Transport in Porous Media

, Volume 129, Issue 3, pp 779–810 | Cite as

An Analytical Solution and Nonlinear Regression Analysis for Sandface Temperature Transient Data in the Presence of a Near-Wellbore Damaged Zone

  • Filippo PaniniEmail author
  • Mustafa Onur
  • Dario Viberti


This study presents an approximate analytical solution for predicting drawdown temperature transient behaviors of a fully penetrating vertical well in a two-zone radial composite reservoir system. The inner zone may represent a damaged (skin) zone, and the outer (non-skin) zone represents an infinitely extended reservoir. The analytical solution is obtained by solving the decoupled isothermal pressure diffusivity equation and temperature equation for the inner and outer zones with the Boltzmann transformation. The convection, transient adiabatic expansion and Joule–Thomson heating effects are all accounted for in the solution. The developed solution compares well with the results of a thermal numerical simulator. The analytical solution is used as a forward model for estimating the parameters of interest by nonlinear regression built on a gradient-based maximum likelihood estimation (MLE) method. A methodology, based on semilog analyses of pressure and temperature data as well as log–log diagnostic plots of pressure- and temperature-derivative data, is proposed to obtain good initial guesses of parameters, which derive the MLE objective function to have reliable optimized estimates. The statistical measures such as estimated standard deviation of noise in pressure and temperature data, confidence intervals for parameters and correlation coefficients between parameter pairs are used to evaluate the goodness of fit and reliability of the estimated parameters from history matching pressure and/or temperature data sets. The results show that the rock, fluid and thermal properties of the skin zone and non-skin zone can be reliably estimated by regressing on temperature transient data jointly with pressure transient data in the presence of noise.


Temperature transient analysis Nonlinear regression Analytical solution Boltzmann transformation Skin zone Radial composite reservoir model 

List of Symbols

\(\alpha \)

Thermal diffusivity constant (\(\hbox {m}^2\)/s)

\(\eta \)

Hydraulic diffusivity constant (\(\hbox {m}^2\)/s)

\(\gamma \)

Euler’s constant (0.577215)

\(\kappa \)

Thermal conductivity (J/m s K)

\(\lambda \)

Mobility (\(\hbox {m}^2\)/Pa s)

\(\mu \)

Viscosity of fluid (Pa s)

\(\phi \)

Porosity (fraction)

\(\rho \)

Density (kg/\(\hbox {m}^3\))

\(\sigma \)

Throttling coefficient (J/kg Pa)

\(\mathbf m \)

Model parameter vector

\(\mathbf m ^*\)

Optimized model parameter vector

\(\varepsilon _\mathrm{JT}\)

Joule–Thomson expansion coefficient (K/Pa)

\(\varphi \)

Adiabatic expansion coefficient (K/Pa)


Formation volume factor (\(\hbox {m}^3\)/\(\hbox {Sm}^3\))


Isothermal compressibility (\(\hbox {Pa}^{-1}\))


Specific heat capacity (J/kg K)


Ratio of volumetric heat capacity of fluid to volumetric capacity of fluid-saturated porous medium


Exponential integral


Reservoir thickness (m)


Effective permeability (\(\hbox {m}^2\))


Pressure (Pa)


Volumetric flow rate at standard conditions (\(\hbox {Sm}^3\)/s)


Radius (m)


Wellbore radius (m)


Skin factor (unitless)


Saturation (fraction)


Temperature (K)


Drawdown time (s)


Velocity of convective heat transfer (m/s)


Darcy’s velocity (m/s)


Boltzmann transformation (unitless)





