## 5 Spatiotemporal Characteristics of Existing Pipeline's Deformation  
### Three-Dimensional Deformation with Tunnel Construction and Grouting Processes  
To study the spatiotemporal characteristics of the existing pipeline's deformation induced by the new shield tunnel parallel excavating beneath, Figure 17 draws the three-dimensional deformation of the existing pipeline and the ground with shield tunnel advance. The \(X\)-axis is the pipeline's length from \(-50\) m to \(50\) m, and the \(Y\)-axis is the tunnel face's location from \(-30\) m to \(30\) m. During the shield tunnel excavation, assume the coefficient of subgrade reaction is \(k\) before the tunnel face and increases to \(2k\) behind the tunnel face because the backfill grouting is carried out. The \(Z\)-axis is the resultant settlements according to the construction and grouting processes.  
Figure 16: Longitudinal profiles of pipeline’s deformation with different equivalent bending stiffness.  
Figure 15: Longitudinal profiles of pipeline’s deformation with different torsional forces.  
The front view shows the envelope of the longitudinal deformation curves (LDC) of the overall existing pipeline, which consists of several S-curves (growth curves). For instance, the solid purple line is the pipeline's LDC when the new shield tunnel excavates to 0 m. The LDC with different shield tunnel advances is plotted in Figure 18a. As the shield tunnel constructs forward, the LDC with similar profiles moves forward as a whole. Following the published conclusions [31, 48], the deformation is categorised into four stages: advance stage, rapid stage, gradual stage, and steady stage. Compared with the perpendicular undercrossing project, the maximum deformation of the existing pipeline is roughly around a specific constant with the tunnel advance. Therefore, the desired pipeline's LDC at a certain moment of excavation can be predicted based on the measured LDC, all else being equal. When the actual monitored LDC changes significantly, the predicted LDC can be used as an early warning value to alert engineers to control deformation.  
The side view shows the envelope of the deformation development curves (DDC) of all points on the pipeline. The envelope consists of several S-curves (growth curves). For instance, the orange dotted line is the DDC for the monitored point at \(x=0\) as the shield tunnel advances. The DDC for the different points on the existing pipeline is shown in Figure 18b. Consistent with the conclusion proposed by [PERSON] et al. [18], the deformation changes rapidly within a certain range. The magnitude of deformation variation at each point of the existing pipeline is essentially the same for the parallel undercrossing project, whereas it is different for the perpendicular undercrossing project. Thus, the DDC of all the points on the pipeline can be easily extrapolated from the DDC of two monitored points, all other conditions being controlled.  
Figure 17: Three-dimensional deformation with shield tunnel construction and grouting processes.  
The top view shows the existing pipeline's settlement contours due to a new shield tunnel parallel excavating beneath. The settlement at any point and any moment during the tunnel advance can be obtained through the top view. The settlement changes significantly around the diagonal, symmetrical about the diagonal.