## 1 Introduction  
With the ongoing development of China's economy and society, railway and highway network systems have been continuously improved and extended into the western region. However, the mountainous terrain in China's western regions is undulating and has complex geologic conditions. Railroad and highway lines through these areas commonly require the construction of deeply buried tunnels. If groundwater is abundant in these areas, the construction and operation of deep tunnels also face the threat of high external water pressure [1]. High water pressure not only causes water inflow in the tunnel but also leads to significant damage to the lining during tunnel operation. For example, after the completion of the Meihuashan Tunnel (Liupanshui, China) along the Guikun Railway, the high water pressure caused cracks in the lining, and a large amount of water passed through the cracks [2]. The drainage system of the Xiushan Tunnel (Chongqing, China) gradually aged during operation, resulting in a reduced drainage capacity, causing water pressure surges that led to tunnel lining cracking [3] The Sanxia Jijiapo Tunnel (Yichang, China), Yuanliangshan Tunnel (Chongqing, China), and QiyueshanTunnel (Enshi, China) have all undergone damage from water pressure [3]. Therefore, reducing damage caused by high water pressure during tunnel operation has become a necessity. Water pressure on a lining will increase during tunnel operation due to an inadequate drainage system design or aging and clogging of the drainage system, resulting in lining damage and cracking. To address such problems, designing tunnels and parallel adits using a collaborative drainage scheme is effective [4]. A parallel adit is generally parallel to the main tunnel and can not only assist in tunnel construction but also affect the seepage field around the tunnel through drainage during operation. Water pressure on the tunnel lining is also released. In recent years, scholars have analyzed drainage schemes for tunnels and parallel adits. In the construction of the No. 2 Branch Tunnel of the Northwest Water Supply Project, [PERSON] et al. [5] added a parallel adit to the side of the main tunnel for drainage, solving the problem of water inflow and collapse. During the construction of the Zhengpantai Tunnel, [PERSON] et al. [6] designed auxiliary pits, such as parallel adits and inclined shafts, in order to reduce water pressure in response to a water surge problem. All of the above studies have empirically investigated the drainage scheme for tunnels and parallel adits but lack quantitative analysis. [PERSON]. et al. [7; 8] studied the effectiveness of various advanced drainage schemes (including drainage through an external parallel adit) on the stabilization of tunnel faces. Several other scholars have also focused on advanced drainage's effect on tunnel face stability [9; 10]. These studies are mainly concerned with the influence of parallel adit drainage during the construction phase. [PERSON]. et al. [11] took the Gongbei Tunnel (Zhuhai, China) as the research object and discussed the waterproofing and drainage system designs. In addition, they analyzed the relationship between the lining external water pressure and drainage volume and the distribution of water pressure around the tunnel with an indoor simulation test and field test. [PERSON] et al. [12] used the numerical software FLAC-3D to quantify the effects of water head, envelope rock level, and relative position of the parallel adit and tunnel cross-section on tunnel water pressure considering parallel adit drainage. However, they did not analyze key factors such as the drainage volume of the parallel adit and tunnel. An analytical model for a tunnel and parallel adit drainage scheme in tunnel operation has yet to be reported. Analytical and semi-analytical methods not only verify the results of numerical simulations but also avoid the complex process of numerical modeling. The water pressure at the tunnel boundary can be easily obtained by entering the relevant parameters.