Outer zone




Skin zone











  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55. Dover Publications, New York (1972)Google Scholar
  2. App, J.F.: Nonisothermal and productivity behavior of high-pressure reservoirs. SPE J. 15(01), 50–63 (2010)CrossRefGoogle Scholar
  3. Bourdet, D., Ayoub, J., Pirard, Y.: Use of pressure derivative in well test interpretation. SPE Form. Eval. 4(02), 293–302 (1989)CrossRefGoogle Scholar
  4. Chekalyuk, E.: Thermodynamics of Oil Formation, vol. 1. Nedra, Moscow (1965)Google Scholar
  5. Chevarunotai, N., Hasan, A., Kabir, C., Islam, R.: Transient flowing-fluid temperature modeling in reservoirs with large drawdowns. J. Pet. Explor. Prod. Technol. 8(3), 799–811 (2018)CrossRefGoogle Scholar
  6. CMG (2015) CMG-STARS Version 2015.10.5715.22942, Advanced Process and Thermal Reservoir Simulator. Computer Modelling Group, CalgaryGoogle Scholar
  7. Duru, O.O., Horne, R.N.: Joint inversion of temperature and pressure measurements for estimation of permeability and porosity fields. In: SPE Annual Technical Conference and Exhibition, Florence, Italy, 19–22 September. Society of Petroleum Engineers (2010a)Google Scholar
  8. Duru, O.O., Horne, R.N.: Modeling reservoir temperature transients and reservoir-parameter estimation constrained to the model. SPE Reserv. Eval. Eng. 13(06), 873–883 (2010b)CrossRefGoogle Scholar
  9. Duru, O.O., Horne, R.N.: Combined temperature and pressure data interpretation: applications to characterization of near-wellbore reservoir structures. In: SPE Annual Technical Conference and Exhibition, Denver, Colorado, USA, 30 October–2 November. Society of Petroleum Engineers (2011a)Google Scholar
  10. Duru, O.O., Horne, R.N.: Simultaneous interpretation of pressure, temperature, and flow-rate data using Bayesian inversion methods. SPE Reserv. Eval. Eng. 14(02), 225–238 (2011b)CrossRefGoogle Scholar
  11. Earlougher, R.C.: Advances in Well Test Analysis. Henry L. Doherty Memorial Fund of AIME, New York (1977)Google Scholar
  12. Galvao, M.D.S.C.: Analytical Models for Thermal Wellbore Effects on Pressure Transient Testing. Master’s thesis, Pontificia Universitade Catolica do Rio de Janeiro, Rio de Janeiro, Brasil (2018)Google Scholar
  13. Hawkins, M.F.: A note on the skin effect. J. Pet. Technol. 8(12), 65–66 (1956)CrossRefGoogle Scholar
  14. Hurst, W.: Establishment of the skin effect and its impediment to fluid flow into a well bore. Pet. Eng. 25(11), B6–B16 (1953)Google Scholar
  15. Kappa: Ecrin Version 4.30.09, Integrated Software Platform for Dynamic Flow Analysis. Kappa Engineering, Sophia Antipolis (2015)Google Scholar
  16. Kuchuk, F.J., Onur, M., Hollaender, F.: Pressure Transient Formation and Well Testing: Convolution, Deconvolution and Nonlinear Estimation, vol. 57. Elsevier, Amsterdam (2010)Google Scholar
  17. Mao, Y., Zeidouni, M.: Analytical solutions for temperature transient analysis and near wellbore damaged zone characterization. In: SPE Reservoir Characterisation and Simulation Conference and Exhibition, Abu Dhabi, UAE, 08–10 May. Society of Petroleum Engineers (2017)Google Scholar
  18. Mao, Y., Zeidouni, M., et al.: Accounting for fluid-property variations in temperature-transient analysis. SPE J. 23(03), 868–884 (2018)CrossRefGoogle Scholar
  19. Moore, W.J.: Physical Chemistry. Prentice Hall, Englewood Cliffs (1972)Google Scholar
  20. Muradov, K.M., Davies, D.R.: Temperature transient analysis in a horizontal, multi-zone, intelligent well. In: SPE Intelligent Energy International. Society of Petroleum Engineers (2012)Google Scholar
