### 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 90276^{1} 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. |