# Optimal Selection of Stimulation Wells Using a Fuzzy Multicriteria Methodology

Authors: Fanhui Zeng; Dayong Wang; Jianchun Guo; Shouren Zhang; Ping Zhang; Bing Zhang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Choosing the most appropriate wells for stimulation is one of the greatest challenges affecting reservoir production and economic development. In this paper, an integrated fuzzy decision-making methodology is developed to choose the most appropriate stimulation wells based on a number of criteria. The presented approach is able to determine the final decision priorities for multiple candidate wells considering uncertainty, incomplete information, and the large number of factors involved. The proposed modelling framework works as a stepwise procedure to fuse several different methods with combined benefits. The analytic hierarchy process (AHP) method is adopted to build the factor framework. Grey theory is introduced to determine the influence weights of factors, and combined with the fuzzy logic concept, these values are used to rank candidate wells. A field application shows that the presented method is able to identify the wells with the highest potential for enhanced production.


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## Body

## 1. Introduction

The availability of energy is a critical factor for the development of economies and societies. Notably, China has attempted to decrease its dependence on imported energy to increase its standard of living. China has become the largest crude oil importer worldwide over the past two decades [1]. The International Energy Agency (IEA) predicts Chinese net oil imports to reach 13.1 million barrels per day in 2030 [2, 3]. As part of an effort to meet the national energy needs, China has turned to the previously neglected region of Africa. Although domestic oil and natural gas reserves are plentiful in China, production has been restricted by the lack of means of selecting the most economically suitable wells for stimulation [4–7]. Selecting among candidate wells for stimulation is a complex process involving several different criteria and experts with different backgrounds [8–15]. In addition, policies for selecting among candidate wells generally have a number of objectives that should be stated clearly. It is also essential that the following questions be considered: What are the criteria or factors that should be taken into account? What standards are to be applied when choosing candidate wells? Which of the candidate wells are most appropriate for stimulation?Several major factors are involved in this process, including geological factors, well reserves, and hydraulic fracturing and construction parameters [16]. The relationship between being a good candidate well for stimulation and these parameters is highly nonlinear and complex [17, 18]. Many approaches have been developed for candidate well selection, including neural networks, support vector machines, deep learning, the analytic hierarchy process (AHP), and the tree-like structure technique (Sellitto M. A., 2018). However, each method has its inherent limitations. Neural networks are based on identifying statistical patterns and do not consider the knowledge on the underlying physical processes associated with fracturing. Consequently, neural networks require a large amount of high-quality data, and errors in these data can lead to inaccurate results [19]. The AHP method provides a framework for making pairwise comparisons and derives ratio scales from a set of paired comparisons by making numerical trade-offs to arrive at a conclusion [20]. One deficiency of the AHP method is that it is unable to operate in the presence of fuzzy or incomplete comparisons [21, 22]. Another deficiency is that when one removes or introduces a new criterion in the AHP, it may cause an inversion of priorities in some circumstances.There is no doubt that stimulation well selection is a problem fraught with error, uncertainty, and fragile correlations between datasets [18]. These uncertainties come from the challenge of designing and building sensors to measure complex geological factors as well as well reserves in hostile environments [23]. It is obvious that the variations used in neural networks and the AHP method are simplifying hypotheses in consideration of extreme or mean values, which cannot deal with the uncertainties associated with vague or imprecise information. Fuzzy logic is better able to tackle system uncertainty and fuse subjective perceptions into evaluations than are the methods discussed above [24, 25]. Different from classical logic, which is based on crisp sets of “true and false,” fuzzy logic views problems as a degree of “truth,” or “fuzzy sets of true and false.” The fuzzy set concept allows us to express uncertainty in a different way from classical probability theory based solely on the randomness concept. The key point in fuzzy logic is to find a membership function to transform fuzzy scales into crisp scales for the computation of a single parameter, the fuzzy probability. Grey system theory is an interdisciplinary scientific area that was first introduced by Deng [26] that requires a limited amount of data to estimate the behaviour of unknown systems and obtains an unbiased as well as consistent point estimator. It is widely employed to determine the influence weights of factors [27]. Combined with grey theory, these fuzzy possibilities can be combined harmonically to provide a combined fuzzy possibility [23, 28].All in all, choosing a stimulation well involves multiple criteria and multiple expert group decision-making problems. A single tool or discipline cannot provide all the information necessary for the complete characterization of a good candidate well, but by integrating geological, geophysical, petrophysical, drilling, and reservoir simulation data, as well as the relevant concepts, appropriate results can be obtained [29]. Hence, a fuzzy multicriteria decision-making approach is necessary to accommodate different perspectives and provide a supportive framework with appropriate alternatives. These issues are addressed in the remainder of this paper, which is organized as follows: Section 2 proposes a new hierarchical approach and evaluation criteria to overcome the identified shortcomings and select the most appropriate wells for stimulation. Section 3 describes the implementation of the proposed method for choosing among alternative wells for stimulation. Section 4 summarizes the conclusions.


## 2. Mathematical Models

### 2.1. Determining the Factor Hierarchical Structure

According to the Gas Research Institute, the greatest benefit for the petroleum industry lies in the selection of candidate wells for stimulation [30]. Selecting target stimulation wells from huge numbers of producing wells in a reservoir is a difficult task involving different domains, attributes, and features. Xiong and Holditch [31] used nine fuzzy variables to assess candidate wells for hydraulic fracturing. Yang [5] studied the process of selecting candidate wells and considered nine influential factors to rank the wells. Yin and Wu [32] studied a range of multiattribute decision-making alternatives using seven parameters. Zoveidavianpoor [6, 33] considered eight drivers to choose the most suitable wells for fracturing. The selection criteria identified in these studies are listed in Table 1.Table 1
Ranking criteria for candidate well selection.
Authors
Input variables
Xiong and Holditch [31]
Permeability/viscosity ratio, skin factor, net pay thickness, water saturation, formation pressure gradient, wellbore condition, drainage area, formation depth, and porosity
Yang [5]
Formation pressure, permeability, recovery percent of reserves, water saturation, production rate of the well, recovery percentage of reserves, flow coefficient, vertical heterogeneity coefficient, and producing pressure drop
Yin and Wu [32]
Skin factor, permeability, fluidity, water cut, reservoir pressure, fracture width, and fracturing effects
Zoveidavianpoor [6]
Permeability, skin, water saturation, porosity, net pay thickness, drainage area, formation pressure, and formation depthSome of the parameters identified in Table1 are not always available, such as the skin factor, recovery percentage of reserves, heterogeneity coefficient, fracture width, fracturing effects, and drainage area. Conversely, several significant parameters are not included in Table 1, such as the information from the gamma ray log, density log, and neutron porosity log [33], structure and lithology [34, 35], and hydraulic fracturing and construction data [36].Because the production following hydraulic fracturing is highly influenced by the reserve capacity, deliverability, and hydraulic efficiency, we use these three main criteria to determine the importance of candidate wells. These criteria are further divided into 14 subcriteria, which are listed in Table2.Table 2
Criteria for the selection of the best candidate well.
Main criteria
C1: reserve capacity
C2: deliverability
C3: hydraulic efficiency
Subcriteria
C11: net pay thickness (TH)
C12: porosity (POR)
C13: gas saturation (SGT)
C21: structure position (SPI)
C31: discharge rate (DCR)
C14: natural gamma porosity (GR)
C22: lithology (LTG)
C32: prepad volume (PRE)
C15: neutron porosity (NR)
C23: lateral resistivity difference (LRDD)
C33: slurry volume (SLU)
C16: density (DEN)
C34: sand volume (SAN)
C17: sonic (SON)The selection hierarchy for choosing the most suitable candidate wells for stimulation is illustrated in Figure1.Figure 1
The most appropriate candidate well selection hierarchy.None of the previous studies used a broader set of selection criteria than that used in our model. This richness in the representation of each well makes it possible to perform analyses with few example cases in building the model. Furthermore, the presented model combines three types of data: crisp, linguistic, and fuzzy data, while the above studies utilized only one data type.


