On the homogenization of some double porosity models with periodic - PDF document

On the homogenization of some double porosity models with periodic thin structures B. Amaziane ( a ) , L. Pankratov ( b ) 15th October 2008 (a) Laboratoire de Math´ ematiques Appliqu´ ees, CNRS–UMR5142, Universit´ e de Pau, av. de l’Universit´ e, 64000 Pau, France; e–mail: brahim.amaziane@univ-pau.fr (b) Laboratoire de Math´ ematiques Appliqu´ ees, CNRS–UMR5142, Universit´ e de Pau, av. de l’Universit´ e, 64000 Pau, France and D´ epartement de Math´ ematiques, B.Verkin Institut des Basses Temp´ eratures, 47, av. L´ enine, 61103, Kharkov, Ukraine e–mail: leonid.pankratov@univ-pau.fr and pankratov@ilt.kharkov.ua 1

Abstract Models describing global behavior of incompressible flow in fractured media are dis- cussed. A fractured medium is regarded as a porous medium consisting of two super- imposed continua, a continuous fracture system and a discontinuous system of medium- sized matrix blocks. We derive global behavior of fractured media versus different para- meters such as the fracture thickness, the size of blocks and the ratio of the block perme- ability and the permeability of fissures, and oscillating source terms. The homogenization results are obtained by mean of the convergence in domains of asymptotically degenerat- ing measure. 2

1 Setting of the problem 1.1 The geometry of the periodic medium • Ω ⊂ R d ( d = 2 , 3 ) – a bounded connected domain with a periodic structure; • Y =]0 , 1[ d – the reference cell of a fractured porous medium; • we assume that Y is made up of two homogeneous porous media M δ and F δ cor- responding to parties of the domain occupied by the matrix block and the fracture, respectively; • we assume that M δ is an open cube centered at the same point as Y with length equal to (1 − δ ) , where 0 < δ < 1 ; Thus Y = M δ ∪ Γ δ m,f ∪ F δ , where Γ δ m,f denotes the interface between the two media m ∪ Γ ε,δ m,f ∪ Ω ε,δ f , where Γ ε,δ m ∩ ∂ Ω ε,δ (see Figure 1). Then Ω = Ω ε,δ m,f = ∂ Ω ε,δ and the f subscripts m and f refer to the matrix and fracture, respectively (see Figure 2). For the sake of simplicity, we will assume that ∂ Ω ∩ Ω ε,δ m = ∅ . ε,δ ε,δ Ω δ m Ω F f 1 δ/2 δ M 0 1 Figure 1: The reference cell Y . ε εδ Figure 2: The periodic domain Ω . 3

1.2 Permeability and porosity of the porous medium Now let us introduce the permeability coefficient and the porosity of the porous medium Ω . We set m ( x ) + k f 1 ε,δ m ( x ) + ω f 1 ε,δ K ε,δ ( x ) = k m 1 ε,δ ω ε,δ ( x ) = ω m 1 ε,δ f ( x ) and f ( x ) , (1.1) where k f is the permeability or the hydraulic conductivity of fissures, k m is the perme- ability or the hydraulic conductivity of blocks, ω f is the porosity of fissures, ω m is the porosity of blocks; 1 ε,δ = 1 ε,δ f ( x ) and 1 ε,δ m = 1 ε,δ m ( x ) denote the (periodic) characteristic f functions of the sets Ω ε,δ f and Ω ε,δ m , respectively. Here 0 < k f , k m , ω f , ω m < + ∞ . 4

1.3 Assumptions We make the following assumptions on permeabilities of fissures and blocks as well as on the source term. (H.1) Porosities ω f , ω m of fissures and blocks are independent of ε, δ . (H.2) The permeability of blocks is related to the permeability of fissures by r , the per- meability ratio: k m = r k f . (1.2) r is supposed to be small and then defined in the following way: r = ( εδ ) θ , (1.3) where θ > 0 is a parameter. (H.3) The source term is given by f ε,δ ( x ) = ( f 0 + f m )( x ) 1 ε,δ f ( x ) + f m ( x ) 1 ε,δ m ( x ) , (1.4) where f 0 ∈ L 2 (Ω) and f m ∈ C 1 (Ω) . 5

