NTNU

1. NTNU Author: Professor Jon Kleppe Assistant producers: Farrokh Shoaei Khayyam Farzullayev

2. Contineous and discrete systems • Continuous system: • Discrete system:

3. i-1 i i+1 Dx i-1 i-1 i i i+1 i+1 Dx Constant grid block sizes • Forward expansion of pressure: • Backward expansions of pressure: • For i=2 to N-1: Summation of forward and backward equations give us:

4. PL 1 2 Dx N-1 N PR Dx • For i=1: Eq. I • Solving forward equation and Eq. I give us: • For i=N: Eq. II • Solving backward equation and Eq. II give us:

5. i-1 i i+1 Dxi-1 Dx Dxi+1 Variable grid block sizes • Finer description of geometry • Better accuracy in areas of rapid changes in pressures and saturations • Specially useful in the neighborhood of production and injection wells • More realistic grid system • The Taylor expansions:

6. The flow term: Where: • For i=2 to N-1: • Due to the different block sizes, the error terms are of first order only. • Flow equation for i=1 and i=N depends to boundary conditions.

7. PL 1 2 Dx1 Dx2 • Boundary conditions • Pressure condition at the sides of slab: • For i=1: • For i=N: • Same for pressure at the right hand side:

8. QL 1 2 Dx1 Dx2 • Flow rate specified at the sides of slab: • For flow rate at the left hand side: • Same for flow rate at the right hand side:

9. Time discretization • Expansion forward: • Expansion backward:

10. i-1 i i+1 t + Δt i-1 i i+1 t Dx Numerical formulations • Explicit method: • Use the forward approximation of the time derivative at time level t. • The left hand side is also at time level t. • Solve for pressures explicitly. • All parameters except pi at (t+Δt) are in time t and are known , so simply by solving equation you can find pi at (t+Δt). This formulation has limited stability, and is therefore seldom used.

11. i-1 i i+1 t + Δt i-1 i i+1 t Dx • Implicit method: • Use the backward approximation of the time derivative at time level t+Δt. • The left hand side is also at time level t+Δt. • Solve for pressures implicitly. • A set of N equations with N unknowns, which must be solved simultaneously. • For instanceusing the Gaussian elimination method. This formulation is unconditionally stable.

12. Crank-Nicholson method: • Use the central approximation of the time derivative at time level t+Δt / 2. • The left hand side is also at time level t+Δt / 2. • The resulting set of linear equations may be solved simultaneously just as in the implicit case. • The formulation is unconditionally stable, but may exhibit oscillatory behavior, and is seldom used.

13. 10 20 40 100 Sensitivity to number of grid blocks 1 i 10 • 10 grid blocks: 1 i 20 • 20 grid blocks: 1 i 40 • 40 grid blocks: 1 i 100 • 100 grid blocks: 1 1-Sor SW SWir 0 X / L 0 1 • The more grid blocks we have, the smaller are the blocks sizes (Δx), smaller is the numerical dispersion because the discretization error is proportional to Δx2 .

14. Δt = 20 sec Δt = 10 sec Δt = 1 sec Sensitivity to time step 1-Sor SW SWir X / L • The smaller are the time steps (Δt), the smaller is the numerical dispersion due to the discretization where the error is proportional to Δt.

15. production injection 0.7 bar 0.01 bar 10 cm injection production 21 bar 0.7 bar 150 m Capillary and viscous forces • pressure difference across a grid block will be directly proportional to the size of this ( in the flow direction ). • The direct effect of capillary pressure will therefore often be dominating in a core-sized grid block while it is negligible in a full field simulation formation scale grid block. Capillary forces (capillary endpoint pressure) Viscous forces (viscous pressure drop) • Core plug: • Simulation grid block:

16. Upscaling • Geological models may contain millions of grid blocks representing geologically interpolated data (geostatistical realizations). • Numerical simulators cannot handle this level of detail due to cost limitations (applicable with less than millions of grid blocks). • The magnitude of the difference between fine and coarse scales is very significant. • The key problem is how to obtain effective input for the numerical flow simulator from data on finer scales. • This process is called upscaling.

17. High permiability Low permiability • Laminar sclae grid blocks: 1.5 * 0.5 m core plug • Formation scale grid blocks: 1.5 * 0.5 m • Formation scale grid blocks: 12 * 2.5 m • Formation scale grid blocks: 60 * 5 m

18. Fine grids and coarse grids: • Fine grids: • Coarse grids:

19. Fij • Permeability tensor: Cij • The permeability tensor of a porous medium is specified on each fine-scale cell Fij, and must be upscaled or homogenized over each coarse-scale or computational cell Cij

20. Quality control: Fine grid model with original relative permeability Coarse grid model with upscaled relative permeability Recovery Time • Every single upscaling step is quality controlled during the upscaling by comparing recovery from the fine model with the recovery from the upscaled coarse gridded model incorporating the pseudo  curves.

21. Questions 1. Use Taylor series to derive the following approximations (include error terms): • Forward approximation of b) Backward approximation of c) Central approximation of • (constant Dx) • (variable Dx) d) Central approximation of 2. Modify the approximation for grid block 1, if the left side of the grid block is maintained at a constant pressure, PL. 3. Modify the approximation for grid block 1, if grid block is subjected to a constant flow rate, QL. 4. Write the discretized equation on implicit and explicit forms.

22. References • Kleppe J.: Reservoir Simulation, Lecture note 3 • EPS

23. About this module • Title: DISCRETIZATION AND GRID BLOCKS (PDF) • Author: • Name: Prof. Jon Kleppe • Address: NTNU S.P. Andersensvei 15A 7491 Trondheim • Website • Email • Size: 660 KB