Transport in Porous Media

, Volume 92, Issue 1, pp 61–81 | Cite as

Analysis of Pore Pressure Distribution in Shale Formations under Hydraulic, Chemical, Thermal and Electrical Interactions

  • Hamid RoshanEmail author
  • M. A. Aghighi


Change in pore pressure in chemically active rocks such as shale is caused by several mechanisms and numerous studies have been carried out to investigate these mechanisms. However, some important coupling terms or driving forces have been neglected in these studies due to simplifying assumptions. In this study, a hydro-chemo-thermo-electrical model based on finite element method is presented to investigate the change in pore pressure in shale formations resulted from thermal, hydraulic, chemical and electric potential gradients. The change in pore pressure is induced by hydraulic conduction, chemical, electrical and thermal osmotic flow. In order to solve the problem of ion transfer under the influence of an electrical field, the Nernst–Planck equation is used. In addition, ion advection is considered to investigate its possible effect on ion transfer for the range of shale permeability. All equations are derived based on the thermodynamics of irreversible processes in a discontinuous system. The numerical results are compared against existing and derived uncoupled analytical solutions and good agreement is observed. The numerical results showed that the ion transfer and pore pressure are considerably affected by the electric field in the vicinity of the wellbore. It was also found that advection can play a remarkable role in ion transfer in shale formations. It was further shown that the change in pore pressure in shale formation is characterized by the combined effect of hydraulic, chemical, thermal and electro osmotic flow.


Thermodynamics of irreversible processes Shale formation Pore pressure 

List of Symbols


Average solute mass fraction in formation


Average diluent mass fraction in formation


Anions mass fractions


Cations mass fractions


Solute mass fraction of n chemical species


Thermal conductivity


Specific heat capacity


Average solute mass fraction in drilling fluid


Average solute mass fraction in pore fluid


Solute diffusion coefficient of each chemical species


Coefficient of thermal diffusion of each chemical species


Electric potential


Average electric potential of drilling fluid


Average electric potential of pore fluid


Faraday’s constant (96,485 C/mol electrons)




Thermal osmosis coefficient


Electrical osmosis coefficient


Fluid bulk module


Molar mass of the solute


Molar mass of j chemical species


Number of nodes


Pressure shape functions


Electric potential shape functions


Temperature shape functions

\({N_{\rm C}^{S}}\)

Mass fraction shape functions




Universal gas constant






Absolute temperature


Average temperature of drilling fluid


Average temperature of pore fluid


Charge of j chemical species


Coefficient of solute retardation



\({\mathop {\rho _{\rm f}}\limits^- }\)

Average fluid density


Fluid density




Standard solute reflection coefficient


The effective electric conductivity of porous media


The electrical capacitance per unit volume


The coefficient of thermo-electricity (Seebeck effect)


