three-phasedisplacementtheory: …€¦ · krg(v) = gv +(1 g)v2; g > 0 kro(u;v) = (1 u v)(1 u)(1...

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


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


Saturation triangle: 0 0.2

0.4

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0

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O

Reduced saturations:

S̃α :=Sα − Sαi

1−∑3

β=1 Sβi

↓

Renormalized triangle

17.11.04 – p. 6


Saturation triangle: 0 0.2

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0

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W

G

O

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 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


Traditional assumed behavior of fast eigenvectors (r2)

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r2

r2

r2

r2

r2r2

17.11.04 – p. 9


Traditional assumed behavior of fast eigenvectors (r2)

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0

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1

0

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1

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O

r2

r2

r2

r2

r2r2

Elliptic region

17.11.04 – p. 9


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


Proposed behavior of eigenvectors (r1, r2) 0 0.2

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Or1r1

r2

r2

r2

r2

17.11.04 – p. 12


Proposed behavior of eigenvectors (r1, r2) 0 0.2

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Or1r1

r2

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r2

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r2

r2

r2

r2

r2r2

Elliptic region

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 ⇔ 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


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

ive

Per

m

Gas Relative Perm0 0.1 0.2 0.3 0.4

0

0.05

0.1

0.15

0.2

PSfrag replacements

Water Saturation

Gas Saturation

Water Relative Perm

Gas

Rel

ativ

ePer

m

17.11.04 – p. 14


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


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

17.11.04 – p. 18


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


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


Schematic of Riemann solution 0 0.2

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ul

ur

ζ

ul

ur

17.11.04 – p. 20


Schematic of Riemann solution 0 0.2

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O

W1

W2

ul

ur

um

ζ

ul

ur

um

�W1

�

W2

17.11.04 – p. 20

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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H1

H2

17.11.04 – p. 23

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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Inflection locus

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2

5

17.11.04 – p. 25


Genuine nonlinearity: eigenvalues vary monotonically alongintegral curvesThis is not the case in multiphase flow, where each wave mayinvolve rarefactions and shocks


0 0.2

0.4

0.6

0.8 1

0

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1

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Inflection locus

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Inflection locus

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0.5

2

5

17.11.04 – p. 25


Genuine nonlinearity: eigenvalues vary monotonically alongintegral curvesThis is not the case in multiphase flow, where each wave mayinvolve rarefactions and shocks


0 0.2

0.4

0.6

0.8 1

0

0.2

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0.6

0.8

1

0

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0.4

0.6

0.8

1

Inflection locus

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2

5

17.11.04 – p. 25

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∗

17.11.04 – p. 26

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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R1

R2

S1

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ul

ur

(a) S1S2 0 0.2

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(b) S1R2

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ul

ur

(c) S1R2S2

17.11.04 – p. 29

Complete Set of Solutions cont’d 0 0.2

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R1

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S1

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ul

ur

(d)R1S2 0 0.2

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(e)R1R2

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R1

R2

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ul

ur

(f)R1R2S2

17.11.04 – p. 30

Complete Set of Solutions cont’d 0 0.2

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S1

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ul

ur

(g)R1S1S2 0 0.2

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(h)R1S1R2

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(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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R1

R2

S1

S2

ul

ur

umu∗

1

u∗

2

17.11.04 – p. 32

Example 1 cont’d

0 0.1 0

1

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water

gas

oil R1

S1

σ1

ζ

0 10 0

1

PSfrag replacements

gas

oilR2

S2

σ2

ζ

17.11.04 – p. 33

Example 1 cont’d

0 0.1 0

1

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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

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

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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

0 1 2

17.11.04 – p. 35

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.

17.11.04 – p. 36

Streamline Simulation

Figure from Yann Gautier

17.11.04 – p. 37

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

three-phasedisplacementtheory: …€¦ · krg(v) = gv +(1 g)v2; g > 0 kro(u;v) = (1 u v)(1 u)(1...

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