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

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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.

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Abbreviations

α :

Biot coefficient

α f :

volumetric thermal expansion coefficient of the pore fluid

α s :

Linear thermal expansion coefficient of solid matrix

δ ij :

Kronecker delta

ε ij :

Strain components

ζ :

Increment of fluid content

κ :

Bulk thermal conductivity

μ :

Coefficient of friction

µ f :

Pore fluid viscosity

ν :

Drained Poisson’s ratio

ρ :

Mass density

σ H :

Maximum horizontal principal stress

σ ij :

Stress

σ kk :

Bulk stress

σ r :

Radial stress

σ θ :

Hoop stress

σ rθ :

Shear stress

σ n :

Total normal stress on the preexisting fracture

σ z :

Axial stress

τ :

Shear stress on the preexisting fracture

φ :

Porosity

a :

Wellbore radius

c :

Hydraulic diffusivity

c h :

Bulk thermal diffusivity

h :

Heat flux

k :

Intrinsic permeability

p :

Pore pressure

p 0 :

Hydrostatic pore pressure

p w :

Fluid pressure at the wellbore

q :

Fluid flux

q w :

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

K 0 :

Modified Bessel function of second kind of order zero

K 1 :

Modified Bessel function of second kind of first order

K 2 :

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

S hmin :

Minimum horizontal in situ stress

S Hmax :

Maximum horizontal in situ stress

S V :

Vertical stress

Q :

Volumetric injection rate

T :

Temperature change

Ts:

Slip tendency

T 0 :

Native fluid temperature

T t0 :

Tensile strength

T w :

Injected fluid temperature

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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.

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  1. Desert Research Institute, Reno, NV, 89512, USA

    Zhiqiang Fan & Rishi Parashar

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  1. Zhiqiang Fan
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  2. Rishi Parashar
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Correspondence to Zhiqiang Fan.

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Fan, Z., Parashar, R. Analytical Solutions for a Wellbore Subjected to a Non-isothermal Fluid Flux: Implications for Optimizing Injection Rates, Fracture Reactivation, and EGS Hydraulic Stimulation. Rock Mech Rock Eng 52, 4715–4729 (2019). https://doi.org/10.1007/s00603-019-01867-9

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