Reservoir Depletion

Effects of reservior depletion

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 10.21, pp. 380)

Estimating stress changes in depleting reserviors

$${}$$$$ S_{Hor} = S_{Hmax} = S_{hmin} = \frac{\nu}{1-\nu} S_v + \alpha P_p \left(1 - \frac{\nu}{1-\nu}\right) $$

$$ \frac{{\rm d}S_{Hor}}{{\rm d}P_p} = \alpha \frac{1 - 2 \nu}{\nu - 1} \qquad \mbox{during production} $$

$$ \Delta S_{Hor} = \alpha \frac{1 - 2 \nu}{\nu - 1} \Delta P_p $$

Taking $\nu = \frac{1}{4}$ and $\alpha = 1$

$$ \Delta S_{Hor} \sim \frac{2}{3} \Delta P_p $$

Comparison of theory and observation

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.2, pp. 382)

Production induced faulting

$${}$$$$ \frac{S_v - (Pp - \Delta P_p)}{(S_{hmin}-\Delta S_{hmin}) - (Pp - \Delta P_p)} = (\sqrt{\mu^2+1}+\mu)^2 $$

Simplification leads to

$$ \frac{\Delta S_{Hmin}}{\Delta P_p} = 1 - \frac{1}{(\sqrt{\mu^2+1}+\mu)^2} $$

For $\mu = 0.6$

$$ \frac{\Delta S_{Hmin}}{\Delta P_p} = 0.67 $$

Reservoir space plot

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.4a, pp. 386)

GOM Field X

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.3ab, pp. 383)

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.4b, pp. 387)

Valhall field in North Sea

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.5, pp. 388)

Stress rotations with depletion

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Original	Depleted

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.6ab, pp. 391)

Rotation angle, $\gamma$ near the fault due to depletion

$${}$$$$ \gamma = \frac{1}{2} \tan^{-1}\left( \frac{A q \sin(2 \theta)}{1+A q \cos(2 \theta)} \right) $$

with

$$ A = \frac{\Delta S_{hmin}}{\Delta P_p} $$

and

$$ q = \frac{\Delta P_p}{S_{Hmax} - S_{hmin}} $$

Deformation in depleting reservoirs

Subsidence

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Source: Wikimedia Commons -- Public Domain

Recall: Compaction curves

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.11, pp. 399)

Recall: End-cap models

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 4.19, pp. 119)

$$ p = \frac{1}{3}(S_1 + S_2 + S_3) - P_p $$$${}$$$$ q^2 = \frac{1}{2}\left( (S_1 - S_2)^2 + (S_2-S_3)^2 + (S_1-S_3)^2) \right) $$$${}$$$$ M^2 p^2 - M^2 p_0 p + q^2 = 0 $$

\begin{align} 9 P_p^2 &+ \left(1+ \frac{9}{M^2}\right)(S_v^2+S_{Hmax}^2+S_{hmin}^2) \\ &+ \left(2 - \frac{9}{M^2}\right)(S_v S_{Hmax} + S_v S_{hmin} + S_{Hmax} S_{hmin} ) \\ &+ 9 P_p p_0 - 3(2 P_p+p_0)(S_v + S_{Hmax} + S_{hmin}) = 0 \end{align}$${}$$$$ M = \frac{6 \mu}{3 \sqrt{\mu^2+1}-\mu} \qquad \mbox{From Mohr-Coulomb assuming $C_0$ is negligible} $$

Deformation Analysis in Reservoir Space (DARS) model of GOM Field X

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.12, pp. 402)

Permeability change as a function of porosity change

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.13a, pp. 404)

Modified Kozeny-Carman relationship

$$ \kappa = B \frac{(\phi - \phi_c)^3}{(1+\phi_c-\phi)^2} d^2 $$

For change in permeability

$$ \frac{\kappa}{\kappa_i} = \left( \frac{\phi - \phi_c}{\phi_i - \phi_c} \right)^3 \left( \frac{1+ \phi_c - \phi_i}{1 + \phi_c - \phi} \right)^2 $$

Including a factor for grain size reduction

$$\Gamma = \frac{1-d/d_i}{1-\phi/\phi_i}$$$$ \frac{\kappa}{\kappa_i} = \left( \frac{\phi - \phi_c}{\phi_i - \phi_c} \right)^3 \left( \frac{1+ \phi_c - \phi_i}{1 + \phi_c - \phi} \right)^2 \left(1-\Gamma\left(1-\frac{\phi}{\phi_i}\right) \right) $$

Permeability loss with depletion in GOM Field Z

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.14, pp. 408)

Idealized resevoir study

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.15, pp. 410)

Viscous compaction in GOM Field Z

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.16, pp. 411)

Geertsma (1973) dispacement solution

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$$ \begin{align} u_z(r,0) &= -\frac{1}{\pi} c_m(1-\nu) \frac{D}{\left(r^2 + D^2\right)^{3/2}} \Delta P_p V \\ u_r(r,0) &= +\frac{1}{\pi} c_m(1-\nu) \frac{D}{\left(r^2 + D^2\right)^{3/2}} \Delta P_p V \end{align} $$

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.17, pp. 414)

Subsidence and horizontal dispacement

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.17, pp. 414)

Leeville case study

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© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 12.18, pp. 416)