1.4 A model with two small parameters We consider the following parabolic problem for the function u ε,δ : Q → R :  ω ε,δ ( x ) u ε,δ − div ( K ε,δ ( x ) ∇ u ε,δ ) = f ε,δ ( x ) in Q ;   t ∇ u ε,δ · � ( P ε,δ ) ν = 0 on S Q ; (1.5)   u ε,δ (0 , x ) = 0 in Ω , where Q =]0 , T [ × Ω , � ν is the outward normal vector to Ω , S Q =]0 , T [ × ∂ Ω , T > 0 is given. 6

1.5 A model with one small parameter The reference cell Y is represented as follows: Y = M ε ∪ G ε m,f ∪ F ε , where G ε m,f denotes the interface between the two media and M ε is the cube with length equal (1 − ℓε α 2 ) . The porous medium Ω in this case is defined as follows: Ω = Ω ε m ∪ Γ ε m,f ∪ Ω ε f , where Γ ε m,f = ∂ Ω ε m ∩ ∂ Ω ε f . For simplicity, we will assume that ∂ Ω ∩ Ω ε m = ∅ . The permeability and porosity are given by: K ε ( x ) = k f r ( ε ) 1 ε m ( x ) + k f 1 ε ω ε ( x ) = ω m 1 ε m ( x ) + ω f 1 ε f ( x ) and f ( x ) , (1.6) where 1 ε f = 1 ε f ( x ) and 1 ε m = 1 ε m ( x ) denote characteristic periodic functions of the fissure set Ω ε f and the matrix system Ω ε m , respectively. (H.4) The source term is given by f ε ( x ) = ( f 0 + f m )( x ) 1 ε f ( x ) + f m ( x ) 1 ε with f 0 ∈ L 2 (Ω) , f m ∈ C 1 (Ω) . m ( x ) (1.7)  ω ε ( x ) u ε t − div ( K ε ( x ) ∇ u ε ) = f ε ( x )  in Q ;  ∇ u ε ( t, x ) · � ( P ε ) ν = 0 on S Q ; (1.8)   u ε (0 , x ) = 0 in Ω . 7

1.6 The concepts of convergence From now on |O| denotes the measure of the set O . m ∪ Γ ε,δ m,f ∪ Ω ε,δ with lim δ → 0 lim ε → 0 | Ω ε,δ Definition 1.1 Let Ω = Ω ε,δ f | = 0 . A sequence f { u ε,δ } ⊂ L 2 (Ω ε,δ f ) is said to L ε,δ –converge to a function u ∈ L 2 (Ω) if 1 � u ε,δ − u � 2 δ → 0 lim lim f ) = 0 . L 2 (Ω ε,δ | Ω ε,δ ε → 0 f | Definition 1.2 Let Ω = Ω ε m ∪ Γ ε m,f ∪ Ω ε f with lim ε → 0 | Ω ε f | = 0 . A sequence { u ε } ⊂ L 2 (Ω ε f ) is said to L ε –converge to a function u ∈ L 2 (Ω) if 1 f |� u ε − u � 2 lim f ) = 0 . L 2 (Ω ε | Ω ε ε → 0 8

A double porosity model with thin fissures: ( P ε,δ ) model 2 In this Section we assume that δ → 0 (2.1) and study the asymptotic behavior of the solution of problem (1.5) first as ε → 0 and then as δ → 0 . The measure of F δ , | F δ | , for δ sufficiently small, is calculated as follows � 2 δ + O ( δ 2 ) if d = 2; | F δ | = (2.2) 3 δ + O ( δ 2 ) if d = 3; then the assumption (2.1) implies that ε → 0 | Ω ε,δ lim δ → 0 lim f | = 0 . (2.3) Notation: � in Ω ε,δ ρ ε,δ f ; u ε,δ = σ ε,δ in Ω ε,δ m ; 9

