PSI - Issue 28

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

696

4

where k is the absolute permeability; μ m is the dynamic viscosity of fluid mixture; g is the gravitational acceleration; p is pore pressure. The mixture viscosity μ m and the mixture density ρ m are determined according to the linear mixture rule (Zhong et al. (2020)). The absolute permeability is supposed to be dependent on the temperature T and the porosity n by the following way (Nixon 1991, Saada et al. 2005)

0 m k S

k



,

(5)

1 (

)

n n 

 

0

where k 0 is the initial permeability; n 0 is initial porosity; β is an experimental parameter. A state equation for the porosity n is written according to Coussy (2005), Zhou and Meschke (2013)

0 ) vol n n b N p p      , 0 (

(6)

where b , N are the effective Biot coefficient and tangent modulus, vol  is the volumetric strain, p 0 is the initial pore pressure. The mass balance equation for the grout dissolved in the pore water can be expressed as ( ) div ( ) div grad 0 g g nC C C t      v D , (7) where C is the grout concentration (grout mass per unit volume of fluid); v g is the grout velocity; D g is the hydrodynamic dispersion tensor (Chupin et al. 2009). The velocity v g is described by a modified Darcy law for Bingham fluid (Owayed and Tiab (2008)) 0, grad G p 

    

v



   

   

,

(8)

k

G

g

g

1

(

grad

), grad

p

p

G

g 









gr d a

p



m

where G is the minimum gradient pressure depending on a characteristic pore size of a porous medium and yield stress of fluid. The energy conservation equation is written as



n C C S  v

v

div

grad

( T S 

)

gra

d

eff C T

T

ph Q Q





 



,

(9)

eff

l

l

m

g

g

g

ch

t



where C eff is the effective volumetric heat capacity, λ eff is the effective thermal conductivity; C l , C g are the volumetric heat capacities of the water and the grout; S l is the water saturation, S g is the grout saturation ( S l + S g = S m ); Q ph is the heat source related to the latent heat of the water-ice phase change (Lay et al. (2014)); Q ch is the heat source generated by cement hydration (Wu et al. (2013)). The volumetric heat capacity C eff and the thermal conductivity λ eff are determined as (Mu & Ladanyi 1987). The equilibrium equation is written for the saturated porous media as div 0   σ γ , (10)