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 toensure 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(mα)+∂x(Fα) = 0, α = w, g, o
mα = ραSαφ
Fα = −ραkλα ∂xp
Saturations add up to one:
∑
α=w,g,o
Sα ≡ 1
17.11.04 – p. 4
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
)
=
(
0
0
)
17.11.04 – p. 5
Displacement Theory cont’d
Saturation triangle: 0 0.2
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Reduced saturations:
S̃α :=Sα − Sαi
1−∑3
β=1 Sβi
↓
Renormalized triangle
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Displacement Theory cont’d
Saturation triangle: 0 0.2
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Imm
obile
wate
r (Sw
i)
Immobile gas (Sgi)
Imm
obileoil (S
oi )
Reduced saturations:
S̃α :=Sα − Sαi
1−∑3
β=1 Sβi
↓
Renormalized triangle
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Displacement Theory cont’d
Saturation triangle: 0 0.2
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Imm
obile
wate
r (Sw
i)
Immobile gas (Sgi)
Imm
obileoil (S
oi )
Reduced saturations:
S̃α :=Sα − Sαi
1−∑3
β=1 Sβi
↓
Renormalized triangle
17.11.04 – p. 6
Character of the System
The character of the system
∂t
(
u
v
)
+ vT ∂x
(
f
g
)
=
(
0
0
)
⇔ ∂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 isdiagonalizable. Strictly hyperbolic: distinct eigenvalues ν1 < ν2.
Elliptic: The eigenvalues are complex.
17.11.04 – p. 7
Character of the System
The character of the system
∂t
(
u
v
)
+ vT ∂x
(
f
g
)
=
(
0
0
)
⇔ ∂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 isdiagonalizable. Strictly hyperbolic: distinct eigenvalues ν1 < ν2.
Elliptic: The eigenvalues are complex.
17.11.04 – p. 7
Character of the System
The character of the system
∂t
(
u
v
)
+ vT ∂x
(
f
g
)
=
(
0
0
)
⇔ ∂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 isdiagonalizable. Strictly hyperbolic: distinct eigenvalues ν1 < ν2.
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 forany 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 saturationtriangle
17.11.04 – p. 8
Character of the System cont’d
It was long believed that the system was strictly hyperbolic forany 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 saturationtriangle
17.11.04 – p. 8
Character of the System cont’d
It was long believed that the system was strictly hyperbolic forany 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 saturationtriangle
17.11.04 – p. 8
Character of the System cont’d
Traditional assumed behavior of fast eigenvectors (r2)
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r2
r2
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r2r2
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Character of the System cont’d
Traditional assumed behavior of fast eigenvectors (r2)
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r2
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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
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 andpore-scale physics
• Umbilic points still act as “repellers” for classical 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:
krα = krα(Sα), α = 1, . . . , 3
• This model is not supported by experiments andpore-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 (r1, r2) 0 0.2
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r2
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Relative Permeabilities
Proposed behavior of eigenvectors (r1, r2) 0 0.2
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r2
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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,uλw
λT
Condition II requires that the gas relative permeability has apositive 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)
0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.5
1
1.5
2
2.5
3x 10−3
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Water Saturation
Gas Saturation Wat
erR
elat
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Gas Relative Perm0 0.1 0.2 0.3 0.4
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0.2
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Water Saturation
Gas Saturation
Water Relative Perm
Gas
Rel
ativ
ePer
m
17.11.04 – p. 14
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
17.11.04 – p. 15
Relative Permeabilities cont’d
Oil isoperms:
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kro = 0.2
kro = 0.4
kro = 0.6
17.11.04 – p. 16
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)
17.11.04 – p. 17
Analytical Solution cont’d
Self-similarity (“stretching”, “coherence”):
0 40
1
PSfrag replacements
x
u
t1
u(x, t) = U (ζ), ζ :=x
∫ t
0 vT (τ) dτ
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Analytical Solution cont’d
Self-similarity (“stretching”, “coherence”):
0 40
1
PSfrag replacements
x
u
t1 t2
u(x, t) = U (ζ), ζ :=x
∫ t
0 vT (τ) dτ
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Analytical Solution cont’d
Self-similarity (“stretching”, “coherence”):
0 40
1
PSfrag replacements
x
u
t1 t2
u(x, t) = U (ζ), ζ :=x
∫ t
0 vT (τ) dτ
17.11.04 – p. 18
Analytical Solution cont’d
Using self-similarity, the Riemann problem is a boundary valueproblem:
(A(U)− ζI)U ′ = 0, −∞ < ζ <∞
with boundary conditions
U(−∞) = ul, U(∞) = ur
Strict hyperbolicity −→ wave separation:
ulW1−→ um
W2−→ ur
17.11.04 – p. 19
Analytical Solution cont’d
Schematic of Riemann solution 0 0.2
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ul
ur
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ul
ur
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Analytical Solution cont’d
Schematic of Riemann solution 0 0.2
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W1
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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
