SUMMARYThis paper presents the development of a computationally efficient finite element tool for the analysis of 3D steady state heat flow in geothermal heating systems. Emphasis is placed on the development of finite elements for vertical borehole heat exchangers and the surrounding soil layers. Three factors have contributed to the computational efficiency: the proposed mathematical model for the heat exchanger, the discretization of the spatial domain using the Petrov-Galerkin method and the sequential numerical algorithm for solving the resulting system of non-linear equations. These have contributed in reducing significantly the required number of finite elements necessary for describing the involved systems. Details of the mathematical derivations and some numerical examples are presented.

SUMMARYThis paper presents an extension to the work presented in Part I of this series of two articles to the transient case. Emphasis is placed on the development of a new model for heat flow in a double U-shape vertical borehole heat exchanger and its thermodynamic interactions with surrounding soil mass. The discretization of the spatial-temporal domain of the heat pipe model is done by the use of the space-time finite element technique in conjunction with the Petrov-Galerkin method and the finite difference method. The paper shows that the proposed model and the choice of the discretization technique, in addition to the utilization of a sequential numerical algorithm for solving the resulting system of non-linear equations, have contributed in reducing significantly the required number of finite elements necessary for describing geothermal heating systems. Details of the mathematical derivations and comparison to experimental data are presented.

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Presented and discussed in this paper is an exact analytical solution of the nonhomogeneous partial differential equation governing the conventional one-dimensional consolidation under haversine repeated loading. The derived analytical solution to the 1D consolidation equation is compared with the numerical solution of the same consolidation problem via FEM. The series solution takes into account the frequency of repeated loading through a dimensionless time factor T 0 . The paper reveals that an increase in the frequency of imposed repeated haversine loading (a decrease in period of repeated loading) causes an increase in the number of cycles required to achieve the steady state, whereas the effect of frequency on the maximum excess pore water pressure at the bottom of a clay layer with permeable top and impermeable bottom for the range of frequencies studied is generally insignificant. The effective stress at the bottom of the clay deposit with permeable top and impermeable bottom increases with time but with some fluctuations without changing the sign. These fluctuations become more pronounced for increasing values of T 0 . An increase in T 0 also causes an increase in maximum effective stress.

A nonlinear theory of consolidation is presented which takes into account secondary compression. The theory is incorporated into a weak form of equilibrium that is suitable for a finite element procedure. The model is used to interpret Crawford's experimental data on Leda clay. Limitations of the model are discussed, and a few thoughts on the effects of temperature on the evaluation of model parameters are briefly presented.Résumé : On présente ici une théorie non linéaire de la consolidation qui prend en compte la compression secondaire. La théorie est introduite dans une forme faible d'équilibre compatible avec un traitement par éléments finis. Le modèle est utilisé pour interpréter les résultats expérimentaux de Crawford sur l'argile Leda. Les limites d'application du modèle sont discutées et on présente brièvement quelques idées sur les effets de la température sur l'évaluation des paramètres du modèle.