Stone-Marchesin Model Equations of Three-Phase Flow in Oil Reservoir - PDF document

Stone-Marchesin Model Equations of Three-Phase Flow in Oil Reservoir Simulation Fumioki ASAKURA, Osaka Electro-Communication Univ. JAPAN asakura@isc.osakac.ac.jp 1

1. Introduction Oil Producer WAG Injector Oil Gas Water-Gas Water Shale Figure 1: WAG Enhanced Oil Recovery (schematic picture) 2

Primary Recovery By the underground pressure, usually about 20% of the oil in an oil reservoir can be extracted. Secondary recovery By injecting water and gas (air or CO 2 ), generally 25% to 35% of the oil can be extracted. Water-Alternating-Gas (WAG) Enhanced Oil Re- covery: Water Injection Good sweep efficiency, but 40 to 60% of the original oil on-site is left behind. Gas Injection Good displacement efficiency, but an expen- sive operation. WAG Injection More efficient than injection of water or gas alone. 3

Overview: Marchesin D. & Plohr B. (2001), Theory of Three- Phase Flow Applied to Water-Alternating-Gas Enhanced Oil Recovery, Proceedings of the 8th International Conference in Magdeburg , Vol.II, Birkh¨ auser Verlag, 693–702. Plan of this Talk: • Model equations • Hyperbolicity, elliptic region • Geometry of characteristic field, 2-phase like flow curves • Compressive, under and overcompressive shock waves • Geometry of Hugoniot curves • Entropy functions 4

Stone’s Model [3]: Neglect the gravity. Assume: the medium is homogeneous and the flow is incompressible and immiscible. water gas oil Volume Fractions: s w = u s g = v s o = 1 − u − v Permeability Functions: k w k g k o Fluid Viscosity: µ w µ g µ o v w v g v o Fluid Velocity: Pressure: p w p g p o Capillary pressure: p c = p non wetting phase − p wetting phase . Water-oil interface: water is the wetting phase. Gas-oil interface: oil is the wetting phase. p ow = p o − p w , p go = p g − p o . 5

Leverett’s assumption: p ow is a decreasing function of u = s w , and p go is an increasing function of v = s g . Darcy’s Law v i = − k i ∇ p i , i = w, g, o. µ i Mass Conservation Laws: ∂s i ∂t + ∇ · v i = 0 , i = w, g, o. By eliminating ∂p o ∂x and denoting k j � D = µ j j = w, g, o 6

� k w ∂s w ∂t + ∂ ∂ � k w 1 − k w � ∂p wo ∂x − k g ∂p go  � �� �� = ,   ∂x µ w D ∂x µ w µ w D µ g D ∂x  � k g ∂s g ∂t + ∂ ∂ � k g − k w ∂p wo 1 − k g � ∂p go � � � �� = ∂x +    ∂x µ g D ∂x µ g µ w D µ g D ∂x Stone’s assumption: The water and gas permeability functions depend only on the water and gas volume fraction k w = k w ( u ) , k g = k g ( v ) . If the capillary pressure is negligible, by using relative perme- ability functions f ( u ) = k w ( u ) g ( v ) = k g ( v ) h ( u, v ) = k o ( u, v ) , µ w µ g µ o 7

∂u ∂t + ∂ f ( u ) � � Water: = 0 , (1) ∂x f ( u ) + g ( v ) + h ( u, v ) ∂v ∂t + ∂ g ( v ) � � Gas: = 0 (2) ∂x f ( u ) + g ( v ) + h ( u, v ) for ( u, v ) ∈ Ω : 0 < u + v < 1 , u, v > 0 . If w = 0 , 2-phase flow is governed by the Buckley-Leverett equation ∂u ∂t + ∂ � f ( u ) � = 0 . ∂x f ( u ) + g (1 − u ) N.B. 1 An example of a single conservation law with a non-convex flux function. 8

