Transport in Porous Media

, Volume 124, Issue 3, pp 825–860 | Cite as

Evaluation of Non-Fourier Heat Transfer on Temperature Evolution in an Aquifer Thermal Energy Storage System

  • Abiola D. ObembeEmail author
  • M. Enamul Hossain
  • Sidqi A. Abu-Khamsin


The heat transport in fractured porous rocks is classically modeled employing the advection–dispersion equation (ADE). However, the nature of heat transfer in fractured reservoir rocks may not be represented by the effective medium properties when the ADE formulation is adopted. In this study, a modified mathematical model describing non-Fourier heat transport in an aquifer thermal energy storage (ATES) system is proposed employing the fractional calculus theory. This mathematical model incorporates the effect of heat losses to the surrounding impermeable rock formations. The Laplace transformation method is applied to derive the semi-analytical solutions describing the dimensionless temperature evolution in the confined aquifer, and the surrounding impermeable rocks (i.e., underlying and overlying rocks). Detailed parametric studies are performed to investigate the role of the introduced parameters, i.e., the fractional order of differentiation, generalized friction coefficient and aquifer pseudo-effective thermal conductivity on the propagation of heat within the ATES system. Computations performed on the derived solutions demonstrate that the temperature profiles in the confined aquifer and the surrounding rocks are influenced by the magnitude of the respective fractional exponents. In addition, observation of the temperature profiles within the thermally perturbed zones demonstrates that larger values of the fractional order of differentiation lead to efficient heat transfer within the ATES system. Furthermore, analysis of the results indicates that the impact of the aquifer pseudo-effective thermal conductivity on the temperature propagation in the ATES system is limited to the aquifer only. The derived solutions will find widespread application in designing and simulating the heat injection performance in an ATES system and assessing the influence of non-Fourier heat transport and geological parameters on temperature transients through porous media.


Heat transport Fractured porous rocks Non-Fourier Fractional derivative Laplace transformation 

List of symbols

\( A_{r} \)

\( 2\pi rh_{m} \); cross-sectional area of rock perpendicular to the flow of flowing fluid in r direction (m2)

\( g \)

Gravitational acceleration (m s−2)

\( h_{m} \)

The aquifer height (m)

\( h_{1} \)

The underlying rock height (m)

\( h_{2} \)

The overlying rock height (m)

\( h_{1D} \)

\( \frac{{h_{2} }}{{h_{m} }} \); dimensionless height of fractured underlying rock

\( h_{2D} \)

\( \frac{{h_{2} }}{{h_{m} }} \); dimensionless height of fractured overlying rock

\( k_{m\gamma ,e} \)

Aquifer system effective pseudo-conductivity (W s1−Ɣ m−1 K−1)

\( k_{\gamma ,w} \)

Water pseudo-conductivity (W s1−Ɣ m−1 K−1)

\( k_{\gamma ,R} \)

Aquifer rock pseudo-conductivity (W s1−Ɣ m−1 K−1)

\( k_{1\gamma ,e} \)

Effective pseudo-conductivity of underlying rock (W s 1−Ɣ m−1 K−1)

\( k_{2\gamma ,e} \)

Effective pseudo-conductivity of overlying rock (W s1−Ɣ m−1 K−1)

\( k_{1D\gamma } \)

\( \frac{{k_{1\gamma ,e} }}{{\chi_{m1} \times k_{m\gamma ,e} }} \); dimensionless effective pseudo-thermal conductivity of underlying rock; Eq. (A-9)

\( k_{2\gamma D} \)

\( \frac{{k_{2\gamma ,e} }}{{\chi_{m2} \times k_{m\gamma ,e} }} \); dimensionless effective pseudo-thermal conductivity of overlying rock; Eq. (A-9)

\( \left( {\rho C_{p} } \right)_{mb} \)

Volumetric thermal capacity of the aquifer system (J m−3 K−1)

\( \left( {\rho C_{p} } \right)_{1b} \)

Volumetric thermal capacity of the underlying rock (J m−3 K−1)

\( \left( {\rho C_{p} } \right)_{2b} \)

Volumetric thermal capacity of the overlying rock (J m−3 K−1)

\( q_{1}^{2} \)

\( \frac{{s^{{\gamma_{m} }} }}{{\alpha_{1D} }} \)

\( q_{2}^{2} \)

\( \frac{{s^{{\gamma_{m} }} }}{{\alpha_{1D} }} \)

\( Q \)

Injection rate (m3 s−1)

