A New Formulation of Immiscible Compressible Two-Phase Flow in - PowerPoint PPT Presentation

A New Formulation of Immiscible Compressible Two-Phase Flow in Porous Media Via the Concept of Global Pressure Brahim Amaziane 1 Mladen Jurak 2 1 Universit´ e de Pau, Math´ ematiques, LMA CNRS-UMR 5142, France 2 Department of Mathematics, University of Zagreb, Croatia Scaling Up and Modeling for Transport and Flow in Porous Media Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 1 / 24

Outline Two-phase immiscible, compressible flow equations Fractional flow formulation New global formulation Simplified global formulation Numerical comparison of the coefficients Conclusion Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 2 / 24

Two-phase immiscible, compressible flow equations Compressible two-phase flow We consider two-phase isothermal, compressible, immiscible flow through heterogeneous porous medium. For example, water and gas. Assumptions: ◮ Incompressible fluid: water, ρ w = const. ◮ Compressible fluid: gas, ρ g = c g p g . ◮ Viscosities µ w and µ g are constant. ◮ No mass exchange between the phases; ◮ The temperature is constant; Note that the assumptions on the form of mass densities are not essential. Independent variables : water saturation S w and gas pressure p g ( p w = p g − p c ( S w ), S g = 1 − S w ). Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 3 / 24

Two-phase immiscible, compressible flow equations Flow equations Mass conservation: for α ∈ { w , g } , Φ ∂ ∂ t ( ρ α S α ) + div( ρ α q α ) = 0 , The Darcy-Muscat law: for α ∈ { w , g } , q α = − K kr α ( S α ) ( ∇ p α − ρ α g ) , µ α Capillary law: p c ( S w ) = p g − p w , No void space. S w + S g = 1 . Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 4 / 24

Fractional flow formulation Fractional flow formulation Goal: ◮ Reformulate flow equations in a form giving less tight coupling between the two differential equations, allowing a sort of IMPES (implicit in pressure and explicit in saturation) numerical treatment. There are two approaches: 1. Introduce total velocity: Q t = q w + q g : leads to non-conservative form of the equations 2. Introduce total flow: Q t = ρ w q w + ρ g q g : leads to conservative form of the equations. We work with total flow. Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 5 / 24

Fractional flow formulation Fractional flow formulation: equations Total flow: Q t = − λ ( S w , p g ) K ( ∇ p g − f w ( S w , p g ) ∇ p c ( S w ) − ¯ ρ ( S w , p g ) g ) , Total mass conservation: Φ ∂ ∂ t ( S w ρ w + (1 − S w ) ρ g ( p g )) + div ( Q t ) = 0 , Water mass conservation: ∂ S w Φ ρ w ∂ t + div( f w ( S w , p g ) Q t + K g b g ( S w , p g )) = div( K a ( S w , p g ) ∇ S w ) . Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 6 / 24

Fractional flow formulation Fractional flow formulation: coefficients λ w ( S w ) = kr w ( S w ) λ g ( S w ) = kr g ( S w ) phase mobilities , , µ w µ g total mobility λ ( S w , p g ) = ρ w λ w ( S w ) + ρ g ( p g ) λ g ( S w ) , f w ( S w , p g ) = ρ w λ w ( S w ) water fractional flow λ ( S w , p g ) , ρ ( S w , p g ) = λ w ( S w ) ρ 2 w + λ g ( S w ) ρ g ( p g ) 2 mean density ¯ , λ ( S w , p g ) b g ( S w , p g ) = ρ w ρ g ( p g ) λ w ( S w ) λ g ( S w ) ”gravity” coeff. ( ρ w − ρ g ( p g )) , λ ( S w , p g ) a ( S w , p g ) = − ρ w ρ g ( p g ) λ w ( S w ) λ g ( S w ) p ′ ”diffusivity” coeff. c ( S w ) . λ ( S w , p g ) Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 7 / 24

Fractional flow formulation Decoupling of the system In total flow we want to eliminate saturation gradient: � � Q t = − λ ( S w , p g ) K ∇ p g − f w ( S w , p g ) p ′ c ( S w ) ∇ S w − ¯ ρ ( S w , p g ) g , ◮ Idea: introduce a new pressure-like variable that will eliminate ∇ S w term ( Chavent-Jaffr´ ee: Mathematical Models and Finite Elements for Reservoir Simulation ) ◮ Find a new pressure variable p , called global pressure, and a function ω ( S w , p ) such that: ∇ p g − f w ( S w , p g ) p ′ c ( S w ) ∇ S w = ω ( S w , p ) ∇ p (1) Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 8 / 24

Fractional flow formulation Decoupling of the system In total flow we want to eliminate saturation gradient: � � Q t = − λ ( S w , p g ) K ∇ p g − f w ( S w , p g ) p ′ c ( S w ) ∇ S w − ¯ ρ ( S w , p g ) g , ◮ Idea: introduce a new pressure-like variable that will eliminate ∇ S w term ( Chavent-Jaffr´ ee: Mathematical Models and Finite Elements for Reservoir Simulation ) ◮ Find a new pressure variable p , called global pressure, and a function ω ( S w , p ) such that: ∇ p g − f w ( S w , p g ) p ′ c ( S w ) ∇ S w = ω ( S w , p ) ∇ p (1) Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 8 / 24

