Post on 24-Jun-2018
TECHNICAL PAPER
A method for reduction of uncertainties in reservoir model usingobserved data: application to a complex case
Celio Maschio • Denis Jose Schiozer
Received: 20 June 2013 / Accepted: 27 December 2013 / Published online: 6 February 2014
� The Brazilian Society of Mechanical Sciences and Engineering 2014
Abstract The aim of this paper is to show a methodology
to reduce uncertainties in complex reservoir models using
observed dynamic data. The basic idea is the use of the
difference between observed and simulated data to con-
strain the probability redistribution of the uncertain attri-
butes, reducing the spread of the posterior distribution and,
as a consequence, reducing the dispersion of the reservoir
response and mitigating risk. To capture the influence
between attributes and reservoir responses, an influence
matrix is proposed. The method deals with discrete and
continuous attributes, which permits an adequate repre-
sentation of the several types of uncertainties. The meth-
odology was applied to a complex case and promising
results are shown.
Keywords Reservoir simulation � History matching �Reduction of uncertainty
List of symbols
dobs Observed data
dsim Simulated data
M Misfit (difference between observed and
simulated data)
ML Misfit computed for each uncertain level
MLW Weighted misfit per level
MLN Normalized misfit for each uncertain level
n Number of model per level
pk Posterior probability of level k
ps Probability of each level for a given pair
attribute/data series
(pk)prior Prior probability of level k
R2 Coefficient of determination
Rc2 Cut-off value for coefficient of determination
S Total number of data series considered in the
analysis
Si Number of influenced data series
UL Number of uncertain levels per attribute
wd Weight factor (0 or 1) to indicate data series with
wh \ whc
wh Indicator of influence for discrete attributes
whc Cut-off value for the indicator of influence for
discrete attributes
x Number of observed data
1 Introduction
Uncertainties are always present in reservoir characteriza-
tion because it is not possible to measure properties, such
as porosity, permeability, fluid saturations etc., by direct
methods in all extension of the reservoir. Observed data,
such as rate and pressure measured in the wells during the
production, can be used indirectly to reduce uncertainties.
The traditional way of incorporating dynamic data in res-
ervoir characterization is called history matching, which
normally seeks a unique matched model. The problem is
that a single history-matched reservoir model is usually
insufficient to address risk and uncertainty issues in res-
ervoir management. Due to several sources of uncertainties
Technical Editor: Celso Kazuyuki Morooka.
C. Maschio (&) � D. J. Schiozer
DEP/FEM/UNICAMP/CEPETRO,
Caixa Postal 6122, Campinas, Sao Paulo 13.083-970, Brazil
e-mail: celio@dep.fem.unicamp.br
D. J. Schiozer
e-mail: denis@dep.fem.unicamp.br
123
J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918
DOI 10.1007/s40430-013-0126-7
related to reservoir characterization and modeling and
measure error in observed data, history matching is an ill-
posed inverse problem with multiple solutions.
The process of uncertainty reduction using dynamic data
in complex cases, for instance, reservoir with many wells,
complex interactions among attributes, complex fluid flow
patterns etc., is a very difficult task. Many efforts have
been made in the last years to develop efficient methods to
deal with this problem; however, many challenges are still
present in this process.
There are, in the literature, several approaches to deal
with the problem. Reis et al. [1], Lisboa and Duarte [2] and
Becerra et al. [3] showed a methodology that uses a cut-off
value in terms of objective function (that measures the
quality of the history matching) to filter the better adjusted
models. In these works, the history objective function was
evaluated by proxy models.
Several authors have used stochastic optimization
methods to quantify uncertainty with the incorporation of
production data. Li and Reynolds [4] used stochastic
Gaussian search direction (SGSD) algorithm for automatic
history matching and uncertainty quantification. Mohamed
et al. [5] carried out a comparative study to investigate the
efficiency of three stochastic algorithms (Hamiltonian
Monte Carlo, Particle Swarm Optimization and Neighbor-
hood Algorithm) in the generation of multiple history-
matched models. The use of an evolutionary algorithm to
obtain several history-matched reservoir models was pre-
sented by Al-Shamma and Teigland [6]. They also used a
proxy model based on Latin Hypercube Design to assess
uncertainty in cumulative oil prediction related to the mat-
ched models. Abdollahzadeh et al. [7] presented a Bayesian
optimization algorithm applied to uncertainty quantifica-
tion. Hajizadeh et al. [8] presented the application of ant
colony optimization for history matching and uncertainty
quantification. Martınez et al. [9] evaluated the performance
of a family of Particle Swarm Optimization (PSO) methods.