\( r \)

Radial distance (m)

\( r_{D} \)

\( \frac{2r}{{h_{m} }} \); dimensionless distance at any point along r direction

\( r_{w} \)

The radius of the injection well (m)

\( r_{wD} \)

\( \frac{{2r_{w} }}{{h_{m} }} \); dimensionless wellbore radius

\( s \)

Laplace variable

\( t \)

Time (s)

\( t_{h} \)

\( \left[ {\frac{{\left( {h_{m} } \right)^{2} }}{{4\alpha_{m\gamma } }}} \right]^{{\frac{1}{{\gamma_{m} }}}} \); characteristic heat transfer time (s)


Dimensionless time

\( T_{m} \left( {r,t} \right) \)

Temperature in the aquifer (K)


\( \frac{{T_{m} - T_{m0} }}{{T_{\text{inj}} - T_{m0} }} \)

\( T_{m0} \)

Initial temperature in the aquifer (K)

\( T_{1} \left( {r,z,t} \right) \)

Temperature in the underlying rock (K)

T1 D

\( \frac{{T_{1} - T_{m0} }}{{T_{inj} - T_{m0} }} \)

\( T_{10} \)

Initial temperature in the underlying rock (K)

\( T_{1D}^{\prime } \)

\( T_{1D} \left( {r_{D} ,z_{2D} ,t_{D} } \right) - T_{10D} \)

\( T_{2} \left( {r,z,t} \right) \)

Temperature in the overlying rock (K)

\( T_{2D} \)

\( \frac{{T_{2} - T_{m0} }}{{T_{\text{inj}} - T_{m0} }} \)

\( T_{20} \)

Initial temperature in the overlying rock (K)

\( T_{2D}^{\prime } \)

\( T_{2D} \left( {r_{D} ,z_{2D} ,t_{D} } \right) - T_{20D} \)

\( T_{\text{inj}} \)

Injection hot water temperature (K)

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u}_{\gamma } \)

Volumetric flux media in r direction (m s−1)

\( u^{*} \)

Dimensionless volumetric flux in r direction

\( w \)

\( \frac{Q}{{4\pi \times \phi \times \zeta_{\gamma } \times \alpha_{m\gamma } \times h_{m} }} \); dimensionless convective parameter

\( z \)

Vertical distance along the injection well (m)

\( z_{1D} \)

\( \frac{ - 4z}{{h_{m} }} \); dimensionless thickness in the underlying rock

\( z_{2D} \)

\( \frac{{4\left( {z - h_{m} } \right)}}{{h_{m} }} \); dimensionless thickness in the overlying rock

Greek symbols

\( D_{t}^{1 - \gamma } \)

Caputo time fractional operator

\( \alpha_{m\gamma } \)

\( k_{m\gamma ,e}{/}\left({\rho C_{p} } \right)_{mb} \); aquifer system pseudo-thermal diffusivity (m2 s−Ɣ)

\( \alpha_{1\gamma } \)

\( k_{1\gamma ,e}{/}\left({\rho C_{p} } \right)_{1b} \); pseudo-thermal diffusivity of underlying rock (m2 s−Ɣ)

\( \alpha_{2\gamma } \)

\( k_{2\gamma ,e}{/}\left({\rho C_{p} } \right)_{2b} \); pseudo-thermal diffusivity of overlying rock (m2 s−Ɣ)

\( \alpha_{1\gamma D} \)

\( \frac{{4\alpha_{1\gamma } }}{{\alpha_{m\gamma } \times \chi_{m1} }} \); dimensionless pseudo-thermal diffusivity of underlying rock

\( \alpha_{2\gamma D} \)

\( \frac{{4\alpha_{2\gamma } }}{{\alpha_{m\gamma } \times \chi_{m2} }} \); dimensionless pseudo-thermal diffusivity of overlying rock

\( \gamma \)

Fractional order of differentiation, dimensionless

\( \zeta_{\gamma } \)

Generalized friction term (sɣ−1)

\( \rho_{R} \)

Sold rock matrix density (kg m−3)

\( \rho_{w} \)

Water density (kg m−3)

\( \phi_{m} \)

Porosity, fraction

\( \Gamma \)

Gamma function





Underlying rock layer


Overlying rock layer


Initial conditions



The authors would like to acknowledge the support provided by King Abdulaziz City for Science and Technology (KACST), through the Science & Technology Unit at King Fahd University of Petroleum & Minerals (KFUPM), for funding this work through project No. 11-OIL1661-04, as part of the National Science, Technology and Innovation Plan (NSTIP).