### 2.2. Weighting Factor Determination

To reflect the real importance of certain parameters in the choice of candidate wells, it is vital to rationally calculate the parameter weights. The factor priorities are determined using grey theory. The application of grey theory consists of a number of substeps, which are detailed below.


#### 2.2.1. Comparative Series

An information series withm components or decision factors can be expressed as xj. If all information series are comparable, then the m information series can be described by the following matrix [27]:(1)x=x1⋮xz⋮xl=x11⋯x1j⋯x1m⋮⋮⋮⋮⋮xz1⋯xzj⋯xzm⋮⋮⋮⋮⋮xl1⋯xlj⋯xlm,z=1,2,…,l;j=1,2,…,m,where xzj denotes the jth factor of sample z and m is the criteria number of each individual well sample under consideration.


#### 2.2.2. Standard Series

A standard series describes the degree of relationship between two series and is expressed as follows:(2)x0=x01⋮x0z⋮x0l=q01⋮q0z⋮q0l,z=1,2,…,l.In our research, a standard series may represent the initial production of all considered sample wellsl.


#### 2.2.3. Difference between a Comparative Series and a Standard Series

The difference between the values of the decision factors and the norm of the standard series is determined as(3)D0=Δ011⋯Δ01j⋯Δ01m⋮⋮⋮⋮⋮Δ0z1⋯Δ0zj⋯Δ0zm⋮⋮⋮⋮⋮Δ0l1⋯Δ0lj⋯Δ0lm,z=1,2,…,l;j=1,2,…,m,where Δ0zj=x0j−xzj.


#### 2.2.4. Grey Relational Coefficient

The local relationships can be constructed, and the relational coefficient can be expressed as(4)γx0j,xzj=Δmin+ζΔmaxΔ0zj+ζΔmax,z=1,2,…,l;j=1,2,…,m,(5)Δ0zj=x0j−xzj,(6)Δmin=∀z∈lmin∀jminx0j−xzj,(7)Δmax=∀z∈lmax∀jmaxx0j−xzj,where x0j is the standard series, xzj is the comparative series, and ζ is an identifier that is generally set to 0.5.


#### 2.2.5. Weighting Factor Determination

Weights are distributed among the parameters to reflect their importance for well selection.(8)aj=γx0j,xzj∑j=1mγx0j,xzj,j=1,2,…,m,where aj represents the weight degree of the jth factor relation for the candidate well and m is the number of factors for the individual wells under consideration.


### 2.3. Single-Factor Evaluation Model

Consider a problem in which each candidate well hasm criteria uj∣j=1,2,…,m and n records in the output vi∣i=1,2,…,n. The main objective is to determine which input variables uj have the most significant impact on the output. A fuzzy rule is used for each point in the space; hence, one can obtain n fuzzy rules for the m points [6, 32]:(9)uid=e−d−a/b2,where d is the value of the evaluation factor, i represents each reviewer’s grade, and a and b are indicator parameters.For the same review, the membership function is determined as follows:(10)ujidj=e−dj−a/b2,where dj represents the value of the jth evaluation factor and ujidj is the dj membership of review i.From formula (10), when dj=a, ujia=1, which indicates the largest membership. Therefore, a represents a level of expectation as follows:(11)a=d1+d22,where d1 and d2 are the maximum and minimum values of the ith review, respectively.The value at these points is equal to the neighbouring review memberships, and such points are called transition points. The degree of membership of a transition point falls at the midpoint of the range of membership values and is equal to 0.5 for transition points connected to edge points [37]:(12)ujid1=e−d1−d2/2b2=0.5.Thus,(13)b=d1−d22ln2.The membership function relationships can be determined after the parametersa and b are determined. These values represent the ith review level, which can be determined by statistical correlation analysis. The fuzzy evaluation set of the jth factor is as follows:(14)Uj=uj1,…,uji,…,ujnT,j=1,2,…,m;i=1,2,…,n.The fuzzy possibilityUj is based on a single parameter and is unable to reflect the combined effects of all factors. With m parameters and n reviews, one can obtain the following matrix:(15)U=ujidj=u11Lu1iLu1nMMuj1LujiLujnMMum1LumiLumn,j=1,2,…,m;i=1,2,…,n.


### 2.4. Fuzzy Multicriteria Evaluation

The purpose of this step is to aggregate individual criteria into a group preference for each factor. The fuzzy multicriteria evaluation scores are obtained using a matrix multiplication summation algorithm. The fuzzy vectorB is then formed and combines the contributions of all the factors:(16)B=bi=b1,…,bi,…,bn=A∘U=a1,…,aj,…,am∘u11u12…u1nu21u22…u2n…………um1um2…umn,j=1,2,…,m;i=1,2,…,n,where bi is the evaluation index, which is the jth factor membership degree of the evaluation object when all input factors are taken into consideration. According to the principle of the maximum membership degree [28], the maximum bi is used for final decision-making.


### 2.5. AHP Combined with Fuzzy Multicriteria Evaluation

Human judgement plays a decisive role throughout the process of candidate well selection. Some of the advantages of computer-based models are that they do not suffer from subjective biases and they give consistent results. However, they have limited flexibility in dealing with new conditions. Meanwhile, human operators can be highly adaptable in applying their knowledge and experience but can be inconsistent in their evaluations. Hence, it is desirable to combine the advantages of both approaches in decision-making. The correspondents include geologists, engineers, and managers. They applied their expert judgements in the evaluation as follows [18]:(1)
Within the supporting framework of the AHP, human experts make judgements about which factors are most significant within the areas of reserve capacity, deliverability, and construction factors. These can be expressed verbally, graphically, or numerically.(2)
Decision-makers then decide whether each criterion is positively or negatively correlated with the well output after fracturing. Then, grey theory is used to determine the relative importance of the individual factors and assign a weighting factor to each one. The results are then used for well selection.(3)
There is a further opportunity for the human validation and revision of the membership functions and fuzzy rules. They can also verify that the computer-based selection is consistent with their knowledge and intuition.