Becerra et al. [3] presented a comparative study between
the two previous approaches and applied both to a real
field. The authors pointed out the advantages and disad-
vantages of each method.
The use of Bayesian statistics as a framework to incor-
porate dynamic data in the uncertainty quantification has
been increased in the last years. Subbey et al. [10] used the
Neighborhood Approximation sampling algorithm com-
bined with a Bayesian framework to generate posterior
probability distribution of the uncertain attributes. The use
of Bayesian techniques to generate probabilistic production
forecasts is also in the work of Schaaf et al. [11] and Slotte
and Smørgrav [12]. More recently, some authors have also
presented the Ensemble Kalman Filter as an attractive
method to integrate the history matching with uncertainty
analysis [13, 14].
In practical history matching cases, there is a trade-off
between physics and statistics in the solution of this com-
plex inverse problem. It is not feasible to perform a rig-
orous (full) statistical analysis combined with the full
physics of the problem. For instance, it is unfeasible to
perform a Monte Carlo evaluation, for example, with ten
(or more) thousands of reservoir simulations of complex
fields. Therefore, in principle, it is necessary to simplify the
physics or the statistics. In this text, the simplification of
the physics means the representation of the reservoir model
by a simplified (or approximated) model that is known in
the literature as proxy models. The advantage of proxy
model is that it permits fast evaluation of the objective
function, allowing an exhaustive exploration of the search
space. However, proxy models are not able to capture the
full physics of the problem, especially in cases with high
nonlinearities. Most of the works previously mentioned use
the first option, where proxy models are used to emulate
reservoir simulator behavior.
The present paper is in the context of a procedure that
uses the second option. Instead of using proxy models,
complete reservoir simulations are run in the proposed
method, incorporating the full physics of the problem. All
assessment of the reservoir response and computation of
the objective function are done with the reservoir simula-
tor. From statistics point of view, instead of performing an
exhaustive sampling in a single iteration, gradual sampling
of the search space is performed, allowing an iterative
process.
The first ideas were proposed by Moura Filho [15]. In
this work, the first attempt was carried out with a simple
reservoir model with only four uncertain attributes. Mas-
chio et al. [16] applied the same methods proposed by
Moura Filho to other models and also compared the results
with conventional history matching. Becerra [17], Maschio
et al. [18] and Becerra et al. [19] presented some advances
of the methodology. However, in these works, the use of
the derivative tree as statistical technique to compose the
uncertainties represents a limitation for the study of cases
with a high number of uncertain attributes. The derivative
tree can be unfeasible, depending on the case, for cases
with more than 7 or 8 attributes.
To circumvent the limitation of the derivative tree,
Maschio et al. [20] proposed the use of a sampling tech-
nique (Latin Hypercube) to combine the uncertainties,
which allowed the study of cases with more attributes. The
authors studied two cases, one of them with 8 and the other
with 16 uncertain attributes.
More recently, Silva and Schiozer [21] applied the
methodology to a very complex case and they have
detected some limitations. One of them is the manner of
composing the objective function. The case studied by the
authors has 50 wells (32 producers and 18 injectors) with a
902 J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918
123
high variability in the magnitude of the production data
among the wells and very complex interactions between
the attributes and the reservoir response. The authors pro-
posed a method for probability redistributions based on an
average weighted by sensitivity indexes that represent the
influence of the attributes on the reservoir responses. For
each attribute, the maximum and minimum values are
evaluated against each simulation output (well water rate
and well pressure). In this way, the most influenced
response has a higher weight in the probability redistribu-
tion of a given attribute.
