study of a new refinement criterion for the use of adaptive mesh refinement in sagd modelling
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Study of a new refinement criterion for the use of adaptive mesh refinement in SAGD modelling. Magnolia Fatchi-Mamaghani Claire Chainais, Guillaume Enchéry. The SAGD recovery process. Thermal process based on steam injection Used for heavy-oils recovery ( μ 10 3 to 10 6 cPo) Objective: - PowerPoint PPT PresentationTRANSCRIPT
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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
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia3
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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
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia4
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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
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia5
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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
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia6
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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
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia7
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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
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia8
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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
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia9
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Choice of a variable oil saturation
Definition of a new refinement criterion 1/5
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia10
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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
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia11
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Model simplification In the flow interface :
Two-phase flow Constant ρo
No source terms
Definition of a new refinement criterion 3/5
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gP
KSkdivS o
ooroot
0),(,,( txStxFdivS oot
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ooo
roooot QgPk
KdivS
)( sP
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia12
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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
Definition of a new refinement criterion 4/5
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IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia13
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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}
Definition of a new refinement criterion 5/5
njlx )(
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia14
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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
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia15
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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)
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia16
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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
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia17
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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
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia18
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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
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia19
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Thank you for your attention.
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia20
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Heterogeneous media
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia21
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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
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia22
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Annexes !
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia23
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IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia24
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Flow evolution during the process
Oil saturation Temperature
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia25
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A dynamic sub-gridding approach
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia26
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New criterion cartography
A new refinement criterion
The maximal value of the new criterion does not change
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia27
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3D case
Results with a refinement in all the directions
a cumulated gain of 2
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia28
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Study case
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia29
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2D Heterogeneous case
2 facies
permeable
impermeable
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia30
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2D Heterogeneous case
Oil saturation : Criterion based on Temperature
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia31
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2D Heterogeneous case
Oil saturation : Criterion based on Space
IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia32
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2D Heterogeneous case
Oil saturation : Criteria based on Time and Space