## 2.1. Determining the Factor Hierarchical Structure

According to the Gas Research Institute, the greatest benefit for the petroleum industry lies in the selection of candidate wells for stimulation [30]. Selecting target stimulation wells from huge numbers of producing wells in a reservoir is a difficult task involving different domains, attributes, and features. Xiong and Holditch [31] used nine fuzzy variables to assess candidate wells for hydraulic fracturing. Yang [5] studied the process of selecting candidate wells and considered nine influential factors to rank the wells. Yin and Wu [32] studied a range of multiattribute decision-making alternatives using seven parameters. Zoveidavianpoor [6, 33] considered eight drivers to choose the most suitable wells for fracturing. The selection criteria identified in these studies are listed in Table 1.Table 1
Ranking criteria for candidate well selection.
Authors
Input variables
Xiong and Holditch [31]
Permeability/viscosity ratio, skin factor, net pay thickness, water saturation, formation pressure gradient, wellbore condition, drainage area, formation depth, and porosity
Yang [5]
Formation pressure, permeability, recovery percent of reserves, water saturation, production rate of the well, recovery percentage of reserves, flow coefficient, vertical heterogeneity coefficient, and producing pressure drop
Yin and Wu [32]
Skin factor, permeability, fluidity, water cut, reservoir pressure, fracture width, and fracturing effects
Zoveidavianpoor [6]
Permeability, skin, water saturation, porosity, net pay thickness, drainage area, formation pressure, and formation depthSome of the parameters identified in Table1 are not always available, such as the skin factor, recovery percentage of reserves, heterogeneity coefficient, fracture width, fracturing effects, and drainage area. Conversely, several significant parameters are not included in Table 1, such as the information from the gamma ray log, density log, and neutron porosity log [33], structure and lithology [34, 35], and hydraulic fracturing and construction data [36].Because the production following hydraulic fracturing is highly influenced by the reserve capacity, deliverability, and hydraulic efficiency, we use these three main criteria to determine the importance of candidate wells. These criteria are further divided into 14 subcriteria, which are listed in Table2.Table 2
Criteria for the selection of the best candidate well.
Main criteria
C1: reserve capacity
C2: deliverability
C3: hydraulic efficiency
Subcriteria
C11: net pay thickness (TH)
C12: porosity (POR)
C13: gas saturation (SGT)
C21: structure position (SPI)
C31: discharge rate (DCR)
C14: natural gamma porosity (GR)
C22: lithology (LTG)
C32: prepad volume (PRE)
C15: neutron porosity (NR)
C23: lateral resistivity difference (LRDD)
C33: slurry volume (SLU)
C16: density (DEN)
C34: sand volume (SAN)
C17: sonic (SON)The selection hierarchy for choosing the most suitable candidate wells for stimulation is illustrated in Figure1.Figure 1
The most appropriate candidate well selection hierarchy.None of the previous studies used a broader set of selection criteria than that used in our model. This richness in the representation of each well makes it possible to perform analyses with few example cases in building the model. Furthermore, the presented model combines three types of data: crisp, linguistic, and fuzzy data, while the above studies utilized only one data type.


## 2.2. Weighting Factor Determination

To reflect the real importance of certain parameters in the choice of candidate wells, it is vital to rationally calculate the parameter weights. The factor priorities are determined using grey theory. The application of grey theory consists of a number of substeps, which are detailed below.


### 2.2.1. Comparative Series

An information series withm components or decision factors can be expressed as xj. If all information series are comparable, then the m information series can be described by the following matrix [27]:(1)x=x1⋮xz⋮xl=x11⋯x1j⋯x1m⋮⋮⋮⋮⋮xz1⋯xzj⋯xzm⋮⋮⋮⋮⋮xl1⋯xlj⋯xlm,z=1,2,…,l;j=1,2,…,m,where xzj denotes the jth factor of sample z and m is the criteria number of each individual well sample under consideration.


### 2.2.2. Standard Series

A standard series describes the degree of relationship between two series and is expressed as follows:(2)x0=x01⋮x0z⋮x0l=q01⋮q0z⋮q0l,z=1,2,…,l.In our research, a standard series may represent the initial production of all considered sample wellsl.


### 2.2.3. Difference between a Comparative Series and a Standard Series

The difference between the values of the decision factors and the norm of the standard series is determined as(3)D0=Δ011⋯Δ01j⋯Δ01m⋮⋮⋮⋮⋮Δ0z1⋯Δ0zj⋯Δ0zm⋮⋮⋮⋮⋮Δ0l1⋯Δ0lj⋯Δ0lm,z=1,2,…,l;j=1,2,…,m,where Δ0zj=x0j−xzj.


### 2.2.4. Grey Relational Coefficient

The local relationships can be constructed, and the relational coefficient can be expressed as(4)γx0j,xzj=Δmin+ζΔmaxΔ0zj+ζΔmax,z=1,2,…,l;j=1,2,…,m,(5)Δ0zj=x0j−xzj,(6)Δmin=∀z∈lmin∀jminx0j−xzj,(7)Δmax=∀z∈lmax∀jmaxx0j−xzj,where x0j is the standard series, xzj is the comparative series, and ζ is an identifier that is generally set to 0.5.


### 2.2.5. Weighting Factor Determination

Weights are distributed among the parameters to reflect their importance for well selection.(8)aj=γx0j,xzj∑j=1mγx0j,xzj,j=1,2,…,m,where aj represents the weight degree of the jth factor relation for the candidate well and m is the number of factors for the individual wells under consideration.


## 2.2.1. Comparative Series

An information series withm components or decision factors can be expressed as xj. If all information series are comparable, then the m information series can be described by the following matrix [27]:(1)x=x1⋮xz⋮xl=x11⋯x1j⋯x1m⋮⋮⋮⋮⋮xz1⋯xzj⋯xzm⋮⋮⋮⋮⋮xl1⋯xlj⋯xlm,z=1,2,…,l;j=1,2,…,m,where xzj denotes the jth factor of sample z and m is the criteria number of each individual well sample under consideration.


## 2.2.2. Standard Series

A standard series describes the degree of relationship between two series and is expressed as follows:(2)x0=x01⋮x0z⋮x0l=q01⋮q0z⋮q0l,z=1,2,…,l.In our research, a standard series may represent the initial production of all considered sample wellsl.


## 2.2.3. Difference between a Comparative Series and a Standard Series

The difference between the values of the decision factors and the norm of the standard series is determined as(3)D0=Δ011⋯Δ01j⋯Δ01m⋮⋮⋮⋮⋮Δ0z1⋯Δ0zj⋯Δ0zm⋮⋮⋮⋮⋮Δ0l1⋯Δ0lj⋯Δ0lm,z=1,2,…,l;j=1,2,…,m,where Δ0zj=x0j−xzj.


## 2.2.4. Grey Relational Coefficient

The local relationships can be constructed, and the relational coefficient can be expressed as(4)γx0j,xzj=Δmin+ζΔmaxΔ0zj+ζΔmax,z=1,2,…,l;j=1,2,…,m,(5)Δ0zj=x0j−xzj,(6)Δmin=∀z∈lmin∀jminx0j−xzj,(7)Δmax=∀z∈lmax∀jmaxx0j−xzj,where x0j is the standard series, xzj is the comparative series, and ζ is an identifier that is generally set to 0.5.


## 2.2.5. Weighting Factor Determination

Weights are distributed among the parameters to reflect their importance for well selection.(8)aj=γx0j,xzj∑j=1mγx0j,xzj,j=1,2,…,m,where aj represents the weight degree of the jth factor relation for the candidate well and m is the number of factors for the individual wells under consideration.