The method proposed by Silva and Schiozer [21] was an
evolution of the previous mentioned works in the context
of this methodology. However, the sensitivity index only
captures the isolated influence of an attribute, because in
the sensitivity analysis, each limit is varied one at a time
and for one attribute at a time. Therefore, the sensitivity
analysis does not capture the cross effects among attributes
when several combinations are generated. Silva and
Schiozer [21] also used derivative tree to combine the
uncertain attributes.
The main objective of the present work is to show other
improvements of the methodology. The two main innova-
tive aspects treated in this work that represent an evolution
of the previous works are:
1. The use of an influence matrix to capture the influence
between attributes and reservoir responses, and;
2. The treatment of attributes with continuous and
discrete characteristics, allowing improvements in the
representation of the uncertainty using a higher
number of levels.
In addition, following the same idea presented by
Maschio et al. [20], in this work, sampling techniques were
employed to combine the uncertain attributes, with the
objective of circumventing the limitation of the derivative
tree technique, allowing the study of a higher number of
attributes.
2 Methodology
The methodology presented in this work is inserted in a
general context of integrating uncertainty analysis and
history matching. The main idea is the use of observed data
to reduce uncertainty in the reservoir attributes, treating the
history matching problem in a probabilistic context. The
first step of the process is the definition of the reservoir
uncertainties. This comprises reservoir characterization
tasks using all available data and the definition of prior
probability distributions for the uncertain attributes. The
second step consists of combining the uncertainties defined
in the first Step. This combination is carried out using a
given sampling technique. Each combination generated in
the second step corresponds to a reservoir model, which is
submitted to the flow simulator. The subsequent steps
comprise the use of the information provided by the flow
simulations to infer posterior probability distributions
based on the values of the misfit (difference between the
simulated and observed data).
The general steps of the methodology (Fig. 1) are
summarized below:
1. Characterization of the reservoir attributes, defining
the prior distribution, variation range and number of
uncertain levels (in the case of discrete attributes).
2. Combination of the uncertainties using a statistical
technique. In this work, Latin Hypercube Sampling is
used.
3. Run simulation models.
4. Reading of output simulator and computation of the
misfit for all models.
5. Composition of the influence matrix.
6. Computation of the posterior probability distributions.
7. Evaluation of the final probability density function: in
this step, a number of models are generated by
Fig. 1 General methodology flowchart
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123
sampling the posterior pdf and submitting them to the
flow simulator to evaluate the dispersion of the curves.
The assessment of the results is shown schematically
in Fig. 2.
The main contributions of the present work are related
to the Steps 5 and 6, which are described in the next sec-
tions. All steps, except Step 1, are automated in a program
written in MatLab. This process can be repeated until the
quality of the results reaches the criterion based on the
judgment of the professional involved. This criterion can
be based, for instance, on the dispersion of the production
and pressure curves with respect to the history.
2.1 Influence matrix
The influence matrix proposed is a robust way to capture
the relationship between attributes and reservoir responses.
It is composed by indicators computed for discrete (wh) and
continuous (R2) attributes separately. Next, the description
of the methods developed to compute both types of indi-
cators is presented.
2.1.1 Discrete attributes
For discrete attributes, an indicator (wh) that captures the
influence of a given attribute in a given component of the
objective function is proposed. Suppose two discrete
attributes, A and B (Fig. 3) each of them with three
uncertain levels. The combination of the three uncertain
levels of the two attributes results in nine models, as shown
in Fig. 4. For each model, the misfit (M) is calculated
according the Eq. 1.
M ¼Xx
i¼1
diobs � di
sim
� �2 ð1Þ
where diobs and di
sim are observed and simulated data,
respectively and x is the number of observed data.
Let n the number of models corresponding to each level
(in the example in Fig. 4, n = 3). The misfit per level (ML)
is computed as follow (Eq. 2):
ML ¼Xn
j¼1
Mj ð2Þ
Based on Fig. 4, the sum of misfit of the three red curves
corresponds to ML1, the sum of misfit of the three green
curves corresponds to ML2 and the sum of misfit of the
three blue curves corresponds to ML3. The normalized
misfit for each uncertain level (MLN)k is computed
according to the Eq. 3:
ðMLNÞk ¼1
ML
� �
kPULk¼1
1
ML
� �
k
ð3Þ
where UL is the number of levels for each attribute.
