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Rock Mechanics and Rock Engineering

, Volume 52, Issue 11, pp 4715–4729 | Cite as

Analytical Solutions for a Wellbore Subjected to a Non-isothermal Fluid Flux: Implications for Optimizing Injection Rates, Fracture Reactivation, and EGS Hydraulic Stimulation

  • Zhiqiang FanEmail author
  • Rishi Parashar
Original Paper
  • 118 Downloads

Abstract

Hydraulic stimulation in enhanced geothermal systems (EGS) involves massive injection of cold fluid into target hot geothermal reservoir through a long open hole section to trigger slip on preexisting fractures and enhance permeability. Fluid injection is typically conducted at a specified rate in a step-increasing manner until the pore pressure exceeds the minimum principal stress. During each step, the injection rate is kept constant. This paper presents analytical solutions for a wellbore subjected to cooling and a constant fluid flux on borehole wall and far field in situ stress in a thermoporoelastic medium with applications to hydraulic stimulations in EGS. The temporal-spatial distribution of temperature, pore pressure and stress are obtained by means of Laplace transform and load decomposition. The results show that for granite and a typical fluid injection scenario, thermal effect is pronounced in the vicinity of the wellbore. At early time, cooling-induced pore pressure/hoop stress counteract the injection induced pore pressure/hoop stress. With increasing time, the induced pore pressure and hoop stress result predominantly from fluid injection, and cooling plays a marginal role.

Keywords

Thermoporoelastic Hydraulic stimulation Enhanced geothermal system Fluid flux boundary Analytical solution 

List of Symbols

α

Biot coefficient

αf

volumetric thermal expansion coefficient of the pore fluid

αs

Linear thermal expansion coefficient of solid matrix

δij

Kronecker delta

εij

Strain components

ζ

Increment of fluid content

κ

Bulk thermal conductivity

μ

Coefficient of friction

µf

Pore fluid viscosity

ν

Drained Poisson’s ratio

ρ

Mass density

σH

Maximum horizontal principal stress

σij

Stress

σkk

Bulk stress

σr

Radial stress

σθ

Hoop stress

σrθ

Shear stress

σn

Total normal stress on the preexisting fracture

σz

Axial stress

τ

Shear stress on the preexisting fracture

φ

Porosity

a

Wellbore radius

c

Hydraulic diffusivity

ch

Bulk thermal diffusivity

h

Heat flux

k

Intrinsic permeability

p

Pore pressure

p0

Hydrostatic pore pressure

pw

Fluid pressure at the wellbore

q

Fluid flux

qw

Constant flux at the wellbore

s

Laplace transform variable

B

Skempton’s coefficient

C

Specific heat capacity

CFS

Coulomb failure stress

E

Young’s modulus

G

Shear modulus

H(t)