## 2.3. Single-Factor Evaluation Model

Consider a problem in which each candidate well hasm criteria uj∣j=1,2,…,m and n records in the output vi∣i=1,2,…,n. The main objective is to determine which input variables uj have the most significant impact on the output. A fuzzy rule is used for each point in the space; hence, one can obtain n fuzzy rules for the m points [6, 32]:(9)uid=e−d−a/b2,where d is the value of the evaluation factor, i represents each reviewer’s grade, and a and b are indicator parameters.For the same review, the membership function is determined as follows:(10)ujidj=e−dj−a/b2,where dj represents the value of the jth evaluation factor and ujidj is the dj membership of review i.From formula (10), when dj=a, ujia=1, which indicates the largest membership. Therefore, a represents a level of expectation as follows:(11)a=d1+d22,where d1 and d2 are the maximum and minimum values of the ith review, respectively.The value at these points is equal to the neighbouring review memberships, and such points are called transition points. The degree of membership of a transition point falls at the midpoint of the range of membership values and is equal to 0.5 for transition points connected to edge points [37]:(12)ujid1=e−d1−d2/2b2=0.5.Thus,(13)b=d1−d22ln2.The membership function relationships can be determined after the parametersa and b are determined. These values represent the ith review level, which can be determined by statistical correlation analysis. The fuzzy evaluation set of the jth factor is as follows:(14)Uj=uj1,…,uji,…,ujnT,j=1,2,…,m;i=1,2,…,n.The fuzzy possibilityUj is based on a single parameter and is unable to reflect the combined effects of all factors. With m parameters and n reviews, one can obtain the following matrix:(15)U=ujidj=u11Lu1iLu1nMMuj1LujiLujnMMum1LumiLumn,j=1,2,…,m;i=1,2,…,n.


## 2.4. Fuzzy Multicriteria Evaluation

The purpose of this step is to aggregate individual criteria into a group preference for each factor. The fuzzy multicriteria evaluation scores are obtained using a matrix multiplication summation algorithm. The fuzzy vectorB is then formed and combines the contributions of all the factors:(16)B=bi=b1,…,bi,…,bn=A∘U=a1,…,aj,…,am∘u11u12…u1nu21u22…u2n…………um1um2…umn,j=1,2,…,m;i=1,2,…,n,where bi is the evaluation index, which is the jth factor membership degree of the evaluation object when all input factors are taken into consideration. According to the principle of the maximum membership degree [28], the maximum bi is used for final decision-making.


## 2.5. AHP Combined with Fuzzy Multicriteria Evaluation

Human judgement plays a decisive role throughout the process of candidate well selection. Some of the advantages of computer-based models are that they do not suffer from subjective biases and they give consistent results. However, they have limited flexibility in dealing with new conditions. Meanwhile, human operators can be highly adaptable in applying their knowledge and experience but can be inconsistent in their evaluations. Hence, it is desirable to combine the advantages of both approaches in decision-making. The correspondents include geologists, engineers, and managers. They applied their expert judgements in the evaluation as follows [18]:(1)
Within the supporting framework of the AHP, human experts make judgements about which factors are most significant within the areas of reserve capacity, deliverability, and construction factors. These can be expressed verbally, graphically, or numerically.(2)
Decision-makers then decide whether each criterion is positively or negatively correlated with the well output after fracturing. Then, grey theory is used to determine the relative importance of the individual factors and assign a weighting factor to each one. The results are then used for well selection.(3)
There is a further opportunity for the human validation and revision of the membership functions and fuzzy rules. They can also verify that the computer-based selection is consistent with their knowledge and intuition.


## 3. An Illustrative Application

The case study’s objective is to demonstrate how the proposed method works for candidate well selection using fourteen input variables.


### 3.1. Source Information Analysis

Based on AHP principles and the actual candidate situation, a hierarchy composed of four levels, three main criteria, fourteen subcriteria, andn candidate wells was obtained, as shown in Figure 1. The criteria system is shown in Table 3.Table 3
Input data used in the sample case.
Code name
Reserve capacity
Deliverability
Hydraulic efficiency
Goal
TH
POR
SGT
GR
NR
DEN
SON
SPI
LTG
LRDD
DCR
PRE
SLU
SAN
IP
Well 1
9.0
6.0
38
78.0
9.8
2.51
64.1
1.77
2.63
18.3
3.1
184.2
76.7
39.6
2.4
Well 2
12.0
8.5
42
64.5
14.5
2.47
66.0
1.77
2.18
7.0
3.2
152.2
69.3
31.2
5.1
Well 3
6.0
6.3
32
80.8
14.8
2.48
68.2
1.77
2.63
8.8
3.0
103.6
77.8
17.4
3.0
Well 4
7.5
7.0
53
69.5
11.5
2.47
69.8
1.77
2.63
16.0
3.2
106.7
61.0
23.3
5.2
Well 5
13.3
6.7
44
77.8
13.4
2.48
71.0
1.77
2.63
10.7
3.5
215.1
106.4
50.5
5.8
Well 6
5.8
9.7
54
87.8
13.4
2.46
71.0
1.77
2.63
10.7
3.5
92.8
45.9
20.0
4.0
Well 7
12.0
9.5
48
70.8
13.3
2.48
70.8
3.42
2.63
28.5
3.0
238.8
142.7
48.0
5.3
Well 8
17.1
5.9
40
80.5
11.4
2.5
67.8
1.77
2.18
12.5
3.5
187.4
98.3
41.0
3.4
Well 9
18.9
3.9
45
78.2
6.9
2.54
67.5
3.42
2.63
28.0
3.5
226.5
100.8
49.1
2.6
Well 10
20.5
10.8
54
63.3
14.3
2.4
72.3
1.77
2.63
7.6
3.6
512.5
278.4
112.8
8.6
Well 11
22.0
9.5
52
61.3
17.5
2.43
71.0
1.77
2.18
7.0
3.9
449.8
224.6
101.2
6.5
Well 12
6.0
6.3
32
82.0
14.8
2.48
68.2
1.77
2.63
8.8
3.0
67.9
33.3
21.4
3.0
Well 13
17.9
9.0
44
69.5
14.5
2.43
71.5
1.77
2.18
10.3
3.5
106.9
62.5
21.5
6.7
Well 14
16.8
9.7
29
88.0
16.7
2.44
71.5
1.77
2.63
5.8
3.2
77.8
43.4
16.8
4.8
Well 15
12.0
9.2
49
70.8
12.6
2.45
72.1
3.42
2.63
14.5
3.2
121.1
63.2
26.4
Well 16
17.9
9.0
44
69.5
14.5
2.42
71.5
1.77
2.63
10.3
3.5
103.3
60.4
21.5
Well 17
7.5
7.0
53
69.5
11.5
2.46
69.8
1.77
2.63
16.0
3.2
106.7
61.0
23.3
Well 18
15.0
9.6
46
81.0
11.9
2.48
69.5
1.77
2.18
16.0
3.2
189.2
98.8
42.0
Well 19
12.8
9.1
36
78.0
14.7
2.42
71.5
1.77
2.63
5.8
3.3
94.8
46.3
20.5The datasets used in this study came from fourteen gas wells, which were drilled in a complex fractured reservoir located in the Sichuan Basin in West China. The wells were named “well 1” through “well 14.” According to geologists, all the wells were drilled in a similar depositional environment. Hence, the properties of one well can be used to infer the properties of other wells. In each well, data for fourteen factors, including geological, drilling, well log, and corresponding initial production information, are available. The input variables used for the analysis were the pay zone thickness (TH), effective porosity (POR), gas saturation (SGT), natural gamma (GR), compensating neutron porosity (NR), rock density (DEN), acoustic time difference (SON), well position in the trap (SPI), lithography (LTG), difference in lateral and vertical resistivity (LRDD), construction rate (DCR), volume of the prepad (PRE), slurry volume (SLU), and sand volume (SAN). These input variables are relatively easy to obtain and are sufficient for estimating the initial production (IP) of each candidate well. To evaluate the performance of the proposed method, wells 15 to 19 were used to test predictions that were made on the basis of the data from the fourteen other wells. The predictions were then verified by comparing the corresponding values with the actual values.


