6.9 Problems

1. Using equations of stresses around a cylindrical cavity, calculate near-wellbore effective radial $\sigma _{rr}$ and hoop $\sigma _{\theta \theta }$ stresses for a vertical well 8 in diameter in the directions of $S_{hmin}$ (4,500 psi – acting E-W) and $S_{Hmax}$ (6,000 psi) up to 3 ft of distance considering that $P_p$ = 320 0psi and
   1. $P_W$ = 3,200 psi
   2. $P_W$ = 4,000 psi
   The result should be presented as plots of stresses ( $\sigma _{rr}$, $\sigma _{\theta \theta }$) for $\theta = 0$ and $\theta=90^{\circ}$ as a function of distance ($r \geq a$) from the center of the wellbore.

2. Effect of overpressure: Consider the problem solved in class (Wellbore: vertical; Site: onshore, 7,000 ft of depth, $S_v$ = 7,000 psi, $S_{hmin}$ = 4,300 psi, $S_{Hmax}$ = 6,300 psi; Rock properties: $UCS$ = 3,500 psi, $\mu = 0.6$, $T_s$ = 800 psi).
   1. Calculate wellbore pressure and corresponding mud weight for (i) $w_{BO}=70^{\circ}$, (ii) $w_{BO} \sim 0^{\circ}$ ( $P_{Wshear}$), and (iii) for inducing tensile fractures ($P_b$) for $\lambda_p= 0.52$ and $\lambda_p= 0.60$. Compare with $\lambda_p= 0.44$ solved in class. How does the drilling mud window change with varying pore pressure?
   2. Assume horizontal stress directions near Dallas-Forth Worth region. What would the azimuth of breakouts and drilling induced fractures be? https://www.nature.com/articles/s41467-020-15841-5/figures/1

3. Effect of stress anisotropy (differential stress) $(\sigma_{Hmax} - \sigma_{hmin})$: Consider the following problem, Wellbore: vertical; Site: onshore, 2 km of depth, $dS_v/dz = 23$ MPa/km, $\lambda_p= 0.44$, $\sigma_{hmin} = 0.4 \: \sigma_v$; Rock properties: $UCS$ = 7 MPa, $q=3.9$, $T_s$ = 2 MPa. Calculate wellbore pressure and corresponding mud weight for (i) $w_{BO}=45^{\circ}$, (ii) $w_{BO} \sim 0^{\circ}$, and (iii) for inducing tensile fractures for
   1. $\sigma_{Hmax} = 0.6 \: \sigma_v$
   2. $\sigma_{Hmax} = 0.8 \: \sigma_v$
   3. $\sigma_{Hmax} = 1.0 \: \sigma_v$
   4. How does the drilling mud window change with $(\sigma_{Hmax} - \sigma_{hmin})$?

4. Offshore: Consider an offshore vertical wellbore being drilled at 2 km of total vertical depth, with 500 m of water, hydrostatic pore pressure, $\sigma_{hmin} = 0.4 \: \sigma_v$, $\sigma_{Hmax} = 0.8 \: \sigma_v$. The rock properties are $UCS$ = 7 MPa, $q=3.9$, $T_s$ = 2 MPa. Calculate wellbore pressure and corresponding mud weight for (i) $w_{BO}=45^{\circ}$, (ii) $w_{BO} \sim 0^{\circ}$, and (iii) for inducing tensile fractures.

5. Horizontal wells: Evaluate wellbore stability for horizontal wells that you will need to exploit in a gas reservoir subjected to a strike-slip stress environment.
   1. Consider two wellbores: one drilled parallel to $S_{hmin}$ and another drilled parallel to $S_{Hmax}$. Draw cross-sections of these two wells, identify involved stresses, and clearly mark expected positions of tensile fractures and wellbore breakouts.
   2. The horizontal wells lie at about 8,000 ft depth where it is estimated that $S_{hmin}$ = 50 MPa, $S_{Hmax}$ = 70 MPa, $dS_v/dz = 1$ psi/ft and $\lambda_p=0.6$. The unconfined compressive strength of the rock is 8,500 psi, the rock internal friction coefficient is $\mu_i=1.0$, and tensile strength is about $T_s = 0$ psi given the large density of natural fractures. Determine the mechanical stability limits on wellbore pressure for both horizontal well directions considered. Assume a safe breakout angle to calculate the lower bound for the mud window.
   3. Determine mud density window appropriate for these wells (keep in mind potential lost circulation).
   4. Which one appears to have a wider mud window? Justify