MLN is a quantity between 0 and 1 that is inversely
proportional to the misfit per level. The indicator wh is then
computed as follows (Eq. 4):
wh ¼XUL
k¼1
ðMLNÞk �1
UL
����
���� ð4Þ
If MLN is close to 1/UL, expressing the condition by
which no influence occurs, wh is close to zero. The
meaning of the indicator wh is illustrated in Fig. 4, in which
there are three hypothetical producers wells. The curves are
grouped according to the uncertain levels (L1, L2 and L3) of
the attribute A. Suppose, for example, that the attribute is
the absolute permeability and the uncertain levels are 500,
1,000 and 1,500 mD. In the case of wells 1 and 2, three
distinct groups of curves are noted, separated by colors.
One can note that the curves that contain the level L1
(permeability equals 500 mD) are close to the history. On
Fig. 2 Assessment of the posterior probability distribution
Fig. 3 Example of two discrete probability distributions
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the other hand, in the case of well 3 there is not a defined
group of curves, that is, the curves are mixed, indicating
that this well is not influenced by the attribute A. This is
denoted by the low value of wh (0.07). On the other hand,
the higher values of wh for wells 1 and 2 indicate the
influence of the attribute A on these wells. The same
procedure is done for attribute B.
2.1.2 Continuous attributes
Although the objective is the same as in the case of discrete
attributes, that is, to capture the influence of the each
attribute with respect to the several component of the
objective function, for continuous attributes, the approach
is different.
Consider the hypothetical reservoir in Fig. 5 and three
regions of permeability (kx1, kx2 and kx3) defined around
three wells. Each region of permeability has an average
represented by the probability density functions, also
depicted in the Fig. 5. Using these pdf, n models are
sampled using a given sampling technique (Latin Hyper-
cube, for example). For each simulation model generated
from the sampling, the misfit of each well is computed.
Using the vector of kx1 and the vector of normalized misfit
of the well 1, the coefficient of determination of a poly-
nomial fitting, ðR2Þkx1;w1, is generated as shown in Fig. 6.
The coefficient of determination (R2) is an indicator of the
influence of the attribute on a given reservoir response. The
same procedure is done for kx1 with respect to well 2, and
so on, until all possible combinations have been done. In
this example, nine values of R2, supposing that the data
analyzed is only the water rate of the three wells, are
generated.
Low values of R2 denote no trend on the data, indicating
any influence of the attribute. The value of R2 is used as
filter to consider only the elements of the matrix that rep-
resent an influence of the attribute over the data series. For
each simulation model, the total number of R2 is equal to
the number of continuous attributes times the number of
data series considered. A theoretical example of data trend
can be seen in Fig. 7.
Theoretically, the criteria applied to continuous attri-
butes can also be applied to discrete ones with a number of
Fig. 4 Schematic
representation of the meaning of
the wh indicator
Fig. 5 A hypothetical reservoir with three regions of permeability
J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918 905
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levels sufficient for an adequate fitting. In other words, a
continuous attribute can be represented by a discrete
probability distribution (known as probability mass func-
tion) with a large number of levels (intervals). Attributes
such as porosity, for example, can be considered continu-
ous or discrete. However, under a practical point of view,
some attributes are discrete in nature, such as for example,
a PVT table, and normally, a great number of levels are not
available for this kind of attribute.
Generalizing this procedure for any number of attribute
(continuous and discrete) and reservoir response, the gen-
eral matrix (Fig. 8) is obtained. The total number of
coefficients in the matrix is equal to the number of attri-
butes (discrete plus continuous) times the number of data
series. An example is shown in the Results section.
2.2 Posterior probability distributions
The influence matrix described earlier is applied to filter
the reservoir response that is used to constrain the posterior
probability distribution. For discrete attributes, the weigh-
ted misfit per level, MLW (Eq. 5) is calculated for those
reservoir responses with wh greater than a given cut-off
value (whc):
MLW ¼ wd
XS
d¼1
ðMLÞd ð5Þ
where wd is a weight factor that can assumes the value 0 or 1.
wd = 0 states for the case where all the series have wh \ whc.
