Capillarity Approximation of Conservation Laws with Discontinuous - PowerPoint PPT Presentation

Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Lorenzo di Ruvo Department of Mathematics University of Bari Via E. Orabona 4 70125 Bari (Italy) EMAIL: diruvo@dm.uniba.it Padova 2012 joint work with Prof. Giuseppe Maria Coclite (University of Bari) Prof. Siddhartha Mishra (ETH Zürich) Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 1 / 20

Two–Phase Flow Motion in a one dimensional porous medium A one dimensional porous medium. Flow of two phases in this porous medium. Fluids are immiscible. Fluids are incompressible. The density of the two fluids is constant. Wetting phase w . Nonwetting phase n . The mass flow rate per unit volume injected or produced is zero. The mass conservation for each phase: ∂ t ( φ S w ) + ∂ x ( v w ) = 0 , t > 0 , x ∈ R , ∂ t ( φ S n ) + ∂ x ( v n ) = 0 , t > 0 , x ∈ R . Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 2 / 20

φ ≡ porosity. S w , S n ≡ saturations of the two phases. v w , v n ≡ velocities. The saturation defines the fraction of the pore volume occupied by a phase. S l ( t , x ) = V l ( t , x ) , l = w , n . V p Therefore, S w + S n = 1 . The porosity denotes the fraction of the volume available for flow. It depends on the pressure. φ = φ 0 [ 1 + c R ( p − p 0 )] . Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 3 / 20

c R ≡ rock compressibility. p ≡ pressure. φ 0 ≡ the porosity at the pressure p 0 . We assume φ = 1 . The mass conservation, and the saturation conservation give v w + v n = q . q ≡ total flow rate (specified, for instance, by boundary conditions). We are able to describe the motion of the system. unknowns: S w , S n , v w , v n . equations: 4. Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 4 / 20

Darcy’s Law The phase velocities are modelled on Darcy’s Law. For the motion of the single-flow in a porous medium, in 1856, Darcy discovered the following law: Darcy’s Law The volumetric flow rate of a homogeneous fluid through a porous medium is proportional to the pressure gradient and to the cross-sectional area normal to the direction of flow and inversely proportional to the viscosity of the fluid. v = − K µ ( ∂ x p − ρ g ) . K ≡ rock permeability. (It can vary (discontinuously) in space) µ ≡ viscosity of fluid g ≡ accelaration due to gravity Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 5 / 20

Two-Phases Flow. v w = K λ w ( ∂ x p w − ρ w g ) , v n = K λ n ( ∂ x p n − ρ n g ) . λ l = k l r ( S l ) l = w , n : phase mobility. µ l k l r ≡ relative permeability. k l r ≤ K l = w , n , k l r = k l r ( S w ) l = w , n . Adding v w and v n , we have K λ w ( ∂ x p w − ρ w g ) + K λ n ( ∂ x p n − ρ n g ) = q . Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 6 / 20

Capillary pressure Capillary Pressure The capillary pressure is the difference between the pressure p w and p n , on the contact surface of the two fluids. p c = p w − p n . p c is a decreasing function of S w Following the model proposed in: R. Helmig, A. Weiss and B. I. Wohlmuth Dynamic capillary effects in heterogeneous porous media. Comp. Geosci., 11 (2007), 261-274. Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 7 / 20

we consider p dc = p sc ( S w ) + τ∂ t S w . p dc ≡ dynamic capillary pressure. p sc ≡ static capillary pressure. τ ≥ 0 dynamic factor. p sc ( S w ) : [ 0 , 1 ] → [ 0 , ∞ ) . p sc ( S w ) is continuously differentiable. p sc ( S w ) is strictly decreasing. p dc depends on the dynamics of moving fronts. Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 8 / 20

PDE in S w τ = 1. Thanks to the mass conservation, Darcy’s Law, and the dynamic capillary pressure, we have λ w + λ n − λ w λ n K ( ρ w − ρ n ) g λ w q ∂ t S w + ∂ x � � λ w + λ n � K λ w λ n ∂ x p sc � � tx S w � K λ w λ n ∂ 2 = − ∂ x − ∂ x . The capillary pressure is supposed to be very small. We can rescale the previous PDE with a small scale ν . λ w + λ n − λ w λ n K ( ρ w − ρ n ) g λ w q ∂ t S w + ∂ x � � λ w + λ n � K λ w λ n ∂ x p sc � � tx S w � − ν 2 ∂ x K λ w λ n ∂ 2 = − ν∂ x . Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 9 / 20

PDE in u S w = u , 0 ≤ u ( t , x ) ≤ 1 . we denote λ w + λ n − λ w λ n K ( ρ w − ρ n ) g λ w q f ( k , u ) = , λ w + λ n g ( l , u ) = − K λ w λ n dp sc du , h ( m , u ) = − K λ w λ n . PDE in u . ∂ t u + ∂ x f ( k , u ) = ν∂ x ( g ( l , u ) ∂ x u ) + ν 2 ∂ x ( h ( m , u ) ∂ 2 tx u ) . Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 10 / 20

