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Submarine Gas Hydrate Reservoir Simulations – A Gas/ Liquid Fluid Flow Model for Gas Hydrate Containing Sedimen ts
Stefan Schlüter, Georg Janicki, Torsten Hennig, Görge DeerbergFraunhofer UMSICHT, Oberhausen (Germany)
COMSOL Conference 2014, Cambridge (UK)
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What are gas hydrates?
Gas molecules
Water molecules
Water
Tem
pera
ture
Gas
© J
en
s G
rein
ert IFM GEOMAR
Petrobras
Fraunhofer UMSICHT
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Idea of the SUGAR project
© IFM
GEO
MA
R
• CO2 hydrate stable at lower pressure / higher temperature
• replacing CH4 by CO2
• simultaneous production of CH4
and storage of CO2
• sustainable energy supply system
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Reservoir model principles
Phases
• 3 solid phases: sediment, methane hydrate, CO2 hydrate
• 2 fluid phases: gas phase, water (liquid) phase
• (1 supercritical phase: supercritical CO2 – not implemented yet)
Components
• 2 gas phase components: methane, carbon dioxide
• 3 components solved in water: sea salt, methane, carbon dioxide
Pressure equation: advanced 2-phase Darcy model
Energy equation: flow through porous solid, hydrate extensions, latent heats, pressure work
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Reservoir Model – Pressure/Saturation Equation
Continuity equations:
Saturation:
Convection splitted form:
( ) ( )
( ) ( )
( )
( )
G G G G G
L L L L L
MH MH MH
CH CH CH
S st
S st
S st
S st
φ ρ ρ
φ ρ ρ
φ ρ
φ ρ
∂+∇⋅ =
∂∂
+∇⋅ =∂∂
=∂∂
=∂
u
u
1S sS
t t
ρ ρφ φ
ρ ρ ρ
∂ ∂ ∇+ +∇⋅ + ⋅ =
∂ ∂u u
1
j j
j
S
jS
ε ε
ε φ= = =−
volume fraction of phase
sediment free volume fraction
Euler/Euler
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Reservoir Model – Pressure/Saturation Equation
Density derivatives:
General form:
Darcy equation:
Phase summation:
, , ,
1 1 1, ,
k k
k
kT y P y P TP T y
ρ ρ ρχ β ϕ
ρ ρ ρ
∂ ∂ ∂ = = − = ∂ ∂ ∂
1,k
k k kk k
yP TP T y
t t t t
ρ ρχ β ϕ χ β ϕ
ρ ρ
∂∂ ∂ ∂ ∇= − + = ∇ − ∇ + ∇
∂ ∂ ∂ ∂∑ ∑
( ) ( )with , ,rel
f rel H L
kP k f S SΛ ρ Λ
η= − ∇ + = =u K g
k
k k kk k
yS P T sS P T y
t t t tφ φ χ β ϕ χ β ϕ
ρ
∂∂ ∂ ∂ + − + +∇⋅ + ⋅ ∇ − ∇ + ∇ = ∂ ∂ ∂ ∂ ∑ ∑u u
1 , 0j
jj j
SS
tφ∂
= =∂
∑ ∑
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Reservoir Model – Pressure/Saturation Equation
Insertion in general form and summation over phases leads to general pressure equation:
Capillary pressure:
Calculation pressure:
( )( )
( ) , ,
,
, , ,
j
j j f j j jj j
f j j j j j k j k j jjk
j k j
j j k j f j j j k j k jj j jkj
PS P
t
P T y P
q yTS T y
t t
φ χ Λ ρ
Λ χ ρ β ϕ
φ β ϕ Λ ρ β ϕρ
∂+∇⋅ − ∇ +
∂
− ∇ + − ∇ + ∇ ∇ =
∂ ∂ + − − ∇ − ∇ ∂ ∂
∑ ∑
∑ ∑
∑ ∑ ∑ ∑
K g
K g
K g
with ( , )C G L C H LP P P P f S S= − =
( )1 1 1
,2 2 2L G L C G C
P P P P P P P P P= + → = − = +
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Reservoir Model – Pressure/Saturation Equation
COMSOL coefficient form PDE:
( )( )
( ) ( ) ( )
0,
0 2,
2
L L G G MH CH H
L L L
C
f G L L L G G
f L G
f L C
f L L
L
f
P S S S Su d
S S
P
cP
φ χ χ χ
φ χ φ
Λ Λ ρ Λ ρ ΛΛ Λ
γΛ ε
Λ ρ
Λ χ
β
+ + + = =
∇ − − + + + = = ∇ − − +
+−
=
K gK
KK g
K( )( )( )( )
( )( )
,
,
0,
0
L C L L S L S
G G C G G C G C
f L L C L L
P P T c
P P T y f
P P T
ρ β ϕ
Λ χ ρ β ϕ
Λ χ ρ β
∇ −∇ + − ∇ + ∇ + ∇ +∇ + − ∇ + ∇ = − ∇ −∇ + − ∇
g
g
K g
…
…
( )2
2
u ue d c u u u au f
ttα γ β
∂ ∂+ +∇⋅ − ∇ − + + ∇ + =∂∂
artificial diffusion for SL
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Reservoir Model – Energy Equations
Summation over single phase energy equations with
Pressure work:
Latent heats: (heats of formation)
( ) ( )
( ) ( )( )
( ) ( )( )
( ) ( )( )
, ,
, ,
, ,
,1 1 0
G
G G P G G G G f G G G G P G G P
L
L L P L L L L f L L L L L P L L L
H
MH MH P MH CH CH P CH MH MH CH CH H H
S
S P S S S
TS c S T P c T q
t
TS c S T P S c T q
t
TS c S c S S T q
t
Tc T
t
φ ρ φ Λ ρ ρ
φ ρ φ Λ ρ ε ρ
φ ρ ρ φ
φ ρ φ
∂+∇⋅ − ∇ − ∇ + ∇ =
∂
∂+∇⋅ − ∇ − ∇ + + ∇ ∇ =
∂
∂+ −∇⋅ + ∇ =
∂
∂− +∇⋅ − − ∇ =
∂
K g
K g
ɺ
ɺ
ɺ
λλλλ
λλλλ
