Stability of Finite Difference Method for Frost Heaving Calculation in 2D Space

摘要

Designing of gas-pipe and condensate lines and lines in conditions of water-inundated soil, high water-level, boggy ground with no runoff, seasonal freezing ground or negative temperature of the pumped product has a high risk of frost heaving of soils. Frosty heaving is the process of increasing of soil volume during the frigidness. Increasing of the volume is partially connected with an increasing of the volume of pore water during the frigidness. But mostly the volume increase due to migration of liquid water from the underlying layers of soil into the freezing zone. Water freezes and increases in volume after crossing the freezing front. The pressure capable of balancing the movement of liquid water is more than 1.2 MPa/°C, that significantly exceeds the base load-bearing capacity. Therefore, the migration of water and frost heaving does not stop at zone with negative temperature and source of liquid water. Pipelines is deforming by expanding soils in short sections, that leads to an emergency depressurization of the internal space. Authors developed a mathematical model of frost heaving to design engineering protection. This model consist of system of nonlinear parabolic differential equations. Authors encountered the problem of instability of finite differences method on an implicit difference scheme, even with a convergent iterative process. Using the spectral condition of stability, authors derived expressions for correlation between the time step and other parameters of the problem (step in space, capacitive and conducting properties of the medium) for the 2D case. Expressions show that the implicit scheme can be unstable under certain combinations of problem parameters. The greatest influence on the stability has the term responsible to the volume source of heat or matter.