Analytical Solution to Study Depletion/Injection Rate on Induced Wellbore Stresses in an Anisotropic Stress Field

Abstract

During production or injection, the state of stresses within the reservoir as well as around the wellbore changes. It is therefore important to evaluate the impact of the induced stresses on stability of wellbores. This study proposes an analytical solution to estimate the influence of production or injection rate on stresses around a wellbore in an anisotropic stress field. For considering the effect of production/injection rate, pseudo steady state flow was assumed within the reservoir that occurs more frequently than transient or steady state flow in a reservoir with an expanding drainage radius. Until now stress equations have not been developed in which the effect of production/injection rate being included. The results showed that the impact of the depletion and injection rate on induced stresses around the wellbore is significant and should be considered in the geomechanical analysis especially in low permeability reservoirs. Application of the proposed solution on a typical sandstone reservoir showed that in relatively high production rate, the radial and tangential stresses near wellbore decrease whereas the vertical and pore pressure increase. An opposite trend was observed for the effect of injection rate on near wellbore stresses. This results are consistent with field observation and means that in rapid production, the vertical stress will increase and horizontal stresses will decrease leading to surface subsidence.

Appendix

The stress Eqs. (7)–(9) around a wellbore with satisfying equilibrium equation, Hooke’s law and boundary conditions proposed by Fjar et al. (2008).

$$\begin{aligned} \sigma_{r} & = a_{1} \left( {1 - \frac{{R_{w}^{2} }}{{r^{2} }}} \right) + a_{2} \left( {1 + 3\frac{{R_{w}^{4} }}{{r^{4} }} - 4\frac{{R_{w}^{2} }}{{r^{2} }}} \right)\cos 2\theta \\ & \quad + \tau_{xy}^{^\circ } \left( {1 + 3\frac{{R_{w}^{4} }}{{r^{4} }} - 4\frac{{R_{w}^{2} }}{{r^{2} }}} \right)\sin 2\theta \\ & \quad + \frac{2\eta }{{r^{2} }}\left( \begin{array}{l} \mathop \int \limits_{{R_{w} }}^{r} r^{\prime}\Delta P_{f} (r^{\prime})dr^{\prime} \hfill \\ - \frac{{r^{2} - R_{w}^{2} }}{{R_{0}^{2} - R_{w}^{2} }}\mathop \int \limits_{{R_{w} }}^{{R_{0} }} r^{\prime}\Delta P_{f} (r^{\prime})dr^{\prime} \hfill \\ \end{array} \right) \\ & \quad + P_{w} \frac{{R_{w}^{2} }}{{r^{2} }}, \\ \end{aligned}$$

(7)

$$\begin{aligned} \sigma_{\theta } & = a_{1} \left( {1 + \frac{{R_{w}^{2} }}{{r^{2} }}} \right) - a_{2} \left( {1 + 3\frac{{R_{w}^{4} }}{{r^{4} }}} \right)\cos 2\theta \\ & \quad - \tau_{xy}^{^\circ } \left( {1 + 3\frac{{R_{w}^{4} }}{{r^{4} }}} \right)\sin 2\theta \\ & \quad - \frac{2\eta }{{r^{2} }}\left( \begin{array}{l} \mathop \int \limits_{{R_{w} }}^{r} r^{\prime}\Delta P_{f} (r^{\prime})dr^{\prime} - r^{2} \Delta P_{f} \left( r \right) \hfill \\ + \frac{{r^{2} + R_{w}^{2} }}{{R_{0}^{2} - R_{w}^{2} }}\mathop \int \limits_{{R_{w} }}^{{R_{0} }} r^{\prime}\Delta P_{f} (r^{\prime})dr^{\prime} \hfill \\ \end{array} \right) \\ & \quad - P_{w} \frac{{R_{w}^{2} }}{{r^{2} }}, \\ \end{aligned}$$

