Three-Phase Displacement Theory: Hyperbolic Models and Analytical - - PowerPoint PPT Presentation

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

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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

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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

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Displacement Theory cont’d

Mass conservation for each phase: ∂t(mα)+∂x(Fα) = 0, α = w, g, o mα = ραSαφ Fα = −ραkλα ∂xp Saturations add up to one:

• α=w,g,o

Sα ≡ 1

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Displacement Theory cont’d

If the fractional flow approach is used:

• “Pressure equation”

∂x(vT ) = 0 vT = − 1 φkλT ∂xp

• System of “saturation equations”

∂t

• Sw

Sg

• + vT ∂x
• fw

fg

• =
• 17.11.04 – p. 5

Displacement Theory cont’d

Saturation triangle:

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

W G O

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Displacement Theory cont’d

Saturation triangle:

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

W G O

Immobile water (Swi) Immobile gas (Sgi) Immobile oil (Soi)

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Displacement Theory cont’d

Saturation triangle:

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

W G O

Immobile water (Swi) Immobile gas (Sgi) Immobile oil (Soi)

Reduced saturations:

˜ Sα := Sα − Sαi 1 − 3

β=1 Sβi

↓

Renormalized triangle

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Character of the System

The character of the system ∂t

• u

v

• + vT ∂x
• f

g

• =
• ⇔ ∂tu + vT ∂xf = 0

is determined by the eigenvalues (ν1, ν2) and eigenvectors (r1, r2) of the Jacobian matrix: A(u) := Duf =

• f,u

f,v g,u g,v

• 17.11.04 – p. 7

Character of the System

The character of the system ∂t

• u

v

• + vT ∂x
• f

g

• =
• ⇔ ∂tu + vT ∂xf = 0

is determined by the eigenvalues (ν1, ν2) and eigenvectors (r1, r2) of the Jacobian matrix: A(u) := Duf =

• f,u

f,v g,u g,v

• Hyperbolic: The eigenvalues are real and the Jacobi matrix is
• diagonalizable. Strictly hyperbolic: distinct eigenvalues ν1 < ν2.

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Character of the System

The character of the system ∂t

• u

v

• + vT ∂x
• f

g

• =
• ⇔ ∂tu + vT ∂xf = 0

is determined by the eigenvalues (ν1, ν2) and eigenvectors (r1, r2) of the Jacobian matrix: A(u) := Duf =

• f,u

f,v g,u g,v

• Hyperbolic: The eigenvalues are real and the Jacobi matrix is
• diagonalizable. Strictly hyperbolic: distinct eigenvalues ν1 < ν2.

Elliptic: The eigenvalues are complex.

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Character of the System cont’d

It was long believed that the system was strictly hyperbolic for any set of relative permeability functions

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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, ...

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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
• n the edges

− “Zero-derivative” conditions − “Interaction” conditions

• Infer loss of strict hyperbolicity inside the saturation

triangle

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Character of the System cont’d

Traditional assumed behavior of fast eigenvectors (r2)

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O r2 r2 r2 r2 r2 r2

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Character of the System cont’d

Traditional assumed behavior of fast eigenvectors (r2)

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O r2 r2 r2 r2 r2 r2

Elliptic region

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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

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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: krα = krα(Sα), α = 1, . . . , 3

• This model is not supported by experiments and

pore-scale physics

• Umbilic points still act as “repellers” for classical waves

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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

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Relative Permeabilities

Proposed behavior of eigenvectors (r1, r2)

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O r1 r1 r2 r2 r2 r2

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Relative Permeabilities

Proposed behavior of eigenvectors (r1, r2)

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O r1 r1 r2 r2 r2 r2

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

W G O r2 r2 r2 r2 r2 r2 Elliptic region

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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 ⇔ λg,u = 0 II g,v − f,u > 0 ⇔ λg,v > λw,u − λT,u λw

λT

Condition II requires that the gas relative permeability has a positive derivative at its endpoint saturation.

