A Multiphysics Approach for the Simulation of Multiphase Flow Processes in Porous Media. · 2019-03-07 · J. Fritz et. al (2010): Multiphysics Modeling of Advection - Dominated Two

Department of Hydromechanics and Modeling of Hydrosystems

A Multiphysics Approach for the Simulation of Multiphase

Flow Processes in Porous Media.

24/03/11

Benjamin Faigle, B. Flemisch,

R. Helmig.


2Motivation: Scale of interest usually large!

Detailed resolution is not always possible:– Computational demands:

• Large domain size.

• Large timespan of interest.

– Scarcity and uncertainty of data:

• Monte-Carlo and sensivity-analysis with multiple runs? BGR (2010)

=> We need efficient models!=> We need efficient models!


3Motivation: Natural systems are complex!

In most environmental applications of flow and transport in porous media (Remediation, CO

2-sequestration, ..), we observe

– Complex physics:

• Compressible, partly miscible substances

• Multi-phase behaviour, capillarity

– Differing processes over space & time:

• Complex multi-phase locally vs. single-phase in far-field.

• ….

dissolved contaminant

multiphase flow

Fritz (2010)

=> We need good models!=> We need good models!


4Motivation: Plenitude of Models

Predictive modelling can best be made with the appropriate model for the problem to solve:

Miscible two-phase Numerical model 2p2c

Two-phase displacement Numerical model 2p


5Motivation: Plenitude of Models

Predictive modelling can best be made with the appropriate model for the problem to solve:

Miscible two-phase Numerical model 2p2c

Two-phase displacement Numerical model 2p

3p3c

2p2cni

2p2c

1p2c

www.dumux.com/

=> Select the right model an apply it locally!=> Select the right model an apply it locally!


6Outline

• Motivation

• Numerical model: 2p2c– Mathematical formulation

– Example: Simple Injection

• Multi-physics concept– Definition of “sub-domains”.

– Large-scale example: CO2 – sequestration.

• Outlook


7Framework: 2p2c Formulation

Model Requirements:– Compressible Flow.

– Compositional phases: small solubility.

– Capillary Pressure and Gravity.

– Fast and efficient.

>> no fully implicit (coupled) solution procedure.

>> no standard fractional flow formulation feasible if non-iterative scheme is desired.

>> formulation based on volume balance.


8Framework: 2p2c Derivation

Pressure equation (implicit):

– If we use pn as primary variable:

water

gas+ =

water

gasAcs et. al (1985)

ctotal@p

@t+X

·

@vtotal@C·

~r ¢ÃX

®

X·®%®v®

!=X

·

@vtotal@C·

q·+ ";

vw = ¡¸wK (r pn ¡r pc ¡ %wg);

vn = ¡¸nK (r pn ¡ %ng);


9Framework: 2p2c Discretization

Discretized pressure equation:

ctotalpt ¡ pold

¢t+

X

faces=i

AiniK

Ãpt ¡ pti¡neighbor

¢x

X

®

%®¸®X

·

X·®

@vtotal@C·

+pc ¡ pc;i¡neighbor

¢x¸w%w

X

·

X·®

@vtotal@C·

¡X

®

%2®¸®gX

·

@vtotal@C·

X·®

!

¡X

subContV ol=j

VjwjKX

·

@vtotal;j@C·

j¡ @vtotal;j¡neighbor

@C·j¡neighbor

¢x

"pt ¡ pti¡neighbor

¢x

X

®

¸®X·®

+pc;j ¡ pc;j¡neighbor

¢x%w¸wX

·w ¡

X

®

%2®¸®gX·®

#

=X

·

@vtotal@C·

q·;


10Framework: 1p Discretization

Discretized pressure equation (single phase):

ctotalpt ¡ pold

¢t+

X

faces=i

AiniK

Ãpt ¡ pti¡neighbor

¢x

X

®

%®¸®X

·

X·®

@vtotal@C·

+pc ¡ pc;i¡neighbor

¢x¸w%w

X

·

X·®

@vtotal@C·

¡X

®

%2®¸®gX

·

@vtotal@C·

X·®

!

¡X

subContV ol=j

VjwjKX

·

@vtotal;j@C·

j¡ @vtotal;j¡neighbor

@C·j¡neighbor

¢x

"pt ¡ pti¡neighbor

¢x

X

®

¸®X·®

+pc;j ¡ pc;j¡neighbor

¢x%w¸wX

·w ¡

X

®

%2®¸®gX·®

#

=X

·

@vtotal@C·

q·;


11Framework: 2p2c Transport

Transport Equation (explicit)

– Defines size of the time step.

– Equilibrium (Flash-) Calculation

@C·

@t= ¡r ¢

ÃX

®

X·®%®v®

!+ q·;


12Examples: 1D Testcase

Testcase:– 1D-case, injection of water into a rectangular domain filled with gas.

– Simulated fully implicit vs. sequential model (IMPEC)

– Compositional two-phase

– Sketch:

– Material data:

Porosity Permeability Entry Pressure BC-lambda0.2 10¡7 500 Pa 2

Dirichlet BC:Sw = 1pw = 2e5 Pa

Neumann BC:qn = 0:01 kg/m2sqw = free out°ow


13Examples: 1D Testcase

Results after 4000s of injection: Fully Implicit vs. Sequential

Ph

ase P

ressu

re [

Pa]


14Outline

• Motivation

• Numerical model: 2p2c– Mathematical formulation

– Example: Simple Injection

• Multi-physics concept– Definition of “sub-domains”.

– Large-scale example: CO2 – sequestration.

• Outlook


15Framework: Multiphysics

a) Why different models?– e.g. remediation scenario

2p2c Conditions

1p2c Conditions



a) Why different models?– e.g. remediation scenario

2p2c Conditions

1p2c Conditions



b) Separation of the Models:– Definition of subdomains.

Numerical model 1p2c




c) Use “safety zone” around complex sub-domains




19Framework: Adaptive Multiphysics

d) Update it after each time step:

Transport step Adaption of sub-domains

Next time step


20

InjectionWell

Examples: Large-scale Benchmark

Benchmark: Injection of CO2:

– Injection of CO2 for 25 years.

– Simulation of 50 years.

– 54756 cells.

– Vertically exaggerated by factor 10.

– Boundary Conditions:

• Hydrostatic pressure

• Temperature gradient

Class et. al (2009)


21Examples: Large-scale Benchmark

Complex sub-domain:Numerical model 2p2c




Sub-domain borders:– Soluted CO

2 can be transported over sub-domain boundary.



6 years 22 years


24Examples

Multi-physics concept also applied for

- 3p simulation:

Subdomain: 1p3c, 2p3c, 3p3c;

- Non-isothermal cases:


25Outlook

Sequential IMPEC:– Most time spent for solution of the

pressure equation.

– Most accuracy needed in the pressure field.

=> local refinement of the grid promising.


26

Thank you for your attention!

Funding by the German Research Foundation

ReferencesG. Acs et. al (1985): General Purpose Compositional Model. Society of Petroleum Engineers Journal, 25:543 552.–

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.


27Framework: 2p2c Derivation

Derivation– Volume constraint

– Taylor expansion:

– 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·+ "


28Framework: Sequential Formulation

Solution Scheme:

Initialization

Transport Estimate

Pressure Equation

Transport Step

Flash Calculation

Upate, Output

End

Time-step

Volumederivatives

Documents

A Multiphysics Approach for the Simulation of Multiphase Flow Processes in Porous Media. · 2019-03-07 · J. Fritz et. al (2010): Multiphysics Modeling of Advection - Dominated Two