Ninth International Conference on Permafrost ... - IARC Research

A Soil Freeze-Thaw Model Through the Soil Water Characteristic CurveStefano EndrizziDepartment of Civil and Environmental Engineering, University of Trento, ItalyRiccardo RigonDepartment of Civil and Environmental Engineering, University of Trento, ItalyMatteo DallAmicoDepartment of Civil and Environmental Engineering, University of Trento, ItalyIntroductionThe mathematical description of the freeze-thaw behaviorof soil mainly depends on its texture. Usually coarsegrainedsoils follow a moving boundary with phase change,commonly referred to as the Stefan problem, whereas finegrainedsoils show the existence of a “frozen-fringe” (Fowler& Krantz 1994) of partially frozen soil between frozen andunfrozen regions. Therefore, the possibility of includingthe freezing characteristic of soil in a model (Hansson et al.2004) appears promising, as it allows the description of thefreezing/thawing cycles of natural soils.The goal of this work is to propose a freezing soil schemebased on soil freezing characteristic curves and to test itagainst the analytical solution of the Neumann problem.The Freeze-Thaw ModelThe freezing soil model can be explained by the followingequations: ∂TC T∂t + L f∂W∂t= ∂ ∂z λ T∂T∂z∂θρ w∂θw=−ρ ii= ∂W ∂t ∂t ∂tThe first term represents the energy budget equationwhere θ wand θ i(-) are the volumetric content of waterand ice, respectively; n is the porosity, ρ wand ρ i(Kg/m 3 )are the density of water and ice, respectively; W (Kg/m 3 )is the quantity of water in the control volume subject tophase change; L f(J/Kg) is the latent heat of fusion; T (°C)is the temperature; and z (m) is the soil depth coordinate.C T=C s(1-n)+c wθ wρ w+c iθ iρ i(J m -3 K -1 ) is the total thermalcapacity of the soil, where C sis the volumetric heat capacityof the soil, and c wand c i, the mass heat capacity of liquid(1-n) θw θiand ice, respectively. λ T=K e(λ sλ wλ i)+(1-K e)λ dry(W m -1K -1 ) represents the thermal conductivity of the soil matrixand follows the formulation of Farouki (1981), where λ s, λ w,and λ iare the thermal conductivities of soil, water, and ice,respectively; λ dryis the thermal conductivity of the dry soil(Johansen 1975); and K eis the Kersten number.The second equation in (1) is a closure relation and isusually known as the “no flow condition”; that is, the waterflux during phase change may be neglected (Fuchs et al.1978). The left-hand side (LHS) of the second equationrepresents the freezing/thawing rate and may be describedby proper equilibrium and closure relations as follows:(1) 1g ⋅ p Lw=f⋅ Tρ wg⋅ 273.15 =ψ (T) eqθ w−θ r( n −θ r ) = [ 1+ ( −α ⋅ψ eq) β] − ( 1− 1 β)The first equation represents the thermodynamic equilibriumduring phase change, when pressure head ψ (m) and T (°C)satisfy the Clapeyron equation (Christoffersen & Tulazczyk2003). The second equation represents the unsaturatedsoil condition by using the soil water characteristic curve(SWCC) according to the Van Genuchten (1980) model,where α (m -1 ) and β (-) are Van Genuchten parameters and θ ris the residual water content.Eventually, the allowed unfrozen water content can beplotted against the temperature, and becomes the soil freezingcharacteristic curve (SFCC) (see Fig. 1). Applying the chainrule to ∂θ w/∂t=∂θ w/∂ψ•∂ψ/∂T•∂Τ/∂t the first equation in (1)may be rewritten as:L 2 f C T+ ρ wC H ∂T g⋅ 273.15∂t = ∂ ∂z λ ∂T T ∂z where C H=∂θ w/∂ψ (m -1 ) is the hydraulic capacity of thesoil. The term in squared brackets is often referred to asthe apparent heat capacity (Williams & Smith 1989) andaccounts for both sensible and latent heat capacity of soils.The numerical schemeThe numerical model follows a finite differencediscretization with a Crank Nicholson scheme. At the firstiteration, the system is solved neglecting phase change inorder to calculate the temperature at the time step k+1 (T k+1 ):if T k+1 0, then the freezing soil module is activated.Given the unfrozen water content θ ueqat the equilibrium, fouralternatives may occur: (a) T k+1 θ ueq: cooling soiland excess water content; (b) T k+1 >T k and θ w>θ weq: warmingsoil and excess water content; (c) T k+1