

## HIGHLIGHTS

Each slow closing time corresponds to a single optimal convex law.

As the slow closing time increases, the range of optimal laws decreases and the appearance of maximum loads is delayed.

The evolution of the optimal pressure at the valve is governed by two models: exponential and linear.

The characteristic method with mixed scheme is used for the simulation of the transient flow of a long buried pipe.

## INTRODUCTION

In hydraulic systems, transient flows occur when there is a disturbance in the initial conditions of steady flow, particularly during valve closure, whether it is slow or rapid. Numerous studies have been conducted in this field. Maurice Gariel's research focused on developing practical rules for calculating the pressure surge induced by water hammer and modeling valve closure to reduce the effects of water hammer, as presented in Remenieras (1961). The study demonstrates the transmission of water hammer along the pipeline, the impact of valve closing time, and the relationship between maximum pressure and the degree of linear valve closure. Hayashi & Eansfoed (1960) studied the effects of sudden opening and closing of fictitious diaphragms located downstream of a very long and highly elastic pipeline. They demonstrated that pressure losses significantly influence the transient phenomenon, resulting in profound changes.

Thanks to the advancements in computing, research has focused on the numerical resolution of the equations governing transient flow, taking into account aspects that can influence this flow. Thirriot (1967) examined approximate methods for calculating water hammer in relatively long pipelines. Streeter & Wylie (1983) and Almeida & Koelle (1992) investigated suitable schemes when friction is the predominant factor. They emphasized the importance of mixed schemes in solving water hammer equations.

The method of characteristics (MOC) has been used for transient analysis (Chaudhry & Hussaini 1985; Chaudhry 2013), where the equations governing the phenomenon are transformed into ordinary differential equations that can be solved through characteristic lines using finite-difference approximations. In their study, Tijsseling & Vardy (2015) examined the application of finite volume and finite difference methods in solving equations related to transient flows. They also highlighted the importance of the MOC for analyzing water hammer phenomena. In fact, the MOC has recently been employed to solve water hammer equations in various types of pipelines, including steel pipelines (studied by BenIffa & Triki (2019)), pipelines with different elastic modulus (studied by Kandil *et al.* (2019)), and glass fiber–reinforced plastic (GRP) pipelines (examined by Rahul & Arun (2020)).

The effect of conduit nature and network configuration has been duly studied. Trabelsi & Triki (2019) used flexible rubber bypass tubes to mitigate water hammer in pressurized steel pipeline systems. Their method involved inserting these tubes upstream of a control valve, which reduced the surge height (Δ*H*) and water hammer celerity, even with shorter tubes. In their study, BenIffa & Triki (2019) conducted research to improve water distribution systems that utilize steel pipes. They proposed the addition of two short sections of plastic conduits, namely high-density polyethylene (HDPE) and low-density polyethylene (LDPE). Through numerical simulations, they demonstrated that the composite configuration (HDPE–LDPE) offered an optimal compromise in terms of pressure surge attenuation and limitation of oscillation waves. Compared to the double configuration (LDPE–LDPE), this composite configuration showed better ability to attenuate pressure surges.

A study conducted by Rahul & Arun (2020) analyzed transient pressure generation in metal-viscoelastic pipelines using both experimental tests and numerical simulations. Two types of pipe materials, mild steel (MS) and GRP, were evaluated. The results indicated that the GRP pipeline was more effective in attenuating transient pressure caused by the water hammer. In cases where the exclusive use of the GRP pipeline was limited due to manufacturing constraints, the use of a combination of GRP + MS pipeline (with GRP for the initial part) proved to be a good solution for reducing transient pressure. Kandil *et al.* (2019, 2021) studied the influence of pipe material properties on hydraulic water hammers and noted that materials with low elasticity modulus had a lower risk of causing water hammer compared to those with a high elasticity modulus. Toumi & Remini (2022) examined pressure, flow, and velocity variations in a pumping system consisting of two and three storage tanks and found that fluctuations were mainly observed in the pipe connected to the tank with a low piezometric head. In addition, the research by Mery *et al.* (2021) and Mahmoudi-Rad & Najafzadeh (2023) focused on various configurations of air chambers to reduce the water hammer effect. These studies demonstrated that the geometry and configuration of the air chamber, including the diameter of the inlet pipe and the diameter of the air chamber, had a significant impact on this phenomenon.

