stability & convergence of sequentially coupled flow-deformation models in porous media
DESCRIPTION
Stability & Convergence of Sequentially Coupled Flow-Deformation Models in Porous Media. By Paul Delgado. Outline. Motivation Flow - Deformation Equations Discretization Operator Splitting Multiphysics Coupling Fixed State Splitting Other Splitting Conclusions. Motivation. - PowerPoint PPT PresentationTRANSCRIPT
Stability & Convergence of Sequentially Coupled Flow-Deformation Models in
Porous MediaBy Paul Delgado
•Motivation•Flow-Deformation Equations•Discretization •Operator Splitting•Multiphysics Coupling•Fixed State Splitting•Other Splitting•Conclusions
Outline
(Quasi-Static) Poroelasticity Equations
Using constitutive relations, we obtain a fully coupled system of equations in terms of pore pressure (p) and deformation (u)
How hard could it be to solve these equations?
Motivation
f
dm f
dtw f S f
Courtesy: Houston Tomorrow
Mechanics Flow
gf b
sfb )1( 00
mf = variation in mass flux relative to solidwf = mass flux relative to solidSf = mass source term
σ = Total Stress Tensorf = body forces per unit area
0 ffm
fff vw 0
DiscretizationdHU and LP Let ))(()( 12
UspanU and PspanP Let ihih
f
nf :
DeformationStron
g form
Weak form
Flow
dm f
dtw f S f
hP
Strong
form
fff Sw
dtdm
hP
Weak form
111:
nnnnf Backward
Euler Form
11
1n
fn
f
nf
nf Sw
tmm
Backward Euler Form
If constitutive relations are non-linear, => Non linear system 0),(
0),(
puNpuN
f
d
Multiphysics Solvers
Simultaneous coupling between flow & deformation at each time step
•Computationally expensive•Code Intrusive•high order approximations are difficult to achieve•Strong numerical stability & consistency properties
Iteration between physics models within a single time step
•computationally cheap•Enables code reuse•Easier to achieve higher order accuracy•Variable convergence properties
We will examine the strategies for sequential coupling and their convergence propertiesWe summarize the work of Kim (2009, 2010) illustrating iterative coupling strategies.
Operator SplittingBased on Kumar (2005)
0),(0),(
hhf
hhd
puNpuN
Newton-Raphson at
time t
kt
kt
k
k
f
dkk
fffd
dfddk
xxx
bb
pu
JJJJ
xJ
111
Rewrite the Jacobian matrix as:
00
00 df
fffd
dd JJJ
J
Until convergence
1
11
t
tt
pu
x
Jdd= mechanical equation with fixed pressureJfd + Jff = flow equation with solution from Jdd
Rewrite Newton-Raphson as Fixed Point Iteration kk
dfk
f
dk
fffd
dd
puJ
bb
pu
JJJ
0000 11
In operator splitting, we apply this technique to the discrete (linear) operators governing the continuous system of equations.
Drained SplitAlgorithm:1. Hold pressure constant2. Solve deformation first3. Solve flow second4. Repeat until convergence
Iteration
Deformation 0p
Flow
t t+1
If converged
If not converged
1
11
t
tA
t
tA
t
t
pu
pu
pu f
drd
dr
)0( pfA mdr
)0( uSvdtdmA f
fdf
0u
State variables are held constant alternately
How else can we decompose the operator?