2. Geometry of Characteristic Vector Field Hyperbolicity: Denote F ( U ) = t � � D , g f D = f ( u ) + g ( u ) + h ( u, v ) , . D Hyperbolic F ′ ( U ) has real eigenvalues λ 1 ( U ) , λ 2 ( U ) for any U ∈ Ω . Strictly Hyperbolic Eigenvalues are distinct : λ 1 ( U ) < λ 2 ( U ) . Umbilical Point U ∗ λ 1 ( U ∗ ) = λ 2 ( U ∗ ) and F ′ ( U ) is diago- nalizable, hence a scalar matrix. The eigenvalue λ j ( U ) is called the j th characteristic speed and corresponding right eigenvector R j ( U ) is called the j th characteristic vector field . 9

2-Phase Like Flow Curves (Medeiros [16]): � f ′ ( g + h ) − fh u − f ( g ′ + h v ) F ′ ( U ) = 1 � − g ( f ′ + h u ) g ′ ( f + h ) − gh v D 2 � 1 � 0 � � � � 1 R u = , R v = , R w = , respectively, are 0 1 − 1 characteristic vectors at v = 0 , u = 0 , w = 0 , respectively. 2-phase like flow curves L 1 , L 2 , L 3 are defined by ( f ′ − g ′ ) h + ( f + g )( h u − h v ) = 0 , L 1 : g ′ + h v = 0 , f ′ + h u = 0 . L 2 : L 3 : where R w , R v , R u , respectively are characteristic vectors on L 1 , L 2 , L 3 , respectively. 10

u = 0 u + v = 1 L 1 R R v R R w L 2 U* R R R L 3 v = 0 R u Figure 2: 2-Phase Like Curves (Quadratic Marchesin’s Model) 11

Introduce the 2-phase like variable: ξ = ( f ′ − g ′ ) h − ( f + g )( h u − h v ) , η = g ′ + h v , ζ = f ′ + h u . Lemma 1 The discriminant has the form: 1 � { f ′ ( g + h ) − g ′ ( f + h ) − fh u + gh v } 2 D char = D 4 + 4 fg ( f ′ + h u )( g ′ + h v )] 1 ( ξ − fη + gζ ) 2 + 4 fgηζ � � = D 4 1 ξ 2 − 2 ξ ( fη − gζ ) + ( fη + gζ ) 2 � � = . D 4 12

Theorem 1 1. The system is hyperbolic in the following 3 regions: (1) ηζ > 0 , (2) ξ > 0 , η > 0 , ζ < 0 , (3) ξ < 0 , η < 0 , ζ > 0 2. Elliptic regions appear in the following 2 regions: (1) ξ < 0 , η > 0 , ζ < 0 , (2) ξ > 0 , η < 0 , ζ > 0 3. If ξ = η = ζ = 0 at U ∗ , then U ∗ is an umbilical point . 13

ξ < 0 η > 0 elliptic region L 2 L 1 ξ > 0 η < 0 ζ < 0 L 3 ζ > 0 Figure 3: Existence of Elliptic Region ( ξ > 0 , η < 0 , ζ > 0) 14

Integral Curves of Characteristic Vector Fields: Let A be a 2 × 2 matrix. Note that t x ⊥ A x = 0 , ( t x ⊥ x = 0) . x : an eigenvectors of A ⇔ The integral curve of the characteristic vector fields: solutions to the differential equation t ˙ U ⊥ F ′ ( U ) ˙ U = 0 . The equation of the trajectory: gζdu 2 + ( ξ − fη + gζ ) dudv − fηdv 2 = 0 . (3) Equivalently √ dv 1 � � du = ξ − fη + gζ ± ∆ 2 fη or √ du dv = − 1 � � ξ − fη + gζ ± ∆ 2 gζ 15

where ξ 2 − 2 ξ ( fη − gζ ) + ( fη + gζ ) 2 � � ∆ = . Notice that √ | ξ − fη + gζ | < ∆ , if ηζ > 0 , √ | ξ − fη + gζ | > ∆ , if ηζ < 0 . Marchesin’s model: Marchesin, Paes-Leme, Schaeffer & Shearer [17]. Theorem 2 (Existence of Umbilical Point) Assume: h ( u, v ) = h (1 − u − v ) , f (0) = g (0) = h (0) = 0 , f ′′ ( u ) , g ′′ ( v ) , h ′′ ( w ) > 0 . Then the system of equations (1) , (2) is hyperbolic and has a unique umbilical point in Ω . 16

