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IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

Study of a new refinement criterion for the use of adaptive mesh refinement in SAGD modelling

Magnolia Fatchi-Mamaghani

Claire Chainais, Guillaume Enchéry

IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia2

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The SAGD recovery process Thermal process based on steam injection Used for heavy-oils recovery (μ 103 to 106 cPo) Objective:

Increase the reservoir temperature Decrease the oil viscosity


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Issues in SAGD modelling Flow interface not wide compared to reservoir dimensions Need of fine mesh discretization for good forecasts of oil

production

A difference at 3000 days:107 000 us.bl 7.5% of the OOIP

Fine mesh discretization high number of cells Long CPU times

185

AMR method good compromise between accuracy and CPU times fine mesh in the flow interface, coarser cells outside

At 500 days

At 1500 days

At 3000 days

Coarse grid 19.5 sec 33.1 sec 77 sec

Fine grid 1535 sec 7263.7 sec 14272.8 sec


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Table of contents Classical refinement strategies for a SAGD

problem

Definition of a new refinement criterion

Results obtained with the new criterion

Conclusions and perspectives


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Table of contents

Classical refinement strategies for a SAGD problem





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Classical refinement strategies in SAGD problem 1/2

Flow interface located, in practice, by: Threshold values of the temperature

S. Lacroix, G. Renard, P. Lemonnier and C. Taïeb. 2003. Gradients of temperatures

J.R. Christensen, G. Darche, B. Déchelette, H. Ma and P.H. Sammon. 2004.

Gradients of temperatures and saturations X-H. Wang, M. Quintard and G. Darche. 2006.

Speed-ups Following the temperature front

Speed-up of 2 to 3 without loss of accuracy in 2D and 3D


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Classical refinement strategies in SAGD problem 2/2

Recall of results Threshold temperatures as refinement criterion

Problem: widening of the temperature frontnot well adapted in heterogeneous media

2D Number of cells

Reduction of thenumber of cells

CPU Time Speed-up

Fine Grid 5832 0% 14272 s 1

Dynamically Refined Grid(Temperature criterion)

58.2%(average) 4423s 3.23


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Table of contents






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Choice of a variable oil saturation

Definition of a new refinement criterion 1/5


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Definition of a new refinement criterion 2/5 The main idea: a criterion based on an error estimate

Close to the oil saturation variations Close to the solution given by the numerical scheme

Reservoir model:

Numerical scheme: 5-points finite-volume scheme, fully implicit

Known error estimates for finite-volume schemes for hyperbolic equations:

An a priori 'h1/4' error estimate C. Chainais, 1999

An a posteriori error estimate D. Kröner and M. Ohlberger, 2000

0),(,,( txutxFdivut

)()0,( 0 xuxu )(P


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Model simplification In the flow interface :

Two-phase flow Constant ρo

No source terms


0

gP

KSkdivS o

ooroot

0),(,,( txStxFdivS oot

)()0,( 0 xSxS oo

ooo

roooot QgPk

KdivS

)( sP


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a posteriori error estimate (Kröner-Ohlberger) Local a posteriori error estimators

discretization on the initial condition

discrete derivative in time

discrete derivative in space

example:

B0, Bt and Bx only depend on the problem data


),,(

)(

),,(

)(

),,(

)(0

000

0

nxxx

njlx

nttt

njt

j

ETtolfB

MTtolfB

MTtolfB

n

lon

joNjljl

njlx SShht 1)()(

xt

K

hxnjlxt

njtj toltoltoluuBandBB 000 )()(,)(


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From the space-estimator our new criterion

New test in our AMR algorithm

From activated cells New adaptive mesh according to the grid hierarchy

For each cell Tf of the finest grid { if ( ) Tf is activated}


njlx )(


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Table of contents






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Results obtained with the new criterion 1/2 Validation on an homogeneous 2D case

Finest mesh zones of deep fronts of So

At 3000 days:a speed-up of 4( recall: speed up of 3 with threshold temperatures )

COP (hm3)


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Validation on an homogeneous 3D case Finest mesh zones of deep fronts of the So

At 2000 days:a speed-up of 20

Results obtained with the new criterion 2/2


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Table of contents Classical refinement strategies for a SAGD

problem





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Conclusions and perspectives Following the temperature front: not optimal

Design of a new refinement criterion Based on an a posteriori error estimate for finite-

volume schemes for hyperbolic equations Applied on the oil saturation Computational gains in 2D

A cumulated gain of 4 at 3000 days Computational gains in 3D

A cumulated gain of 20 at 2000 days

Ongoing works: SAGD problem in heterogeneous media Mathematical analysis of the a posteriori error

estimators


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Thank you for your attention.


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Heterogeneous media


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Dead-oil model3 phases (S/O/W) - 2 components (W/O)a nonlinear problem with 4 unknowns

Mass conservation law of water

Mass conservation law of oil

Conservation law of energy

Darcy’s law

Balance equations

Conservation law of the porousmedia


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Annexes !


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