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Particle-based mesoscopic methods for Fluid Dynamics
2) Lattice-Boltzmann Equation (LBE)
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Mesoscopic methods – OPTIONAL exercise / homework
• Assemble in groups and review sections in paper • J. Crawshaw and E.S.Boek, “Multi-scale imaging and simulation of structure,
flow and reactive transport for CO2 storage and EOR in carbonate reservoirs." Reviews in Mineralogy & Geochemistry 77, 431-458 (2013).
1. Molecular Dynamics p. 447 - 448 2. Dissipative Particle Dynamics p. 448 - 450 3. Stochastic Rotation Dynamics p. 450 - 451 4. Lattice Gas and lattice-Boltzmann models p. 451 – 453
• Present to your peers 2
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Molecular dynamics in multiphase fluid flow on the pore scale
5thMarseilleWinterSchoolonMul2-ScalePorousMaterials
January26,2017
Konstan'nosKeremidisVolunteer#1
Crawshaw, J.P., Boek, E.S., “Mul'-scale Imaging andSimula'on of Structure, Flow and Reac've TransportforCO2StorageandEORinCarbonateReservoirs”,pp447-448.
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Individual molecules as the smallest unit
MOLECULAR DYNAMICS APPROACH
Numerical integration of Newton’s equation of motion (e.g. Verlet algorithm)
!(#+Δt)=r(t)+v(t)Δt+1/2 ((#)Δt↑2 +*( Δt↑3 )
!(#−Δt)=r(t)−v(t)Δt+1/2 ((#)Δt↑2 +*( Δt↑3 )
((#)=− 1/+ ∗[,-(!(#))− .↓01# ]
!(#+Δt)=2r(t)−!(#−Δt)+((#)Δt↑2
Time and length scales ranging from nano- to micro- scales
Ensemble averages to calculate average properties (diffusion, viscocity, e.g.)
Source:h*p://www.bgce.de
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MOLECULAR DYNAMICS – APPLICATION TO MULTIPHASE FLUID FLOW ON THE PORE SCALE
Limitations
01 on the order of 23↑ on the order of 23↑−24 sec in typical atomistic simulations
Duration of the simulations on the order of nano- or micro- seconds
Applications
Can be used in multiphase fluid flow simulations on the pore scale provided an accurate scale model
Not efficient to consider the effects of solid boundaries on the fluid
Dimensionless ratios same between MD and real system
Important in studying the behavior of special systems (fluids near solid surfaces, etc)
Hybrid models - MD for fluid near solid, continuum for fluid phase
Continuum inadequate near solid!
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LBE simulation of Hydrodynamics • LBE in a nut shell • Simple and Complex Fluids • Lattice Gas Models; • Boundary conditions; • Single component single phase (SCSP) flow; • Single component Multi phase (SCMP) • Multi component
• Applications: • Non-ideal gases; liquid-vapour phase separation; Laplace law and single
component multiphase models; Wetting and interactions with solid surfaces; Body forces and gravity; Oil-water systems and multi-component models; Solute and heat transport; Flow in Porous media
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Lattice Boltzmann (LB) in a nut-shell
– microscopic kinetic theory: fluid is represented by real-valued distribution functions fi
– solve discrete lattice Boltzmann Equation (LBE):
• Ω is collision operator: • single time relaxation τ • viscosity ν
1. advection step: fi move to adjacent sites 2. collision step: fi relax to equilibrium value
→ macroscopic Navier-Stokes equations are recovered !
0.53
τν
−=
2-D lattice:
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Simple fluids? • Simple Fluids
• Described by a few parameters, e.g. viscosity • Microscopic physics not important for macroscopic properties • Hydrodynamic simulation via “conventional” CFD (FE / FD)
• Complex Fluids • Microscopic structure directly influences bulk properties • E.g., mixtures, suspensions, liquid crystals, etc… • Hydrodynamics required, but often additional physics required
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The Lattice Gas Cellular Automaton • Some simple assumptions
• Particles on a lattice with discrete velocity • Free streaming (or propagation step) • Collisions (conserve particle number, momentum)
• Navier-Stokes fluid!! • However, some problems
• In 2-d collision complexity O(26) while in 3-d O(224)
Frisch et al. Phys Rev Lett (1986)
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Lattice Boltzmann Fluid
• Abandon “Boolean” LGCA • Introduce continuous distribution function • Density of fictitious particles with discrete velocity
• Obeys discrete Boltzmann equation
• Propagation and collision steps • Local equilibrium depends on local density, velocity • Complexity now O(242) !
