Geotechnical and Geological Engineering

, Volume 33, Issue 2, pp 291–306 | Cite as

Numerical Modeling of Vertical Geothermal Heat Exchangers Using Finite Difference and Finite Element Techniques

  • Tolga Y. Ozudogru
  • Omid Ghasemi-Fare
  • C. Guney Olgun
  • Prasenjit Basu
Original paper


This paper presents the development of a 2D finite difference modelling approach and a 3D finite element numerical model for simulating vertical geothermal heat exchangers (GHEs), explaining the theory governing the thermal processes, element discretization and the selection of the appropriate boundary conditions. Both of these models provide fully coupled solutions for the fluid flow in the circulation pipes and the thermal processes between the fluid and solid domains (pipes, grout and soil). The numerical models are verified with a field test and subsequently they are utilized to simulate the thermal performance of a borehole heat exchanger integrated with a single U-tube. Two different thermal operation cases are analyzed; a constant rate heat injection and a fluid injection at a constant temperature. A model validation study is also carried out for the constant rate heat injection case by comparing the numerical results with the available analytical solution for a finite line source. Furthermore, effective thermal conductivity of the ground back-calculated from the results of the numerical analyses is compared with the value used in the numerical models. Comparison of the results obtained from both numerical models and validating model predictions with the analytical solution confirms that both FE and FD models can accurately simulate the heat transfer mechanisms governing the thermal performance of GHE systems.


Geothermal heat exchanger Numerical modeling Finite difference analysis Finite element analysis Finite line source 

List of symbols


Cross-section area (m2)


Specific heat capacity (J kg−1 K−1)


Diameter (m)


Hydraulic diameter of pipe (m)


Darcy friction factor


Heat transfer coefficient (W m−2 K−1)


Length of the heat exchanger (m)


Thermal conductivity (W m−1 K−1)


Slope of the temperature versus log time curve


Nusselt number


External heat exchange through pipe wall (W)


Heat (W)


Radial coordinate (m)


Pipe radius (m)


Shank spacing between two pipe legs (m)


Temperature (K)


Time (s)


Flow velocity (m s−1)


Volumetric flow rate (m3 s−1)


Velocity field (m s−1)


Depth (m)

Greek symbols


Thermal diffusivity (m2 s−1)


Applied temperature difference at time t (K)


Density (kg m−3)








i, j

Node index









The first and third authors would like to express their gratitude for the support by the National Science Foundation under Grants No. CMMI-0928807 and CMMI-1100752. The authors would also like to gratefully acknowledge the financial support by the Mid-Atlantic Universities Transportation Center (MAUTC) under Grant No. 415-77 76R20. The first author is funded as a visiting scholar by the Turkish Council on Higher Education and Istanbul Technical University. These funding supports are greatly appreciated.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Tolga Y. Ozudogru
    • 1
    • 2
  • Omid Ghasemi-Fare
    • 3
  • C. Guney Olgun
    • 2
  • Prasenjit Basu
    • 3
  1. 1.Department of Civil EngineeringIstanbul Technical UniversityIstanbulTurkey
  2. 2.Charles E. Via, Jr. Department of Civil and Environmental EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  3. 3.Department of Civil and Environmental EngineeringThe Pennsylvania State UniversityPennsylvaniaUSA

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