modelling two-phase flow in porous media using hmm · overview 1 introduction 2 governing equations...

Modelling Two-Phase Flow in Porous Media UsingHMM

Elin Solberg

27 January 2011

Elin Solberg Modelling Two-Phase Flow in Porous Media Using HMM 27 January 2011 1 / 33

Overview

1 Introduction

2 Governing Equations

3 Modelling with Multiple Scales

4 Multi-Scale-Upscaling Solution Strategy

5 Results and Conclusion

Overview

1 Introduction

The Complex Reality

Three-phase-three-component (3p3c) processes in a heterogenousporous medium

Simplifications

Two-phase-two-component (2p2c) processes in a heterogenousporous medium (2D model)

Assumptions:1 Incompressible fluids (water and air)2 No capillary pressure3 No gravity4 No external sinks/sources

Simplifications

Two-phase-two-component (2p2c) processes in a heterogenousporous medium (2D model)

Assumptions:1 Incompressible fluids (water and air)2 No capillary pressure3 No gravity4 No external sinks/sources

Overview

1 Introduction

Two-Phase Model (Fully Coupled Formulation)

mass-conservationequations

∂Sα∂t

+∇ · vα = 0, α = w ,n

momentum-balanceequations (Darcy’s law)

vα = −krα

µαK¯̄· ∇p, α = w ,n

Sw + Sn = 1

α = w : wetting phaseα = n: non-wetting phaseSα saturation of phase αvα velocityp pressure (pw = pn = p)krα relative permeabilityµα viscosityK¯̄

absolute permeability

Choice of Relative Permeability Parameterization

1 Linear approach:

krw (Sw ) = Sw

krn(Sn) = krn(1− Sw ) = 1− Sw

2 Non-linear approach (Brooks and Corey):

krw (Sw ) = S(2+3λBC)/λBCw

krn(Sn) = krn(1− Sw ) = (1− Sw )2(1− S(2+3λBC)/λBCw )

Choice of Relative Permeability Parameterization

1 Linear approach:

krw (Sw ) = Sw

krn(Sn) = krn(1− Sw ) = 1− Sw

2 Non-linear approach (Brooks and Corey):

krw (Sw ) = S(2+3λBC)/λBCw

krn(Sn) = krn(1− Sw ) = (1− Sw )2(1− S(2+3λBC)/λBCw )

Two-Phase Model (Fractional Flow Formulation)

pressure equation

∇ · [λK¯̄∇p] = 0

saturation equation

∂t+ v · ∇fw = 0

λα = krα/µα phase mobilitiesλ = λw + λn total mobilityv = vw + vn total velocityfw = λw/λ fractional flow

of wetting phase

Two-Phase-Two-Component Model

Concentration equations

∂Cκ

∂t+ v · ∇

fαCκα = 0, κ = 1,2

Cκα = ραXκ

Cκ = SwCκw + SnCκ

κ = 1: water componentκ = 2: air componentCκ total concentration of component κCκα concentration of component κ in phase α

ρα density of phase αXκα = mκ

m1α+m2

αmass fraction of component κ in phase α

Only one concentration equation is solved. Sw can be computed fromC1 and p using ”flash computations”.

Two-Phase-Two-Component Model

Concentration equations

∂Cκ

∂t+ v · ∇

fαCκα = 0, κ = 1,2

Cκα = ραXκ

Cκ = SwCκw + SnCκ

κ = 1: water componentκ = 2: air componentCκ total concentration of component κCκα concentration of component κ in phase α

ρα density of phase αXκα = mκ

m1α+m2

αmass fraction of component κ in phase α

Only one concentration equation is solved. Sw can be computed fromC1 and p using ”flash computations”.

Overview

1 Introduction

Why Multiple Scales?

Want to compute coarse scale saturations (and velocity fields)These are influenced by

1 heterogeneities in the porous medium (everywhere)2 mass transfer processes (locally)

which can only be resolved on a fine scaleIn order to reduce computational cost and amount of data to becollected an upscaling of the saturation equation is appliedEffect of fine-scale heterogeneities captured by macrodispersionterm in the upscaled saturation equationOne fine-scale concentration equation is solved locally and usingthe solution a sink/source term for the upscaled saturationequation is computed, to make it mass conservative

Solution Strategy Overview

Upscaling of Saturation Equation (1/5)

Let D be a coarse-grid block. Express the unknowns as:

v = v̄ + v ′, Sw = S̄w + S′w (1)

with ·̄ =1|D|

∫D·(x , t)dx

What we want to achieve (linear relative permeability case):

