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SPE 50643
Application of Novel Upscaling Approaches to the Magnus and
Andrew Reservoirs M.J. King, D.G. MacDonald, S.P. Todd, H. Leung,
SPE, BP Exploration Operating Co. Ltd.
Copyright 1998, Society of Petroleum Engineers, Inc. This paper
was prepared for presentation at the 1998 SPE European Petroleum
Conference held in The Hague, The Netherlands, 20-22 October 1998.
This paper was selected for presentation by an SPE Program
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Abstract Cases studies from three North Sea turbidite reservoirs
will be presented, which together demonstrate our current
understanding of permeability and relative permeability upscaling.
The three formations, the Magnus, Magnus Sand Member (MSM), the
Magnus, Lower Kimmeridge Clay Formation (LKCF), and the Andrew
reservoir each provide distinct challenges for reservoir modelling,
either because of reservoir complexity, the fluids in place, or the
phase of field life. To meet these challenges, several novel
upscaling approaches have been developed. Their use will be
explored and current best practice delineated. This best practice
differs significantly from previous definitions of effective
permeability by placing more emphasis on extracting multiple
properties from the fine scale geologic models. Distinct upscaling
calculations are required to assess (i) the quality of sands, (ii)
the quality of barriers, and (iii) the tortuosity of flow around
these barriers. Similarly, when constructing upscaled relative
permeabilities, the effective curves are distinguished from the
pseudo curves. The former describe the physical displacement of
fluids, while the latter include the additional numerical
dispersion corrections required when implementing the relative
permeability functions within a coarsely gridded full field
simulator.
Introduction Three dimensional geologic modelling escaped from
the laboratory approximately three years ago, taking with it a
variety of geostatistical and upscaling tools. Along the way, it
acquired 3D visualisation and a graphical user interface, making it
widely
accessible to asset based non-specialists. A result of this
proliferation has been a change in working practice, where the same
asset-based teams which build the geologic models now upscale them
and use them for well planning and reservoir simulation purposes.
This has highlighted limitations within the standard approach to
upscaling [1, 2] and has accelerated the development of new
upscaling methodologies [3, 4].
Perhaps we should remind ourselves of why we are building these
large geologic models, and why we need to upscale them. The second
question is easier to answer than the first. 1 - 20 - 1000 million
cell local area and full field three dimensional geologic models
are being built, some fairly routinely. Black oil simulation for
routine engineering calculations limits us to a maximum of about
100,000 active cells. Upscaling is the process whereby the very
detailed geologic model is reduced to the coarser flow simulation
model.
Why do we need such detailed models? Sometimes we dont: coarse
grid mass balance style calculations are often adequate for many
operational decisions. But, when we wish to improve our mechanistic
understanding of the reservoir, or to explore the dynamic
implications of different geologic concepts, then these detailed
models provide insight that otherwise would be difficult to
obtain.
Industry experience in the full life cycle of these upscaled
geologic models is still extremely limited. We are still learning
why and how to build them, what value they provide, and how to use
them in combination with more conventional techniques. The
literature includes references where these models have worked
extremely well [5, 6]. However, there are many other (unpublished)
examples in which the performance prediction from the upscaled
model was significantly different than the performance of the
reservoir and/or to the performance prediction of sector models
drawn from portions of the original detailed geologic models.
This paper is organised around four case studies, chosen to
demonstrate different combinations of successful and unsuccessful
upscaling calculations. Three of these case studies, and most of
the material within this paper, emphasise permeability upscaling.
Multiphase upscaling (pseudoisation) is
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2 M.J. KING, D.G. MACDONALD, S.P. TODD, H. LEUNG SPE 50643
included as an alternative approach for the Magnus MSM. The
first three case studies are preceded by a discussion of effective
permeability and followed by a list of elements that define best
practice, as we currently understand it. The multiphase upscaling
example is much more exploratory, and should be considered work in
progress. For convenience all of the formal derivations are
relegated to the Appendices.
Effective Permeability What is effective permeability? A cynic
could describe it in terms of putting incorrect information into
the wrong model, to get the right answer. The wrong model is that
carried in a typical full field simulator: homogeneous blocks of
numerical rock that extend for perhaps a hundred metres or more
laterally, and which may be tens of metres thick. The incorrect
information is the effective property. But, what the cynic would
emphasise is that we can only define this property based on our
expectation of the right answer.
Consider the simplified sketch of a simple sand/shale reservoir
zone, in Figure 1a, and focus your attention onto the coarse cell
in the centre of the figure. What is the vertical permeability of
the cell? With the standard flow based computation of effective
permeability [1], the upscaling region is treated as if it were a
laboratory coreflood, Figure 1b. The sides of the system are
sealed, a pressure drop is exerted vertically, and the pressures
and flux are determined numerically. The volumetric flux defines
the effective permeability, according to Darcys Law.
(Q )A
K PL
EFF= = 1 .................................................(1) In
this particular case the vertical permeability is zero. It
corresponds to the right answer in which two of the shales stretch
off to infinity. Vertical flow is not possible within the
reservoir.
This is certainly not the only possible right answer. Another is
sketched in Figure 1c, in which the upscaling volume is embedded
within a region of uniform (but unknown) permeability. It is not
necessary to explicitly solve the coupled equations as one can show
that they reduce to a linear pressure drop on the boundary of the
upscaling volume. This effective medium approach is discussed in
[4]. The calculated vertical permeability is positive. Streamlines
can run from bottom to top of the upscaling volume, spreading
beyond the volume to avoid the shales.
Periodic boundary conditions are used in the volume averaging
literature [7, 8]. Here the upscaling volume is embedded in
multiple replicates of itself, stretching off to infinity. As in
the effective medium approach, the numerical calculation can be
re-stated as a local calculation on the upscaling volume. In this
case the calculated vertical
permeability would be zero. Finally, another possible flow
picture is sketched in Figure
1d. Following Begg et.al., [9], the computational region is
extended far beyond the upscaling volume. The total vertical flux
is positive, due to tortuous flow paths which stretch the full
width of the shales. The vertical volumetric flux is only summed
within the upscaling volume, although the pressure solution is
determined on the entire computational domain. This is a more
expensive calculation than any of the local calculations described
previously.
Which value of effective permeability should be used? The answer
is that any of these values may be the right one
to use, depending upon the flow pattern which one considers to
be important. Hopefully this will become clear after examining the
three case studies. However, a few additional comments before
examining these examples.
The standard upscaling approach is a local approximation.
Although the upscaling region is physically embedded in a larger
region, upscaling is typically performed without access to this
extended information. Hence, the sand and shale patterns in Figures
1a-c are identical within the upscaling region. However, within a
local approximation, there are some rigorous results. The sealed
side calculation will always give a lower bound, and the linear
pressure boundary conditions will give an upper bound (over the
natural set of boundary conditions). If the results of these two
local calculations differ significantly, then the pre-dominate flow
direction is to leave the cell. Both periodic boundary conditions
and the use of a wide computational region will give intermediate
values.
If the upper and lower bounds are significantly different, as
they are in this case, then the cynics perspective is correct.
