Stability analysis of a super deep petroleum well drilled in strike-slip fault zones in the Tarim Basin, NW China

Abstract

The deeply buried petroleum reservoirs are usually associated with fault zones resulting from substantial tectonic activities. Thus, the issue of wellbore stability is particularly important since these natural fractures are quite abundant in fault zones. In this study, we focus on the wellbore stability of a deeply buried petroleum well located in the Tarim area of China. The well was drilled into an Ordovician limestone formation with a buried depth of 8000 m. Laboratory tests were conducted on rock samples to characterize the mineral compositions, micro-structures, and strength properties. Dual-porosity theories of poromechanics were employed to derive stress and pore pressure distributions in the limestone formation surrounding the wellbore. The risk of wellbore instability was analyzed accordingly. Our results show that stress distribution is susceptible to borehole azimuth. In addition, effective stresses in the rock matrix and fractures surrounding the borehole were derived and analyzed separately, where two failure criteria were applied to rock matrix and fracture, respectively. Given the in situ stress conditions, the importance of selecting the optimum well trajectory is highlighted. Time-dependent solutions were used to show the importance of including fracture strength in stability analysis of wellbores drilled in fractured porous media.

Appendices

Appendix A

The solution of ${\sigma}_{\mathrm{r}\mathrm{r}}^{(1)}-{\sigma}_{\mathrm{r}\mathrm{r}}^{(2)}-{\sigma}_{\mathrm{r}\mathrm{r}}^{(3)},{\sigma}_{\uptheta \uptheta}^{(1)}-{\sigma}_{\uptheta \uptheta}^{(2)}-{\sigma}_{\uptheta \uptheta}^{(3)},{\sigma}_{\mathrm{r}\theta}^{(3)},{\mathrm{P}}^{\mathrm{I}(2)},{\mathrm{P}}^{\mathrm{I}(3)},{\mathrm{P}}^{\mathrm{I}\mathrm{I}(2)}$ and P^II(3) in the Laplace domain are given as (Abousleiman and Nguyen 2005):

$$ {\overline{\sigma}}_{rr}^{(1)}=\left({\sigma}_m-{p}_w\right)\frac{R^2}{r^2}H(t) $$

(A-1)

$$ {\overline{\sigma}}_{\theta \theta}^{(1)}=-\left({\sigma}_m-{p}_w\right)\frac{R^2}{r^2}H(t) $$

(A-2)

$$ {\overline{\mathrm{P}}}^{\mathrm{I}(2)}=-\frac{1}{s}\left(\frac{p_0-{p}_w}{m^I-{m}^{II}}\right)\times \left\{\begin{array}{c}\left(1-{m}^{II}\right)\kern0.5em \Phi \left[{\xi}^{\mathrm{I}}\right]\\ {}\begin{array}{cc}-\left(1-{m}^I\right)& \Phi \left[{\xi}^{\mathrm{I}\mathrm{I}}\right]\end{array}\end{array}\right\} $$

(A-3)

$$ {\overline{\mathrm{P}}}^{\mathrm{I}\mathrm{I}(2)}=-\frac{1}{s}\left(\frac{p_0-{p}_w}{m^I-{m}^{II}}\right)\times \left\{\begin{array}{c}{m}^I\left(1-{m}^{II}\right)\kern0.5em \Phi \left[{\xi}^{\mathrm{I}}\right]\\ {}\begin{array}{cc}-{m}^{II}\left(1-{m}^I\right)& \Phi \left[{\xi}^{\mathrm{I}\mathrm{I}}\right]\end{array}\end{array}\right\} $$

(A-4)

$$ {\overline{\sigma}}_{rr}^{(2)}=-\frac{1\ }{s}\frac{2G}{K_v}\left(\frac{p_o-{p}_w}{m^I-{m}^{II}}\right)\times \left\{\begin{array}{c}{h}^I\left(1-{m}^{II}\right)\Lambda \left[{\xi}^{\mathrm{I}}\right]\\ {}-{h}^{II}\left(1-{m}^I\right)\Lambda \left[{\xi}^{\mathrm{I}\mathrm{I}}\right]\end{array}\right\} $$

