1
A015 CLASSIFICATION OF PSEUDO FUNCTIONS
USING STATISTICAL ANALYSES
1
2
3
Pierre SAMIER , Eline BAGNOLET , Philippe DUPOUY , Jean Paul VALOIS
1
1 TOTAL CSTJF Avenue Larribau 641018 PAU Cedex – France
2 IMPERIAL COLLEGE Earth Science and Engineering – London - UK
3 TOTAL YEMEN – SANA’A - YEMEN
Abstract
In reservoir engineering, although upscaling absolute permeability is now a well-established technique, upscaling
relative permeability and capillary pressures remains still a complicated issue.
Pseudo relative permeabilities and capillary pressures are used in reservoir simulation in an attempt to capture the
effects on multi-phase fluid flow of heterogeneities not represented on the coarse simulation grid and to compensate
for numerical dispersion effects. Dynamic pseudo relative permeabilities are generated from the results of a fine grid
simulation model as opposed to those calculated under an assumption of capillary and gravity equilibrium.
Numerous pseudo relative permeability methods have been published and a few industrial tools do exist to compute
them. Applied to a full field reservoir model, it ends up to a huge amount of pseudos (3 curves for each gridblock
(one per direction) and for each fluid phase).
Among all these pseudos curves, some display unphysical shapes or are defined with a small variation of the fluid
phase saturation.
The paper describes a methodology to sort all these pseudo curves in a limited amount of classes using a statistical
tool and to retain a suitable amount of pseudos. In order to keep a link with the geological model, the classification
is linked to the regional area associated to the lithology and the rocktype.
Introduction
Relative permeability curves are usually measured in the laboratory on small cores which are
generally homogeneous. Then they are used in multiphase flow simulations where they
determine the relative flow rates of each phase for coarse grid block that are typically tens or
hundred of meters in areal extent and more than one meter in vertical extent. The volume
associated to each grid block of the dynamic simulation model is certainly not homogeneous.
It is therefore necessary to apply some upscaling procedures which generate pseudo relative
permeabilities. Many techniques 1,2,3,5,6,16,17 have been proposed for upscaling relative
permeabilities, each having its own particular assumptions, approximations and range of
applicability. Reference 3 indicates a review of these techniques.
In the present study, pseudos are generated with methods based on fine scale simulations. One of
these methods is the Pore Volume Weighted method. This method is derived from the Kyte and
Berry method7 and is available in a commercial software PSEUDO TM 18.
This specific method and most of the other pseudoisation techniques generate three sets of
pseudos for each active coarse grid block inside the simulation model (one per direction).
Generally, among all the generated pseudos curves, some are not useable inside the simulation.
It occurs in zones where the saturation of one phase does not vary in the fine scale simulations.
Grouping pseudos
In this paper, we examine the problem of grouping pseudos in an a posteriori manner. In
practice, it is not possible to work with all the different sets of generated pseudos. For a standard
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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medium full field reservoir model using around 100 000 active cells, the upscaling of the
relative permeabilities is performed on one or several representative sector models which are a
reduced part of the full field. The process ends up with a total of about 15 000 sets of pseudos
(5000 sets of Krw, Krg, Kro, Pcw and Pcg curves per direction).
It is not realistic to use all the generated sets in the final full field simulation model.
The objective of this paper is to examine solutions for efficiently grouping theses curves and
reducing the number of sets to a suitable value ranging from 3 to 100. A number of 10 to 20 sets
per direction is a usual practice. In order to keep a link with the geological model, the grouping
is linked to the regional area associated to the lithology and the rock type. Instead of indifferently
grouping all the pseudos, the process will start with a partition of all the pseudos in 3 N groups
where N is the number of rock types of the fine grid model (geological or petrophysical model).
The grouping is based on the dominant rock type (also named dominant facies) method which is
the simplest upscaling method for relative permeability. If a given grid block of the dynamic
model is composed of 5 different rock types, the principle is to characterize the heterogeneous
grid block by the rock type number associated to the higher pore volume percentage. This
characterization method is available in all commercial geomodelers (discrete mean “upscaling”
method).
With this method, a dominant rock type number is assigned to each heterogeneous grid block of
the simulation model. Therefore, all the pseudos will be partitioned into 3N groups. The
grouping process starts with 3 N individual files of pseudos curves
Previous Work
We are aware of only three previous papers that have addressed the subject of a grouping of
pseudos 8,9,10. The first two papers make suggestions for how to perform the grouping but neither
presents any results indicating the usefulness of these suggestions. The third paper introduced a
cluster analysis method with a global set of parameterized pseudos. All the previous papers start
the grouping using all the pseudos. It seems that the initial partition into 3 N groups of pseudos
where N is the initial number of rock types for 3D problem has not been yet published.