Effective equations for ( P ε,δ ) when θ = 2 2.1 We study the asymptotic behavior of u ε,δ as ε, δ → 0 . We will show that, for any fixed δ , problem (1.5) admits (as ε → 0 ) a homogenization problem and that the homogenized solution converges, as δ → 0 , to the solution of the following effective problem:  t − K f ( d ) ∆ ρ ∗ = ( f 0 + f m )( x ) + S ( ρ ∗ ) ω f ρ ∗  in Q ;  ∇ ρ ∗ · � ν = 0 on S Q ; (2.4)   ρ ∗ (0 , x ) = 0 in Ω , where � k f / 2 if d = 2; K f ( d ) = (2.5) 2 k f / 3 if d = 3; and the additional source term S ( ρ ∗ ) is given by: � � � t ρ ∗ S ( ρ ∗ ) = − 2 k f ω m t ( x, τ ) t k f √ π √ t − τ dτ + 4 f m ( x ) . (2.6) πω m 0 The following convergence result is valid. Theorem 2.1 Let u ε,δ = � ρ ε,δ , σ ε,δ � be the solution of (1.5) and let θ = 2 in (1.3). Then, under assumptions (H.1)–(H.3) , for any t ∈ ]0 , T [ , (I) the function σ ε,δ , as well as the function u ε,δ , converges to ( tf m ( x )) , namely: � � � � � 2 � 2 � ω m σ ε,δ ( t ) − t f m � ω ε,δ u ε,δ ( t ) − t f m lim δ → 0 lim m ) = lim δ → 0 lim L 2 (Ω) = 0; (2.7) L 2 (Ω ε,δ ε → 0 ε → 0 (II) the function ρ ε,δ L ε,δ –converges to ρ ∗ the solution of a global model (2.4) with the additional source term (2.6) and the fracture porosity as effective porosity. 10

Effective equations for ( P ε,δ ) when θ > 2 2.2 In this case the homogenized problem has the form:  t − K f ( d )∆ ρ ∗ = ( f 0 + f m )( x ) ω f ρ ∗ in Q ;   ∇ ρ ∗ · � ν = 0 on S Q ; (2.8)   ρ ∗ (0 , x ) = 0 in Ω , where the coefficient K f ( d ) is defined in (2.5). The following convergence result is valid. Theorem 2.2 Let u ε,δ = � ρ ε,δ , σ ε,δ � be the solution of (1.5) and let θ > 2 in (1.3). Then, under assumptions (H.1)–(H.3) , for any t ∈ ]0 , T [ , (I) the function σ ε,δ , as well as the function u ε,δ , converges to ( tf m ( x )) , namely: � � � � � 2 � 2 � ω m σ ε,δ ( t ) − t f m � ω ε,δ u ε,δ ( t ) − t f m δ → 0 lim lim m ) = lim δ → 0 lim L 2 (Ω) = 0; (2.9) L 2 (Ω ε,δ ε → 0 ε → 0 (II) the function ρ ε,δ L ε,δ –converges to ρ ∗ the solution of the effective model (2.8). 11

Effective equations for ( P ε,δ ) when 0 < θ < 2 2.3 In this case the following convergence result is valid. Theorem 2.3 Let u ε,δ = � ρ ε,δ , σ ε,δ � be the solution of (1.5) and let θ < 2 in (1.3). Then, under assumptions (H.1)–(H.3) , for any t ∈ ]0 , T [ , (I) the function σ ε,δ , as well as the function u ε,δ , converges to ( tf m ( x )) , namely: � � � � � 2 � 2 � ω m σ ε,δ ( t ) − t f m � ω ε,δ u ε,δ ( t ) − t f m δ → 0 lim lim m ) = lim δ → 0 lim L 2 (Ω) = 0; (2.10) L 2 (Ω ε,δ ε → 0 ε → 0 (II) the function ρ ε,δ L ε,δ –converges to ( t ω − 1 m f m ( x )) . 12