Hydraulic Fracture: multiscale processes and moving

interfaces

Anthony PeirceDepartment of Mathematics

University of British Columbia

Nanoscale Material Interfaces:Experiment, Theory and Simulation

Singapore10-14 January 2005

Collaborators:

• Jose` Adachi (SLB, Houston)• Rachel Kuske (UBC)• Sarah Mitchell (UBC)• Ed Siebrits (SLB, Houston)

2

Outline

• What is a hydraulic fracture?

• Scaling and special solutions for 1D models

• Numerical modeling for 2-3D problems

• Conclusions

3

HF Examples - block caving

4

HF Example – caving (Jeffrey, CSIRO)

5

HF Examples – well stimulation

FracturingFluid

Proppant

6

An actual hydraulic fracture

7

HF experiment (Jeffrey et al CSIRO)

8

1D model and physical processes

FractureEnergy

Breaking rock

Leak-offViscous energy

loss

9

Scaling and dimensionless quantities

• Rescale:• Dimensionless quantities:

• Governing equations become:

10

Two of the physical processes

• Large toughness:

• Large viscosity:

11

Experiment I (Bunger et al 2004)

Theory

Large toughness:

(Spence & Sharp)

12

Experiment II (Bunger et al 2004)

TheoryLarge viscosity:

(Desroches et al)

13

The coupled EHF equations in 2D

• The elasticity equation – balance of forces

• The fluid flow equation – mass balance

14

Boundary & propagation conditions

• Boundary conditions

• Propagation condition

15

The elasticity equation

x,m,k

y,n,l

z

16

3 Layer Uniform Asymptotic Solution

-4 -3 -2 -1 0 1 2 3-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6 Sample spectral coefficient for a touching element

log10

(k)

Al=4

,Ps1

(k)

Actual spectral coefficient Uniform asymptotic solution Infinite space spectral coefficient

17

Crack cutting an interface

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

2

Nor

mal

ized

cra

ck o

peni

ng d

ispl

acem

ent:

w

G1/p

1

Crack opening displacement comparison with Erdogan and Biricikoglu

Erdogan & Biricikoglu b2/b

1 =0.05

Erdogan & Biricikoglu b2/b

1 =1.00

Erdogan & Biricikoglu b2/b

1 =2.00

MLAYER2D b2/b

1 =0.05

MLAYER2D b2/b

1 =1.00

MLAYER2D b2/b

1 =2.00

18

A penny crack cutting 2 interfaces

Lin and Keer three layer test

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 1

y/r

Gw

/(rp

)

MLAYER3D (eta = 2)

MLAYER3D (eta = 4)

MLAYER3D (eta = 10)

Lin & Keer (eta = 2)

Lin & Keer (eta = 4)

Lin & Keer (eta = 10)

19

A crack touching an interface

0.08 0.09 0.1 0.11 0.12 0.13 0.140.34

0.345

0.35

0.355

0.36

y (m)

wid

th w

(m

m)

ABAQUS 4000 elements ABAQUS 16000 elements DIGSMM linear elements BHP 15 quadratic elements MLAYER2D 20 elements MLAYER3D 20 vertical elements

20

Eulerian approach & Coupled HF Equations

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1

23

45

67

9

10

8

11

12

131415

16

17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1

23

4

5

67

9

10

8

11

12

131415

16

17

1 2 3 4 5 6 7 8 9

1

2

3

4

5

6

7

8

9

21

MG preconditioning of• C coarsening using dual mesh

• Localized GS smoother

i,j

i,j+1

i,j-1

i-1,j

i-1,j-1

i-1,j+1

i+1,j

i+1,j+1

i+1,j-1

12345

112345

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

22

Performance of MG Preconditioner

0 1000 2000 3000 4000 5000 60000

0.5

1

1.5

2

2.5

3 Cumulative solution times BiCGSTAB vs BiCGSTAB-MG

Number of active elements (N)

Cu

mu

lativ

e S

olu

tion

Tim

e (

ho

urs

)

BiCGSTAB : Total solution time 2.58 hours BiCGSTAB-MG V(0,2) cycle : Total solution time 0.27 hours BiCGSTAB-MG V(1,2) cycle : Total solution time 0.31 hours

1 1.5 2 2.5 3 3.5 40.5

1

1.5

2

2.5

3

log10(N)

log 1

0(Nu

mb

er

of

itera

tion

s)

Average iteration counts for BiCGSTAB and BiCGSTAB-MG

BiCGSTAB with number of iterations ~ 0.69 N0.72

BiCGSTAB-MG V(0,2) cycle BiCGSTAB-MG V(1,2) cycle

~9.5 x

23

Front evolution via the VOF method

0.75 0.15

1.0

1.0 1.0

0.78

0.3

0.06

24

Time stepping and front evolution

Time step loop:

VOF loop:

Coupled Solution

end

next VOF iteration

next time step

25

Radial Solution

Radial fracture ISTEP = 19

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

1.0E-02

0 25 50 75 100 125

y (m)

Fra

ctu

re w

idth

(m

)

NumericalExact

26

Fracture width for modulus contrast

150 200 250 300 350 400 450 500 550 6000

1

2

3

4

5

6

7

8

9x 10

9

y (m)

Youngs Modulus (MPa)

Modulus Changes

Source

27

Fracture width for stress jump

28

Pressure and width evolution

29

Channel fracture and breakout

30

Hourglass HF with leakoff

31

Concluding remarks

• Examples of hydraulic fractures

• Scaling and physical processes in 1D models Tip asymptotics: & Experimental verification

• Numerical models of 2-3D hydraulic fractures 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