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Acar Y.B., Gale R.J., Alshawabkeh A.N., Marks R.E., Puppala S., Bricka M., Parker R.: Electrokinetic remediation: basics and technology status. J. Hazard. Mater. 40(2), 117–137 (1995)CrossRefGoogle Scholar
  2. Alshawabkeh A.N., Acar Y.B.: Electrokinetic remediation. II: theoretical model. J. Geotech. Eng. 122(3), 186–196 (1996)CrossRefGoogle Scholar
  3. Bader S., Kooi H.: Modelling of solute and water transport in semi-permeable clay membranes: comparison with experiments. Adv. Water Res. 28(3), 203–214 (2005)CrossRefGoogle Scholar
  4. Bear J.: Dynamics of fluids in porous media. Dover, New york (1988)Google Scholar
  5. Bear J., Bachmat Y.: Theory and application of transport in porous media, vol. 4. Kluwer Academic Publishers, Dordrecht (1991)Google Scholar
  6. Chenevert M.E.: Shale control with balanced-activity oil-continuous muds. SPE J. Petroleum Technol. 22(10), 1309–1316 (1970). doi: 10.2118/2559-pa Google Scholar
  7. Cooper GA, Roy S (1994) Prevention of bit balling by electro-osmosis. Paper presented at the SPE western regional meeting, Long Beach, Mar 23 1994Google Scholar
  8. Detournay E., Cheng A.H.D.: Poroelastic response of a borehole in a non-hydrostatic stress field. Int. J. Rock Mech. Min. Sci. & Geomech Abstract 25(3), 171–182 (1988)CrossRefGoogle Scholar
  9. Ghassemi A., Diek A.: Porothermoelasticity for swelling shales. J. Petroleum Sci. Eng. 34(1-4), 123–135 (2002)CrossRefGoogle Scholar
  10. Ghassemi A., Diek A.: Linear chemo-poroelasticity for swelling shales: theory and application. J. Petroleum Sc. Eng. 38(3-4), 199–212 (2003)CrossRefGoogle Scholar
  11. Ghassemi A., Tao Q., Diek A.: Influence of coupled chemo-poro-thermoelastic processes on pore pressure and stress distributions around a wellbore in swelling shale. J. Petroleum Sci. Eng. 67(1-2), 57–64 (2009)CrossRefGoogle Scholar
  12. Haase R.: Thermodynamics of irreversible processes. Dover, New York (1990)Google Scholar
  13. Hanshaw B.B., Zen E.-A.: Osmotic Equilibrium and Overthrust Faulting. Geol. Soc. Am. Bull. 76(12), 1379–1385 (1965). doi: 10.1130/0016-7606(1965)76[1379:oeaof];2 CrossRefGoogle Scholar
  14. Hariharan PR, Cooper GA, Hale AH (1998) Bit balling reduction by electro-osmosis while drilling shale using a model BHA (Bottom Hole Assembly). Paper presented at the IADC/SPE drilling conference, Dallas, 03 Feb 1998Google Scholar
  15. Heidug W.K.: A thermodynamic theory of fluid-infilterated porous media undergoing large deformations and change of phase. Brown University, Providence (1985)Google Scholar
  16. Heidug W.K., Wong S.-W.: Hydration Swelling of Water-Absorbing Rocks: A Contitutive Model. Int. J. Numer. Anal. Methods Geomech. 20(6), 403–430 (1996)CrossRefGoogle Scholar
  17. Kooi H., Garavito A.M., Bader S.: Numerical modelling of chemical osmosis and ultrafiltrationacross clay formations. J. Geochem. Explor. 78(79), 333–336 (2003)CrossRefGoogle Scholar
  18. Lal M (1999) Shale stability: drilling fluid interaction and shale strength. Paper presented at the SPE Asia pacific oil and gas conference and exhibition, Jakarta, Apr 20 1999Google Scholar
  19. Lewis R.W., Nithiarasu P., Seethararmu K.N.: Fundamentals of the finite element method for heat and fluid flow, p. 356. Wiley, Hoboken (2004)CrossRefGoogle Scholar
  20. Moyne C., Murad M.: A two-scale model for coupled electro-chemo-mechanical phenomena and onsagers reciprocity relations in expansive clays: I homogenization analysis. Transp. Porous Med. 62, 333–380 (2006)CrossRefGoogle Scholar
  21. Narasimhan B., Sri Ranjan R.: Electrokinetic barrier to prevent subsurface contaminant migration: theoretical model development and validation. J. Contam. Hydrol. 42(1), 1–17 (2000)CrossRefGoogle Scholar
  22. Onsager L.: Reciprocal relations in irreversible processes. I. Phys. Rev. 37(4), 405–426 (1931)Google Scholar
  23. Revil A, Leroy P (2001) Hydroelectric coupling in a clayey material. Geophys. Res. Lett. 28. doi: 10.1029/2000gl012268
  24. Revil A, Pessel M (2002) Electroosmotic flow and the validity of the classical Darcy equation in silty shales. Geophys. Res. Lett. 29. doi: 10.1029/2001gl013480
  25. Roshan H, Rahman SS (2010) A fully coupled chemo-poroelastic analysis of pore pressure and stress distribution around a wellbore in water active rocks. Rock Mech. Rock Engng. doi: 10.1007/s00603-010-0104-7
  26. Roy S., Cooper G.A.: Prevention of bit balling in shales—preliminary results. SPE Drill. Complet. 8(3), 195–200 (1993). doi: 10.2118/23870-pa Google Scholar
  27. Soler J.M.: The effect of coupled transport phenomena in the Opalinus Clay and implications for radionuclide transport. J. Contam. Hydrol. 53(1-2), 63–84 (2001)CrossRefGoogle Scholar
  28. Srivastava R.C., Jain A.K., Upadhyay S.K.: Electro-osmosis of water through a collodion membrane. J. Non-Equilib. Thermodyn. 3(2), 83–92 (2009). doi: 10.1515/jnet.1978.3.2.83 CrossRefGoogle Scholar
  29. van Oort E., Hale A.H., Mody F.K., Roy S.: Transport in shales and the design of improved water-based shale drilling fluids. SPE Drill. Complet. 11(3), 137–146 (1996). doi: 10.2118/28309-pa Google Scholar
  30. Wang M., Chen S.: Electroosmosis in homogeneously charged micro- and nanoscale random porous media. J. Colloid and Interface Sci. 314(1), 264–273 (2007). doi: 10.1016/j.jcis.2007.05.043 CrossRefGoogle Scholar
  31. Wang M., Kang Q.: Modeling electrokinetic flows in microchannels using coupled lattice Boltzmann methods. J. Computational Phys. 229(3), 728–744 (2010). doi: 10.1016/ CrossRefGoogle Scholar
  32. Wang M., Revil A.: Electrochemical charge of silica surfaces at high ionic strength in narrow channels. J. Colloid and Interface Sci. 343(1), 381–386 (2010). doi: 10.1016/j.jcis.2009.11.039 CrossRefGoogle Scholar
  33. Wang Y., Dusseault M.B.: A coupled conductive-convective thermo-poroelastic solution and implications for wellbore stability. J. Petroleum Sci. Eng. 38(3-4), 187–198 (2003)CrossRefGoogle Scholar
  34. Yeung A.T.: Coupled flow equations for water, electricity and ionic contaminants through clayey soils under hydraulic, electrical and chemical gradients. J. Non-Equilib. Thermodyn. 15(3), 247–268 (2009). doi: 10.1515/jnet.1990.15.3.247 CrossRefGoogle Scholar
  35. Zienkiewicz O.C., Taylor R.L.: The finite element method—basic formulation and linear problems, vol 1. 5th edn. Butterworth-Heinemann, Oxford (2000)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.School of Petroleum EngineeringUniversity of New South WalesSydneyAustralia
  2. 2.Department of Mining and Petroleum EngineeringIK International UniversityQazvinIran

Personalised recommendations