∂S̄w

∫∂D

v̄jnj S̄wdl =1|D|

∫∂D

[∫ t

0v ′i (x)v ′j (x(τ))dτ

]ni∇j S̄w (x , t)dl

Let D be a coarse-grid block. Express the unknowns as:

v = v̄ + v ′, Sw = S̄w + S′w (1)

with ·̄ =1|D|

∫D·(x , t)dx

What we want to achieve (linear relative permeability case):

∂S̄w

∫∂D

v̄jnj S̄wdl =1|D|

∫∂D

[∫ t

Assuming linear relative permeabilities and µw = µn = 1⇒ fw (Sw ) = Sw .Inserting (1) in the saturation equation and averaging over D gives theaveraged saturation equation:

∂S̄w

∂t+ v̄ · ∇S̄w + v ′ · ∇S′w = 0 ⇐⇒ (2)

∂S̄w

∫∂D

(v̄ · n)S̄wdl +1|D|

∫∂D

(v ′ · n)S′wdl = 0 (3)

Subtract (2) from the fine saturation equation to obtain the fluctuatingequation:

∂S′w∂t

+ v̄ · ∇S′w + v ′ · ∇S̄w + v ′ · ∇S′w = v ′ · ∇S′w (4)

∂S̄w

∂t+ v̄ · ∇S̄w + v ′ · ∇S′w = 0 ⇐⇒ (2)

∂S̄w

∫∂D

(v̄ · n)S̄wdl +1|D|

∫∂D

(v ′ · n)S′wdl = 0 (3)

∂S′w∂t

+ v̄ · ∇S′w + v ′ · ∇S̄w + v ′ · ∇S′w = v ′ · ∇S′w (4)

∂S̄w

∂t+ v̄ · ∇S̄w + v ′ · ∇S′w = 0 ⇐⇒ (2)

∂S̄w

∫∂D

(v̄ · n)S̄wdl +1|D|

∫∂D

(v ′ · n)S′wdl = 0 (3)

∂S′w∂t

+ v̄ · ∇S′w + v ′ · ∇S̄w + v ′ · ∇S′w = v ′ · ∇S′w (4)

Projecting (4) onto coarse-grid streamlines dx/dt = v̄ , using

dS′w (x(t), t)dt

=∂S′w∂t

+ v̄ · ∇S′w

givesdS′w (x(t), t)

dt+ v ′j∇j S̄w + v ′j∇jS′w = v ′j∇jS′w (5)

Integrating (5) over (0, t) yields, with x = x(t),

S′w (x , t) =−∫ t

[v ′j (x(τ))∇j S̄w (x(τ), τ) + v ′j (x(τ))∇jS′w (x(τ), τ)

0v ′j (x(τ))∇jS′w (x(τ), τ))dτ (6)

Projecting (4) onto coarse-grid streamlines dx/dt = v̄ , using

dS′w (x(t), t)dt

=∂S′w∂t

+ v̄ · ∇S′w

givesdS′w (x(t), t)

dt+ v ′j∇j S̄w + v ′j∇jS′w = v ′j∇jS′w (5)

Integrating (5) over (0, t) yields, with x = x(t),

S′w (x , t) =−∫ t

[v ′j (x(τ))∇j S̄w (x(τ), τ) + v ′j (x(τ))∇jS′w (x(τ), τ)

0v ′j (x(τ))∇jS′w (x(τ), τ))dτ (6)

Multiplying (6) by v ′i (x) and integrating over the boundaries of acoarse-grid block D further yields∫

∂DS′w (x , t)v ′i (x)nidl =

−∫∂D

0v ′i (x)v ′j (x(τ))ni∇j S̄w (x(τ), τ)dτdl

−∫∂D

0v ′i (x)v ′j (x(τ))ni∇jS′w (x(τ), τ)dτdl︸︷︷︸

neglected

∫∂D

0v ′i (x)v ′j (x(τ))ni∇jS′w (x(τ), τ)dτdl︸︷︷︸

Substituting the approximation∫∂D

S′w (x , t)v ′i (x)nidl ≈ −∫∂D

0v ′i (x)v ′j (x(τ))ni∇j S̄w (x(τ), τ)dτdl

into (3) renders (almost) the final equation:

∂S̄w

∫∂D

v̄jnj S̄wdl =

∫∂D

0v ′i (x)v ′j (x(τ))ni∇j S̄w (x(τ), τ)dτdl ≈

∫∂D

[∫ t

Overview

1 Introduction

Discretization

Space discretization: Discontinuous GalerkinNot too much numerical diffusionFlexible use for different types of equations

