Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Loss of strict hyperbolicity in Riemann solutions for vertical three-phase flow in porous media Panters Rodriguez Bermudez Dan Marchesin Instituto Nacional de Matemática Pura e Aplicada Rio de Janeiro, Brazil. Work supported by CNPq-TWAS-IMPA. HYP2012, June 25 t h
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Historical Review The horizontal two-phase flow injection problem was solved by Buckley and Leverett in 1942. The B-L equation for two-phase flow with gravity is solved through Oleinik’s construction, ( ex. , Proskurowski (1981)). Isaacson, Marchesin, Plohr, Temple, Paes Leme, Seabra, De Souza, Furtado, etc, contributed to solve the R-P for immiscible horizontal three-phase flow. We present a class of Riemann solutions for immiscible three-phase flow with gravity.
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Physical Problem and Simplifications The porosity φ and the absolute permeability of the rock k are constant. The fluids are immiscible and Gravity there is no mass interchange between phases. The flow occurs uniformly in the vertical direction filling the Interface entire porous medium. The fluids are incompressible. There are no sources or sinks.
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Mass Conservation and Darcy’s Law. Conservation of mass ∂ t φ s i + ∂ ∂ ∂ x u i = 0 i = 1 , 2 , 3 , (1) where s i denotes saturation and u i is the velocity of each phase. Darcy’s law u i = − k k r , i ∂ p ❙ ✓ ∂ x − ρ i g ✓ i = 1 , 2 , 3 , � � (2) ❙ ✓ ❙ µ i p is the pressure, k r , i is the relative permeability, µ i is the viscosity and ρ i is the density for each phase i .
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Deriving the System of Conservation Laws. u i = u f i + k Λ i f j ρ ij g , i = 1 , 2 , 3 , � (3) j � = i where ρ ij = ρ i − ρ j . Mobilities and fractional flow functions Λ i Λ i = k r , i /µ i , f i = i = 1 , 2 , 3 . , � 3 j = 1 Λ j We use the quadratic Corey permeability model k r , i ( s i ) = s 2 i . (4)
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow System of Conservation Laws Dimensionless equations for vertical three-phase flow ∂ s i ∂ t + ∂ α f i ( s 1 , s 2 ) + G i ( s 1 , s 2 ) i = 1 , 2 , 3 , � � = 0 , (5) ∂ x where u u µ ref α = = K ref ρ ref g u ref is the convection/gravity ratio (later we will set α = 0). Since � s i = 1 there is a redundant equation. Gravitational Fluxes G 1 k Λ 1 ( 1 − f 1 ) ρ 13 + f 2 ρ 32 � � = , G 2 k Λ 2 ( 1 − f 2 ) ρ 21 + f 3 ρ 13 � � = , G 3 k Λ 3 ( 1 − f 3 ) ρ 32 + f 1 ρ 21 � � = .
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Umbilic and quasi-umbilic points Umbilic point A coinc. point S ∗ is an umbilic point of S t + [ F ( S )] x = 0 if (H1) dF ( S ∗ ) is diagonalizable. (H2) There is a neighborhood V of S ∗ such that dF ( S ) has distinct eigenvalues for all S ∈ V − S ∗ . Quasi-umbilic point A coincidence point S ∗ is a quasi-umbilic point ⇔ (H2) holds but (H1) fails. Coincidence diagonalization curve A coincidence diagonalization curve is a curve of coincidence points along which condition (H2) fails but condition (H1) holds.
