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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.
Department of Hydromechanics and Modeling of Hydrosystems
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!
Department of Hydromechanics and Modeling of Hydrosystems
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!
Department of Hydromechanics and Modeling of Hydrosystems
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
Department of Hydromechanics and Modeling of Hydrosystems
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!
Department of Hydromechanics and Modeling of Hydrosystems
6Outline
• Motivation
• Numerical model: 2p2c– Mathematical formulation
– Example: Simple Injection
• Multi-physics concept– Definition of “sub-domains”.
– Large-scale example: CO2 – sequestration.
• Outlook
Department of Hydromechanics and Modeling of Hydrosystems
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.
Department of Hydromechanics and Modeling of Hydrosystems
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);
Department of Hydromechanics and Modeling of Hydrosystems
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·;
Department of Hydromechanics and Modeling of Hydrosystems
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·;
Department of Hydromechanics and Modeling of Hydrosystems
11Framework: 2p2c Transport
Transport Equation (explicit)
– Defines size of the time step.
– Equilibrium (Flash-) Calculation
@C·
@t= ¡r ¢
ÃX
®
X·®%®v®
!+ q·;
Department of Hydromechanics and Modeling of Hydrosystems
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
Department of Hydromechanics and Modeling of Hydrosystems
13Examples: 1D Testcase
Results after 4000s of injection: Fully Implicit vs. Sequential
Ph
ase P
ressu
re [
Pa]
Department of Hydromechanics and Modeling of Hydrosystems
14Outline
• Motivation
• Numerical model: 2p2c– Mathematical formulation
– Example: Simple Injection
• Multi-physics concept– Definition of “sub-domains”.
– Large-scale example: CO2 – sequestration.
• Outlook
Department of Hydromechanics and Modeling of Hydrosystems
15Framework: Multiphysics
a) Why different models?– e.g. remediation scenario
2p2c Conditions
1p2c Conditions
Department of Hydromechanics and Modeling of Hydrosystems
16Framework: Multiphysics
a) Why different models?– e.g. remediation scenario
2p2c Conditions
1p2c Conditions
Department of Hydromechanics and Modeling of Hydrosystems
17Framework: Multiphysics
b) Separation of the Models:– Definition of subdomains.
Numerical model 1p2c
Numerical model 2p2c
Department of Hydromechanics and Modeling of Hydrosystems
18Framework: Multiphysics
c) Use “safety zone” around complex sub-domains
Numerical model 2p2c
Numerical model 1p2c
Department of Hydromechanics and Modeling of Hydrosystems
19Framework: Adaptive Multiphysics
d) Update it after each time step:
Transport step Adaption of sub-domains
Next time step
Department of Hydromechanics and Modeling of Hydrosystems
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)
Department of Hydromechanics and Modeling of Hydrosystems
21Examples: Large-scale Benchmark
Complex sub-domain:Numerical model 2p2c
Numerical model 1p2c
Department of Hydromechanics and Modeling of Hydrosystems
22Examples: Large-scale Benchmark
Sub-domain borders:– Soluted CO
2 can be transported over sub-domain boundary.
Department of Hydromechanics and Modeling of Hydrosystems
23Examples: Large-scale Benchmark
6 years 22 years
Department of Hydromechanics and Modeling of Hydrosystems
24Examples
Multi-physics concept also applied for
- 3p simulation:
Subdomain: 1p3c, 2p3c, 3p3c;
- Non-isothermal cases:
Department of Hydromechanics and Modeling of Hydrosystems
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.
Department of Hydromechanics and Modeling of Hydrosystems
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.
Department of Hydromechanics and Modeling of Hydrosystems
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·+ "
Department of Hydromechanics and Modeling of Hydrosystems
28Framework: Sequential Formulation
Solution Scheme:
Initialization
Transport Estimate
Pressure Equation
Transport Step
Flash Calculation
Upate, Output
End
Time-step
Volumederivatives