However, if these two values are very close, then you have a
pleasant surprise: a representative effective permeability. This
happens more often for the horizontal permeability than for the
vertical. In this particular case, the upscaled horizontal
permeability is very close to the net-to-gross of the upscaling
volume, times the permeability of the sand, irrespective of the
manner of calculation.
Case Studies: Effective Permeability Models of three submarine
fan turbidite reservoirs will be examined in this section. Two of
these are sector models and one is a full field model. The fine and
coarse three dimensional grids and model sizes are listed in Table
1. All three models have been constructed and upscaled using
Smedvigs IRAP Reservoir Modeling System (IRMS version 4.0.7) [10].
The basic upscaling approaches within IRMS follow the methods of
[1, 8], which will be demonstrated to be inadequate. Instead, we
have needed to make extensive use of the IRMS command language to
extend the upscaling toolkit and to provide the results described
herein.
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SPE 50643 APPLICATION OF NOVEL UPSCALING APPROACHES TO THE
MAGNUS AND ANDREW RESERVOIRS 3
Magnus Reservoir, Magnus Sand Member. The Magnus MSM is a large
Upper Jurassic, sand dominated, turbidite reservoir deposited above
a field wide shale bed, the B shale, Figure 2 [11]. It was
discovered in 1974, commenced production in August of 1983, and was
on plateau until January of 1995. Decline has been managed since
then with an active infill drilling programme, which has also been
extremely instructive as to the complexity of the reservoir
[12].
The MSM has an average net-to-gross of 76% and an average net
porosity of 21%. The reservoir has been depleted with a edge-drive
waterflood, with a plateau production rate of ~140 mbd. There is no
gas cap and limited aquifer support. The oil is undersaturated,
with a gravity of 39 degrees API, a viscosity of 0.48 cp and a GOR
of 775 scf/stb. The waterflood is dynamically stable with a
mobility ratio close to unity, and a frontal mobility ratio of
about 0.3.
The continued infill drilling in Magnus has placed increased
emphasis on both the areal and vertical sweep patterns behind the
waterflood flood front. Surveillance has shown that in some sands 5
- 10 metres of oil are unswept, in immediate proximity to sands
which are being effectively drained. It is for the purpose of
understanding these remaining oil targets that detailed reservoir
modelling has been under-taken, including the three dimensional
cross-sectional model described in [4]. The geocellular resolution
of this model was 0.5m vertically, and 30m x 30m areally. The model
covered the four reservoir zones above the B shale, each of which
was eroded by subsequent reservoir zones. (Of the ~750,000 cells,
only ~400,000 were active.) The model extended 2970m from below the
OOWC to the crest of the field, and had a width of 630m. Details of
the sand and mud facies, and their properties, can be found in that
reference.
The simulation was performed using the streamline based Frontsim
simulator [13] because of its speed and lack of numerical
dispersion. Without upscaling, the convergence of the simulator was
impaired because of the large number of pinched-out cells in the
geologic model. Even after resampling, simulation took several cpu
days when in a history match mode, because of the limited time-step
sizes.
To reduce the simulation time to several hours, the decision was
made to upscale the model. Working within IRMS, the model was first
resampled into a shoebox computational domain of absolutely uniform
thickness, which was then upscaled. Upscaled properties were
subsequently transferred back to a coarse proportional simulation
grid. Local flow directionality and the dip of the geologic layers
within each reservoir zone were preserved in the resampling steps
using the transmissibility construction derived in [4], and the
different tensor representations of permeability within a geologic
model and a flow simulator.
The resampling step took the model from a 99 x 21 x 352 3D grid
with pinch-out, to a 99 x 21 x 375 layer proportional model. As
this computational domain was completely featureless, porosity was
converted to cell pore volumes, and directional permeabilities to
cell transmissibilities. Over-sampling was utilised to preserve
geologic continuity, especially of muds, as shown in Figure 3.
Warren and Price flow based upscaling was performed using a uniform
3 x 3 x 3 element to provide a coarse 33 x 7 x 125 layer model. The
average cell thickness was less than 1.5m, and so it was believed
that this was still a high resolution, mechanistic model.
Validation of the upscaling was performed using Time of Flight
streamline-based flow visualisation [14, 4].
Was this a successful upscaling calculation? Based on the large
scale waterflood performance, as validated using Time of Flight,
the answer was yes. However, what about the original question:
definition of the remaining oil targets? Unfortunately, inspection
of the upscaled permeability patterns showed that lateral
dimensions of the muds had increased, sometimes leading to multiple
muds merging, generating significantly larger trapped oil
accumulations in the upscaled model, than in the original geologic
model, Figure 4.
In our experience, this is the single most common error
introduced when using flow based upscaling. Another example is
given in Figure 5, with more emphasis placed on the loss of sand
channel continuity upon upscaling. The sealed sides systematically
bias the permeability of sand and mud mixtures towards muds. In a
high net-to-gross system, we had not expected this to be a
significant effect, and in fact, it does not have an impact on the
gross performance of the waterflood. However, as our calculations
were intended to understand the detailed habitat of the remaining
oil, it was realised that these upscaled models were
inadequate.
Andrew Reservoir. The Andrew reservoir was discovered in 1974,
and came on production in June of 1996 [15]. The reservoir consists
of a low relief structure with four way dip closure, and a
relatively thin oil column. The oil is overlain by a gas cap, and
is in contact with extensive water bearing sands, which may provide
an effective aquifer. The oil is contained within distal Palaeocene
turbidite sands, composed mainly of fine to medium grained, clean
to moderately poorly cemented sandstones. The reservoir has a high
net-to-gross (0.8 to 0.99), medium porosities (16 to 22%), and
medium permeabilities. The oil is saturated with gas, 40 degrees
API gravity, with a GOR of 871 scf/bbl.
Detailed three dimensional geologic modelling is being used on a
fieldwide basis as the fundamental static description of the
reservoir. Local area models are extracted, screened and simulated
without upscaling, to understand individual well performance. The
geologic model is also being upscaled to
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4 M.J. KING, D.G. MACDONALD, S.P. TODD, H. LEUNG SPE 50643
construct the full field simulation model. Variations in
geologic description are being explored as part of the history
match process.
The field is being developed with horizontal wells. Production
will be gas constrained, so it is extremely important to understand
the extent and continuity of shales, as they will be the major
control on the vertical flow of gas within the reservoir.
The Andrew reservoir model was constructed and upscaled using
IRMS [10]. Each zone of the reservoir was upscaled separately. The
upscaling step reduced the model from approximately 3.8 million
cells (140 x 161 x 169) to 74,000 (36 x 44 x 47). After some
experimentation it was decided to use proportional layering for the
geologic model as it provided better behaved upscaled
permeabilities. Because of this simple geologic grid structure,
there is no need for a resampling step, as there was for the Magnus
MSM.
In the first attempt, permeability was upscaled using the flow
base full tensor calculation within IRMS, although only the
diagonal terms were retained. Water production is shown for a
validation run on a sector model in Figure 6. The fine scale model
has the slowest increase in watercut, and the upscaled model has
the fastest. Inspection of the three dimensional saturation
profiles indicates that the upscaling process has smeared out the
shales. Although volumetrically insignificant, the shales are
extremely important for understanding the RFT response from the
field, and for predicting horizontal well performance. Without the
shales to act as barriers, water and gas are free to move
vertically throughout the reservoir. The intermediate curve was
obtained after modifying the vertical transmissibility multipliers
by hand to ensure that the largest (deterministic) shales were
correctly modelled as barriers to flow.