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Coincidence Points V 2 α = 0, ρ 1 > ρ 2 > ρ 3 V i are umbilic points, Q i are � Q � 1 1 3 Q 3 � Q 2 quasi-umbilic points. V Q V 3 2 1 � 2 α = 0, ρ 1 = ρ 2 � = ρ 3 V 2 V 3 is an umbilic point, Q 1 , Q 2 are Q � 1 � 1 3 quasi-umbilic points, ∂ 3 is a coincidence diagonalization line. V Q V 3 2 � 1 2 V 2 α � = 0 small, ρ 1 > ρ 2 > ρ 3 V i and U ∗ α are umbilic points, Q α � 1 � 3 � i Q 1 � Q 3 � Q are quasi-umbilic points. 2 U * � V Q � V 3 2 1 � 2
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Coincidence Points V 2 α = 0, ρ 1 > ρ 2 > ρ 3 V i are umbilic points, Q i are � Q � 1 1 3 Q 3 � Q 2 quasi-umbilic points. V Q V 3 2 1 � 2 α = 0, ρ 1 = ρ 2 � = ρ 3 V 2 V 3 is an umbilic point, Q 1 , Q 2 are Q � 1 � 1 3 quasi-umbilic points, ∂ 3 is a coincidence diagonalization line. V Q V 3 2 � 1 2 V 2 α � = 0 small, ρ 1 > ρ 2 > ρ 3 V i and U ∗ α are umbilic points, Q α � 1 � 3 � i Q 1 � Q 3 � Q are quasi-umbilic points. 2 U * � V Q � V 3 2 1 � 2
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Coincidence Points V 2 α = 0, ρ 1 > ρ 2 > ρ 3 V i are umbilic points, Q i are � Q � 1 1 3 Q 3 � Q 2 quasi-umbilic points. V Q V 3 2 1 � 2 α = 0, ρ 1 = ρ 2 � = ρ 3 V 2 V 3 is an umbilic point, Q 1 , Q 2 are Q � 1 � 1 3 quasi-umbilic points, ∂ 3 is a coincidence diagonalization line. V Q V 3 2 � 1 2 V 2 α � = 0 small, ρ 1 > ρ 2 > ρ 3 V i and U ∗ α are umbilic points, Q α � 1 � 3 � i Q 1 � Q 3 � Q are quasi-umbilic points. 2 U * � V Q � V 3 2 1 � 2
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Schaeffer-Shearer Cone and Deviator operator X Y + Z Dev ( dF ( s 1 , s 2 )) = dF − 1 � � 2 tr ( dF ) I = (6) Y − Z X ( s 1 , s 2 ) − → ( X , Y , Z ) Umbilic point Quasi-umbilic point Coinc. diag. line
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Wave groups and Riemann solutions The Riemann solutions consist of two wave groups separated by a constant state. Each wave group consists on a sequence of rarefaction waves and adjacent shock waves. Shock waves must satisfy the Generalized Lax conditions. No 1-wave is preceded by a 2-wave. Parameterized by wave curves (Liu). Saturation x g Figure: Wave groups in Riemann solutions
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Generalized Lax Shock Waves and TSR Generalized Lax 1-shock and 2-shock waves 1-shock: λ 1 ( S + ) ≤ σ ≤ λ 1 ( S − ) , and σ < λ 2 ( S + ) . (7) 2-shock: λ 2 ( S + ) ≤ σ ≤ λ 2 ( S − ) , and λ 1 ( S − ) < σ. (8) At most one equality in (7), (8). Triple Shock Rule (TSR) Assume that the states S 1 , S 2 and S 3 satisfy S 2 ∈ H ( S 1 ) , S 3 ∈ H ( S 2 ) and σ ( S 1 , S 2 ) = σ ( S 2 , S 3 ) then S 3 ∈ H ( S 1 ) and σ ( S 1 , S 3 ) = σ ( S 1 , S 2 ) = σ ( S 2 , S 3 ) .
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Some bifurcation manifolds i -Inflection manifold States S such that ∇ λ i ( S ) · r i ( S ) = 0 i -Boundary extension manifold States S for which exist S ′ such that with S ′ on the boundary and S ∈ H ( S ′ ) λ i ( S ) = σ ( S , S ′ ) . ( i , j ) -Double contact manifold States S for which exist states S ′ such that S ′ ∈ H ( S ) λ i ( S ) = σ ( S , S ′ ) = λ j ( S ′ ) , with
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Simplified cases Buoyancy driven flow α = 0 Three distinct densities ELD EHD Solutions for general buoyancy-driven problem (three fluids with distinct densities) are “superpositions” of the solutions of simplified cases EHD and ELD.
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Rarefaction curves for EHD: α = 0, ρ 1 = ρ 2 > ρ 3 V V 2 2 Q Q B 1 3 1 V V V V Q 3 Q 3 1 1 2 2 Figure: Integral curves for both families. The arrows indicate increasing characteristic speed. The dots represent the Inflection curves.
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Wave curve for EHD. Left state in ∂ 3, right state V 3 . P 1 S L satisfies σ ( S L , P 1 S L ) = λ 1 ( P 1 S L ) < 0. P 1 S M satisfies 1-wave curve σ ( P 1 S M , S M ) = λ 1 ( P 1 S M ) > 0. σ ( P 1 S M , S M ) < σ ( S M , V 3 ) B 2-wave curve (speed compatibility!!!) Dominant phase in S L remains “dominant” along the solution: 1-INF The triangles V 3 – B 3 – V 2 and V 3 – B 3 – V 1 are invariant in the solution.
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Example of Riemann solution in EHD t ρ = ρ > ρ s P s U P U M M U L U R � V U 3 L x Saturation 1 M 1 � s 2 L s 2 g blue fluid dominant M s 2 � s /µ > s /µ x 0 g
Introduction The model Loss of Hyperbolicity RP General Theory Simplified Problems Buoyancy Driven Flow Wave curves in ELD: α = 0, ρ 3 = ρ 2 < ρ 1
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