Controlled CO 2 | Diversified Fuels | Fuel-efficient Vehicles | Clean Refining | Extended Reserves © IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium 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

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 2 The SAGD recovery process Thermal process based on steam injection Used for heavy-oils recovery (μ  10 3 to 10 6 cPo) Objective: Increase the reservoir temperature Decrease the oil viscosity

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 3 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:  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 grid19.5 sec33.1 sec77 sec Fine grid1535 sec sec sec

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 4 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

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 5 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

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 6 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 Gradients of temperatures  J.R. Christensen, G. Darche, B. Déchelette, H. Ma and P.H. Sammon Gradients of temperatures and saturations  X-H. Wang, M. Quintard and G. Darche Speed-ups Following the temperature front  Speed-up of 2 to 3 without loss of accuracy in 2D and 3D

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 7 Classical refinement strategies in SAGD problem 2/2 Recall of results Threshold temperatures as refinement criterion Problem:widening of the temperature front not well adapted in heterogeneous media 2DNumber of cells Reduction of the number of cells CPU TimeSpeed-up Fine Grid 58320%14272 s1 Dynamically Refined Grid (Temperature criterion)  58.2% (average) 4423s3.23

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 8 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

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 9 Choice of a variable  oil saturation Definition of a new refinement criterion 1/5

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 10 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 'h 1/4 ' error estimate  C. Chainais, 1999 An a posteriori error estimate  D. Kröner and M. Ohlberger, 2000

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 11 Model simplification In the flow interface :  Two-phase flow  Constant ρ o  No source terms  Definition of a new refinement criterion 3/5

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 12 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: B 0, B t and B x only depend on the problem data Definition of a new refinement criterion 4/5

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 13 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 T f of the finest grid { if ( ) T f is activated } Definition of a new refinement criterion 5/5

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 14 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

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 15 Results obtained with the new criterion 1/2 Validation on an homogeneous 2D case Finest mesh  zones of deep fronts of S o At 3000 days: a speed-up of  4 ( recall: speed up of 3 with threshold temperatures ) COP (hm 3 )

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 16 Validation on an homogeneous 3D case Finest mesh  zones of deep fronts of the S o At 2000 days: a speed-up of  20 Results obtained with the new criterion 2/2

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 17 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

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 18 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

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 19 Thank you for your attention.

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 20 Heterogeneous media

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 21 Dead-oil model 3 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 porous media

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 22 Annexes !

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

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 24 Flow evolution during the process Oil saturationTemperature

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 25 A dynamic sub-gridding approach

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 26 New criterion cartography A new refinement criterion The maximal value of the new criterion does not change

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 27 3D case Results with a refinement in all the directions a cumulated gain of 2

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 28 Study case

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 29 2D Heterogeneous case 2 facies permeable impermeable

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 30 2D Heterogeneous case Oil saturation : Criterion based on Temperature

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 31 2D Heterogeneous case Oil saturation : Criterion based on Space

© IFP IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium September 2009, Canberra, Australia 32 2D Heterogeneous case Oil saturation : Criteria based on Time and Space