One dimensional models of hydraulic fracture

Anthony Peirce (UBC)

Collaborators: Jose` Adachi (SLB) Shira Daltrop (UBC)Emmanuel Detournay (UMN) WITS University

1 April 2009

2

Outline

• The HF problem and 2D models

• Slender geometries -> the possibilities for 1D models

• The classic PKN model – porous medium eq – limitations

• Extension of PKN to include toughness - 1D integro-PDE

• P3D model – PKN methodology ->pseudo 3D

• Conclusions

3

HF Examples - block caving

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HF Example – caving (Jeffrey, CSIRO)

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Oil well stimulation

FracturingFluid

Proppant

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Lab test with stress contrast (Bunger)

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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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The PKN Model

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Assumptions behind the PKN Model

• Pressure independent of :

• Vertical sections approximately in a state of plane strain:

• PKN model:

Local elasticity equation

Can’t model pressure singularities for example when

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PKN – Averaging the Lubrication Eq

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Scaled lubrication & similarity solution

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Numerical soln of a finger-like frac

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Width and scaled pressures

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Asymptotics of numerical soln

1.E-02

1.E-01

1.E+00

1.E-03 1.E-02 1.E-01 1.E+00

1 - x/L

Nor

mal

ized

frac

ture

wid

th (w

/wo)

Planar3D

(1-x/L) (̂1/3)

(1-x/L) (̂2/3)

H/L = 0.13L = 85.0 m

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Extended PKN:2D integral eq 1D integral eq

• Integral eq for a pressurized rectangular crack

• Re-scale variables:

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Asymptotic behaviour of the kernel

~ Hilbert Transform

PKN

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Assumed behaviour of

Power law in the tip regions:

Analytic away from the tips:

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Outer expansion

Hilbert Transform Region

PKN Region

PKN

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Outer expansion

Test function:

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Inner expansion:

Hilbert Transform Region

PKN Region

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Inner Expansion

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Discretizing the Elasticity Equation

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Discretizing the Fluid Flow Equation

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EPKN Tip solutions – viscosity

Close to the tip

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Locating the tip position

Viscosity Dominated:

Toughness Dominated:

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Numerical Results K=0

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Numerical Results K=1

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Numerical Results K=5

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P3D models

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Scaled elasticity equations

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Averaged equations

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Scaled lubrication equation

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Collocation scheme to solve ODE

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Numerical Results

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Footprint comparison with 3D solution

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Width comparison with 3D solution

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Concluding remarks

• The HF problem and 2D models• Slender geometries -> the possibilities for 1D models• The classic PKN model – porous medium eq - limitations• Extension of PKN to include toughness - 1D integro-PDE

Reduction of the 2D integral to a 1D nonlocal equation in the small aspect ratio limit

Asymptotics of the 1D kernel Asymptotic analysis to determine the action of the integral

operator in different regions of the domain:Outer region: 1D integral equation - local PDE Inner tip region: 1D integral equation - Hilbert Transform

Tip asymptotics and numerical solution• P3D model – PKN methodology ->pseudo 3D

Plane strain width solution and averaging Averaging and scaling the lubrication Reduction to a convection diffusion equation Numerical solution via collocation Results and comparison with 3D solution

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PKN Traveling Wave solution