1. Dual Mesh Method in Upscaling Pascal Audigane and Martin Blunt Imperial College London SPE Reservoir Simulation Symposium, Houston, 3-5 February 2003

2. Fine Mesh Large memory and time requirements Coarse Mesh Upgridding + Upscaling As the techniques for reservoir modeling are improved, very refined meshes to represent the reservoir can be generated. However, time and memory constraints often prevent flow simulation on this grid, forcing the use of an upscaled, coarser grid.

3. Using a coarser grid introduces errors in predictions of recovery An example of the difference obtained for waterflooding simulation performed on a fine mesh (30 x 30 cells) and a coarse mesh (10 x 10 cells). Geometric averaging of absolute permeability. fine coarse The problem with upscaling

4. The aim of the Dual Mesh Method To reduce this error so that the simulation performed with the coarse mesh will provide a good estimate in little time How? By reintroducing the fine mesh information In this study, only waterflooding simulation is considered.

5. Previous work 1. Verdière and Guérillot (1996): dual mesh method with geometric averaging for upscaled properties. 2D, no wells, no gravity. 2. Guedes and Schiozer (1999): 2D with gravity. 3. Gautier and Blunt (1999): nested gridding in the context of streamlines. 3D with gravity and wells. This work: -3D with gravity and wells. -Different upscaling methods. -First step towards a general multi-scale upscaling technique.

6. Uses a standard IMPES approach to reservoir simulation, but…. Solving the Pressure field with an implicit scheme is the most time consuming step of the simulation, hence : 1- Pressure field is solved on the coarse mesh Saturation is updated with an explicit scheme and can be easily handle on grids containing large number of grid-blocks: 2- Saturation field is updated on the fine mesh The method is implemented in 3 steps… The Dual Mesh Method:

7. STEP 1 The pressure field is found on the coarse mesh with all the properties upscaled using any type of upscaling technique. Fine Mesh Coarse Mesh Upscaling Pressure Velocity Darcy’s law

8. 1- Coarse pressure value as a Dirichlet condition on the centre of the coarse cell 2- Coarse flux weighted by interblock transmissivities on the fine cell faces Coarse Mesh A refinement technique is used to compute the total velocity on the fine mesh using coarse scale information STEP 2 For each coarse grid block, the following pressure solve is done using:

9. Performing STEP 2 for all coarse grid blocks, the velocity field can be recomputed on the entire fine grid, and the saturation field can be up-dated on the fine grid. Then the simulation can be performed for the next time step with the new saturation field (STEP 1). STEP 3 Darcy flow field used to update saturation explicitly at the fine scale: For the fine cells in contact with coarse cell face, the weighted flux is applied. Inside each coarse grid block, the flux across each fine cell face is deduced from the pressure gradient. Mass balance is ok

10. 1.Pressure solver method (PSM): For each coarse grid block each effective transmissivity component is deduced from the flux obtained with a pressure drop applied on two opposite faces of the cell and no flux for the other faces 2.Geometrical average (GA): perform on the product between absolute permeability and total mobility P i (pressure drop) T eff i = Q cal / P i Dir i One Coarse Grid block: No flux Two different upscaling techniques:

11. Heterogeneous Medium, 2D Horizontal, 2 Phases The watercut estimates are better using the dual mesh method in comparison with the simulation performed on the coarse mesh only Permeability : 80 % 100 mD (Black) 20 % 1 mD (White) Porosity: 0.2 Fine mesh: 30 x 30 Coarse mesh:10 x 10 Reservoir size: 300 m x 300 m x 1m Water injection: 5 m^3 / day Fluid production: 76 bars fine coarse Watercut comparisons Dual Mesh Method x y

12. Saturation Field Comparison after 300 days Fine Grid PSM case Coarse Grid GA case Lack of precision for the simulation performed only on the coarse mesh Improvement of the saturation field simulation especially for the PSM case. With the dual mesh method