### 3.2. Input Variable Analysis

We used the sample data in Table3 (well 1 to well 14) to apply linear regression analysis and determine the effect of each parameter on IP. Based on Figure 2, we found that IP increases with TH, POR, SGT, NR, SON, DCR, PRE, SLU, and SAN and that GR, DEN, SPI, LTG, and LRDD are negatively correlated with IP.Figure 2
Testing the initial production relationship data with input variables.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)Using this knowledge in combination with the parameters in Table3, we obtained ratings for all the input variables separately, as shown in Tables 4 and 5. Each input variable is divided into four equally spaced categories with grades “I,” “II,” “III,” and “IV.” Grade “I” represents the input variables with the largest positive correlations with IP, and grade “IV” represents those with the smallest correlations, as shown in Tables 4 and 5.Table 4
Review grades of reserve capacity factor.
Review grade
TH
POR
SGT
GR
NR
DEN
SON
I
22.00–17.94
10.83–9.09
54.00–47.75
61.29–67.97
17.5–14.84
2.39–2.43
72.29–70.25
II
17.94–13.89
9.09–7.36
47.75–41.50
67.97–74.64
14.84–12.19
2.43–2.47
70.25–68.20
III
13.89–9.84
7.36–5.63
41.50–35.25
74.64–81.32
12.19–9.54
2.47–2.50
68.20–66.15
IV
9.84–5.79
5.63–3.90
35.25–29.00
81.32–87.99
9.54–6.89
2.50–2.54
66.15–64.10Table 5
Review grades of deliverability and hydraulic efficiency factors.
Review grade
Deliverability
Hydraulic efficiency
SPI
LTG
LRDD
DCR
PRE
SLU
SAN
I
1.76–2.18
2.17–2.28
5.75–11.43
3.89–3.67
512.5–401.33
278.39–217.11
112.75–88.76
II
2.18–2.59
2.28–2.40
11.43–17.12
3.67–3.44
401.33–290.17
217.11–155.83
88.76–64.77
III
2.59–3.00
2.40–2.51
17.12–22.81
3.44–3.22
290.17–179.01
155.83–94.55
64.77–40.78
IV
3.00–3.41
2.51–2.63
22.81–28.50
3.22–2.99
179.01–67.85
94.55–33.27
40.78–16.80


### 3.3. Fuzzy Weights

The relative importance of each input variable and the corresponding effect on the selection of candidate wells are critical when several possible input variables are considered. In this method, we use a grey theory tool to rank the parameters given in Table3. Based on equations (1) to (8), we can obtain the reserve capacity, deliverability, and hydraulic efficiency criteria weights as 0.5117, 0.2005, and 0.2878, respectively. Similarly, Table 6 shows each factor weight value for the subcriteria (grade II).Table 6
Factor criteria weights.
Main criteria weights
Subcriteria weights
Factor
Weight
Factor
Correlation coefficient
Weight
Reserve capacity
0.5117
TH
0.7086
0.1445
POR
0.7373
0.1503
SGT
0.6936
0.1414
GR
0.6714
0.1369
NR
0.7324
0.1493
DEN
0.6688
0.1364
SON
0.6929
0.1413
Deliverability
0.2005
SPI
0.6705
0.3489
LTG
0.6614
0.3441
LRDD
0.5900
0.3070
Hydraulic efficiency
0.2878
DCR
0.6852
0.2484
SAR
0.6662
0.2415
PADR
0.7354
0.2666
SANDR
0.6719
0.2436
PADR is the ratio of the prepad fluid volume to the injected fluid volume. SANDR is the ratio of the prepad sand volume to the slurry volume.When considering specific criteria weights in the assessment system, one can multiply the main criteria weights by the subcriteria weights. The calculation results show that the most important factors in the sequence are as follows: (1) POR, (2) SLU, (3) NR, (4) TH, (5) SGT, (6) SON, (7) DCR, (8) SAN, (9) GR, (10) SPI, (11) DEN, (12) PRE, (13) LTG, and (14) LRDD.Figure3 shows that the input variable weights vary between 0 and 1. Clearly, POR is found to be the highest-ranked driver in this example.Figure 3
Ranking the input drivers between 0 and 1.


### 3.4. Single-Criterion Candidate Well Selection

Table7 provides the fuzzy scores of fourteen evaluation parameters with four review grades for well 15 using equations (10) to (14). Based on the maximum membership degree principle, the rating of input variable TH is closest to review grade “III,” so we can say that the TH (net pay thickness) of well 15 is grade “III.” POR is grade “I.” As previously mentioned, the candidate well selection results are affected by all the relevant criteria, and it is essential to combine the factors to tackle the task in a comprehensive way.Table 7
Input values of well 15 and the fuzzy scores.
Input variables
Value
Fuzzy scores
I
II
III
IV
TH
12.0
0.0000
0.0625
1.0000
0.0625
POR
9.2
0.5892
0.4145
0.0011
0.0000
SGT
49
0.7792
0.2570
0.0003
0.0000
GR
70.8
0.0976
0.9810
0.0389
0.0000
NR
12.6
0.0074
0.7340
0.2923
0.0005
DEN
2.45
0.0625
1.0000
0.0230
0.0002
SON
72.1
0.6319
0.0043
0.0000
0.0000
SPI
3.42
0.4665
0.0016
0.0000
0.0000
LTG
2.63
0.0000
0.0000
0.0006
0.4545
LRDD
14.5
0.0497
0.9957
0.0775
0.0000
DCR
3.2
0.0000
0.0028
0.5000
0.5000
SAR
121.1
0.0000
0.0000
0.0556
0.9988
PADR
63.2
0.0000
0.0000
0.0587
0.9997
SANDR
26.4
0.0000
0.0000
0.1019
0.9766


