1
A018 A NEW WAY OF LOOKING AT UPSCALING
G.E. Pickup, H. Monfared, P. Zhang and M.A. Christie
Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh, EH14 4As, Scotland, UK.
Abstract
The current trend in reservoir simulation, is to generate a large number of models (sometimes
many thousands) in order to investigate the effects of uncertainty in model parameters. For
speed, often only coarse-scale models are created. However, these may over-look the effects of
finer-scale structure. On the other hand, generating fine-scale models presents problems because
it is time-consuming, and the models have to be upscaled to reduce the number of cells for flow
simulation.
Upscaling in a heterogeneous model is often inaccurate. Usually only single-phase upscaling is
performed, although often we are concerned with two phases (e.g. a water flood). The effect of
upscaling is to narrow the probability density function (pdf) of the permeability distribution. As
a result, the dispersion of the flood front due to heterogeneities is reduced. A coarse-scale flow
simulation will therefore tend to have a sharper breakthrough than a fine-scale one. (Although
numerical diffusion at the coarse-scale will counteract this.) In addition, the mode of the pdf
distribution at the coarse-scale may not match that of the fine-scale model.
We have used a new approach to analyse the effects of upscaling and characterising the errors
which arise. Coarse-scale models were generated using two alternative approaches. The first
was a conventional single-phase upscaling (pressure solution, no-flow boundary conditions), and
in the second procedure coarse-scale permeabilities were calculated by history-matching a twophase flow simulation. (In the history matching process, the misfit function used total oil
recovery and average pressure.) The standard deviations from history-matched coarse-scale
models are closer to the standard deviations of fine-scale models. These results suggest a new
alternative approach to upscaling and indicate that, in some cases, sampling from a geostatistical
description may be better than conventional upscaling.
Introduction
In order to investigate uncertainty more thoroughly, engineers have started to run large numbers
of flow simulations with a range of parameters. The models for these runs must be small (e.g. a
few tens of thousands of grid cells), so that the simulations are fast. If the coarse-scale models
are based on fine-scale models, a large upscale factor is necessary and the fine-scale detail is
lost. In additional during the history-matching process, the permeabilities may be altered
significantly, so that the final distribution is quite different from the original one. This suggests
that the effort expended in fine-scale modelling and upscaling is sometimes not worthwhile.
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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To overcome these problems, there has recently been a move to start by generating coarse-scale
models. This is reasonable, because at the start of field development the model is highly
uncertain. At a later stage, models may be refined as more data becomes available, and as more
detailed models are required for in-fill drilling or EOR.
The aim of this paper is to investigate the errors in upscaling. In addition to comparing fine- and
coarse-scale results, we also compare upscaled permeabilities with permeabilities calculated by
history matching. These investigations lead us to suggest a new way of generating coarse-scale
models without upscaling.
Background
Permeability upscaling in hydrocarbon reservoirs poses a considerable challenge to engineers,
because of the multiphase nature of the fluids. It is relatively easy to upscale absolute
permeability, and a variety of techniques ranging from simple averaging of the permeabilities to
pressure solution methods have been developd (e.g. Renard and Marsily, 1997). However, when
there are two phases flowing, the relative permeabilities should also be upscaled to take account
of the interaction of fine-scale heterogeneity and the fluid forces (viscous, capillary and gravity).
A number of methods have been developed for two-phase upscaling (see for example Christie,
2001). These include steady-state and dynamic techniques. Steady-state methods are feasible,
particularly the capillary equilibrium method, which is useful for upscaling small-scale
heterogeneity (Pickup and Stephen, 2000). On the other hand, dynamic methods, which should
be capable of dealing with any force balance, are time-consuming, and suffer from a number of
drawbacks, as discussed by Barker and Thibeau (1997). For this reason, engineers often only
use single-phase upscaling.
In this paper, we are interested in viscous-dominated floods. An injected fluid will tend to travel
faster in the high permeability regions, leading to a dispersion of the flood front (e.g. Zhang and
Tchelepi, 1999). The amount of dispersion will depend on the viscous cross-flow between high
and low permeability regions (Artus et al., 2004; Noetinger et al., 2004). If only single-phase
upscaling is applied, this dispersion will be lost, and breakthrough in the upscaled model will be
later and more rapid than in the fine-scale model, leading to an overestimate of the recovery.
