Transport coefficients and pressure conditions for growth of ice lens in frozen soil

Abstract

In this paper, the transport of sub-cooled water across a partially frozen soil matrix (frozen fringe) caused by a temperature difference over the fringe, is described using non-equilibrium thermodynamics. A set of coupled transport equations of heat and mass is presented; implying that, in the frozen fringe, both driving forces of pressure and temperature gradients simultaneously contribute to transport of water and heat. The temperature-gradient-induced water flow is the main source of frost heave phenomenon that feeds the growing ice lens. It is shown that three measurable transport coefficients are adequate to model the process; permeability (also called hydraulic conductivity), thermal conductivity and a cross coupling coefficient that may be named thermodynamic frost heave coefficient. Thus, no ad hoc parameterizations are required. The definition and experimental determination of the transport coefficients are extensively discussed in the paper. The maximum pressure that is needed to stop the growth of an ice lens, called the maximum frost heave pressure, is predicted by the proposed model. Numerical results for corresponding temperature and pressure profiles are computed using available data sets from the literature. Frost heave rates are also computed and compared with the experimental results, and reasonable agreement is achieved.

Freezing‐induced stiffness and strength anisotropy in freezing clayey soil: Theory, numerical modeling, and experimental validation

“…55 Second, in the balance principle of mass, we consider the pore-fluid mass flux to be related to the gradient of temperature under the non-isothermal condition, which represents a phenomenon analogous to the Ludwig-Soret effect. 56,57 Consequently, the balance of linear momentum, mass, and energy constitute the governing equations for the strong form, with the displacement field, pore liquid pressure, and temperature chosen to be the nodal variables interpolated. Physical quantities, such as the saturation of ice and liquid, ice pressure, Darcy flux, and stress are evaluated at the Gauss point through the corresponding constitutive laws discussed in Section 2.…”

“…where d 𝛼 (⋅)∕d𝑡, 𝛼 = S, L, C are the material time derivatives of the variable (⋅) following the motion of phase 𝛼, 𝒗 𝛼 is the velocity of phase 𝛼, ṁL→C is the rate of liquid water mass crystallizing into ice crystal, the term 𝑠 T ∇𝑇 in Equation ( 22b) is the flux induced by the gradient of temperature, [56][57][58] and 𝑠 T is the Soret coefficient. In the freezing soil, the Soret coefficient corresponds to the liquid flux that flows from higher to lower temperature.…”

“…First, heat generated from the plastic dissipation is incorporated in the balance of energy and is controlled by the Taylor–Quinney coefficient 55 . Second, in the balance principle of mass, we consider the pore‐fluid mass flux to be related to the gradient of temperature under the non‐isothermal condition, which represents a phenomenon analogous to the Ludwig–Soret effect 56,57 . Consequently, the balance of linear momentum, mass, and energy constitute the governing equations for the strong form, with the displacement field, pore liquid pressure, and temperature chosen to be the nodal variables interpolated.…”

Formation and growth of multiple, distinct ice lenses in frost heave

“…with 𝜌 ′ 𝑤𝑇 = 𝜕𝜌 𝑤 ∕𝜕𝑇 and 𝜌 ′ 𝑖𝑇 = 𝜕𝜌 𝑖 ∕𝜕𝑇. According to Kjelstrup et al, 18 the flux of water in a partially frozen soil is affected not only by pressure gradient, but also by temperature gradient, and can be described using the following coupled equation:…”

“…The efforts made to model the frost heave may be classified into three different approaches: semi-empirical models, [8][9][10] phenomenological models, [11][12][13][14] and theoretical models. [15][16][17][18][19] F I G U R E 1 A partially frozen soil sample exposed to a temperature gradient: an ice lens is permitted to grow, imposing a material discontinuity in the system Among the semi-empirical approaches, the segregation potential of Konrad 5,10,20 is widely known by geotechnical engineers. Despite the beauty and simplicity of their model, and the theoretical support provided, the nature of a semiempirical approach would normally limit its application.…”