PSI - Issue 28

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

695

3

Newtonian behavior. The grout is characterized as Bingham fluid. To describe the grout flow, a Reynolds type equation is adopted. The present paper is devoted to a numerical study of cement grout propagation, heat transfer, mechanical response of a shaft lining, unfrozen and frozen soils during the grout injection under pressure. For this purpose, a fully coupled THM model of soil freezing and thawing presented in Panteleev et al. (2017) has been extended to consider transport of the injected grout and a stress-strain state of a shaft lining. The model has been derived within a framework of the theory of porous media. Earlier it has been shown (Panteleev et al. (2017), Kostina et al. (2019)) that the model is able to describe distribution of stress and strain, water velocity and temperature in soil during AGF. To extend the model to take into account a grout flow the governing equations are supplemented by the mass balance equation for the water dissolved component. Non-Newtonian behavior of the grout is included by a modified Darcy law for Bingham fluid flow in a porous media. Besides state equations accounting for an influence of pore pressure and volumetric strain on porosity and a permeability evolution are used. A stress-strain state of the shaft lining is estimated based on the linear elasticity. The numerical study based on the model has been conducted for the cement grouting procedure upon hydrogeological conditions in the Petrikov potash deposit. Various grout injection regimes have been investigated. 2. Mathematical model Mathematical formulation of cement grout injection in saturated soil is performed based on the theory of porous media. It is assumed that the grout and saturated water is to miscible. The saturated soil is modeled as a three-phase porous media consisting of solid particles (index s), fluid mixture phase (index m), and ice (index i). Fluid mixture is the pore water (index l) and the dissolved grout (index g). The governing equations of the grouting process include the mass balance equations for the fluid mixture and the grout, the energy conservation equation and the equilibrium equations for the saturated porous medium and the shaft lining. The equations are derived using the macroscopic continuum mechanics approach. The pore space of the soil is assumed to be fully saturated by the fluid mixture and the ice. Therefore, the following equation can be written 1 m i S S   , (1) where S m is the fluid mixture saturation and S i is the ice saturation. The ice saturation S i is expressed by a power function of the temperature T (Tice et al. 1976)



  

T T T T  

1 1 ( 

 

)

T T

  

ph

ph

,

S

 

,

(2)

i

0



ph

where T is the temperature; T ph is the freezing temperature of pore water; α is an experimental parameter. The mass balance equation for the fluid mixture can be written as ( ) div 0 m m m n t       v , where ρ m is the fluid mixture density; n is the porosity; v m is the fluid mixture velocity. The velocity v m is governed by the Darcy law (3)

k

v

) g ,

(grad

p  

 

(4)

m

m



m