  1. Abdulla, A., Reddy, K.S.: Effect of operating parameters on thermal performance of molten salt packed-bed thermocline thermal energy storage system for concentrating solar power plants. Int. J. Therm. Sci. 121, 30–44 (2017)CrossRefGoogle Scholar
  2. Bandai, T., Hamamoto, S., Rau, G.C., Komatsu, T., Nishimura, T.: The effect of particle size on thermal and solute dispersion in saturated porous media. Int. J. Therm. Sci. 122, 74–84 (2017)CrossRefGoogle Scholar
  3. Berkowitz, B., Cortis, A., Dentz, M., Scher, H.: Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys. 44, RG2003 (2006)Google Scholar
  4. Biemans, B.C.B.: The influence of fractures on geothermal heat production in the Roer Valley Graben. Msc. Thesis, Department of Geoscience & Engineering, Delft University of Technology, Delft, Netherlands (2014)Google Scholar
  5. Cattaneo, C.: Sulla conduzione del calore. Atti Semin. Mat. Fis. Univ. Modena 3, 83–101 (1948)Google Scholar
  6. Cattaneo, C.: Sur une forme de lequation de la chaleur eliminant le paradoxe dune propagation instantanee. Comptes Rendus Hebd. Des Seances L Acad. Des Sci. 247, 431–433 (1958)Google Scholar
  7. Cortis, A., Berkowitz, B.: Anomalous transport in “classical” soil and sand columns. Soil Sci. Soc. Am. J. 68, 1539–1548 (2004)CrossRefGoogle Scholar
  8. Cortis, A., Birkholzer, J.: Continuous time random walk analysis of solute transport in fractured porous media. Water Resour. Res. (2008). Google Scholar
  9. Day, W.A.: The Thermodynamics of Simple Materials with Fading Memory. Springer, New York (2013)Google Scholar
  10. Dentz, M., Cortis, A., Scher, H., Berkowitz, B.: Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport. Adv. Water Resour. 27, 155–173 (2004)CrossRefGoogle Scholar
  11. Emmanuel, S., Berkowitz, B.: Continuous time random walks and heat transfer in porous media. Transp. Porous Media 67, 413–430 (2007)CrossRefGoogle Scholar
  12. Faulkner, D.R., Jackson, C.A.L., Lunn, R.J., Schlische, R.W., Shipton, Z.K., Wibberley, C.A.J., Withjack, M.O.: A review of recent developments concerning the structure, mechanics and fluid flow properties of fault zones. J. Struct. Geol. 32, 1557–1575 (2010)CrossRefGoogle Scholar
  13. Geiger, S., Emmanuel, S.: Non-Fourier thermal transport in fractured geological media. Water Resour. Res. 46(7), 759–768 (2010)CrossRefGoogle Scholar
  14. Hejazi, H., Moroney, T., Liu, F.: A finite volume method for solving the two-sided time-space fractional advection-dispersion equation. Open Phys. 11, 1275–1283 (2013)CrossRefGoogle Scholar
  15. Ishteva, M.: Properties and applications of the Caputo fractional operator. Msc. Thesis, Department of Mathematics, Universität Karlsruhe (TH), Sofia, Bulgaria (2005)Google Scholar
  16. Kaviany, M.: Principles of Heat Transfer in Porous Media. Springer, New York (1995)CrossRefGoogle Scholar
  17. Li, K.-Y., Yang, S.-Y., Yeh, H.-D.: An analytical solution for describing the transient temperature distribution in an aquifer thermal energy storage system. Hydrol. Process. 24, 3676–3688 (2010)CrossRefGoogle Scholar
  18. Luchko, Y., Punzi, A.: Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations. GEM Int. J. Geomath. 1, 257–276 (2011)CrossRefGoogle Scholar
  19. Malate, R.C.M., O’sullivan, M.J.: Modelling of chemical and thermal changes in well PN-26 Palinpinon geothermal field, Philippines. Geothermics 20, 291–318 (1991)CrossRefGoogle Scholar
  20. Nield, D.A.: Modelling fluid flow and heat transfer in a saturated porous medium. Adv. Decis. Sci. 4, 165–173 (2000)Google Scholar
  21. Nield, D.A., Bejan, A.: Heat transfer through a porous medium. In: Convection in Porous Media, pp. 37–55. Springer, Cham (2017)Google Scholar
