Three-Phase Displacement Theory: Hyperbolic Models and Analytical Solutions Ruben Juanes † and Knut–Andreas Lie ‡ † Petroleum Engineering, School of Earth Sciences, Stanford University ‡ SINTEF ICT, Dept. Applied Mathematics 17.11.04 – p. 1
Contents Discussion of general conditions for relative permeabilities to ensure hyperbolicity Presentation of the analytical solution to the Riemann problem Implementation in a front-tracking method Streamline simulation results 17.11.04 – p. 2
Displacement Theory Assumptions: • One-dimensional flow • Immiscible fluids • Incompressible fluids • Homogeneous rigid porous medium • Multiphase flow extension of Darcy’s law • Gravity and capillarity are not considered • Constant fluid viscosities 17.11.04 – p. 3
Displacement Theory cont’d Mass conservation for each phase: ∂ t ( m α )+ ∂ x ( F α ) = 0 , α = w, g, o m α = ρ α S α φ F α = − ρ α kλ α ∂ x p Saturations add up to one: � S α ≡ 1 α = w,g,o 17.11.04 – p. 4
Displacement Theory cont’d If the fractional flow approach is used: • “Pressure equation” ∂ x ( v T ) = 0 v T = − 1 φkλ T ∂ x p • System of “saturation equations” � � � � � � 0 S w f w ∂ t + v T ∂ x = S g f g 0 17.11.04 – p. 5
Displacement Theory cont’d Saturation triangle: G 0 1 0.2 0.8 0.4 0.6 0.6 0.4 0.8 0.2 1 0 0.2 0.4 0.6 0.8 0 1 W O 17.11.04 – p. 6
Displacement Theory cont’d Saturation triangle: G 0 1 0.2 0.8 0.4 0.6 Immobile water ( S wi ) Immobile oil ( S oi ) 0.6 0.4 0.8 0.2 1 Immobile gas ( S gi ) 0 0.2 0.4 0.6 0.8 0 1 W O 17.11.04 – p. 6
Displacement Theory cont’d Saturation triangle: Reduced saturations: G S α − S αi ˜ 0 S α := 1 1 − � 3 β =1 S βi 0.2 ↓ 0.8 Renormalized triangle 0.4 0.6 Immobile water ( S wi ) Immobile oil ( S oi ) 0.6 0.4 0.8 0.2 1 Immobile gas ( S gi ) 0 0.2 0.4 0.6 0.8 0 1 W O 17.11.04 – p. 6
Character of the System The character of the system � � � � � � 0 u f + v T ∂ x = ⇔ ∂ t u + v T ∂ x f = 0 ∂ t 0 v g is determined by the eigenvalues ( ν 1 , ν 2 ) and eigenvectors ( r 1 , r 2 ) of the Jacobian matrix: � � f ,u f ,v A ( u ) := D u f = g ,u g ,v 17.11.04 – p. 7
Character of the System The character of the system � � � � � � 0 u f + v T ∂ x = ⇔ ∂ t u + v T ∂ x f = 0 ∂ t 0 v g is determined by the eigenvalues ( ν 1 , ν 2 ) and eigenvectors ( r 1 , r 2 ) of the Jacobian matrix: � � f ,u f ,v A ( u ) := D u f = g ,u g ,v Hyperbolic: The eigenvalues are real and the Jacobi matrix is diagonalizable . Strictly hyperbolic: distinct eigenvalues ν 1 < ν 2 . 17.11.04 – p. 7
Character of the System The character of the system � � � � � � 0 u f + v T ∂ x = ⇔ ∂ t u + v T ∂ x f = 0 ∂ t 0 v g is determined by the eigenvalues ( ν 1 , ν 2 ) and eigenvectors ( r 1 , r 2 ) of the Jacobian matrix: � � f ,u f ,v A ( u ) := D u f = g ,u g ,v Hyperbolic: The eigenvalues are real and the Jacobi matrix is diagonalizable . Strictly hyperbolic: distinct eigenvalues ν 1 < ν 2 . Elliptic: The eigenvalues are complex . 17.11.04 – p. 7
Character of the System cont’d It was long believed that the system was strictly hyperbolic for any set of relative permeability functions 17.11.04 – p. 8
Character of the System cont’d It was long believed that the system was strictly hyperbolic for any set of relative permeability functions However: existence of elliptic regions proved 1985–95 • Bell et al., Fayers, Guzmán and Fayers, Hicks and Grader, .. • Shearer and Trangenstein, Holden et al, ... 17.11.04 – p. 8
Character of the System cont’d It was long believed that the system was strictly hyperbolic for any set of relative permeability functions However: existence of elliptic regions proved 1985–95 • Bell et al., Fayers, Guzmán and Fayers, Hicks and Grader, .. • Shearer and Trangenstein, Holden et al, ... Approach in the existing literature: • Assume “reasonable” conditions for relative permabilities on the edges − “Zero-derivative” conditions − “Interaction” conditions • Infer loss of strict hyperbolicity inside the saturation triangle 17.11.04 – p. 8
Character of the System cont’d Traditional assumed behavior of fast eigenvectors ( r 2 ) G 0 1 0 . 2 0.8 r 2 r 2 0 . 4 0.6 0 . 6 0.4 r 2 r 2 0 . 8 0.2 r 2 r 2 1 0 0 2 4 6 8 1 W O . . . . 0 0 0 0 17.11.04 – p. 9