Single-phase upscaling is reasonably accurate for small scale-up factors (Durlofsky, 2003), and
the accuracy may be improved by carefully choosing the coarse grid (e.g. Li et al., 1997;
Durlofsky et al.,1997). The results of the 10th SPE Comparative Solution Project on Upscaling
(Christie and Blunt, 2001), however, show that large errors may arise if only single-phase
upscaling is used with a large scale-up factor and a uniform coarse grid.
Upscaling Tests
We have carried out several sets of tests on single-phase upscaling. The first two sets of tests
were based on the 3D model used in the 10th SPE Comparative Solution Project (Christie and
Blunt, 2001). The third example consists of a set of 2D stochastic models with a range of
correlation lengths and standard deviations.
The model for the 10th SPE Comparative Solution Study model consists of 60 x 220 x 85 cells,
each of 20ft x 10ft x 2ft. The oil and water relative permeabilities were modelled using a Corey
exponent of 2 and the viscosities were µo = 3.0 and µw = 0.3, giving an endpoint mobility ratio
of 10 (unstable flood). There were 5 wells in the model, a water injector at the centre and a
producer in each corner. This model was upscaled, using a range of different scale-up factors.
3
The pressure solution method was used, with no-flow boundaries applied locally to each coarse
grid cell.
Single-Phase Tests
To test single-phase upscaling, we compared upscaling in a single step to upscaling in two
stages, as shown in Figure 1. If upscaling is accurate, we should obtain a similar answer. As a
measure of the accuracy of the upscaling, we defined the effective permeability ratio, EPR, as
2
, to the one-stage result, k1eff , as follows:
the ratio of the two-stage result, k eff
k2
EPR = eff
k1eff
(1)
fine-scale model
single cell
k1eff
k 2eff
Figure 1
Schematic diagram illustrating the method for testing single-phase upscaling.
Figure 2 shows the value of EPR for different coarse-scale models. NX, NY and NZ represent
the number of coarse-scale cells in the x-, y- and z-directions, respectively. One might expect
that EPR would approach 1 for small scale-up factors, i.e. as the number of cells in the coarse
model increases. This occurs in the cases where the scale-up factor in the z-direction was 5, (NZ
= 17), and when NY is large (Figure 2b). However, when the scale-up factor in the z-direction
was 17 (NZ = 5), the value of EPR tended to 2 as the NX increased (Figure 2a), suggesting that
some important detail had been lost.
b) NZ = 17
a) NZ = 5
1.4
2.5
1.2
2
1
EPR
EPR
1.5
1
NY=11
NY=22
NY=44
NY=55
NY=110
0.8
0.6
0.4
0.5
0.2
0
0
0
5
10
15
20
25
30
No. of Coarse Grid Cells in x-direction
35
0
5
10
15
20
25
30
35
No. of Coarse Grid Cells in x-direction
Figure 2
Effective permeability ratio (EPR) as a function of the number of coarse cells in the x-direction
(NX), for different values of NY. a) For NZ = 5, and b) for NZ = 17.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Two-Phase Tests
The next set of tests was carried out with two-phase flow. A waterflood was simulated in each
of the coarse-scale models. The results were compared with the fine-scale simulation performed
in the SPE 10 study (Christie and Blunt, 2001). In each case, we compared the fine- and the
coarse-scale results for total oil recovery and average field pressure. We computed the relative
RMS error in the total recovery as follows:
n
2
∑ (R ci − R fi )
RMS = i =1
n
÷ max(R fi ) × 100% ,
(2)
where R is the cumulative recovery. The subscripts c and f refer to “coarse” and “fine”, and i =
1, 2, ..n refers to time steps. Contour plots of the RMS error in recovery are shown in Figure 3,
in terms of the scale-up factors in x, y and z: SX, SY and SZ, respectively. As expected, the
errors tend to be larger for larger scale-up factors. However, in Figure 3b, it can be seen that the
largest errors are for intermediate values of the scale-up factor in y (SY).