13. Injection Production 2D with Gravity Fine: 60 x 60 Coarse: 10 x 10 Gravity number : N g = gravity / viscous = 0.3 Permeability:80 % 100 mD (White) 20% 0.1 mD (Red) Porosity: 0.2 x z The dual mesh method using the pressure solve method (psm) is virtually identical to the fine mesh simulation coarse fine Watercut comparisons Dual Mesh Method

14. Three different scenarios have been set up based on the SPE Comparative project on upscaling (Christie & Blunt, 2001). The original grid contains more than one million grid blocks but the size of the problem has been reduced to a fine mesh: 20x55x85 = 93,500 cells and a coarse mesh: 5x11x5= 275 cells. upperlower Porosity scale: blue = 0.2 3D Results

15. Permeability: dual facies, 80% of high permeable zone of 100mD and 20% at 1mD. 1 injector and 1 producer well. The same as 1 but with 1 central injector well and 4 producers at each corner. Permeability and Porosity created for the SPE Comparative Project 2001: 1 central injector and 4 producers at each corner Case 1 Case 2 Case 3

16. Case 1 The Dual Mesh Method is very good at matching the fine mesh simulation especially with the use of the pressure solve method for upscaling (psm) fine coarse Dual Mesh Method

17. Case 2 fine coarse

18. Case 3 fine coarse fine coarse fine coarse fine coarse

19. Cumulative oil Produced Comparison (case 3) 5 x 11 x 6 SPE 5 x 11 x 5 coarse 5 x 11 x 5 ga 5 x 11 x 5 psm 30 x 55 x 17 SPE 20 x 55 x 85 fine 60 x 220 x 85 SPE

20. Time of Calculation 2D2Dv3DSPE Fine mesh30x3060x6030x30x3020x55x85 Coarse mesh10x1010x1010x10x105x11x17 DMM0.92.12.73.9 NESTED0.61.31.33.1 COARSE3.62894432 Speed Factor: time(fine) / time(method) ( > 1 ?), Gautier et al, 1999: 3 to 4 (in 3D with gravity) Verdiere and Guerillot, 1996: from 4 to 7 (in 2D, no gravity)

21. FineCoarse Upscal ing Coarse Next? A fully decoupled approach Step 1 Step 1- The same as before: upscaling from the fine to the coarse mesh Step 2 Step 2- The Pressure field AND THE SATURATION FIELD are updated on the coarse mesh. Step 3 Step 3- The saturation field is recomputed on the fine mesh inside each coarse grid block separately rather than on the whole fine grid.Why? Less time-step restriction. Only perform fine grid simulation on selected cells. Save CPU time and memory Coarse The whole simulation is done on the coarse mesh: pressure and sat update

22. Conclusions Dual Mesh Method has been implemented in 3D with gravity effects and well boundary conditions. This method is able to reduce considerably the error between a conventional simulation performed on the chosen coarse mesh and the one performed on the initial fine mesh. The accuracy of the dual mesh method depends on the complexity of the reservoir heterogeneity Different up-scaling techniques can be applied. The time of calculations is reduced compared with the simulation performed on the fine mesh.

23. References -Christie, M A. and Blunt, M J., (2001), “Tenth SPE Comparative Solution Poject: A comparison of Upscaling Techniques”, SPE 66599, presented at the SPE Reservoir Simulation Symposium, Houston, Texas, 11-14 February 2001 - Gautier, Y. and Blunt, M J.,(1999), “ Nested Gridding and Streamline- Based Simulation for Fast Performance Prediction”, Computational Gesosciences 3, 295-320 -Guedes, S S. and Schiozer, D J., (1999), “An Implicit Treatement of Upscaling in Numerical Reservoir Simulation”, SPE 51937, presented at the SPE Reservoir Simulation Symposium, Houston, Texas, 14-17 February 1999 -Verdière, S. and Guérillot, D., (1996), “Dual Mesh Method in Heterogeneous Porous Media”, 5 th European Conference on the Mathematics of Oil Recovery, Leoben (September).