by paul delgado. outline motivation heterogeneous multiscale framework fluid flow example...
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Generalization of Heterogeneous Multiscale Models:
Coupling discrete microscale and continuous macroscale representations of physical laws in porous media
By Paul Delgado
Outline•Motivation•Heterogeneous Multiscale Framework•Fluid Flow Example•Generalization for Potential Fields•Steady State Applications•Multiscale-Multiphysics•Challenges
Flow in Porous MediaMicroscopic PhysicsMacroscopic Demand 2µm
5cm10km
•DNS not scalable
Navier StokesHigh Detail – Low Efficiency
Darcy’s LawLow Detail – High Efficiency
Multiscale ModelHybrid Detail-Efficiency
•Goldilocks Problem
Heterogeneous Multiscale Method (HMF) E & Engquist (1994)• Incomplete continuum scale model• Microscale models supplement missing information at continuum scale• Iteration between scales until convergence is achieved
Current work• Based off of Chu et al. (2012) for steady state flow • Discrete microscale constitutive relations with macroscale conservation
laws• Multiscale convergence established for certain non-linear conductance
relations.• Higher Dimensional framework established
Multiscale Framework
Microscale ModelPore Network (Fatt, 1956)•Discrete void space inside porous medium•Network of chambers (pores) and pipes (throats)•Prescribed Hydraulic Conductance•Heuristic Rules for unsteady/multiphase flow•Log-normally distributed throat radii
)( ijijij Pgq
fpC
Network ModelPressure-Flux Equations(potentially non-linear)
Flow Rules
0][
iKj
ijq
g may not be linear
Courtesy: Houston Tomorrow
Macroscale ModelFinite Volume Method
v S(x)
v d V S(x)
dV
dVxSdAnvd
)(
v v(x,P,P)
Fi12 F
i 12Si
Fn Fs Fe Fw Si
No explicit form of v is assumed
1D 2D
),(
0
)0,,(),,(),,(
1
1
21
21
21
21
iii
ii
l
x
lxl
PPKPP
fP
PxFPPxF
dPx
PxdF
By mean value theorem,
Iterative Coupling
Assume when .0),,( xPPxF 0xP
where fi 12F
i 12(Pi,Pi1)
Let be the characteristic length of the microscale model.
dF(x,P,)dPx
F(x,P,Px ) F(x,P,0)
Px 0 [0,Px ]
xx PdPx
PxdFPPxF
),,(),,(
Estimate
Hence
Fi12(Pi,Pi1)
dF(x,P,)dPx
P Ki12(Pi,Pi1)D
[Pi]
Chu et al, (2012)
Multiscale Coupling
Iterative Coupling:
xQPDPPKD
PPfPDPxF
ln
ln
ln
ll
nl
nll
nl
n
ll
)1()(1
)(
)(1
)()()(
,
,ˆ)(,,
21
21
21
21
QpK n ˆ)1( Macroscopic
Microscopic
NlfpG n
l
n
l ,..,0,)()(
21
21
Chu, et al. (2011b)
Numerical Analysis
Chu et al. (2012) examined numerical properties of this micro-macro iteration scheme
•Existence •Uniqueness•Consistency•No stability conditions required•Order of convergence•Source terms•Multidimensional and anisotropic cases
Steady State Physics
Conservation LawConstitutive RelationSteady State Equation
S
K
SK
Classical Continuum Mechanics
Heterogenous Multiscale Approach
Coupling Relation
Micro-Conservation Law
Micro-Constitutive Relation
MM S
),( mMF
m k (v k )
Macro-Conservation Law
f ijk 0
f ijk g()
Microscale models are discrete projections of macroscale relations
Example 1Discrete Microscale
Model
fpC )(
qij
jK [ i]
0
Conservation Law
System of Equations
Courtesy: University of Manchester
v v(x,P,P)
dVxSdAnvd
)(
Continuous Macroscale Model
Constitutive Relation
)(1)()()( ,ˆ)(,,21
21
21
nl
nll
nl
n
ll PPfPDPxF
Continuum Scale Equation
Microscale Equations
Control Volume
Multiscale Coupling
•Pressure centered control volumes•Flux at boundaries evaluated using microscale network models•Iteration between scales to convergence
)( Pgq Constitutive Relation
)(xSv Conservation Law
xQPDPPKD ln
ln
ln
ll
)1()(1
)( ,21
System of Equations
Flow in Porous Media
Example 2
xSTDTTKD
TTfTDTxF
ln
ln
ln
ll
nl
nll
nl
n
ll
)1()(1
)(
)(1
)()()(
,
,ˆ)(,,
21
21
21
21
Continuum Scale Equation
Microscale Equations
Control Volume
•Temperature centered control volumes•Flux at boundaries evaluated using microscale network models•Iteration between scales to convergence
fTC
)(
qij
jK [ i]
0Conservation Law
System of Equations
Courtesy: University of Manchester
)( Tgq Constitutive Relation
Conservation Law
Heat Transfer in Porous Media
),,( TTxQQ
dVxSdAnQd
)(
Constitutive Relation
)(xSQ System of Equations
Discrete Microscale Model
Continuous Macroscale Model
Multiscale Coupling
Example 3
derived... beingcurrently
Continuum Scale Equation
Microscale Equations
Control Volume
Multiscale Coupling
•Displacement centered control volumes•Forces at boundaries evaluated using microscale spring system models•Iteration between scales to convergence
Discrete Microscale Model
fxK )(
0][
iKj
ijF
Conservation Law
System of Equations
Courtesy: University of Manchester
)( xkF
Constitutive Relation
Conservation Law
Linear Elasticity in Porous Media
),,( uux
v n dA
d S(x)
dV
Continuous Macroscale Model
Constitutive Relation
)(xS System of Equations
Models
Microscale Flow Microscale Deformation
Continuum Flow Continuum Deformation
Biot (1941), Kim (2010)
Darcy’s Law (1856)
Chu et. al. 2012
Zienkiewicz et. Al. (1947)
Current Work
Courtesy: Georgia College
Courtesy: Symscape
Fatt et. al. (1956)
Courtesy: Miehe et. Al. (2002)
Courtesy: Dostal et. Al. (2005)
Current Work
Current Direction
Uniphysics multiscale models withmicroscale muliphysics coupling
•Interscale communication for all physics•Interphysics communication at microscale only.
+ Consistent with HMM Framework+ Amenable to C2 non-linear microscale models for all physics
Micro-Flow Micro-Deformation
Macro-Flow Macro-Deformation
Challenges
•Microscale multiphysics coupling•Non-overlapping microscale models•Deformation mechanics multiscale coupling•Lagrangian & Eulerian Reference Frames•Iterative multiphysics coupling between timesteps
•Working Paper: A discrete microscale model coupling flow and deformation mechanics
•Working Paper: A generalization of the HMM framework coupling continuous macroscale and discrete microscale models of steady state uniphysics for porous media.
ModelsMicroscale Multiphysics Model Prototype I:
•Iterative coupling between physics•Flow first, deformation second•Solid Matrix pinned at center•Horizontal linear elasticity only•Modeled as Hooke springs.
Observations: •Unrestricted deformation near inlet•Deformation steady near outlet•Pressure at P2 approaches outlet pressure as inlet throat widens•No time dependendent terms introduced in model