Modeling and numerical approximation of multi-component anisothermal flows in porous media Modeling and numerical approximation of multi-component anisothermal flows in porous media M. Amara ∗ , D. Capatina ∗ , L. Lizaik ∗ ,⊲ , P. Terpolilli ⊲ ∗ Laboratory of Applied Mathematics, CNRS UMR 5142, University of Pau ⊲ TOTAL, CST Jean Feger, Pau Journ´ ee des Doctorants, 18 avril 2008
Modeling and numerical approximation of multi-component anisothermal flows in porous media Motivations Motivations Optical fiber Send a light source Detect a backscattering light The time for the backscattered signal gives distance along fiber The ratio of wave lengths gives temperature Possible applications Estimate virgin reservoir temperature Predict flow profiles and the flow rate of each layer
Modeling and numerical approximation of multi-component anisothermal flows in porous media Motivations Motivations Optical fiber Send a light source Detect a backscattering light The time for the backscattered signal gives distance along fiber The ratio of wave lengths gives temperature Possible applications Estimate virgin reservoir temperature Predict flow profiles and the flow rate of each layer
Modeling and numerical approximation of multi-component anisothermal flows in porous media Coupling of monophasic reservoir and wellbore models with heat transfer Coupling of monophasic reservoir and wellbore models with heat transfer Resevoir model : r φ ∂ρ ∂ t + div ( r G ) = 0 ρ − 1 ( µ K − 1 G + F | G | G ) + ∇ p = − ρ g ∂ t + r ρ − 1 ( ρ c ) f G · ∇ T − div ( r q ) − r φβ T ∂ p r ( ρ c ) ∗ ∂ T ∂ t − r ρ − 1 ( β T − 1) G · ∇ p = 0 1 λ q − ∇ T = 0 ρ = ρ ( p , T ) Wellbore model ∂ ∂ t ( r ρ ) + ∇ · ( r ρ u ) = 0 ∂ t ( r ρ u r ) + ∇ · ( ru r ρ u ) + r ∂ p ∂ ∂ r − ∂ ∂ r ( r τ rr ) − ∂ ∂ z ( r τ zr ) + τ θθ + r κρ | u | u r = 0 ∂ t ( r ρ u z ) + ∇ · ( ru z ρ u ) + r ∂ p ∂ ∂ z − ∂ ∂ r ( r τ rz ) − ∂ ∂ z ( r τ zz ) + r ρ g + r κρ | u | u z = 0 ∂ ∂ t ( r ρ E ) + ∇ · ( r ( ρ E + p ) u ) − ∇ · ( r τ u ) − ∇ · ( r λ ∇ T ) + r ρ gu z = 0 ρ = ρ ( p , T ) ∗ M. Amara, D. Capatina and L. Lizaik, Coupling of a Darcy-Forchheimer model and compressible Navier-Stokes equations with heat transfer , Accepted in SIAM J. Sci. Comp. 2008. ∗ M. Amara, D. Capatina and L. Lizaik, Numerical coupling of 2.5D reservoir and 1.5D wellbore models in order to interpret thermometrics , Int. J. Numer. Meth. Fluids, Vol. 56, No. 8, pp. 1115-1122, 2008.
Modeling and numerical approximation of multi-component anisothermal flows in porous media Outline Outline Physical modeling Primary and secondary variables Boundary conditions Numerical scheme Numerical simulations
Modeling and numerical approximation of multi-component anisothermal flows in porous media Physical modeling Physical modeling Three phases ( p ) : water( w ), oil( o ) and gas ( g ) w ¯ n 1 n 2 . . . . . . n h n c components: water, heavy hydrocarbons, w × light hydrocarbons, methan.... o × × × × × g × × × × × n h hydrocarbon components ( n h = n c − 1 ) 3D / Porous media Ω with n W wells Gridding Cartesian rectangular mesh The code is able to interface with any gridding software by reading some necessary informations
Modeling and numerical approximation of multi-component anisothermal flows in porous media Physical modeling Physical modeling Three phases ( p ) : water( w ), oil( o ) and gas ( g ) w ¯ n 1 n 2 . . . . . . n h n c components: water, heavy hydrocarbons, w × light hydrocarbons, methan.... o × × × × × g × × × × × n h hydrocarbon components ( n h = n c − 1 ) 3D / Porous media Ω with n W wells Gridding Cartesian rectangular mesh The code is able to interface with any gridding software by reading some necessary informations
Modeling and numerical approximation of multi-component anisothermal flows in porous media Physical modeling Physical modeling Three phases ( p ) : water( w ), oil( o ) and gas ( g ) w ¯ n 1 n 2 . . . . . . n h n c components: water, heavy hydrocarbons, w × light hydrocarbons, methan.... o × × × × × g × × × × × n h hydrocarbon components ( n h = n c − 1 ) 3D / Porous media Ω with n W wells Gridding Cartesian rectangular mesh The code is able to interface with any gridding software by reading some necessary informations
