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The purpose of this paper is to introduce a spectral model capable of simulating fully transient conductive‐convective heat transfer processes in an axially‐symmetric shallow geothermal system consisting of a borehole heat exchanger embedded in a soil mass.

The proposed model combines the exactness of the analytical methods with important extent of generality in describing the geometry and boundary conditions of the numerical methods. It calculates the temperature distribution in all involved borehole heat exchanger components and the surrounding soil mass using the discrete Fourier transform, for the time domain, and the Fourier‐Bessel series, for the spatial domain.

The paper calculates the temperature distribution in all involved borehole heat exchanger components and the surrounding soil mass in a robust and computationally very efficient procedures. Analysis which might take long time in a work station, if use is made of standard numerical procedures, takes only 1 second in an Intel PC with the proposed model.

The model is capable of simulating fully transient heat transfer in a shallow geothermal system subjected to short and long‐term time varying boundary conditions. The CPU time for calculating temperature distributions in all involved components; pipe‐in, pipe‐out, grout, and soil, using 2048 FFT samples, for the time domain, and 100 Fourier‐Bessel series samples, for the spatial domain, was in the order of 1 second in an Intel PC. The accuracy and computational efficiency of the model makes it, if elaborated, vital for engineering practice.

The proposed model is original and generic. The idea behind it is new and has not been utilized in this field of application. The model can be extended easily to include other types of borehole heat exchangers embedded in multi‐layer systems.

This paper aims to present a framework for deriving analytical and semi‐numerical models for coupled conductive‐convective heat transfer processes in a borehole heat exchanger subjected to general initial and boundary conditions.

The discrete Fourier transform and the spectral element method have been utilized for deriving two spectral models for a single U‐tube borehole heat exchanger in contact with a soil mass.

Verification and numerical examples have shown that the developed models are accurate and computationally very efficient. It is illustrated that one spectral element is capable of producing results which are more accurate than those produced by 200 finite elements.

The gained computational efficiency and accuracy will boost considerably the possibilities for more insight into geothermal analysis, which will improve the procedure for designing competitive energy extraction systems.

The models are capable of calculating exactly the temperature distribution in all involved borehole heat exchanger components and their thermal interactions.

This paper aims to present a computationally efficient finite element model for the simulation of isothermal immiscible two-phase flow in a rigid porous media with a particular application to CO₂ sequestration in underground formations. Focus is placed on developing a numerical procedure, which is effectively mesh-independent and suitable to problems at regional scales.

The averaging theory is utilized to describe the governing equations of the involved unsaturated multiphase flow. The level-set (LS) method and the extended finite element method (XFEM) are utilized to simulate flow of the CO₂ plume. The LS is employed to trace the plume front. A streamline upwind Petrov-Galerkin method is adopted to stabilize possible occurrence of spurious oscillations due to advection. The XFEM is utilized to model the high gradient in the saturation field front, where the LS function is used for enhancing the weighting and the shape functions.

The capability of the proposed model and its features are evaluated by numerical examples, demonstrating its accuracy, stability and convergence, as well as its advantages over standard and upwind techniques. The study showed that a good combination between a mathematical model and a numerical model enables the simulation of complicated processes occurring in complicated and large geometry using minimal computational efforts.

A new computational model for two-phase flow in porous media is introduced with basic requirements for accuracy, stability, and convergence, which are met using relatively coarse meshes.