Improvements to the fuzzy mathematics comprehensive quantitative method for evaluating fault sealing
Improvements to the fuzzy mathematics comprehensive quantitative method for evaluating fault sealing
Da-Wei Dong 0 1 2 3 4
Ji-Yan Li 0 1 2 3 4
Yong-Hong Yang 0 1 2 3 4
Xiao-Lei Wang 0 1 2 3 4
Jian Liu 0 1 2 3 4
0 Working Stations for Post Doctors , Dongying 257000, Shandong , China
1 Research Institute of Petroleum Exploration and Development, Shengli Oilfield Company , Dongying 257000, Shandong , China
2 Shengli College of China University of Petroleum , Dongying 257061, Shandong , China
3 Edited by Jie Hao
4 Oil Production Plant of Dongxing, Shengli Oilfield Company , Sinopec, Dongying 257061, Shandong , China
Fuzzy mathematics is an important means to quantitatively evaluate the properties of fault sealing in petroleum reservoirs. To accurately study fault sealing, the comprehensive quantitative evaluation method of fuzzy mathematics is improved based on a previous study. First, the single-factor membership degree is determined using the dynamic clustering method, then a single-factor evaluation matrix is constructed using a continuous grading function, and finally, the probability distribution of the evaluation grade in a fuzzy evaluation matrix is analyzed. In this study, taking the F1 fault located in the northeastern Chepaizi Bulge as an example, the sealing properties of faults in different strata are quantitatively evaluated using both an improved and an un-improved comprehensive fuzzy mathematics quantitative evaluation method. Based on current oil and gas distribution, it is found that our evaluation results before and after improvement are significantly different. For faults in ''best'' and ''poorest'' intervals, our evaluation results are consistent with oil and gas distribution. However, for the faults in ''good'' or ''poor'' intervals, our evaluation is not completely
Fault sealing property; Fuzzy mathematics; Dynamic clustering method; Quantitative study
consistent with oil and gas distribution. The improved
evaluation results reflect the overall and local sealing
properties of target zones and embody the nonuniformity of
fault sealing, indicating the improved method is more
suitable for evaluating fault sealing under complicated
Throughout geological history, faults have played a very
important role in hydrocarbon migration and
accumulation. Large faults, which form the boundaries of oil and
gas fields, can generally control hydrocarbon
(Agosta et al. 2012; Allan 1989; Balsamo et al.
2010; Bouvier et al. 1989; Braathen et al. 2009; Brogi
and Novellino 2015; Choi et al. 2015; Collettini et al.
2014; Fisher and Jolley 2007)
. However, small faults
generally separate oil and gas, resulting in an increase in
well spacing that poses challenges in oil and gas
exploration. Thereby, studies of fault sealing have
become a very important focus for oil and gas
exploration and development
(Ciftci et al. 2013; Collettini
et al. 2014; Davatzes and Aydin 2005; Fachri et al.
. Studies of fault sealing have evolved from
trap theory, which is used to identify hydrocarbon
sealing mechanism(s) and physical parameters of faults
(Hubbert 1953; Smith 1980)
. Based on different
mathematical principles and modes, the sealing properties of
one or more given faults are quantitatively identified and
predicted by considering various sealing mechanisms and
(Childs et al. 2009; Egholm et al. 2008; Fachri
et al. 2011; Faulkner et al. 2003; Fisher and Knipe 1998;
Wu et al. 2010; Knipe et al. 1997)
With constant improvements in petroleum geology,
scholars in China and abroad have gradually realized that
fault sealing is jointly controlled by multiple factors, rather
than just one or two single factors
(Lu¨ et al. 2007; Knott
1993; Knipe et al. 1997; Faulkner et al. 2010; Gibson 1994;
Zhang et al. 2013; Pei et al. 2015)
. With a deepening of
quantitative single-factor studies of fault sealing, many
scholars tried to comprehensively evaluate fault sealing
using mathematical theory methods, including nonlinear
mapping methods, gray relational, logical information,
fault connectivity probabilistic methods and fuzzy
comprehensive evaluation methods
(Fu et al. 2005, 2008, 2012;
Lu¨ et al. 1995; Lu¨ and Fu 2002; Zhang et al. 2007; Jiang
et al. 2008; Li 2009; Zhang et al. 2015)
. However, for all
these methods, some key parameters are difficult to obtain
and have regional limitations, so that human factors are
generally introduced for value assignment or adjustment.