λ λλ λλ λλ λ
λλλλ
( )( )1G
P G G G f G G G G G G
Pq S T P T P
tφ β Λ ρ β
∂= + ∇ + − ∇
∂K gɺ
( )H MH MH CH CHq R h R h= − ∆ + ∆ɶ ɶɺ
G L H ST T T T= = =
real gasbehaviour
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Reservoir Model – Single Component Equations
Gas phase component, molar fraction conservative form
Liquid phase component, molar concentration conservative form
( ), ,
1
1
( )
1( )
effi
G G i G G G i G G i G i G i G
n
G G k n k
G G G G kkG G
yS y S S y y q
t t
S P M M yTS S
t S t t t M t
φ ρ φ ρ φ ρ ρ
φ ρ φ ρ χ β ϕ−
=
∂ ∂+ +∇⋅ − ∇ + =
∂ ∂
∂ ∂ − ∂∂ ∂ = + − + − ∂ ∂ ∂ ∂ ∂ ∑
uɶ ɶ ɶ ɶ ɶ
ɶ ɶ
ɶ ɶɶ
δδδδ
( )( )( ),
, , , , ,
i L effL
L i L L i L i L f L L L L i L i L
c SS c S c P S c q
t tφ φ φ Λ ρ ε∂ ∂
+ +∇⋅ − ∇ − ∇ + + ∇ =∂ ∂
K gδδδδ ɶ
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Reservoir Model – Gas Hydrate Equations
Methane and Carbon Dioxide hydrate saturation
Hydrate kinetics (linearized partial pressure kinetic)
MH G C MH
MH MH MH
MH
CH G C CH
CH CH CH
CH
S P P sTS
t t t t
S P P qTS
t t t t
φ φ χ βρ
φ φ χ βρ
∂ ∂ ∂ ∂ + − − = ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ + − − = ∂ ∂ ∂ ∂
( )
( )
*
*
1
1
MH
MH MH MH M G MH
CH
CH CH CH C G CH
NR k a y P P
V t
NR k a y P P
V t
∂= = −
∂
∂= = −
∂
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COMSOL Implementation
COMSOL Multiphysics implementation essentials
� 3D and 2D axisymmetric models
� Coefficient Form PDE + Heat Transfer in Porous Media
� Discretization/Stabilization as default
� Fully Coupled solution, Direct linear solver (PARDISO)
� Pressure Dirichlet boundary condition given as Weak Constraint
� Initial time step 10 s, maximum time step 5 · 106 s
� maximum mesh size = 15 m, fine meshing at the well
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Methods for gas hydrate decomposition
Depressurization
Thermal stimulation
Additives
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Field production plan
1000 m 1000 m 1000 m 1000 m
1000 m
1000 m
1000 m
1000 m
500 m
500m
1/8 well
1/8 well
Injection well
Production well
45°
45°
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t = 15 years
Case Study I – Methane production by DepressurizationP0 = 92 bar T0 = 10,0°C Sh0 = 0,40 Hres = 20 m
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Case Study I – Methane production by Depressurization
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Case Study II – Depressurization of Multi-Layer Reser voir
smoothing
5 Layers a‘ 4 m in height
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Case Study II – Depressurization of Multi-Layer Reser voir
„Fingering“ Effect
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Case Study II – Depressurization of Multi-Layer Reser voir
„Fingering“ Effect
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SUGAR project – case studies
More case studies developed with COMSOL:
� Injection of Carbon Dioxide with parallel Methane production (2 wells)
� Reservoir simulations for the Ulleung Basin, South Korea (UGBH 2.6)
� Simulation cases for process safety and reservoir integrity issures
� Production upriser pipe simulations (1D / 2Ph Euler Equations)
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Summary
State
• Development of a gas hydrate reservoir model in COMSOL Multiphysics
• Usage of the Coefficient Form PDE tool of the Mathematics branch
• Highly nonlinear model, needs the fully coupled approach with direct solution
• Simulation of important reservoir production cases were successful
Outlook
• New 3. project phase starts in October 2014
• Field development simulations in preparation of a real field test
• Production safety simulations
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FRAUNHOFER UMSICHTProcess Technology / Modeling & Simulation
Thank you for your attention!
Contact:Fraunhofer UMSICHTOsterfelder Strasse 3D-46047 Oberhausen (Germany)E-Mail: [email protected]: http://www.umsicht.fraunhofer.de
Dr.-Ing. S. SchlüterTelefon: +49 208 8598 1126E-Mail: [email protected]