New global formulation New global formulation We introduce unknown function π such that p g = π ( S w , p ) , where p is global pressure to be defined. Then (1) reads ∇ p g = ω ( S w , p ) ∇ p + f w ( S w , π ( S w , p )) p ′ c ( S w ) ∇ S w , or, ∂π ( S w , p ) ∇ S w + ∂π ∂ p ( S w , p ) ∇ p ∂ S w = ω ( S w , p ) ∇ p + f w ( S w , π ( S w , p )) p ′ c ( S w ) ∇ S w . Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 9 / 24

New global formulation New global formulation We introduce unknown function π such that p g = π ( S w , p ) , where p is global pressure to be defined. Then (1) reads ∇ p g = ω ( S w , p ) ∇ p + f w ( S w , π ( S w , p )) p ′ c ( S w ) ∇ S w , or, ∂π ( S w , p ) ∇ S w + ∂π ∂ p ( S w , p ) ∇ p ∂ S w = ω ( S w , p ) ∇ p + f w ( S w , π ( S w , p )) p ′ c ( S w ) ∇ S w . Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 9 / 24

New global formulation New global formulation Since p and S w are independent variables we must have: ∂π ( S w , p ) = f w ( S w , π ( S w , p )) p ′ c ( S w ) (2) ∂ S w ∂π ∂ p ( S w , p ) = ω ( S w , p ) . (3) Conclusion : 1. To calculate π ( S w , p ) solve the Cauchy problem:  d π ( S , p ) ρ w λ w ( S ) p ′ c ( S )  = ρ w λ w ( S ) + c g λ g ( S ) π ( S , p ) , 0 < S < 1  dS  π (1 , p ) = p .  2. Get ω ( S w , p ) from (3). Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 10 / 24

New global formulation New global formulation: Remarks 1. It is more natural to replace saturation S w with capillary pressure u = p c ( S w ) as an independent variable. Then: ρ w ˆ  d ˆ π ( u , p ) λ w ( u ) = , u > 0   ρ w ˆ λ w ( u ) + c g ˆ du λ g ( u )ˆ π ( u , p )  π (0 , p ) = p . ˆ  and π ( S w , p ) = ˆ π ( p c ( S w ) , p ) [ Hat denotes the change of variables. ] 2. ω is strictly positive and less than 1: � p c ( S w ) � c g ρ w ˆ λ w ( u )ˆ � λ g ( u ) ω ( S w , p ) = exp − π ( u , p )) 2 du , ( ρ w ˆ λ w ( u ) + c g ˆ λ g ( u )ˆ 0 3. p ≤ π ( S w , p ) ≤ p + p c ( S w ) and therefore p w ≤ p ≤ p g . Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 11 / 24

New global formulation New global formulation: Remarks 1. It is more natural to replace saturation S w with capillary pressure u = p c ( S w ) as an independent variable. Then: ρ w ˆ  d ˆ π ( u , p ) λ w ( u ) = , u > 0   ρ w ˆ λ w ( u ) + c g ˆ du λ g ( u )ˆ π ( u , p )  π (0 , p ) = p . ˆ  and π ( S w , p ) = ˆ π ( p c ( S w ) , p ) [ Hat denotes the change of variables. ] 2. ω is strictly positive and less than 1: � p c ( S w ) � c g ρ w ˆ λ w ( u )ˆ � λ g ( u ) ω ( S w , p ) = exp − π ( u , p )) 2 du , ( ρ w ˆ λ w ( u ) + c g ˆ λ g ( u )ˆ 0 3. p ≤ π ( S w , p ) ≤ p + p c ( S w ) and therefore p w ≤ p ≤ p g . Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 11 / 24

New global formulation New global formulation: Remarks 1. It is more natural to replace saturation S w with capillary pressure u = p c ( S w ) as an independent variable. Then: ρ w ˆ  d ˆ π ( u , p ) λ w ( u ) = , u > 0   ρ w ˆ λ w ( u ) + c g ˆ du λ g ( u )ˆ π ( u , p )  π (0 , p ) = p . ˆ  and π ( S w , p ) = ˆ π ( p c ( S w ) , p ) [ Hat denotes the change of variables. ] 2. ω is strictly positive and less than 1: � p c ( S w ) � c g ρ w ˆ λ w ( u )ˆ � λ g ( u ) ω ( S w , p ) = exp − π ( u , p )) 2 du , ( ρ w ˆ λ w ( u ) + c g ˆ λ g ( u )ˆ 0 3. p ≤ π ( S w , p ) ≤ p + p c ( S w ) and therefore p w ≤ p ≤ p g . Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 11 / 24

New global formulation New global formulation: Flow equations Total flow: Q t = − λ n ( S w , p ) K ( ω ( S w , p ) ∇ p − ¯ ρ n ( S w , p ) g ) . where the superscript n stands for new . Total mass conservation: Φ ∂ ∂ t ( S w ρ w + c g (1 − S w ) π ( S w , p )) + div Q t = 0 . Water mass conservation: ∂ S w ∂ t + div( f n w ( S w , p ) Q t + K g b n g ( S w , p )) = div( K a n ( S w , p ) ∇ S w ) . Φ ρ w Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 12 / 24