Thus, scholars manage to optimize these parameters using
a variety of mathematical approaches. Among them, the
fuzzy comprehensive evaluation method, which is strongly
systematic and provides definite results, is accepted by
many scholars and has been gradually improved over the
In this paper, based on the principles of fuzzy
comprehensive evaluation, the single-factor membership degree
was established using the dynamic clustering method.
Then, a single-factor evaluation matrix was constructed
using the continuous grading function in order to optimize
the probability distribution of the evaluation grade in the
fuzzy comprehensive evaluation method. Next, we
considered a fault in the Chepaizi Bulge in the northwestern
margin of the Junggar Basin as an example, and the sealing
properties in the vertical and strike directions were
evaluated. Finally, our results are compared with oil and gas
2 Un-improved fuzzy mathematics evaluation
of fault sealing
The physical principles that affect fault sealing include
principal stress sealing, lithological allocation sealing,
shale smear sealing, time allocation sealing and occurrence
. Single factors include fault
properties, fault plane pressure, lithological allocation,
fault dip angle, shale smear and fault active time. Thus,
being affected by numerous factors, fault sealing shows
very strong complexity and randomness. Fortunately, the
fuzzy evaluation method can take various types of single
factors that affect fault sealing into consideration, and
hence realize a comprehensive evaluation using the
principle of fuzzy transformation and maximum membership
degree. Using this mathematical method, fault sealing
properties can be effectively identified. The process is as
follows: First, based on regional geology and fault
development, the single factors that affect fault sealing are
selected and quantified in order to obtain a single-factor
quantization matrix Un 1 (n is the number of single
factors). Then, based on the general division of fault sealing
and regional requirements, the number of evaluation grades
is divided (m). Finally, a single-factor membership degree
matrix is constructed based on the number of evaluation
grades and the oil and gas exploration results of the work
area, V1 m, based on which, Un 1 is evaluated in order to
construct a fuzzy evaluation matrix Rn m ¼ Un 1 V1 m.
Since each single factor has different contributions to fault
sealing, once the weighted matrix of each single factor is
provided, W1 n, the fuzzy evaluation matrix B1 m ¼ W1 n
Rn m can be obtained. Finally, the maximum value of the
fuzzy evaluation matrix B19m is taken as the corresponding
However, there are various methods that can be used to
acquire the parameters mentioned above, both qualitatively
and quantitatively, and which are constantly optimized
and Fu 2002; Russell et al. 2003; Fu et al. 2012; Li et al.
. There are many single factors that affect fault
sealing, generally including fault properties, fault plane
pressure, fault lithological allocation and shale smear.
Evaluation grades can be divided into five ranks: good,
better, moderate, fairly poor and poor. The single-factor
membership degree can be obtained via a discrete function
method and/or a continuous function method. The former
applies to qualitative single-factor membership degree,
such as fault properties, while the latter applies to
quantitative single-factor membership degree, such as fault plane
pressure. Weighting coefficients can be obtained via the
expert survey method, analytic hierarchy process (AHP),
Delphi method and/or the weight matrix method
Jin 2012; Sun and Wu 1995; Qiu et al. 2007)
. The larger
the weighting coefficient of a single factor, the greater the
impact of the single factor on fault sealing. The
mathematical model of fuzzy comprehensive evaluation includes
the weighted-average type, the main factor highlight type
and the main factor determination type
(Sun et al. 2010)
The weighted-average type can take a variety of single
factors into consideration in order to avoid information
loss. The main factor highlight type and the main factor
determination type emphasize the main controlling factors
and prevent interference factors. In the evaluation process
of fault sealing, many scholars use the weighted-average
type. However, determining the above parameters and
effectively predicting fault sealing must be based on the
actual petroleum geology of the study area and from
suggestions made by local experts in order to reduce
3.2 The establishment of a single-factor evaluation
matrix using the continuous grading function
3 Improved fuzzy mathematics of fault sealing
3.1 The establishment of the single-factor
membership degree using the dynamic
In this study, the membership degree of each single
factor is established by applying the dynamic clustering
method. Based on a large sample dataset,
pre-classification is roughly performed. Then, gradual adjustment is
made until a reasonable classification is obtained. The