### 3.5. Fuzzy Multicriteria Candidate Well Selection

The fuzzy comprehensive evaluation vectorB, which considers the contributions of all the factors, is derived a follows:(17)B=A∘U=b1,…bi,…,bn,=0.07390.07490.07050.06820.07440.06800.07040.06810.06720.05990.06960.06770.07470.0683∘0.00000.06251.00000.06250.58920.41450.00110.00000.77920.25700.00030.00000.09760.98100.03890.00000.00740.73400.29230.00050.06251.00000.02300.00020.63190.00430.00000.00000.46650.00160.00000.00000.00000.00000.00060.45450.04970.99570.07750.00000.00000.00280.50000.50000.00000.00000.05560.99880.00000.00000.05870.99970.00000.00000.10190.9766,=0.16210.31210.15190.3176,where bi reflects the value set of grades “I,” “II,” “III,” and “IV.” Equation (17) shows that the fuzzy scores of well 15 for grades “I,” “II,” “III,” and “IV” are 0.1621, 0.3122, 0.1519, and 0.3176, respectively. According to the maximum membership degree principle, the rating of the well is closest to grade “IV,” so we can say that candidate well 15 belongs to review grade “IV.” Similarly, the results for the other four wells are shown in Table 8.Table 8
Measured data and single-criterion fuzzy scores of wells.
Input variables
Well 16
Well 17
Well 18
Well 19
Value
Fuzzy scores
Value
Fuzzy scores
Value
Fuzzy scores
Value
Fuzzy scores
I
II
III
IV
I
II
III
IV
I
II
III
IV
I
II
III
IV
TH
17.9
0.4657
0.5350
0.0024
0.0000
7.5
0.0000
0.0000
0.0299
0.9576
15.0
0.0131
0.8409
0.2102
0.0002
12.8
0.0001
0.1696
0.8950
0.0185
POR
9.0
0.4300
0.5733
0.0030
0.0000
7.0
0.0003
0.2490
0.7896
0.0098
9.6
0.8881
0.1735
0.0001
0.0000
9.1
0.5080
0.4920
0.0019
0.0000
SGT
44
0.0349
0.9727
0.1058
0.0000
53
0.7258
0.0069
0.0000
0.0000
46
0.1851
0.8744
0.0161
0.0000
36
0.0000
0.0051
0.6701
0.3445
GR
69.5
0.2291
0.8162
0.0115
0.0000
69.5
0.2291
0.8162
0.0115
0.0000
81.0
0.0000
0.0029
0.5674
0.4349
78.0
0.0000
0.0612
1.0000
0.0633
NR
14.5
0.3353
0.6818
0.0054
0.0000
11.5
0.0002
0.2013
0.8528
0.0141
11.9
0.0008
0.3571
0.6551
0.0047
14.7
0.4288
0.5744
0.0030
0.0000
DEN
2.42
0.8409
0.2102
0.0000
0.0000
2.46
0.0131
0.8409
0.1458
0.0020
2.48
0.0002
0.2102
0.9259
0.0625
2.42
0.8409
0.2102
0.0000
0.0000
SON
71.5
0.9654
0.0329
0.0000
0.0000
69.8
0.2370
0.8040
0.0106
0.0000
69.5
0.1240
0.9513
0.0283
0.0000
71.5
0.9654
0.0329
0.0000
0.0000
SPI
1.77
0.0000
0.0000
0.0020
0.5333
1.77
0.0000
0.0000
0.0020
0.5333
1.77
0.0000
0.0000
0.0020
0.5333
1.77
0.0000
0.0000
0.0020
0.5333
LTG
2.63
0.0000
0.0000
0.0009
0.5000
2.63
0.0000
0.0000
0.0009
0.5000
2.18
0.6288
0.0072
0.0000
0.0000
2.63
0.0000
0.0000
0.0009
0.5000
LRDD
10.3
0.7778
0.2584
0.0003
0.0000
16.0
0.0089
0.7751
0.2602
0.0003
16.0
0.0089
0.7751
0.2602
0.0003
5.8
0.5122
0.0021
0.0000
0.0000
DCR
3.5
0.0112
0.8534
0.1910
0.0003
3.2
0.0000
0.0014
0.3798
0.6231
3.2
0.0000
0.0014
0.3798
0.6231
3.3
0.0000
0.0331
0.9498
0.1363
SAR
103.3
0.0000
0.0000
0.0209
0.9131
106.7
0.0000
0.0000
0.0255
0.9391
189.2
0.0000
0.0041
0.6298
0.3789
94.8
0.0000
0.0000
0.0125
0.8320
PADR
60.4
0.0000
0.0000
0.0451
0.9909
61.0
0.0000
0.0000
0.0477
0.9938
98.8
0.0000
0.0034
0.5980
0.4071
46.3
0.0000
0.0000
0.0101
0.7954
SANDR
21.5
0.0000
0.0000
0.0347
0.9721
23.3
0.0000
0.0000
0.0528
0.9975
42.0
0.0000
0.0098
0.7894
0.2492
20.5
0.0000
0.0000
0.0272
0.9467Finally, all the candidate well fuzzy scores and ranking results are listed in Table9. ‘‘Well 19,” which has the maximum fuzzy score values closest to review grade “I,” is determined to be the best candidate for hydraulic fracturing. The ranking of the candidate wells can be summarized in the following order: well 19, well 16, well 18, well 15, and well 17. The field application shows that the wells with high fuzzy scores consistently have high initial production levels after hydraulic fracturing.Table 9
Rating and ranking of other wells.
Candidate alternatives
Fuzzy scores
Rating
Ranking
IP (104 m3/d)
I
II
III
IV
Well 15
0.1621
0.3121
0.1519
0.3176
IV
4
4.62
Well 16
0.3274
0.3556
0.0028
0.0028
II
2
6.74
Well 17
0.1244
0.2567
0.1902
0.3555
IV
5
3.21
Well 18
0.1730
0.3007
0.3443
0.1674
III
3
5.84
Well 19
0.2689
0.1204
0.2525
0.2436
I
1
7.25


## 3.1. Source Information Analysis

Based on AHP principles and the actual candidate situation, a hierarchy composed of four levels, three main criteria, fourteen subcriteria, andn candidate wells was obtained, as shown in Figure 1. The criteria system is shown in Table 3.Table 3
Input data used in the sample case.
Code name
Reserve capacity
Deliverability
Hydraulic efficiency
Goal
TH
POR
SGT
GR
NR
DEN
SON
SPI
LTG
LRDD
DCR
PRE
SLU
SAN
IP
Well 1
9.0
6.0
38
78.0
9.8
2.51
64.1
1.77
2.63
18.3
3.1
184.2
76.7
39.6
2.4
Well 2
12.0
8.5
42
64.5
14.5
2.47
66.0
1.77
2.18
7.0
3.2
152.2
69.3
31.2
5.1
Well 3
6.0
6.3
32
80.8
14.8
2.48
68.2
1.77
2.63
8.8
3.0
103.6
77.8
17.4
3.0
Well 4
7.5
7.0
53
69.5
11.5
2.47
69.8
1.77
2.63
16.0
3.2
106.7
61.0
23.3
5.2
Well 5
13.3
6.7
44
77.8
13.4
2.48
71.0
1.77
2.63
10.7
3.5
215.1
106.4
50.5
5.8
Well 6
5.8
9.7
54
87.8
13.4
2.46
71.0
1.77
2.63
10.7
3.5
92.8
45.9
20.0
4.0
Well 7
12.0
9.5
48
70.8
13.3
2.48
70.8
3.42
2.63
28.5
3.0
238.8
142.7
48.0
5.3
Well 8
17.1
5.9
40
80.5
11.4
2.5
67.8
1.77
2.18
12.5
3.5
187.4
98.3
41.0
3.4
Well 9
18.9
3.9
45
78.2
6.9
2.54
67.5
3.42
2.63
28.0
3.5
226.5
100.8
49.1
2.6
Well 10
20.5
10.8
54
63.3
14.3
2.4
72.3
1.77
2.63
7.6
3.6
512.5
278.4
112.8
8.6
Well 11
22.0
9.5
52
61.3
17.5
2.43
71.0
1.77
2.18
7.0
3.9
449.8
224.6
101.2
6.5
Well 12
6.0
6.3
32
82.0
14.8
2.48
68.2
1.77
2.63
8.8
3.0
67.9
33.3
21.4
3.0
Well 13
17.9
9.0
44
69.5
14.5
2.43
71.5
1.77
2.18
10.3
3.5
106.9
62.5
21.5
6.7
Well 14
16.8
9.7
29
88.0
16.7
2.44
71.5
1.77
2.63
5.8
3.2
77.8
43.4
16.8
4.8
Well 15
12.0
9.2
49
70.8
12.6
2.45
72.1
3.42
2.63
14.5
3.2
121.1
63.2
26.4
Well 16
17.9
9.0
44
69.5
14.5
2.42
71.5
1.77
2.63
10.3
3.5
103.3
60.4
21.5
Well 17
7.5
7.0
53
69.5
11.5
2.46
69.8
1.77
2.63
16.0
3.2
106.7
61.0
23.3
Well 18
15.0
9.6
46
81.0
11.9
2.48
69.5
1.77
2.18
16.0
3.2
189.2
98.8
42.0
Well 19
12.8
9.1
36
78.0
14.7
2.42
71.5
1.77
2.63
5.8
3.3
94.8
46.3
20.5The datasets used in this study came from fourteen gas wells, which were drilled in a complex fractured reservoir located in the Sichuan Basin in West China. The wells were named “well 1” through “well 14.” According to geologists, all the wells were drilled in a similar depositional environment. Hence, the properties of one well can be used to infer the properties of other wells. In each well, data for fourteen factors, including geological, drilling, well log, and corresponding initial production information, are available. The input variables used for the analysis were the pay zone thickness (TH), effective porosity (POR), gas saturation (SGT), natural gamma (GR), compensating neutron porosity (NR), rock density (DEN), acoustic time difference (SON), well position in the trap (SPI), lithography (LTG), difference in lateral and vertical resistivity (LRDD), construction rate (DCR), volume of the prepad (PRE), slurry volume (SLU), and sand volume (SAN). These input variables are relatively easy to obtain and are sufficient for estimating the initial production (IP) of each candidate well. To evaluate the performance of the proposed method, wells 15 to 19 were used to test predictions that were made on the basis of the data from the fourteen other wells. The predictions were then verified by comparing the corresponding values with the actual values.


