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Hydraulic Fracture: multiscale processes and moving interfaces

Hydraulic Fracture: multiscale processes and moving interfaces

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Hydraulic Fracture: multiscale processes and moving interfaces

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  1. Hydraulic Fracture: multiscale processes and moving interfaces Anthony Peirce Department of Mathematics University of British Columbia • Collaborators: • Jose` Adachi (SLB, Houston) • Rachel Kuske (UBC) • Sarah Mitchell (UBC) • Ed Siebrits (SLB, Houston) WORKSHOP ON ROCK MECHANICS AND LOGISTICS IN MINING February 26, 2007

  2. Outline • What is a hydraulic fracture? • Mathematical models of hydraulic fracture • Scaling and special solutions for 1-2D 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. Model EQ 1: Conservation of mass

  7. 1-2D model and physical processes Viscous energy loss Leak-off Fracture Energy Breaking rock

  8. Scaling and dimensionless quantities • Rescale: • Dimensionless quantities: • Governing equations become:

  9. Toughness dominated propagation • Asymptotic behaviour of the Hilbert transform • Large toughness:

  10. Viscosity dominated propagation • Asymptotic behaviour of the Hilbert transform • Large viscosity:

  11. HF experiment (Bunger& Jeffrey CSIRO)

  12. 2-3D HF Equations • Elasticity • Lubrication • Boundary conditions at moving front (non-locality) (non-linearity) (free boundary)

  13. Coupled equations – a model problem Elasticity Fluid Flow n-2 n-1 n n+1 n+2

  14. Conditioning of the Jacobian • Evolution equation for w: • Eigenvalues of AC: Explicit Scheme Use implicit time stepping

  15. General first order iterative method • Consider solving: using the iterative method • Residual errors are damped according to

  16. Damping factor Poorly Damped Well Damped

  17. Multigrid Methods

  18. MG approach for coupled HF Equations

  19. MG preconditioning of • C coarsening using dual mesh • Localized GS smoother

  20. Performance of MG Preconditioner ~9.5 x

  21. function [A] = lu(A) [n,n]=size(A); for j=1:n for k=1:j-1 for i=k+1:n A(i,j)=A(i,j)-A(i,k)*A(k,j); end end for i=j+1:n A(i,j)=A(i,j)/A(j,j); end end return function [A] = basicILUc(A) [n,n]=size(A); for j=1:n for k=1:j-1 if A(k,j) ~=0, for i=k+1:n if A(i,j) ~=0, A(i,j)=A(i,j)-A(i,k)*A(k,j); end end end end for i=j+1:n if A(i,j) ~=0, A(i,j)=A(i,j)/A(j,j); end end end return Incomplete LU factorization

  22. Local Jacobian & Fourier Analysis Elasticity Fluid Flow n-2 n-1 n n+1 n+2

  23. Spectrum of AC (ACLoc)-1

  24. Eigenvalues of preconditioners

  25. Numerical results for stress jump

  26. Numerical results for stress jump

  27. Front evolution via the VOF method 0.75 0.15 1.0 0.78 0.06 1.0 1.0 0.3

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

  29. Radial Solution

  30. Fracture width for modulus contrast Modulus Changes Source

  31. Fracture width for stress jump

  32. Pressure and width evolution

  33. Channel fracture & pinch region

  34. Hourglass HF with leakoff

  35. Concluding remarks • Examples of hydraulic fractures • Scaling and physical processes in 1D models • Tip asymptotics: & • Numerical models of 2-3D hydraulic fractures • The non-local, nonlinear & free boundary problem • An Eulerian approach and the coupled equations • A multigrid algorithm for the coupled problem • ILU Factorization for the Localized Jacobian • Front evolution via the VOF method • Numerical results