maximum and minimum values of each class constitute
the range of membership degrees. For fault sealing, the
single-factor membership degree is established as
follows: Firstly, a large number of samples of one single
factor of the work area and their corresponding oil–gas
show are constructed. Then, taking best, good, poor and
poorest oil and gas show as the standard, the range of
the membership degrees of the fault gouge ratio is
determined. This means that the fault gouge ratio of a
given area is clustered. In the process of clustering, the
maximum and minimum values are the range limits of
membership degree. The so-called dynamic clustering
means the gradual adjustment of a value to a reasonable
range using the dynamic clustering principle. This
method has the advantages of incorporating the oil and
gas geology of the study area, so that it has regional
eigenvalues. It is closely related to the fault sealing
properties of the study area and hence has a certain
degree of predictability. The advantage of this method is
that the regional petroleum geology has a regional
characteristic value which is related to fault sealing, and
After the single-factor membership degree is determined
via the dynamic clustering method, fuzzy evaluation should
be carried out for all single factors in order to establish a
single-factor evaluation matrix. The division of the
singlefactor membership degree SðiÞ only gives evaluation
results of good and poor intervals, and it does not provide
probabilistic evaluation results of each interval. For
example, the range of ‘‘good’’ and ‘‘poor’’ is (1–0.5) and
(0.5–0). If the single factor value is 0.7, it cannot be
regarded simply as ‘‘good,’’ thus implying that a further
probabilistic evaluation is needed. Hence, the definition
‘‘good’’ accounts for 70% good, and ‘‘poor’’ accounts for
30%. eðiÞ is the boundary value of the re-probabilistic
evaluation. The precise method followed is as
follows: First, a range of values for fault sealing is determined
using single factors, SðiÞ. Then, the threshold value of each
adjacent class is determined, namely the classification
representative value eðiÞ, which is determined as per the
eð1Þ ¼ Sð1Þ >9
eð2Þ ¼ ½Sð1Þ þ Sð2Þ =2 =>
eð4Þ ¼ Sð3Þ
eð3Þ ¼ ½Sð2Þ þ Sð3Þ =2 >>
Fault properties, fault gouge ratios and fault sealing are
not always linearly related with each other. However, they
have a piecewise function relation based on single-factor
membership degree. According to Newton’s iteration
principle, it can be assumed that the single-factor
evaluation criteria rðvÞ are a linear piecewise function, which can
be solved by successive approximation. Then, fuzzy
subsets of the fault evaluation criteria can be determined using
the following methods:
8> 1 r1ðvÞ
>< eð3Þ v
Taking the fault F1 located in the northeastern Chepaizi
Bulge as an example (Fig. 1), the fault sealing properties of
the main target zones were evaluated using both the
improved and un-improved methods. This approach was
used to determine whether the improved comprehensive
evaluation method for fault sealing is more reasonable. The
Chepaizi Bulge is located at the southern end of the
northwestern margin of the Junggar Basin. It is a secondary
structural unit in the western uplift of the Junggar Basin,
adjacent to the Changji Sag and the Zhongguai bulge to the
east, Sikeshu Sag to the south and Zaire Mountain to the
northwest (Fig. 1). Conditions in the Chepaizi Bulge make
this a good place for hydrocarbon accumulation. The main
reservoirs are Cretaceous (K) and Neogene Sha1 Member
(N1s1). The F1 fault in the northeastern Chepaizi Bulge is a
very important oil control fault, and its sealing properties
directly affect hydrocarbon accumulation in this area. The
fault has NNE-trending, EW-trending compression in the
Yanshanian Period and NWW-trending extension in the
Himalayan Period. Vertically, there are reverse faults in the
Cretaceous strata and basement and normal faults in
Cenozoic strata, which represent a negative inverted
structure as a whole.
4.1 Single-factor selection and quantitative
Since the study area has undergone multiple phases of
tectonic evolution, various types of faults with different
occurrences have developed. There are many single factors
that affect fault sealing in response to tectonic evolution, so
that the weighting coefficients of the single factors are
different. However, in this paper, and taking fault F1 as an
example, the sealing properties in the vertical and strike
directions were analyzed. Four single factors with
comparable significance were selected, including fault surface
normal stress (r), fault properties, sand/formation ratio (N/
G) and the shale content of fault zone fillings. In addition,
four seismic profiles vertical to fault F1 were selected (the
location is shown in Fig. 1), which were converted into
four geological sections via a time-depth conversion. Then,
lithological sections were recovered using well logs in
order to obtain the quantitative and qualitative values of
each single factor (Table 1).
