Dissipation characteristics of excess pore-water pressure around tunnels in viscoelastic foundation using a fractional-derivative model

The dissipation characteristics of the excess pore-water pressure around tunnels in viscoelastic foundation are investigated by using a fractional-derivative model. Firstly, the fractional-derivative Merchant model is introduced to describe the rheological behavior of the saturated soft foundation around tunnels, and its compliance function is deduced by means of the Laplace transform and its inverse transform. Secondly, using the methods of conformal mapping and variable separating, two independent equations only including the variable of time and only including the variable of space are derived from the partial differential equation governing the dissipation of the excess pore-water pressure during two-dimensional consolidation process, and the analytical solution of the excess pore-water pressure is also formulated in the Laplace domain. Thirdly, the numerical method for the excess pore-water pressure in the time domain is presented based on the Crump method. The developed solution is simplified into two special cases for the elastic foundation and viscoelastic foundation of integer order, and its correctness is validated against the existing solutions of these two special cases. Finally, using the developed solution, the dissipation characteristics of the excess pore-water pressure around tunnels are investigated, and the influences of fractional order, modulus ratio, viscosity coefficient and boundary condition are discussed. The results show that the influences of the fractional order and viscosity coefficient on the dissipation of the excess pore-water pressure have two distinct stages. In the early stage, larger fractional order and larger viscosity coefficient bring about faster dissipation of the excess pore-water pressure; while at the later stage, they lead to slower dissipation of the excess pore-water pressure. Larger modulus ratio means softer foundation, and it also results in slower dissipation of the excess pore-water pressure. In addition, at the middle stage, the modulus ratio has a much greater influence. In the early stage, the effect of the inner boundary condition on the excess pore-water pressure mainly emerges on the regions close to the tunnels, and then this effect is spread far away.