SPE10 and Upscaling
Also see:
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 finescale
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 paper^{2} 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:
Company 
Model 
Grid 
Cells 
Chevron 
CHEARS 
finegrid 
1,100,000 


22x76x42 
70,000 




Coats Engineering 
Sensor 
30x55x85 
141,000 


10x20x10 
2,000 


3x5x5 
75 




GeoQuest 
Frontsim 
finegrid 
1,100,000 

Eclipse 
15x55x17 
14,000 




Landmark 
VIP 
finegrid 
1,100,000 


5x11x17 
935 




Phillips Petroleum 
Sensor 
11x19x11 
2,300 




Roxar 
Nextwell 
15x55x22 
40,000 


(plus lgr's) 





Streamsim 
3DSL 
finegrid 
1,100,000 


30x110x85 
280,000 


60x220x17 
224,000 


30x110x17 
56,000 


12x44x17 
9,000 




TotalFinaElf 
3DSL 
finegrid 
1,100,000 

Eclipse 
10x37x13 
4,800 
At the time of the project, the finegrid 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 finegrid model, Landmark and Chevron using their parallel finite difference
models, and GeoQuest, Streamsim, and Total using streamline models.
All finegrid solutions showed very good agreement, and the Landmark finegrid solution was chosen as the reference for comparison of the upscaled
solutions.
Since we were not able to run the finegrid model, we
first used conventional singlephase flowbased upscaling by a factor of 2
in the xdirection and 4 in the ydirection to obtain a 30x55x85
'intermediate' grid, results from which were considered to be correct for
use in further flowbased 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 singlephase flowbased 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 finegrid results". The figures
below present results for our three solutions (corresponding to those given
for the others) along with the reference finegrid 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 finegrid 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 finegrid 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

Conventional singlephase flowbased
upscaling of up to an order of magnitude with no pseudoization (our
'intermediate' 141,000 cell case) can
accurately reproduce production predicted by the finescale model.

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 singlephase flowbased upscaling and pseudoization
techniques, can accurately reproduce production predicted by the
extremely heterogeneous finescale model having ten orders of magnitude
permeability variation.

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

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,
308317, (2001). 