Large deformation modelling in geomechanics

LBM-DEM, MPM, and LEM


Krishna Kumar, kks32@cam.ac.uk
University of Cambridge

Kenichi Soga, soga@berkeley.edu
University of California, Berkeley.




PUC Rio, Brasil - 2 June 2017
Shell - Cambridge - PUC

Cambridge-Berkeley Computational Geomechanics

  • Lattice-Boltzmann + Discrete Element Method
  • Finite Element Method - Thermo-Hydro Mechanical Coupling
  • Material Point Method
  • Lattice Element Method
View the CB-Geo website for more information and software tools

Global landslide hazard

Fatalities due to landslides, 2007 - 2013 (Source: Nasa, 2015).

Aerial landslides

Afghanistan landslide - 2014 (Source: Boston Globe, 2014).

Samarco dam collapse (2015)

Satellite image from 12 November 2015, one week after the disaster.
© CNES 2015 Distribution Airbus DS

Granular column collapse

Experimental results (Lube et al 2005)

Multiscale modelling in geomechanics

Discrete Element Method

  • Particle level interaction based on Newton's equation of motion

  • The contact force is computed as:

    • $F_n=\left\{ \begin{matrix} \text{ }0\text{ },\text{ }{{\delta }_{n}}>0 \\ -{{k}_{n}}{{\delta }_{n}}-{{\gamma }_{n}}\frac{d{{\delta }_{n}}}{dt},\text{ }{{\delta }_{n}}<0 \\ \end{matrix} \right.$

  • The Newton's equation of motion

    • $F_n =m \times a $

Mesh-based vs Mesh-free techniques

Material Point Method

Porosity in MPM

Material Point Method

Micro to Macro

Simple shear test
Critical state friction angle

MPM v DEM column collapse

a = 0.4
a = 6

DEM column collapse

MPM v DEM column collapse

Run-out v aspect ratio

MPM v DEM column collapse

a = 0.4
a = 6

MPM slope failure

MPM v DEM uniform impact (200 J)

MPM
DEM

MPM v DEM run-out slope v collapse

Run-out v aspect ratio

MPM slope failure: pore pressure changes

Selborne case study (Alonso et al., 2016)

Possible boundary conditions of submarine run‐out

  • Presence of ambient water (larger drag force & less gravity).
  • Water entrainment.
  • Pore pressure does not dissipate.

Submarine landslides

MPM submarine landslide

(Taka., 2012)

Mechanism of submarine landslides

Modelling Test at 1g Condition

  • Material type influences the mode of the flow.
  • Target: Clay‐rich flow (Less diffusive, Hydroplaning).

MPM submarine landslide: Water entrainment

(Taka., 2012)

Mechanism of submarine runout

LBM - DEM simulation of granular collapse in fluid




aspect ratio 'a' of 6

Lattice Boltzmann - MRT

Real Fluid vs LBM Idealisation
LBM D2Q9 Model

\[f_{i}(x + dx, t +\Delta t) - f_{i}(x, t) = -S_{\alpha i}( f_{i}(x, t) - f_{i} ^ {eq}(x, t))\]
  • $S_{\alpha i}$ is the collisional matrix.
  • Probability density of finding a particle : $f(x,\varepsilon, t) $, where, x is position, $\varepsilon$ is velocity, and t is time.
Streaming
Collision

LBM-DEM fluid-solid coupling

$$\Delta t_{s}=\frac{\Delta t}{\mathit{n}_{s}} \qquad (\mathit{n}_{s}=[\Delta t/ \Delta t_{D}]+1) $$
  • At every fluid iteration, $\mathit{n}_{s}$ sub-steps of DEM iterations are performed using the time step $\Delta t_{s}$.
  • The hydrodynamic force is unchanged during the sub-cycling.

LBM laminar & turbulent flows

Lattice Boltzmann

CFD
Poiseuille Flow

Smagorinsky model (LES):

$\nu_{s}(x,t)=(C_s \Delta)^2\sqrt{S_{ij}S_{ij}} \mbox{ ; } S_{ij}=\frac{1}{2}(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})$
Karman Vortex Street

Collapse in fluid

Collapse in fluid ('a'=0.8)

Granular collapse in fluid: Effect of aspect ratio



aspect ratio 'a' of 0.4

aspect ratio 'a' of 4

Collapse in fluid: Runout evolution

a = 0.4
a = 4

Critical time $\tau_c=\sqrt{H/g}$ (Staron and Hinch, 2005)
where, H = Height of the granular pile.

LBM - DEM simulation of granular collapse in fluid




aspect ratio 'a' of 8

Runout: dry vs. fluid

Collapse in fluid: Effect of permeability


Reduction ‘r’=0.7R

Reduction ‘r’=0.9R

Collapse in fluid: Effect of permeability

Dirichlet boundary conditions constrain the pressure/density at the boundaries (Zou and He, 1997)
$\rho_0=\sum_{a}f_{a} \mbox{ and } \textbf{u}=\frac{1}{\rho_0}\sum_{a}f_{a}$


Reduction in radius
LBM-DEM Permeability and Theoretical Solutions

Collapse in fluid: Effect of permeability


Reduction ‘r’=0.7R

Reduction ‘r’=0.9R

Collapse in fluid: Effect of permeability

Effect of permeability: stress

Effect of permeability: effective stress

Runout: effect of permeability

aspect ratio 0.8

Effect of permeability: runout

Effect of permeability: kinetic energy

Effect of permeability: runout

Runout: effect of permeability

aspect ratio 0.8 (loose)

Collapse on an inclined plane




aspect ratio 'a' of 6 on a slope of 5*

CPU v GPU

GPU programming

LBM - DEM a = 0.8 & 10,000 partilces



  • LBM Nodes = 50 Million : DEM grains = 10000 discs
  • Real-time = 2 seconds
  • Run-time = 4 hours
  • Speedup = 25x on a Tesla K20

2D to 3D

LBM multi-component multi-phase


Lattice Element Method

LEM Tension test

LEM: Tension test (uniform)

LEM: Tension test (Log-Normal 1.0)

LEM Tension test

Uniform
LogNormal 1.0

Lattice Element Method - Fluid coupling

  • First assume injection pressure $P_{in}$ and injection rate $Q_{in}$ at injection point
  • Solve fluid pressure at each fluid node
  • Convert pressure to node force and solve LEM to update fracture aperture
  • Repeat the above process until convergence
$$q = - \frac{h^3}{12\mu}\frac{dp}{dx}$$

LEM hydraulic fracturing

John Wong, University of Cambridge

Thank you!


Krishna Kumar, kks32@cam.ac.uk

www.cb-geo.com