(A-5)

$$ {\overline{\sigma}}_{\theta \theta}^{(2)}=\frac{1}{s}\frac{2G}{K_v}\left(\frac{p_o-{p}_w}{m^I-{m}^{II}}\right)\times \left\{\begin{array}{c}{h}^I\left(1-{m}^{II}\right)\left(\Phi \left[{\xi}^{\mathrm{I}}\right]+\Lambda \left[{\xi}^{\mathrm{I}}\right]\right)\\ {}-{h}^{II}\left(1-{m}^I\right)\left(\Phi \left[{\xi}^{\mathrm{I}\mathrm{I}}\right]+\Lambda \left[{\xi}^{\mathrm{I}\mathrm{I}}\right]\right)\end{array}\right\} $$

(A-6)

$$ {\overline{\mathrm{P}}}^{\mathrm{I}(3)}=\frac{\sigma_d}{2s}{K}_v\mathit{\cos}\left[2\left(\theta -{\theta}_r\right)\right]\times \left\{{C}_1{K}_2\left[{\upxi}^{\mathrm{I}}\mathrm{r}\right]-{C}_2{K}_2\left[{\upxi}^{\mathrm{I}\mathrm{I}}\mathrm{r}\right]+{f}^1{C}_3\frac{R^2}{r^2}\right\} $$

(A-7)

$$ {\displaystyle \begin{array}{c}{\overline{\mathrm{P}}}^{\mathrm{I}\mathrm{I}(3)}=\frac{\sigma_d}{2s}{K}_v\cos \left[2\left(\theta -{\theta}_r\right)\right]\\ {}\times \left\{{m}^I{C}_1{K}_2\left[{\upxi}^{\mathrm{I}}\mathrm{r}\right]-{m}^{II}{C}_2{K}_2\left[{\upxi}^{\mathrm{I}\mathrm{I}}\mathrm{r}\right]+{f}^{II}{C}_3\frac{R^2}{r^2}\right\}\end{array}} $$

(A-8)

$$ {\overline{\sigma}}_{rr}^{(3)}=\frac{\sigma_d}{s}\cos \left[2\left(\theta -{\theta}_r\right)\right]\times \left\{\begin{array}{c}G\left({h}^I{C}_1\Theta \left[{\upxi}^{\mathrm{I}}\right]-{h}^{II}{C}_2\Theta \left[{\upxi}^{\mathrm{I}\mathrm{I}}\right]\right)\\ {}-\left( Gh+{K}_v\right){C}_3\frac{R^2}{r^2}-3{C}_4\frac{R^4}{r^4}\end{array}\right\} $$

(A-9)

$$ {\overline{\sigma}}_{\theta \theta}^{(3)}=-\frac{\sigma_d}{s}\cos \left[2\left(\theta -{\theta}_r\right)\right]\times \left\{G\left({h}^I{C}_1\Omega \left[{\upxi}^{\mathrm{I}}\right]-{h}^{II}{C}_2\Omega \left[{\upxi}^{\mathrm{I}\mathrm{I}}\right]\right)-3{C}_4\frac{R^4}{r^4}\right\} $$

(A-10)

$$ {\overline{\sigma}}_{r\theta}^{(3)}=\frac{\sigma_d}{s}\sin \left[2\left(\theta -{\theta}_r\right)\right]\times \left\{\begin{array}{c}2G\left({h}^I{C}_1\uppsi \right)\left[{\upxi}^{\mathrm{I}}\right]-{h}^{II}{C}_2\uppsi \left[{\upxi}^{\mathrm{I}\mathrm{I}}\right]\\ {}-\frac{\left( Gh+{K}_v\right)}{2}{C}_3\frac{R^2}{r^2}-3{C}_4\frac{R^4}{r^4}\end{array}\right\} $$