The upscaling calculation was unsuccessful, essentially because
of the difficulty of modelling the shales. On balance, both
deterministic and stochastic shales are important in the Andrew
reservoir, e.g., some shales can be mapped over inter-well
distances, while others cannot. Nonetheless, most of the stochastic
shales are wider than single columns of the coarse simulation
model, allowing vertical flow in the reservoir, but requiring that
the flow be tortuous.
The upscaling calculation has another feature: volumetrically
insignificant shales which form vertical barriers will be modelled
as cells with zero permeability, PERMZ = 0. This over-estimates the
volume of shale, and also over-estimates its flow impact, as
indicated in Figure 7. In a finite difference calculation, we can
visualise flow as running along pipes from cell centre to cell
centre [16]. With this image in mind, we see that zero permeability
not only prevents flow through a cell, but it also prevents flow
through the half cells on either side. The spatial resolution can
be doubled, and the volumetric impact of shales removed, if instead
shale barriers are modelled as
transmissibility barriers, MULTZ = 0. With this construction in
mind, a half cell upscaling
approach was developed for the vertical permeabilities: (1)
Upscale using the full tensor permeability algorithm, into a
3D grid identical to the full field model in X and Y, but with
twice as many layers;
(2) Each cell in the FFM now has two corresponding cells on this
grid, and two permeability values. Select the greater of the two
PERMZ values for the FFM;
(3) Upscale using the sealed side flow based algorithm, into a
3D grid shifted up one-half cell from the original grid, and
centred on the faces of the FFM;
(4) Calibrate the vertical inter-cell transmissibility
multiplier from the permeability on the face (from Step 3), and the
harmonic average of vertical transmissibilities from the two
adjacent cells (from Step 2):
( ) ( )( ) ( )
KZDZ
MULTZKZ DZ KZ DZ
KZ DZ KZ DZkk
k
k k
=
++
+
+1 21
1
2
/
k ......... (2)
The harmonic average is the standard expression for inter-cell
transmissibility [16, 17], here simplified since the
cross-sectional area is in common throughout the coarse column.
Values of MULTZ typically vary from zero to unity. When sand is
juxtaposed again shale, then MULTZ = 0. In regions of uniform
properties, MULTZ 1 . Otherwise, MULTZ takes on intermediate
values, depending upon the local contrast in vertical
permeabilities.
The results of this extra effort have been quite gratifying. In
Figure 6, the water-cut prediction of the upscaled model now
follows the fine scale prediction extremely closely. In Figure 8, a
cross-section through the FFM is shown, with a clearly resolved gas
under-run indicated. Finally in Figure 9, the actual gas production
of well A04 tracks the most likely prediction.
Magnus Reservoir, Lower Kimmeridge Clay Formation. The Magnus
reservoir consists of two units: the MSM described earlier, and a
lower unit, the mud-dominated Lower Kimmeridge Clay Formation
(LKCF) [18], Figure 2. The LKCF is a geologically complex generally
low net-to-gross reservoir, consisting mainly of sequences of
thinly bedded sandstones and mudstones inter-bedded on the
centimetre to metre scale. The field average net-to-gross is less
than 25%, but it varies from ~65% in the crest of the field, to
essentially zero on the margins.
The dominant reservoir facies is structureless, fine to coarse
grained sandstone, with permeabilities up to 800 mD. These high
density turbidites are often deformed by water escape structures,
injected sands, and slumping.
As with the MSM, a three dimensional model was constructed to
explore different geologic concepts of the LKCF, and to support
LKCF development planning. A waterflood pilot in the southern basin
of the LKCF has been successful, but its
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SPE 50643 APPLICATION OF NOVEL UPSCALING APPROACHES TO THE
MAGNUS AND ANDREW RESERVOIRS 5
extension across the field is problematic. Fieldwide RFT
response indicates that pressure communication exists within the
LKCF, and between the LKCF and the MSM. However, it is not clear
what kind of oil recovery efficiencies are likely, nor how the
recovery factor will vary with well spacing.
The LKCF sector model was constructed below the crest of the
MSM, Figure 2, because of the availability of LKCF core data, and
as a compromise between calibration with field performance of the
southern basin waterflood, and prediction of crestal recoveries.
This model covered a significantly greater gross volume than the
MSM model, covering an area of 3.2km x 3.2km, and all six reservoir
zones. The geocellular resolution was 50m x 50m areally, and as
before, 0.5m vertically. The model consisted of approximately 1.7
million cells, of which about half were active. An upscaled model
was required in order to perform well spacing studies in reasonable
amounts of simulation time.
A typical geocellular model is shown in Figure 10. The reservoir
is mud dominated, and in the model, all sands are in immediate
proximity to the muds. As discussed earlier, such sand and mud
mixtures are expected to upscale to mud, destroying the reservoir
quality. After the success of the half cell upscaling approach for
Andrew, it was both simplified and extended for the LKCF.
The extension has to do with the complexity of fluid flow within
the LKCF. In Andrew, half cell upscaling was used to improve the
vertical resolution of the coarse model. In the LKCF it is as
important to retain lateral flow: the sketch of Figure 5 is
expected to be typical of the sands within the LKCF. Hence, the
approach was extended to all three directions.
It was also possible to simplify the half cell upscaling
approach described earlier. Steps (1) and (2) together were used to
calculate a cell permeability which was rarely zero, and that only
in regions of uniform thick shales. Step (3) independently
calculated the inter-cell transmissibility with a second upscaling
calculation, which was then encoded as a multiplier with respect to
the cell permeabilities in Step (4). However, as long as the cell
permeabilities did not vanish, then these inter-cell
transmissibilities in no way depended upon the cell permeabilities.
Hence, we may avoid the expense of a flow based calculation for the
cell permeabilities, and instead replace their determination with a
simpler method.
We have chosen to calculate the cell permeabilities using a well
productivity based upscaling approach described in the Appendix. In
other words, we ask that the productivity of a perforation within
each coarse cell of the model, be identical to the average of the
productivities within all of the corresponding fine cells. As these
hypothetical perforations can have three possible orientations, we
obtain three equations for KXEFF, KYEFF, and KZEFF. This may either
be viewed as an artifice to calculate non-zero permeabilities, or,
one may recognise that
permeabilities enter into the calculation of well connection
factors, just as they do the inter-cell transmissibilities [19].
Either equation may be used to calibrate the permeabilities. As
discussed earlier, the simulator has higher spatial resolution if
transmissibilities are modelled as face properties, instead of cell
properties. The advantage of this approach is obvious in Figure 11,
where in contrast to Figure 5, the continuity of the sands are
preserved.
