Hydraulic Fracture multiscale processes and moving interfaces Anthony

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) Nanoscale Material Interfaces: Experiment, Theory and Simulation Singapore 10 -14 January 2005

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

HF Examples - block caving 3

HF Example – caving (Jeffrey, CSIRO) 4

HF Examples – well stimulation 5

An actual hydraulic fracture 6

HF experiment (Jeffrey et al CSIRO) 7

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

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

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

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

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

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

Boundary & propagation conditions • Boundary conditions • Propagation condition 14

The elasticity equation 15

3 Layer Uniform Asymptotic Solution 16

Crack cutting an interface 17

A penny crack cutting 2 interfaces 18

A crack touching an interface 19

Eulerian approach & Coupled HF Equations 20

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

Performance of MG Preconditioner ~9. 5 x 22

Front evolution via the VOF method 0. 75 0. 15 1. 0 0. 78 1. 0 0. 06 0. 3 23

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

Radial Solution 25

Fracture width for modulus contrast Modulus Changes Source 26

Fracture width for stress jump 27

Pressure and width evolution 28

Channel fracture and breakout 29

Hourglass HF with leakoff 30

Concluding remarks • Examples of hydraulic fractures • Scaling and physical processes in 1 D models Ø Tip asymptotics: Ø Experimental verification & • Numerical models of 2 -3 D hydraulic fractures Ø Ø Ø 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 31