adaptive methods for multi-phase flow: grid-adaptivity · 0.15 10¡13m2 500 pa 2 dirichlet bc: sw =...

Adaptive methods for multi-phase flow:Grid-adaptivity

Benjamin Faigle,I. Aavatsmark, B. Flemisch, R. Helmig

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Department of Hydromechanics and Modeling of Hydrosystems

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a) Efficiency reasons:– Most time spent for the solution of

the pressure equation• Grid resolution affects

assembling and solution time!

Why adaptive grids?

25,0%

45,1%

22,7%0,4% PE assemble

PE solveTransportMuPhy

Sinsbeck (2011)

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b) Qualitative reasons:– Sequential: Solution not

converged but approximated: Pressure field should be as good as possible.

– The finer the grid, the less nummerical diffusion.

– Global refinement not always possible.

Why adaptive grids?

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Numerical Scheme

Mass conservation:– For phases ® and components ·, for each component:

– Solution strategies:• Fully implicit• Sequential

– Summation yields one pressure equation.– Transport equation– -Flash calculations

v® = ¡¸®K(r p® ¡ %®g)

C·=X

®

%®S®X·®

X

®

@ÁS®%®X·®

@t+ r ¢

ÃX

®

X·®%®v® + J·®

!+X

®

X·®%®q

·= 0

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Outline

• Introduction– Why using adaptive grids?

• Simulation on adaptive grids– How to adapt

• Representation of fluxes near refined cells

– Numerical Results

• Outlook

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How to refine the grid?

Finite Volume context:– Refine with closure

» Problem:

» e.g. Johannsen: Cell 80m x 10m

– Refine with irregular edges Hanging node

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Fluxes: Tpfa

– Standard approach to approximate flux:

Use a Two-Point flux approximation

– Problem:

6=

r© ¼ ©j ¡ ©i¢x

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Example Simulation 1

– 2 D, injection of CO2 into a rectangular domain filled with brine.

– Comparison of fully refined vs. adaptive grids.

– Compositional two-phase system.

– Material data:

Neumann BC:qn = ¡0:2 Mt/m yr.qw = free °ow

Porosity Permeability Entry Pressure BC-lambda0.15 10¡13m2 500 Pa 2

Dirichlet BC:Sw = 1pn = 2:5e7Pa +%gz

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Fluxes: Tpfa is inaccurate

– Standard approach to approximate flux:

Use a Two-Point flux approximation

– Problem:• Coloured cells expect flux to the left!

r© ¼ ©j ¡ ©i¢x

Height: ©w;i = pw;i + %w;igzi

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Fluxes: Mpfa

Multi-point flux approximation (Mpfa)a) Define an “interaction region”.

b) Introduce new points on interface.

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Fluxes: Mpfa



c) Approximate flux with new points.

Area of left inter-action region

Value at cell center of cell i

Rotated vector connecting the points i,k

f°;i = ¡nT°KirUi Value on introduced point k

rUi =1

T

2X

k=1

ºk(u¤k ¡ u0)

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Fluxes: Mpfa




d) Approximate flux as seen from all cells.

f°;j = ¡n2TK31

T3º5(u

¤2 ¡ u3)

¡n2TK31

T3º6 (uj ¡ u3

+1

TiºT7 Rºi(u

¤1 ¡ ui)

+1

TiºT7 Rºj(u

¤2 ¡ ui)

¶

f°;i = : : :

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Fluxes: Mpfa




d) Approximate flux as seen from all cells.

e) Write everything into a large equationsystem of form

where T contains the „transmissibility coefficients“ (introduced points already eradicated), and u is the vector of unknown cell values

f = Tu

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Fluxes with Mpfa

Results: Flux error in horizontal direction

z ¡ Coordinate: ©w;i = pw;i + %w;igxi

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Fluxes with Mpfa

Results after 5 years of injection

reference

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Example Simulation 2

More challenging example:– Tilted domain.

– No quadratic cells.

– Two periods• Injection phase: Pressure gradient drives flow.• Post-injection phase: Gravity drives flow.

Neumann BC:qn = ¡0:2 Mt/m yr.qw = free °ow

Dirichlet BC:Sw = 1pn = 2:5e7Pa +%gz

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Saturation at the end of the injection period (2 years):

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Saturation at the end of the post-injection period (5 years):

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Fluxes with MPFA – Improvements

Flux through the edge:– Twice the flux of first half-edge of the interaction volume.

– Construct second interaction region:

• Strongly dependent onsurrounding cells.

• Expensive way to “search”the region.

• Is it “worth the effort”?

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Accurate result with second half edge!

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Mpfa – Programming Obstacles

Larger stencil: More inteaction– Many iterators used to

come from intersection to all 3 neighbours and point connecting point.

Per sub-face required:– 3 neighbour elements

– Point in the middle

– All surrounding Elements??

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Summary

Mpfa on adaptive grids “does not look so bad”!

With adaptive grids simulation can be fast and still accurate:– Tpfa on static, fine grid: 75s

– Tpfa (3 levels): 16s

– Mpfa (3 levels) 16.3s

– Mpfa ('', 2 half-edges) 16.9s

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Future work

– Investigate Refinement Indicators• Application of several indicators.• Dependence of refinement criteria on solution scheme.• Influence of Refinement on the solution.