So, in summary, the half cell upscaling approach for the LKCF
consists of: (1) Upscale all three directional cell permeabilities
using the
algebraic well productivity algorithm; (2) Upscale across each
face of the coarse 3D grid, using the
sealed side flow based algorithm; (3) Calibrate the three
inter-cell transmissibility multipliers from
the permeability on the face (from Step 2), and the harmonic
average of directional transmissibilities from the two adjacent
cells (from Step 1).
There is one additional implementation detail for the LKCF which
was not required for Andrew. The LKCF model is constructed of
geologic layers which erode at the zone boundaries. It is important
that the spatial correlation of sands within the geologic model be
preserved when IRMS performs the flow based upscaling calculations,
even in the presence of erosion of these layers. This was ensured
by reconstructing the eroded portions of each zone so that each
coarse cell consisted or apparently non-eroded fine cells. Although
each zone was upscaled separately, this reconstruction accessed
fine scale permeability information from the adjacent zone before
initiating the flow based upscaling calculation.
The results are shown in Figures 12 - 15. In Figure 12, a three
dimensional four well flooding pattern through the 1.7 million cell
(64 x 64 x 452) geologic model of Figure 10 is shown. The Time of
Flight flow visualisation shows the flood progressing from the two
downdip injectors to the two updip producers in a fence diagram.
Figure 13 shows the same pattern after a 2 x 2 x 6 half cell
upscaling, which reduced the model to approximately 70,000 cells
(32 x 32 x 72). Both of these calculations had flow rates specified
for the four wells. Additional validation is obtained in Figure 14,
where the pressure gradients calculated in Figure 12 are applied to
the 70,000 cell, half cell upscaled, model. The flooding pattern is
extremely similar. Pressures are presented also, and show that
internal permeability and flow is being preserved. In contrast,
Figure 15 shows a similar flow and pressure visualisation when
conventional sealed side flow based upscaling was used. The total
flow rate is now only 5% of the previous figure. The pressures
indicate that the model has been upscaled into a fairly uniform,
extremely low quality reservoir.
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6 M.J. KING, D.G. MACDONALD, S.P. TODD, H. LEUNG SPE 50643
Best Practice: Effective Permeability What have we learned from
these three case studies? The most important lesson is that
upscaling is not simply a mathematical transformation of one three
dimensional model into another. Instead, it is very much as
described by the cynic: one must decide upon the important flow
solution, and select numerical techniques that extract the
appropriate effective properties. This is very different than in
the upscaling of porosity or net-to-gross, in which there is a
physically conserved property (volume) and an obvious manner of
calculating its average (addition). Although the lack of an unique
upscaling approach may be viewed as an issue, it is in fact an
opportunity: we select an upscaling method to extract the
information of choice from the detailed geologic model.
The lack of a unique upscaling approach does not mean that there
are no guidelines that can be developed. Here is a checklist of six
items that have been explored through these examples.
Layering. It is extremely important to preserve the geologic
correlations of the fine scale models while upscaling, especially
in cross-section. In the Magnus MSM, this rule was violated, but
was compensated for by using over-sampling. Andrew explicitly used
proportional layering for the geologic model as this fine grid
nestled simply into the coarse grid. In the Magnus LKCF, the eroded
geologic layers were reconstructed for each of the active coarse
cells before upscaling.
In our experience, if this step is not handled carefully, then
the geologic model may be randomised upon upscaling. Effective
spatial correlation lengths may be limited to fractions of the
coarse cell size. Most importantly, shale continuity may be
compromised, and upscaled Kv/Kh ratios may approach unity.
Preserve Sand Quality. The most common error in permeability
upscaling is to not distinguish between the different uses of
permeability within a model. In particular, the sealed side flow
based effective permeability calculation of Warren and Price is
used far more frequently than it should be. Mathematically it
provides a lower bound on the effective permeability. Physically it
reduces the permeability of sand and mud mixtures, thickening
shales, and narrowing and disconnecting sand channels.
Typically, the productivity of a well in the field starts high
and then diminishes. The early production is due to the fluids in
immediate proximity to the well, and the quality of the near-by
sands. Sustained production requires that these fluids be
replenished. The well productivity based upscaling approach
calibrates the cell permeabilities to correctly model the first
flush of production. The inter-cell transmissibility controls the
sustained production.
Preserve Barriers. An excellent use of the Warren and Price
upscaling approach is to determine the continuity of reservoir
baffles and barriers, especially at the scale of the coarse grid
block. In Andrew, it was important to determine if gas could cone
directly down from the OGOC, or whether the path of the gas would
be far more tortuous and would under-run the shales.
The half cell upscaling approach takes advantage of the
redundancy of permeability and transmissibility multipliers within
the flow simulators, to model both reservoir quality and barriers
simultaneously. Further, modelling barriers as inter-cell
transmissibility modifiers has doubled the effective spatial
resolution of the simulation, compared to grid based
permeabilities. As this benefit is obtained in all three
directions, the improvement in volumetric resolution is a factor of
2**3 = 8.
Preserve Flow Around Barriers. In the Andrew and Magnus LKCF
examples, inter-cell transmissibilities were calibrated using the
Warren and Price algorithm. Is this always the best approach? In
the discussion of [9], vertical permeability is thought of as a
global property, although its value is placed in individual coarse
cells. As the vertical flow of fluid becomes more tortuous, as in
Figure 1d, then the tendency of the sealed side calculation will be
to still under-estimate flow. As the coarse cell gets thicker
compared to the vertical shales and muds, then the calculated
permeability may drop to zero, completely removing pressure support
from a large portion of the reservoir.
In such a case it is almost certainly preferable to calibrate
the vertical transmissibility using the non-local computational
region of Figure 1d. However, by so doing, we are in the realm of
the cynic, as this contradicts the recommendation to preserve
barriers. Again, decide upon which element of the reservoirs
performance is most important, and then extract the appropriate
upscaled properties.
Validation and Iterations. In our experience, no results from an
upscaling calculation have ever been correct the first time.
Sometimes it is the upscaling calculation itself which has
performed in a manner other than expected. Other times it is the
geologic model. More positively, after performing an upscaling
calculation one is in a position to ascertain the dynamic
consequences of the static model. For example, for the Magnus LKCF,
the forward prediction of the geologic model lead to a
re-evaluation of the core data, and a decrease in importance of the
injected sands.
Some of the validation is conceptual, as in a review of the
underlying geologic concepts. However, other forms of validation
are purely numerical, as in the comparison of fine and coarse scale
simulation of sector models (Andrew) or in the Time of Flight flow
visualisation (Magnus MSM and LKCF). In one way or another,
distinct forms of validation should be
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SPE 50643 APPLICATION OF NOVEL UPSCALING APPROACHES TO THE
MAGNUS AND ANDREW RESERVOIRS 7
utilised, corresponding to the distinct uses of permeability
already listed.
Still An Approximation! Finally, it should be remembered that
upscaling is still an approximation. There will always be
permeability distributions which are difficult to upscale. As an
example, examine Figure 16, where sands on a mud background are
upscaled onto two coarse cells. Most upscaling approaches will
merge the two disconnected sands upon upscaling. This problem is
most effectively handled by re-gridding the problem. Options are to
shift the coarse grid to prevent merger, or to re-grid to a
slightly finer coarse mesh, or sometimes to re-grid to a much
coarser mesh in which the sands as recognised as isolated.
