Page 20 - Transmissibility Corrections and Grid Control for Shale Gas Numerical Simulation

VA - DF: Transmissibility Corrections and Grid Control for Shale Gas Numerical Simulation p 19/23
Above, k and are the porosity and permeability of the formation, is the fluid viscosity and ct
is the total compressibility of the system.
We implemented this solution, taking half the investigation radius as the guess for the size of
first cells. As can be seen from Figure 22, the results are quite consistent with the
expectations.
Figure 22 – Transient numerical results for various ‘time resolution’ choices,
with the automatic grid setting procedure – Linear PVT
One can notice that the curve obtained with the 1hr choice for the resolution even matched the
ultrafine grid much earlier. This is because we want to ensure the conformity of fine and
coarse grids. As a consequence, we do not have full flexibility on the first cell size, and the
safest choice is taken. In this example, the finest grid contained 22442 cells, against 13742
cells for the 1hr resolution grid. The 100hr resolution grid involved 9542 cells.
These very robust results were obtained because the PVT used was linear. If we use a real,
non-linear gas PVT and produce with a large pressure drop (from 5000 to 250 psi), the results
are still qualitatively acceptable. However, the effective resolution of the simulation is
somewhat lower than expected (Figure 23). This can be explained by the fact that the
compressibility and viscosity of the fluid change significantly within the space occupied by the
first cells. This effect cannot be corrected by the upstream scheme. Hence, the pressure at
which compressibility and viscosity are evaluated when deriving the investigation radius should
be carefully chosen, as some average values are not sufficient. As shown on Figure 23, the
consequences are not dramatic with gas, even with a large pressure drop. When dealing with
shale oil, however, one should ensure that the viscosity of the oil does not encounter severe
variation within the pressure range corresponding to the first cells, otherwise the automatic
refinement may not be suited, and even the cumulative may start deviating from the fine
simulation.
In this study, we limited our analysis to very simple assumptions regarding the geometry of
the fractures. For this reason, we didn’t encounter any relevant inaccuracy problem associated
with the coexistence of gridblocks of very different volumes (fracture vs. matrix blocks). This,
however, may become more problematic when we extend this work to the simulation of
natural networks of fractures [11]. In this case, other simple quality control indicators may
have to be derived.
1E-4 1E-3 0.01 0.1 1 10 100 1000
Time [hr]
1
10
100
1000
Pressure [psi]
Ultra-fine grid
t=10hr
t=1hr
t=100hr