1. Dual Mesh Method in Dynamic Upscaling
Pascal Audigane, Martin Blunt
(Imperial College, 2001)

2. Summary Dynamic Up-scaling Dual Mesh Method Description
Test Case in 2D
3D Implementation
Test Case in 3D
Future work

3. Dynamic Up-scaling
Three distinct length scales in reservoir simulations :
- Core scale: pore scale network modelling (0.001 – 10 mm)
Core-to-simulation grid block scale: standard grid based finite difference method (0.1 – 10 m)
Field scale: finite difference, streamline method (100 – 10,000 m)
(Capillary forces)
(Viscous, capillary and gravity forces)
(Viscous and gravity forces)

4. Traditional ‘static’ linking of length-scales
Pore
Scale
Core
Scale
Laboratory
Upscaling
Core
Scale
Field
Scale
-Predictions depend on the applied boundary conditions
-Performed separately

5. New ‘dynamic’ linking of length-scales
Simulations at different scales
are performed simultaneously
Capillary forces
Network simulation
(relative permeability
estimates)
Viscous, capillary and gravity forces
Finite difference simulation
Viscous and gravity forces Streamline simulation
Up-scaling
Up-scaling

6. Objectives To build a tool able to:
simulate water injection in a 3D heterogeneous medium with 3 phases present
up-scale reservoir properties
Up-scaling

7. Dual Mesh Method? Fine Mesh Coarse Mesh prop PROP (up-scaled)
High CPU Time
Lots of Memory
Whole simulation on a coarse mesh with up-scaled properties

8. Error Precision

9. Dual Mesh Method
Instead of performing the simulation on the coarse mesh with averaged properties, try to find a way to keep the fine mesh information during the simulation.

10. How? IMPES method is used to perform water flooding simulation.
Pressure field solved during IMPLICIT STEP
(high cpu time and lots of memory)
COARSE MESH
Saturation update performed during EXPLICIT STEP
FINE MESH

11. DMM in IMPES method Fine Mesh Initial properties Absolute Permeability
Porosity
Saturation
Viscosity …

12. DMM in IMPES method Fine Mesh fluid Initial properties
Absolute Permeability
Porosity
Saturation
Viscosity …
Boundary conditions
Wells
Injection rate
Bottom hole pressure
water

13. DMM in IMPES method
Fine Mesh
Coarse Mesh
Up-scaling

14. DMM in IMPES method Coarse Mesh
Pressure field is solved using implicit method on the coarse mesh

15. DMM in IMPES method Coarse Mesh
From this pressure field and the up-scaled properties can be deduced a “coarse velocity field”

16. DMM in IMPES method From coarse to fine grid Coarse Mesh
Coarse pressure value as a Dirichlet condition on the centre of the coarse cell
Coarse flux weighted by interblock transmissivities on the fine cell faces

17. DMM in IMPES method Coarse Mesh
The pressure field at the fine scale is built coarse grid block by coarse grid block

18. DMM in IMPES method Fine Mesh
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 (red).
Inside each coarse grid block, the flux across each fine cell face is deduced from the pressure gradient (blue).
Mass balance is ok

19. Up-scaling 1. Pressure solver method (nested*): No flux
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 (dmm**):
perform on the product between absolute permeability and total mobility
No flux
* Gautier et al., 1999
Coarse
Grid
block
Dir i
Pi (pressure drop)
** Verdiere and Guerillot, 1996
Teffi = Qcal / Pi

20. Test Cases Heterogeneous medium, 2D, 2 phases (water-oil) Production76
bars
Fine mesh:
30 by 30 cells
Absolute permeability:
100 mD (black)
1 mD (white)
Coarse mesh:
10 by 10 cells
Reservoir volume:
300 by 300 by 1 meters
5 m^3
/ day
Injection

21. Error precision

22. Saturation Field Comparison
After 300 days
Fine mesh
Coarse mesh

23. Saturation Field Comparison
After 300 days
Fine mesh
DMM mesh

24. Saturation Field Comparison
After 300 days
Fine mesh
Nested mesh

25. In 3D… Segregation induced by density difference between phases P1, S1
To define the upstream direction considering gravity effects, we can’t use the pressure drop of each phase because the pressure field is reconstructed inside each coarse grid block without considering continuity with the next coarse grid block. The continuity at the fine mesh is respected according the total velocity field only.
So, we decided to use a mid-saturation point to define each phase velocity and to choose the upstream direction in the model
P1, S1
(S1+S2)/2
Vtotal
P2,S2

26. In 3D… Wells boundary conditions Coarse: Fine: Qwc, Twc Qwf , Twf
We used the same method as Gautier et. al, 1999.
The flux from the well is redistributed inside each coarse grid block weighted by the well transmissivity defined by Peaceman 1978
Fine:
Qwf , Twf
Coarse:
Qwc, Twc

27. Test Case 2D vertical Fine: 60 * 60 Coarse: 10 * 10
injection
production
Fine: 60 * 60
Coarse: 10 * 10
Injection on the left, production on the right
Red mD
White 100 mD
Permeability field

28. Test Case 2D vertical

29. Test Case 2D vertical Fine Coarse 1 At the end of the simulation
Saturation
At the end of the simulation 1

30. Test Case 2D vertical Fine Nested 1 At the end of the simulation
Saturation 1
At the end of the simulation

31. Test 3D Fine: 30 * 30 * 30 Coarse: 10 * 10 * 10
Heterogeneous permeability field
White = 100 mD
Red = 0.1 mD
Gravity effects, (Ng = 0.7)
No capillary pressure
Wells boundary
Injection
Production

32. Test 3D

33. Test 3D
Fine
Coarse
Saturation 1
Nb:Z*10

34. Test 3D
Fine
Nested
Saturation 1
Nb:Z*10

35. Test 3D
Fine
DMM
Saturation 1
Nb:Z*10

36. Time of Calculation Speed Factor: time(fine) / time(method)
( > 1 ?),
Gautier et al, 1999,Verdiere and Guerillot, 1996,
from 1 to 10
Method \ Dimension 2D 2Dv 3D
Fine mesh 30*30 60*60 30*30*30
Coarse mesh 10*10 10*10 10*10*10
DMM
NESTED
COARSE

37. Conclusions
Dual Mesh Method has been implemented in 3D with gravity effects and wells boundary conditions.
This method is able to reduce considerably the error precision between a conventional simulation performed on the chosen coarse mesh and the one performed on the initial fine mesh.
Different up-scaling techniques can be applied.
The time of calculations is reduced compared with the simulation performed on the fine mesh (for a “complex” enough problem).

38. Future work 3 phases (gas) Capillary effects
Grid block selection for the up-scaling step
Link this work to the pore scale model using relative permeabilities