Fracturing a subsurface formation based on a probabilistic determination of the required breakdown pressure

1. A method of hydraulic fracturing a subsurface formation, the method comprising:

using a deterministic model implemented on at least one processor to simulate a deviated well comprising a casing and at least one perforation cluster;

determining a different statistical distribution for each of one or more parameters to the deterministic model based on log data from a wellbore, the statistical distributions being represented as a function of a measured depth within the wellbore;

probabilistically solving the deterministic model implemented on the at least one processor to determine a mean and a standard deviation of breakdown pressures for each perforation cluster of the at least one perforation cluster by randomly accounting for uncertainties of the one or more parameters based on the determined statistical distributions for each of the one or more parameters;

processing the mean and the standard deviation of breakdown pressures for each perforation cluster to determine an upper bound breakdown pressure for each perforation cluster based on a confidence level;

drilling and completing a deviated well with a wellhead selected at least in part based on the determined upper bound breakdown pressure for each perforation cluster; and

injecting hydraulic fluid into the deviated well at an injection pressure determined at least in part on the determined upper bound breakdown pressure for each perforation cluster to cause hydraulic fracturing of the subsurface formation.

2. The method of claim 1 , further comprising determining a different mean and standard deviation of breakdown pressures for each fracturing stage of the deviated well.

3. The method of claim 1 , wherein the one or more parameters include an orientation angle of the maximum horizontal stress of the subsurface formation.

4. The method of claim 3 , wherein the statistical distribution of the orientation angle is a normal distribution with a mean and a standard deviation.

5. The method of claim 1 , wherein the one or more parameters include a tensile strength of the subsurface formation.

6. The method of claim 5 , wherein the statistical distribution of the tensile strength is a normal distribution with a mean and a standard deviation.

7. The method of claim 1 , wherein the one or more parameters include both an orientation angle of the maximum horizontal stress of the subsurface formation and a tensile strength of the subsurface formation and the statistical distributions for both the orientation angle and the maximum horizontal stress are normal distributions.

8. The method of claim 1 , wherein the one or more parameters include both a Young's modulus and a Poisson's ratio of the subsurface formation and both the statistical distribution of the Young's modulus and the Poisson's ratio are normal distributions.

9. The method of claim 1 , wherein the one or more parameters include an azimuth angle and a deviation angle of the wellbore.

10. The method of claim 1 , wherein the one or more parameters include a perforation fraction representing a fraction of wellbore pressure transferred to a perforation hole and acting on a perforation wall of the at least one perforation tunnel to counter induced hoop stresses by in-situ stresses.

11. The method of claim 1 , wherein probabilistically solving the deterministic model comprises:

randomly generating a pair of stochastic parameters based on the statistical distributions of each respective parameter of the one or more parameters using a Box-Muller method; and

substituting the pair of randomly generated stochastic parameters into the deterministic model to determine at least part of the mean and standard deviation of breakdown pressures.

12. The method of claim 1 , wherein probabilistically solving the deterministic model comprises:

transforming, with the at least one processor, in-situ stresses from a global coordinate system to a wellbore coordinate system of a deviated portion of the deviated well that comprises at least one perforation tunnel for each perforation cluster; and

transforming, with the at least one processor, the in-situ stresses from the wellbore coordinate system to a perforation coordinate system through at least one rotation matrix,

wherein the breakdown pressures are determined based at least in part on the in-situ stresses in the perforation coordinate system.

13. The method of claim 1 , wherein probabilistically solving the deterministic model comprises performing a Monte Carlo statistical analysis.

14. The method of claim 13 , wherein performing the Monte Carlo statistical analysis involves at least one thousand different variations of the one or more parameters to the deterministic model.

15. The method of claim 1 , wherein processing the mean and the standard deviation of breakdown pressures comprises generating a histogram of the breakdown pressures for each perforation cluster.

16. The method of claim 1 , wherein the breakdown pressures are determined based at least in part on hoop stresses at a perforation tunnel wall of at least one perforation tunnel for each perforation cluster.

17. The method of claim 1 , wherein the confidence level is a 95% confidence level.

18. The method of claim 1 , further comprising determining a pump schedule of an injection pump based at least in part on the determined upper bound breakdown pressure for each perforation cluster.

19. The method of claim 18 , wherein injecting the hydraulic fluid into the deviated well is performed by the injection pump using the determined pump schedule.