improvement of the microstructure-based pfc3d modelfracmeet/ffpm2012/srutarshi.pdf · sintef...

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

SINTEF Petroleum Research

Simple situation

3

Composite material under load

• Strength of the material

• Prediction of failure point

SINTEF Petroleum Research 4

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

SINTEF Petroleum Research 5

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

SINTEF Petroleum Research 6

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

SINTEF Petroleum Research

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)

SINTEF Petroleum Research 8

Prediction of Failure point

Burst or avalanche

Hemmer & Hansen, 1992

2/5)( −∆∝∆D

610 ; 20000N avg= =kk xkNF )1( −+=

SINTEF Petroleum Research 9

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)

SINTEF Petroleum Research 10

Similar crossover behavior in seismic data

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

SINTEF Petroleum Research 11

Prediction of failure point

Divergence of χ and τ

1000;105 4 =×= avgN 710N =

SINTEF Petroleum Research 12

Prediction of Failure point: Overloaded situation

Fiber breaking rate: dt

dUtR t=)(

20ft

t =

(Pradhan & Hemmer; 2009)

SINTEF Petroleum Research 13

Prediction of Failure point: Energy burst

Single sample 710=N

2

1

1 ( )2 iEn k x

∆

= ∑Energy release:

20ft

t =

(Pradhan & Hemmer; 2011)

SINTEF Petroleum Research

Complex situation: Fracturing during fluid injection in porous media

14

SINTEF Petroleum Research

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

SINTEF Petroleum Research 16

TerraTek system

MessTek system

SINTEF Laboratory

Jørn, SINTEF

SINTEF Petroleum Research 17

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)

SINTEF Petroleum Research

Test 1: Red sandston

18

SINTEF Petroleum Research

CT Scan Image

19

SINTEF Petroleum Research

Amp. Vs. time

20

SINTEF Petroleum Research

AE data: Amplitude distribution

21

SINTEF Petroleum Research

AE data: Location

22

SINTEF Petroleum Research

Test 2: Limestone

23

SINTEF Petroleum Research

CT scan image

24

SINTEF Petroleum Research

Amp vs. Time

25

SINTEF Petroleum Research

AE data: Amplitude distribution

26

SINTEF Petroleum Research

Event location

27

SINTEF Petroleum Research 28

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

SINTEF Petroleum Research

Hydraulic fracture: PFC 2D

29

SINTEF Petroleum Research

Fracture modeling at mesoscopic scale Bjørn, NTNU

30

Elastic beam lattice model Fracture pattern and pressure distribution

SINTEF Petroleum Research

Macro-scale modeling: MDEM Haitham, SINTEF

31

SINTEF Petroleum Research

Effect of Poisson’s ratio

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

SINTEF Petroleum Research 33

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.

SINTEF Petroleum Research 34

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

SINTEF Petroleum Research 35

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