This concludes the discussion of single phase upscaling. Of the
cases studied, two have been successful and one has not. We return
to the Magnus MSM example, and consider its upscaling as a
multiphase flow example.
Multiphase Upscaling All of the single phase upscaling case
studies had one important unspoken assumption: that the upscaled
model was to be mechanistic. In other words, it was important that
the detailed heterogeneity of the geologic model be preserved, so
that we could rely upon the upscaled model to describe the
interaction of the displacement mechanisms with this
heterogeneity.
In the Magnus LKCF example, we are very close to violating this
assumption. In the reservoir, and in the reservoir model, few sands
are individually thicker than 2.0m. With the half cell upscaling,
the 3.0m thick cells have an effective vertical resolution of 1.5m,
allowing these sands, and their contrast from the muds, to be
modelled. However, with even a factor of 2 coarser cells, then
there would be no expectation of providing a mechanistic simulation
of the LKCF using solely single phase upscaling techniques.
In such a case, there is little choice but to include finer
scale fluid by-passing and displacement mechanisms within the
upscaled effective relative permeabilities, i.e., to use pseudo
relative permeabilities. There is an extensive literature on
pseudos, which we will not attempt to summarise. However, we do
recommend two recent reviews of dynamic pseudos [20, 21] and the
literature summarised therein.
The method being presented is a hybrid, with elements selected
from different approaches to minimise known artifacts. It is a
total mobility approach, in which the phase mobilities
, , are analysed in terms of a total mobility,
, and a fractional water flow,
( )W WS ( )O WS( ) ( ) ( ) S SW W W O W= + S
)( ) ( ) (F S S SW W W W W= . Effective fractional flows and
total mobilities will be determined by analysing a suitably
averaged sequence of saturation profiles obtained by direct
simulation.
The effective phase relative permeabilities will be derived
secondarily.
The discussion will also distinguish between the calculation of
effective and pseudo relative permeabilities. The former describe
the physical displacement of fluids, while the latter include the
additional numerical dispersion corrections required when
implementing the relative permeability functions within a coarsely
gridded full field simulator.
We take advantage of the JBN method for interpreting a
laboratory coreflood as it is guaranteed to provide water and oil
relative permeabilities which exactly reproduce the laboratory
experiments [22, 23]. This approach is similar to that of [24].
However, the combination of the treatment of effective total
mobility (as opposed to fractional flow), and numerical dispersion
corrections are new.
Fractional Flow. The derivations follow [22, 23]. The analysis
is based upon the one dimensional Buckley-Leverett equations for
incompressible fluids in a variable cross-section. After
appropriate transformations, the pore volume becomes the equivalent
of spatial position, and the volume of water injected is the
equivalent of time. The JBN approach calculates both the fractional
water flow and the water saturation at the outlet, based on the
average water saturation and its time derivative.
( ) ( )
( ) ( ){ }F T PV
d S TdT
S Td
dT T S T
W OUTW
W OUT W
,
,
=
= 1
11
................................... (3)
A cross-plot of the two provides a single fractional flow curve
that is guaranteed to reproduce the average waterflood performance,
( )S TW , within an analytic calculation. Such an effective
fractional flow curve can also be utilised directly within the
Frontsim simulator, as the latter is free from numerical
dispersion.
How may this result be extended to finite difference
calculations? If one integrates the differential equations over the
space of a grid block and the interval of a time step, then one
obtains a finite difference form for the Buckley-Leverett equations
[25, 26]. The average saturation in a grid block is then exactly
determined from the time averaged fluxes which enter and leave that
block. The order of the numerical scheme is then related to how
this time averaged flux function is calculated from block averaged
saturations [27].
Ignoring for the moment the time discretization, then the
equivalent of the JBN analysis is obtained by calculating the
average saturation within the last (outlet) grid block of the
hypothetical homogeneous solution.
-
8 M.J. KING, D.G. MACDONALD, S.P. TODD, H. LEUNG SPE 50643
( )( ) ( )
PV S T
PV S T PV PV ST PV
PV PV
OUT W OUT
W OUT WOUT
=
....(4)
In other words, the average saturation in a volume adjacent to
the system outlet (a pore volume of ), can be obtained from the
average saturation in the total system at the same time and at the
later time of ( )
PVOUT
( )T PV PV PVOUT . Two limits are easy to understand: the use of
one coarse
simulation cell, , and the use of many cells, PV PVOUT =PV PVOUT
0 . The first is the standard pseudoisation
approach in which the fractional flow at the outlet of the
system is cross-plotted against the average saturation of the total
system, SW OUT W= S [28]. The second limit is that of the analytic
result, S SW OUT W OUT , , Eqn (3). Hence Eqn (4) provides an
interpolant between pseudos for use in fine grid simulation
(analytic result) and in coarse grids. Because of the intrinsic
relationship between Eqns (3) and (4), we will call Eqn (4) the
extended JBN analysis.
The fractional flow function obtained by a cross-plot of and (
)F TW OUT, ( )SW OUT T will include the effects of spatial
truncation, appropriate for a first order finite difference
scheme. This is an exact result for a pseudo fractional flow, in
the limit of very small time steps.
Once a finite time step size is selected, then ( )F TW OUT, may
be averaged over each time interval, providing a result which is
exact for finite time steps. Unfortunately, the simulation time
step size is rarely known when constructing a pseudo, especially as
it may vary during the course of a simulation run. However, the
impact of numerical scheme and time step size are easy to
understand. For an IMPES scheme, the (time averaged) fractional
flow is represented as a function of the initial upstream block
averaged saturation. For a fully implicit representation, the
fractional flow depends upon the final upstream block averaged
saturation [16].
Consider the sketch of Figure 17. The underlying curve is the
effective fractional flow function: a cross-plot of instantaneous
outlet fractional flow versus the instantaneous outlet saturation.
In both numerical schemes, the block averaged saturation is at a
higher water saturation than the instantaneous outlet saturation,
moving a point on the curve to the right. The time averaged
fractional flow is greater than its initial value, but less than
its final value, either moving a point up or down, depending upon
the numerical scheme. For an IMPES method, the spatial and temporal
truncation errors tend to compensate for each other. Although
individual points will shift, as the local
CFL number approaches unity, the overall impact of numerical
dispersion may be negligible. (This may explain why the performance
of multiphase renormalisation improves when no attempt is made to
correct for numerical dispersion [29].) For fully implicit schemes,
the two errors augment, driving the pseudo fractional flow curve
down and to the right.
Total Mobility. None of this development has included discussion
of the effective total mobility. It has relied solely upon
knowledge of the water and oil volumes as functions of time, with
no mention of distributed or averaged pressures. As no total
mobility upscaling technique is rigorous [20, 21], the decision was
made to use a simple approach instead: steady state upscaling [30 -
32]. In particular, if the fine scale (rock) curves do not vary
with facies or poro-fabric, then the upscaled curve is
simply the rock curve, . ( ) OUT W OUTS ,The rock relative
permeability curves are typically
determined by a reservoir condition unsteady state experiment,
and are available in tabular form. The only remaining decision is
how to interpolate between these points, especially in the
saturation interval below the Buckley-Leverett shock saturation.
