# An Analytical Model for the Prediction of Emptying Processes in Single Water Pipelines
###### Abstract  
Air pockets in water distribution networks can cause various operational issues, as their expansion during drainage operations leads to sub-atmospheric conditions that may result in pipeline collapse depending on soil conditions and pipe stiffness. This study presents an analytical solution for calculating air pocket pressure, water column length, and water velocity during drainage operations in a pipeline with an entrapped air pocket and a closed upstream end. The existing system of three differential equations is reduced to two first-order nonlinear differential equations, enabling a rigorous analysis of the existence and uniqueness of solutions. The system is then further reduced to a single second-order  
the valve chamber--air cannot be adequately managed within the hydraulic installation. This may lead to hazardous pressure transients, including extreme pressure surges or undesirable sub-atmospheric conditions, posing a significant risk to system integrity. This latter situation represents one of the most critical scenarios water utilities may face during routine operations [5]. The modelling of air-water interactions during pipeline emptying is inherently complex due to the nonlinear and transient nature of the governing equations [5; 9]. Reference [5] addressed this problem by developing a model to describe the behaviour of an emptying process with an entrapped air pocket. This model is formulated as a system of three equations--two ordinary differential equations (ODEs) coupled with one algebraic equation--characterising the coupled dynamics of air and water. The first equation is derived from the rigid water column model, representing the dynamics of the water phase. The second is based on the polytropic law governing the air phase, while the third describes the conditions at the air-water interface. The model has been validated through experimental measurements conducted by References [5; 10; 11] in several European laboratories across different universities and research institutes. The results confirm that the three-equation system provides an adequate physical representation of the phenomena involved. The system was solved numerically using appropriate computational methods in all these studies. The problem related to emptying processes has been explored by relatively few researchers worldwide. Initially, a semi-empirical approach for computing the emptying process in water pipelines was proposed by [PERSON] et al. [10]. Subsequently, the authors developed a mathematical model based on physical principles [5], which has been solved numerically. Water utilities increasingly focus on creating digital twin models [12; 13] to support decision-makers in managing emptying operations within water distribution networks. In this context, developing alternative approaches to the numerical solution of complex differential equations may contribute to achieving the Sustainable Development Goals (SDGs) [14]. A detailed analysis of the emptying process can support the advancement of digital twins [13; 15]. This concept integrates modelling, the Internet of Things (IoT), an infrastructure layer, sensing technologies, and artificial intelligence (AI) [16]. In this regard, improvements in modelling techniques may enhance the application of AI tools to detect potential failure scenarios--such as pipeline collapse caused by sub-atmospheric pressures--that may occur during emptying operations. This research advances the theoretical understanding of the emptying process involving an entrapped air pocket with the upstream closed (without air valves). The primary objective of this study is to derive an analytical solution to the three governing equations previously described. In this sense, the system is first reduced to two first-order nonlinear ordinary differential equations (ODEs), allowing for an investigation into the existence and uniqueness of the solutions.