This means that the attribute has no influence in any data
series. S is the number of data series (reservoir responses)
considered in the analysis. For example, if a reservoir model
Fig. 6 Schematic representation of the meaning of R2 indicator
Fig. 7 Theoretical example of data trend with qualitative meaning
of R2
Fig. 8 General aspect of the influence matrix
906 J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918
123
has five wells and water rate and bottom-hole pressure are
considered for each well, then S = 10. When wd = 1, there
is at least one data series with wh [ whc.
The new probability (pk) is calculated for each uncertain
level according to the Eq. 6:
pk ¼1
Si
XSi
j¼1ðpsÞj; if wd ¼ 1
ðpkÞprior; if wd ¼ 0
8<
: ð6Þ
where Si is the number of influenced data series. From
Eq. 6, if wd = 0, the probability is equal the prior
probability. ps is the probability of each level for a given
pair attribute/data series given by Eq. 7:
ps ¼1=MLWkPUL
k¼1 1=MLWk
ð7Þ
Figure 9 shows a schematic example for the application
of the method. At the top of the figure is an hypothetical
reservoir model with three producers wells (w1, w2 and w3)
and two uncertain attributes are also indicated in the figure
(kx and MTF). kx states for horizontal permeability and
MTF means fault transmissibility multiplier. Suppose that
Fig. 9 A schematic example for discrete attributes
J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918 907
123
the date series considered in the analysis is the water rate
(Qw). Therefore, in this case S = 3 (total number of data
series). The influence matrix shows the indicators wh
computed for each pair (kx=Qw1, kxQw2
, kxQw3, MTF=Qw1
,
MTF=Qw2and MTF=Qw3
). The three plots on the left side
of the figure represent the pairs kx=Qw1, kxQw2
and
MTF=Qw3, for which (wh)1,1 = 1.06, (wh)1,2 = 1.05
(wh)2,3 = 1.18, respectively. As can be seen, these three
pairs have values of wh close to 1, representing the influ-
ence of the kx in the wells w1 and w2 (Si = 2) and the
influence of MTF in the well w3 (Si = 1). On the other
hand, values close to zero, as in the case of MTF/Qw1 and
MTF/Qw2 denote that the attribute MTF does not have
influence in the wells w1 and w2.
For continuous attributes, the proposed method is based
on the likelihood concept, which is common in statistical
inference. This method consists of computing the likeli-
hood based on the misfit value. This is done using an
exponential function, normally used in the literature [10,
12, 22, 23]. The computation of the likelihood proposed in
this work is conditioned by the values of R2. Suppose that
the two R2 values shown in Fig. 6 are greater than a given
cut-off value (Rc2). In this case, the lower and upper curves
are used to obtain an average curve, as shown in Fig. 10.
This average curve is used as input to constrain the prob-
ability density function a posteriori (Pdf), obtained using a
statistical routine available in MatLab.
Therefore, the posterior probability is a function of the
average misfit (blue curve). The lower the average misfit
(filtered by the condition of R2 greater or equal to Rc2 in the
influence matrix), the higher the probability. Based on the
example shown in Fig. 5, and supposing ðR2Þkx1;w3is less
than the cut-off value, the pdf a posteriori of kx1 is con-
strained by the wells 1 and 2. If none of the data satisfies
this criterion, that is, if the attribute does not influence any
component of the objective function, the prior distribution
is not changed. This means that the initial uncertainty
related to the attribute is kept. A value of Rc2 equals 0 has
the effect of disabling the influence matrix because all
reservoir responses (data series) will be considered. In the
Fig. 10 Probability a posteriori for continuous attribute. Pdf is a
function of the average curve, representing the average misfit. The
lower the average misfit (filtered by the influence matrix), the higher
the probability (color figure online)
Fig. 11 Three-dimensional
distribution of porosity of the
model
908 J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918
123
above example, the pdf a posteriori of kx1 would be con-
strained by the wells 1, 2 and 3. In the results, this situation
is named ‘‘without influence matrix’’.
