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Hydraulic Fractures: multiscale phenomena, asymptotic and numerical solutions
SANUM ConferenceStellenbosch
6-8 April 2009
Anthony Peirce
University of British Columbia
Collaborators:
Emmanuel Detournay (UMN)
Eduard Siebrits (SLB)
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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
FracturingFluid
Proppant
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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
FractureEnergy
Breaking rock
Leak-offViscous energy
loss
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2-3D HF Equations
• Elasticity
• Lubrication
• Boundary conditions at moving front
(non-locality)
(free boundary)
(non-linearity)
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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
.75
1
.15
11
.06.8
.3
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 surfacesThe signed distance function
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Solving the Eikonal BVP
• The Eikonal Equation• First order discretization
• Reduction to a quadraticLet
• 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
1.44 1.09 0.78 0.53 0.41 0.52 0.75 1.05 1.41.09 0.7 0.34 0.05 -0.09 0.03 0.31 0.66 1.050.79 0.35 -0.07 -0.43 -0.59 -0.46 -0.11 0.3 0.740.55 0.07 -0.42 -0.83 -0.97 -0.86 -0.47 0.01 0.50.43 -0.07 -0.57 -0.96 -1.16 -0.99 -0.63 -0.13 0.370.55 0.07 -0.42 -0.83 -0.97 -0.86 -0.47 0.01 0.50.79 0.35 -0.07 -0.43 -0.59 -0.46 -0.11 0.3 0.741.09 0.7 0.34 0.05 -0.09 0.03 0.31 0.66 1.051.44 1.09 0.78 0.53 0.41 0.52 0.75 1.05 1.4
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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 evolutionTime 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 sol’n – 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 evolutionILSAEXACT
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M-vertex radial soln:width & pressure
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M-vertex width
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M-vertex pressure
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K-vertex radial sol’n – 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 soln:width & 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 - movieILSA Linear Uniform field
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Breakthrough: Low to High Stress
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Stress contrast high->low
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Lab Test (Bunger et al 2008)ILSATest
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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