## 1 Introduction
[PERSON] [26] considered the effects of the prism height and soil arching. Then, he proposed an improved three-dimensional wedge-shaped prism model and an analytical formula for the lateral stress ratio between the prism and adjacent soil, which reflects the arching effect of the soil. In a comparison of normalized support pressures \(S_{lim}\) by [PERSON] [27], its tolerance was within \(\pm 0.3\gamma D\). [PERSON] [28] introduced the supporting force of a single anchor in a wedge limit equilibrium model and proposed a limit equilibrium method for analyzing the stability of the tunnel faces of reinforced tunnels in purely cohesive soils without draining shear strength. Based on four typical pipe-slurry-soil contact modes, [PERSON] [29] proposed a calculation method for the jacking force of each pipe section, considering the jacking resistance at the excavation surface and the lateral friction resistance around the pipe. The reliability of the calculation method was verified by comparing the model test results with engineering example data. Numerous scholars have carried out research on the stability of the pipe-jacking tunnel face. The results have been fruitful. However, some aspects are yet to be further deepened. Existing tunnel-face support pressure calculations are analyzed based on the plane strain [30], and the front earth pressure is simplified into a rectangular distribution calculation. This differs from the trapezoidal distribution of soil pressure in front of an actual tunnel face [31]. Moreover, the study above only focused on single pipe jacking, but double-row or even triple-row pipe jacking is widely used in engineering practice [32]. In addition, passive damage to the soil in front of the pipe jacking is more common in actual pipe-jacking projects. If the thickness of the overlying soil layer is not large and the construction involves pipe jacking crossing a river channel, active damage to the soil in front of the pipe jacking must be considered. Therefore, in this study, based on the mud-water balance and considering the conditions of pipe jacking through a river channel, the thin overburden of the river, and multi-line construction with small clearances, the support pressures of the tunnel face have been established. This research can provide a theoretical basis and practical guidance for the construction of pipe jacking under similar engineering conditions, which is of great practical significance. The organization of this paper is as follows: this study firstly investigates the mechanisms of different forms of damage occurring at the tunnel face of a pipe-jacking tunnel. And accordingly, the calculation and analysis model of the tunnel-face support pressure are constructed, respectively. Secondly, based on the principle of mud-water balance and combined with the limit equilibrium method, the upper limit (P\({}_{\text{max}}\)), lower limit (P\({}_{\text{min}}\)), and optimum values (P\({}_{\text{o}}\)) of the support pressure are derived, respectively. Then, the mud-water-balanced pipe-jacking construction is numerically simulated by taking the dependent project as a prototype. Finally, the results of theoretical calculations and engineering measurements are discussed and the main conclusions of this paper are outlined.