• Navier-Stokes isothermal / incompressible flow • In practice exact incompressibility is relaxed • Low Mach number constraint (cf. Courant condition) • But, a purely local operation
McNamara and Zanetti Phys. Rev. Lett. (1988), Higuera and Jimenez Europhys. Lett. (1989)
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Discrete velocity model • on a 3-D regular lattice, discrete velocities are, e.g.,
• hydrodynamic quantities are, e.g.,
McNamara and Zanetti Phys. Rev. Lett. (1988),
Higuera and Jimenez Europhys. Lett. (1989)
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• propagation stage • moves distributions
• easy to assign solid / fluid status • boundary at half-way point
• distributions are `bounced back’ • generally no-slip condition • works very well for arbitrarily
complex shapes – porous media
boundary conditions: solid-liquid
E.g., Ladd JFM (1994)
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Single Component Single Phase (SCSP):
1) High Reynolds Numbers
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Reynolds number • What is the definition?
Re = inertial forcesviscous forces
Re = ρvLµ
ρ = density of the fluid (kg/m3) v = characteristic velocity of the fluid (m/s) L = characteristic linear dimension (m) µ = dynamic viscosity of the fluid (Pa·s).
Sir George Stokes, introduced Reynolds numbers Osborne Reynolds popularised the concept
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Flow Past a Cylinder • The drag force FD is defined in terms of the drag
coefficient CD as
• Gravitational Force
FD = rρu02CD
( )2rLWgF πρ −=
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http://scienceworld.wolfram.com/physics/CylinderDrag.html
Tritton DJ (1988) Physical Fluid Dynamics, 2nd Ed. Oxford University Press, Oxford New York
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Flow Past a Cylinder
Re << 1 (Stokes Flow)
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford University Press, Oxford. 519 pp.
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Re = 0.16
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Eddies and Cylinder Wakes
Streamlines for flow around a circular cylinder at 9 ≤ Re ≤ 10.(g=0.00001, L=300 lu, D=100 lu)
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Eddies and Cylinder Wakes
Streamlines for flow around a circular cylinder at 40 ≤ Re ≤ 50.(g=0.0001, L=300 lu, D=100 lu) (Photograph by Sadatoshi Taneda. Taneda 1956a, J. Phys. Soc. Jpn., 11, 302-307.)
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Re = 41
Taneda S (1956)
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LBE: Re = 41
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Eddies and Cylinder Wakes
Re = 30
Re = 40
Re = 47
Re = 55
Re = 67
Re = 100
Re = 41 Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford University Press, Oxford. 519 pp.
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von Kármán Vortex Street, Re = 105
h / l = (1 /π )sinh−1(l) = 0.281
l
h
Photo by S. Taneda. S. Taneda and the Society for Science on Form, Japan
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LBE: Re = 105 - von Karman street
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von Kármán Vortex Street, Re = 105: LBM Simulation (vorticity)
h / l = (1 /π )sinh−1(l) = 0.281
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Single Component Single Phase (SCSP):
2) Flow in complex geometries
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28
• revolution in understanding and description of flow and transport in porous media.
• ability to image rocks and fluids at the pore scale, coupled with novel predictive computational methods.
• micro-CT imaging: • in situ reservoir-condition imaging • pore-scale modelling: • LBE
Revolution in porous media analysis
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• Image and model fluid displacement at the pore-scale • but experiments are hard, expensive, long… • alternative: use dry pore scale images for flow simulation • Predictive models: from pore-scale processes to
dispersive transport to relative permeability. • “digital rock” technology
• Applications: • CO2 storage • optimize oil recovery in giant mixed-wet fractured
carbonate fields. 29
applications in CO2 storage / fossil fuel recovery
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30
benchmarks – rock “library”
Indiana
Capillary trapping
Dispersion, reaction
Capillary trapping
Relative permeability
Dispersion
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Sandstone (Bentheimer)
Bead pack Carbonate (Portland)
Transport modelling
LBE solver and streamlines through pore-space images. Explore the generic nature of displacement.