The JBN analysis provides the necessary guidance: for one
dimensional incompressible flow, the pressure gradient is
proportional to the inverse of the total mobility, which is what we
use. Further for saturations below the shock saturation, the ( )1
SW curve is evaluated using quadratic interpolation to insure that
these saturations are less mobile than the shock saturation, as is
required physically.
Magnus Reservoir, Magnus Main Sand. We return to the Magnus MSM
example, but now upscale to a coarse grid comparable to the 1997
Magnus Full Field Model (FFM97). The average grid block size is
less than 100m laterally, but there is only a single layer for each
of the reservoir zones. The only possibility of accessing
mechanistic information is through multiphase pseudoisation.
Although in the reservoir the zones communicate, the JBN analysis
can only be used for one dimensional floods. Hence, each of the
four zones are upscaled independently. A wall of injectors is
placed on the downdip face of the geologic model, and a wall of
producers, updip. The multiple perforations are pressure
controlled, although the total injection and production rates are
fixed to give a Darcy velocity of 0.3m/day.
As there is a degradation of reservoir quality downdip, the
flood is initiated at the OWC. Because of erosion of one zone by
the next, each reservoir zone may not be continuous, in which case
an attempt was made to include the largest possible volume. For
convenience, the number of grid blocks along the flow direction was
chosen to be a multiple of three. Transversely the entire width of
the model was included in the simulation.
-
SPE 50643 APPLICATION OF NOVEL UPSCALING APPROACHES TO THE
MAGNUS AND ANDREW RESERVOIRS 9
Three dimensional saturation distributions are determined as a
function of time using the Frontsim simulator. (All pinched-out
cells are now on the boundary of the simulation domain, and so the
convergence difficulties mentioned earlier do not arise.) These
saturation profiles are averaged using the Gridsim post-processor
[13] to obtain a one dimensional saturation profile, at the
resolution of the coarse grid. Pore volumes were upscaled to the
same resolution, providing the necessary spatial coordinates to
apply the extended JBN analysis.
As an example, consider the top reservoir zone of the Magnus
MSM. The simulation was performed on a 39 x 21 x 61 cell 3D grid.
Initial time steps were set at 60 days, increasing gradually to 730
days late in the simulation. Simulation continued until a water-cut
in excess of 98% was achieved. Using Gridsim, the saturation
profiles and pore volumes were coarsened to a 13 x 1 x 1 domain. As
in the above discussion, we will first determine the effective and
then the pseudo fractional flows, followed by total mobility and
pseudo relative permeabilities.
The extended JBN analysis is based upon the average saturation.
In a traditional laboratory experiment the only volume over which
this average is available is the entire system. However, the
simulator provides an entire three dimensional saturation profile,
and so the average can be selected at will. With 13 coarse blocks,
each of which are comparable in length to an FFM97 cell, then we
can support 13 different pseudoisation calculations from this one
data set. We reference the analysis to each of the 13 outlet faces
by using a saturation averaged over the entire upstream profile.
This manipulation of the saturation profiles, and the entire
pseudoisation calculation, are performed using Excel Visual
Basic.
Two features are apparent within these fractional flow curves,
Figure 18. At low fractional flows, the foot of the curve continues
to steepen as the profile evolves. This is a gravity slump, the
importance of which is increasing as the flood proceeds. However,
at late times, the rate of recovery appears to be essentially
independent of cell, i.e., independent of the viscous to gravity
ratio.
Figure 19 shows the spatial numerical dispersion correction,
with a cross-plot of the outlet block saturation, versus the outlet
saturation. For the first series, there is only one coarse block in
the profile, providing a significant difference between these two
saturations. For the last series (series 13), the differences are
quite small, giving essentially a straight line. Once there are
more than six or seven blocks, the numerical dispersion corrections
are quite small. At late times, the saturation profiles are more
uniform, as the flood approaches the irreducible water saturation.
Then, even for the early series, the numerical dispersion
corrections are small.
Figure 20 shows the pseudo fractional flow curve: a cross-plot
of outlet fractional flow versus outlet block saturation.
There is far more spread between these curves, than the
effective fractional flows. Apparently, numerical dispersion is
more important than the physical displacement mechanisms. The
impact of gravity is still present, but not as obvious as in the
effective curves.
Figure 21 is a plot of the inverse mobility function, determined
from the rock curves. There is no evidence that the rock relative
permeabilities vary significantly with facies or with rock fabric
for any of the reservoir facies. With a single set of rock water
and oil relative permeability curves, then the steady state
upscaling approach reproduces this curve. The parabolic fit is used
to simplify the evaluation of the inverse mobility, especially at
saturations below the shock saturation.
Figures 22 and 23 show the result of combining the inverse
mobility function with the pseudo fractional flow curves. The
effects of numerical dispersion and of gravity are both evident in
these final curves. At late times, dispersion is less important but
the gravity slump has had more opportunity to evolve. For early
times dispersion is extremely important but the gravity slump has
had little opportunity to act.
The question remains: which of these 13 pairs of pseudo relative
permeability curves should be used in the full field model? The
answer depends upon the injection - production well spacing. A
typical well distance within FFM97 is about 400m, and so the series
4 set of curves were applied. These pseudos differed significantly
from previous calculations (Dykstra-Parsons, and simulation based)
in predicting that when water first arrived at a producer it would
only provide a limited increase in water-cut, followed by an
extended period of negligible water-cut increase. This is much more
typical of the field response than the predictions based on the
previous pseudos.
Multiphase Discussion The Magnus MSM pseudoisation has
demonstrated the ability to separately resolve physical mechanisms,
e.g., gravity slumping, and numerical dispersion. The resulting
pseudo curves appear reasonably monotonic and can be used in
simulation with little modification. This approach has extended the
classic JBN analysis to include numerical dispersion corrections in
the fractional flow.
The upscaling of total mobility has been significantly
simplified with the use of steady state upscaling, removing much of
the difficulty in its calculation. In a one dimensional model of
fluid flow, the total mobility decouples from the prediction of
saturation, and so the latter cannot be used to invalidate this
simplification. Numerically calculated pressures may potentially be
used for this purpose, but their averaging from a three dimensional
distributed system to a one dimensional pressure profile has
sufficient ambiguity that again may make validation (or
invalidation) of this simplification difficult.
-
10 M.J. KING, D.G. MACDONALD, S.P. TODD, H. LEUNG SPE 50643
One important implementation detail in this approach was the
treatment of total mobility below the Buckley-Leverett shock
saturation. If simple linear interpolation is used, then the total
mobility will be artificially high. This is definitely
non-physical, as the two phase total mobility must decrease as the
phases compete for the pore space. The pseudos calculated in this
manner will be artificially increased in this saturation range,
leading to non-monotonic pseudo relative permeabilities,
potentially with values greater than their endpoint values.
