Large Strain Consolidation of Unsaturated Soil: Model Formulation and Numerical Analysis

Data Availability Statement

Acknowledgments

Notation

A₁, A₂, A₃

coefficients of consolidation for airflow;

a_f

water retention curve fitting parameter;

a_sh

minimum void ratio;

b_sh

slope of line of tangency;

C_a1

matric suction adjustment factor;

C_a2

matric suction adjustment factor;

C_s

swelling index;

C^aw

compressibility of water–air mixture, kPa⁻¹;

C^w

compressibility of water, kPa⁻¹;

c_sh

curvature of shrinkage curve;

c_v

coefficient of consolidation, m²/s;

dM^a

air mass per time, kg/s;

e

soil void ratio;

e₀

initial soil void ratio;

e^a

air void ratio;

$e_0^a$

initial air void ratio;

e^w

water void ratio;

$e_0^w$

initial water void ratio;

g

gravitational acceleration, m/s²;

H

height of soil layer, m;

h_r

residual matric suction;

i

ith node in horizontal direction;

$i_x^w$

hydraulic gradient in horizontal direction, m/m;

$i_{\xi }^w$

hydraulic gradient in vertical direction, m/m;

j

jth node in vertical direction;

k

coefficient of permeability;

$k_d^a$

permeability of air in dry condition, m/s;

$k_{d,x}^a$

horizontal permeability of air in dry condition, m/s;

$k_{d,\xi }^a$

initial permeability of air in dry condition, m/s;

$k_x^a$

horizontal permeability of air, m/s;

$k_{\xi }^a$

vertical permeability of air, m/s;

$k_s^w$

permeability of water in the fully saturated condition, m/s;

$k_{s,x}^w$

horizontal permeability of water in the fully saturated condition, m/s;

$k_{s,\xi }^w$

vertical permeability of water in the fully saturated condition, m/s;

$k_x^w$

horizontal permeability of water, m/s;

$k_{\xi }^w$

vertical permeability of water, m/s;

L

spacing of vertical drains, m;

M

mesh node count in vertical direction;

m_f

water retention curve fitting parameter;

m₁

coefficient of soil volume change with respect to net stress, kPa⁻¹;

m₂

coefficient of soil volume change with respect to matric suction, kPa⁻¹;

m^a

air mass flux through the element, kg/m²/s;

$m_x^a$

air mass flux in horizontal direction, kg/m²/s;

${\dot{m}}_x^a$

air mass flow rate in horizontal direction, kg/s;

$m_{\xi }^a$

air mass flux in vertical direction, kg/m²/s;

${\dot{m}}_{\xi }^a$

air mass flow rate in vertical direction, kg/s;

$m_1^a$

coefficient of air volume change with respect to net stress, kPa⁻¹;

$m_2^a$

coefficient of air volume change with respect to matric suction, kPa⁻¹;

$m_1^w$

coefficient of water volume change with respect to net stress, kPa⁻¹;

$m_2^w$

coefficient of water volume change with respect to matric suction, kPa⁻¹;

N

mesh node count in horizontal direction;

n

porosity;

n_f

water retention curve fitting parameter;

p

surcharge load, kPa;

$q_x^w$

flow rate of water in horizontal direction, m²/s per m run;

$q_{\xi }^w$

flow rate of water in vertical direction, m²/s per m run;

R

universal gas constant, J/mol/K;

S

degree of saturation;

t

elapsed time, s;

Δt

time-step increment, s;

U_avg

average degree of consolidation;

u^a

excess pore air pressure, kPa;

$u_{\mathrm{atm}}^a$

standard atmosphere, kPa;

$u_{\mathrm{pore}}^a$

absolute pore air pressure, kPa;

$u_t^a(i,j)$

excess pore air pressure of node (i, j) at time t, kPa;

$u_0^a$

initial excess pore air pressure, kPa;

u^w

excess pore water pressure, kPa;

$u_{\mathrm{hydro}}^w$

hydrostatic pore water pressure, kPa;

$u_{\mathrm{pore}}^w$

pore water pressure, kPa;

$u_t^w(i,j)$

excess pore water pressure of node (i, j) at time t, kPa;

$u_0^w$

initial pore water pressure, kPa;

${\overline{u}}_t^a$

average pore air pressure at time t, kPa;

${\overline{u}}_t^w$

average pore water pressure at time t, kPa;

V₀

initial volume of soil, m³;

V^a

volume of pore air, m³;

V^s

volume of solids, m³;

V^v

volume of void, m³;

V^w

volume of pore water, m³;

$v_x^{a,\mathrm{act}}$

actual velocity of airflow in horizontal direction, m/s;

$v_{\xi }^{a,\mathrm{act}}$

actual velocity of airflow in vertical direction, m/s;

$v_{\xi }^{s,a}$

velocity of soil particles in vertical direction due to airflow, m/s;

$v_{\xi }^{s,w}$

velocity of soil particles in vertical direction due to water flow, m/s;

$v_x^w$

apparent velocity of water flow in horizontal direction, m/s;

$v_{\xi }^w$

apparent velocity of water flow in vertical direction, m/s;

$v_x^{w,\mathrm{act}}$

actual velocity of water in horizontal direction, m/s;

$v_{\xi }^{w,\mathrm{act}}$

actual velocity of water in vertical direction, m/s;

W₁, W₂, W₃

coefficients of consolidation for water flow;

w_s

saturated gravimetric water content;

x

Abscissa in Lagrangian coordinate;

Δx

mesh size in horizontal direction, m;

y

ordinate in Lagrangian coordinate;

Δy

mesh size in vertical direction, m;

z

time-step number;

Δ

differential change;

α

parameter for time-step increment;

${\epsilon }_f$

final volumetric strain of the layer;

${\epsilon }_t$

volumetric strain of the layer at time t;

${\xi }_{t+1}^a$

elevation of element (x_t, y_t) due to airflow;

${\xi }_{t+1}^w$

elevation of element (x_t, y_t) due to water flow;

θ

temperature in Kelvin scale, K;

λ

parameter for permeability function;

ρ^a

density of air, kg/m³;

ρ

density of element, kg/m³;

σ_x

horizontal total stress, kPa;

σ_y

vertical total stress, kPa;

γ^w

unit weight of water, kN/m³;

ψ

matric suction; and

ω

molecular mass of air, kg/mol.

References