Freezing Soil
Introduction
When wet soil or clay is subjected to freezing temperatures, water in the interstices freezes. Becau water expands when it freezes, the surrounding soil deforms. The deformation changes the pressure in the interstices. The combined impacts of the freezing and the deformation affect the water flow.
This model predicts 3-way interactions between stress and strain, fluid flow, and temperature change. This type of analysis is important in asssments related to road and building construction, freeze-thaw weathering, fluid flow, and a number of
environmental applications. The model that follows comes from a COMSOL client who ud the results to asss thermo-mechanical impacts in a transportation study.
This analysis couples equations that predict what happens when a water-filled soil core freezes from the center outwards. Included in the analysis are:
•Effects of a stepwi change in the thermomechanical properties at the pha
transition temperature
•Porous fluid-flow behavior involving a temperature-driven contribution
•Stress-strain behavior including loads from thermal expansion and fluid flow
•Heat conduction including pha change and the latent heat of freezing
•Coupling effects between the above-mentioned phenomena.
This discussion assumes temperature changes are fully transient with a quasi-static interaction with fluid flow and solid deformation. The model us the Darcy’s Law and the Convection and Conduction application modes from the Earth Science Module.
It also employs a Stress-Strain application mode from the Structural Mechanics
Module, an application mode that automates temperature-deformation coupling.
Model Definition 1
Figure 3-18: Experimental tup for investigations of freezing soil. A symmetry obrvation permits modeling in axisymmetry 2D.
The model geometry is bad on a general test rig often ud for investigating the properties of soil and clay (e Figure 3-18). A cylindrical container with the sample soil has a concentric channel, through which a coolant fluid flows. The initial temperature of the wet soil is +3 °C. As the coolant at −15 °C enters the pipe, a
freezing front travels outward in the soil specimen. Becau the soil is homogeneous, you can take advantage of the geometric symmetry and model the phenomena in 2D.S T R E S S -S T R A I N E Q U A T I O N S
The fundamental Navier’s equation describes a force equilibrium
(3-6)
where σ is the stress tensor and F is a volume force.The entries of the stress tensor for axisymmetry are
1. Model definition and material data courtesy of Dr. J.P.B.N. Derks, Ministry of Transport, Public Work & Water Management, the Netherlands.3D to 2D axial symmetry Coolant flow
0.22 m
r o =0.05 m
r i =0.006 m
Soil or clay fill
∇–σ⋅F =
where τ denotes off-diagonal components of strain or shear.
The equations simplify to two force balances in the r and z directions:
(3-7) (3-8)
where τ denotes off-diagonal components of strain or shear.Flow-to-Structure Coupling
When considering the impacts of fluid flow on structural deformation, the stress tensor decompo into two parts
(3-9)
where σ is the total stress, σ' is the so-called grain stress, and p is the pressure of the fluid moving through a porous sand or clay matrix.
In this analysis
.
Temperature-to-Structure Coupling
The thermomechanical relationship is given by the generalized Hooke’s law for an elastic nonisothermal material as in
. (3-10)
Here D is the elasticity matrix, σ' reprents the elastic stress, ε gives the total strain, and εth is the thermal strain. Further, α (K −1) is the coefficient of thermal expansion, T is the temperature, and T ref is the strain reference temperature.
σσxx
σyy
σzz
τxy
τxz
τyz
=∂σr ∂r --------∂τr ∂r
-------σr σθ–r -----------------F r +++0=∂τrz ∂r ----------∂σz ∂r --------τrz r
------F z +++0=σσ'm p +=m 111000T =σ'D εεth –() ,=εth αT T ref –()=
F L U I D F L O W E Q U A T I O N S
Model the flow with the modified Darcy’s law
(3-11)
where u = (u , v , w ) denotes the vector of fluid velocities in the x , y , and z directions, and is the suction pressure.
Temperature-to-Flow Coupling, Segregation Potential
The fluid flow and temperature relationships couple through the term
,
where SP 0 is the gregation potential (kg·m/(s 2·K)), which is the ratio of the
moisture migration velocity to the temperature gradient in a freezing soil, and κ is the permeability (m 2). The gregation potential SP 0 is a positive constant below the freezing point and 0 above. Experimental obrvations on specimens frozen under a temperature gradient suggest that, even though much of the pore water is frozen, water transport still occurs in the frozen soil past the pore freezing front in respon to temperature-induced unfrozen water content gradients and suction gradients in the unfrozen water films. The migratory water freezes at the gregation freezing temperature, T s , which is lower than the pore freezing temperature T p (Ref. 1). In this model example, it is assumed that the gregation freezing temperature is well below the temperature range of the study.
Given the definition for , Equation 3-11 states that the fluid velocities depend on the pressure gradient and the temperature gradient for conditions below the freezing point.
Structure-to-Flow Coupling
For quasi-steady flow, the following relationship holds:
, (3-12)
where and similar terms are the rates of strain (s −1) from the stress-strain
equations.
Combining Equation 3-11 and Equation 3-12 gives the governing equation
, (3-13)
u κη---∇p φs +()–=φs φs SP 0T κ⁄⋅=φs ∇u ⋅ε·xx ε·yy ε·zz ++()–=ε·xx ∇k η---∇p φs +()–⎝⎠⎛⎞⋅ε·xx ε·yy ε·zz ++()–=
which this example models with the Darcy’s Law application mode.
T E M P E R A T U R E E Q U A T I O N S
This problem us the well-known heat equation to model the transfer of heat. As described in the Earth Science Module Ur’s Guide , the heat transfer equation reads .Results
Figure 3-19: A snapshot of the von Mis stress (surface plot) and displacements (contours) in a column of freezing soil.
ρc p ∂T ∂t
------∇k ∇T –()⋅+Q
=
Figure 3-19 shows the displacements in the solid sample and the von Mis stress after 12 hours of freezing operation. It is also easy to monitor how the physical properties of the sand change with time and space; e Figure 3-20.
Figure 3-20: Thermal conductivity changes in a step at the freezing point. The lower curve
corresponds to 24 minutes, and the upper curve to 7 hours and 12 minutes. Reference
1. Jean-Marie Konrad, “Estimation of the gregation potential of fine-grained soils using the frost heave respon of two reference soils,” Can. Geotech. J., vol. 42, pp. 38–50, 2005.
Modeling in COMSOL Multiphysics
Turning to the COMSOL Multiphysics Structural Mechanics Module, you choo the Axial Symmetry, Stress-Strain application mode to solve Equation 3-7 and Equation 3-8. To account for the fluid pressure according to Equation 3-9, simply add