Furthermore, to reduce surges, Subani & Norsarahaida (2015) studied six different models for valve closure: instantaneous (*m* = 0), linear (*m* = 1), concave (*m* = 0.05), concave (*m* = 0.5), convex (*m* = 5), and convex (*m* = 50). They clearly demonstrated the variation of the pressure wave profile and amplitude for each law and emphasized that the instantaneous and convex closure models produce the minimum and maximum pressures, respectively. The study conducted by Yao *et al.* (2015) examined the influence of valve closing time on the variation of water hammer pressure. The results of this study revealed a negative correlation between closing time and maximum pressure. On the other hand, Twyman (2018a, 2018b) studied the influence of different types of valves (butterfly, gate, circular, square, ball, needle, and globe) and network configurations on transient pressures. They concluded that extreme pressure values were related to the type of valve while the network configuration was not a significant factor in attenuating transient pressure.

Herasymov *et al.* (2019) tested linear closure, closure with a breaking point, intermittent closure, and combined closure. They recommended linear closure with a breaking point to reduce surges. However, Wan *et al.* (2019a, 2019b) emphasized that staggered valve closing times reduce the water hammer phenomenon. According to research conducted by Zhao *et al.* (2020), it has been proven that the flow rate and valve closing speed have a significant influence on the increase in water hammer pressure. The higher the flow rate and valve closing speed, the greater the surge pressure. Arefi *et al.* (2021) studied water transport pipelines, which are typically characterized by large diameter and high flow rates and often pose problems during the negative pressure phase. They analyzed the effects of several factors such as abrupt valve closing time, friction coefficient, wave velocity, viscoelasticity, and valve type. The main results indicate that the exclusive use of air valves is not effective; however, the proper installation of equipment such as water flow control devices and hydropneumatic tanks along the pipeline helps prevent the creation of negative pressure.

Jinhao *et al.* (2021) examined the sensitivity of six key parameters that influence transient flow. The results revealed a positive correlation between the flow rate and Young's modulus with maximum pressure, while pipe diameter, valve closing time, and wall thickness showed a negative correlation. In addition, flow rate, diameter, and valve closing time were identified as key factors impacting water hammer simulation. Yong *et al.* (2022) demonstrated that extending the valve closing time effectively reduces the maximum water hammer pressure. Moreover, they found that pressure fluctuations are influenced by the valve closing speed, where a faster closing speed initially results in higher water hammer pressure. Zheng *et al.* (2022) discovered that the optimal nonlinear closure law of the valve exhibits distinct characteristics during fast and slow closure, which provides guidance for real-time valve control and serves as a reference for the design of valves aimed at attenuating surge waves.

It also appears that improving valve closure strategies and coordinating hydraulic system performance are among the most commonly used optimization methods to control the water hammer phenomenon (Bazargan-Lari *et al.* 2013; Wan & Li 2016; Zhou *et al.* 2017; Wan & Zhang 2018; Wan *et al.* 2019a, 2019b). Researchers have focused on optimizing valve closure strategies to mitigate maximum water hammer pressure, as this pressure primarily depends on the valve closure characteristics. Therefore, implementing an optimal valve closure law (OVCL) is a useful approach to control these extreme pressures and mitigate their impact. An OVCL is necessary to design a programmed closure device to control extreme pressure fluctuations, improve operational reliability, and extend the lifespan of the water distribution system.

The aim of this study is to investigate OVCL taking into account additional factors that have received less attention, including the duration of slow closure and the influence of soil load on the pipeline. The study focuses on the specific scenario of a valve situated at the endpoint of an underground gravity supply pipe. By analyzing these factors, the research aims to understand how the optimal closing law develops in such conditions.

