Homogenization of a Pseudoparabolic System M. Peszy nska, R.E. - PowerPoint PPT Presentation

Introduction The Classical Bimodal System The Highly-Heterogeneous Case Homogenization of a Pseudoparabolic System M. Peszy´ nska, R.E. Showalter, Son-Young Yi Department of Mathematics Oregon State University Dubrovnik, 2008 Dept of Energy, Office of Science, 98089 “Modeling, Analysis, and Simulation of Multiscale Preferential Flow” Peszynska-RES-Yi Dubrovnik, 2008

Introduction The Classical Bimodal System The Highly-Heterogeneous Case Outline Introduction 1 Richards Equation Pseudoparabolic System Asymptotic Expansion The Classical Bimodal System 2 The ε -problem The Partially-Upscaled System The Upscaled System The Highly-Heterogeneous Case 3 The ε -problem The Partially-Upscaled System The Macro-Micro System Peszynska-RES-Yi Dubrovnik, 2008

Introduction Richards Equation The Classical Bimodal System Pseudoparabolic System The Highly-Heterogeneous Case Asymptotic Expansion Richards Equation Two-phase flow through a partially-saturated porous medium with porosity φ ( x ) , permeability K ( x ) , relative permeability k w ( u ) and capillary pressure function P c ( u ) : φ ( x ) ∂ u ∂ t − ∇ · K ( x ) k w ( u ) ∇ ( P c ( u ) + ρ Gd ( x )) = 0 , µ w u ( x , t ) denotes saturation, and gravitational effects depend on depth d ( x ) = x 3 . Peszynska-RES-Yi Dubrovnik, 2008

Introduction Richards Equation The Classical Bimodal System Pseudoparabolic System The Highly-Heterogeneous Case Asymptotic Expansion Dynamic Capillary Pressure Experimental determination of p = P c ( u ) is based on the assumption that this is an instantaneous process. In reality it requires substantial time to approach an equilibrium before measurements can be taken. Hassanizadeh-Gray (1993) model ∂ u P c , dyn ( u ) ≡ P c ( u ) + τ H ∂ t : φ ( x ) ∂ u ∂ t − ∇ · K ( x ) k w ( u ) ∇ ( P c ( u ) + ρ Gd ( x )) µ w − ∇ · K ( x ) k w ( u ) ∂ u ∇ τ H ∂ t = 0 . µ w Peszynska-RES-Yi Dubrovnik, 2008

Introduction Richards Equation The Classical Bimodal System Pseudoparabolic System The Highly-Heterogeneous Case Asymptotic Expansion pseudoparabolic equation Linearize ... the pseudoparabolic equation ∂ u ( t , x )+ τ ( x ) ∂ � � � � φ ( x ) u ( t , x ) −∇· κ ( x ) ∇ ∂ t φ ( x ) u ( t , x ) = 0 ∂ t is distinguished from the usual parabolic equation by τ ( x ) > 0. Porous media applications require that we know how to homogenize such equations. Peszynska-RES-Yi Dubrovnik, 2008

Introduction Richards Equation The Classical Bimodal System Pseudoparabolic System The Highly-Heterogeneous Case Asymptotic Expansion Bensoussan, Lions, and Papanicolaou briefly investigated the homogenization of pseudoparabolic equations as an example for which the limiting problem is of a different type, and perhaps non-local , not even a PDE. We shall see below that this occurs when certain variables are eliminated or hidden . Peszynska-RES-Yi Dubrovnik, 2008

Introduction Richards Equation The Classical Bimodal System Pseudoparabolic System The Highly-Heterogeneous Case Asymptotic Expansion pseudoparabolic system ∂ 1 � � � � φ ( x ) u ( t , x ) + u ( t , x ) − v ( t , x ) = 0 , ∂ t τ ( x ) 1 � � � � − ∇ · κ ( x ) ∇ v ( t , x ) + v ( t , x ) − u ( t , x ) = 0 , x ∈ Ω , τ ( x ) v ( t , s ) = 0 , s ∈ ∂ Ω , φ ( x ) u ( 0 , x ) = φ ( x ) u 0 ( x ) , x ∈ Ω . Peszynska-RES-Yi Dubrovnik, 2008