## 3.2. Input Variable Analysis

We used the sample data in Table3 (well 1 to well 14) to apply linear regression analysis and determine the effect of each parameter on IP. Based on Figure 2, we found that IP increases with TH, POR, SGT, NR, SON, DCR, PRE, SLU, and SAN and that GR, DEN, SPI, LTG, and LRDD are negatively correlated with IP.Figure 2
Testing the initial production relationship data with input variables.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)Using this knowledge in combination with the parameters in Table3, we obtained ratings for all the input variables separately, as shown in Tables 4 and 5. Each input variable is divided into four equally spaced categories with grades “I,” “II,” “III,” and “IV.” Grade “I” represents the input variables with the largest positive correlations with IP, and grade “IV” represents those with the smallest correlations, as shown in Tables 4 and 5.Table 4
Review grades of reserve capacity factor.
Review grade
TH
POR
SGT
GR
NR
DEN
SON
I
22.00–17.94
10.83–9.09
54.00–47.75
61.29–67.97
17.5–14.84
2.39–2.43
72.29–70.25
II
17.94–13.89
9.09–7.36
47.75–41.50
67.97–74.64
14.84–12.19
2.43–2.47
70.25–68.20
III
13.89–9.84
7.36–5.63
41.50–35.25
74.64–81.32
12.19–9.54
2.47–2.50
68.20–66.15
IV
9.84–5.79
5.63–3.90
35.25–29.00
81.32–87.99
9.54–6.89
2.50–2.54
66.15–64.10Table 5
Review grades of deliverability and hydraulic efficiency factors.
Review grade
Deliverability
Hydraulic efficiency
SPI
LTG
LRDD
DCR
PRE
SLU
SAN
I
1.76–2.18
2.17–2.28
5.75–11.43
3.89–3.67
512.5–401.33
278.39–217.11
112.75–88.76
II
2.18–2.59
2.28–2.40
11.43–17.12
3.67–3.44
401.33–290.17
217.11–155.83
88.76–64.77
III
2.59–3.00
2.40–2.51
17.12–22.81
3.44–3.22
290.17–179.01
155.83–94.55
64.77–40.78
IV
3.00–3.41
2.51–2.63
22.81–28.50
3.22–2.99
179.01–67.85
94.55–33.27
40.78–16.80


## 3.3. Fuzzy Weights

The relative importance of each input variable and the corresponding effect on the selection of candidate wells are critical when several possible input variables are considered. In this method, we use a grey theory tool to rank the parameters given in Table3. Based on equations (1) to (8), we can obtain the reserve capacity, deliverability, and hydraulic efficiency criteria weights as 0.5117, 0.2005, and 0.2878, respectively. Similarly, Table 6 shows each factor weight value for the subcriteria (grade II).Table 6
Factor criteria weights.
Main criteria weights
Subcriteria weights
Factor
Weight
Factor
Correlation coefficient
Weight
Reserve capacity
0.5117
TH
0.7086
0.1445
POR
0.7373
0.1503
SGT
0.6936
0.1414
GR
0.6714
0.1369
NR
0.7324
0.1493
DEN
0.6688
0.1364
SON
0.6929
0.1413
Deliverability
0.2005
SPI
0.6705
0.3489
LTG
0.6614
0.3441
LRDD
0.5900
0.3070
Hydraulic efficiency
0.2878
DCR
0.6852
0.2484
SAR
0.6662
0.2415
PADR
0.7354
0.2666
SANDR
0.6719
0.2436
PADR is the ratio of the prepad fluid volume to the injected fluid volume. SANDR is the ratio of the prepad sand volume to the slurry volume.When considering specific criteria weights in the assessment system, one can multiply the main criteria weights by the subcriteria weights. The calculation results show that the most important factors in the sequence are as follows: (1) POR, (2) SLU, (3) NR, (4) TH, (5) SGT, (6) SON, (7) DCR, (8) SAN, (9) GR, (10) SPI, (11) DEN, (12) PRE, (13) LTG, and (14) LRDD.Figure3 shows that the input variable weights vary between 0 and 1. Clearly, POR is found to be the highest-ranked driver in this example.Figure 3
Ranking the input drivers between 0 and 1.


## 3.4. Single-Criterion Candidate Well Selection

Table7 provides the fuzzy scores of fourteen evaluation parameters with four review grades for well 15 using equations (10) to (14). Based on the maximum membership degree principle, the rating of input variable TH is closest to review grade “III,” so we can say that the TH (net pay thickness) of well 15 is grade “III.” POR is grade “I.” As previously mentioned, the candidate well selection results are affected by all the relevant criteria, and it is essential to combine the factors to tackle the task in a comprehensive way.Table 7
Input values of well 15 and the fuzzy scores.
Input variables
Value
Fuzzy scores
I
II
III
IV
TH
12.0
0.0000
0.0625
1.0000
0.0625
POR
9.2
0.5892
0.4145
0.0011
0.0000
SGT
49
0.7792
0.2570
0.0003
0.0000
GR
70.8
0.0976
0.9810
0.0389
0.0000
NR
12.6
0.0074
0.7340
0.2923
0.0005
DEN
2.45
0.0625
1.0000
0.0230
0.0002
SON
72.1
0.6319
0.0043
0.0000
0.0000
SPI
3.42
0.4665
0.0016
0.0000
0.0000
LTG
2.63
0.0000
0.0000
0.0006
0.4545
LRDD
14.5
0.0497
0.9957
0.0775
0.0000
DCR
3.2
0.0000
0.0028
0.5000
0.5000
SAR
121.1
0.0000
0.0000
0.0556
0.9988
PADR
63.2
0.0000
0.0000
0.0587
0.9997
SANDR
26.4
0.0000
0.0000
0.1019
0.9766