In this paper, the MOC with a mixed scheme is employed to simulate transient flow and investigate the evolution of the optimal valve closing law for a valve situated at the end of a relatively long buried supply pipeline in Guelma, Algeria

### Mathematical model

In the above equations, *a* is the wave velocity (m/s); *P* is the pressure (Pa); *ρ* is the density of the liquid, for water *ρ* = 10^{3}kg/m^{3}; *u* is the velocity in the *x* direction (m/s); *g* is the acceleration due to gravity (m/s^{2}); *α* is the pipe inclination angle; *j* is the unit pressure drop; and *x* and *t* denote the spatial and time dimensions, respectively. These equations describe the changes in velocity and pressure with respect to time (*t*) and along the flow direction (*x*) (Meunier 1980; Ouragh 1971; Toumi & Remini 2022). To determine the values of these parameters at any given time and location along a supply pipeline, the MOC was utilized.

### Resolution method

*et al.*2021; Chaudhry 2013; Ouragh 1971; Piskonov 1995; Toumi & Remini 2022).The application of the MOC to the system of equations number one leads to the following equations:where

*S*is the cross section of the pipe (m

^{2}),

*Q*is the flow rate (m

^{3}/s), and

*H*is the hydraulic head (m).

The differential system is solved by discretizing the pipe into multiple nodes, where these nodes are assumed to be sufficiently close to each other. This assumption allows us to consider the following expression: *dF*(*x*, *t*) = *F*(*i* + 1) − *F*(*i*) where *F* represents a function such as *H* or *Q* at consecutive discretization points *i* and (*i* + 1) (Ouragh 1971).

The total load and flow rate along the pipe are assumed to be known at time *t*. By integrating the characteristic equation passing through point *M* with coordinates (*i*, *t*) between time instants *t* and (*t* + Δ*t*), we move from point *i* at time *t* to point (*i* + 1) at time (*t* + Δ*t*). For the positive flow direction, Equation (3) is multiplied by *dt*.

*i*and (

*i*+

*1*), it can be written as follows:

*ρ*being the density of the liquid, for water

*ρ*= 10

^{3}kg/m

^{3};

*E*the Young's modulus of metal (Pa);

_{m}*v*the Poisson's coefficient of the metal;

_{m}*e*the armor thickness (m);

_{m}*K*the modulus of elasticity of water (Pa);

*r*the pipe radius (m);

*P*the hydrostatic pressure (Pa);

*P*the backfill soil load (Pa).

_{a}^{3});

*B*the trench width (m);

_{d}*H*the height of embankment above pipe (m);

*C*the load coefficient.

_{d}*μ′*is the vertical coefficient friction between the backfill and the trench walls and

*K*is the Rankine's coefficient (Moser 2001).

In these equations, the parameter *R* represents the effect of flow resistance, and its elements are defined as follows: *a* represents the wave velocity in meters per second (m/s), *g* represents the acceleration due to gravity in meters per second squared (m/s^{2}), and *S* represents the cross-section area of the pipe in square meters (m^{2}).

### Mixed numerical scheme

Here, *P*(*i* + 1) represents the pressure at the selected node in bar, *H*(*i* + 1) denotes the total load at this node in meters of water column, NGL signifies the Natural Ground Level in meters, *ρ* represents the fluid density in m^{3}/s, and *g* represents the acceleration due to gravity in m/s^{2}.

### Valve closing law

*N*, which corresponds to the valve, is closely connected to the average flow velocity passing through the valve. This velocity can take on a concave, linear, or convex shape and is represented by the following equation:where 0 ≤

*i*≤

*N*, Ct =

*N*.Δ

*t*, 0 ≤

*m*< ∞,

*u*

_{0}is the initial fluid velocity, and

*u*is the fluid velocity at the end of the closing operation.

_{τ}*m*determines the type of valve closing law (Provenzano

*et al.*2011; Subani & Norsarahaida 2015). When the fluid velocity at the end of the valve closing operation is zero (

*u*= 0), Equation (32) can be written as follows:

_{τ}Exponent *m* in the expression for *Q* at node *N* leads to a multitude of *Q* functions, making it a crucial task to determine the optimal value of this exponent for efficient management of hydraulic systems.

