hydraulic fracture: multiscale processes and moving interfaces
DESCRIPTION
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). - PowerPoint PPT PresentationTRANSCRIPT
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