  21. Onur, M., Cinar, M.: Analysis of sandface-temperature-transient data for slightly compressible, single-phase reservoirs. SPE J. 22(04), 1–134 (2017a)Google Scholar
  22. Onur, M., Cinar, M.: Modeling and analysis of temperature transient sandface and wellbore temperature data from variable rate well test data. In: SPE Europec featured at 79th EAGE Conference and Exhibition, 12–15 June, Paris, France. Society of Petroleum Engineers (2017b)Google Scholar
  23. Onur, M., Palabiyik, Y.: Nonlinear parameter estimation based on history matching of temperature measurements for single-phase liquid–water geothermal reservoirs. In: World Geothermal Congress, pp. 19–25. Melbourne, Australia (2015)Google Scholar
  24. Onur, M., Palabiyik, Y., Tureyen, O.I., Cinar, M.: Transient temperature behavior and analysis of single-phase liquid-water geothermal reservoirs during drawdown and buildup tests: part ii. Interpretation and analysis methodology with applications. J. Pet. Sci. Eng. 146, 657–669 (2016a)CrossRefGoogle Scholar
  25. Onur, M., Ulker, G., Kocak, S., Gok, I.M.: Interpretation and analysis of transient-sandface-and wellbore-temperature data. In: SPE Annual Technical Conference and Exhibition, Dubai, UAE, 26–28 September. Society of Petroleum Engineers (2016b)Google Scholar
  26. Palabiyik, Y., Onur, M., Tureyen, O.I., Cinar, M.: Transient temperature behavior and analysis of single-phase liquid–water geothermal reservoirs during drawdown and buildup tests: part i. Theory, new analytical and approximate solutions. J. Pet. Sci. Eng. 146, 637–656 (2016)CrossRefGoogle Scholar
  27. Panini, F.: Parameter Estimation from Drawdown Temperature Transient Test Data. Master’s thesis, Department of Environment, Land and Infrastructure Engineering, Politecnico di Torino, Torino, Italy (2017)Google Scholar
  28. Ramazanov, A.S., Nagimov, V.: Analytical model for the calculation of temperature distribution in the oil reservoir during unsteady fluid inflow. Oil Gas Bus. J. (2007)Google Scholar
  29. Ramazanov, A., Valiullin, R.A., Shako, V., Pimenov, V., Sadretdinov, A., Fedorov, V., Belov, K.: Thermal modeling for characterization of near wellbore zone and zonal allocation. In: SPE Russian Oil and Gas Conference and Exhibition, Moscow, Russia, 26–28 October. Society of Petroleum Engineers (2010)Google Scholar
  30. Ramey, H.: Approximate solutions for unsteady liquidflow in composite reservoirs. J. Can. Pet. Technol. 9(01) (1970)Google Scholar
  31. Riley, K.F., Hobson, M.P., Bence, S.J.: Mathematical Methods for Physics and Engineering. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  32. Sidorova, M., Theuveny, B., Pimenov, V., Shako, V., Guzman-Garcia, A.: Do not let temperature transients hinder your build-up pressure interpretation-proper gauge placement in highly productive reservoirs in well testing operations (Russian). In: SPE Annual Caspian Technical Conference and Exhibition. Society of Petroleum Engineers (2014)Google Scholar
  33. Sui, W., Zhu, D., Hill, A.D., Ehlig-Economides, C.A.: Determining multilayer formation properties from transient temperature and pressure measurements. In: SPE Annual Technical Conference and Exhibition, Denver, Colorado. Society of Petroleum Engineers (2008)Google Scholar
  34. Van Everdingen, A.: The skin effect and its influence on the productive capacity of a well. J. Pet. Technol. 5(06), 171–176 (1953)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Environment, Land and Infrastructure EngineeringPolitecnico di TorinoTurinItaly
  2. 2.McDougall School of Petroleum EngineeringUniversity of TulsaTulsaUSA

Personalised recommendations