## 1 Introduction  
When subsea tunnels pass through unfavorable geological conditions, the stability of the surrounding rock is reduced under the combined action of the seepage and ground stressor fields, and, thus, it could dramatically respond due to the excavation disturbance. In serious cases, collapses, water inrushes, mud bursts, and other disastrous accidents may also be induced, which not only causes great economic losses, but also create delays in constructional works [1; 2]. To ensure successful tunneling through weak zones, ground reinforcement methods including grouting, concrete backfilling, and freezing are necessarily exploited [3]. Among these approaches, grouting is often preferred because of its advantages of being a mature technology, having a low cost, and exhibiting good outcomes [4]. Researchers have shown that ground reinforcement can significantly improve stability and water resistance of the surrounding rock, as a result, construction safety is ensured [5; 6]. Therefore, ground reinforcement has become a strategic technology for subsea tunnels crossing through weak zones, whose design concepts are of particular importance.  
Common approaches to evaluating reinforcement parameters are mainly based on empirical considerations [7], numerical analyses [8], laboratory investigations [9], and analytical methods [10]. However, due to the complexity of tunnel engineering as well as the environmental diversity, application of empirical methods generally depends on the experience of the designer, and these approaches are greatly subjective. Nevertheless, most numerical methods are relatively complex, and they suffer from low transparency, where rationality relies greatly on the theoretical background of the users. Thus far, model tests represent the reference works for estimation of the reinforcement effect, but they need multiple trials to determine optimal reinforcement parameters. Compared with the other three methods, although theoretical analyses are commonly based on certain assumptions, their derivations are usually transparent, and the predicted results are more concise and convenient for practical purposes [11].  
Generally, there exist two complementary theoretical approaches, namely, the equivalent force approach and the equivalent material approach [12; 13; 14]. In the former approach, the reinforcement is equivalent to a support force exerted on the tunnel wall [15]. This approach, however, does not give any information on the stressors and displacements of the reinforcement media, and the determination of the equivalent force is usually empirical. As to the equivalent material approach, the reinforced ground is considered to be a homogeneous medium [16]; hence, the reinforcement effect can be estimated more precisely. Since the reinforcement and seepage effects have not simultaneously considered in the previous models [17], the suggested methods were not comprehensive enough, which did not make them popular and readily applicable to subsea tunnels.  
In the present study, a mechanical model for the surrounding rock by considering both ground reinforcement and seepage effect was suggested and the solutions were appropriately sought. The proposed theoretical solution was then verified by both a numerical analysis and a classical model. Moreover, by combination with the stratified sequencing method and the sensitivity analysis, the multi-objective optimization of the reinforcement parameters is put forward. The proposed approach was applied to the Qingdao Jiaozhou bay tunnel project in China, and it is expected to provide significant insights for similar projects.