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$: Improvement of the microstructure-based PFC3D modelfracmeet/FFPM2012/srutarshi.pdf · SINTEF Petroleum Research . 28. Pore scale modeling: using PFC •PFC (Particle Flow Code) is$
SINTEF Petroleum Research 1

Srutarshi Pradhan

SINTEF Petroleum Research Fracture & Flow meeting, January 2012,

IMSc, Chennai, India

Fracture Modeling: Simple and complex situations

SINTEF Petroleum Research

Dynamics of fracture

2

Crack-growth in nano-materials ……onset of earthquake


Simple situation

3

Composite material under load

• Strength of the material

• Prediction of failure point


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


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


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


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)


Prediction of Failure point

Burst or avalanche

Hemmer & Hansen, 1992

2/5)( −∆∝∆D

610 ; 20000N avg= =kk xkNF )1( −+=


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)


Similar crossover behavior in seismic data

Seismic data prior to a mainshock (Kawamura et al, 2006)


Prediction of failure point

Divergence of χ and τ

1000;105 4 =×= avgN 710N =


Prediction of Failure point: Overloaded situation

Fiber breaking rate: dt

dUtR t=)(

20ft

t =

(Pradhan & Hemmer; 2009)


Prediction of Failure point: Energy burst

Single sample 710=N

2

1

1 ( )2 iEn k x

∆

= ∑Energy release:

20ft

t =

(Pradhan & Hemmer; 2011)


Complex situation: Fracturing during fluid injection in porous media

14


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.

15


TerraTek system

MessTek system

SINTEF Laboratory

Jørn, SINTEF


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)


Test 1: Red sandston

18


CT Scan Image

19


Amp. Vs. time

20


AE data: Amplitude distribution

21


AE data: Location

22


Test 2: Limestone

23


CT scan image

24


Amp vs. Time

25


AE data: Amplitude distribution

26


Event location

27


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


Hydraulic fracture: PFC 2D

29


Fracture modeling at mesoscopic scale Bjørn, NTNU

30

Elastic beam lattice model Fracture pattern and pressure distribution


Macro-scale modeling: MDEM Haitham, SINTEF

31


Effect of Poisson’s ratio

Poisson’s ratio =0.25 Poisson’s ratio=0.4


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.


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


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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