Stress and pressure fields around two wellbores in a poroelastic medium

Abstract

The problem of two circular wellbores of different size in a poroelastic medium is considered in the present work. The constitutive behaviour of the poroelastic medium is assumed to comply with the classical Biot model for isotropic porous materials infiltrated by compressible fluid. The wellbores are assumed infinitely long and the fluid flow is taken stationary, thus making it possible to perform a plane strain analysis. Owing to the geometrical layout of the system, bipolar cylindrical coordinates have been adopted. Three different sets of BCs on the pressure field and on the fluid flux have been considered, founding the corresponding forms of the pressure field. Based on Helmholtz representation, a displacement potential has been introduced, and the corresponding stress field in the poroelastic medium has been assessed. However, such a solution does not satisfy the BCs at the edges of the wells. Then, an auxiliary stress function, which allows accomplishing the BCs, is introduced, leading to the complete solution of the problem. The cases of two coaxial wellbores (eccentric annulus), a single hole bored in a poroelastic half plane and two intersecting holes have been considered also. The proposed approach allows evaluating the pore pressure and the stress and strain fields in the system varying the amplitude of the wells and the physical parameters of the porous material. In particular, the evaluation of the peak values of the stress components around the circular boreholes plays a key role in a variety of engineering contexts, with particular reference to the stability analysis of wellbores and tunnels and failure of vascular vessels in biological tissues.

Appendix

Some differential operators expressed in bipolar coordinates in two dimensions are listed below:

Laplacian operator of a scalar function f(α, β):

$$\nabla^{2} f(\alpha ,\beta ) = \frac{{\left( {{ \cosh}\alpha - \cos\beta } \right)^{2} }}{{a^{2} }}\left( {\frac{{\partial^{2} f}}{{\partial \alpha^{2} }} + \frac{{\partial^{2} f}}{{\partial \beta^{2} }}} \right);$$

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gradient of a scalar function f(α, β):

$${\mathbf{\nabla }}f(\alpha ,\beta ) = \frac{{\left( {{ \cosh}\alpha - \cos\beta } \right)}}{a}\left( {\frac{\partial f}{\partial \alpha }{\hat{\mathbf{\alpha }}} + \frac{\partial f}{\partial \beta }{\hat{\mathbf{\beta }}}} \right);$$

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divergence of a vector field f(α, β) = $f_{\alpha } {\hat{\mathbf{\alpha }}} + f_{\beta } {\hat{\mathbf{\beta }}}$:

$$\nabla \cdot \varvec{f}(\alpha ,\beta ) = \frac{{\left( {{ \cosh}\alpha - \cos\beta } \right)}}{a}\left( {\frac{{\partial \varvec{f}}}{\partial \alpha } + \frac{{\partial \varvec{f}}}{\partial \beta } - \frac{{{ \sinh}\alpha \, \varvec{f}_{\alpha } + { \sin}\beta \, \varvec{f}_{\beta } }}{{\left( {{ \cosh}\alpha - \cos\beta } \right)}}} \right).$$

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The following integrals reported in [31] have been used:

$$\int\limits_{ - \pi }^{\pi } {\frac{{\cos \,n\upbeta\;d\upbeta}}{{\cosh \,\upalpha - \cos\upbeta}}} = \frac{{2\uppi}}{{\sinh\left|\upalpha \right|}} e^{{ - \left| {n\upalpha} \right|}} ,\int_{ - \pi }^{\pi } {\frac{{\cos (m\upbeta)}}{{\left( {\cosh\upalpha - \cos\upbeta} \right)^{2} }}} \, {d\beta } = \frac{{2\uppi}}{{\sinh^{2}\upalpha}}e^{{ - m\left|\upalpha \right|}} (m + \coth \left|\upalpha \right|).$$

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Identity (A4)₁ allows to obtain the following Fourier series expansions [16]:

$$\frac{{\sinh\alpha \sin^{2} \beta }}{\cosh\alpha - \cos\beta } = e^{ - \left| \alpha \right|} \sinh\alpha + \sum\limits_{n = 1}^{\infty } {S_{n} (\alpha )} \;\cos n\beta ,\quad \frac{1 - \cosh\alpha \,\cos\beta }{\cosh\alpha - \cos\beta }\sin\beta = \sum\limits_{n = 1}^{\infty } {T_{n} (\alpha )} \;\sin n\beta ,$$

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being

$$S_{1} (\upalpha) = e^{{ - 2\left|\upalpha \right|}} \sinh\upalpha,\quad S_{n} (\upalpha) = - 2e^{{ - n\left|\upalpha \right|}} \sinh\left|\upalpha \right|\sinh{\alpha ,}\quad (n \ge 2)$$

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$$T_{1} (\upalpha) = e^{{ - \left|\upalpha \right|}} \left( {1 + e^{{ - \left|\upalpha \right|}} \sinh\left|\upalpha \right|} \right),\quad T_{n} (\upalpha) = - 2e^{{ - n\left|\upalpha \right|}} \sinh^{2}\upalpha\quad (n \ge 2).$$

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It can be useful to expand the term ln(coshα − cosβ) in Fourier series as follows [32]:

$$\ln (\cosh\alpha - \cos\beta ) = \left| \alpha \right| \, - \, \ln \, 2 \, - 2 \, \sum\limits_{n = 1}^{\infty } {\frac{{e^{ - n\left| \alpha \right|} }}{n}} \, \cos (n \, \beta ).$$

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