Determining a subterranean formation breakdown pressure

1. A computer-implemented method for determining a breakdown pressure of a subterranean formation, the method comprising:

identifying, with one or more hardware processors, in-situ stresses for a portion of a wellbore formed from a terranean surface into a subterranean formation;

transforming, with the one or more hardware processors, the in-situ stresses from a global coordinate system to a wellbore coordinate system of a deviated portion of the wellbore that comprises at least one perforation tunnel for a hydraulic fracturing treatment;

transforming, with one or more hardware processors, the in-situ stresses from the wellbore coordinate system to a perforation coordinate system through at least one rotation matrix;

determining, with one or more hardware processors, one or more stresses at a wellbore-perforation interface of the perforation tunnel from the in-situ stresses in the perforation coordinate system;

calculating, with one or more hardware processors, one or more hoop stresses at a perforation tunnel wall of the perforation tunnel from the determined one or more stresses on the wellbore-perforation interface; and

determining, with one or more hardware processors, a breakdown pressure for the subterranean formation based at least in part on the calculated one or more hoop stresses.

2. The computer-implemented method of claim 1 , wherein the global coordinate system comprises an x-axis directed toward north, a y-axis directed toward east, and a z-axis directed vertically downwards from the terranean surface, and the wellbore coordinate system is defined by a measured depth, a wellbore deviation, and a wellbore azimuth, and transforming the in-situ stresses from the global coordinate system to the wellbore coordinate system comprises:

rotating the wellbore deviation about the y-axis; and

rotating the wellbore azimuth about the z-axis.

3. The computer-implemented method of claim 1 , wherein transforming the in-situ stresses from the wellbore coordinate system to the perforation coordinate system through the at least one rotation matrix comprises:

rotating a first perforation phase angle about a first axis of the wellbore coordinate system; and

rotating a second perforation phase angle about a second axis of the wellbore coordinate system.

4. The computer-implemented method of claim 1 , wherein determining one or more stresses at the wellbore-perforation interface of the perforation tunnel from the in-situ stresses in the perforation coordinate system comprises:

determining a far-field in-situ stress tensor that acts on a wellbore-perforation interface between the deviated portion of the wellbore and the perforation tunnel wall;

determining a bottom hole pressure that acts on the subterranean formation adjacent the perforation tunnel wall from a fracturing liquid;

determining a pressure within the at least one perforation tunnel that induces stress on the perforation tunnel wall; and

calculating the one or more stresses on the perforation tunnel wall from the determinations of the far-field in-situ stress tensor, the bottom hole pressure, and the pressure within the at least one perforation tunnel.

5. The computer-implemented method of claim 4 , wherein determining the bottom hole pressure that acts on the subterranean formation adjacent the perforation tunnel wall from the fracturing liquid comprises determining a pressure at an interface of the subterranean formation and a cement layer that binds a wellbore casing to the subterranean formation based on a pressure of the fracturing liquid.

6. The computer-implemented method of claim 4 , wherein determining the pressure within the at least one perforation tunnel that induces stress on the perforation tunnel wall comprises determining the pressure within the at least one perforation tunnel by subtracting a perforation pressure loss from the bottom hole pressure.

7. The computer-implemented method of claim 4 , wherein calculating one or more hoop stresses at the perforation tunnel wall of the perforation tunnel from the determined one or more stresses on the wellbore-perforation interface comprises:

summing induced stresses based on the far-field in-situ stress tensor and the bottom hole pressure;

calculating local maximum and minimum stresses at the wellbore-perforation interface based on the far-field in-situ stress tensor and the bottom hole pressure; and

calculating the one or more hoop stresses around the at least one perforation tunnel.

8. The computer-implemented method of claim 7 , wherein determining the breakdown pressure for the subterranean formation based at least in part on the calculated one or more hoop stresses comprises solving for the breakdown pressure with a quadratic equation that comprises the one or more calculated hoop stresses, the bottom hole pressure, and a tensile strength of the subterranean formation.