The integral curves of the characteristic vector fields sketched: fast integral curves slow integral curves Figure 4: Integral Curves N.B. 2 Impossible to construct globally in Ω fast or slow characteristic fields. 17

3. Quadratic Marchesin Model Quadratic Relative Permeability Functions: f ( u ) = αu 2 , g ( v ) = βv 2 , h ( u, v ) = γ (1 − u − v ) 2 , α, β, γ > 0 The model equations: αu 2 ∂u ∂t + ∂ � � = 0 , (4) αu 2 + βv 2 + γ (1 − u − v ) 2 ∂x βv 2 ∂v ∂t + ∂ � � = 0 . (5) αu 2 + βv 2 + γ (1 − u − v ) 2 ∂x A unique umbilical point and the coincident characteristic speed: � β γ � 2 αβγ ( βγ + γα + αβ ) U ∗ = λ ∗ = , αβ 2 γ 2 + βγ 2 α 2 + γα 2 β 2 . α βγ + γα + αβ 18

2-phase like curves: L 1 : αu − βv = 0 � β λ = 2 γαwu � , λ ⊥ = 2 αu � � 1 R ⊥ = , R = D , , α − 1 D 2 L 2 : ( β + γ ) v − γ (1 − u ) = 0 � β + γ � 0 λ = 2 αβuv � , λ ⊥ = 2 βv � D , R ⊥ = , R = , D 2 − γ 1 L 3 : ( α + γ ) u − γ (1 − v ) = 0 � − γ � 1 � � λ = 2 αβuv , λ ⊥ = 2 αu D , R ⊥ = , R = , α + β 0 D 2 Theorem 3 Each 2-phase like flow curve L j , j = 1 , 2 , 3 is a line and the characteristic vector field R is parallel to the direction of the line. 19

Useful lemmas: Along the 2-phase like flow curve L 1 : u = βτ, v = ατ, D = ( α + β )( βγ + γα + αβ ) τ 2 − 2 γ ( α + β ) τ + γ. Hence D τ = 0 , if and only if ( βγ + γα + αβ ) τ = γ. Lemma 2 The quantity D attains its minimum at the um- bilical point. Confine our attention to the 2-phase like flow curve L 1 . At u = v = 0 , we have λ = λ ⊥ = 0 and at u + v = 1 ( w = 0) , λ = 0 , λ ⊥ > 0 . Lemma 3 The characteristic speed λ attains its maximum in the interior of each 2-phase like flow curve. 20

Lemma 4 � ∂λ 2 αβγ ( αβ − βγ − γα ) � = on L 1 � ∂w ( α + β )( βγ + γα + αβ ) D � U = U ∗ � ∂λ = 2 αβ ( βγ − γα − αβ ) � on L 2 � ∂v ( βγ + γα + αβ ) D � U = U ∗ � ∂λ = 2 αβ ( γα − αβ − βγ ) � on L 3 � ∂u ( βγ + γα + αβ ) D � U = U ∗ Study L 1 as a rarefaction or shock curve: 1. If αβ > βγ + γα, then L 1 consists of slow rarefaction curve and fast shock curve. The slow rarefaction curve ends in the interior of the 2-phase like flow curve. 2. If αβ = βγ + γα, then L 1 consists of slow and fast shock curves. 21

3. If αβ < βγ + γα, then L 1 consists of slow shock curve and fast rarefaction curve. The fast rarefaction curve ends in the interior of the 2-phase like flow curve. N.B. 3 Note that: if αβ ≥ βγ + γα, then βγ < γα + αβ, γα < αβ + βγ. 22

αβ > βγ + γα λ λ 2− phase like flow curve U ∗ αβ < βγ + γα λ λ 2− phase like flow curve U ∗ Figure 5: Characteristic Speeds 23