3 Application and results
The proposed method was applied to a synthetic reservoir
model based on a complex off-shore field from Campos
Basin in Brazil (Fig. 11). The numerical simulation model
consists of a corner point grid with 83 9 55 9 14 cells
(49,346 active). The model consists of three blocks (A, B
and C) separated by two main north–south faults (Fig. 12).
There are 50 horizontal wells (32 oil producers and 18
water injectors) distributed in the blocks according to the
Table 1.
The description of the uncertain attributes is shown in
Tables 2 and 3. Table 2 refers to the attributes considered
as continuous in this application. They have 16 levels,
equally spaced between the minimum and maximum val-
ues, that is enough to allow a polynomial fitting. Table 3
refers to the discrete attributes, with three levels each one.
The levels of kr are shown in Fig. 13 and the fault models
are shown in Fig. 14. For PVT, the three levels were
generated according to the data shown in Table 4. The
prior uncertainty of the attributes was represented by uni-
form distributions.
A combination (reference values) was sampled among
the uncertain attributes and a reference model was built.
This model was used to generate a synthetic production
and pressure history. The reason to use a synthetic reser-
voir model (with a known response) concerns the valida-
tion of the procedure. In the following the results are
presented.
Fig. 12 Fault transmissibilities
Table 1 Oil volume and number of wells of the model
Block OOIP (9106 m3) Producers Injectors
A 162 10 6
B 227 11 6
C 148 11 6
Field 537 32 18
Table 2 Description of the attributes with 16 levels
Attribute Description Number of
levels
Min Max
PhiA Porosity multiplier in Block A 16 0.75 1.25
PhiB Porosity multiplier in Block B 16 0.75 1.25
PhiC Porosity multiplier in Block C 16 0.75 1.25
kB Permeability multiplier in
Block B (Top zone)
16 0.70 3.0
ki Permeability multiplier in the
bottom zone
16 0.5 1.5
kv Horizontal and vertical
permeability ratio
16 0.05 0.4
Table 3 Description of the attributes with three levels
Attribute Description Level
1
Level
2
Level
3
FAB Transmissibility multipliers
between Blocks A and B
0.0 0.01 1
Faults Small faults models n1 n2 n3
PVT PVT tables n1 n2 n3
kr Water relative permeability n1 n2 n3
Fig. 13 Uncertain levels (L1, L2 and L3) of the attribute kr
J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918 909
123
3.1 Analysis of the influence matrix
The indicator wh for the attribute FAB is shown in Fig. 15, in
which the water rate curves for well PROC-3, whose value of
wh is 0.012, are also shown. One can note that the curves are
mixed, indicating that there is no predominance of any of the
three uncertain levels. On the other hand, for well PROB-11
(wh = 0.98), it is possible to observe that the curves are
divided in groups, indicating a strong influence of the attri-
bute in this well. Each group of curves is associated to a level
of the attribute. The curves related to Level 0 (red curves) are
closer to the history. Analyzing the position of these wells
with respect to the fault between Blocks A and B (Fig. 16),
one can see that the well PROC-3 is far away of the fault and,
therefore, it is not influenced by this attribute. This means
that well PROC-3 is not used in the reduction of uncertainty
of the attribute FAB. The well PROC-11, however, is close to
the fault and then is strongly influenced by this attribute.
Figure 17 contains a fragment of the influence matrix
with values of R2 for polynomial fitting of order 1 and 2
(named in the figure as R2G1 and R2G2, respectively). High
values indicate strong influence of the attribute. It is
observed that there is a well-defined trend in the data (points
representing the normalized misfit). As the porosity multi-
plier increases, the misfit decreases. The red curve repre-
sents the polynomial fitting of degree 2 with R2 = 0.74.
Figure 18 contains water rate curves for the well PROB-
1. To facilitate the visualization, the curves were grouped
according to two ranges of PhiB. Red curves correspond to
models containing porosity multiplier on Block B varying
between 0.75 and 0.88 and blue curves correspond to
models containing porosity multiplier on Block B varying
between 1.10 and 1.25. It is possible to see two distinct
group of curves related to these two ranges. Also in
Fig. 18, is a similar plot with respect to the attribute PhiA,
in which it is not possible to distinguish the groups, indi-
cating that the attribute PhiA does not influence the well
PROB-1.