  22. Obembe, A.D., Abu-khamsin, S.A., Hossain, M.E.: A review of modelling thermal displacement processes in porous media. Arab. J. Sci. Eng. 41, 4719–4741 (2016)CrossRefGoogle Scholar
  23. Obembe, A.D., Al-Yousef, H.Y., Hossain, M.E., Abu-Khamsin, S.A.: Fractional derivatives and their applications in reservoir engineering problems: a review. J. Petrol. Sci. Eng. 157, 312–327 (2017a)CrossRefGoogle Scholar
  24. Obembe, A.D., Hasan, M., Fraim, M.: A mathematical model for transient testing of naturally fractured shale gas reservoirs. In: SPE Kingdom of Saudi Arabia Annual Technical Symposium and Exhibition, 24–27 April, Dammam, Saudi Arabia. SPE-188058-MS (2017b)Google Scholar
  25. Obembe, A.D., Hasan, M., Fraim, M.: An Anomalous Productivity Model for Naturally Fractured Shale Gas Reservoirs. In: SPE Kingdom of Saudi Arabia Annual Technical Symposium and Exhibition. SPE-188033-MS (2017c)Google Scholar
  26. Obembe, A.D., Hossain, M.E., Abu-Khamsin, S.A.: Variable-order anomalous heat transport mathematical models in disordered and heterogeneous porous media. In: 2nd Thermal and Fluid Engineering Conference, TFEC2017 4th International Workshop on Heat Transfer, IWHT2017, pp. 1–15. Begell House Inc. (2017d)Google Scholar
  27. Obembe, A.D., Hossain, M.E., Abu-Khamsin, S.A.: Variable-order derivative time fractional diffusion model for heterogeneous porous media. J. Petrol. Sci. Eng. 152, 391–405 (2017e)CrossRefGoogle Scholar
  28. Obembe, A.D., Abu-khamsin, S.A., Hossain, M.E.: Anomalous effects during thermal displacement in porous media under non-local thermal equilibrium. J. Porous Media 21, 161–196 (2018a)CrossRefGoogle Scholar
  29. Obembe, A.D., Abu-Khamsin, S.A., Hossain, M.E., Mustapha, K.: Analysis of subdiffusion in disordered and fractured media using a Grünwald-Letnikov fractional calculus model. Comput. Geosci. (2018b). Google Scholar
  30. Ochsner, T.E., Horton, R., Kluitenberg, G.J., Wang, Q.: Evaluation of the heat pulse ratio method for measuring soil water flux. Soil Sci. Soc. Am. J. 69, 757–765 (2005)CrossRefGoogle Scholar
  31. Ozisik, M.N.: Heat Conduction. Wiley, New York (1993)Google Scholar
  32. Podlubny, I.: Fractional Differential Equation. Academic Press, San Diego (1999)Google Scholar
  33. Povstenko, Y.Z.: Fractional heat conduction equation and associated thermal stress. J. Therm. Stress. 28, 83–102 (2004)CrossRefGoogle Scholar
  34. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Philadelphia (1993)Google Scholar
  35. Stehfest, H.: Algorithm 368: numerical inversion of Laplace transforms [D5]. Commun. ACM 13, 47–49 (1970)CrossRefGoogle Scholar
  36. Suzuki, A., Niibori, Y., Fomin, S.A., Chugunov, V.A., Hashida, T.: Analysis of water injection in fractured reservoirs using a fractional-derivative-based mass and heat transfer model. Math. Geosci. 47, 31–49 (2015)CrossRefGoogle Scholar
  37. Vafai, K.: Handbook of Porous Media. CRC Press, London (2015)Google Scholar
  38. Wibberley, C.A.J., Yielding, G., Di Toro, G.: Recent advances in the understanding of fault zone internal structure: a review. Geol. Soc. Lond. Spec. Publ. 299, 5–33 (2008)CrossRefGoogle Scholar
  39. Wu, C.C., Hwang, G.J.: Flow and heat transfer characteristics inside packed and fluidized beds. J. Heat Transf. 120, 667–673 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Abiola D. Obembe
    • 1
    Email author
  • M. Enamul Hossain
    • 2
  • Sidqi A. Abu-Khamsin
    • 3
  1. 1.Technology Transfer, Innovation and EntrepreneurshipKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  2. 2.Department of Petroleum Engineering, School of Mining and GeosciencesNazarbayev UniversityAstanaRepublic of Kazakhstan
  3. 3.Department of Petroleum Engineering, College of Petroleum and GeosciencesKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

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