Introduction Richards Equation The Classical Bimodal System Pseudoparabolic System The Highly-Heterogeneous Case Asymptotic Expansion Asymptotic Expansion Let Y denote the unit cube in R N . Let the Y -periodic functions φ ( y ) , τ ( y ) , κ ( y ) be given and define φ ε ( x ) = φ ( x ε ) , τ ε ( x ) = τ ( x ε ) , κ ε ( x ) = κ ( x ε ) . The corresponding solution u ε , v ε depends on ε . We write these as formal asymptotic expansions ∞ ∞ � � ε p u p ( t , x , y ) , ε p v p ( t , x , y ) , u ε ( t , x ) = v ε ( t , x ) = p = 0 p = 0 y = x ε , with each u p ( t , x , · ) , v p ( t , x , · ) being Y -periodic. Peszynska-RES-Yi Dubrovnik, 2008

Introduction Richards Equation The Classical Bimodal System Pseudoparabolic System The Highly-Heterogeneous Case Asymptotic Expansion Cell problem The effective tensor κ ∗ is obtained in this calculation as � κ ∗ ij = Y κ ( y )( ∇ y ω i ( y ) + e i ) · ( ∇ y ω j ( y ) + e j ) dy , where Periodic Cell Problem : ω j is Y -periodic and −∇ y · κ ( y ) ( ∇ y ω j ( y ) + e j ) = 0 , j = 1 . . . N . Peszynska-RES-Yi Dubrovnik, 2008

Introduction Richards Equation The Classical Bimodal System Pseudoparabolic System The Highly-Heterogeneous Case Asymptotic Expansion partially-upscaled system The leading terms in the expansion satisfy the pseudoparabolic system φ ( y ) ∂ u 0 ( t , x , y ) 1 + τ ( y )( u 0 ( t , x , y ) − v 0 ( t , x )) = 0 , ∂ t � 1 −∇ · κ ∗ ∇ v 0 ( t , x ) + τ ( y )( v 0 ( t , x ) − u 0 ( t , x , y )) dy = 0 , Y together with boundary and initial conditons, v 0 ( t , s ) = 0 , s ∈ ∂ Ω , u 0 ( 0 , x , y ) = u 0 ( x ) . Peszynska-RES-Yi Dubrovnik, 2008

Introduction Richards Equation The Classical Bimodal System Pseudoparabolic System The Highly-Heterogeneous Case Asymptotic Expansion Upscaled pseudoparabolic equation Only if the product φ ( · ) τ ( · ) is constant do we get u 0 ( t , x , y ) = u 0 ( t , x ) independent of y ∈ Y , and in that case we can eliminate v 0 from the system: φ ∗ ∂ u 0 ( t , x ) − ∇ · κ ∗ ∇ u 0 ( t , x ) − ∇ · κ ∗ ∇ φ ∗ τ ∗ ∂ u 0 ( t , x ) = 0 . ∂ t ∂ t NOTE: φ ∗ = � Y φ ( y ) dy is the average � − 1 τ ∗ = �� 1 τ ( y ) dy is the harmonic average Y Peszynska-RES-Yi Dubrovnik, 2008

Introduction The ε -problem The Classical Bimodal System The Partially-Upscaled System The Highly-Heterogeneous Case The Upscaled System Classical Bimodal Medium Unit cube Y is given in open disjoint complementary parts, Y 1 and Y 2 , χ j ( y ) = Y -periodic characteristic function of Y j . Corresponding ε -periodic characteristic functions are � x � χ ε j ( x ) ≡ χ j x ∈ R N , , j = 1 , 2 , ε and these partition the global domain Ω into two sub-domains, Ω ε 1 and Ω ε 2 by x ∈ Ω : χ ε Ω ε � � j ≡ j ( x ) = 1 , j = 1 , 2 . Peszynska-RES-Yi Dubrovnik, 2008