For this problem, the influence matrix has 640 coeffi-
cients (32 wells 9 2 data series—water rate and bottom-
hole pressure—times 10 attributes) for each simulation
model. The advantage of the influence matrix is that it
permits the automation of the process.
3.2 Influence of the indicators
The influence of the cut-off value of the indicators wh and
R2 is depicted in Fig. 19a, b, which correspond to the
Fig. 14 Uncertain levels of the
attribute faults
Table 4 Data for each uncertain PVT level
PVT Level 1 Level 2 Level 3
Psat (kgf/cm2) 301.0 294.1 271.4
Rs @ Psat (sm3/sm3) 102.2 79.2 66.0
l0 @ Psat (cp) 3.2 7.7 13.7
Oil API 21 19 18
910 J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918
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probability distribution of the attribute FAB and ki,
respectively. It can be seen that the distribution related to
R2 C 0.35 is more spread out. Low values of R2 means that
components of objective function less influenced by the
attribute are accepted. In the case of the attribute FAB, as
the minimum value wh increases, the probability of Level 0
increases. This means that wells far from the fault between
blocks A and B, therefore with low influence, are not
considered in the probability computation. To balance the
influence, intermediate values (whc = 0.6 and Rc2 = 0.55),
both shown in Fig. 19, were used as cut-off values to
obtain the posterior distribution, following the procedure
explained in the methodology section.
In the following two sections (Case 1 and Case 2)
analyses regarding the influence of well and field responses
in the posterior probability distribution are presented.
3.3 Case 1
In Fig. 20, the posterior probability distributions of the
six attributes with 16 levels are presented and in the
Fig. 21 the posterior distributions of the four attributes
with three levels are shown. These figures bring a
Fig. 15 Analysis of the indicator wh for the case studied (L1, L2 and
L3 state for the three levels of uncertainty used for the attribute FAB)
Fig. 16 Position of the wells PROB-11 and PROC-3 with respect to
the fault between blocks A and B
Fig. 17 Analysis of the indicator R2 for the case studied (R2G1 and
R2G2 state for polynomial fitting of degree 1 and 2, respectively)
J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918 911
123
comparison of the posterior distributions obtained with
and without the use of the influence matrix. The
obtaining of the posterior distributions without influence
matrix can be interpreted as whc = 0 and Rc2 = 0. This
means that all reservoir response (data series) are used
in the composition of the misfit function applied to
redistribute the new (posterior) probabilities. The blue
bars represent the reference model. One can see that, in
general, there is an improvement of the results, con-
centrating the distribution near the reference value
(except for two global attributes, kr and PVT, for which
the level with higher probability was not close to the
reference value).
Figure 22 depicts the water rate for some wells for the
cases with and without the influence matrix. Red curves
were obtained from combinations sampled from prior dis-
tribution and blue curves were obtained from samples
generated with the posterior distributions. It is clear that the
dispersion of the curves is lower when the influence matrix
is used and more concentrated around the observed data.
The results presented in Fig. 22 are in agreement with the
results presented in Fig. 20. For instance, attribute kB has a
probability distribution relatively more concentrated near
the reference value compared to case without influence
matrix, which leads to models closer to the history (as the
examples of wells PROB-1 and PROB-11 shown in
Fig. 22).
Figure 23 shows a box plot comparing the range of
misfit of the models obtained from prior, posterior
without and posterior with the influence matrix. Besides
the wells PROB-1 and PROB-11 (shown in Fig. 22),
two additional wells (PROA-3 and PROA-4) are shown.