(A-11)

Where s is the Laplace parameter, $\left(\overline{\mathrm{P}}\right)$ denotes the Laplace transformation and K_n is the modified Bessel function of the second kind of nth order. For brevity, remaining expressions and coefficients are shown in

Appendix B

$$ \frac{1}{K}=\frac{1}{K^I}+\frac{1}{K^{II}} $$

(B-1)

$$ \frac{1}{G}=\frac{1}{G^I}+\frac{1}{G^{II}} $$

(B-2)

$$ {K}_v=K+\frac{4}{3}G $$

(B-3)

$$ {\beta}^{(N)}=\frac{\alpha^{(N)}{K}^{(N)}}{K}\ N=I, II $$

(B-4)

$$ \Phi \left[\xi \right]=\frac{K_0\left[\upxi \mathrm{r}\right]}{K_0\left[\upxi \mathrm{R}\right]} $$

(B-5)

$$ \Lambda \left[\xi \right]=\frac{K_1\left[\upxi \mathrm{r}\right]}{{\upxi \mathrm{r}K}_0\left[\upxi \mathrm{R}\right]}-\frac{RK_1\left[\upxi \mathrm{R}\right]}{{\xi r}^2{K}_0\left[\upxi \mathrm{R}\right]} $$

(B-6)

$$ \Theta \left[\upxi \right]=\frac{1}{\xi r}{K}_1\left[\upxi \mathrm{r}\right]+\frac{6}{{\left(\upxi \mathrm{r}\right)}^2}{K}_2\left[\upxi \mathrm{r}\right] $$

(B-7)

$$ \Omega \left[\upxi \right]=\frac{1}{\xi r}{K}_1\left[\upxi \mathrm{r}\right]+\left(1+\frac{6}{{\left(\upxi \mathrm{r}\right)}^2}\right){K}_2\left[\upxi \mathrm{r}\right] $$

(B-8)

$$ \uppsi \left[\upxi \right]=\frac{1}{\xi r}{K}_1\left[\upxi \mathrm{r}\right]+\frac{3}{{\left(\upxi \mathrm{r}\right)}^2}{K}_2\left[\upxi \mathrm{r}\right] $$

(B-9)

ξ^Iand ξ^IIare two positive roots of the following characteristic equation

$$ \left({\upxi}^2-{M}_{11}\right)\left({\kappa}_r{\upxi}^2-{M}_{22}\right)-{M}_{12}{M}_{21}=0 $$

(B-10)

κ_r is mobility ratio which can be found by fluid mobility κ as

$$ {\kappa}_r=\frac{\kappa^{II}}{\kappa^I} and{\kappa}^{(N)}=\frac{k^{(N)}}{\mu^{(N)}},N=I, II $$

(B-11)

in which k^I and k^II are the permeability of matrix and fracture respectively.

M₁₁, M_12,M₂₁ and M_12, are lumped coefficients defined as

$$ {M}_{11}=\frac{s}{\kappa^I}\left(\frac{1}{M^I}+\frac{{\left({\beta}^I\right)}^2}{K_v}\right)+\frac{\Gamma}{\kappa^I} $$

(B-12)

$$ {M}_{12}={M}_{21}=\frac{s}{\kappa^I}\left(\frac{{\left({\beta}^I{\beta}^{II}\right)}^2}{K_v}\right)-\frac{\Gamma}{\kappa^I} $$

(B-13)

$$ {M}_{22}=\frac{s}{\kappa^I}\left(\frac{1}{M^{II}}+\frac{{\left({\beta}^{II}\right)}^2}{K_v}\right)+\frac{\Gamma}{\kappa^I} $$

