Page 18 - Transmissibility Corrections and Grid Control for Shale Gas Numerical Simulation

VA - DF: Transmissibility Corrections and Grid Control for Shale Gas Numerical Simulation p 17/23
Figure 19 – Cumulative production error (relative to final reference cumulative)
with and without transmissibility corrections versus time.
5. Grid refinement control
Analysing Figure 5, we saw that the extremely low value of the permeability in shale gas
context enforces the use of a very fine grid in order to correctly simulate early-time transient
effects. In many cases, however, the early-time behaviour is not essential to the interpretation
and a coarser grid may be sufficient, as long as later time scales are correctly captured. This
can be properly addressed with numerical near-well upscaling [9, 10].
Using this methodology, the well index and the transmissibility values of the coarse grid are
corrected using the results of a steady-state, fine-grid simulation. Figure 20 shows the
resulting cumulative obtained for various grid sizes, for the example of Table 1, with k=1e-4
mD and linear PVT. We see that the overall process is extremely robust, so forecasts match
perfectly. Note that in this example, the gain on the number of cells is reached only by
coarsening the size of the first rings of cells around the fractures (from millimetres to meters
scale) without changing the back grid. Note that the excellent match at late-time is not a
surprise since the original field was homogeneous. However, the overall procedure also gives
very good results at late time, even with near-well heterogeneity.
On the loglog plot (Figure 21), we see that early times are lost after the upscaling process.
This is logical: while the grid is coarsened, near-fracture cells become too large to capture
transient effects correctly. We see that, however, after a given time which depends on the
coarsening level, any curve nicely comes back on the derivative of the finest, reference
simulation. What is important to notice here is that even for a mild coarsening level, the
correct answer is captured only after 100 hr, because of the very low permeability. For a
conventional permeability value, it would only be a matter of minutes even with a larger
upscaling level. This raises a practical problem, as one would like to choose the best
coarsening level depending on the desired time resolution of its analysis, i.e. without always
simulating on the finest grid.