Cauchy Problem Let t ≥ 0 , x ∈ R . ∂ t u ν + ∂ x f ( k ν , u ν ) = ν∂ x ( g ( ℓ ν , u ν ) ∂ x u ν ) + ν 2 ∂ x ( h ( m ν , u ν ) ∂ 2  tx u ν ) ,    ∂ t k ν = ν∂ 2  xx k ν ,      ∂ t ℓ ν = ν∂ 2 xx ℓ ν ,      ∂ t m ν = ν∂ 2  xx m ν ,  u ν ( 0 , x ) = u 0 ,ν ( x ) ,     k ν ( 0 , x ) = k 0 ,ν ( x ) ,       ℓ ν ( 0 , x ) = ℓ 0 ,ν ( x ) ,      m ν ( 0 , x ) = m 0 ,ν ( x ) .  ( u ν , k ν , ℓ ν , m ν ) is a smooth solution, 0 ≤ u ν ≤ 1. Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 11 / 20

0 < ν ≤ 1. f , g , h : R 2 → R are smooth functions. 0 < α ≤ g ( · , · ) , h ( · , · ) . f ( k , · ) is genuinely nonlinear for every k ∈ R . u ∈ [ 0 , 1 ] → f ( k , u ) is not affine on any nontrivial interval for every k ∈ R . Moreover, we assume k 0 ,ν , ℓ 0 ,ν , m 0 ,ν ∈ C ∞ ( R ) ∩ W 1 , 1 ( R ) , u 0 ,ν ∈ C ∞ ( R ) ∩ L 1 ( R ) ∩ L ∞ ( R ) , 0 ≤ u 0 ,ν ≤ 1 , and that there exist k , ℓ, m ∈ BV ( R ) ∩ L 1 ( R ) , u 0 ∈ L 1 ( R ) ∩ L ∞ ( R ) , 0 ≤ u 0 ≤ 1 , Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 12 / 20

such that a.e. and in L p ( R ) u 0 ,ν → u 0 , k 0 ,ν → k , ℓ 0 ,ν → ℓ, m 0 ,ν → m , � u ν, 0 � L 2 ( R ) ≤ � u 0 � L 2 ( R ) , ν, � k 0 ,ν � L ∞ ( R ) ≤ � k � L ∞ ( R ) , � k 0 ,ν � L 2 ( R ) ≤ � k � L 2 ( R ) , � ∂ x k 0 ,ν � L 1 ( R ) ≤ TV ( k ) , ν, � ℓ 0 ,ν � L ∞ ( R ) ≤ � ℓ � L ∞ ( R ) , � ℓ 0 ,ν � L 2 ( R ) ≤ � ℓ � L 2 ( R ) , � ∂ x ℓ 0 ,ν � L 1 ( R ) ≤ TV ( ℓ ) , ν, � m 0 ,ν � L ∞ ( R ) ≤ � m � L ∞ ( R ) , � m 0 ,ν � L 2 ( R ) ≤ � m � L 2 ( R ) , � ∂ x m 0 ,ν � L 1 ( R ) ≤ TV ( m ) , ν, 1 ≤ p < ∞ . Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 13 / 20

Theorem There exist a sequence { ν n } n ⊂ ( 0 , 1 ] , ν n → 0, a sequence { u ν n } n ∈ N of solutions of the previous Cauchy problem, and a distributional solution u of � ∂ t u + ∂ x f ( k ( x ) , u ) = 0 , t > 0 , x ∈ R , u ( 0 , x ) = u 0 ( x ) , x ∈ R . such that in L p u ν n → u , loc (( 0 , ∞ ) × R ) , 1 ≤ p < ∞ , and a.e. in ( 0 , ∞ ) × R . Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 14 / 20

Proof Murat’s Lemma Let Ω be a bounded open subset of R 2 . Suppose the sequence {L n } n ∈ N of distributions is bounded in W − 1 , ∞ (Ω) . Suppose also that L n = L 1 , n + L 2 , n , where {L 1 , n } n ∈ N lies in a compact subset of H − 1 loc (Ω) and {L 2 , n } n ∈ N lies in a bounded subset of M loc (Ω) . Then {L n } n ∈ N lies in a compact subset of H − 1 loc (Ω) . G. M. Coclite, K. H. Karlsen, S. Mishra, and N. H. Risebro Convergence of vanishing viscosity approximations of 2 × 2 triangular systems of multi-dimensional conservation laws. (2009). Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 15 / 20

Let us show that the following family { ∂ t | u ν − c | + ∂ x ( sign ( u ν − c ) ( f ( k , u ν ) − f ( k , c ))) } ν> 0 (1) is compact in H − 1 loc (( 0 , ∞ ) × R ) . c is a fixed real number. Let { ( η ν , Q ν ) } ν> 0 be a family of maps such that Q ν ∈ C ∞ ( R 2 ) , η ν ∈ C ∞ ( R ) , ∂ u Q ν ( k , u ) = ∂ u f ( k , u ) η ′ ν ( u ) , η ′′ ν ≥ 0 , � η ′ � ν − η ′ � � η ν − η 0 � L ∞ ([ 0 , 1 ]) ≤ ν, L 1 ([ 0 , 1 ]) ≤ ν, 0 � � � η ′ � L ∞ ([ 0 , 1 ]) ≤ 1 , η ν ( 0 ) = Q ν ( k , 0 ) = 0 . � ν It is an approximation of (1). Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 16 / 20