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Digitised Bentheimer Sandstone with resolution of 6 µm/pixel
Velocity Distribution
Single Phase flow simulation in Porous Media: sandstone
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Transport modelling
YANG AND BOEK: MOLECULAR PROPAGATOR SIMULATION IN POROUS MEDIA X - 17
Figure 1. Pore space images (left: pore is green, solid is blue) and velocity distributions
(right) of porous media with increasing heterogeneity: 1) bead pack; 2) Bentheimer sandstone;
3) Portland carbonate. Resolutions are 5µm, 4.9µm and 9µm respectively. Regarding the velocity
distributions, red and blue indicate high and low velocities respectively.
D R A F T September 16, 2013, 2:50pm D R A F T
Beadpack
Bentheimer sandstone
Portland carbonate
J.Yang, J.Crawshaw, E.S.Boek, Water Res. Research 49, 1–8 (2013)
Lattice Boltzmann solver and particle tracing in pore space images
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Comparison with NMR experiments: propagators
NMR data: Scheven, et al. (2005), Phys. Fluids, 17 (11), 7107
J.Yang, J.Crawshaw, E.S.Boek, Water Res. Research 49, 1–8 (2013)
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2563 643
Single Phase permeability for different porosity and system sizes
1283
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Permeability as a function of porosity, for different rock sub-samples of different heterogeneity
Single Phase permeability for different porosity and system sizes
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TSM DTC Hydrodynamics: 37
`Multi-Component Single Phase (MCSP):
Solute and heat transport
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TSM DTC Hydrodynamics: 38
Multicomponent Miscible LB Models
Multi- Component Multiphase
Miscible Fluids/Diffusion (No Interaction)
Immiscible Fluids
Single Component Multiphase
Single Phase (No Interaction)
Num
ber o
f C
ompo
nent
s Interaction
Strength
Nat
ure
of
Inte
ract
ion
Attractive
Repulsive
Low High
Inherent Parallelism
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��
1) Heat convection in a square cavity
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Heat Convection in a square cavity, Pr=0.71(Air), Ra=1000,10000,100000
Temperature isotherms
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Heat Convection in a square cavity, Pr=0.71(Air), Ra=1000,10000,100000
Velocity Field
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Heat Convection in a square cavity, Pr=0.71(Air), Ra=1000,10000,100000
Streamlines
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��
Initial configuration for Rayleigh-Benard convection
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Rayleigh-Benard convection, Ra=108 �
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TSM DTC Hydrodynamics: 45
Single Component Multi-Phase (SCMP) LBM
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TSM DTC Hydrodynamics: 46
Multiphase LB Models
Multi- Component Multiphase
Miscible Fluids/Diffusion (No
Interaction)
Immiscible Fluids
Single Component Multiphase
Single Phase (No Interaction)
Num
ber o
f C
ompo
nent
s Interaction
Strength
Nat
ure
of
Inte
ract
ion
Attractive
Repulsive
Low High
Inherent Parallelism
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TSM DTC Hydrodynamics: 47
Equations of State
• Perfect (Ideal) gas law • van der Waals gas law • Molar volumes • Temperature dependence and Critical
Points • Liquid-vapor coexistence and the Maxwell
Construction
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TSM DTC Hydrodynamics: 48
Perfect or ideal gas law (EOS)
• P is pressure (ATM)
• V is volume (L)
• n is number of mols
• R is gas constant (0.0821 L atm mol-1 K-1 / 8.314 J mol-1 K-1)
• T is temperature (K)
⇒= nRTPV
VnRTP =
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TSM DTC Hydrodynamics: 49
Single Component D2Q9 LBM (with c = 1 lu ts-1)
mVRTP =
32 ρρ == scP
So, if 1/Vm (mol L-1) is density, cs2 = RT
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TSM DTC Hydrodynamics: 50
van der Waals EOS
2
⎟⎠
⎞⎜⎝
⎛−−
=Vna
nbVnRTP
• ‘a’ term due to attractive forces between molecules [atm L2 mol-2]
• ‘b’ term due to finite volume of molecules [L mol-1]
• Q: Effect on pressure?
VnRTP =Perfect or ideal gas law (EOS):
van der Waals EOS:
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TSM DTC Hydrodynamics: 51
Molar Volume
21⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
mm Va
bVRTP
van der Waals gas law (EOS):
Vm is the volume occupied by one mole of substance. The gas laws can be re-written to eliminate the number of moles n.
