Hydraulic%20Fractures:%20multiscale%20phenomena,%20asymptotic%20and%20numerical%20solutions

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Hydraulic Fractures multiscale phenomena,
asymptotic and numerical solutions
Anthony Peirce University of British
Columbia Collaborators Emmanuel Detournay
(UMN) Eduard Siebrits (SLB)
SANUM Conference Stellenbosch 6-8 April 2009
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Outline

• Examples of hydraulic fractures
• Governing equations, scaling, asymptotic
  solutions 1-2D and 2-3D
• Solving the free boundary problem
• Existing methods VOF, Level Set and KI matching
• Tip asymptote and Eikonal boundary value problem
• Setting the tip volumes using the tip asymptotes
• Coupled equations
• Numerical examples
• M-vertex and K-vertex
• Viscous crack propagating in a variable in situ
  stress
• Stress jump solutions

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HF Examples - block caving
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Oil well stimulation
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Quarries
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Magma flow
Tarkastad
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Model EQ 1 Conservation of mass
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Model EQ 2 The elasticity equation
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1-2D model and physical processes
Viscous energy loss
Fracture Energy Breaking rock
Leak-off
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2-3D HF Equations

• Elasticity
• Lubrication
• Boundary conditions at moving front

(non-locality)
(non-linearity)
(free boundary)
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Scaling

• Rescale the physical quantities as follows
• Dimensionless groups
• Scaled Equations

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Reduced eq near a smooth front
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Tip asymptotic behaviour

• Elasticity
• Lubrication

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HF experiment (Bunger Jeffrey CSIRO)
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Discretizing the Elasticity Equation

• Piecewise constant DD
• Discrete Elasticity Equation
• Operator form

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Discretizing the Fluid Flow Equation

• The fluid flow equation
• Integrate over cell
• Approximate the integrals
• where
• Operator form

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How do we find the fracture front?

• VOF method

• Level set method
• -move contours of a surface
• - differentiate

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1
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1
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Problem we need an accurate
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Why not use stress intensity factor?

• Move front till SIF
• Find the SIF numerically
• if move forward
• if move back
• if stay
• Problems
• Accurate calculation of
• How far do you move?
• What about viscous and leak-off dominated
  propagation regimes

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Divide elements into tip channel
tip elt
channel elt
At survey points
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Eikonal solution surfaces
The signed distance function
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Solving the Eikonal BVP

• The Eikonal Equation
• First order discretization
• Reduction to a quadratic
• Let
• Solution to the quadratic equation

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Geometric interpretation
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Initializing from tip asymptotes

• General asymptote
• K-vertex
• M-vertex

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Sample signed distance function
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Calculating the tip volume
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Tip volume width as a function of

• Width of the tip element
• Volume of the tip element
• Tip widths

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Tip widths and Pressures
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The coupled equations

• Channel lubrication equation
• Tip lubrication equation
• Elasticity Equation (eliminate channel pressure)
• Coupled system

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Time stepping and front evolution
Time step loop Front iteration loop Coupled
Solution end set in use FMM
to solve Locate front next front
iteration next time step
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M-vertex radial soln footprints
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M-Vertex radial solution front speed
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M-vertex radial solution -
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M-Vertex Footprint evolution
ILSA
EXACT
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M-vertex radial solnwidth pressure
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M-vertex width
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M-vertex pressure
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K-vertex radial soln footprints
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K-Vertex radial solution front speed
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K-vertex radial solution -
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K-vertex convergence rate for
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K-vertex radial solnwidth pressure
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K-vertex width
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K-vertex Pressure
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Linear in-situ stress field - footprints
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Zoom in
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Linear in-situ stress field - movie
ILSA Linear
Uniform field
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Breakthrough Low to High Stress
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Stress contrast high-gtlow
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Lab Test (Bunger et al 2008)
ILSA
Test
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Conclusions

• Governing equations, scaling, asymptotic
  solutions 1-2D and 2-3D
• Solving the free boundary problem
• Existing methods VOF, Level Set and KI matching
• Tip asymptote and Eikonal boundary value problem
• Setting the tip volumes using the tip asymptotes
• Coupled equations
• Numerical examples
• M-vertex and K-vertex
• Viscous crack propagating in a variable in situ
  stress
• Stress jump solutions