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Future work

– Investigate Refinement Indicators• Application of several indicators.• Dependence of refinement criteria on solution scheme.• Influence of Refinement on the solution.

– Extension to three dimensions.

– Applications

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Thank you

for your attention!

ReferencesI. Aavatsmark et. Al (2008): A compact multipoint flux approximation method with improved robustness.

Numerical Methods for Partial Differential Equations, 24:1329-1360, 2008.

J. Fritz et. al (2010): Multiphysics Modeling of Advection - Dominated Two - Phase Compositional Flow in Porous Media. International Journal of Numerical Analysis & Modeling. (accepted)

BGR (2010): Projekt CO2 - Drucksimulation, final report of Federal Institute of Geosciences and Natural Resources.

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Implicit Pressure

Volume balance:

– If we use non-wetting pressure as primary variable

watergas

+ =water

gas

vw = ¡¸wK(r pn ¡r pc ¡ %wg);

vn = ¡¸nK(r pn ¡ %ng);

Acs et. al (1985)

Transport Estimate

Pressure Equation

Transport Step

Flash Calculation

Upate, Output

Volumederivatives

Formulation

ctotal@p

@t+X

·

@vtotal@C·

r ¢ÃX

®

X·®%®v®

!=X

·

@vtotal@C·

q·+ ";

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Solution Scheme: Sequential

Initialization

Transport Estimate

Pressure Equation

Transport Step

Flash Calculation

Upate, Output

End

Time-step

Volumederivatives @vtotal

@C·estimate

C·estimate

C·

C·=X

®

%®S®X·®

S® ; X·®; pc

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Solution Procedure on Adaptive Grids

Initialization

Transport Estimate

Pressure Equation

Transport Step

Flash Calculation

Upate, Output

End

Time-step

Volumederivatives

Adaptive Module

Create Indicator → Criteria

Store solution

Flash calculation

Mark cells, adapt grid

Reconstruct solution

S®; X·®; pc

C·; p

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Implicit Pressure

Discretized (multi-phase):

Formulation

Vict;ipti ¡ pt¡¢ti

¢t

¡X

°ij

A°ijn°ij ¢KX

®

%®¸®X

·

@vt@C·

X·®

µpt®;j ¡ pt®;i

¢x+ %®g(zj ¡ zi)

¶

+ViX

°ij

A°ijUi

KX

®

%®¸®X

·

@vt;j@C·

j¡ @vt;i

@C·i

¢xX·®

µpt®;j ¡ pt®;i

¢x+ %®g(zj ¡ zi)

¶

= ViX

·

@vt@C·

q·i + Vi®rvt ¡ Á

¢t:

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Explicit Step

Transport Equation (explicit):

– Determines size of the time step.

Equilibrium (Flash-) Calculation

@C·

@t= ¡r ¢

ÃX

®

X·®%®v®

!+ q·;

Transport Estimate

Pressure Equation

Transport Step

Flash Calculation

Upate, Output

Volumederivatives

Formulation

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Derivation of the Pressure Equation

– Volume contraint:

– Taylor expansion in time:

– Reordering:

vt = Á

vt (t) + ¢t@vt@t

+ O¡¢t2

¢= Á (t) + ¢t

@Á

@t+ O

¡¢t2

¢:

@vt@t

=@vt@p

@p

@t+X

·

@vt@C·

@C·

@t

@Á

@t=@Á

@p

@p

@t

µ@vt@p

¡ @Á

@p

¶@p

@t+X

·

@vt@C·

@C·

@t=Á¡ vt

¢t

ct@p

@t+X

·

@vt@C·

X

®

r ¢ (v®%®X·®) =

X

·

@vt@C·

q·+ "

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Pressure Equation with MpfaAdaptive Grid

Victotalpti ¡ pt¡¢ti

¢t

¡X

°ij;irregular

X

®

%®¸®X

·

@vtotal@C·

X·®

0@0@t2ipt®;i +

X

j

t2jpt®;j

1A+ %®g

0@t2izi +

X

j

t2jzj

1A1A

¡X

°ij;regular

A°ijn°ij ¢KX

®

%®¸®X

·

@vtotal@C·

X·®

µpt®;j ¡ pt®;i

¢x+ %®g

zj ¡ zi¢x

¶

+ViX

°ij;irregular

X

®

%®¸®X

·

@vt;j@C·

j¡ @vt;i

@C·i

¢xX·®

0@0@t2ipt®;i +

X

j

t2jpt®;j

1A+ %®g

0@t2izi +

X

j

t2jzj

1A1A

+ViX

°ij;regular

A°ijUi

KX

®

%®¸®X

·

@vt;j@C·

j¡ @vt;i

@C·i

¢xX·®

µpt®;j ¡ pt®;i

¢x+ %®g

zj ¡ zi¢x

¶

= ViX

·

@vt@C·

q·i + Vi®rvt ¡ Á

¢t: (1)

adaptive methods for multi-phase flow: grid-adaptivity · 0.15 10¡13m2 500 pa 2 dirichlet bc: sw =...

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