The aim of the 10th SPE Comparative Solution Project is to compare upgridding and upscaling approaches and the ability to predict performance of a waterflood through a million-cell geological model. The second dataset in the project is part of a Brent sequence described on a regular Cartesian grid with 60 x 220 x 85 (1,122,000) cells. The model consists of two formations: a shallow-marine Tarbert formation in the top 35 layers, where the permeability is relatively smooth, and a fluivial Upper-Ness permeability in the bottom 50 layers. Both formations are characterised by large permeability variations, 8--12 orders of magnitude, but are qualitatively different, as can be seen from Figure 1, where the model has been turned upside-down to emphasize the most heterogeneous structure in the Upper-Ness formation. The porosity field is strongly correlated to the permeability, and about 2.5% of the blocks have zero porosity and are therefore considered to be inactive.
The reservoir is produced using a water drive from a vertical well in the centre of the reservoir with a constant constant injection rate of 5000 stb/d and produced from four vertical wells at the corners, each at a bottom-hole pressure of 4000 psi. We simulated 2000 days of production assuming incompressible flow and using 25 time steps, as reported by StreamSim.
Figure 2 shows a comparison of water cuts for the fine-grid reference solution and our results obtained using two different versions of our method on a coarse grid with 10 x 22 x 5 cells. For the least accurate method (using local boundary conditions when generating basis functions), the accuracy is at least as good as the pseudo and nonpseudo upscaling results reported in the benchmark. Using global boundary conditions, the results give an almost perfect match.
The comparison is not entirely fair, since we use the subscale velocities obtained in the multiscale method to simulate transport directly on the fine grid. However, it gives a strong indication of the potential accuracy of the multiscale method. In particular, the method is very robust with respect to coarse grid, which is demonstrated in Figure 3. Here we have picked a horizontal slice of the Upper-Ness formation and compare saturation profiles obtained with transport solvers using the coarse-grid and the subgrid fluxes, respectively. By using only the coarse-grid fluxes, the multiscale method can be viewed as some kind of local-global upscaling procedure.
Fig. 1: Logarithm of horizontal and vertical permeability seen from below.
Fig. 2: Water cut for fine-grid and multiscale solution.
Fig. 3: Comparison of saturation profiles obtained using the coarse-grid fluxes (left column) and subgrid fluxes (right column) for the transport calculation.
Published April 17, 2008
A portfolio of strategic research projects funded by the Research Council of Norway