Many questions remain with this approach. We have separated the
reservoir zones into four distinct calculations. Gravity slumping
is permitted within a zone during the pseudoisation, and between
zones when used in the full field model. Is this a valid
separation? What about the requirement for incompressible flow? It
is definitely required to preserve the scaling behaviour of the
Buckley-Leverett solution. Are other treatments possible when rock
or fluid compressibility are not negligible? Finally, what about
the interaction between upscaled absolute permeability and the
pseudoisation? The current development is suitable for when
permeability is smoothly varying, but is likely to over-emphasise
the impact of heterogeneity otherwise. Extensive use has been made
of the exact nature of the JBN formulation, including the numerical
dispersion corrections. However, the impact of time step size is
difficult to pre-calculate, as time steps in the simulator are
often quite large (fully implicit) and variable in magnitude (IMPES
and fully implicit).
After this work was completed, the authors received a preprint
from Stanford University, which applied similar numerical
dispersion corrections [33]. Their application was to one
dimensional waterflood, which allowed a treatment of total mobility
upscaling identical to that of JBN for laboratory coreflood.
Interestingly, the resulting pseudo curves are much more prone to
non-monotonicity that those obtained in the present study, although
whether this is due to the one dimensional floods, or to the
treatment of total mobility, must be added to the list of open
questions.
Conclusions 1. The upscaling of single phase permeability
and
transmissibility has been shown to be a critical step in
extracting maximum value from detailed geologic models. When done
with understanding, permeability upscaling has been demonstrated in
several case studies to provide excellent reservoir performance
prediction.
2. It is recommended that permeability upscaling be performed
for distinct reservoir properties: sand quality, barriers, and flow
around barriers.
3. Half cell upscaling has been developed to simultaneously
extract as much of this information as possible from the geologic
model. It also provides roughly a factor of 8
improved resolution compared to standard approaches. 4.
Layering, validation, iterations, and managing expectations
are as important elements of upscaling as those listed above. A
checklist of six items has been provided to remind the practitioner
of this.
5. A novel approach to multiphase upscaling has also been
developed. This treatment extends the laboratory-scale JBN approach
to include numerical dispersion effects, and allowed the separate
evaluation of physical and numerical components within the pseudo
relative permeabilities.
Nomenclature = Phase mobility, L3t/m = Viscosity, m/Lt A = Area,
L2 da = Area element, L2 dq = Flux density, L/t dv = Volume
element, L3 DV = Cell volume, L3 DZ = Cell thickness, L F =
Fractional flow, dimensionless K, K[X|Y|Z], PERMZ = Permeability,
L2MULT[X|Y|Z] = Transmissibility modifiers, dimensionless n =
Normal vector, L2
P, P = Pressure, Pressure drop, m/Lt2 PV = Pore Volume, L3 Q =
Volumetric flux, L3/t r = Pressure drop vector, L S = Saturation,
dimensionless T = Volume Injected, L3
= Face Transmissibility, LTFace4t/m
u = Darcy velocity, L/t x = Position vector, L Subscripts &
Superscripts , S = Upscaling surfaces = Upscaling domain EFF =
Effective value h = Horizontal i, j, k = Cell indices O = Oil OUT =
Outlet, Outlet block v = Vertical W = Water Acknowledgements The
material in this paper, and its exposition, were developed in the
course of extensive discussions with many colleagues within BP, too
numerous to list individually. However, special thanks
-
SPE 50643 APPLICATION OF NOVEL UPSCALING APPROACHES TO THE
MAGNUS AND ANDREW RESERVOIRS 11
go to the Magnus Subsurface team, especially Ken Wells and Dave
Richards, for providing the foundations for much of this work, to
Alistair Jones for his review of the multiphase upscaling
literature, and to Paul Bowden and Pat Neeve for their work with
the Andrew simulations. External to BP, special thanks to Lou
Durlofsky, Chris Farmer, Tom Hewett, Lindsay Kaye, Don Peaceman,
Jens Rolfness and Jeb Tyrie.
The authors would like to thank the Magnus License partners and
the Andrew License partners for their permission to publish this
paper.
The views expressed in this paper are those of the authors and
do not necessarily represent those of BP Exploration.
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Appendix: Upscaling Fundamentals In the text, three styles of
single phase upscaling have been described without explicitly
describing the algorithms. Two of them (upscaling for permeability,
and upscaling for transmissibility) can be formulated without
reference to a particular fine scale 3D grid. In contrast,
upscaling for well productivity (well connection factors) relies
explicitly on the coordinate system of the fine grid, and is most
easily used when the fine and coarse 3D grids are in alignment.
Upscaling for Permeability. Consider the upscaling of a region of
heterogeneous porous media, Figure A-1. Flow at each position
within the region is described by Darcys Law
( ) ( ) ( ) ( )u x K x P x= = 1
.................................. (A-1) which we may integrate
over the domain, , to define the effective permeability.
u K P K PEFF = ...................... (A-2) This is a similar
approach to Bears use of a representative element volume [34] to
define a continuum property. In distinction, an effective property
is only representative if the scales of heterogeneity are small
compared to the extent of the averaging region. Otherwise the
results may depend strongly upon the size and detailed placement of
the averaging region. The definition of Eqn (A-2) is suitable in
either case.
The averaged velocity involves a volume integral over the
domain, while the averaged pressure gradient may be converted to
a surface integral [35].
udv K n P daEFF $
........................................... (A-3)
where is the outwardly directed unit normal vector. In general,
the volume averaged Darcy velocity and the volume averaged pressure
gradient need not be aligned, necessitating the use of a full
permeability tensor [8, 36, 37].
$n
The standard upscaling calculation for permeability due to
Warren and Price [1] is a special case of this definition, in which
the numerical calculation is configured to emulate a laboratory
coreflood. In this treatment the domain has a simple rectilinear
shape, with sealed sides and a uniform pressure gradient exerted
along the axis of the system. For incompressible fluids, the
integrals in Eqn (A-3) are simple to evaluate
( )( )udv Q A L A x = $ , and ($ $n P da x P A
= ) , giving an
effective permeability of K Q LXXEFF = P , as expected from
Darcys Law. Upscaling for Transmissibility. The above definition
of effective permeability has attempted to construct an intrinsic
property, i.e., one which does not depend strongly upon the volume
of the upscaling region. Both the velocity and pressure gradient
integrals are performed over the same (fine scale) domain, while
the ratio, the effective permeability, is reasonably defined as a
property on the coarse scale. In contrast, transmissibility is
defined as the volumetric flux per unit pressure drop across a
cross-sectional interfacial area.
Q TFace Face P=
......................................................... (A-4)
Transmissibility explicitly depends upon both the specification of
an interfacial area and a distance over which the pressure drop is
exerted. Hence it is important that both this distance and the
interfacial area be defined identically on the fine and the coarse
scales.
A construction with these elements in sketched in Figure A-2.