Character of the System cont’d Traditional assumed behavior of fast eigenvectors ( r 2 ) G 0 1 0 . 2 0.8 r 2 r 2 0 . 4 0.6 0 . 6 0.4 Elliptic region r 2 r 2 0 . 8 0.2 r 2 r 2 1 0 0 2 4 6 8 1 W O . . . . 0 0 0 0 17.11.04 – p. 9
Character of the System cont’d Consequences of ellipticity of the system: • Flow depends on future boundary conditions • The solution is unstable : arbitrarily close initial and injected saturations yield nonphysical oscillatory waves 17.11.04 – p. 10
Character of the System cont’d Consequences of ellipticity of the system: • Flow depends on future boundary conditions • The solution is unstable : arbitrarily close initial and injected saturations yield nonphysical oscillatory waves However: • The elliptic region can be shrunk to an umbilic point only if interaction between phases is ignored: k rα = k rα ( S α ) , α = 1 , . . . , 3 • This model is not supported by experiments and pore-scale physics • Umbilic points still act as “repellers” for classical waves 17.11.04 – p. 10
Relative Permeabilities Juanes and Patzek – New approach: • Assume the system is strictly hyperbolic • Infer conditions on relative permeabilities Key observation: • Whenever gas is present as a continuous phase, its mobility is much higher than that of the other two fluids • Fast paths ← → changes in gas saturation 17.11.04 – p. 11
Relative Permeabilities Proposed behavior of eigenvectors ( r 1 , r 2 ) G 0 1 0 . 2 0.8 r 2 r 2 0 . 4 0.6 0 . 6 0.4 r 2 r 2 0 . 8 0.2 r 1 r 1 1 0 0 2 4 6 8 1 O W . . . . 0 0 0 0 17.11.04 – p. 12
Relative Permeabilities Proposed behavior of eigenvectors ( r 1 , r 2 ) G G 0 1 0 0.2 1 0.8 r 2 r 2 0.4 0.6 0 . 2 0.6 0.8 Elliptic region 0.4 r 2 r 2 r 2 0.8 r 2 0.2 0 . 4 r 2 r 2 1 0.6 0 0 0.2 0.4 0.6 0.8 1 O W 0 . 6 0.4 r 2 r 2 0 . 8 0.2 r 1 r 1 1 0 0 2 4 6 8 1 O W . . . . 0 0 0 0 17.11.04 – p. 12
Relative Permeabilities cont’d Two types of conditions: • Condition I. Eigenvectors are parallel to each edge • Condition II. Strict hyperbolicity along each edge In particular, on the OW edge: Condition Frac. flows Mobilities I g ,u = 0 ⇔ λ g,u = 0 λ g,v > λ w,u − λ T,u λ w II g ,v − f ,u > 0 ⇔ λ T Condition II requires that the gas relative permeability has a positive derivative at its endpoint saturation. 17.11.04 – p. 13
Relative Permeabilities cont’d Remarks: • Necessary condition for strict hyperbolicity • Can be justified from pore-scale physics (bulk flow vs. corner flow) • Supported by experimental data (Oak’s steady-state) −3 x 10 3 0.2 Water Relative Perm 2.5 Gas Relative Perm 0.15 2 1.5 0.1 1 0.05 0.5 0 0 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.1 0.2 0.3 0.4 Water Saturation Gas Saturation 17.11.04 – p. 14
Relative Permeabilities cont’d A simple model: k rw ( u ) = u 2 β g v + (1 − β g ) v 2 � � k rg ( v ) = β g > 0 , k ro ( u, v ) = (1 − u − v )(1 − u )(1 − v ) with reasonable values of viscosities: µ w = 1 , µ g = 0 . 03 , µ o = 2 cp and a small value of the endpoint slope: β g = 0 . 1 17.11.04 – p. 15
Relative Permeabilities cont’d Oil isoperms: G 0 1 0.2 0.8 k ro = 0 . 2 0.4 0.6 k ro = 0 . 4 0.6 0.4 k ro = 0 . 6 0.8 0.2 1 0 0 0.2 0.4 0.6 0.8 1 W O 17.11.04 – p. 16
Analytical Solution Riemann problem: Find a weak solution to the 2 × 2 system ∂ t u + v T ∂ x f = 0 , −∞ < x < ∞ , t > 0 � if x < 0 u l u ( x, 0) = if x > 0 u r Previous work: • Sequence of two successive two-phase displacements (Kyte et al., Pope, ..) • Triangular systems (Gimse et al., ..) New results by Juanes and Patzek: • A complete classification all wave types • Solution of Riemann problem (structure of waves) 17.11.04 – p. 17
Analytical Solution cont’d Self-similarity (“stretching”, “coherence”): 1 t 1 u 0 0 4 x 17.11.04 – p. 18
Analytical Solution cont’d Self-similarity (“stretching”, “coherence”): 1 t 1 t 2 u 0 0 4 x 17.11.04 – p. 18
Analytical Solution cont’d Self-similarity (“stretching”, “coherence”): 1 t 1 t 2 u 0 0 4 x x u ( x, t ) = U ( ζ ) , ζ := � t 0 v T ( τ ) dτ 17.11.04 – p. 18
Analytical Solution cont’d Using self-similarity, the Riemann problem is a boundary value problem: ( A ( U ) − ζ I ) U ′ = 0 , −∞ < ζ < ∞ with boundary conditions U ( −∞ ) = u l , U ( ∞ ) = u r Strict hyperbolicity − → wave separation: W 1 W 2 − → u m − → u r u l 17.11.04 – p. 19
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