An analytical model of a non-stationary temperature field in a reservoir with a hydraulic fracturing

Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy


2021. Vol. 7. № 2 (26)

An analytical model of a non-stationary temperature field in a reservoir with a hydraulic fracturing

For citation: Sharafutdinov R. F., Davletshin F. F. 2021. “An analytical model of a non-stationary temperature field in a reservoir with a hydraulic fracturing”. Tyumen State University Herald. Physical and Mathematical modeling. Oil, Gas, Energy, vol. 7, no. 2 (26), pp. 75-94. DOI: 10.21684/2411-7978-2021-7-2-75-94

About the authors:

Ramil F. Sharafutdinov, Dr. Sci. (Phys.-Math.), Professor, Department of Geophysics, Bashkir State University (Ufa);

Filyus F. Davletshin, Postgraduate Student, Department of Geophysics, Bashkir State University (Ufa);


At the present stage of development of the oil and gas industry, considerable attention is paid to methods of increasing oil recovery of productive reservoirs. One of the most popular methods of intensifying oil production today is hydraulic fracturing. The efficiency and success of hydraulic fracturing largely depends on the parameters of the formed fracture; in this regard, the development of methods for evaluating the parameters of hydraulic fracturing fractures is an urgent task. Non-stationary thermometry is a promising area for monitoring the quality of hydraulic fracturing. To date, thermometry is used to localize the locations of multiple fractures in horizontal wells. In this paper, we study the application of non-stationary thermometry for estimating the parameters of a vertical hydraulic fracturing fracture.

An analytical model of non-isothermal single-phase fluid filtration in a reservoir with a vertical fracture is developed. To calculate the temperature field in the formation and the fracture, the convective heat transfer equation is used, taking into account the thermodynamic effects (Joule — Thomson and adibatic), for the fracture, the heat and mass transfer between the fracture and the formation area is also taken into account. To assess the correctness of the model, the analytical solution is compared with the results of numerical modeling in the Ansys Fluent software package.

The nonstationary temperature field is calculated for the constant sampling mode. It is established that at the initial moment of time after the well start-up, a negative temperature anomaly is formed due to the adiabatic effect, the value of which increases with a decrease in the fracture width. Over time, the temperature of the fluid flowing into the well increases due to the Joule — Thomson effect, and the value of the positive temperature anomaly increases as the width and permeability of the fracture decreases due to an increase in the pressure gradient in it.

The developed analytical model can be used to solve inverse problems for estimating hydraulic fracturing parameters based on non-stationary temperature measurements in the wellbore of producing wells.


  1. Bulgakova G. T., Sharifullin A. R., Sitdikov M. R. 2020. “Mathematical modeling heat and mass transfer in a vertical hydraulic fracture crack during inflation and cleaning”. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 6, no. 2 (22), pp. 41-62. DOI: 10.21684/2411-7978-2020-6-2-41-62 [In Russian]

  2. Valiullin R. A. Sharafutdinov R. F., Fedotov V. Ya., Zakirov M. F., Sharipov A. M., Akhmetov K. R., Azizov F. F. 2015. “The use of non-stationary thermometry for diagnosing the state of wells”. Oil Industry, no. 5, pp. 93-96. [In Russian]

  3. Gilmiev D. R., Shabarov A. B. 2013. “Modeling of non-isothermal flooding of an oil reservoir with hydraulic fracturing cracks”. Innovations and Investments, no. 7. pp. 32‑38. [In Russian]

  4. Ramazanov A. SH., Islamov D. F. 2017. “Analytical model of non-stationary temperature in heterogeneous reservoir”. Bulletin of the Tomsk Polytechnic University. Geo Аssets Engineering, vol. 328, no. 5, pp. 39-48. [In Russian]

  5. Ramazanov A. Sh., Sharipov A. M. 2016. “Assessment of the influence of the heat capacity of a hydraulic fracture on measurements of unsteady temperature in a well”. Scientific and Technical Bulletin “Karotazhnik”, no. 5 (263), pp. 81-86. [In Russian]

  6. Fakhreeva R. R., Zarafutdinov I. A., Pityuk Yu. A. 2019. “Numerical modeling of pressure and temperature changes in a reservoir with a positive and negative skin factor”. Bulletin of the Bashkir University, no. 2. pp. 272-277. [In Russian]

  7. Khasanov M. M., Golovneva O. Yu. 2016. “Determination of the flow rate of vertical wells with hydraulic fracturing on the unsteady filtration mode”. Oil Industry, no. 12, pp. 64-68. [In Russian]

  8. Sharafutdinov R. F. Sadretdinov A. A., Sharipov A. M. 2017. “Numerical study of the temperature field in a reservoir with hydraulic fracturing”. Applied Mechanics and Technical Physics, no. 4, pp. 153-162. [In Russian]

  9. Sharipov A.M., Sharafutdinov R. F., Ramazanov A. SH., Valiullin R. A. 2017. “Study of the temperature recovery process in a well after stop of water injection in a reservoir with hydraulic fracturing”. Bulletin of the Bashkir University, no. 2, pp. 315-319. [In Russian]

  10. Economides M., Olini R., Valko P. Unified design of hydraulic fracturing: from theory to practice 2007. Мoscow-Izhevsk, Institute of Computer Research, 2007. 236 pp. [In Russian]

  11. Kamphuis H., Davies D. R., Roodhart L. P. 1993. “A new simulator for the calculation of the in situ temperature profile during well stimulation fracturing treatments”. Journal of Canadian Petroleum Technology, vol. 32, no. 5, pp. 38-47. DOI: 10.2118/93-05-03

  12. Mao Y., Zeidouni M., Godefroy C., Gysen M. 2019. “Fracture diagnostic using distributed temperature measurements during stimulation fluid flow-back”. SPE Oklahoma City Oil and Gas Symposium (9-10 April, Oklahoma City, Oklahoma, USA). Paper SPE-195221-MS. DOI: 10.2118/195221-MS

  13. Meyer B. R. 1989. “Heat transfer in hydraulic fracturing”. SPE Production Engineering. vol. 4, no. 4, pp. 423-429. DOI: 10.2118/17041-PA

  14. Onay M. E. 2020. “Analytical solutions for predicting fracture outlet temperature of produced fluid from enhanced geothermal systems with different well-completion configurations”. SPE Annual Technical Conference & Exhibition originally scheduled to be held (5-7 October 2020, Denver, Colorado, USA). Paper SPE-204274-STU. DOI: 10.2118/204274-STU

  15. Pityuk Yu. A. Davletbayev A. Ya., Musin A. A., Marin D. F., Seltikova E. V., Zarafutdinov I. A., Kovaleva L. A., Fursov G. A., Nazargalin E. R., Mustafin D. A. 2016. “Three-dimensional numerical simulation of pressure and temperature dynamics in a fractured well”. SPE Russian Petroleum Technology Conference and Exhibition held (24-26 October, Moscow, Russia). Paper SPE-181971-MS. DOI: 10.2118/181971-MS

  16. Ribeiro P. M., Horne R. N. 2013. “Pressure and temperature transient analysis: hydraulic fractured well application”. SPE Annual Technical Conference and Exhibition (30 September — 2 October, New Orleans, Louisiana, USA). Paper SPE-166222-MS. DOI: 10.2118/166222-MS

  17. Yoshida N., Hill A. D., Zhu D. 2018. “Comprehensive modeling of downhole temperature in a horizontal well with multiple fractures”. SPE Journal, no. 10, pp. 1580-1602.