Fault surface normal stress, which is an important
parameter used to characterize the fault opening degree, is
crucial to fault sealing
(Fu et al. 2005)
. Typically, this
value is the vector sum of the gravitational force imparted
by the overlying strata, the regional principal compressive
stress, the regional principal stress as well as the fault dip
angle. Fault F1 is a shovel fault, where the fault plane
normal stress increases from the bottom to the top, where
the fault sealing properties become better. However, the
Chepaizi Bulge is located in a slope belt of a foreland
basin, where the buried depth is shallow, so that the fault
surface pressure is small.
In general, the sealing properties of compressional, shear
and compressional-shear faults are good, while the sealing
properties of extensional and extensional-shear faults are
poor. Fault F1 is a negative inverted fault, and it is an
extensional fault that formed during the Neogene. However,
the fault occurrence changes significantly, which is steep in
the upper part and gentle in the lower part. That is, regional
tensile stress can only cause the Neogene ‘‘steep’’ fault
section to extend, but cannot completely make the Cretaceous
‘‘gentle’’ fault section extend. In addition, the pressure
experienced by the Cretaceous fault plane is significantly
larger than that of the Neogene fault plane, which can also
prove this change. After undergoing compressional
deformation, the Cretaceous fault section has better sealing
properties than the Neogene extensional fault section due to its
fault structure and mudstone smearing. Therefore, the
Neogene fault section has been assigned as ‘‘extensional’’, while
the Cretaceous fault section is assigned as ‘‘compressional’’.
Sand shale contraposition is an important process for oil
and gas lateral sealing. In fault dislocation, if sand shale
contraposition occurs in one interval, the fault in this
interval is closed, while if multiple sand layers are
connected in one interval, the fault in this interval is open. The
parameter characterizing possible sand shale contraposition
is the sand/formation ratio, which is the ratio between the
sandstone and layer formation thicknesses, N/G. When
N/G is large, the possibility of a sand–sand connection is
large, and hence the fault sealing ability is poor. When the
N/G ratio is small, the fault sealing ability is in fact good.
However, this single factor does not take the fault throw
generated by fault slipping into consideration, Therefore,
the following single factor is added.
Faults will form fault zones during displacement and
dislocation processes. When a given fault zone is filled
with shale, it can seal oil and gas laterally due to the plastic
flow and compaction of shale. When a fault zone is filled
with sandstone, it can act as a hydrocarbon migration
pathway since sandstone compacts poorly and has good
porosity and permeability. Accordingly,
Fu et al. (2012)
improved the fault gouge ratio (SGR) proposed by
et al. (1997)
and proposed to characterize the sealing
property of faults using the ratio between the mudstone
thickness of a fault belt and the sum of the fault throw and
faulted formation thicknesses, namely the shale content of
the fillings in a given fault belt. Due to the negative
inversion of fault F1, Cretaceous shale smeared the fault
surface repeatedly. Therefore, the shale in the Cretaceous
fault zone fillings was formed in compressional and
extensional stages. Hence, the faults that formed in
compressional and extensional stages were obtained from
structural section restoration.
4.2 Single-factor weighting coefficients
and membership degree
Single-factor weighting coefficients are significantly
different in different areas. The numerous methods that are
used to determine these coefficients are all based on the
expert investigation method. Therefore, in this paper, the
weighting coefficients of four single factors, including the
normal stress of the fault plane (w1), fault properties (w2),
sand/formation ratio (w3) and the shale content of the
fillings in the fault belt (w4), were determined using the expert
investigation method (Table 2). It is worth mentioning that
the formation pressure of the oil and gas discovery wells in
this area is abnormal, while the formation pressure of dry
wells and water wells is normal, meaning pressure is
sensitive to oil and gas accumulation so that the weighting
coefficient of fault plane normal stress is the largest.
The statistics used in this study were applied to the well
logging data of 85 wells around fault F1 and the
corresponding oil and gas shows. Moreover, the values of four
single factors were calculated, which were classified into
four categories using the dynamic clustering method: best,
good, poor and poorest. The maximum and minimum
values of each category were taken as the boundary values
of each membership degree. Based on the advice of oilfield
experts, membership degree was then slightly adjusted.
