Hydraulic fracturing to determine $S_3$

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Hydraulic fracture initiation in vertical well

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$$ \sigma_{\theta \theta}^{min} = 3 S_{hmin} - S_{Hmax} - 2 P_p - \Delta P - \sigma^{\Delta T} = -T_0 $$

Leakoff test (mini-frac, FIT, XLOT)

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 7.2, pp. 211)

$S_3$ from instantaneous shut-in pressure

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 7.3, pp. 213)

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 7.3, pp. 213)

Step-rate test

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 7.5, pp. 216)

Be careful!

When $S_3 \sim S_v$ integrate density logs

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 7.4, pp. 214)

What about $S_{Hmax}$?

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$$ \Delta P = P_b - P_p $$

so

$$ S_{Hmax} = 3 S_{hmin} - P_b - P_p + T_0 $$

or

$$ S_{Hmax} = 3 S_{hmin} - P_b(T=0) - P_p $$

Does it work?

Consider a system with compressibility $\beta_s$

$$ \beta_s = \frac{\Delta V_s}{V_s} \frac{1}{\Delta P} $$
$$ \Delta P = \frac{1}{\beta_s V_s} \Delta V_s $$
$$ \frac{\Delta P}{\Delta t} = \frac{1}{\beta_s V_s} \frac{\Delta V_s}{\Delta t} $$

Answer: Not very well.