Page 212 - Reservoir Geomechanics

193 Compressive and tensile failures in vertical wells

mud is cooler than the formation (the usual case at the bit) thermal stresses make the
stress concentration around a well more tensile at all azimuths in the same manner as
increasing mud pressure.
The effect of temperature is time-dependent, in the sense that the longer the rock
is in contact with the wellbore fluid the further away from the hole the temper-
ature perturbation will propagate. To simplify this problem, one can assume that
the material is impermeable, and relatively simple integral equations can be writ-
ten for the magnitudes of σ θθ and σ rr as a function of radial position r and time t
(Stephens and Voight 1982). Although the exact solution for the temperature distribu-
tion near a constant-temperature wellbore is a series expansion (Ritchie and Sakakura
1956), solutions which approximate the temperature using the first two terms of the
expansion give sufficiently accurate results close to the hole, where the stresses are
given by:

α t E
T 1 1 −1 1 1

σ θθ = − − ln ρ I 0 − + (6.20)
1 − ν 2ρ 2 2 2ρ
α t E
T 1 1 −1 1 1

σ rr = − + − lnρ I 0 − − (6.21)
1 − ν 2ρ 2 2 2ρ
0+ z
1 e [4τ z /σ 2 ]
I −1 = dz
0
2πi zlnz
−∞
Once steady state has been reached, the change in the hoop stress is given by

α t E
T

T
σ θθ = (6.22)
1 − ν
where α t is the linear coefficient of thermal expansion and E is the static Young’s
modulus. Figure 3.14 shows the coefficient of thermal expansion for different rock
types. As shown, α t is a strong function of the silica content because the coefficient of
thermal expansion of quartz is an order of magnitude higher than other common rock
forming minerals.
◦
Figure 6.13b incorporates the effect of wellbore cooling of 25 Con the formation
of drilling-induced tensile fractures described by equation (6.17). As seen through
←
Figure 6.13. Polygons showing the possible values of S hmin and S Hmax at a given depth and pore
pressure that are constructed in the manner of Figure 4.28.A coefficient of friction of 0.6 for faults
in the crust is assumed. In addition, the equation describing the magnitude S Hmax ,asa function of
S hmin that is required to cause drilling-induced tensile fractures in a vertical well. (a) No cooling
stress and no excess pore pressure are considered. (b) When the mud is 25 cooler than the
◦
formation, drilling-induced tensile fractures can be induced at a slightly lower value of S Hmax for a
given value of S hmin because the thermal stress slightly decreases the σ θθ . (c) When there is 6 MPa
of excess mud weight, tensile wall fractures occur at still lower values of S Hmax .