Table 6. Summary of the variables and...

Table 6.

Summary of the variables and symbols featured in the equations

Equation	Parameter	Unit	Symbol	Eq. nos
Water retention curve	Degree of liquid and gas saturation	—	$S_{l}$ ⁠, $S_{g}$	1, 11
Water retention curve	Gas pressure	MPa	$P_{g}$	1, 20
	Liquid pressure	MPa	$P_{l}$	1, 20
	Shape parameter	—	$ψ$	1, 3
	Porosity-dependent shape parameter	—	$ψ (ϕ)$	3
	Air entry pressure	MPa	$P_{0}$	1
	Porosity-dependent air entry pressure	MPa	$P_{0} (ϕ)$	2
	Strain-dependent air entry pressure	MPa	$P_{0} (ε)$	4
	Porosity	—	$ϕ$	2, 3, 5, 13
	Parameter in $P_{0} (ϕ)$	—	C	2
	Parameter in $λ (ϕ)$	—	D	3
Permeability	Intrinsic permeability of gas	m²	$k_{int}$	4, 8
	Porosity-dependent permeability	m²	$k_{Kozeny}$	5, 8
	Relative permeability in phase α (l: liquid, g: gas)	—	$k_{rl}$ ⁠, $k_{rg}$ ⁠, $k_{r α}$	6, 11, 16
	Relative permeability constant	—	A	6, 11
	Relative permeability power	—	n	6, 11
	Effective permeability of gas and liquid phase	m²	$k_{effl}, k_{effg}$	7, 12
Cubic law	Strain-dependent gas permeability	m²	$k_{Cubic}$	8, 9
	Initial aperture	m	$b_{0}$	9
	Spacing of the fractures	m	$a$	9
	Reference strain to calculate aperture variations	—	$ε_{0}$	10
	Maximum aperture	m	$b_{\max}$	10
Fick’s law	Non-advective mass flux of species $i$ in phase α (air or water vapour)	kgs⁻¹ m⁻²	$i_{α}^{i}$	13
	Phase density	kgm⁻³	$ρ_{α}$	13, 14, 16
	Tortuosity coefficient	—	$τ$	13
	Diffusion coefficient of the phase α	m²s⁻¹	$D_{α}^{i}$	13
	Mass fraction of species $i$ in phase α	—	$ω_{α}^{i}$	13, 14
	Longitudinal dispersivity	m	$d_{l}$	14
	Transversal dispersivity	m	$d_{t}$	14
	Hydrodynamic dispersion coefficient	m²s⁻¹	$D_{α}^{'}$	14
Darcy	Advective flux in phase α	ms⁻¹	$q_{α}$	16
	Intrinsic permeability of the phase α	m²	$k$	16
	Dynamic viscosity	Pa s	$μ_{α}$	16
	Gravitational constant	ms⁻²	$g$	16
Mechanical model	Elastic and plastic volumetric deformation	—	$ε_{v}^{e}$ ⁠, $ε_{v}^{p}$	17, 18
	Volumetric stiffness parameter under changes of mean stress	—	$κ_{i}$	18, 19, 23
	Volumetric stiffness parameter under changes of suction	—	$κ_{s}$	18
	Suction	MPa	$s$	18, 22
	Void ratio	—	$e$	18, 19
	Preconsolidation stress for saturated conditions	MPa	$p_{0}^{*}$	19
	Preconsolidation stress	—	$p_{0}$	23
	Mean effective stress	MPa	$p^{'}$	18, 20
	Mean net stress	MPa	$p$	20
	Slope of void ratio – mean stress curve at zero suction	—	$λ (0)$	19, 23
	Effective stress	MPa	$σ^{'}$	20, 24
	Deviatoric stress	MPa	$q$	21
	Non-associativity parameter	—	$α$	21
	Tensile strength in saturated conditions	MPa	$p_{s}$	21
	Critical state line parameter	—	$M$	21
	Plastic potential	—	$G$	21
	Parameter defining the maximum soil stiffness	—	$r$	22
	Parameter controlling the rate of increase of the soil stiffness with suction	MPa⁻¹	$β$	22
	Reference stress	MPa	$p^{c}$	23

