PSI - Issue 28

M. Zhelnin et al. / Procedia Structural Integrity 28 (2020) 693–701 Author name / Structural Integrity Procedia 00 (2019) 000–000

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where σ is the total stress tensor, γ is the unit weight. According to the theory of poroelasticity and the effective stress principle the total stress can be expressed by the following way: : el bp   σ C ε I , (11)

where I is the identity tensor, C is the stiffness tensor for isotropic material, ε el is the elastic strain tensor. In the proposed model the elastic strain ε el is determined as

el ph th    ε ε ε ε ,

(12)

where ε is the total strain; th s T T T    ε is the thermal strain, where α s , T is the thermal expansion coefficient of the soil, T 0 is the initial temperature. The total strain ε is defined through the geometric relation   T 1 grad grad 2   u u ε , (13) where u is the displacement vector. A stress-strain state of the shaft lining during the grout injection is estimated from the following equilibrium equation written for an isotropic linear elastic material div 0 l l   σ γ , (14) of the lining. The equations (1) – (14) were implemented in the Comsol Multiphysics® software. The governing equations (3), (10), (11), (10), (14) were solved numerically relative to the pore pressure p , the grout concentration C , the temperature T and the displacement vector u using the finite element method. 3. Results of numerical simulations The proposed model was applied to numerical simulation of a cement grouting of a soil stratum during a mine shaft construction using the AGF. The simulation was performed for a silty sand stratum laid at the depth of 74.5 – 76 m in the Petrikov potash deposit. The grouting parameters corresponded to technical documentations accepted for building vertical mine shafts in the Republic of Belarus. The computation domain and a scheme of boundary conditions are presented in Fig. 1 (a). The domain is a thick walled ring consisting of saturated soil and the shaft lining. The height of the domain is 5 m. Outer radius is 8.25 m, inner radius is 4.74 m. The radius of the injection boreholes is 0.028 m. The shaft lining includes the concrete shell of 0.45 m thickness and the cast iron tubing of 0.06 m thickness. The top surface is loaded by the overburden pressure P ob of 0.745 MPa. The displacement u on the outer boundary of the domain is restricted in horizontal direction because the ice-soil wall restrains the effect of the rock pressure and has a higher stiffness compared to the unfrozen zone. On the bottom boundary the vertical displacement u z is fixed. The lateral side of the shaft lining is free on any loads or displacement constraints. According to technical documentation it is assumed that the grout injection is carried out through bottom boreholes. The top boreholes are used only for monitoring purposes. On the injection boreholes the pressure p bh of 1 MPa and the temperature T bh of the fresh grout of 20 ◦ C are given. The cement grouting is studied for a time period of 24 hours for injection through two and three boreholes. The ice soil wall is considered at 95-th day of the freezing. During the injection the freezing is not performed. Initial distributions of the temperature and the porosity in the soil stratum are presented in Fig. 1 (b,c). The distributions 0.09 ph  ε i nS I is the strain caused by the phase transition of water into ice; , 0 ( ) where γ l is the unit weight, σ l is the stress for the lining such that : l  σ C ε , where C l is the stiffness tensor for the el l l lining materials, el l ε is the elastic strain. The elastic strain el l ε is defined as el l l   ε ε ε , where th th l ε is the thermal strain