1. One dimensional models of hydraulic fracture Anthony Peirce (UBC) Collaborators: Jose` Adachi (SLB) Shira Daltrop (UBC) Emmanuel Detournay (UMN) WITS University 1 April 2009

2. 2 Outline The HF problem and 2D models Slender geometries -> the possibilities for 1D models The classic PKN model – porous medium eq – limitations Extension of PKN to include toughness - 1D integro-PDE P3D model – PKN methodology ->pseudo 3D Conclusions

3. 3 HF Examples - block caving

4. 4 HF Example – caving (Jeffrey, CSIRO)

5. 5 Oil well stimulation

6. 6 Lab test with stress contrast (Bunger)

7. 7 2-3D HF Equations Elasticity Lubrication Boundary conditions at moving front (non-locality) (free boundary) (non-linearity)

8. 8

9. 9 The PKN Model

10. 10 Assumptions behind the PKN Model Pressure independent of : Vertical sections approximately in a state of plane strain: PKN model:  Local elasticity equation  Can’t model pressure singularities for example when

11. 11 PKN – Averaging the Lubrication Eq

12. 12 Scaled lubrication & similarity solution

13. 13 Numerical soln of a finger-like frac

14. 14 Width and scaled pressures

15. 15 Asymptotics of numerical soln

16. 16 Extended PKN: 2D integral eq 1D integral eq Integral eq for a pressurized rectangular crack Re-scale variables:

17. 17 Asymptotic behaviour of the kernel ~ Hilbert Transform PKN

18. 18 Assumed behaviour of  Power law in the tip regions:  Analytic away from the tips:

19. 19 Outer expansion Hilbert Transform Region PKN Region PKN

20. 20 Outer expansion Test function:

21. 21 Inner expansion: Hilbert Transform Region PKN Region

22. 22 Inner Expansion

23. 23 Discretizing the Elasticity Equation

24. 24 Discretizing the Fluid Flow Equation

25. 25 EPKN Tip solutions – viscosity Close to the tip

26. 26 Locating the tip position Viscosity Dominated: Toughness Dominated:

27. 27 Numerical Results K=0

28. 28 Numerical Results K=1

29. 29 Numerical Results K=5

30. 30 P3D models

31. 31 Scaled elasticity equations

32. 32 Averaged equations

33. 33 Scaled lubrication equation

34. 34 Collocation scheme to solve ODE

35. 35 Numerical Results

36. 36 Footprint comparison with 3D solution

37. 37 Width comparison with 3D solution

38. 38 Concluding remarks The HF problem and 2D models Slender geometries -> the possibilities for 1D models The classic PKN model – porous medium eq - limitations Extension of PKN to include toughness - 1D integro-PDE  Reduction of the 2D integral to a 1D nonlocal equation in the small aspect ratio limit  Asymptotics of the 1D kernel  Asymptotic analysis to determine the action of the integral operator in different regions of the domain:  Outer region: 1D integral equation - local PDE  Inner tip region: 1D integral equation - Hilbert Transform  Tip asymptotics and numerical solution P3D model – PKN methodology ->pseudo 3D  Plane strain width solution and averaging  Averaging and scaling the lubrication  Reduction to a convection diffusion equation  Numerical solution via collocation  Results and comparison with 3D solution

39. 39 PKN Traveling Wave solution