↗ ↖
eigenvalue (νp) eigenvector (rp)
A smooth solution(rarefaction) must lieon a curve whose tan-gent is in the direction ofthe eigenvector (integralcurve)
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1-family (slow paths)2-family (fast paths)
17.11.04 – p. 21
Wave Structure: Rarefactions cont’d
Admissibility of a rarefaction wave• To avoid a multiple-valued solution, νp must increase
monotonically along the curve
ulRp
−→ ur
• Thus, rarefaction curves Rp are subsets of integralcurves Ip
17.11.04 – p. 22
Wave Structure: Shocks
If the solution is discontinuous, it mustsatisfy the Rankine-Hugoniot jumpcondition: ζ
u−
u+
- σ
f(u+)− f(u−) = σ ·[
u+ − u−
]
The set of states which can be con-nected satisfying the jump condition iscalled the Hugoniot locus
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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
17.11.04 – p. 24
Wave Structure: Rarefaction-Shocks
Genuine nonlinearity: eigenvalues vary monotonically alongintegral curves
This is not the case in multiphase flow, where each wave mayinvolve rarefactions and shocks
Inflection locus: set of points at which eigenvalues attain alocal maximum when moving along integral curves
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5
17.11.04 – p. 25
Wave Structure: Rarefaction-Shocks
Genuine nonlinearity: eigenvalues vary monotonically alongintegral curvesThis is not the case in multiphase flow, where each wave mayinvolve rarefactions and shocks
Inflection locus: set of points at which eigenvalues attain alocal maximum when moving along integral curves
0 0.2
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0.6
0.8 1
0
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0.8
1
0
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1
Inflection locus
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Wave Structure: Rarefaction-Shocks
Genuine nonlinearity: eigenvalues vary monotonically alongintegral curvesThis is not the case in multiphase flow, where each wave mayinvolve rarefactions and shocks
Inflection locus: set of points at which eigenvalues attain alocal maximum when moving along integral curves
0 0.2
0.4
0.6
0.8 1
0
0.2
0.4
0.6
0.8
1
0
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0.4
0.6
0.8
1
Inflection locus
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1Inflection locus
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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:
ulRp
−→ u∗
Sp
−→ ur
0 1 0
1
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u0
f
f ′′ < 0
f ′′ > 0
−→
←−
0 1
0
1
0
1
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u0
ff ′′ < 0
f ′′ > 0
−→←−
W
G
O
Inflection locus
R1
S1
ulur 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
ulRp
−→ u∗
• The shock must satisfy the extended-Lax entropycondition:
νp(u∗)= σp > νp(ur)
17.11.04 – p. 27
Wave Structure cont’d
Complete set of solutions: 9 different wave configurations
ul
S
R
RS
umSR
RS
SR
RS
SR
RS
ur
S1S2
S1R2
S1R2S2
R1S2
R1R2
R1R2S2
R1S1S2
R1S1R2
R1S1R2S2
17.11.04 – p. 28
Complete Set of Solutions 0 0.2
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(b) S1R2
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(c) S1R2S2
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Complete Set of Solutions cont’d 0 0.2
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(d)R1S2 0 0.2
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Complete Set of Solutions cont’d 0 0.2
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(g)R1S1S2 0 0.2
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ur
(h)R1S1R2
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ul
ur
(i)R1S1R2S2
17.11.04 – p. 31
Example 1
Injection of water and gas into an oil-filled core (with somemobile 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
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0
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1 0
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W
G
O
R1
R2
S1
S2
ul
ur
umu∗
1
u∗
2
17.11.04 – p. 32
Example 1 cont’d
0 0.1 0
1
PSfrag replacements
water
gas
oil R1
S1
σ1
ζ
0 10 0
1
PSfrag replacements
gas
oilR2
S2
σ2
ζ
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Example 1 cont’d
0 0.1 0
1
PSfrag replacements
water
gas
oil R1
S1
σ1
ζ 0 10 0
1
PSfrag replacements
gas
oilR2
S2
σ2
ζ
17.11.04 – p. 33
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)-raysWhile t < tend:
track discontinuitiessolve Riemann problems
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t
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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
0 1 2
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Streamline Methods
Interpret the saturation equation φ∂tS + v · ∇f(S) = 0 as anequation along streamlines using
v
|v|=[dx
ds,dy
ds,dz
ds
]T
or v·∇ = |v|∂
∂s
Transformation using time-of-flight τ
|v|∂
∂s= φ
∂
∂τ
dx
dsdy
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 0 1 2 3 4
• 30 × 110 × 15 upscaledsample from Tarbertformation
• Initial composition:(Sw, Sg) = (0.0, 0.2)
• 2000 days of production
• Either: continuous waterinjection
• Or: water-alternating-gasevery 200 day
Data reduction is used to speed up front-tracking solution: weakinteractions are treated as being of type S1S2.
17.11.04 – p. 38
Example 4 cont’d
Oil production:
0 200 400 600 800 1000 1200 1400 1600 1800 20000
5
10
15
20
25
30
35
40
45
50
days
m3 /d
ays
Oil production rate
EclipseSL: 200 daysSL: 50 days
0 200 400 600 800 1000 1200 1400 1600 1800 20000
5
10
15
20
25
30
35
40
45
50
days
m3 /d
ays
Oil production rate
EclipseSL: 50 daysSL: 25 daysSL: 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