Factors Affecting Frost Heave of Chilled Gas Pipelines

Acknowledgments

Notation

a

soil constant;

C_a

apparent volumetric heat capacity of soil;

c

cohesion of unfrozen soil;

c_f

cohesion of frozen soil;

c_i

specific heat of ice;

c_s

specific heat of soil particle;

c_w

specific heat of water;

D

pipe diameter;

E

Young’s modulus of unfrozen soil;

E_f

Young’s modulus of frozen soil;

gradT

temperature gradient;

gradT_ff

temperature gradient in frozen fringe;

H

burial depth;

h

frost heave;

k

constant for thawing with temperature increase;

L

latent heat of water;

n

porosity;

P

percentage of total water content that remains unfrozen well below 0°C;

P_e

stress parameter for segregation potential;

p

effective mean stress;

Q

factor controls unfrozen water content with temperature;

q

heat flux;

q_x, q_y

heat flux in x- and y-directions;

R

parameter related to P;

r

radial distance from the current pipe center;

r₀

radial distance from the initial pipe center;

SP₀

segregation potential at zero applied pressure;

T

temperature;

T_f

in situ freezing temperature;

T_g

initial ground temperature;

T_p

pipe temperature;

T_s

ground surface temperature;

T_sf

segregation freezing temperature;

T_sm

mean annual ground surface temperature;

T_sv

seasonal ground surface temperature;

T₅₀

temperature at the highest rate of thawing;

t

time;

t_ff

thickness of frozen fringe;

v_m

migrated water influx;

W_u

fraction of total water that remains unfrozen;

w_u

unfrozen water content basis of the dry mass of soil;

w₀

soil water content basis of the dry mass of soil before freezing;

x, x₀

horizontal distance measured from pipe center;

Y₀

depth of frost front from initial pipe invert position;

y

vertical distance measured from current pipe center;

y₀

vertical distance measured from initial pipe center;

α

rate of increase of cohesion of frozen soil with ice content;

β

angle from the y-axis to a point with respect to the current pipe center;

β₀

angle from the y-axis to a point with respect to the initial pipe center;

φ

direction of heat flux with respect to the y-axis;

λ_e

equivalent thermal conductivity of soil;

λ_i

thermal conductivity of ice;

λ_s

thermal conductivity of soil particle;

λ_w

thermal conductivity of water;

θ_i

volumetric fraction of ice;

θ_is

volumetric fraction of segregate ice;

θ_{is_max}

maximum volumetric fraction of segregate ice lens prior to start of thawing;

θ_s

volumetric fraction of soil particle;

θ_w

volumetric fraction of unfrozen water;

ρ_d

dry density of soil;

ρ_i

density of ice;

ρ_s

density of soil particle;

ρ_w

density of water;

σ_n

effective normal stress;

σ_ov

effective overburden pressure;

ν

Poisson’s ratio of unfrozen soil;

ν_f

Poisson’s ratio of frozen soil;

ϕ

angle of internal friction of unfrozen soil;

ϕ_f

angle of internal friction of frozen soil;

ψ

dilation angle of unfrozen soil; and

ψ_f

dilation angle of frozen soil.

References