One might assume that coarse models where the single-phase upscaling is reasonably accurate
(EPR ~ 1) would also reproduce the two-phase flow accurately. However, Figure 4 shows that
this is not the case. We have marked the models which gave a good prediction of total oil
recovery (FOPT) and average field pressure (FAP) on the EPR diagram, for models with NZ =
17, and NY = 11, 22 and 44. While some of the points with EPR close to one gave good
matches, others did not. On the other hand, some of the points with a good match to FOPT had
values of EPR significantly different from 1 (e.g. NY=11 and NX=30). The reason is because
single-phase upscaling is inadequate for two-phase systems, because it cannot reproduce the
fine-scale dispersion in the flood front.
a) SZ=5
b) SZ=17
20
10SY
20
4-5
13-14
3-4
10SY 12-13
2-3
11-12
1-2
10-11
0-1
10
6
5
4
SX
3
5
2
9-10
10
6
4
SX
3
5
2
Figure 3
Error in total oil recovery as a function of scale-up factor in the x- and y-directions. a) SZ = 5
and b) SZ = 17.
5
1.4
1.2
EPR
1
0.8
NY=11
NY=22
NY=44
FAP match
FOPT match
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
NX
Figure 4
The EPR diagram showing the upscaling factors which produced good matches with the finescale model.
Additional Tests
Additional tests were carried out using 2D stochastic models. The fine-scale models consisted of
64 x 64 cells, and had ln-normal permeability distributions, with standard deviations ranging
from 0.2 to 2.5, and with correlation lengths ranging from 0.0 to 0.3 (in terms of the system
length). A quarter 5-spot well pattern was simulated, with an injector and a producer at
diagonally opposite corners. The relative permeabilities for oil and water were generated using a
Corey exponent of 2, as in the SPE 10 model. The viscosities were µw = 0.3 for water and either
µo = 0.3, or 3.0 for oil, giving endpoint mobility ratios of 1 (stable) or 10 (unstable),
respectively.
The models were upscaled using different scale-up factors. Again, the pressure solution method
was used, with no-flow boundary conditions applied locally to each coarse grid cell. The RMS
error in recovery was calculated using Equation 2. A diagram of the errors as a function of
standard deviation and correlation length is shown in Figure 5, for an upscaling factor of 4. It
can be seen that the errors increase with standard deviation, and that the error for the unstable
case is much larger than for the stable case.
a) Diagonal, Stable
b) Diagonal, Unstable
0.3
Correl. Length
Correl. Length
0.3
0.2
0.1
0.0
0.0
0.5
1.0
1.5
2.0
0.2
0.1
0.0
0.0
2.5
0.5
Standard Deviation
1.0
1.5
2.0
2.5
Standard Deviation
Percentage Error
0.00
0.75
1.50
2.25
3.00
Figure 5
Errors in upscaling 2D models.
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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History Matching Coarse-Scale Permeabilities
As observed in the previous section, accurate single-phase upscaling does not necessarily
provide an accurate reproduction of two-phase flow. In this section, we describe a method for
obtaining coarse-scale permeabilities by history matching.
Methodology
In order to history-match the coarse model, we used the pilot-point method (e.g Cuypers et al.,
1998). In this method, a set of points (pilot points) is selected, and during the history matching
process, the absolute permeability is altered only at these points. The permeabilities of the
remaining cells are calculated by interpolation. The history matching was carried out using the
Neighbourhood Approximation (NA) algorithm (Christie and Subbey, 2002).
This is a
stochastic algorithm which identifies regions of parameter space which give good history
matches, and then preferentially samples in these regions to obtain better matches. The
algorithm uses Voronoi cells to represent the parameter space.
This approach was used to obtain a coarse-scale permeability distribution for a grid of 5 x 11 x 6
with 11 pilot points, distributed random throughout the coarse model. A summary of the
procedure is shown in Figure 6. We describe the steps in more detail below.
