1. PETE 603 Lecture Session #30 Thursday, 7/29/10

2. 30.1 Accuracy of Solutions Material Balance Error Nonlinear Error Instability Error Truncation Error Roundoff Error Numerical Dispersion Grid Orientation

3. 30.2 Material Balance Error Over a single timestep Over entire simulation

4. 30.3 Material Balance Error Causes of material balance error: - non-conservative equation formulation - error in solution of nonlinear equations (Newton-Raphson) - error in matrix solutions - roundoff errors (numerical precision) - data errors (negative compressibilities) - program bugs

5. 30.4 Nonlinear Error Interpolating PVT functions linearly is a source of error. Data points in PVT table Pressure PVT quantity True PVT data

6. 30.5 Nonlinear Error Nonlinear errors are especially a problem near the bubblepoint due to discontinuity in total compressibility. Chord slopes in IMPES Solutions: –take smaller timesteps –iterate on chord slope (IMPES)

7. 30.6 Instability Error SwSw Time If a simulation run becomes unstable erroneous saturations will occur.

8. 30.7 Instability Error Caused by taking timesteps which are too large in an IMPES method. Solution: –Take smaller timesteps –Use fully implicit method

9. 30.8 Truncation Error Truncating the Taylor series expansions is a source of error. The size of this error depends on the grid size and the timestep.

10. 30.9 Truncation Error Refining the grid and/or reducing the timestep reduces the truncation error.

11. 30.10 Numerical Dispersion Numerical dispersion causes fluid fronts to arrive sooner than they should. Fronts that should be sharp become smeared.

12. 30.11 Numerical Dispersion Solution: –Use smaller gridblocks –Upstream relative permeabilities also help minimize numerical dispersion. Multipoint upstream sometimes used in tracer studies –Use pseudorelative permeabilities (pseudofunctions) –Choose timestep wisely in IMPES (maximum stable timestep)

13. 30.12 Numerical Dispersion Do we want to remove numerical dispersion altogether? Buckley-Leverett flow does not apply for many field cases. –Capillary pressure spreads out fluid front, especially in low permeability reservoirs. –Permeability heterogeneity smears fluid front (fingering). Some numerical dispersion may be acceptable.

14. 30.13 Grid Orientation The orientation of the grid with respect to the well locations can influence the simulation results. Fluid moves preferentially along the grid lines. If the red dot represented a water injector, the water would break through fastest in the diagonal grid (right) - even if the grid dimensions were the same.

15. 30.14 Grid Orientation To overcome the grid orientation effect a different form of the finite-difference method can be used. The usual method is called the “five-point” approach. The “nine-point” approach to the flow equations removes the grid orientation effect, however more computational work is required. The “Control Volume Finite Element” approach often uses triangular gridblocks (full permeability tensor required).

16. 30.15 Grid Orientation

17. 30.16 Grid Orientation

18. 30.17 Data Modification - 5 spot Injector Producer

19. 30.18 Data Modification - 5 spot Simulator: Quarter 5-spot

20. 30.19 Data Modification - 5 spot Data Modification: Quarter 5-spot

21. 30.20 Data Modification - 5 spot Final Result: Quarter 5-spot