Equation	Parameter	Unit	Symbol	Eq. nos
Water retention curve	Degree of liquid and gas saturation	—	$S_{l}$ , $S_{g}$	1, 11
Water retention curve	Gas pressure	MPa	$P_{g}$	1, 20
	Liquid pressure	MPa	$P_{l}$	1, 20
	Shape parameter	—	$ψ$	1, 3
	Porosity-dependent shape parameter	—	$ψ (ϕ)$	3
	Air entry pressure	MPa	$P_{0}$	1
	Porosity-dependent air entry pressure	MPa	$P_{0} (ϕ)$	2
	Strain-dependent air entry pressure	MPa	$P_{0} (ε)$	4
	Porosity	—	$ϕ$	2, 3, 5, 13
	Parameter in $P_{0} (ϕ)$	—	C	2
	Parameter in $λ (ϕ)$	—	D	3
Permeability	Intrinsic permeability of gas	m²	$k_{int}$	4, 8
	Porosity-dependent permeability	m²	$k_{Kozeny}$	5, 8
	Relative permeability in phase α (l: liquid, g: gas)	—	$k_{rl}$ , $k_{rg}$ , $k_{r α}$	6, 11, 16
	Relative permeability constant	—	A	6, 11
	Relative permeability power	—	n	6, 11
	Effective permeability of gas and liquid phase	m²	$k_{effl}, k_{effg}$	7, 12
Cubic law	Strain-dependent gas permeability	m²	$k_{Cubic}$	8, 9
	Initial aperture	m	$b_{0}$	9
	Spacing of the fractures	m	$a$	9
	Reference strain to calculate aperture variations	—	$ε_{0}$	10
	Maximum aperture	m	$b_{\max}$	10
Fick’s law	Non-advective mass flux of species $i$ in phase α (air or water vapour)	kgs⁻¹ m⁻²	$i_{α}^{i}$	13
	Phase density	kgm⁻³	$ρ_{α}$	13, 14, 16
	Tortuosity coefficient	—	$τ$	13
	Diffusion coefficient of the phase α	m²s⁻¹	$D_{α}^{i}$	13
	Mass fraction of species $i$ in phase α	—	$ω_{α}^{i}$	13, 14
	Longitudinal dispersivity	m	$d_{l}$	14
	Transversal dispersivity	m	$d_{t}$	14
	Hydrodynamic dispersion coefficient	m²s⁻¹	$D_{α}^{'}$	14
Darcy	Advective flux in phase α	ms⁻¹	$q_{α}$	16
	Intrinsic permeability of the phase α	m²	$k$	16
	Dynamic viscosity	Pa s	$μ_{α}$	16
	Gravitational constant	ms⁻²	$g$	16
Mechanical model	Elastic and plastic volumetric deformation	—	$ε_{v}^{e}$ , $ε_{v}^{p}$	17, 18
	Volumetric stiffness parameter under changes of mean stress	—	$κ_{i}$	18, 19, 23
	Volumetric stiffness parameter under changes of suction	—	$κ_{s}$	18
	Suction	MPa	$s$	18, 22
	Void ratio	—	$e$	18, 19
	Preconsolidation stress for saturated conditions	MPa	$p_{0}^{*}$	19
	Preconsolidation stress	—	$p_{0}$	23
	Mean effective stress	MPa	$p^{'}$	18, 20
	Mean net stress	MPa	$p$	20
	Slope of void ratio – mean stress curve at zero suction	—	$λ (0)$	19, 23
	Effective stress	MPa	$σ^{'}$	20, 24
	Deviatoric stress	MPa	$q$	21
	Non-associativity parameter	—	$α$	21
	Tensile strength in saturated conditions	MPa	$p_{s}$	21
	Critical state line parameter	—	$M$	21
	Plastic potential	—	$G$	21
	Parameter defining the maximum soil stiffness	—	$r$	22
	Parameter controlling the rate of increase of the soil stiffness with suction	MPa⁻¹	$β$	22
	Reference stress	MPa	$p^{c}$	23

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