Multi-Scale Finite-Volume (MSFV) method for elliptic problems

1. Multi-Scale Finite-Volume (MSFV) method for elliptic problems Subsurface flow simulation Mark van Kraaij, CASA Seminar Wednesday 13 April 2005

2. Overview • Introduction • Flow problem • Solution method (MSFV) • Numerical results • Conclusions Multi-scale finite-volume method for elliptic problems in subsurface flow simulation P.Jenny, S.H.Lee, H.A. Tchelepi Journal of Computational Physics 187, 47-67 (2003) A multiscale finite element method for elliptic problems in composite materials and porous media Thomas Y. Hou and Xiao-Huis Wu Journal of Computational Physics 134, 169-189 (1997)

3. L e Introduction Flow problem with different scales Problem The level of detail exceeds computational capability Goal Obtain the large scale solution accurately and efficiently without resolving the small scale details

4. Flow problem Incompressible flow in porous media mobility permeability tensor fluid viscosity pressure source term flux velocity outward normal

5. Solution methods • Homogenization/Upscaling (First four presentations by Yves, Miguel, Heike and Matthias) • Periodicity restrictions • Solving problems with many separate scales is expensive • Multi-scale approaches (Last two presentations by Nico and Mark) • Random coefficients on fine grid • Solving problems with continuous scales is no problem

6. Multi-scale approaches • Multi-Scale Finite Element Method • Homogeneous elliptic problems with special oscillatory boundary conditions on each element • Small-scale influence captured with basis functions • Small-scale information brought to large scales through the coupling of the global stiffness matrix

7. Multi-Scale Finite-Volume (MSFV) • Based on ideas from Flux-Continuous Finite Difference and Finite Element Method • Allows for computing effective coarse-scale transmissibilities • Conservative at the coarse and fine scales • Computationally efficient and well suited for massively parallel computation

8. Finite-volume formulation • Partition domain into smaller rectangular volumes , i.e. the coarse grid Challenge Find a good approximation for the flux at that captures the small scale information for each volume

9. In general the flux is expressed as a linear combination of the pressure values at the coarse grid with the effective transmissibilities • By definition, the fluxes are continuous across the interfaces and as a result the finite-volume method is conservativeat the coarse grid

10. 3 4 1 2 3 4 1 2 Construction of transmissibilities • Construct a dual grid by connecting the mid-points of four adjacent grid-blocks • Define four local elliptic problems • Solutions are the dual basis functions for

11. 3 4 1 2 • Pressure field within can be obtained as a function of the coarse-volume pressure values by super- position of the dual basis functions • Compute effective transmissibilities by assembling integral flux contri- butions across volume interfaces

12. Construction of fine-scale velocity field • Dual basis functions cannot be used to reconstruct fine-scale velocity field because of • large errors in divergence field • violation local mass balance • A second set of local fine-scale basis functions is constructed that is • consistent with fluxes across volume interfaces • conservative with respect to fine-scale velocity field

13. 7 8 9 C D 4 5 6 A B 1 2 3 • Focus on mass balance in : • Define nine local elliptic problems with prescribed flux on derived from pressure field (take ) • Solutions are the fine-scale basis functions for Coarse grid (bold solid lines) Dual grid (bold dashed lines) Underlying fine grid (fine dotted lines)

14. Fine-scale pressure field within can be obtained as a function of the coarse-volume pressure values by superposition of the fine-scale basis functions • Compute conservative fine-scale velocity field from fine-scale pressure and permeability field

15. Compute 2nd set of fine-scale basis functions: Solve finite volume problem on coarse grid: Reconstruct fine-scale velocity field in (part of) the domain: Compute transmissibilities from 1st set of basis functions: Computational efficiency # adjacent coarse volumes to a coarse node # adjacent coarse volumes to a coarse volume CPU time to solve linear system with n unknowns CPU time for one multiplication # volumes fine grid # volumes coarse grid # nodes coarse grid # time steps

16. Example: fine grid coarse grid

17. Numerical results Configuration Injection rate = −1 Production rate = +1 Tracer particles at initial time

18. MS solution on a 5x5 coarse grid (reconstructed fine-scale velocity field not divergence free!) Permeability field Fine solution on 30x30 fine grid MS solution on 5x5 coarse grid 1. Random permeability field with random variable equally distributed between 0 and 1

19. Permeability field Fine solution on 30x30 fine grid MS solution on 5x5 coarse grid 2. Permeability field with isotropic correlation structure Geostatistically generated permeability field with and of . Correlation lengths: .

20. Permeability field Fine solution on 30x30 fine grid Fine solution on 30x30 fine grid MS solution on 5x5 coarse grid MS solution on 5x5 coarse grid 3. Permeability field with anisotropic correlation structure Geostatistically generated permeability field with and of . Correlation lengths: .

21. Conclusions • Multi-Scale Finite-Volume (MSFV) method for elliptic problems describing flow in porous media • Conservative on coarse and fine grid • Transmissibilities account for the fine-scale effects • Parallel computations Possible extensions • Unstructured grids (oversampling technique) • Multi-phase flow (saturation)