5.4 WP7: Rock Failure and Application to Wellbore Stability

5.4.1 Exercise 1: Determination of shear and tensile strength

The following data set has experimental data of shear strength for the Darley Dale Sandstone measured in an axisymmetric triaxial cell.

Figure 5.1: Peak stress measured from triaxial tests (image source: Jaeger et al. 2007 - FRM 4th Ed).

where $S_1$ is the maximum total principal stress, $S_3$ is the minimum total principal stress, and $P_{pore \: fluid}$ is the pore pressure in the rock.

a.

Determine the best fitting parameters of a Coulomb criterion: $\sigma_1 = UCS + q \sigma_3$.

b.

Determine the corresponding values of cohesive strength $S_0$ and friction angle $\varphi$ from point (1) and draw the shear failure line together with Mohr circles at failure.

c.

Determine the corresponding fitting parameters for a Drucker-Prager criterion and plot experimental data in a $I_1$ v.s. $(J_2)^{1/2}$ space.

d.

Compute the corresponding tensile strength assuming a reasonable ratio of unconfined compression strength to tensile strength (browse Jaeger’s book).

5.4.2 Exercise 2: Shear and tensile failure in vertical and deviated wellbores

For this assignment you have to be able to calculate principal stresses on the wall of a wellbore at an arbitrary deviation and azimuth (See Chapter 6 of my notes “Wellbore stability” (https://dnicolasespinoza.github.io/) and Zoback's book Ch. 8).

Develop a script to compute and illustrate graphically (lower hemisphere projection):

a.

The likelihood of tensile fractures considering wellbore mud pressure is equal to pore pressure in the formation $P_W = P_p$ and a simple tensile strength criterion (independent of intermediate and maximum principal stresses), i.e., required $T_s$.

b.

The likelihood of breakouts considering wellbore mud pressure is equal to pore pressure in the formation $P_W = P_p$ and a simple unconfined compression strength criterion, i.e., required $UCS$.

c.

The wellbore breakout angle $w_{BO}$ considering a Mohr-Coulomb shear failure criterion and the properties of the Darley Dale Sandstone from Exercise 1.

d.

The wellbore breakout angle $w_{BO}$ considering a linear $I_1$ v.s. $(J_2)^{1/2}$ shear failure criterion and the properties of the Darley Dale Sandstone from Exercise 1.

e.

The wellbore breakout angle $w_{BO}$ considering a Modified Lade shear failure criterion (Zoback's book, p. 100) and the properties of the Darley Dale Sandstone from point 1.

Limit your wellbore breakout angle plots to $0^{\circ} < w_{BO} < 60^{\circ}$ because predictions with linear elasticity over $w_{BO} > 60^{\circ}$ are likely highly inaccurate. Test your script with the three examples given in Zobacks's book in Figures 8.2 and 10.4 (normal faulting, strike-slip faulting, and reverse faulting). When computing breakouts assume $P_W$ = 32 MPa (NF), $P_W$ = 40 MPa (SSF), and $P_W$ = 52 MPa (RF). Use a Poisson ratio of 0.25 for stress calculations.

Figure 5.2: Wellbore stability results plotted as lower hemisphere projections (image source: Zoback 2013 - RG). Note: $P_{mud} = P_W$ in the figure above.