Hydraulic Fracture: multiscale processes and moving interfaces Anthony Peirce Department of Mathematics University of British Columbia Nanoscale Material Interfaces: Experiment, Theory and Simulation Singapore January 2005 Collaborators: Jose` Adachi (SLB, Houston) Rachel Kuske (UBC) Sarah Mitchell (UBC) Ed Siebrits (SLB, Houston)

2 Outline What is a hydraulic fracture? Scaling and special solutions for 1D models Numerical modeling for 2-3D problems Conclusions

3 HF Examples - block caving

4 HF Example – caving (Jeffrey, CSIRO)

5 HF Examples – well stimulation

6 An actual hydraulic fracture

7 HF experiment (Jeffrey et al CSIRO)

8 1D model and physical processes Fracture Energy Breaking rock Leak-off Viscous energy loss

9 Scaling and dimensionless quantities Rescale: Dimensionless quantities: Governing equations become:

10 Two of the physical processes Large toughness: Large viscosity:

11 Experiment I (Bunger et al 2004) Theory Large toughness: (Spence & Sharp)

12 Experiment II (Bunger et al 2004) Theory Large viscosity: (Desroches et al)

13 The coupled EHF equations in 2D The elasticity equation – balance of forces The fluid flow equation – mass balance

14 Boundary & propagation conditions Boundary conditions Propagation condition

15 The elasticity equation

16 3 Layer Uniform Asymptotic Solution

17 Crack cutting an interface

18 A penny crack cutting 2 interfaces

19 A crack touching an interface

20 Eulerian approach & Coupled HF Equations

21 MG preconditioning of C coarsening using dual mesh Localized GS smoother

22 Performance of MG Preconditioner ~9.5 x

23 Front evolution via the VOF method

24 Time stepping and front evolution Time step loop: VOF loop: Coupled Solution end next VOF iteration next time step

25 Radial Solution

26 Fracture width for modulus contrast Modulus Changes Source

27 Fracture width for stress jump

28 Pressure and width evolution

29 Channel fracture and breakout

30 Hourglass HF with leakoff

31 Concluding remarks Examples of hydraulic fractures Scaling and physical processes in 1D models  Tip asymptotics: &  Experimental verification Numerical models of 2-3D hydraulic fractures  The non-local elasticity equation  An Eulerian approach and the coupled equations  FT construction of the elasticity influence matrices  A multigrid algorithm for the coupled problem  Front evolution via the VOF method Numerical results