Numerical Modelling of Fracture Propagation and Fragmentation in Brittle Rocks  Dr Adriana Paluszny, Morteza Nejati, Philipp S. Lang Prof Robert W. Zimmerman, Rock Mechanics Chair Dept Earth Science and Engineering Imperial College London November 12, 2015 - Dept Geotechnical Engineering and Geosciences Universitat Politecnica de Catalunya, UPC-BarcelonaTech

Flow in Fractured Porous Media

Kárahnjúkar tunnel, Iceland, therobbinscompany.com

The total flow is related to the pressure gradient by the effective, or equivalent, permeability of the fracture-matrix ensemble

Quantifying Fractures in Rocks for Flow Modelling

Analytical Fracture network statistics • density • orientation distribution • length distribution • aperture distribution • Limitations

•  often 2D •  impermeable/uniform matrix •  restrictions on nature of

distributions

Numerical Fracture network models • discrete fracture networks (DFN) models, matrix assumed impermeable or not contributing • discrete fracture and matrix models (DFM), i.e., fractured porous media models

DFM Models in 2D

Tracing fractures: Outcrops allow to create models with geologically realistic fracture geometries

Zahm & Hennings, 2009

DFM Models in 2D

Zahm & Hennings, 2009

statoil.com 5 m

Flow in Fractured Porous Media

Flow varies heterogeneously throughout the ensemble of fractures and rock, depending on •  connectivity •  conductivity of fractures and whether the network allows for viscous pressure gradients to exist across matrix blocks. An ensemble throughput will result. The proportionality factor that relates this flow to the hydraulic gradient is referred to as effective of equivalent permeability

Permeability of Fratured Porous Media 2D vs. 3D

[Lang et al. JGR, 2014]

•  2D underestimates consistently, up to orders of magnitude

•  Retardation of percolation in 2D with respect to 3D

•  Largest uncertainty around 2D percolation density

Context

-  Numerical simulation of fracture and fragmentation (in 3D) -  Fracture growth -  Fragmentation dynamics -  Fracture pattern formation

-  Development of C++ based numerical simulator (uses: CSMP++, SAMG solver library, and non-interactive 3D Tetra Meshing)

CSMP++ and the Geomechanics Toolkit

• C++ based numerical library for finite element & finite volume methods • Unstructured grids • Discrete fracture representation • THMC applications • Core developers (~2-3) • Numerical methods developers (~4-6) • Application programmers (~17) • Users (~100)

Two-phase flow Numerical methods Core development

Black-oil Parallelization

Computational mechanics

Reactive, compositional high temperature transport

Numerical (forward) models for fracture propagation

Fracture Growth (2D) - Principles

(1)

(2)

(3)

Initial flaws

Separate Geometry from Mesh

Growth Laws (energy-based) -  Failure -  Propagation -  Angle

[Paluszny & Matthai, IJSS, 2009]

Fracture pattern (2D)

[Paluszny & Matthai, IJSS, 2009]

Polygonal patterns

Polygonal fracture growth due to shrinkage of the matrix. Mean stress contours are plotted with the polygonal fracture pattern. Stress concentrates ahead of the fracture tips.

Stress intensity factor computation: RVIT technique

[Paluszny & Zimmerman, CMAME, 2011]

[Nejati et al., IJSS & EFM, 2015]

1960-2010

2010-2014

2015

Reduced virtual integration (Disk)

SIF Errors are quantified

Fracture Growth (3D)

[Paluszny & Zimmerman, CM, 2013]

Fracture: direct growth of fracture NURBS

[Paluszny & Zimmerman, Engineering Fracture Mechanics, 2013]

Fracture set growth

[Paluszny & Zimmerman, Engineering Fracture Mechanics, 2013]

Context: Interaction at growth (1)

Context: Interaction at growth (2)

Fragmentation context: FEM with impulse-based dynamics

Draw point analysis

-  A set of initial flaws grows as a function of stress relief at the draw point

-  Growth induces fragmentation

[Paluszny & Zimmerman, ARMA, 2012]

Primary fragmentation and flow

[Paluszny & Zimmerman, ARMA, 2012]

Secondary fragmentation: Fractures drive fragmentation

Fracture propagation is evaluated for each fragment as a function of impact, with fracture growth based on local modal stress intensity factors.

Fracture-driven Fragmentation

[Paluszny & Zimmerman, Computational Mechanics, 2013]

Fracture-driven fragmentation (3D)

Velocity-dependent fragmentation pattern

[Paluszny & Zimmerman, Computational Mechanics, 2013]

Increasing impact velocity

Fracture-driven fragmentation

[Paluszny & Zimmerman, Computational Mechanics, 2013]

Multiple fracture growth – coarse mesh but accurate SIFs

- Peak: 320k nodes - Runtime: 10 hours

(minutes to run)

All simulations run on a Dell Precision Workstation (2013) with a maximum of 8 cores dedicated to one job.

Mechanical Variables Poisson’s ratio Density Young’s Modulus UCS Fault Friction

Caving

Realistic extraction schedule: Caved elements

Initial fracture geometry (1)

Growth

Growth

Growth

Growth during caving (1)

Displacement contours around horizontal fractures before caving

Displacements detail Details of the displacement pattern from a perspective view. Fractures and model boundaries are represented with a wireframe mesh, while effective stresses are depicted by ghosted contours

Initial fracture geometry (2)

Growth during caving (2)

Effect of the fractures of the prediction of cave displacement

(a) initial state (b) three orthogonal sets (developed)

(c) single set (onset) (d) single set (developed)

Cave shape

Cave shapes for month 12, for (a) a model without fractures and (b) a model with ninety fractures, for (s2) boundary conditions

(a)

(b)

Related Publications

Nejati, M., Paluszny, A., Zimmerman, R.W. (2015) “A disk-shaped domain integral method for the computation of stress intensity factors using tetrahedral meshes”, Int. J. Solids Struct., 69-70, 230-251.

Nejati, M., Paluszny, A., Zimmerman, R.W. (2015) “On the use of quarter-point tetrahedral finite elements in linear elastic fracture mechanics”, Eng. Fract. Mech., 144, 194-221.

Tang, X.H., Paluszny, A., Zimmerman, R.W. (2014) “An impulse-based energy tracking method for collision resolution”, Comput. Meth. Appl. Mech. Eng., 278, 160-185.

Paluszny, A., Tang, X.H., Zimmerman, R.W. (2013) “Fracture and impulse based finite-discrete element modeling of fragmentation”, Comput. Mech., 52(5), 1071-1084.

Paluszny A, Zimmerman RW (2013) Numerical fracture growth modeling using smooth surface geometric deformation", Engineering Fracture Mechanics (available online).

Nejati M, Paluszny A, Zimmerman RW (2013) Theoretical and Numerical Modeling of Rock Hysteresis Based on Sliding of Microcrack" 47th U.S. Rock Mechanics / Geomechanics Symposium (ARMA), San Francisco, USA, 23-26 June.

Zimmerman RW, Paluszny A (2012) Some New Developments in Modelling the Failure, Fracture and Fragmentation of Rocks", 7th Asian Rock Mechanics Symposium, Invited Paper, Seoul, Korea, 15-19 October.

Paluszny A, Zimmerman RW (2011) Numerical simulation of multiple 3D fracture propagation using arbitrary meshes", Computer Methods in Applied Mechanics and Engineering, Vol:200, Pages:953-966.