Saad et al.8 suggest grouping the pseudos either according to functional models fitted to the
curved or according to the endpoint relative permeabilities. In their paper, only the latter method
is used: seven groups were created by dividing the range spanned by the water endpoint relative
permeability into seven equally spaced bins. A curve with endpoint relative permeability equal to
the value at the middle of each bin was selected as the representative curve for that bin. The
authors do not present any results obtained using just a few sets of pseudos grouped in this way.
Use of endpoint relative permeability values alone seems to us to be inadequate, as all
information about the shape of the curves will be lost.
Christie9 suggests a parameterization of each set of two-phase pseudo relative permeability
curves in a more physically meaningful manner. The pseudos are parameterized using three
parameters indicated below and computed according to the Buckley –Leverett solution.
Simple grouping method (SGM)
In our attempts at grouping discretized sets of pseudos, we start with a simple classification
method. The main idea is to select one pseudo per dominant rock type and per direction. The
main advantage is to keep the regional area (dominant rock type) associated to the coarse
dynamic model and to use it as a new saturation region for the pseudos. This regional array can
be computed easily by any commercial geomodeler or grid generator. The N initial sets of
3
relative permeability associated to the fine grid model are substituted by 3 N new sets of pseudos
associated to the coarse grid model. The numbering of the saturation region remains unchanged,
The saturation regions 1 to N associated to the dominant facies correspond to the pseudos in the
X direction, the regions N+1 to 2N in the same order correspond to the Y direction and the
region 2N+1 to 3N corresponds to the Z direction.
Two sorting criteria are chosen by the user: the number Nc of classes and the type of table used
in the sorting (either krg-krw, krog-krow or Pcw-Pcg). A simultaneous sorting on both krg/krogkrw/krow could be an extension.
The main steps of the algorithm are described below. The sorting is performed independently on
the 3 N groups of pseudos (indicated before).
The pseudos associated to a given dominant rock type and a given direction, are first normalized
and discretized in 11 samples. Linear interpolation is used if necessary (e.g. if there are missing
values, more than the sample step between two values). An arithmetic average based on all
pseudos of the group is then calculated to get the average curve. The distance of each pseudo to
this average curve is computed. The square root of the difference of squares is currently used but
other distances could be considered. The pseudo closest to the average curve is selected as the
reference curve. Pseudos are then sorted into Nc classes according to their distance to the
reference curve. The class with the greater number of samples is selected and, in this class, the
curve with the higher saturation range is selected. This curve becomes the new reference curve.
The algorithm is run till the reference curve remains the same for two successive iterations. This
is repeated for the three directions and for all the dominant rock type.
This algorithm has many advantages. It converges in only two or three steps on our examples.
The reference curve is one pseudo curve. The selected class contains lots of curves, thus
avoiding marginal classes, and the selected curve lots of points, thus avoiding working with too
few information. Finally, the algorithm can easily be extended to the selection of several
representative pseudos. The main assumption of this simple sorting method (SGM) is that a
representative pseudo curve among all the set of pseudos can be uniformly associated to all the
grid blocks having the same dominant facies.
Cluster Analysis
In this type of grouping, we will obtain a larger number of pseudos than the initial number of
rocktype curves. The regional array (saturation region) associating the pseudo curve number to
the coarse grid block needs to be renumbered.
The main idea behind this study is thus to use a class of multivariate statistical techniques known
as Cluster Analysis 11 to perform the grouping. Several such techniques are available in
commercial software packages: The SASTM program12 is used here. The aim of these techniques
is to divide a set of objects into groups or clusters in a manner suggested by the data rather than
one define in advance. Objects in the same cluster should be in some sense similar, while objects
in different clusters should be dissimilar.
In our case the objects are the sets of pseudo relative permeability curves. Each set is represented
by a finite number of values which is the values of krw or krg and kro at certain values. Theses
values define the position of each set in a N dimensional space and the similarity between two
sets is then measured by the distance between them in space.
At this stage, two options are possible:
1. Parameterized sets of pseudos: Once the pseudos have been calculated, the corresponding
fractional flow and total mobility are computed and the Buckley-Leverett solution is
found by constructing the appropriate tangent to the fractional flow curve. The pseudos
curves can be parametrized with for example 3 parameters as suggested by Christie9 (the
shock front saturation (i.e. the saturation at the tangent point), the slope of the fractional
flow curve at the tangent point, and the minimum value of the total mobility curve).
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With this parametrization, two sets of pseudos would be considered to be similar if they
have the same (or nearly the same) values of these three parameters, even if their shapes
are otherwise quite different. This option has been addressed in a previous paper 10
.
2. Discretized sets of pseudos: The saturation range is the same for all tables. It was divided
in 10 intervals and values were extracted by linear interpolations at 11 water saturations.