### Program presentation

The calculation code used in this work is the result of original programming performed in a Fortran language environment.

*x*, is equal to 0.972 m, and the time step, Δ

*t*, is equal to 0.001 s. The stability and convergence of the scheme are verified by the Courant–Friedrichs–Lewy (CFL) condition. This computer code is designed to consider both pumping and gravity-fed scenarios. The originality of this code lies in the introduction of the valve closure function, which allows the deduction of the optimal law to minimize extreme pressures for any closure time. Figure 1 presents a flowchart outlining the calculation steps, while Tables 1 and 2 provide details on the pipe characteristics and the transported liquid, respectively.

Kind of materials . | Fretted concrete . |
---|---|

Elasticity modulus in Pascal | E_{FC} = 1.962 × 10^{10} |

Pipe Poisson's ratio | ν = 0.20 _{m} |

Coefficient of vertical friction between the backfill and the walls of the trench | μ' = 0.50 |

Rankine's coefficient | K = 0.33 |

Soil density (KN/m^{3}) | γ = 19,000 |

Height of embankment above pipe (m) | H = 1.1 |

Trench width (m) | B = 2 _{d} |

Exterior diameter in (mm) | d_{ext} = 1,170 |

Internal diameter in (mm) | d_{int} = 1,000 |

Pipe thickness e_{1} in (mm) | e_{1} = 85 |

Pipe roughness ε in (mm) | ε = 0.1 |

Pipe length (m) | L = 11,498 |

Kind of materials . | Fretted concrete . |
---|---|

Elasticity modulus in Pascal | E_{FC} = 1.962 × 10^{10} |

Pipe Poisson's ratio | ν = 0.20 _{m} |

Coefficient of vertical friction between the backfill and the walls of the trench | μ' = 0.50 |

Rankine's coefficient | K = 0.33 |

Soil density (KN/m^{3}) | γ = 19,000 |

Height of embankment above pipe (m) | H = 1.1 |

Trench width (m) | B = 2 _{d} |

Exterior diameter in (mm) | d_{ext} = 1,170 |

Internal diameter in (mm) | d_{int} = 1,000 |

Pipe thickness e_{1} in (mm) | e_{1} = 85 |

Pipe roughness ε in (mm) | ε = 0.1 |

Pipe length (m) | L = 11,498 |

Fluid elasticity modulus (Pa) | K = 2.07 × 10^{9} |

Fluid temperature (°C) | 15 |

Kinematic viscosity (m^{2}/s) | 1.1445 × 10^{−6} |

Density of the liquid (kg/m^{3}) | ρ = 10^{3} |

Fluid elasticity modulus (Pa) | K = 2.07 × 10^{9} |

Fluid temperature (°C) | 15 |

Kinematic viscosity (m^{2}/s) | 1.1445 × 10^{−6} |

Density of the liquid (kg/m^{3}) | ρ = 10^{3} |

### Problem statement and boundary conditions

#### Problem statement

#### Boundary and stability conditions

*Q*,

*H*) plane is represented as

#### The convergence and stability of MOC

*x*or

*t*) becomes smaller (Allaire 2016; Sebastien & Victor 2020).

The stability of a scheme refers to its ability to maintain a consistent behavior even in the presence of numerical disturbances, such as calculation errors in boundary conditions or initial conditions. A stable scheme attenuates disturbances, while an unstable scheme amplifies them over time or in space (Allaire 2016; Sebastien & Victor 2020).

*et al.*(1928)). In the case of linear systems, the stability is studied through the Fourier expansion of the error. The CFL condition is given by stability criterion

*S*,where

_{c}*a*is the wave velocity in (m/s), Δ

*t*is the time step, and Δ

*x*is the space step.

The MOC gives exact numerical results when *S _{c}* = 1.0 (Twyman 2018a, 2018b; Goldberg & Benjamin 1983). Recently, authors such as Pal

*et al.*(2021) and Izquierdo & Iglesias (2002) have used this stability criterion. In this study, the stability criterion was verified and taken equal to one (

*S*= 1).

