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

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:

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

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  1. Study of a new refinement criterion for the use of adaptive mesh refinement in SAGD modelling Magnolia Fatchi-Mamaghani Claire Chainais, Guillaume Enchéry

  2. 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, 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: • 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 IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 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 IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 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 IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 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. 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, 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 IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 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 IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

  9. Definition of a new refinement criterion 1/5 • Choice of a variable  oil saturation IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 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 'h1/4'error estimate • C. Chainais, 1999 • An a posteriori error estimate • D. Kröner and M. Ohlberger, 2000 IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

  11. Definition of a new refinement criterion 3/5 • Model simplification • In the flow interface : • Two-phase flow • Constant ρo • No source terms IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

  12. Definition of a new refinement criterion 4/5 • 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 IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

  13. Definition of a new refinement criterion 5/5 • 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 } IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 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 IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 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 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, Australia

  16. Results obtained with the new criterion 2/2 • Validation on an homogeneous 3D case • Finest mesh  zones of deep fronts of the So • At 2000 days: • a speed-up of 20 IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 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 IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 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 IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

  19. Thank you for your attention. IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

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

  21. 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 Dead-oil model 3 phases (S/O/W) - 2 components (W/O)a nonlinear problem with 4 unknowns IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

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

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

  24. Oil saturation Temperature Flow evolution during the process IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

  25. A dynamic sub-gridding approach IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

  26. A new refinement criterion • The maximal value of the new criterion does not change New criterion cartography IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

  27. 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, Australia

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

  29. 2D Heterogeneous case 2 facies permeable impermeable IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

  30. 2D Heterogeneous case Oil saturation : Criterion based on Temperature IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

  31. 2D Heterogeneous case Oil saturation : Criterion based on Space IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

  32. 2D Heterogeneous case Oil saturation : Criteria based on Time and Space IEA Collaborative Project on EOR - 30th Annual Workshop and Symposium - 21-23 September 2009, Canberra, Australia

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