Coupled Two-Phase Flow and Elastodamage Modeling of Laboratory and In Situ Gas Injection Experiments in Saturated Claystone as a Potential Host Rock for Nuclear Waste Repository

Acknowledgments

Notation

A_k

empirical parameter;

a

fracture spacing;

a_ijkl

stiffness tensor of intact material;

b

fracture aperture;

$C_{ijkl}^{*}$

homogenized stiffness tensor;

C_s

specific storage coefficient;

c

drained compressibility of FPM;

c_p

drained compressibility of matrix;

c_s

drained compressibility of solid grains;

${\mathcal{D}}_d$

energy dissipation for damage evolution;

${\mathcal{D}}_f$

energy dissipation for crack propagation;

d

damage variable;

E

Young’s modulus;

e_xij

macroscopic strain tensor;

e_yij

microscopic strain tensor;

G_s

thermodynamic potential;

${\mathcal{G}}^{\epsilon }$

energy release rate at crack tip;

$\hat{H}$

Heaviside step function;

$\mathcal{H}$

history maximum value of $Y_d^{+}$;

$\mathbf{I}$

identity tensor;

K₀

reference stress intensity factor;

K_I

model-I stress intensity factor;

${\mathbf{k}}_f$

intrinsic permeability tensor of fractures;

${\mathbf{k}}_{in}$

intrinsic permeability tensor;

${\mathbf{k}}_m$

intrinsic permeability tensor of matrix;

k_rπ

relative permeability of fluid $\pi (\pi =g,w)$;

L_c

size of macrostructure;

L(d)

correction function of permeability;

l

microcrack length;

l_c

size of locally periodic microstructure;

l_q

characteristic length;

M

molar mass of gas;

m

model parameter;

$\mathbf{N}$

unit vector normal to crack face;

n

van Genuchten model parameter;

$\overrightarrow{n}$

unit vector normal to fracture plane;

n_d

subcritical growth index;

p_c

capillary pressure;

p_cr

critical pressure;

p_g

gas pressure;

p_gev

gas entry value;

p_ig

gas injection pressure;

p_w

water pressure;

$p_f^{\epsilon }$

fluid pressure field of heterogeneous porous medium;

$p_f^{(0)}$

Y-periodic function of fluid pressure field at order 0;

$p_g^{\epsilon }$

gas pressure field of heterogeneous porous medium;

$p_w^{\epsilon }$

water pressure field of heterogeneous porous medium;

p₀

initial air entry value;

p₁

pore fluid pressure;

p₂

crack fluid pressure;

p₃

confining pressure;

${\mathbf{q}}_{g0}$

mass flux at gas outlet boundary;

q_ig

applied gas injection rate;

R

universal gas constant;

S_e

effective saturation degree;

$S_{\pi }$

saturation degree for fluid $\pi (\pi =g,w)$;

T

absolute temperature;

${\mathbf{u}}^{\epsilon }$

displacement field of heterogeneous body in vector format;

$u_i^{\epsilon }$

displacement field of heterogeneous body;

${\mathbf{u}}^{(r)}$

Y-periodic function of displacement field at order r_;

V_ig

gas injector volume;

v

crack velocity;

${\mathbf{v}}_{\pi }^D$

Darcy’s velocity for fluid $\pi (\pi =g,w)$;

v₀

reference crack velocity;

$\mathbf{x}$

macroscopic coordinate in vector format;

x_i

macroscopic coordinate;

Y_d

damage energy release rate;

$Y_d^{+}$

positive part of damage energy release rate;

$\mathbf{y}$

microscopic coordinate in vector format;

y_i

microscopic coordinate;

α₁

Biot coefficient for porous matrix;

α₂

Biot coefficient for fractures;

β_t

empirical parameter;

δ_ij

Kronecker delta;

ɛ

scale parameter;

ɛ_V

total volumetric strain;

${\mu }_{\pi }$

dynamic viscosity of fluid $\pi (\pi =g,w)$;

ν

Poisson’s ratio;

${\mathbf{\xi }}^{ij}$

characteristic function of elementary deformation mode in vector format;

ρ_g

gas density;

ρ_w

water density;

${\mathrm{\Sigma }}_{ij}^{(0)}$

macroscopic total stress tensor;

$\overline{\sigma }$

mean stress;

σ_cr

critical tensile stress;

σ_i

principal stress (i = 1, 2, 3);

${\sigma }_{ij}^{(r)}$

Y-periodic function of total stress at order r (r = −1, 0, 1 …);

${\mathbf{\sigma }}^{(r)}$

Y-periodic function of total stress at order r (r = −1, 0, 1 …) in matrix format;

${\mathbf{\sigma }}^{\epsilon }$

total stress of heterogeneous body in matrix format;

${\sigma }_{ij}^{\epsilon }$

total stress of heterogeneous body;

${\sigma }_{pi}^{\mathrm{\prime }}$

principal effective stress (i = 1, 2, 3); and

ϕ

porosity.

References