Modeling and numerical approximation of multi-component anisothermal flows in porous media Physical modeling Governing equations Mass conservation equation for each component c : � � ∂ � F c = ∂ t ( φ S p ρ p y c , p ) + ∇ · ( ρ p u p y c , p ) = 0 p = o , g , w u p is given by the generalized Darcy law : u p = − k rp µ − 1 p K ( ∇ p p − ρ p g ) Energy equation : � � � � F T = ∂ � � ( φ S p ρ p H p − p p ) + 1 − φ ρ s H s + ∇ · ( φ S p ρ p H p u p ) ∂ t p = o , g , w p = o , w , g � −∇ · ( λ ∇ T ) + u p · ∇ p p = 0 p = o , g , w H p enthalpy of phase p T temperature � s g × φ λ = ( λ s )( 1 − φ ) × ( λ w ) s w × φ × ( λ o ) s o × φ × � λ equivalent thermal conductivity λ g Take into account convective, diffusive, compressibility and viscous dissipation effects
Modeling and numerical approximation of multi-component anisothermal flows in porous media Physical modeling Capillary pressure constraints : p c , ow = p o − p w (oil-water capillary pressure) p c , go = p g − p o (gas-oil capillary pressure) Capillary pressures are measured in laboratories Saturation constraint : n p � S p = 1 p = 1 Component mole fraction constraints : n c � ∀ p = w , o , g y c , p = 1 c = 1 Phase equilibrium relation for each hydrocarbon component c in oil and gas phases: F e = f c , o − f c , g = 0 f c , o and f c , g are the fugacities of hydrocarbon component c in oil and gas phases respectively, calculated from the Peng Robinson equation of state
Modeling and numerical approximation of multi-component anisothermal flows in porous media Primary and secondary variables ∗ Number of equations : Type Number Mass conservation n h + 1 Energy equation 1 Capillary pressure constraints 2 Saturation constraint 1 Component mole fraction constraints 2 Equilibrium relation equations n h Total 2 n h + 7 Primary and secondary variables According to Gibb’s phase rule, the number of primary variables is equal to : ( n c + 2 − n phase ) + ( n phase − 1) = n c + 1 Use linear constraint equations to remove two pressures, one saturation and two component mole fractions −→ 2 n h + 2 number of non-linear equations and variables is left
Modeling and numerical approximation of multi-component anisothermal flows in porous media Primary and secondary variables ∗ Number of equations : Type Number Mass conservation n h + 1 Energy equation 1 Capillary pressure constraints 2 Saturation constraint 1 Component mole fraction constraints 2 Equilibrium relation equations n h Total 2 n h + 7 Primary and secondary variables According to Gibb’s phase rule, the number of primary variables is equal to : ( n c + 2 − n phase ) + ( n phase − 1) = n c + 1 Use linear constraint equations to remove two pressures, one saturation and two component mole fractions −→ 2 n h + 2 number of non-linear equations and variables is left
Modeling and numerical approximation of multi-component anisothermal flows in porous media Primary and secondary variables Multiple choices for the selection of primary variables and equations leading to different models Coats Model Primary equations are the n c + 1 mass and energy balance equations ( F p = {F c , F T } ) Equations left over are the secondary equations ( F s = {F e } ) Primary variables X p are : 1 p g , T , S g , S o , y c , g ; c = 3 ... n h when both oil and gaz phases are present 2 p o , T , S o , y c , o ; c = 1 ... n h when gaz phase is not present 3 p g , T , S g , y c , g ; c = 1 ... n h when oil phase is not present Adjacent gridblocks may have different sets of primary variables −→ need to switch variables when a hydrocarbon phase disappears or reappears Phase equilibrium relations are used to eliminate the secondary variables from the primary equations After solving primary variables, secondary variables are updated explicitly gridblock by gridblock
Modeling and numerical approximation of multi-component anisothermal flows in porous media Primary and secondary variables Multiple choices for the selection of primary variables and equations leading to different models Coats Model Primary equations are the n c + 1 mass and energy balance equations ( F p = {F c , F T } ) Equations left over are the secondary equations ( F s = {F e } ) Primary variables X p are : 1 p g , T , S g , S o , y c , g ; c = 3 ... n h when both oil and gaz phases are present 2 p o , T , S o , y c , o ; c = 1 ... n h when gaz phase is not present 3 p g , T , S g , y c , g ; c = 1 ... n h when oil phase is not present Adjacent gridblocks may have different sets of primary variables −→ need to switch variables when a hydrocarbon phase disappears or reappears Phase equilibrium relations are used to eliminate the secondary variables from the primary equations After solving primary variables, secondary variables are updated explicitly gridblock by gridblock
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