The grouping and the distance is computed based on these 11 values per curve. Irregular
(non monotonous) pseudos are corrected in substituting krj to the previous kri and a
cumulative error indicator is associated to these curves. If the error indicator is too high,
the curve is removed and replaced by the rocktype curve of the dominant facies.
The present paper retains the second option which removes any possible loss of information
caused by the parameterization. The second option allows also to separate pseudos with different
shapes if the values of the 3 parameters are similar. Capillary pressures (Pcw and Pcg) can also
be more easily incorporated in the second approach. Unlike the parameterized approach which is
based on the physics of flow, this approach becomes purely statistical. The resulting parameters
are called “principal components”. This method has been considered as an alternative to the
computationally expensive grouping of sampled curves.
A particular class of cluster analysis method is called AHC “Agglomerative Hierarchical
Clustering”. The various methods of this type differ in how the distance between two clusters is
measured. The method used is the minimum variance method of Ward 13,14. In this method, the
distance between two clusters whose centroids lie at the point xa and xb and which contains Na
and Nb elements, respectively, is defined as:
x −x 2
Dab =
(
a
b
1
1
+
Na Nb
)
This method was chosen as it tends to join clusters containing a small number of objects and
hence is bases towards producing clusters with roughly the same number of objects. Several
standard way of monitoring the quality of the clustering as a function of the number of groups
are possible12. The simplest is the inertia, the sum of the distances between all pairs of elements.
Since the number of pseudos can be very large (more than 5000 sets), the AHC method (limited
to 1000 sets) cannot be applied directly. Different methods can be used to reduce the number of
values. One alternative is to perform a principal Component Analysis (PCA) of the discretized
data. We use the standard method KMEANS15 available in SAS. The clustering is applied to the
reduced set of curves.
A hierarchy of partitions is obtained in a step by step process where the closest elements of the
population are grouped by pairs to form a new element for the next step. The result appears as a
tree (Fig. 1). The possible classes are obtained by cutting the branches of the tree: the further
from the root, the more homogeneous and numerous classes. Figure 1 represents the proximity
between classes. The number of samples and the number of each class are indicated on the left.
On the example, a convenient choice of 6 classes is represented by the vertical line (thick line).
The quality of a partition depends on the homogeneity within the classes and the heterogeneity
between the classes. The algorithm minimizes the inertia within each class and maximizes the
inertia between the classes. These two indicators are represented on Fig. 2 where the vertical axis
indicates the distance between nodes of the tree versus the number of classes. On the example, a
choice of 6 classes allows a good compromise between a high level of details and synthetic
vision of the population.
The analysis of these curves allows retaining the best compromise between the number of classes
and the quality of the partition.
5
Class #
#of samples
Figure 1: Classification tree
Figure 2: Distance versus number of classes
THE TEST CASES
The validation study is conducted on SPE1 and SPE10 model. These models are simple and
provide good example to test the method. Moreover, the results of the study could be clearly
understood and compared. The main characteristics for our study are described below. Their
whole description is given in the SPE papers 21,22.
CASE 1: from SPE10 model 1
The fine scale grid (Fig. 3) is a vertical cross section of 100*1*20 with uniform size cells. The
reservoir is initially saturated with dead oil. We added 15% of connate water and dummy
krw/krow curves to run the model with a standard black-oil simulator and to test our sorting
software on a basic 3 phase model. The absolute permeability distribution is a correlated
geostatistically generated field. There is no capillary pressure effect. Gas is injected from the left
of the model and oil is produced from the right of the model.
Figure 3: Fine scale grid: permeability distribution and well locations
The aggregation rate (20*1*4) is uniform, the upscaling results in a coarse grid of 5*1*5 cells.
There is only one rock type and one set of relative permeability at the fine scale level. Pseudos
will correct from numerical diffusion.
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CASE 2 : from SPE1 model
The SPE1 model is a 10*10*3 grid and consists in our coarse grid. The reservoir is stratified in
three layers. The corresponding fine grid dimensions are 50*50*12. Each layer of the coarse grid
is refined into 4 layers according to Table 1. The reservoir is initially filled with 88% oil and
12% of connate water. For this case, in order to test the influence of several rock types, the layers
of the fine scale grid are populated by four different facies (4 sets of relative permeability
curves) as indicated in Fig. 4. There is no capillary pressure effect. Gas is injected from the left
corner of the model and oil is produced from the opposite right corner.