2
⎟⎠
⎞⎜⎝
⎛−−
=Vna
nbVnRTP
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TSM DTC Hydrodynamics: 52
CO2: P-V Space
0
50
100
150
200
250
300
0 0.1 0.2 0.3 0.4 0.5Vm (L)
P (a
tm)
Ideal, 298K
Supercritical, 373K
Critical, 304K
Subcritical, 200K
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TSM DTC Hydrodynamics: 53
Liquid-Vapor Coexistence: CO2, P-V Space
6061626364656667686970
0 0.1 0.2 0.3Vm (L)
P (a
tm)
Subcritical, 293KVapor Pressure
Liquid Molar Volume
Vapor Molar Volume
Vapor Pressure
a = 3.592 L6 atm mol-2
b = 0.04267 L3 mol-1
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TSM DTC Hydrodynamics: 54
Maxwell Construction: CO2, P-V Space
6061626364656667686970
0 0.1 0.2 0.3Vm (L)
P (a
tm)
Subcritical, 293KVapor Pressure
A
B
Maxwell Construction: Area A = Area B
Liquid Molar Volume
Vapor Molar Volume
)( ,,,
,lmgm
V
V m VVPPdVgm
lm−=∫
Vapor Pressure
Flat interfaces only!
Gives: --vapor pressure --densities of coexisting liquid and vapor
Exercise: Maxwell construction for vdW
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TSM DTC Hydrodynamics: 55
Liquid - Vapour phase separation immiscible liquid / vapour introduces an interface:
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TSM DTC Hydrodynamics: 56
Laplace Law • surface tension σ between a liquid and its vapor • how do we get σ?
y = 2.003x - 3E-05R2 = 0.9983
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
1/r (lu-1)
ΔP
(mu
ts-2
)
rP σ=Δ
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Multi Component Multi-Phase (MCMP) LBM
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Fluid-fluid interfaces
• A mixture of two fluids • Introduces an interfacial tension
• LB can use two distribution functions • Density of “water” and density of “air” Shan and Chen
Phys. Rev. E (1993)• Density and composition Swift et al. Phys. Rev. Lett. (1994)
• Interfaces evolve naturally on the lattice • No explicit interfacial tracking required • Continuity of velocity field at interface automatic
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Free energy • Symmetric pair of fluids
• Free energy
E.g., Kendon et al. JFM (2001)
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Binary Fluid Immiscible fluid / fluid mixture introduces an interface:
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��
12
12cosσσσ
θ SS −=
Solid Surfaces – contact angle and wetting - Young’s law
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2D immiscible displacement: drainage
• 2-D Berea micromodel
• rock is water wet
• oil displacing water
• qualitative observation: pore body filling, snap off
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3D Primary Drainage – Sandstone – Unsteady State
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Capillary pressure – Bentheimer sandstone
Bentheimer sandstone - 512x256x256 sample, voxel size = 4.9µm, porosity = 0.23; single phase permeability = 4755 mD
E S Boek, J Yang, EM Chapman, JP Crawshaw, SCA2014-060
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3D LB simulation of drainage in sandstone
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Non-wetting phase distribution at different saturations in Bentheimer sandstone: (a) Sw = 0.2, (b) Sw = 0.4, (c) Sw = 0.6, (d) Sw = 0.8
Steady-State Relative Permeability – Bentheimer sandstone
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• Comparison to experimental data - sandstone [1]
[1] T.Ramstad et al., Transport in Porous media 2012, DOI 10.1007/s11242-011-9877-8
the Holy Grail: relative permeability prediction
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Multi-phase flow in pore space images: drainage and imbibition
Process based algorithm to establish initial wettability Bentheimer sandstone: drainage
Slide68
Bentheimer: imbibition
E S Boek, J Yang, EM Chapman, JP Crawshaw,SCA2014-060
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• Sample initially fully saturated with wetting phase
• Drainage followed by a forced imbibition
• Measure residual non-wetting phase clusters number and sizes
• Calculate cumulative cluster size distribution
• Calculate exponential factor and compare to experiment
S(s) = iin(s)i=s
∞
∑
n(s) = N(s) / Nv N(s) = number of clusters of size s Nv = number of pore space voxels
2( )S s s τ− +∝
Residual cluster size distribution
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Typical shape of big (left), medium (middle) and small (right) clusters
• Big clusters span several pores
• Medium size clusters fill only one to three pores
• Small clusters fill only single pores.
Residual cluster size distribution
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Cluster size distribution vs. Cluster size:
LB simulation and experiment.
