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