9. The computer-implemented method of claim 1 , further comprising:

generating, with the one or more hardware processors, data that comprises a graphic that represents the determined breakdown pressure for the subterranean formation; and

presenting the generated data through a graphical user interface (GUI).

10. A system, comprising:

one or more memory modules; and

one or more hardware processors communicably coupled to the one or more memory modules, the one or more hardware processors configured to execute instructions stored on the one or more memory modules to perform operations comprising:

identifying in-situ stresses for a portion of a wellbore formed from a terranean surface into a subterranean formation;

transforming the in-situ stresses from a global coordinate system to a wellbore coordinate system of a deviated portion of the wellbore that comprises at least one perforation tunnel for a hydraulic fracturing treatment;

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

determining one or more stresses at a wellbore-perforation interface of the perforation tunnel from the in-situ stresses in the perforation coordinate system;

calculating one or more hoop stresses at a perforation tunnel wall of the perforation tunnel from the determined one or more stresses on the wellbore-perforation interface; and

determining a breakdown pressure for the subterranean formation based at least in part on the calculated one or more hoop stresses.

11. The system of claim 10 , wherein the global coordinate system comprises an x-axis directed toward north, a y-axis directed toward east, and a z-axis directed vertically downwards from the terranean surface, and the wellbore coordinate system is defined by a measured depth, a wellbore deviation, and a wellbore azimuth, and transforming the in-situ stresses from the global coordinate system to the wellbore coordinate system comprises:

rotating the wellbore deviation about the y-axis; and

rotating the wellbore azimuth about the z-axis.

12. The system of claim 10 , wherein the operation of transforming the in-situ stresses from the wellbore coordinate system to the perforation coordinate system through the at least one rotation matrix comprises:

rotating a first perforation phase angle about a first axis of the wellbore coordinate system; and

rotating a second perforation phase angle about a second axis of the wellbore coordinate system.

13. The system of claim 10 , wherein the operation of determining one or more stresses at the wellbore-perforation interface of the perforation tunnel from the in-situ stresses in the perforation coordinate system comprises:

determining a far-field in-situ stress tensor that acts on a wellbore-perforation interface between the deviated portion of the wellbore and the perforation tunnel wall;

determining a bottom hole pressure that acts on the subterranean formation adjacent the perforation tunnel wall from a fracturing liquid;

determining a pressure within the at least one perforation tunnel that induces stress on the perforation tunnel wall; and

calculating the one or more stresses on the perforation tunnel wall from the determinations of the far-field in-situ stress tensor, the bottom hole pressure, and the pressure within the at least one perforation tunnel.

14. The system of claim 13 , wherein the operation of determining the bottom hole pressure that acts on the subterranean formation adjacent the perforation tunnel wall from the fracturing liquid comprises determining a pressure at an interface of the subterranean formation and a cement layer that binds a wellbore casing to the subterranean formation based on a pressure of the fracturing liquid.

15. The system of claim 14 , wherein the operation of determining the pressure within the at least one perforation tunnel that induces stress on the perforation tunnel wall comprises determining the pressure within the at least one perforation tunnel by subtracting a perforation pressure loss from the bottom hole pressure.

16. The system of claim 13 , wherein the operation of calculating one or more hoop stresses at the perforation tunnel wall of the perforation tunnel from the determined one or more stresses on the wellbore-perforation interface comprises:

summing induced stresses based on the far-field in-situ stress tensor and the bottom hole pressure;

calculating local maximum and minimum stresses at the wellbore-perforation interface based on the far-field in-situ stress tensor and the bottom hole pressure; and

calculating the one or more hoop stresses around the at least one perforation tunnel.

17. The system of claim 16 , wherein the operation of determining the breakdown pressure for the subterranean formation based at least in part on the calculated one or more hoop stresses comprises solving for the breakdown pressure with a quadratic equation that comprises the one or more calculated hoop stresses, the bottom hole pressure, and a tensile strength of the subterranean formation.

18. The system of claim 10 , further comprising a graphical user interface (GUI) communicably coupled to the one or more hardware processors, and the operations further comprise:

generating data that comprises a graphic that represents the determined breakdown pressure for the subterranean formation; and

presenting the generated data through the GUI.