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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)

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.5 1 1.5 2 2.5 3 x 10

−3

Water Saturation Water Relative Perm

0.1 0.2 0.3 0.4 0.05 0.1 0.15 0.2

Gas Saturation Gas Relative Perm

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Relative Permeabilities cont’d

A simple model: krw(u) = u2 krg(v) =

• βgv + (1 − βg)v2

, βg > 0 kro(u, v) = (1 − u − v)(1 − u)(1 − v) with reasonable values of viscosities: µw = 1, µg = 0.03, µo = 2 cp and a small value of the endpoint slope: βg = 0.1

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Relative Permeabilities cont’d

Oil isoperms:

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

W G O

kro = 0.2 kro = 0.4 kro = 0.6

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Analytical Solution

Riemann problem: Find a weak solution to the 2 × 2 system ∂tu + vT ∂xf = 0, −∞ < x < ∞, t > 0 u(x, 0) =

• ul

if x < 0 ur if x > 0 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)

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Analytical Solution cont’d

Self-similarity (“stretching”, “coherence”):

4 1

x u t1

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Analytical Solution cont’d

Self-similarity (“stretching”, “coherence”):

4 1

x u t1 t2

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Analytical Solution cont’d

Self-similarity (“stretching”, “coherence”):

4 1

x u t1 t2 u(x, t) = U(ζ), ζ := x t

0 vT (τ) dτ

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Analytical Solution cont’d

Using self-similarity, the Riemann problem is a boundary value problem: (A(U) − ζI)U ′ = 0, −∞ < ζ < ∞ with boundary conditions U(−∞) = ul, U(∞) = ur Strict hyperbolicity − → wave separation: ul

W1

− → um

W2

− → ur

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Analytical Solution cont’d

Schematic of Riemann solution

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O

ul ur

ζ

ul ur

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Analytical Solution cont’d

Schematic of Riemann solution

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O

W1 W2 ul ur um

ζ

ul ur um ✙ W1 ✢ W2

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Wave Structure: Rarefactions

If the solution is smooth, it satisfies: (A(U)−ζI)U ′ = 0 ր տ

eigenvalue (νp) eigenvector (rp)

A smooth solution (rarefaction) must lie

• n

a curve whose tan- gent is in the direction of the eigenvector (integral curve)

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O 1-family (slow paths) 2-family (fast paths)

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Wave Structure: Rarefactions cont’d

Admissibility of a rarefaction wave

• To avoid a multiple-valued solution, νp must increase

monotonically along the curve ul

Rp

− → ur

• Thus, rarefaction curves Rp are subsets of integral

curves Ip

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Wave Structure: Shocks

If the solution is discontinuous, it must satisfy the Rankine-Hugoniot jump condition:

ζ

u− u+ ✲ σ

f(u+) − f(u−) = σ ·

• u+ − u−
• The set of states which can be con-

nected satisfying the jump condition is called the Hugoniot locus

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O H1 H2

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Wave Structure: Shocks cont’d

Admissibility of a shock wave

• Not every discontinuity satisfying the jump condition is a

valid shock

• Characteristics of the p-family must go into the shock (Lax

entropy condition): νp(u−) > σp > νp(u+),

• Thus, shock curves Sp are subsets of Hugoniot loci Hp

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Wave Structure: Rarefaction-Shocks

Genuine nonlinearity: eigenvalues vary monotonically along integral curves

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Wave Structure: Rarefaction-Shocks

Genuine nonlinearity: eigenvalues vary monotonically along integral curves This is not the case in multiphase flow, where each wave may involve rarefactions and shocks

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Wave Structure: Rarefaction-Shocks

Genuine nonlinearity: eigenvalues vary monotonically along integral curves This is not the case in multiphase flow, where each wave may involve rarefactions and shocks Inflection locus: set of points at which eigenvalues attain a local maximum when moving along integral curves

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Inflection locus

W G O . 2 . 5 . 1 . 2 0.5

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Inflection locus

W G O . 2 0.5 2 5

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Wave Structure: Rarefaction-Shocks cont’d

Properties of the inflection loci:

• Single curves, transversal to integral

curves

• Correspond to maxima of eigenvalues

Consequences:

