a method for reduction of uncertainties in reservoir … · a method for reduction of uncertainties...

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: [email protected]

D. J. Schiozer

e-mail: [email protected]

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

J Braz. Soc. Mech. Sci. Eng. (2014) 36:901–918 903

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


123

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


123

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


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


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


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


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


123

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)


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)


123

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)


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


123

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)


123

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)


123

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)


123

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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