The final results are shown in Table 3.
4.3 Optimized fuzzy evaluation matrix
The advantage of the dynamic clustering method in
comprehensive fault sealing property evaluation is to establish
a single-factor evaluation matrix. In order to illustrate the
establishment process of the un-improved and improved
single-factor evaluation matrices, the establishment of the
single-factor matrix in the first member of the Shawan
Formation in profile I was analyzed.
Before improvement, a single value was directly
assigned to a single-factor membership degree evaluation
matrix using the discrete function and continuous function
methods. When the N/G of the first member of the Shawan
Formation was 0.56, (i.e., the grade of the maximum
membership degree is ‘‘poor’’), this grade was directly
assigned a value of 0.5 when using the discrete function
method and 0.3 when using the continuous function
method. The single-factor membership degree evaluation
matrix is RN=G ¼ ð0:5Þ or ð0:3Þ. However, although a
value of 0.56 implies a grade of ‘‘poor,’’ which is close to a
grade of ‘‘good,’’ a value of 0.56 should have some
probability evaluation in the grade ‘‘good.’’ However, the
unimproved single-factor membership degree evaluation
matrix does not embody the probability in this transitional
Instead, a single-factor evaluation matrix that is
constructed by the continuous grading membership function
can properly address the problem mentioned above. Its
calculation process after improvement is as follows: First,
SðiÞ is determined based on N/G, namely S 1
ð Þ ¼ 0:3,
ð Þ ¼ 0:55, Sð3Þ ¼ 0:8. Then, the threshold value of each
adjacent evaluation level is calculated, i.e., grading
representative value eðiÞ:
> eð1Þ = Sð1Þ = 0:3
< eð2Þ ¼ ½Sð1Þ þ Sð2Þ =2 ¼ 0:425
>> eð3Þ ¼ ½Sð2Þ þ Sð3Þ =2 ¼ 0:675
: eð4Þ ¼ Sð3Þ ¼ 0:8
Finally, when the N/G of the first member of the Shawan
Formation is 0.56, the single-factor evaluation matrix is
calculated: eð2Þ \0:56 \eð3Þ,
> r1 ¼ 0
<> r2 ¼ ½eð3Þ 0:56 =½eð3Þ eð2Þ ¼ 0:46
>> r3 ¼ 1 r2 = 0:54
: r4 ¼ 0
After improvement, the evaluation matrix of the
sand/formation ratio of the first member of the Shawan
Formation is RN=G ¼ ð 0 0:46 0:54 0 Þ.
The comprehensive evaluation matrix of the four factors
of the first member of the Shawan Formation is:
2 0:00 0:51 0:49 0:00 3
6 0:00 0:00 0:80 0:20 7
R ¼ 64 0:00 0:46 0:54 0:00 75
0:00 0:00 0:00 1:00
The fuzzy evaluation matrix of the first member
of the Shawan Formation is B ¼ W R ¼
ð 0 0:27 0:44 0:29 Þ. According to the principle of the
maximum membership degree, the evaluation result of the
first member of the Shawan Formation is poor. However,
the probability distribution of this evaluation grade is
scattered, indicating the fuzziness of the fault sealing
property is large. Conversely, if the probability distribution
of the evaluation grade is concentrated, the evaluation
result of fault sealing property is obvious.
4.4 Fuzzy evaluation results and analysis
Using the improved fuzzy evaluation method described
above, the sealing properties of fault F1 in the vertical and
strike directions were evaluated. The results are shown in
Table 4. Cross sections I, II, III and IV are well-tied cross
sections that span from south to north along the fault strike.
Cretaceous (K), the first member of Shawan Formation
(N1s1), the second member of Shawan Formation (N1s2)
and the third member of Shawan Formation (N1s3)
constitute intervals from top to bottom in the vertical
direction. The evaluation results show that the valuation
result of Cretaceous faults is ‘‘good,’’ but the maximum
probability is only 0.58, indicating oil and gas are sealed
when they migrate to the footwall, but it may be oil and gas
bearing in the hanging wall. The valuation result of the first
member of the Shawan Formation is ‘‘poor’’ or ‘‘poorest.’’
