Large deformation modelling in geomechanics

Lattice-Boltzmann and Discrete Element Method


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

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




NMG2017, Hamburg, Germany - 27 September 2017

Possible boundary conditions of submarine run‐out

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

Mechanism of submarine landslides

Modelling Test at 1g Condition

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

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.

Granular column collapse

Experimental results (Lube et al 2005)

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.

Runout: dry vs. fluid

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

Collapse on an inclined plane




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

Collapse of a dense column on an inclined plane

aspect ratio 'a' of 0.8 on a slope of 5* (dense)

Collapse of a dense column on an inclined plane

aspect ratio 'a' of 0.8 on a slope of 5* (dense)

Collapse of a dense column on slopes: runout

aspect ratio 'a' of 0.8 (dense)

Collapse of a loose column on slopes: runout

aspect ratio 'a' of 0.8 (loose)

Collapse on slopes: loose v dense

Loose
Dense

Kinetic energy evolution

Loose v dense: Initiation phase

initial runout evolution ('a' of 0.8)

Loose v dense: Initiation phase

Loose
Dense

Pore-pressure distribution along the failure plane during initiation.

Loose v dense: Runout phase

Attack angle ('a' of 0.8) $t = 3 \tau_c $

Loose v dense: Runout phase

Loose
Dense

Water entrainment front (~15d length) at a slope of 5*

Loose v dense: Runout phase

Froude's number - hydroplaning ('a' of 0.8)

Loose v dense: Settlement phase

volume evolution ('a' of 0.8)

Collapse on slopes: loose v dense

runout evolution ('a' of 0.8)

LBM - DEM a = 0.8 & 10,000 partilces



  • LBM Nodes = 50 Million : DEM grains = 7500 discs
  • Real-time = 2 seconds
  • Run-time = 4 hours
  • Speedup = 25x on a Tesla K20

Thank you!


Krishna Kumar, kks32@cam.ac.uk

www.cb-geo.com