COMSOL Implementation for Two-Phase Immiscible Flows in ...

COMSOL Implementation for Two-Phase

Immiscible Flows in Layered Reservoir

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Xuan Zhang

Center for Energy Recourses Engineering

Technical University of Denmark

COMSOL 2010, France, Nov. 17, 2010

Presented at the COMSOL Conference 2010 Paris

http://www.comsol.com/conf_cd_2011_eu

Outline

� Introduction

� Implementation

� Results

Conclusion

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� Conclusion


Introduction

, , , ,h k s sφ

� Waterflooding

� Layered model of reservoir

� Non-/Communicating layers

� Anisotropy parameter

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1 1 1 1 1, , , ,

wi orh k s sφ

2 2 2 2 2, , , ,wi orh k s sφ

…

…

, , , ,n n n win orn

h k s sφ

� Anisotropy parameter

2 2

0 0/y xE k x k y=

:Length

:Height

:Permeability

0x

0y

kScheme of layered reservoir


Introduction

U U∂ ∂

( ) ( )0X YU F s U F ss

T X Yφ

∂ ∂∂+ + =

∂ ∂ ∂

� Mass conservation

� Incompressible flow

U: total flow velosity

F: fractional flow of water

Φ: porosity

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0X YU U

X Y

∂ ∂+ =

∂ ∂

� Darcy’s law

X X Y Y

P PU U E

X Y

∂ ∂= −Λ ⋅ = − Λ ⋅

∂ ∂

Φ: porosity

Sw: water saturation

Λ: mobility

2 2

0 0/y xE k x k y= - anisotropy ratio


Implemention in COMSOL

( ) ( )0X Y

U F s U F ss

T X Yφ

∂ ∂∂+ + =

∂ ∂ ∂

0X YU U

X Y

∂ ∂+ =

∂ ∂

5/15COMSOL 2010, France, Nov. 17, 2010

T X Y∂ ∂ ∂

X X

Y Y

PU

X

PU E

Y

∂= −Λ ⋅

∂

∂= − Λ ⋅

∂

Implemention in COMSOL

� Initial conditions

� Boundary conditions

COMSOL 2010, France, Nov. 17, 2010 6/15

Results-2 layers


Results- 2 layers

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Water saturation profile of 2D waterflooding simulation in COMSOL, at

time=0.25 PVI (Pore Volume Injected). Left: M<1, right: M>1


Comparision with analytical solution

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M<1 M>1


Average water saturation profile of 2D waterflooding, at time=0.25 PVI

(Pore Volume Injected). Left: M<1, right: M>1

Analytical derivation

Anisotropy parameter E

� Small E – Poorly communicating layers

� Large E – Well communicating layers

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Asymptotic approximation – assumption for perfectly communicating layers:

Consequence: Pressure gradient across the layers is negligebly small!


Analytical derivation

� Pressure P is constant in y-direction

For the ith layer 1( ) ( )0i i Xi Y i Y i

i

i

s FU FU FU

T X α−∂ ∂ −

Φ + + =∂ ∂

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,

, 1

0

X i

X i

X

U QdY

Λ=

Λ∫Q: injection rate

( )

0, 1

0

Y i

X

Y i

X

dYQU

E X dY

Λ∂ = − ∂ Λ

∫

∫

Y(i): height from top

to ith layer

,X iU

,Y iU

, 1Y iU −


Results: log-normal distributed permeability

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M>1M<1


(Pore Volume Injected). Left: M<1, right: M>1


Results:Levels of communication between layers


Results: Involving gravity

( )Y Y Y oY

PU E EG A

Y

∂= − Λ − Λ − Λ

∂: Gravity ratioG : Density ratioA



(Pore Volume Injected). 1 layer. G=0.02, A=0.8.

Conclusions

� 2D simulation of waterflooding in oil recovery

� Show the effect of crossflow

� When E increases, inter-layer communication increases

� Gravity and cappilary can be added easily

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� Gravity and cappilary can be added easily

� Artificial diffusion is needed


COMSOL Implementation for Two-Phase Immiscible Flows in ...

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