Analytical method for surrounding rock reinforced by bolts-grouting in tunnel under seepage

Bolts-grouting is a common reinforcement and waterproofing method in water-rich tunnel engineering. At present, there is still a lack of analytical theories that can simultaneously consider the mechanical interaction between rockbolt and surrounding rock, as well as the waterproof and impermeable effects of grouting. Based on Darcy's law and "smeared" method, a differential equation of rock bolts-grouting composite surrounding rock was established by substituting the pore water pressure in the surrounding rock after grouting reinforcement and the interaction force between the rockbolt and the surrounding rock into the stress equilibrium equation of the reinforced surrounding rock. By solving the proposed differential equation, the closed-form solutions for pore pressure, displacement, and effective stress on bolts-grouting composite surrounding rock, and stress on the rockbolt are obtained and compared with numerical solutions and the solutions of other scholars. The comparison results show that the analytical results are consistent with the finite element solution; and bolts-grouting reinforcement can improve the stress state of surrounding rock compared with grouting only. Furthermore, based on our closed-form solutions, and the effect of bolts-grouting reinforcement parameters, support pressure, impermeability pressure of lining, and diameter of the tunnel on safety factor is elaborated. These laws indicate that with the increases of bolts-grouting range, rockbolts density factor and supporting pressure, the safety factor increases in the ways of first fast then slow, quasi-linear and first slow then fast, respectively. The increase of impermeability pressure on secondary lining can reduce the water inflow, but it is not conducive to the stability of the tunnel. Finally, based our closed-form solutions, we propose the method of parameter optimization for the bolts-grouting reinforcement in the tunnel as well as case study.