19. An apparatus comprising a non-transitory, computer readable, storage medium that stores instructions that, when executed by at least one processor, cause the at least one processor to perform operations comprising:

identifying in-situ stresses for a portion of a wellbore formed from a terranean surface into a subterranean formation;

transforming the in-situ stresses from a global coordinate system to a wellbore coordinate system of a deviated portion of the wellbore that comprises at least one perforation tunnel for a hydraulic fracturing treatment;

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

determining one or more stresses at a wellbore-perforation interface of the perforation tunnel from the in-situ stresses in the perforation coordinate system;

calculating one or more hoop stresses at a perforation tunnel wall of the perforation tunnel from the determined one or more stresses on the wellbore-perforation interface; and

determining a breakdown pressure for the subterranean formation based at least in part on the calculated one or more hoop stresses.

20. The apparatus of claim 19 , wherein the global coordinate system comprises an x-axis directed toward north, a y-axis directed toward east, and a z-axis directed vertically downwards from the terranean surface, and the wellbore coordinate system is defined by a measured depth, a wellbore deviation, and a wellbore azimuth, and transforming the in-situ stresses from the global coordinate system to the wellbore coordinate system comprises:

rotating the wellbore deviation about the y-axis; and

rotating the wellbore azimuth about the z-axis.

21. The apparatus of claim 19 , wherein the operation of transforming the in-situ stresses from the wellbore coordinate system to the perforation coordinate system through the at least one rotation matrix comprises:

rotating a first perforation phase angle about a first axis of the wellbore coordinate system; and

rotating a second perforation phase angle about a second axis of the wellbore coordinate system.

22. The apparatus of claim 19 , wherein the operation of determining one or more stresses at the wellbore-perforation interface of the perforation tunnel from the in-situ stresses in the perforation coordinate system comprises:

determining a far-field in-situ stress tensor that acts on a wellbore-perforation interface between the deviated portion of the wellbore and the perforation tunnel wall;

determining a bottom hole pressure that acts on the subterranean formation adjacent the perforation tunnel wall from a fracturing liquid;

determining a pressure within the at least one perforation tunnel that induces stress on the perforation tunnel wall; and

calculating the one or more stresses on the perforation tunnel wall from the determinations of the far-field in-situ stress tensor, the bottom hole pressure, and the pressure within the at least one perforation tunnel.

23. The apparatus of claim 22 , wherein the operation of determining the bottom hole pressure that acts on the subterranean formation adjacent the perforation tunnel wall from the fracturing liquid comprises determining a pressure at an interface of the subterranean formation and a cement layer that binds a wellbore casing to the subterranean formation based on a pressure of the fracturing liquid.

24. The apparatus of claim 23 , wherein the operation of determining the pressure within the at least one perforation tunnel that induces stress on the perforation tunnel wall comprises determining the pressure within the at least one perforation tunnel by subtracting a perforation pressure loss from the bottom hole pressure.

25. The apparatus of claim 22 , wherein the operation of calculating one or more hoop stresses at the perforation tunnel wall of the perforation tunnel from the determined one or more stresses on the wellbore-perforation interface comprises:

summing induced stresses based on the far-field in-situ stress tensor and the bottom hole pressure;

calculating local maximum and minimum stresses at the wellbore-perforation interface based on the far-field in-situ stress tensor and the bottom hole pressure; and

calculating the one or more hoop stresses around the at least one perforation tunnel.

26. The apparatus of claim 25 , wherein the operation of determining the breakdown pressure for the subterranean formation based at least in part on the calculated one or more hoop stresses comprises solving for the breakdown pressure with a quadratic equation that comprises the one or more calculated hoop stresses, the bottom hole pressure, and a tensile strength of the subterranean formation.

27. The apparatus of claim 19 , wherein the operations further comprise:

generating data that comprises a graphic that represents the determined breakdown pressure for the subterranean formation; and

presenting the generated data through a graphical user interface (GUI).