_{c}## RESULTS AND DISCUSSION

*m*< 1), linear functions (

*m*= 1), and convex functions (

*m*> 1). The objective was to identify the closing law that minimizes the overpressure. Figure 5(a)–5(d) depict the variations of the overpressure in relation to the exponent

*m*for four different closing times: 0.55

*t*

_{4}, 0.60

*t*

_{4}, 0.65

*t*

_{4}, and 0.70

*t*

_{4}. Each curve exhibits a distinct minimum value, representing a unique value of the exponent

*m*that defines an optimal closure law, leading to the minimum overpressure.

*t*= 26.0304 s (

*t*= 0.51

*t*

_{4}) and

*t*= 473.28 s (

*t*= 10

*t*

_{4}) using a time step of 0.01

*t*

_{4}(resulting in 950 points). Figure 6 illustrates the changes in the optimal values of the exponent

*m*for each time instant

*t*. For each instant, the value of

*m*was determined to minimize the overpressure.

For valve closing times ranging from 0.51*t*_{4} to 10*t*_{4} (corresponding to *t* = 26.0304 s to *t* = 473.28 s) and using a time step of 0.01*t*_{4}, the optimum values of the exponent m range from 1.126 to 1.446. These optimal values of *m* correspond to convex closing laws (*m* > 1), indicating that a single optimum closing law exists for each time interval, resulting in the minimum overpressure. It can be observed that the optimal values of the exponent *m* vary periodically as the closure time changes. The amplitude of variation is larger for shorter closure times and decreases as the closure time increases. A larger amplitude indicates a wider range of optimal closure laws, while a smaller amplitude narrows down the range of optimal laws. In addition, within each period, multiple optimal closing laws are repeated, but with a progressively narrower distribution.

*m*of the optimal closure law falls within the interval defined by the two straight lines. This variation can be expressed as

*m*, leading to a higher number of optimal valve closing laws (averaging 14 laws).

*t*

_{4}), the pressure deviation between the quadratic law and the optimal laws diminishes, resulting in a diminishing effect of the closing time.

## CONCLUSION

Transient flow is a complex phenomenon characterized by significant pressure fluctuations that can pose serious risks to hydraulic systems. Numerous parameters influence this phenomenon, and it is crucial to minimize the potential risks associated with overpressure, such as pipe rupture, valve deformations or failures, leaks, and equipment malfunctions. Achieving this requires determining the optimal valve closing law, considering various often overlooked or less studied parameters, such as the interaction between the valve closing law and the closing delay, as well as the effect of ground load on the pipe. To investigate this, we conducted a study on a real case involving a long buried gravity pipe that supplies water to the Guelma region in Algeria.

Based on the results, it is evident that convex closure laws (*m* > 1) are superior, resulting in lower overpressure compared to other laws. The optimal closing law depends on the specific characteristics of the hydraulic system. For each closing time, there exists a unique optimal closure law that minimizes overpressure. Furthermore, the optimal values of the exponent *m* vary periodically over time. When the closure time is short, the variation in *m* is significant, but it decreases as the closure time increases. Larger amplitudes lead to a broader dispersion of the *m* coefficient and a wider range of optimal closure laws, whereas smaller amplitudes narrow down the margin of optimal laws. Over time, it appears that the variation in the optimal law tends to stabilize toward a single value.

Simulation of valve overpressure over time demonstrates a smooth regression curve with occasional pressure peak protrusions. Similarly, the simulation of valve depression over time exhibits an increasing curve with peaks of pressure. These curves align well within the envelope of the overpressure and underpressure curves of the quadratic valve closure law. As the slow-closing time exceeds 4*t*_{4}, the pressure deviation between the maximum pressure of the quadratic law and the minimum pressure of the optimal law decreases, and the impact of the slow-closing time becomes less pronounced.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*Calcul du Coup de Bélier Dans Les Conduites Enterrées*

*,*

Transport and Distribution. Numerical Methods. 41 pages. Available from: https://www.utc.fr/mecagom4/MECAWEB/EXEMPLE/FICHES/F3/CVGAF3.htm.