SPE10 and Upscaling

Also see:

•   SPE10, Capillary Pressure, and Streamlines

•   The Importance of Capillary Pressure Inclusion and Accuracy

•  Gridding and Upscaling

Model 2 of the Tenth SPE Comparative Solution Project^1,2 was designed to compare the ability of upscaling approaches used by various participants to predict the performance of a waterflood in a simple but highly heterogeneous black oil reservoir that is described by a fine-scale 1.1 million cell (60x220x85) Cartesian geological model.  The problem statement specified that the intent and the basis for competition was to "compare accuracy of solution with cost, which will be measured by the size of the coarse model rather than by cpu time" ¹.  But the published paper² gave no discussion or conclusions regarding that comparison, and overlooked what were by far the most significant results that were submitted.

SPE10 Model 2 participants and solutions were:

At the time of the project, the fine-grid 1.1 million cell geological model required too much memory to run serially on available (32 bit) hardware.  Five of the participants were able to run the fine-grid model, Landmark and Chevron using their parallel finite difference models, and GeoQuest, Streamsim, and Total using streamline models.  All fine-grid solutions showed very good agreement, and the Landmark fine-grid solution was chosen as the reference for comparison of the upscaled solutions.

Since we were not able to run the fine-grid model, we first used conventional single-phase flow-based upscaling by a factor of 2 in the x-direction and 4 in the y-direction to obtain a 30x55x85 'intermediate' grid, results from which were considered to be correct for use in further flow-based upscaling to10x20x10 and 3x5x5 coarse grids.  We applied pseudo water relative permeabilities to our 2000 and 75 cell upscaled cases in matching the intermediate solution, with reported adjusted values of Nw = 1.28 and Nw=1.2, respectively, where Nw is the water relative permeability exponent.  Our solutions are characterized by this single parameter (which is possible in this relatively simple case because of the single rock type using relative permeability functions), and by well PI adjustments in the coarse grids to match the intermediate solution production well PI ratios and field average pressure (described in data files given below).   The Phillips solution was almost as simple as ours (they used single-phase flow-based upscaling and adjusted both the water and oil relperm exponents).  The other participants used relatively complex upscaling methods, which prevents a complete description of their solutions.

Although we submitted complete results for our 75 block case, only Producer 1 oil rate results were presented (in Fig. 15 of the paper).  The only discussion of our 75 block case in the paper is "Both Coats upscaled solutions using pseudo relative permeabilities provide good predictions of the fine-grid results".  The figures below present results for our three solutions (corresponding to those given for the others) along with the reference fine-grid solution.  Figure numbers are those of the paper.

Comparison of the first five figures below (field oil rate, well P1 oil rate, well P3 oil rate, well P1 cumulative oil, and well P3 water cut) with those from the paper indicates that all three of our upscaled solutions are here about as good as or better than any of the others.

For well P1 water cut (the 6^th figure below), it appears that four of the participants (using from 1 to 3 orders of magnitude more cells) obtained a better match than our 75 cell solution, but these watercut figures are somewhat misleading.  Water rate might have been a better reporting variable.  For example, the Roxar (40,000 cells), Total (4,800 cells), and Geoquest (14,000 cells) P1 oil rate curves seem to have the largest errors of all participants, but the Roxar, Total, and Geoquest P1 water cut curves seem to give three of the four best matches.  These solutions have significant error in predicted water rate, particularly in the first half of the run, that is obscured through the choice of water cut as the reported variable, which reflects error in both oil rate and water rate.

Of all the results presented, the greatest deviations from fine-grid results were observed for field average pressure.  There is a good reason for that.  Average pressure is very sensitive to the upscaled values of the production well PI’s (values for the injector have little effect within reasonable range of adjustment due to small buildup pressure).  Because of the extreme permeability heterogeneity in this case (ten orders of magnitude variation), those are highly dependent on the choice of the upscaled grid (grouping of fine cells).  The match of field average pressure has little meaning or impact on production results in this simple case having nearly incompressible fluids with virtually no pressure dependence of properties. Landmark, Total, and Chevron obtained excellent matches of field average pressure.  All had the advantage of knowing the fine-grid solution.  If pressure behavior is known, it is a fairly simple matter to adjust the well PI’s to match it, as doing so does not have a significant effect on production (we neglected to make this adjustment in our 75 cell case, which is using the tuned 2000 cell case value of .7 for the global PI multiplier - use of .6 instead results in a good match of average pressure with our other 2 cases without affecting production).

So, how did accuracy of the solutions compare with cost, as measured by the size of the coarse model?  No other solution came close to comparing with our 75 cell case.  All of the others used 1 to 3 orders of magnitude more cells, and were either less accurate, or showed little or no improvement in accuracy.  And what if we added complexity of solution to the cost basis?  What if we added total time to solution, or run cpu time?  Our 75 cell solution runs in 0.06 seconds on our old (2004) 2.8 GHz desktop.  Our 2000 cell solution runs in 1.5 seconds.  The intermediate case runs in about 45 minutes.

Conclusions

1. Conventional single-phase flow-based upscaling of up to an order of magnitude with no pseudoization (our 'intermediate' 141,000 cell case) can accurately reproduce production predicted by the fine-scale model.

2. A very coarse model (our 75 cell case) , with over four orders of magnitude less cells and only four orders of magnitude permeability variation, constructed through conventional single-phase flow-based upscaling and pseudoization techniques, can accurately reproduce production predicted by the extremely heterogeneous fine-scale model having ten orders of magnitude permeability variation.

3. In terms of the intended basis of comparison, the Coats Engineering 75 cell solution is the clear winner, by an order of magnitude.

4. Here, as is usually the case, the simplest approach that is sufficient is by far the most efficient.

Observation

Any upscaled solution that must be tuned using the results of a run or runs made on a finer grid in order to adequately match the finer grid production behavior has value in practice only if it also adequately matches the finer grid behavior under other operating conditions or well placements or other specifications within the range of intended investigation.  It might be interesting to see if the results of some optimizations using the upscaled models, with no further tuning, would be sufficiently accurate.

Sensor Data and Output Files

30x55x85      spe10_case2.dat             spe10_case2.inc                spe10_case2.out

10x20x10      spe10_case2_2000.dat  spe10_case2_2000.inc     spe10_case2_2000.out

3x5x5             spe10_case2_75.dat                                                     spe10_case2_75.out

(For the .inc files, please remove the added .dat extension after downloading)

1.   http://www.spe.org/csp/

2.  Christie, M.A., and Blunt, M.J., "Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques", SPE Reservoir Engineering and Evaluation, 4, 308-317, (2001).