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Intermezzo
• Computationally intensive calculations • Speed up by running on GPU instead of CPU • but CUDA programming is tricky…
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MATLAB code on the GPU • Same MATLAB code on CPU and GPU
• 200+ MATLAB functions supported on the GPU
• Additional support in toolboxes
Random number generation FFT Matrix multiplications
Solvers Convolutions Min/max
SVD Cholesky and LU factorization
Image Processing Morphological filtering, 2-D filtering
Communications Turbo, LDPC Viterbi decoders
Signal Processing Cross correlation FIR filtering
Requires NVIDIA GPUs with Compute Capability 2.0 or higher. See a complete listing at www.nvidia.com/object/cuda_gpus.html
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MATLAB Code on CPU and GPU Solving 2D Wave Equation
0
10
20
30
40
50
60
70
80
0 512 1024 1536 2048
Tim
e (s
econ
ds)
Grid size
18 x faster
23x faster
20x faster
GPU
NVIDIA Tesla K20c 706MHz
2496 cores memory bandwith 208 Gb/s
CPU
Intel(R) Xeon(R) W3550 3.06GHz
4 cores memory bandwidth 25.6 Gb/s
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Criteria for Good Problems to Run on a GPU
• Massively parallel: • Calculations can be broken into hundreds
or thousands of independent units of work • Problem size takes advantage of GPU cores
• Computationally intensive: • Computation time significantly exceeds CPU/GPU data
transfer time
• Algorithm consists of supported functions: • Growing list of toolboxes with built-in support
• Parallel Support in Toolboxes (pdf) •
• Subset of core MATLAB for gpuArray, arrayfun, bsxfun • MATLAB functions with gpuArray arguments (doc) • Run element-wise MATLAB code on a GPU (doc)
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Speed up MATLAB code with NVIDIA GPUs
4x speedup in adaptive filtering (part of acoustic tracking algorithm)
14x speedup in template matching (part of cancer cell image analysis)
4x speedup in wave equation solving (part of seismic data processing algorithm)
10x speedup in data clustering via K-means clustering algorithm
17x speedup in simulating the movement of 3072 celestial objects
20x speedup in wind tunnel acoustic data analysis (NASA Langley Research Center)
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Reaction – diffusion example
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Reactive flow in porous media for CO2 storage
78
• Understand how flow and chemical system influence structural changes
• Develop predictive simulation model
• Extract parameters for use in reservoir simulation (upscaling)
ORNL Review http://www.ornl.gov/info/ornlreview/v33_2_00/research.htm
• Inject CO2 into a brine aquifer • Produces carbonic acid: • CO2 + H2O −> H+ + HCO3—
• Flow and diffusion of H+
• Reacts with CaCO3 • Dissolution - altering pore-
structure
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Reactive flow in porous media: Dissolution
79
1) Flow: • Calculate flow field with Lattice
Boltzmann 2) Convection-Diffusion: • Transport of H+ using
convection-diffusion eqn. • Finite Volume Method 3) Reaction: 4) Update flow field …
Sphere R=25 in a square channel of 1003 lattice units. Flow pressure boundary condition.
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Dissolution Regimes in carbonate rock
• Acid dissolution experiments show dependence on flow rate
Pe < 1 Pe >>1
“face dissolution” “wormholing”
Can we obtain this from direct simulation?
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Dissolution Regimes in carbonate rock
• Compute dissolution in porous medium by HCl at different flow rates
Ketton geometry at 15.8µm resolution and flow field
Inject HCl
• Inject HCl at 1.5 µLmin-1 (Pe = 0.8) and 150 µLmin-1 (Pe = 80)
• In both cases Da > 1 so we expect transport controlled
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82
Dissolution Regimes: Pe < 1
• Inject HCl at 1.5 µLmin-1 (Pe = 0.8)
• Face dissolution – reactant used up before it is transported into the pore-space
Concentration field of H+
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83
Dissolution Regimes: Pe > 1
• Inject HCl at 150 µLmin-1 - Pe = 80
• Wormhole formation – reactant is transported into the pore-space and through high permeability paths
Concentration field of H+
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TSM DTC Hydrodynamics: 84
Mesoscale simulation of hydrodynamics using particle-
based methods – Lecture1:
• Stochas'cRota'onDynamics(SRD)• Dissipa'vePar'cleDynamics(DPD)
– Lecture2:• LaVceBoltzmann(LB)• Mul'-phase Reac've Flow in porousmedia for CO2 storage / hydrocarbonrecoveryapplica'ons
• Ques'ons?