(B-14)

in which the material coefficients can be founded by

$$ \frac{1}{{\mathrm{M}}^{\mathrm{I}}}=\left(\frac{\alpha^{\mathrm{I}}-{\phi}^{\mathrm{I}}}{{\mathrm{K}}_{\mathrm{s}}}+\frac{\phi^{\mathrm{I}}}{{\mathrm{K}}_{\mathrm{f}}}\right) $$

(B-15)

$$ \frac{1}{{\mathrm{M}}^{\mathrm{II}}}=\left(\frac{\alpha^{\mathrm{II}}-{\phi}^{\mathrm{II}}}{{\mathrm{K}}_{\mathrm{s}}}+\frac{\phi^{\mathrm{II}}}{{\mathrm{K}}_{\mathrm{f}}}\right) $$

(B-16)

$$ {h}^{(N)}={\beta}^I+{\beta}^{II}{m}^{(N)},N=I, II $$

(B-17)

$$ h={\beta}^I{f}^I+{\beta}^{II}{f}^{II}-1 $$

(B-18)

$$ {m}^{(N)}=\frac{\left[{\left({\upxi}^{(N)}\right)}^2-{M}_{11}\right]}{M_{12}},N=I, II $$

(B-19)

$$ \left\{\begin{array}{c}{f}^I\\ {}{f}^{II}\end{array}\right\}=\frac{s}{\kappa^I{K}_v}{\left[\begin{array}{c}{M}_{11}\kern0.5em {M}_{12}\\ {}\begin{array}{cc}{M}_{21}& {M}_{22}\end{array}\end{array}\right]}^{-1}\left\{\begin{array}{c}{\beta}^I\\ {}{\beta}^{II}\end{array}\right\} $$

(B-20)

$$ {C}_1=4{\upxi}^{\mathrm{I}}\mathrm{R}\frac{D_2}{D_4},\kern0.5em {C}_2=4{\upxi}^{\mathrm{I}\mathrm{I}}\mathrm{R}\frac{D_1}{D_4},\kern0.5em {C}_3=4\frac{D_3}{D_4} $$

(B-21)

$$ {C}_4=\frac{1}{D_4}\left[\begin{array}{c}2{Gh}^{II}{D}_1\left({K}_1\left[{\upxi}^{\mathrm{I}\mathrm{I}}R\right]+\frac{4}{\upxi^{\mathrm{I}\mathrm{I}}R}{K}_2\left[{\upxi}^{\mathrm{I}\mathrm{I}}R\right]\right)\\ {}-2{Gh}^I{D}_2\left({K}_1\left[{\upxi}^{\mathrm{I}}R\right]+\frac{4}{\upxi^{\mathrm{I}}R}{K}_2\left[{\upxi}^{\mathrm{I}}R\right]\right)-\left( Gh+{K}_v\right){D}_3\end{array}\right] $$

(B-22)

where

$$ {D}_1={\upxi}^{\mathrm{I}}\left({f}^{II}-{f}^I{m}^I\right){K}_2\left[{\upxi}^{\mathrm{I}}R\right] $$

(B-23)

$$ {D}_2={\upxi}^{\mathrm{II}}\left({f}^{II}-{f}^I{m}^{II}\right){K}_2\left[{\upxi}^{\mathrm{II}}R\right] $$

(B-24)

$$ {D}_3={\upxi}^{\mathrm{I}}{\upxi}^{\mathrm{I}\mathrm{I}}\mathrm{R}\left({m}^I-{m}^{II}\right){K}_2\left[{\upxi}^{\mathrm{I}}R\right]{K}_2\left[{\upxi}^{\mathrm{I}\mathrm{I}}R\right] $$

(B-25)

$$ {D}_4=2\mathrm{G}\left({h}^{II}{D}_1{\mathrm{K}}_1\left[{\upxi}^{\mathrm{I}\mathrm{I}}R\right]-{h}^I{D}_2{\mathrm{K}}_1\left[{\upxi}^{\mathrm{I}}R\right]\right)+\left( Gh+{K}_v\right){D}_3 $$

(B-26)