Heaviside function

K

Bulk modulus of fluid-saturated rock

K0

Modified Bessel function of second kind of order zero

K1

Modified Bessel function of second kind of first order

K2

Modified Bessel function of the second kind of order two

L

Length of open hole section

M

Biot modulus

N

Total number of terms in the Stehfest series

Shmin

Minimum horizontal in situ stress

SHmax

Maximum horizontal in situ stress

SV

Vertical stress

Q

Volumetric injection rate

T

Temperature change

Ts

Slip tendency

T0

Native fluid temperature

Tt0

Tensile strength

Tw

Injected fluid temperature

Notes

Acknowledgements

The first author would like to acknowledge the support from Nell J. Redfield foundation. The authors would like to thank two anonymous reviewers and the associate editor for their critical but helpful comments.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. Abousleiman Y, Chen S (2010) Poromechanics response of an inclined borehole subject to in situ stress and finite length fluid discharge. J Mech Mater Struct 5:47–66CrossRefGoogle Scholar
  2. Abousleiman Y, Cui L (1998) Poroelastic solutions in transversely isotropic media for wellbore and cylinder. Int J Solid Struct 35:4905–4929CrossRefGoogle Scholar
  3. Abousleiman Y, Ekbote S (2005) Solutions for the inclined borehole in a porothermoelastic transversely isotropic medium. J Appl Mech 72:102–114CrossRefGoogle Scholar
  4. Bai B, Li T (2009) Solutions for cylindrical cavity in saturated thermoporoelastic medium. Acta Mech Solida Sin 22:85–94CrossRefGoogle Scholar
  5. Chen SL (2019) Three-dimensional analytical poromechanical solutions for an arbitrarily inclined borehole subjected to fluid injection. Proc R Soc A Math Phy Eng Sci 475(2221):20180658CrossRefGoogle Scholar
  6. Chen S, Abousleiman Y (2016) Stress analysis of borehole subjected to fluid injection in transversely isotropic poroelastic medium. Mech Res Commun 73:63–75CrossRefGoogle Scholar
  7. Cheng AH-D (2016) Poroelasticity. Theory and applications of transport in porous media, vol 27. Springer, BerlinGoogle Scholar
  8. Clauser C (1992) Permeability of crystalline rocks. Eos Trans Am Geophys Union 73:233–238CrossRefGoogle Scholar
  9. Detournay E, Cheng A-D (1988) Poroelastic response of a borehole in a non-hydrostatic stress field. Int J Rock Mech Min Sci Geomech Abstr 25:171–182CrossRefGoogle Scholar
  10. Evans KF, Genter A, Sausse J (2005) Permeability creation and damage due to massive fluid injections into granite at 3.5 km at Soultz: 1. Borehole observations. J Geophys Res Solid Earth 110:B04203Google Scholar
  11. Fan Z, Eichhubl P, Gale JFW (2016) Geomechanical analysis of fluid injection and seismic fault slip for the Mw4.8 Timpson, Texas, earthquake sequence. J Geophys Res Solid Earth 121:2798–2812.  https://doi.org/10.1002/2016JB012821 CrossRefGoogle Scholar
  12. Fan Z, Parashar, R (2018) Effect of coupled porothermoelastic stress on shear stimulation of enhanced geothermal systems. In: 43rd workshop on geothermal reservoir engineering, Stanford University, Stanford, California, February 12–14 2018Google Scholar
  13. Gao J, Deng J, Lan K, Song Z, Feng Y, Chang L (2017) A porothermoelastic solution for the inclined borehole in a transversely isotropic medium subjected to thermal osmosis and thermal filtration effects. Geothermics 67:114–134CrossRefGoogle Scholar
  14. Ghassemi A (2012) A review of some rock mechanics issues in geothermal reservoir development. Geotech Geol Eng 30:647–664CrossRefGoogle Scholar
  15. Kanfar MF, Chen Z, Rahman S (2016) Fully coupled 3D anisotropic conductive-convective porothermoelasticity modeling for inclined boreholes. Geothermics 61:135–148CrossRefGoogle Scholar
  16. Kurashige M (1989) A thermoelastic theory of fluid-filled porous materials. Int J Solid Struct 25:1039–1052CrossRefGoogle Scholar
  17. Li X, Cui L, Roegiers J-C (1998) Thermoporoelastic modelling of wellbore stability in non-hydrostatic stress field. Int J Rock Mech Min Sci 35:584CrossRefGoogle Scholar
  18. McClure MW, Horne RN (2014) An investigation of stimulation mechanisms in Enhanced Geothermal Systems. Int J Rock Mech Min Sci 72:242–260.  https://doi.org/10.1016/j.ijrmms.2014.07.011 CrossRefGoogle Scholar
  19. McTigue D (1986) Thermoelastic response of fluid-saturated porous rock. J Geophys Res Solid Earth 91:9533–9542CrossRefGoogle Scholar
  20. McTigue D (1990) Flow to a heated borehole in porous, thermoelastic rock: analysis. Water Resour Res 26:1763–1774CrossRefGoogle Scholar
  21. Morris A, Ferrill DA, Henderson DB (1996) Slip-tendency analysis and fault reactivation. Geology 24:275–278CrossRefGoogle Scholar
  22. Nield DA, Bejan A (2006) Convection in porous media. Springer, BerlinGoogle Scholar
  23. Pine R, Batchelor A (1984) Downward migration of shearing in jointed rock during hydraulic injections. Int J Rock Mech Min Sci Geomech Abstr 21:249–263CrossRefGoogle Scholar
  24. Rajapakse R (1993) Stress analysis of borehole in poroelastic medium. J Eng Mech 119:1205–1227CrossRefGoogle Scholar
  25. Tao Q, Ghassemi A (2010) Poro-thermoelastic borehole stress analysis for determination of the in situ stress and rock strength. Geothermics 39:250–259CrossRefGoogle Scholar
  26. Tester JW et al (2006) The future of geothermal energy: impact of enhanced geothermal systems (EGS) on the United States in the 21st century. Massachusetts Institute of Technology, CambridgeGoogle Scholar
  27. Wang HF (2000) Theory of linear poroelasticity. Princeton University Press, PrincetonGoogle Scholar
  28. Wang Y, Dusseault MB (2003) A coupled conductive–convective thermo-poroelastic solution and implications for wellbore stability. J Pet Sci Eng 38:187–198CrossRefGoogle Scholar
  29. Wang Y, Papamichos E (1994) Conductive heat flow and thermally induced fluid flow around a well bore in a poroelastic medium. Water Resour Res 30:3375–3384CrossRefGoogle Scholar
  30. Xie L, Min K-B (2016) Initiation and propagation of fracture shearing during hydraulic stimulation in enhanced geothermal system. Geothermics 59:107–120CrossRefGoogle Scholar
  31. Zoback M (2010) Resevoir geomechanics. Cambridge University Press, CambridgeGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Desert Research InstituteRenoUSA

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