Determining a characteristic associated with a reservoir
Techniques for determining a reservoir characteristic include determining, with an analytical solution, a change to at least one control point of a boundary of a control volume defined in a subterranean formation, the change to the at least one control point caused by a hydraulic fracture formed in or adjacent the subterranean formation; determining, with a numerical solution, a fluid pressure change of the control volume based on the change to the at least one control point; and determining, with a mechanical model based on the analytical solution and the numerical solution, at least one reservoir characteristic based at least in part on the determined fluid pressure change of the control volume.
1. A computer-implemented method, comprising:
determining, with an analytical solution executed by one or more hardware processors, a change to at least one control point of a boundary of a control volume defined in a subterranean formation, the change to the at least one control point caused by a hydraulic fracture formed in or adjacent the subterranean formation;
determining, with a numerical solution executed by the one or more hardware processors, a fluid pressure change of the control volume based on the change to the at least one control point; and
determining, with a mechanical model generated by the one or more hardware processors based on the analytical solution and the numerical solution, at least one reservoir characteristic based at least in part on the determined fluid pressure change of the control volume.
2. The computer-implemented method of claim 1 , wherein the change to the at least one control point comprises a stress field.
3. The computer-implemented method of claim 2 , wherein determining the fluid pressure change of the control volume based on the change to the at least one control point comprises:
evaluating, with the one or more hardware processors, a stress tensor of the stress field; and
determining, with the one or more hardware processors, the fluid pressure change of the control volume based on the evaluation of the stress tensor.
4. The computer-implemented method of claim 2 , wherein the at least one control point defines at least one stress on the boundary of the control volume.
5. The computer-implemented method of claim 4 , wherein the at least one control point comprises a plurality of control points that define the stress field.
6. The computer-implemented method of claim 1 , wherein the change to the at least one control point comprises a strain field.
7. The computer-implemented method of claim 6 , wherein determining the fluid pressure change of the control volume based on the change to the at least one control point comprises:
evaluating, with the one or more hardware processors, a strain tensor of the strain field; and
determining, with the one or more hardware processors, the fluid pressure change of the control volume based on the evaluation of the strain tensor.
8. The computer-implemented method of claim 6 , wherein the at least one control point defines at least one strain on the boundary of the control volume.
9. The computer-implemented method of claim 8 , wherein the at least one control point comprises a plurality of control points that define the strain field.
10. The computer-implemented method of claim 1 ; wherein the change to the at least one control point comprises a traction field.
11. The computer-implemented method of claim 10 , wherein determining the fluid pressure change of the control volume based on the change to the at least one control point comprises:
evaluating, with the one or more hardware processors, a traction vector of the traction field; and
determining, with the one or more hardware processors, the fluid pressure change of the control volume based on the evaluation of the traction vector.
12. The computer-implemented method of claim 10 , wherein the at least one control point defines at least one traction on the boundary of the control volume.
13. The computer-implemented method of claim 12 , wherein the at least one control point comprises a plurality of control points that define the traction field.
14. The computer-implemented method of claim 1 , wherein the change to the at least one control point comprises a displacement field.
15. The computer-implemented method of claim 14 , wherein determining the fluid pressure change of the control volume based on the change to the at least one control point comprises:
evaluating, with the one or more hardware processors, a displacement vector of the displacement field; and
determining, with the one or more hardware processors, the fluid pressure change of the control volume based on the evaluation of the displacement vector.
16. The computer-implemented method of claim 14 , wherein the at least one control point defines at least one displacement on the boundary of the control volume.
17. The computer-implemented method of claim 16 , wherein the at least one control point comprises a plurality of control points that define the displacement field.
18. The computer-implemented method of claim 1 , wherein the control volume comprises at least a portion of a wellbore formed from a terranean surface to the subterranean formation, and the portion of the wellbore is fluidly sealed from the hydraulic fracture.
19. The computer-implemented method of claim 18 , wherein the at least one control point comprises a plurality of control points representative of a plurality of displacements on a boundary of the portion of the wellbore.
20. The computer-implemented method of claim 18 , wherein the wellbore comprises a first wellbore, and the hydraulic fracture formed in or adjacent the subterranean formation emanates from a second wellbore different than the first wellbore.
21. The computer-implemented method of claim 18 , wherein the at least one reservoir characteristic comprises a reservoir effect that comprises at least one of undrained rock compression or fluid migration.
22. The computer-implemented method of claim 21 , further comprising determining at least a part of a hydraulic fracture geometry of the hydraulic fracture or a fracture growth rate of the hydraulic fracture, or both based on the determination of the undrained rock compression effect.
23. The computer-implemented method of claim 21 , further comprising determining an effective hydraulic connectivity in the subterranean formation or a rate of leak-off of a treatment fluid that forms the hydraulic formation into the subterranean formation based on the determination of the fluid pressurization effects.
24. The computer-implemented method of claim 1 , wherein the hydraulic fracture emanates from a first wellbore formed in the subterranean formation, and the control volume comprises a sealed section of a second wellbore formed in the subterranean formation that is different than the first wellbore.
25. The computer-implemented method of claim 24 , wherein the at least one control point comprises at least one displacement representative of at least one of a displacement, a stress tensor, a strain tensor, or a traction vector on a boundary of the sealed section.
26. The computer-implemented method of claim 24 , wherein the at least one dimension of the hydraulic fracture comprises at least one of a half-length of the hydraulic fracture, a length of the hydraulic fracture, a half-height of the hydraulic fracture, or a height of the hydraulic fracture.