Undrained SplitAlgorithm:1. Hold mass constant2. Solve deformation first3. Solve flow second4. Repeat until convergence
Iteration
Deformation
€
m = 0
Flow
t t+1
If converged
If not converged
€
ut
pt
⎡
⎣ ⎢
⎤
⎦ ⎥→Adr
d ut +1
pt + 12
⎡
⎣ ⎢
⎤
⎦ ⎥→Adr
f ut +1
pt +1
⎡
⎣ ⎢
⎤
⎦ ⎥
€
Adrm ≡ ∇ ⋅σ = f (δm = 0)
€
Adff ≡ dm
dt+ ∇ ⋅v = S f (δε = 0)
€
ε 0
Conservation variable are held constant alternately
Deformation solution produces pressure adjustment before solving flow equations
Fixed Strain SplitAlgorithm:1. Hold strain constant2. Solve flow first3. Solve deformation second4. Repeat until convergence
Iteration
Flow
€
˙ σ = 0
Deformation
t t+1
If converged
If not converged
€
ut
pt
⎡
⎣ ⎢
⎤
⎦ ⎥→Adr
dut + 1
2
pt +1
⎡
⎣ ⎢
⎤
⎦ ⎥→Adr
f ut +1
pt +1
⎡
⎣ ⎢
⎤
⎦ ⎥
€
Adrm ≡ ∇ ⋅σ = f (δm = 0)
€
Adff ≡ dm
dt+ ∇ ⋅v = S f (δε = 0)
€
p = 0
State variables are held constant alternately
Flow solution produces strain adjustment before solving deformation equations
€
12
Fixed Stress SplitAlgorithm:1. Hold stress constant2. Solve flow first3. Solve deformation second4. Repeat until convergence
Iteration
Flow
€
˙ σ = 0
Deformation
t t+1
If converged
If not converged
€
ut
pt
⎡
⎣ ⎢
⎤
⎦ ⎥→Adr
dut + 1
2
pt +1
⎡
⎣ ⎢
⎤
⎦ ⎥→Adr
f ut +1
pt +1
⎡
⎣ ⎢
⎤
⎦ ⎥
€
Adrm ≡ ∇ ⋅σ = f (δ ˙ σ = 0)
€
Adff ≡ dm
dt+ ∇ ⋅v = S f (δσ '= 0)
€
0
Conservation variable are held constant alternately
Flow solution produces strain adjustment before solving deformation equations
€
12
ClassificationFixed State Fixed
ConservationDeform 1st Drained Split Undrained SplitFlow 1st Fixed Strain
SplitFixed Stress Split
Courtesy: Kim (2010)
Numerical AnalysisKim et al. (2009)Derived stability criteria for all four operator splitting schemes using Fourier Analysis for the linear systems.
Kim (2010) Tested operator splitting strategies on a variety of 1D & 2D cases•Fixed number of iterations per time step => fixed state methods are inconsistent!•Fixed conservation methods => consistent even with a single iteration!•Undrained split suffers from numerical stiffness more than fixed-stress.•Fixed Stress method => fewer iterations for same accuracy compared to undrained
Fixed Stress Method is highly recommended for •Consistency•Stability•Efficiency
More SplittingLoose CouplingMinkoff et al. (2003)
•Special case of sequential coupling•Solid mechanics equations not updated every timestep.•Extremely computationally efficient•Linear elasticity & porosity-pressure dependency leads to good convergence.•Approximate rock compressibility factor in flow equations to compensate for non-linear elasticity in staggered coupling•Heuristics to determine when to update elasticity equations.
Flow + Deform
Flow Flow … Flow Flow + Deform
t t+1 t+2 t+k-1
t+k
Future DirectionMicroscale PoroelasticityContinuum scale models assume fluid and solid occupy same space, in different volume fractions.
For microscale models: •Non-overlaping flow-deformation domains•Discrete conservation laws and constitutive equations•Discrete flow-deformation coupling relations•Fixed Stress Operator Splitting Method???
Wu et al. (2012)
ReferencesKim J. et al. (2009) Stability, Accuracy, and Efficiency of Sequential Methods for Coupled Flow and Geomechanics, SPE Reservoir Simulation Symposium Feb. 2009.
Kim, J. (2010) Sequential Formulation of Coupled Geomechanics and Multiphase Flow, PhD Dissertation, Stanford University
Kumar, V. (2005) Advanced Computational Techniques for Incompressible/Compressible Fluid-Structure Interactions. PhD Disseration, Rice University
Wu, R. et al. (2012) Impacts of mixed wettability on liquid water and reactant gas transport through the gas diffusion layer of proton exchange membrane fuel cells. International Journal of Heat and Mass Transfer 55 (9-10), p. 2581-2589