Panfilov(2001)_Introduction to Differential Equations of One and Two Variables
S. OLADYSHKIN, M. PANFILOV
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Transcript of S. OLADYSHKIN, M. PANFILOV
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S. OLADYSHKIN, M. PANFILOVS. OLADYSHKIN, M. PANFILOV
Laboratoire d'Énergétique et de Mécanique Théorique et Appliquée Ecole Nationale Supérieure de GéologieInstitut National Polytechnique de Lorraine
STREAMLINE SPLITTING THE STREAMLINE SPLITTING THE THERMO- AND HYDRODYNAMICS THERMO- AND HYDRODYNAMICS
IN COMPOSITIONAL FLOW THROUGH POROUS IN COMPOSITIONAL FLOW THROUGH POROUS MEDIAMEDIA
APPLICATION TO HAPPLICATION TO H22-WATER IN RADIOACTIVE WASTE -WATER IN RADIOACTIVE WASTE DEPOSITSDEPOSITS
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SommaireP
r e s
e n
t a t
I o n
Flow Model
Limit compositional model
Streamline HT-splitting
Introduction
Validation to the limit thermodynamic modelValidation to the limit thermodynamic model
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IntroductionIntroductionPhysical descriptionPhysical description
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Hydrogen generation in a radioactive waste deposit
Gas generation:
Waste storage
Storage pressure growth : - Initial : 100 bar - Increased by H2 : 300 bar
Monitoring problem :H2 transport through porous media
accompanied with radionuclides
H2 + CO2 + N2 + O2 + …
Corrosion in storage tank
underground: 900 - 1100m
Water
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Fluid structure
Phases :
Components :
Gas Liquid
H2CO2N2O2H20…
Gas
Liquid
2 phases
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Similar phenomena in an underground H2 storage
Well Well
GAS and LIQUID
H20 + H2 + CO2 + CH4 + …
Hydrogen storage
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Initial stateL
L + G
G
Phase behaviour
Critical point
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Flow ModelFlow Model
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2 phases (gas & liquid)N chemical components
Compositional model
Mass balance for each chemical component k :
Momentum balance for each phase (the Darcy law)
Phase equilibrium :
Phase state :
( = the chemical potential)
or
Closure relationships:or
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Limit contrast Limit contrast compositional modelcompositional model
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Canonical dimensionless form of the compositional
modelgas flow
liquid flow
transport of basicchemical components
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Mathematical type of the system
Parabolic equation
Hyperbolic equation
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gas flow
liquid flow
transport of basicchemical components
Characteristic parameters of a gas-liquid
system
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Perturbation parameter:
Parameter of relative phase mobility:
Perturbation propagation timeReservoir depletion time
Characteristic parameters of the system
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Limit behaviour
Semi-stationarity : p and C(k) are steady-state, while s is non stationary
gas flow
liquid flow
transport of basicchemical components
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Streamline HT-splittingStreamline HT-splitting
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Integration of the transport subsystem
gas flow
liquid flow
transport of basicchemical components
This subsystem can be integrated along streamlines :
Asymptotic contrast compositional model :
A differential thermodynamic system
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Hydrodynamic subsystem (limit hydrodynamic model):
Thermodynamic subsystem (limit thermodynamic model):
HT-splitting
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variation of the total composition in an open system
The thermodynamic independent system is monovariant: all the thermodynamic variables depend on
pressure only The new thermodynamic model is valid along streamlines
Split Thermodynamic Model
Properties
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Due to the monovariance, the thermodynalmic differential equations may be simplified to a “Delta-law”:
Thermodynamic “Delta-law”
“Delta-law”
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Individual gas volume
Individual condensate volume
Interpretation of the delta-law
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gas flow
liquid flow
Split Hydrodynamic Model
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Validation to the limit Validation to the limit thermodynamic modelthermodynamic model
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These functions have been calculated using Eclipse simulation data for a
dynamic system
F1 F2
Validation of the Delta-law
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Phase plot
Fluid compositionCH4
H2
C10H22
Initial conditions:P0 = 315 barT = 363 K
T
P
Flow simulation: Fluid properties
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Well
Flow simulation: Flow problem
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Validation of the Delta-law
F1
F2
These functions have been calculated using the Eclipse simulation data
“Delta-law”
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Liquid mole fractions
Gas mole fractions
Validation of the total limit thermodynamic model
Composition variation in an open thermodynamic system
Compositional Model (Eclipse) - points; Limit thermodynamic model - solid Compositional Model (Eclipse) - points; Limit thermodynamic model - solid curvescurves
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FinitaFinita