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SINTEF Petroleum Research 1
Srutarshi Pradhan
SINTEF Petroleum Research Fracture & Flow meeting, January 2012,
IMSc, Chennai, India
Fracture Modeling: Simple and complex situations
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Dynamics of fracture
2
Crack-growth in nano-materials ……onset of earthquake
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Simple situation
3
Composite material under load
• Strength of the material
• Prediction of failure point
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Fiber bundle model (FBM)
First used in textile engineering (Peirce, 1926)
Statistical analysis (Daniels,1945)
Different class of FBM
Static and dynamic FBM
Different load-sharing rules: ELS, LLS, mixed-mode, hierarchical
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Failure dynamics
The Recursion relation
)/(11 tt UPU σ−=+ ∫=y
dxxpyP0
)()(
NNU tt /=Uniform dist. 1 1 /t tU Uσ+ = −
Fixed point 0*2* =+− σUU
Solution
(Pradhan, Bhattacharyya & Chakrabarti; 2001-02)
*1 UUU tt ==+
2/1)(21)( σσσ −±=∗
cU
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Critical behavior
Order parameter
Susceptibility 2/1)()( −∗
−≈= σσσσχ cd
dU
Relaxation time 2/1)( −−∝ σστ c
(Pradhan, Bhattacharyya & Chakrabarti; 2001-02)
2/1* )()()( σσσσ −≈−= ∗ccUUO
tttt U
UUUdt
dU σ+−=−= + 11Differential form
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Failure time
Above critical stress
)(σft is the failure time
Below critical stress
)(σft is the time to reach the fixed point
2/12 )()( −−= cft σσσ π
2/14
)ln( )()( −−= σσσ cN
ft(Pradhan & Hemmer; 2007)
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Prediction of Failure point
Burst or avalanche
Hemmer & Hansen, 1992
2/5)( −∆∝∆D
610 ; 20000N avg= =kk xkNF )1( −+=
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Prediction of Failure point
Crossover behavior
610 ; 50000N avg= =
Single sample 710N =
)1()( /2/5 ceD ∆∆−− −∆∝∆
20
18( )c
cx x∆ =
−
(Pradhan, Hemmer & Hansen; 2005)
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Similar crossover behavior in seismic data
Seismic data prior to a mainshock (Kawamura et al, 2006)
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Prediction of failure point
Divergence of χ and τ
1000;105 4 =×= avgN 710N =
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Prediction of Failure point: Overloaded situation
Fiber breaking rate: dt
dUtR t=)(
20ft
t =
(Pradhan & Hemmer; 2009)
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Prediction of Failure point: Energy burst
Single sample 710=N
2
1
1 ( )2 iEn k x
∆
= ∑Energy release:
20ft
t =
(Pradhan & Hemmer; 2011)
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Complex situation: Fracturing during fluid injection in porous media
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Methods: 1) Fracturing test on core sample 2) PFC at pore scale 3) Beam-lattice model at mesoscopic scale 4) MDEM at macro scale (even resorvoir scale) Application: 1) Planning safe & efficient drilling (geothermal, shale-gas) 2) Reservoir characterisation (CO2 storage) 3) Prediction of well collapse & leakage 4) Enhance production by increasing permeability.
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TerraTek system
MessTek system
SINTEF Laboratory
Jørn, SINTEF
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Triaxial cell instrumentation
Sample
Directions of radial strain measurements
Steel piston
Acoustic transducer Sintered plate
Sleeve
Axial strain LVDT
Acoustic transducer
Pore pressure (fluid flow)
Pore pressure (fluid flow)
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Test 1: Red sandston
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CT Scan Image
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Amp. Vs. time
20
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AE data: Amplitude distribution
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AE data: Location
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Test 2: Limestone
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CT scan image
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Amp vs. Time
25
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AE data: Amplitude distribution
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Event location
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Pore scale modeling: using PFC
• PFC (Particle Flow Code) is a code based on the Discrete Element Method (DEM).
• PFC solves the equations of motion directly.
• In each time step, the movements of all the particles are calculated according to the motion law, and the forces at all the contacts are calculated according to a contact law.
ITASCA, USA
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Hydraulic fracture: PFC 2D
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Fracture modeling at mesoscopic scale Bjørn, NTNU
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Elastic beam lattice model Fracture pattern and pressure distribution
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Macro-scale modeling: MDEM Haitham, SINTEF
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Effect of Poisson’s ratio
Poisson’s ratio =0.25 Poisson’s ratio=0.4
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Conclusions
Failure in FBM
Dynamics of failure can be formulated by recursion relation.
Failure dynamics shows critical behavior.
Time to failure/reach fixed point shows 2-sided divergence.
Quasi-static loading: Crossover behavior in burst statistics.
Loading by discrete steps: Divergence of susceptibility & relaxation time.
Overloaded situation: Breaking rate and Energy burst have a minimum at the half way to complete failure.
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Conclusions
Lab test on core sample: Fracture pattern depends on strength & brittleness of the cores.
PFC: Ratio between tensile & shear strength influnces fracture pattern & numbers.
Beam-lattice: Disorder in strength and pressure distribution are responsible for different fracture pattern.
MDEM: Number of fracture branches depends on Poisson ratio.
Effect of pre-existing ferctures ?? & temperature ?
Fluid injection in porous media
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FUNDING: RESEARCH COUNCIL OF NORWAY (NFR)
THROUGH CLIMIT PROGRAMME
Collaborators
Jørn Stenebråten, Haitham Alassi, SINTEF Alex Hansen, Per C. Hemmer, Bjørn Skjetne, NTNU Bikas K. Chakrabarti, P. Bhattacharyya, SINP
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