## 1 Introduction  
The slurry pressure balance shield tunnel is widely adopted in practice, and ground movement is inevitably induced during tunnel excavation. The induced ground movement may cause serious damage to existing buildings and subterranean facilities. Therefore, the estimation of ground movement is important in the design stage of tunnelling. Many researchers have contributed to analyzing the ground movement by different methods. Those methods can be divided into three categories: the analytical method [1, 2, 3, 4, 5, 6, 7, 8], the empirical method [9, 10, 11], and numerical simulation [12, 13, 14].  
Volume loss plays an important role in the analysis of ground movement. The effect of the tail grout on the volume loss was studied [15]. Different ways for determining the volume loss are summarized [11]. Furthermore, many researchers have investigated the volume loss in two-dimensional problems by assuming a zero-volume change in soil (i.e., undrained volume loss). The undrained volume loss is equal to ground loss, which is defined as the ratio of shrinkage deformation to the area of the tunnel (sometimes the undrained volume loss is called \"ground loss\", in some literature). The analytical solution for the tunnel excavation as the function of the ground loss and the tunnel ovalization parameter are developed [16]. The deformation was divided into the ground loss, ovalization, and vertical movement [17]. A lot of researchers have investigated undrained volume loss with gap parameters, which was developed by considering the 3D elastoplastic deformation at the tunnel face [18]. The procedure for estimating the gap parameter was supported with 14 case monitoring data [19]. [PERSON] and [PERSON] [3] incorporated the gap parameter into the closed-form solution derived by [PERSON] and [PERSON] [16]. Based on the work performed by [PERSON] and [PERSON] [3], A solution for predicting the ground movement in multi-layered clayed soil was proposed [20]. The above research assumed a plane strain condition, which was limited to a two-dimension solution.  
Some researchers have devoted their study to investigating the ground deformation and volume loss before the tunnel face. Based on the field monitoring data, Different relationships between the stability number, which is used in the stability analysis for tunnelling, and the volume loss before the tunnel face are established [21; 22; 23; 24].  
In recent years, the energy method has been developed rapidly in the analysis of ground deformation due to tunnelling. The two-dimensional ground movement was predicted with the balance between energy and work, ignoring soil volume change [25]. Based on the principle of energy conservation, an innovative method to predict the two-dimensional undrained volume loss were proposed [26]. [PERSON] and [PERSON] [27] gave a solution for estimating volume loss in the elastic ground on the basis of the reciprocal theorem. As for volume loss at the tunnel face, [PERSON] and [PERSON] [28] estimated the face 'take' undrained volume loss with the use of a mobilizable strength design (MSD).  
In this paper, we develop a method to predict volume loss at the tunnel face based on the principle of minimum total potential energy. Moreover, a new deformation mechanism is created based on a proposed deformation mechanism [29] to consider the soil volume change. Finally, we analyze the influence of support pressure supplied by slurry, tunnel diameter, and tunnel depth on volume loss. The proposed method provides a more direct way of estimating volume loss at the tunnel face from the energy point of view. The soil volume changes during ground movement are considered in the proposed method.