The probability distribution of the evaluation grade is
dispersed, showing the ambiguity of footwalls and hanging
walls on its oil-bearing properties. The evaluation result of
the second member of the Shawan Formation is ‘‘good.’’ A
regional survey showed that this interval is a stably
distributed shale bed, namely regional caprock. The
evaluation result of the third member of the Shawan Formation is
‘‘poor,’’ but mud logging results show this interval is
conglomerate. There were no oil and gas shows discovered
in this interval in the work area.
Next, we compared the variations between the improved
and un-improved fuzzy evaluations and their impact on the
evaluation results. In this paper, a single-factor
membership degree evaluation matrix was directly assigned a value
using the discrete function method. The assigned values of
evaluation grades R ¼ ðbest, good, poor, poorest) = (1, 0
:66; 0:33; 0Þ and the fuzzy evaluation results are shown in
Table 4. Upon inspection, the fuzzy evaluation values of
the first and second members of the Shawan Formation
change significantly, but they do have consistent results.
However, the fuzzy evaluation values and evaluation
‘‘–’’ in the table means there are no drilling test data or wells are not drilled in this horizon
Poorest Best Poor Best
Water – Oil Oil
results of the first member of the Shawan Formation
change significantly before and after improvement, and the
evaluation results before improvement are not well
matched with the oil and gas shows in footwall and hanging
wall. Taking cross section III as an example, the evaluation
results before improvement are poor, but the footwall and
hanging walls are both oil bearing. After improvement, the
probability distribution of the evaluation grades is
scattered, indicating oil and gas may be trapped locally after
having continuously migrated from the footwall to the
hanging wall in the local open interval (Fig. 2). The above
analysis shows that the improved method is more
advantageous for regions with larger degrees of vagueness of
their fault sealing properties, and hence higher exploration
The evaluation of fault sealing properties is an important
part of oil and gas exploration and development. Fuzzy
comprehensive evaluation is a systematic evaluation
method for fault sealing. Since traditional methods are
greatly affected by human factors during the establishment
of the single-factor membership degree, the assignment
method results in fuzzy evaluation values, so that the result
only reflects the target interval. In this paper, the dynamic
clustering method was introduced to determine the
singlefactor membership degree. Then, a single-factor evaluation
matrix was constructed using the continuous grading
function in order to determine the optimum comprehensive
evaluation matrix and make a fuzzy evaluation of the fault
sealing properties. A comparison of the fuzzy evaluation
and its result before and after improvement, combined with
current oil and gas distribution regularity, showed that the
evaluation results before and after improvement are
significantly different. For faults designated as ‘‘best’’ and
‘‘poorest,’’ the evaluation results are consistent with oil and
gas distribution. However, for faults designated as ‘‘good’’
or ‘‘poor,’’ the evaluation results of sealing property are not
completely consistent with the oil and gas distributions.
The improved results reflect the overall and local sealing
properties of target zones and embody the fuzziness of fault
sealing, indicating the improved method is more precise for
evaluating fault sealing properties under complicated
However, the improved method still has its limitations.
First of all, this method still cannot solve the multi-scale
fault sealing problem. As mentioned above, the N1s1
section in profile III displays strong heterogeneity, and
although the evaluation results of large scale are ‘‘poor,’’
the fault sealing properties exist at small scales. Secondly,
this method can only explain the sealing properties of the
faults in profile. When oil and gas migrate in multiple
directions, the evaluation results will not be consistent with
drilling results. For example, the evaluation results of
section K in the III and IV profiles are good, but the fault is
oil bearing in the hanging wall. This phenomenon is related
to a two-way oil supply. Oil source comparison shows that
the oil and gas migration in the study area is from east to
west in the Changji Sag and from south to north in the
(Song et al. 2007)
. Therefore, this method is
only applicable for studies of single-scale fault sealing
properties. In addition, the direction of oil and gas
migration should be vertical to the fault plane.
Acknowledgements This study is supported by the Science and
Technology Project of Universities and Colleges in Shandong
Province ‘‘Investigation on diagenetic environment and transformation
pattern of red-bed reservoirs in the rift basins’’ (No. J16LH52).
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creative
commons.org/licenses/by/4.0/), which permits unrestricted use,
distribution, and reproduction in any medium, provided you give appropriate
credit to the original author(s) and the source, provide a link to the
Creative Commons license, and indicate if changes were made.
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