(8)

$$\begin{aligned} \sigma_{z} & = \sigma_{z}^{^\circ } - 4\nu \left( {a_{2} \frac{{R_{w}^{2} }}{{r^{2} }}\cos 2\theta + \tau_{xy}^{^\circ } \frac{{R_{w}^{2} }}{{r^{2} }}\sin 2\theta } \right) \\ & \quad + 2\eta \, \Delta P_{f} \left( r \right) - 4\eta \frac{\nu }{{R_{0}^{2} - R_{w}^{2} }}\mathop \int \limits_{{R_{w} }}^{{R_{0} }} r^{\prime}\Delta P_{f} (r^{\prime})dr^{\prime} \\ \end{aligned}$$

(9)

In the Eq. (10), the variable used in the Eqs. (7)–(9) are defined.

$$\begin{aligned} & a_{1} = \left( {\frac{{\sigma_{x}^{^\circ } + \sigma_{y}^{^\circ } }}{2}} \right), \quad a_{2} = \left( {\frac{{\sigma_{x}^{^\circ } - \sigma_{y}^{^\circ } }}{2}} \right), \quad \eta = \frac{1 - 2v}{2(1 - v)}\alpha \\ & \Delta P_{f} (r) = P_{f} (r) - P_{{f_{0} }} , \\ & \sigma_{x}^{^\circ } = I_{{xx^{\prime}}}^{2} \sigma_{H} + I_{{xy^{\prime}}}^{2} \sigma_{h} + I_{{xz^{\prime}}}^{2} \sigma_{v} , \\ & \sigma_{y}^{^\circ } = I_{{yx^{\prime}}}^{2} \sigma_{H} + I_{{yy^{\prime}}}^{2} \sigma_{h} + I_{{yz^{\prime}}}^{2} \sigma_{v} , \\ & \sigma_{z}^{^\circ } = I_{{zx^{\prime}}}^{2} \sigma_{H} + I_{{zy^{\prime}}}^{2} \sigma_{h} + I_{{zz^{\prime}}}^{2} \sigma_{v} , \\ & \tau_{xy}^{^\circ } = I_{{xx^{\prime}}} I_{{yx^{\prime}}} \sigma_{H} + I_{{xy^{\prime}}} I_{{yy^{\prime}}} \times \sigma_{h} + I_{{xz^{\prime}}} I_{{yz^{\prime}}} \sigma_{v} , \\ & {\text{where}} \\ & I_{{xx^{\prime}}} = \cos a \times \cos i; \quad I_{{yx^{\prime}}} = - \sin a;\quad I_{{zx^{\prime}}} = \cos a \times \sin i, \\ & I_{{xy^{\prime}}} = \sin a \times \cos i; \quad I_{{yy^{\prime}}} = \cos a; \quad I_{{zy^{\prime}}} = \sin a \times \sin i, \\ & I_{{xz^{\prime}}} = - \sin i; \quad I_{{yz^{\prime}}} = 0;\quad I_{{zz^{\prime}}} = \cos i, \\ \end{aligned}$$

(10)

In Eq. (10); σ_H is the maximum horizontal stress, σ_h the minimum horizontal stress, R_w the radius of the well, R₀ the drainage radius, P_f the pore pressure in the distance r from the wellbore center, P _f0 the reservoir pressure, $\vartheta$ the Poisson’s ratio and α is the Biot coefficient. According to Fig. 5, the angles $a$ and i are used for transformation of stresses from the vertical wellbore to deviated wellbore (Heidarian et al. 2014; Al-Shaaibi et al. 2013).

Fig. 5

(reproduced with permission from Al-Shaaibi et al. 2013)

The angles a and i for transformation of stresses from the vertical well to deviated well used in the Eqs. 8–10

Assuming that pore pressure changes are constant with radius and time, ($\frac{\partial p}{\partial r} = 0$,$\frac{\partial p}{\partial t} = 0$) solution of the Eqs. (7–10) was presented in Fig. 6 (Fjar et al. 2008).

Fig. 6

(reproduced with permission from Fjar et al. 2008)

Analytical solution of the Eqs. 7–10 for P_f = constant (elastic stress solution)