By Paul Delgado
•Motivation
•Flow-Deformation Equations
•Discretization
•Operator Splitting
•Multiphysics Coupling
•Fixed State Splitting
•Other Splitting
•Conclusions
(Quasi-Static) Poroelasticity Equations
Flow
Mechanics
   f
σ = Total Stress Tensor
f = body forces per unit area

f  b g

b  0  f  (1  0 )  s
dm f
dt
   wf  Sf
mf = variation in mass flux relative to solid
wf = mass flux relative to solid
Sf = mass source term
m f   f 0
Courtesy: Houston Tomorrow
w f   f 0 v f
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?
Let P  L2 () and U  ( H 1 ()) d
Let Ph  spani   P and U h  span i   U
Deformation
Strong
form
Weak
form
    f
Strong
form
  :    f     n



Backward
Euler Form
  :  

   f 
n 1


   n
n 1

Backward
Euler Form
dt
dm f


  Ph
n 1

Weak
form
dm f
Flow
dt
   wf  Sf
     w f   S f


  Ph


mf
n 1
 mf
t
n
     w f n1   S f n1
If constitutive relations are non-linear, => Non linear system


 
N d (u , p)  0
 
N f (u , p)  0
Simultaneous coupling between flow &
deformation at each time step
Iteration between physics models within a
single time step
•Computationally expensive
•Code Intrusive
•high order approximations are difficult to
achieve
•Strong numerical stability & consistency
properties
•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 properties
We summarize the work of Kim (2009, 2010) illustrating iterative coupling strategies.
Based on Kumar (2005)
 
N d (uh , ph )  0
 
N f (uh , ph )  0
Newton-Raphson
at time t
 J dd
Jxk  
 J fd
t 1
xk 1
Rewrite the Jacobian matrix as:
  J dd

 J fd

0  0 J df  




J ff  0 0  
J df 
J ff 
 xk
k
bd 
u 

b 
p 
 
 f
t 1
k
k
Until convergence
 xk
x
t 1
 u t 1 
  t 1 
p 
Rewrite Newton-Raphson as Fixed Point Iteration
 J dd
J
 fd
0  u 
J ff  p 
k 1
bd 
 
b f 
Jdd= mechanical equation with fixed pressure
Jfd + Jff = flow equation with solution from Jdd
In operator splitting, we apply this technique to the discrete (linear)
operators governing the continuous system of equations.
k 1
0 J df  u 

 p 
0
0

  
k
k
Algorithm:
1. Hold pressure constant
2. Solve deformation first
3. Solve flow second
4. Repeat until convergence
State variables
are held constant
alternately
Iteration
 u t  Adr u t 1  Adr  u t 1 
 t    t    t 1 
p   p  p 
d
Deformation
p  0
Adr      f
m
Adf 
f
Flow
u  0
If not converged
If converged
t
t+1
How else can we decompose the operator?
dm
 v  S f
dt
f
(p  0)
(u  0)
Algorithm:
Iteration
1. Hold mass constant
2. Solve deformation first
3. Solve flow second
4. Repeat until convergence
Conservation
variable are held
constant
alternately
u t A dr d u t 1 A dr f u t 1 
 t   t  12   t 1
p  p  p 
Deformation
m  0

Adr      f
m

Flow
  0
If not converged


Adf
f
dm

   v  Sf
dt
If converged
t
(m  0)
t+1

Deformation solution produces pressure adjustment before
solving flow equations
(  0)
Algorithm:
Iteration
1. Hold strain constant
2. Solve flow first
3. Solve deformation second
4. Repeat until convergence
State variables
are held constant
alternately
u t A dr d u t  12 A dr f u t 1 
 t   t 1  t 1
p  p  p 
Flow
Ý 0


m


Deformation
p  0
If not converged

Adr      f
 12

Adf
f
dm

   v  Sf
dt
If converged
t
(m  0)
t+1

Flow solution produces strain adjustment before solving
deformation equations
(  0)
Algorithm:
Iteration
1. Hold stress constant
2. Solve flow first
3. Solve deformation second
4. Repeat until convergence
Conservation
variable are held
constant
alternately
u t A dr d u t  12 A dr f u t 1 
 t   t 1  t 1
p  p  p 
Flow
Ý 0


m


Deformation
  0
If not converged

Adr      f
 12

Adf
f
dm

   v  Sf
dt
If converged
t
t+1
Flow solution produces strain adjustment before solving
deformation equations
Ý 0)
(
( ' 0)
Fixed State
Fixed Conservation
Deform 1st
Drained Split
Undrained Split
Flow 1st
Fixed Strain Split
Fixed Stress Split
Courtesy: Kim (2010)
Kim 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
Loose Coupling
Minkoff 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.
t
Flow +
Deform
t+1
Flow
t+k-1
t+2
Flow
…
Flow
t+k
Flow +
Deform
Microscale Poroelasticity
Continuum 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)
Kim 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 FluidStructure 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