Consider first a planar area representing the coarse cell
interface. (For a corner point cell, the planar approximation is
typically taken from the center of the cell face [16], although
generalisations are possible [38].) Define the fine scale
interface, S, from the perpendicular projection of this area onto
the boundary of the upscaling region, . From the divergence theorem
applied to the volume subtended by the cell face and S, it follows
that
( )$ $n n da nFaceS
Face = .................................................. (A-5)
indicating that the coarse interfacial area may be obtained from an
integral over the fine, so long as the fine area is reduced by
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SPE 50643 APPLICATION OF NOVEL UPSCALING APPROACHES TO THE
MAGNUS AND ANDREW RESERVOIRS 13
its perpendicular projection. The total flux through S may be
calculated in an almost identical fashion,
( )Q n nFace FaceS
= $ $ dq ................................................. (A-6)
where now dq dq
dada=
is the outwardly directed flux density
on S. To complete the construction for transmissibility, it
is
necessary to define the pressure drop across the upscaling
region. Figure A-2 includes a reference vector, r , which for the
half cell transmissibility, points from the centre of the coarse
cell to the centre of the interfacial area. With this vector, the
pressure drop may be calculated from the average pressure gradient,
defined as in the construction of effective permeability, ( )
P r P r n P da dv = $
. The resulting
expression for effective transmissibility
( ) ( )$ $ $n n dq T r n P da dvFaceS
Face =
Q
........... (A-7)
only includes terms evaluated on the fine scale. With all the
apparent complexity of this expression, for the
Warren and Price calculation each of the three integrals are
simple to evaluate: , ( )$ $n n dqFaceS
=( ) ( ) = r n Pda L P A$
, and . The resulting
transmissibility is given by
(dv L A = )
T Q PX = , again as expected. Upscaling for Well Productivity.
Following [19], the productivity of a vertical well is proportional
to the horizontal permeability, times the gross length of the
wellbore.
Well Pi KX KY DZ~ ..........................................
(A-8) Consider the stack of fine cells of Figure 3. We can drop a
vertical well through each of the nine fine columns, and calculate
the total productivity of each. On the possibility that different
columns have different cross-sectional areas, we can define the
average productivity as the areal weighted sums of the PIs. This in
turn defines an average horizontal permeability. Similarly,
consideration of well productivities for horizontal wells in the i
and j directions, provides two other averages.
{ }{ }{ }
KX KY DV KX KY DV
KX KZ DV KX KZ DV
KY KZ DV KY KZ DV
EFF EFFi j k
i j k i j ki j k
EFF EFFi j k
i j ki j k
i j k
EFF EFFi j k
i j k i j ki j k
=
=
=
, ,, ,
, ,, ,
, ,, ,
, ,, ,
, ,, ,
, ,, ,
.. (A-9)
Once these averages have been taken, they can be converted to
three directional permeabilities algebraically. The three
directional permeabilities are defined with respect to the fine
grid. If the coarse grid has a different orientation, then the
elements of the resulting permeability tensor will need to be
evaluated in the coarse coordinate system before use.
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14 M.J. KING, D.G. MACDONALD, S.P. TODD, H. LEUNG SPE 50643
Figure 1 - (a) A simple sand/shale reservoir zone with three
different calculations for effective vertical permeability (b) With
sealed sides, the vertical permeability vanishes (c) With linear
pressure boundary conditions the vertical permeability is positive
(d) With a wide computational region the vertical permeability is
also positive as flow diverts around the shales.
Figure 2 - Magnus reservoir stratigraphy showing the MSM and
LKCF sector model locations
Figure 3 - Resampling of the Magnus MSM geologic model, 99 x 21
x 352 to 99 x 21 x 375, showing the 3 X 3 fine columns which
upscale to one coarse column.
Figure 4 - Detailed examination of the Magnus MSM permeability
pattern shows that mud dimensions have increased at the expense of
sand channels, leading to an overly optimistic estimate of
remaining oil targets.
Figure 5 - Loss of sand permeability and channel continuity when
upscaling using the standard flow based (sealed sides) upscaling
algorithm. The dot signifies a well, whose performance is
significantly degraded within the upscaled model.
Figure 6 - Contrast of the Andrew reservoir performance
prediction calculated from a fine scale sector model, that model
upscaled using conventional techniques, and after adjusting some of
the vertical transmissibility multipliers by hand. The half cell
technique gives answer indistinguishable from the fine scale
prediction.
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SPE 50643 APPLICATION OF NOVEL UPSCALING APPROACHES TO THE
MAGNUS AND ANDREW RESERVOIRS 15
Figure 7- Two representations of shale within a flow simulator.
In the first, PERMZ=0, and the effective thickness of the shale is
2*DZ. In the second, MULTZ=0, and the shale now has no volume but
impacts the flow as if it had the minimum thickness of DZ.
Figure 8 - Performance of the Andrew FFM, showing the resolution
of gas under-run, after use of the vertical half cell upscaling
approach.
Figure 9 - Successful prediction of GOR in the Andrew A04 well.
The most likely case corresponds to the simulation of the previous
figure, while the P90 case follows from the use of conventional
upscaling.
Figure 10 - LKCF geocellular model, showing sands (yellow) and
regions of injected sands (red) within a mud background.
Figure 11 - Well Productivity based permeability upscaling,
preserving the sand channel of Figure 5. Inter-cellular barriers
are now modelled as transmissibility multipliers on the face of the
cells. Colour coding in this figure is only approximate.
Figure 12 - Time of Flight visualisation of the flooding pattern
in the 1.7 million cell (64 x 64 x 452) LKCF geologic model, with
two downdip injectors and two updip producers. The flow calculation
is in three dimensions but the results are presented as a fence
diagram.
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16 M.J. KING, D.G. MACDONALD, S.P. TODD, H. LEUNG SPE 50643
Figure 13 - Time of Flight visualisation of the flooding pattern
in the 70,000 cell upscaled LKCF model (32 x 32 x 72), using the
half cell upscaling approach. As in the previous figure, the flood
is in three dimensions.
Figure 14 - Additional validation, showing pressures and time of
flight for the 70,000 cell model. In this calculation pressures are
specified at the wells, instead of the fluxes, as in the previous
two figures.
Figure 15 - Pressure and time of flight information for the
70,000 cell model upscaled using conventional sealed side flow
based upscaling. This model has only 5% of the effective
connectivity of the model in the previous figure.
Figure 16 - A two cell coarse model whose effective horizontal
permeability is very difficult to represent. Most calculations
would merge the non-communicating sands into one larger sand.
Figure 17 - Pseudoisation of Effective Fractional Flow for IMPES
and Fully Implicit numerical schemes
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SPE 50643 APPLICATION OF NOVEL UPSCALING APPROACHES TO THE
MAGNUS AND ANDREW RESERVOIRS 17
Figure 18 - Effective Fractional Flows for the 13 coarse cells
of the Magnus MSM simulation
Figure 19 - The outlet block water saturation plotted against
the outlet water saturation. With few cells the dispersion
correction is important. At late times, saturation profiles are
more uniform, and dispersion corrections are not important.
Figure 20 - Pseudo fractional flow curves. Dispersion provides
more of an impact on the different curves than the physical
mechanisms
Figure 21 - Rock curve inverse total mobility function, showing
the quadratic fit used, especially below the shock saturation
Figure 22 - Pseudo relative permeability to water
Figure 23 - Pseudo relative permeability to oil
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18 M.J. KING, D.G. MACDONALD, S.P. TODD, H. LEUNG SPE 50643
Figure A-1 - Upscaling of effective permeability
Figure A-2 - Upscaling of effective transmissibility