Introduction The ε -problem The Classical Bimodal System The Partially-Upscaled System The Highly-Heterogeneous Case The Upscaled System Coefficients Given φ j ( · , · ) , κ j ( · , · ) , τ j ( · , · ) ∈ L ∞ (Ω; C ( Y j )) , define Y -periodic functions in L ∞ (Ω; L 2 # ( Y )) by φ ( x , y ) ≡ φ j ( x , y ) , y ∈ Y j , j = 1 , 2 , x ∈ Ω , similarly κ ( x , y ) and τ ( x , y ) . Corresponding functions on Ω ε j are x , x x , x x , x � � � � � � φ ε κ ε τ ε j ( x ) ≡ φ j , j ( x ) ≡ κ j , j ( x ) ≡ τ j , ε ε ε and coefficients for the pseudoparabolic system are φ ε ( x ) ≡ χ ε 1 ( x ) φ ε 1 ( x ) + χ ε 2 ( x ) φ ε 2 ( x ) , κ ε ( x ) ≡ χ ε 1 ( x ) κ ε 1 ( x ) + χ ε 2 ( x ) κ ε 2 ( x ) , τ ε ( x ) ≡ χ ε 1 ( x ) τ ε 1 ( x ) + χ ε 2 ( x ) τ ε 2 ( x ) . Peszynska-RES-Yi Dubrovnik, 2008

Introduction The ε -problem The Classical Bimodal System The Partially-Upscaled System The Highly-Heterogeneous Case The Upscaled System The ε - problem u ε ( · ) ∈ H 1 (( 0 , T ); L 2 (Ω)) and v ε ( · ) ∈ L 2 (( 0 , T ); H 1 0 (Ω)) φ ε ( x ) ∂ u ε ( t , x ) 1 u ε ( t , x ) − v ε ( t , x ) � � + = 0 , x ∈ Ω , ∂ t τ ε ( x ) 1 � κ ε 1 ( x ) ∇ v ε ( t , x ) � � v ε ( t , x ) − u ε ( t , x ) � = 0 , x ∈ Ω ε −∇ · + 1 , τ ε 1 ( x ) 1 κ ε 2 ( x ) ∇ v ε ( t , x ) v ε ( t , x ) − u ε ( t , x ) = 0 , x ∈ Ω ε � � � � −∇ · + 2 , τ ε 2 ( x ) γ ε 1 v ε ( t , s ) = γ ε 2 v ε ( t , s ) , κ ε 1 ( s ) ∇ v ε ( t , s ) · ν = κ ε 2 ( s ) ∇ v ε ( t , s ) · ν, s ∈ Γ ε , boundary condition v ε ( t , s ) = 0 , s ∈ ∂ Ω , and the initial condition u ε ( 0 , x ) = u 0 ( x ) , x ∈ Ω , independent of ε . Peszynska-RES-Yi Dubrovnik, 2008

Introduction The ε -problem The Classical Bimodal System The Partially-Upscaled System The Highly-Heterogeneous Case The Upscaled System two-scale limit LEMMA 1: For each ε > 0, let u ε ( · ) , v ε ( · ) denote the unique solution to the pseudoparabolic ε -problem. There exist (i) a function U in L 2 � ( 0 , T ) × Ω; L 2 � # ( Y ) , (ii) a function v in L 2 � ( 0 , T ); H 1 � 0 (Ω) , (ii) a function V in L 2 � ( 0 , T ) × Ω; H 1 � # ( Y ) / R , and a subsequence which two-scale converges 2 u ε → U ( t , x , y ) , 2 v ε → v ( t , x ) , 2 ∇ v ε → ∇ v ( t , x ) + ∇ y V ( t , x , y ) . Peszynska-RES-Yi Dubrovnik, 2008

Introduction The ε -problem The Classical Bimodal System The Partially-Upscaled System The Highly-Heterogeneous Case The Upscaled System The effective tensor κ ∗ is given by � κ ∗ ij ( x ) = κ ( x , y )( ∇ y ω i ( x , y ) + e i ) · ( ∇ y ω j ( x , y ) + e j ) dy . Y where each ω k is the solution of the periodic cell problem ω k ∈ L 2 (Ω; H 1 # ( Y )) : � � � κ ( x , y ) ∇ y ω k ( x , y ) + e k · ∇ y Ψ( x , y ) dy = 0 Y for all Ψ ∈ L 2 (Ω; H 1 # ( Y )) . � (Let’s ask that Y ω k ( x , y ) dy = 0 to fix the constant.) Peszynska-RES-Yi Dubrovnik, 2008