SPE 90276

Claims have been made in the literature of the superiority of higher-order finite-element (FE) methods over finite difference (FD) methods for compositional reservoir simulation.  The authors of SPE 90276¹ claim  "Even in a structured grid, the proposed algorithm is far superior to the FD schemes, as we will see shortly.”  All example problems given eliminate the normal levels of physical dispersion (mixing) in real reservoirs due to multidimensional, multiphase flow, residual saturations, heterogeneity, and gravity and capillary forces.  The example problems given in SPE 90276  showing multiple compositional fronts for highly theoretical 1D and 2D areal cases do not exist in real reservoirs, because the large level of true physical dispersion (compared to numerical dispersion, or numerical discretization error) prevents their formation.  Simple, theoretical example problems are highly desirable in demonstrations, but only if their solutions are significant with respect to real problems.  Some of the examples do demonstrate reduced numerical dispersion given by higher order methods, but it is not significant in real cases having normal levels of physical dispersion.  That is why there are no realistic 3D flow problems that have been used in the literature to show any advantages of FE over FD models, such as the most basic (or modified) industry compositional benchmarks, SPE3 or SPE5 (see our Benchmarks page).

Sensor data and results for SPE 90276 Example 1a, using 50 and 500 cells, are spe90276_1a_50.dat, spe90276_1a_50.out, spe90276_1a_500.dat, and spe90276_1a_500.out.  Cpu times for the 50 and 500 cell cases are 0.02s and 0.4s, respectively.   Search the .out files for the final printout of 1D MAP at time 40 days (.68 pv injected).  The fronts in both bases are little more than one cell wide (1m in 50 cell case, .1m in 500 cell case).  These results are as good or better than the FE 50 cell cases reported, and run in a fraction of the cpu time.  Reported FE model timing for the 1a 50 cell case is .37s.  According to our benchmarks, our smaller cases were running about twice as long 10 years ago as they run today.  So the Sensor 50 cell case is about 10 times more efficient than the reported FE model solution.  Almost any desired amount of numerical dispersion can be introduced into the FD model solution through selection of timestep size.  The Sensor cases use a timestep size computed to give exactly 1 block pv injected per timestep.

Any claimed advantages of the use of unstructured grids in any model should include demonstration of the correct solution to the unidirectional flow problem, completely independent of the value(s) of Ky (using values of 0 and effectively-infinite) , given here.  It has never been demonstrated for other than a Cartesian grid.  Unstructured grids can cause high numerical dispersion, as indicated by any sensitivity of the solution to Ky (level of anisotropy) in that example.  An upscaling solution should also be included (no averaging technique is sufficient, and flow based upscaling is impossible because the coarse and fine-scale block boundaries do not coincide - see Gridding and Upscaling).

¹ Hoteit, H. and Firoozabadi, A., "Compostional Modeling by the Combined Discontinous Galerkin and Mixed Methods", SPE Journal, March 2006, p. 19-34.