A Mathematical Model of Thaw-Freeze Cycles Beneath Drilling Rigs and Production Platforms in Cold Regions
Authors: Kazemi Hossein , Perkins T.K. 
Published: December 2007

The earth in cold regions, such as the North Slope of Alaska, undergoes periodic thaw and freeze cycles. In such a process, the depth of a frozen-unfrozen interface is a function of time and is controlled by a combined mechanism of heat transfer by conduction, convection, and radiation. Discussed here is a mathematical model to simulate this mechanism.

Introduction
The surface layer of the earth's crust on the North Slope of Alaska undergoes periodic thaw-freeze cycles. This layer, consisting of a mixture of water and solid material, varies in composition, depending upon depth and location. As a result of the recent oil exploration and exploitation activities in this area, the technology of production engineering encounters new situations. For instance, during the thaw period the exposed surface of the earth is softened. To guard against this softening under the drilling rigs and the production platforms, engineers have suggested banking up a few platforms, engineers have suggested banking up a few feet of low-water-content gravel on the locations of interest. An embankment of the proper height can prevent or minimize thawing in the original earth prevent or minimize thawing in the original earth layer and offer a rigid support for the drilling and production facilities. production facilities. In a thaw-freeze cycle, the seasonal temperature rise causes initiation of melting at the exposed surface of the earth. The melting front moves downward until inadequate seasonal heat supply causes refreezing of the earth at the exposed surface. This frozen front also moves downward into the earth. At about this time, two fronts are in motion; one is freezing downward and the other (which used to be a melting front) becomes a freezing front and moves upward. Eventually the two meet, and the entire surface layer is completely frozen again. The heat transfer mechanism is a combination of heat conduction, convection, and radiation. For a better understanding of the details of the process, refer to the mathematical formulation of the problem that appears later. The most notable part of the mathematical problem is the classical Stefan's moving-interface problem is the classical Stefan's moving-interface problem of conduction heat transfer with phase change. problem of conduction heat transfer with phase change. The nonlinearity of the problem and the nature of the initial and boundary conditions dictated the use of a numerical implicit finite-difference method to obtain a solution. The method is similar to that of Douglas and Gallie, who have studied the stability of the numerical technique with respect to small errors. In a more recent paper, Cannon et al. have proven a theorem on the global existence and uniqueness of the solution to a multiboundary Stefan problem; this proof should add strength to our mathematical results. proof should add strength to our mathematical results. From a practical standpoint, we hope that the basic approach of this paper can be used to evaluate problems associated with foundation design of production and drilling facilities on the North Slope.

Mathematical Formulation
Two sample cases are considered:
a thaw-freeze cycle in a combined layer of man-made gravel embankment overlying a surface layer of earth crust; and the same situation with the addition of a production platform. production platform.
JPT
P. 381