First, the spatial structure of the fine-scale model was characterised using the semivariogram.
The average semivariogram in each of the 6 coarse layers was calculated, and fitted to a
spherical model. These semivariograms were used in the kriging of the coarse-scale
permeabilities.
The starting point for the history match was the upscaled coarse model, described above. During
the history matching, permeability multipliers were applied at the pilot points to adjust the
horizontal permeability (kx and ky). The permeability in the z-direction was fixed, as was the
permeabilities at the wells. The same permeability multiplier was applied to each of the 6 layers.
The NA algorithm was run for 500 iterations. For the first iteration, 40 values of the multipliers
were chosen from a uniform distribution between 0.01 and 20. In subsequent iterations, 22
models were generated and 11 of the Voronoi cells were re-sampled (Christie and Subbey,
2002). A waterflood was simulated in each model, and the cumulative oil production and
average field pressure were compared with the fine-scale values, using the following misfit
function:
n
2
n
M = w1 ∑ ( FOPTf − FOPTc ) + w 2 ∑ ( FAPf − FAPc )
i =1
i =1
2
(3)
where w1 and w2 are weights which were adjusted to vary the relative influence of cumulative
recovery (FOPT) and average pressure (FAP). The current results used w1 = 1/30 and w2 = 1.
Although the NA algorithm generates multiple history-matched models, in this case we selected
a single model with the smallest misfit.
Results
Figure 7 shows a comparison of the cumulative oil recovery for the fine-scale model, the
upscaled model, the best history-matched model, and a kriged model, obtained by kriging the
upscaled values at the pilot points (before history matching was carried out). It can be seen that
both the upscaled model and the kriged model overestimate the recovery, but the historymatched model reproduces the fine-scale model very well.
7
NA
Algorithm
pilot
points
variogram
kriging
flow
simulation
new perm
field
misfit
Figure 6
Schematic diagram of history matching procedure.
A comparison of the probability density function for the fine, the upscaled and the historymatched models is shown in Figure 8. This graph is plotted in terms of ln(permeability). The
fine-scale distribution is bimodal, but this has been lost in the upscaling process and the upscaled
pdf is relatively narrow. However, the pdf for the best-matched model is broader, so there is
more dispersion in two-phase flow and the recovery curve is close to that of the fine-scale value.
0.4
Probability density function
Cumulative Recovery (stb)
5.0E+06
4.0E+06
3.0E+06
2.0E+06
fine
upscaled
best matched
kriged
1.0E+06
0.3
fine
0.2
upscaled
0.1
best
matched
0.0
0.0E+00
-5
0
500
1000
1500
2000
-0
5
10
ln(permeability)
Time (days)
Figure 7
Comparison of cumulative recovery vs. time
for fine- and coarse-scale models.
Figure 8
Probability distribution functions for the fineand coarse-scale models.
Conclusions
In this paper we have examined the accuracy of single-phase upscaling when applied to twophase problems. Firstly, we have shown that accurate single-phase upscaling (effective
permeability ratio, EPR ~ 1) does not guarantee that the coarse-scale model will accurately
reproduce two-phase flow (Figure 4). On the other hand, when EPR ≠ 1, a good match may still
be achieved to the production data. For an accurate match to the average pressure, however,
EPR must be close to 1.
We have also calculated coarse-scale permeabilities using history matching in order to
understand what kind of coarse-scale model produces a good match to production and pressure
data. The results show that the permeabilities in the history-matched model have a different
mean and a wider pdf than the upscaled permeability distribution. These results suggest that it
may be better to generate models directly at the coarse-scale, using statistics from the fine scale.
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Acknowledgements
We should like to thank NIOC for sponsoring Hashem Mofared, and the following companies
for sponsoring the Upscaling Project at Heriot-Watt University: Anadarko, BG, BP, DTI,
JOGMEC (formerly JNOC) and Petronas. We should also like to thank Schlumberger for the
use of the Eclipse Reservoir Simulator, and Vasily Demyanov and Sam Subbey (now with the
Institute for Marine Research, Bergen, Norway) for useful discussions.
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