## 1 Introduction  
The shield method was widely used to build tunnels in densely urban areas due to its fast construction speed and high safety. As more and more shield tunnels are built, some new tunnels will be inevitably excavated beneath some existing pipelines. To ensure the safety and serviceability of the existing pipelines, some researchers have systematically studied the mechanical behaviours of existing pipelines. The researchers commonly used field measurements [1, 2, 3, 4, 5], laboratory test methods [6, 7, 8], numerical simulations [9, 10, 11], and analytical methods. The analytical method is convenient and straightforward to study the tunnel-soil-pipeline interaction (TSPI). The main analytical method consists of the elastic continuum approach [12, 13, 14, 15] and the subgrade reaction analysis [16, 17, 18, 19, 20, 21]. In these methods, the existing pipelines are regarded as equivalent beams resting on elastic foundation models (Winkler model, Pasternak model, Kerr model). These foundation models consist of closely spaced linear springs and shear layers to model the TSPL. A series of springs provide subgrade reaction forces to prevent the existing pipeline from deforming. The subgrade reaction forcesdepend on the tunnelling-induced ground movements that are significant for predicting the existing pipeline's deformation. The empirical and analytical methods have been commonly employed to estimate the tunnelling-induced ground movements. [PERSON] [22] proposed an empirical solution to represent the tunnelling-induced ground deformation trough. Based on [PERSON]'s solution, relative researchers have proposed modified solutions, considering the influences of ground conditions and buried depths. Compared with the empirical solution, the analytical solution has a rigorous framework. [PERSON]'s solution [23], virtual image method [24], and complex variable solution [25] are commonly used to predict the ground movements. These analytical methods have inspired further investigations about tunnelling-induced ground movements [26; 27; 28; 29; 30; 31]. [PERSON] et al. [32; 33] proposed analytical solutions of ground movements due to curved shield tunnelling. Most previous studies have focused on the final deformation of ground and existing pipelines, regardless of the deformation in the construction process with spatiotemporal characteristics. As a result, the analytical predictions cannot be used during tunnel construction to suggest appropriate measures to control the structural deformation. Moreover, the applied forces of the shield machine to the surrounding soil are also frequently ignored in analytical methods. The tunnel construction process, applied forces, and backfill grouting should be considered to analyse the spatiotemporal deformation of existing pipelines. This paper aims to analyse the spatiotemporal characteristics of an existing pipeline' deformation using the virtual image method, [PERSON]'s solution, and subgrade reaction analysis. The virtual image method and [PERSON]'s solution calculate the ground deformation induced by the ground volume loss, the supporting pressure at tunnel face, the frictional force of shield shell, and the torsional force of cutter head. Subsequently, the subgrade reaction analysis calculates the deformation of existing pipelines. The spatiotemporal deformation considers the shield tunnel construction and grouting processes. Moreover, the mechanical behaviour of the existing pipelines due to the shield tunnelling parallel beneath in Nanchang subway Line 3 is investigated to validate our proposed method. The influences of the shield tunnel construction factors on the existing pipeline's deformation are carried out. Besides, we provide the longitudinal deformation curve (LDC), deformation development curve (DDC), and grouting reinforcement curve (GRC). The spatiotemporal characteristic curves combined with LDC, DDC, and GRC can give feasible, effective, and economical construction and grouting schemes to control the deformation.