• At most one rarefaction and one shock
• Rarefaction is always slower than shock:

ul

Rp

− → u∗

Sp

− → ur

1 1

u0 f

f′′ < 0 f′′ > 0

− → ← −

1 1 1

W G O Inflection locus R1 S1 ul ur u∗

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Wave Structure: Rarefaction-Shocks cont’d

Admissibility of a rarefaction-shock wave

• Eigenvalue νp must increase monotonically along the curve

ul

Rp

− → u∗

• The shock must satisfy the extended-Lax entropy

condition: νp(u∗) = σp > νp(ur)

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Wave Structure cont’d

Complete set of solutions: 9 different wave configurations

ul

S R RS

um

S R RS S R RS S R RS

ur S1S2

S1R2 S1R2S2 R1S2 R1R2 R1R2S2 R1S1S2 R1S1R2 R1S1R2S2

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Complete Set of Solutions

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O S1 S2 ul ur

(a) S1S2

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O R2 S1 ul ur

(b) S1R2

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O R2 S1 S2 ul ur

(c) S1R2S2

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Complete Set of Solutions cont’d

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O R1 S2 ul ur

(d) R1S2

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O R1 R2 ul ur

(e) R1R2

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O R1 R2 S2 ul ur

(f) R1R2S2

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Complete Set of Solutions cont’d

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O R1 S1 S2 ul ur

(g) R1S1S2

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O R1 R2 S1 ul ur

(h) R1S1R2

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1 . 2 . 4 . 6 . 8 1

W G O R1 R2 S1 S2 ul ur

(i) R1S1R2S2

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Example 1

Injection of water and gas into an oil-filled core (with some mobile water) Problem of great practical interest

Injected saturation Initial saturation

Sg = 0.5 Sg = 0 So = 0 So = 0.95 Sw = 0.5 Sw = 0.05

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

W G O R1 R2 S1 S2 ul ur um u∗

1

u∗

2

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Example 1 cont’d

10 1

gas

• il

R2 S2

σ2

ζ

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Example 1 cont’d

0.1 1

water gas

• il

R1 S1

σ1

ζ

10 1

gas

• il

R2 S2

σ2

ζ

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The Cauchy Problem: Front Tracking

first interaction second interaction

Start (t = 0): piecewise constant inital data − → sequence of local Riemann problems − → p.w discontinuities between (x,t)-rays While t < tend: track discontinuities solve Riemann problems

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x t 17.11.04 – p. 34

Example 2

• Initially, reservoir filled with 80% oil and 20% gas
• Alternate cycles of water and gas injection
• Front-tracking solution (with ∆u = 0.005
• Half a million Riemann solves ∼ 5 sec on a desktop PC

1 2

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Streamline Methods

Interpret the saturation equation φ∂tS + v · ∇f(S) = 0 as an equation along streamlines using v |v| = dx ds , dy ds, dz ds T

• r

v·∇ = |v| ∂ ∂s Transformation using time-of-flight τ |v| ∂ ∂s = φ ∂ ∂τ

dx ds dy s(x,y)

gives a family of 1-D transport equations along streamlines ∂tS + ∂τf(S) = 0.

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Streamline Simulation

Figure from Yann Gautier

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Example 4: SPE 10 Tarbert Formation

10 10 10 10 10 10 10

−2 −1 1 2 3 4

• 30 × 110 × 15 upscaled

sample from Tarbert formation

• Initial composition:

(Sw, Sg) = (0.0, 0.2)

• 2000 days of production
• Either: continuous water

injection

• Or:

water-alternating-gas every 200 day

Data reduction is used to speed up front-tracking solution: weak interactions are treated as being of type S1S2.

17.11.04 – p. 38

Example 4 cont’d

Oil production:

200 400 600 800 1000 1200 1400 1600 1800 2000 5 10 15 20 25 30 35 40 45 50 days m3/days Oil production rate Eclipse SL: 200 days SL: 50 days 200 400 600 800 1000 1200 1400 1600 1800 2000 5 10 15 20 25 30 35 40 45 50 days m3/days Oil production rate Eclipse SL: 50 days SL: 25 days SL: 12.5 days

water injection water alternating gas injection

Runtimes: 8 hr 20 min for Eclipse, 2 hr 13 min for streamlines

17.11.04 – p. 39