This figure reflects the results shown in Fig. 22. It can
be seen that the blue bars, corresponding to the case
where the influence matrix was used, is significantly
smaller. The hachured band corresponds to misfit values
related to a deviation lower than 20 % with respect to
the history data. One can observe that the dispersion
obtained with the proposed method is very smaller than
that obtained with the original method (named in the
figure as ‘‘Posterior without’’). This result highlights the
Fig. 18 Water rate of well PROB-1 grouped according to two ranges
of the attributes PhiA and PhiBFig. 19 Influence of cut-off values of the indicators wh (a) and R2 (b)
912 J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918
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contribution of the influence matrix to the reduction of
uncertainty.
3.4 Case 2
The analysis of the results of the first run (Case 1) showed
that the probability distribution of the attributes with global
influence, such as relative permeability (kr) and PVT, did
not converged to the reference value, e.g., the level with
higher probability was different from the reference value.
Figure 24 shows the analysis of kr with respect to four
wells (two of Block B and two of Block C). It is possible to
observe that the Level 3 is closer to the history, which leads
to a higher probability for this level. However, it is known
that the correct is the Level 2. Analyzing the influence of
the attribute kr on the field response (Fig. 25) one can note
Fig. 20 Comparison of the a posteriori pdf obtained with and without the use of the influence matrix (6 attributes with 16 levels)
J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918 913
123
Fig. 21 Comparison of the a
posteriori pdf obtained with and
without the use of the influence
matrix (4 attributes with 3
levels)
Fig. 22 Dispersion of water rate curves with and without influence matrix
914 J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918
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a trend of concentration of curves related to Level 2 around
the history. Based on this observation, it was decided to
carry out a second run (Case 2) considering global func-
tions for the attributes kr and PVT. This means that for the
probability redistribution of these attributes, the field
response is taken into account (in this case, field water
rate). For the other eight attributes, the same functions were
used as in the Case 1. The new probabilities for these two
attributes are shown in Fig. 26. The higher probability is
now in agreement with the reference value. The posterior
probability of the attributes with 16 levels was similar to
the results of Case 1.
After Case 2, some levels with very low probability
were eliminated. In addition, the uncertain levels of attri-
butes kr were redefined near the most probable level
obtained in Case 2. The Step 7 (Evaluation) of the general
Fig. 23 Box plot comparing the
range of misfit of models
obtained from prior, posterior
without and posterior with the
influence matrix
Fig. 24 Analysis of the influence of the attribute kr (wells)
J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918 915
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methodology was then carried out, producing results
named Case 2E.
A comparison of water rate and average pressure for the
field is shown in Figs. 27 (Case 1), 28 (Case 2) and 29
(Case 2E). A total of 900 simulations were used. The
improvement of the results can be seen. In Fig. 30, the
results for four wells for the Case 2E are depicted, also
Fig. 25 Influence of the attribute kr on the field response
Fig. 26 New probabilities for the attributes kr and PVT
Fig. 27 Field water rate and average pressure obtained from prior
and posterior distributions (Case 1)
Fig. 28 Field water rate and average pressure obtained from prior
and posterior distributions (Case 2)
916 J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918
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showing a significant reduction of uncertainty, denoted by
the reduction of dispersion of the curves. It is important to
highlight that the procedure shown in this paper seeks the
reduction of dispersion of the responses in local (wells) and
global (field) scale.
4 Conclusions
A methodology for reduction of uncertainties in reser-
voir attributes using observed data was presented in this
paper. The robustness of the method was shown through
the application to a complex case. The application of a
sampling technique, as a way of circumventing the
limitation of the derivative tree, allowed the study of a
higher number of uncertain attributes. The proposed
method allows the treatment of attributes with contin-
uous and discrete characteristics. The influence matrix
permitted to capture automatically the influence of each
attribute in each component of the objective function,
allowing better results. This work also showed that in
complex cases, it is difficult to solve the problem in a
single step. It is necessary to divide the problem in
stages. Finally, a robust method was presented which
has potential to deal with complex real field
applications.Fig. 29 Field water rate and average pressure obtained from prior
and posterior distributions (Case 2E)
Fig. 30 Well water rate obtained from prior and posterior distributions (Case 2E)
J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918 917
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Acknowledgments The authors wish to thank PETROBRAS
(REDE SIGER), UNISIM, CEPETRO and the Department of Petro-
leum Engineering for the support to this work.
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