Time discretization (saturation and concentration equations):Runge-Kutta

Time explicit⇒ must meet CFL conditions⇒{Macro time steps for upscaled saturation equationMicro time steps for fine-scale concentration equation

Discretization

Space discretization: Discontinuous GalerkinNot too much numerical diffusionFlexible use for different types of equations

Time discretization (saturation and concentration equations):Runge-Kutta

Time explicit⇒ must meet CFL conditions⇒{Macro time steps for upscaled saturation equationMicro time steps for fine-scale concentration equation

Compression and Reconstruction Operators

S̄w = QSSw :=1|D|

v̄ = Qv v :=1

∫∂E

v dl , + linear interpolation between edges

Φ(x) = RΦ̄(x) := Φ̄(x), ∀x ∈ T (for Φ = Sw , v)

Solution Strategy (1/2)

1 Solve fine-scale pressureequation in T⇒ p, v

2 Volume average v (with Qv )⇒ v̄ , v ′

3 Solve one fine-scaleconcentration equation in U⇒ C1

4 Compute fine-scale SCw in U

from C1 and p, using flashcalculations⇒ SC

5 Volume average SCw (with QS)

⇒ S̄Cw ,S′w C

6 Compute sink/source term forupscaled saturation equation:

q̄w =S̄S

w − S̄Cw

7 Solve upscaled saturationequation in T⇒ S̄S

⇒ S̄Cw ,S′w C

q̄w =S̄S

w − S̄Cw

⇒ S̄Cw ,S′w C

q̄w =S̄S

w − S̄Cw

Overview

1 Introduction

Limitations of Result Presentation

No error estimatesNo comparison to reference solutionNo run time comparisons

Possibly because focus is on the overall concept

Test Case Setup

µw = µn = 1.0 kg/ms, ρw = 1000.0 kg/m3, ρn = 0.9 kg/m3, λBC = 2.0

Results: Linear Heterogeneous Case

Results: Linear Homogeneous Case

Results: Non-Linear Heterogeneous Case

Results: Non-Linear Homogeneous Case

Conclusion

A modell for 2p2c processes in heterogeneous porous media waspresentedThe saturation equation was upscaled, but one term withfine-scale velocities remained, taking into account fine-scaleheterogeneitiesAn HMM approach was used to account for local fine-scale masstransfer processesResults were presented but were difficult to interpret due to lack ofreference solutionsThe presented algorithm is ”one first step” towards an efficientmodel framework - much work is yet to be done

Conclusion

Flash Calculations (1/2)

Method to compute Sw given C1 and p using the relations

X 1α + X 2

α = 1

x1α + x2

α = 1

Xκα =

xκαMκ

x1αM1 + x2

αM2 (7)

xκn = K κxκw

K κ mole equilibrium ratio, known function of pxκα mole fraction of component κ in phase αMκ molar mass of component κ

Flash calculations (2/2)

Compute1 mole equilibrium ratios:

K 1 = const1p , K 2 = 1

const2·p2 mole fractions:

x1w = 1−K 2

K 1−K 2 , x2w = 1− x1

w , x1n = x1

wK 1, x2n = 1− x1

3 mass fractions: from mole fractions, using relation (7)4 mass equilibrium ratios:Kκ = Xκ

5 wetting phase saturation:Sw = C1·K1−C1·K2−K1·ρn+K1·K1·ρn

(K2−1)(K1·ρn−ρw )

K 1 = const1p , K 2 = 1

x1w = 1−K 2

K 1−K 2 , x2w = 1− x1

w , x1n = x1

wK 1, x2n = 1− x1

(K2−1)(K1·ρn−ρw )

K 1 = const1p , K 2 = 1

x1w = 1−K 2

K 1−K 2 , x2w = 1− x1

w , x1n = x1

wK 1, x2n = 1− x1

(K2−1)(K1·ρn−ρw )

K 1 = const1p , K 2 = 1

x1w = 1−K 2

K 1−K 2 , x2w = 1− x1

w , x1n = x1

wK 1, x2n = 1− x1

(K2−1)(K1·ρn−ρw )

K 1 = const1p , K 2 = 1

x1w = 1−K 2

K 1−K 2 , x2w = 1− x1

w , x1n = x1

wK 1, x2n = 1− x1

(K2−1)(K1·ρn−ρw )

modelling two-phase flow in porous media using hmm · overview 1 introduction 2 governing equations...

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