27. The computer-implemented method of claim 1 , wherein the analytical solution comprises u i (x)=f(Dim treatfrac ,vec),
where u i (x) is the displacement field that comprises the at least one control point, and is a function of one or more dimensions of the control volume (Dim ev ), one or more dimensions of the treatment fracture (Dim treatfrac ), and a vector between the control volume and the treatment fracture (vec).
28. The computer-implemented method of claim 27 , wherein the analytical solution further comprises u i (x)=f(Dim treatfrac ,vec,rot,geo),
where u i (x) is the displacement field that comprises the at least one control point, and is a function of one or more dimensions of the control volume (Dim ev ), one or more dimensions of the treatment fracture (Dim treatfrac ), a vector between the control volume and the treatment fracture (vec), a rotation of the control volume relative to the treatment fracture (rot), and one or more geologic properties of the subterranean formation (geo).
29. The computer-implemented method of claim 1 , wherein the analytical solution comprises a modified Eshelby solution.
30. The computer-implemented method of claim 29 , wherein the modified Eshelby solution comprises one or more equations that determines the at least one control point based at least in part on a plurality of parameters that are associated with the control volume and the hydraulic fracture.
31. The computer-implemented method of claim 30 , wherein the plurality of parameters comprise at least two dimensions of the control volume, at least two dimensions of the hydraulic fracture, and at least three dimensions that represent a vector between the control volume and the hydraulic fracture.
32. The computer-implemented method of claim 31 , wherein the plurality of parameters further comprise at least three dimensions that represent an axis of rotation between the control volume and the hydraulic fracture and an angle of rotation about the axis of rotation.
33. The computer-implemented method of claim 30 , wherein the plurality of parameters further comprise one or more geologic characteristics of the subterranean formation.
34. The computer-implemented method of claim 30 , wherein at least one of the equations comprises:
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ψ
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ϵ
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,
where u i (x) represents the displacement field that comprises the at least one control point, ϵ* is the Eigenstrain, ν is Poisson's ratio, and ψ and Φ are given by
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and
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.
35. The computer-implemented method of claim 1 , wherein determining, with a numerical solution executed by the one or more hardware processors, a fluid pressure change of the control volume based on the change to the at least one control point, comprises:
calculating, with the numerical solution executed by the one or more hardware processors, a pressure transfer function on the control volume based on the fluid pressure change on the control volume.
36. A distributed computing system, comprising:
one or more memory modules; and
one or more hardware processors communicably coupled to the one or more memory modules and configured to execute instructions stored in the one or more memory modules to perform operations comprising:
determining, with an analytical solution, a change to at least one control point of a boundary of a control volume defined in a subterranean formation, the change to the at least one control point caused by a hydraulic fracture formed in or adjacent the subterranean formation;
determining, with a numerical solution, a fluid pressure change of the control volume based on the change to the at least one control point; and
determining, with a mechanical model based on the analytical solution and the numerical solution, at least one reservoir characteristic based at least in part on the determined fluid pressure change of the control volume.
37. An apparatus that comprises a non-transitory, tangible computer readable media that stores instructions that, when executed by one or more computing systems that comprise one or more hardware processors, cause the one or more computing systems to perform operations comprising:
determining, with an analytical solution, a change to at least one control point of a boundary of a control volume defined in a subterranean formation, the change to the at least one control point caused by a hydraulic fracture formed in or adjacent the subterranean formation;
determining, with a numerical solution, a fluid pressure change of the control volume based on the change to the at least one control point; and
determining, with a mechanical model based on the analytical solution and the numerical solution, at least one reservoir characteristic based at least in part on the determined fluid pressure change of the control volume.
38. A computer-implemented method, comprising:
determining, with an analytical solution executed by one or more hardware processors, a change to at least one control point of a boundary of a control volume defined in a subterranean formation, the change to the at least one control point caused by a hydraulic fracture formed in or adjacent the subterranean formation;
determining, with a numerical solution executed by the one or more hardware processors, a fluid pressure change of the control volume based on the change to the at least one control point, wherein the determining with the numerical solution comprises calculating, with the numerical solution executed by the one or more hardware processors, a pressure transfer function on the control volume based on the fluid pressure change on the control volume; and
determining, with a mechanical model generated by the one or more hardware processors based on the analytical solution and the numerical solution, at least one reservoir characteristic based at least in part on the determined fluid pressure change of the control volume.
39. A computer-implemented method, comprising:
determining, with an analytical solution executed by one or more hardware processors, a change to at least one control point of a boundary of a control volume defined in a subterranean formation, the change to the at least one control point caused by a hydraulic fracture formed in or adjacent the subterranean formation, wherein the hydraulic fracture emanates from a first wellbore formed in the subterranean formation, and the control volume comprises a sealed section of a second wellbore formed in the subterranean formation that is different than the first wellbore;
determining, with a numerical solution executed by the one or more hardware processors, a fluid pressure change of the control volume based on the change to the at least one control point; and
determining, with a mechanical model generated by the one or more hardware processors based on the analytical solution and the numerical solution, at least one reservoir characteristic based at least in part on the determined fluid pressure change of the control volume, the at least one reservoir characteristic comprising at least one dimension of the hydraulic fracture that comprises at least one of a half-length of the hydraulic fracture, a length of the hydraulic fracture, a half-height of the hydraulic fracture, or a height of the hydraulic fracture.