1,00
Rock type
number
1
2
2
2
3
4
3
3
3
3
4
3
Table 1: Model 2 layering
0,80
Set 1 krg
Set 1 krog
Set 2 krg
Set 2 krog
Set 3 krg
Set 3 krog
Set 4 krg
Set 4 krog
0,60
Kr
Corresponding
coarse grid layer Fine grid layer
1
2
3
4
1
5
6
7
8
2
9
10
11
12
3
0,40
0,20
0,00
0,00
0,20
0,40
0,60
0,80
1,00
Sg
Figure 4: Krg/krog curves for the 4 facies
RESULTS CASE 1 :
Pseudos using the Pore volume Weighted method have been generated for every coarse grid
block and the two flow directions (2D problem). The total number of generated pseudos for this
oil-gas problem is 42 (18 for the X direction and 24 in the Z direction). The total number of cells
is 25. Six grid-blocks among the 25 have no pseudo in the X direction (boundary cells) and 1
grid block has no pseudo in the Z direction. Among these 42 pseudos, we eliminate 9 pseudos
(unrealistic values, non monotonous values). The grouping procedure starts with a total of 33
pseudos.
Since there is only one rock type, and 2 active directions, the pseudos are partitioned in 2 groups.
The cluster analysis has been carried out with 4 and 2 classes. Figure 5 indicates the results on
the cluster analysis with 2 classes. The red dots correspond to the pseudo of the class #1 and the
blue dots to the pseudo associated to the class #2.
Figure 6 compares the cumulative oil production given by the following methods:
• the fine grid or reference solution (black line)
• the coarse grid with all the 33 pseudos (red line),
• the coarse grid with the fine grid rock type (no upscaling) – green line
• a manual sorting choosing an arbitrary pseudo (blue line)
• the simple grouping method (SGM) with 1 pseudo per direction (yellow curve)
the cluster method (orange line) with 2 pseudos per direction.
Due to numerical dispersion effects, the simulation using the rock type curve (no upscaling)
overestimates the oil production. It can be noticed that for this case, the pseudoisation method
PVW gives production results very close to the fine grid solution.
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Figure 5: Results of the cluster analysis Figure 6: Cumulative oil production for different
grouping of pseudos
The manual grouping method is worse than using the fine scale rock type. The SGM and the
cluster method give similar results close to the fine scale solution. The SGM method seems
slightly better. For this case with a small amount of pseudos, the cluster method which has been
tailored for a high number of pseudos (more than 5000) is not the most adequate method.
Also, grouping pseudos is better than keeping all pseudos, the grouping methods eliminate the
pseudos with bad shape or with a small range in saturation..
RESULTS CASE 2 :
Pseudos using the Pore volume Weighted method have been generated for every coarse grid
block and the three flow directions. The total number of generated pseudos for this oil-gas
problem is 756 (264 for the X and Y directions and 225 in the Z direction). The total number of
cells is 300. The grouping procedure starts with a total of 756 pseudos. The number of rock
types at the fine grid level is four but the number of dominant facies is 2 since two minor facies
(#1 and #4) have disappeared in the upscaling process. According to the methodology described
before, all the pseudos are partitioned into 6 groups (3 directions and 2 dominant facies).
The cluster analysis has been carried out with 3 classes. Figure 7 indicates the results of the
cluster analysis. The red dots correspond to the pseudo of the class #1, the blue dots to the
pseudo associated to the class #2 et the yellow dots to the class #3.
Figure 8 compares the cumulative oil production given by the following methods:
• the fine grid or reference solution (black line)
• the coarse grid with all the pseudos (red line),
• the coarse grid with the fine grid rock type (no upscaling) – green line
• the simple grouping method (SGM) with 1 pseudo per direction (yellow curve)
• the cluster method (orange line) with 3 pseudos per direction
Without pseudoisation, results between the fine grid and the coarse grid are quite different. The
pseudoisation with the PVW method improves the coarse grid results but the coarse model still
overestimates the oil production. It could be interesting for this case with several rocktypes to try
other pseudoisation techniques. As previously, the two grouping methods give results very close
to the simulation using all pseudos. The SGM method is closer to the “all pseudos” simulation
while the cluster method is closer to the fine scale solution.
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Figure 7: Cluster analysis with 3 classes
Figure 8: comparison of cumulative production
Conclusions
In this paper we have examined the question of grouping pseudo relative permeabilities curves
from an a posteriori point of view, i.e. once a set of pseudos has been generated for each grid
block and each flow direction.
For the two simple examples we tried, we found that:
• the cluster analysis technique can efficiently replace all the pseudos by a small amount of
classes.
• the simple grouping method retaining only one pseudo per direction and per dominant
facies is also efficient.
• it is necessary to use different pseudos for each direction of the flow to approach the
production results obtained with all the pseudos.
The comparison between the fine grid and the coarse grid results has been performed only on
production results. A more detailed comparison of the difference in flow pattern could be done
using streamlines simulation 20 and could be helpful to better make the difference between the
two grouping methods.
The next step is to apply the methodology on a real field case.
Acknowledgment
The authors thank John Barker and Lisette Quettier from Total in Pau from many helpful
discussions and are grateful to TOTAL and Imperial College for permission to publish this work.
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