Predicting Thaw Degradation in Algid Climates along Highway Embankments using a Boundary Element Method
A.N. Johnson, T.V. Hromadka II, M. Phillips, and J. Williams
Introduction
The Complex Variable Boundary Element Method or CVBEM is used to develop a computer model for estimat- ing the location of the freezing front in soil-water phase change problems. The CVBEM based computer program is based on the following major assump- tions:
(i) the problem is two-dimensional,
(ii) the entire soil system is homoge- neous and isotropic,
(iii) the problem thermal boundary con- ditions are constant values of tem- perature (or stream function),
(iv) soil-water flow effects are neglected (the problem is strictly geothermal),
(v) all heat flow from the freezing front is within the control volume; there is no heat flux associated with the freezing front from exterior of the control volume,
(vi) the freezing front movement is suf- ficiently slow such that heat flux along the moving boundary can be determined by assuming steady state heat flow conditions.
The CVBEM is used to model the thermal regime of the soil system. The theory and development of the CVBEM is given in Hromadka (1984). Because the numerical technique is a boundary integral approach, the control volume thermal regime is a boundary integral approach, the control volume thermal regime is modeled with respect to the boundary values and, therefore, the com- puter program data entry requirements are significantly less than that usually required of domain methods such as finite-differences or finite-elements.
Soil-water phase change along the freezing front is modeled as a simple balance between computed heat flux and the evolution of soil-water volumetric latent heat of fusion. Because of the dominating effects of phase change of
water in the heat balance equations, the governing heat transport equations are simplified to assume steady state heat transport conditions above and below the freezing front. The CVBEM is applied independently in the thawed and in the frozen regions of the problem domain, with heat flux along the freez- ing front calculated by the CVBEM as the stream function that is conjugate to the potential function (temperature). Freezing and thawing is modeled as the net balance between the freezing and thawing heat flux over small modeling time steps. To model the displacement of the freezing front, the computer model provides two options:
(i) Displace the freezing front coordi- nates with respect to changes along the y-coordinates only, or
(ii) Displace the freezing front coordi- nates with respect to a vector normal to the freezing front boundary.
Objectives of Documentation Report
The ongoing work for this project is to be completed during the summer 2016 session.
The objectives of the Documentation report are:
(1) Provide background information regarding the CVBEM and the soil- water phase change model used in program CVBFR1;
(2) Provide documentation for the data entry sequence associated with pro- gram CVBFR1; and
(3) Provide FORTRAN computer code as an Appendix.
Computational Mathematical Model Description
The problem domain under study is assumed to have a well-defined freezing
front where soil water arriving at the freezing front location due to soil water saturated and unsaturated transport is subject to isothermal phase change. Further phase change effects beyond the phase change of arriving soil water at the front results in movement of the freez- ing front. Such displacement of the front redefines the problem of heat transport domains in the thawed and frozen soil regions, respectively (Figure 1).
Figure 2 depicts the relative vicinity of the frozen soil located above the freezing front and the thawed soils located below the front. In the model, an arbitrary temperature of 0.1 degrees centigrade is assumed applicable for the thawed soils that are subject to phase change due to periodic freezing and thawing cycles. The CVBEM is the boundary element technique used to develop heat transport quantities along the freezing front, where phase change effects result in slow rates of displacement of the freezing front. Under such conditions, the Laplace equation is modeled in each of the thawed and frozen soil regions, respectively.
Figure 3 shows a typical CVBEM mod- eling node layout. As soil water freezes or thaws, the location of the freezing front changes, with displacement occurring orthogonal to the freezing front contour (Figure 4). Figures 5 and 6 show the CVBEM model boundary value problem specification used for each of the thawed and frozen soil regions, respectively. The relationship between freezing front displacement and net heat flux at the front is depicted in Figure 7.
Figure 8 shows details of the freezing front displacement geometric algorithm used. To validate the CVBEM model, comparison with a two-dimensional finite element method model of a road- way embankment in algid soils is made, with the FEM model discretized as shown in Figure 9.
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