## 3.5. Fuzzy Multicriteria Candidate Well Selection

The fuzzy comprehensive evaluation vectorB, which considers the contributions of all the factors, is derived a follows:(17)B=A∘U=b1,…bi,…,bn,=0.07390.07490.07050.06820.07440.06800.07040.06810.06720.05990.06960.06770.07470.0683∘0.00000.06251.00000.06250.58920.41450.00110.00000.77920.25700.00030.00000.09760.98100.03890.00000.00740.73400.29230.00050.06251.00000.02300.00020.63190.00430.00000.00000.46650.00160.00000.00000.00000.00000.00060.45450.04970.99570.07750.00000.00000.00280.50000.50000.00000.00000.05560.99880.00000.00000.05870.99970.00000.00000.10190.9766,=0.16210.31210.15190.3176,where bi reflects the value set of grades “I,” “II,” “III,” and “IV.” Equation (17) shows that the fuzzy scores of well 15 for grades “I,” “II,” “III,” and “IV” are 0.1621, 0.3122, 0.1519, and 0.3176, respectively. According to the maximum membership degree principle, the rating of the well is closest to grade “IV,” so we can say that candidate well 15 belongs to review grade “IV.” Similarly, the results for the other four wells are shown in Table 8.Table 8
Measured data and single-criterion fuzzy scores of wells.
Input variables
Well 16
Well 17
Well 18
Well 19
Value
Fuzzy scores
Value
Fuzzy scores
Value
Fuzzy scores
Value
Fuzzy scores
I
II
III
IV
I
II
III
IV
I
II
III
IV
I
II
III
IV
TH
17.9
0.4657
0.5350
0.0024
0.0000
7.5
0.0000
0.0000
0.0299
0.9576
15.0
0.0131
0.8409
0.2102
0.0002
12.8
0.0001
0.1696
0.8950
0.0185
POR
9.0
0.4300
0.5733
0.0030
0.0000
7.0
0.0003
0.2490
0.7896
0.0098
9.6
0.8881
0.1735
0.0001
0.0000
9.1
0.5080
0.4920
0.0019
0.0000
SGT
44
0.0349
0.9727
0.1058
0.0000
53
0.7258
0.0069
0.0000
0.0000
46
0.1851
0.8744
0.0161
0.0000
36
0.0000
0.0051
0.6701
0.3445
GR
69.5
0.2291
0.8162
0.0115
0.0000
69.5
0.2291
0.8162
0.0115
0.0000
81.0
0.0000
0.0029
0.5674
0.4349
78.0
0.0000
0.0612
1.0000
0.0633
NR
14.5
0.3353
0.6818
0.0054
0.0000
11.5
0.0002
0.2013
0.8528
0.0141
11.9
0.0008
0.3571
0.6551
0.0047
14.7
0.4288
0.5744
0.0030
0.0000
DEN
2.42
0.8409
0.2102
0.0000
0.0000
2.46
0.0131
0.8409
0.1458
0.0020
2.48
0.0002
0.2102
0.9259
0.0625
2.42
0.8409
0.2102
0.0000
0.0000
SON
71.5
0.9654
0.0329
0.0000
0.0000
69.8
0.2370
0.8040
0.0106
0.0000
69.5
0.1240
0.9513
0.0283
0.0000
71.5
0.9654
0.0329
0.0000
0.0000
SPI
1.77
0.0000
0.0000
0.0020
0.5333
1.77
0.0000
0.0000
0.0020
0.5333
1.77
0.0000
0.0000
0.0020
0.5333
1.77
0.0000
0.0000
0.0020
0.5333
LTG
2.63
0.0000
0.0000
0.0009
0.5000
2.63
0.0000
0.0000
0.0009
0.5000
2.18
0.6288
0.0072
0.0000
0.0000
2.63
0.0000
0.0000
0.0009
0.5000
LRDD
10.3
0.7778
0.2584
0.0003
0.0000
16.0
0.0089
0.7751
0.2602
0.0003
16.0
0.0089
0.7751
0.2602
0.0003
5.8
0.5122
0.0021
0.0000
0.0000
DCR
3.5
0.0112
0.8534
0.1910
0.0003
3.2
0.0000
0.0014
0.3798
0.6231
3.2
0.0000
0.0014
0.3798
0.6231
3.3
0.0000
0.0331
0.9498
0.1363
SAR
103.3
0.0000
0.0000
0.0209
0.9131
106.7
0.0000
0.0000
0.0255
0.9391
189.2
0.0000
0.0041
0.6298
0.3789
94.8
0.0000
0.0000
0.0125
0.8320
PADR
60.4
0.0000
0.0000
0.0451
0.9909
61.0
0.0000
0.0000
0.0477
0.9938
98.8
0.0000
0.0034
0.5980
0.4071
46.3
0.0000
0.0000
0.0101
0.7954
SANDR
21.5
0.0000
0.0000
0.0347
0.9721
23.3
0.0000
0.0000
0.0528
0.9975
42.0
0.0000
0.0098
0.7894
0.2492
20.5
0.0000
0.0000
0.0272
0.9467Finally, all the candidate well fuzzy scores and ranking results are listed in Table9. ‘‘Well 19,” which has the maximum fuzzy score values closest to review grade “I,” is determined to be the best candidate for hydraulic fracturing. The ranking of the candidate wells can be summarized in the following order: well 19, well 16, well 18, well 15, and well 17. The field application shows that the wells with high fuzzy scores consistently have high initial production levels after hydraulic fracturing.Table 9
Rating and ranking of other wells.
Candidate alternatives
Fuzzy scores
Rating
Ranking
IP (104 m3/d)
I
II
III
IV
Well 15
0.1621
0.3121
0.1519
0.3176
IV
4
4.62
Well 16
0.3274
0.3556
0.0028
0.0028
II
2
6.74
Well 17
0.1244
0.2567
0.1902
0.3555
IV
5
3.21
Well 18
0.1730
0.3007
0.3443
0.1674
III
3
5.84
Well 19
0.2689
0.1204
0.2525
0.2436
I
1
7.25


## 4. Conclusions

Appropriate well selection for stimulation is one of the most important decisions taken by oilfield engineers, and it incorporates geological parameters, construction parameters, and economic effects. Uncertainty is an inherent condition in the candidate well selection decision-making process. This paper presents a modified fuzzy AHP multiple-criteria decision-making approach developed to choose the most appropriate stimulation wells based on a number of criteria. The proposed methodology allows hydraulic fracturing data to be used in a flexible way by exploiting linguistic terms, fuzzy numbers, precise numerical values, and ranges of numerical values. The main advantage of the method introduced in this paper is that it requires a relatively small number of samples to identify the wells most likely to benefit from a fracture treatment without human interference. Notably, the method uses the available data for each well, including the reservoir parameter information, fracture treatment information, and production history, to find the most promising candidate well. A field application shows that the presented model based on the proposed complex evaluation system provides satisfactory candidate wells with high production levels. This study is based on actual field data, and there are no a priori assumptions or simulated data used in the verification of the method. This approach provides well-structured comprehensive information at different levels to aid in decision-making.



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*Source: 4084982-2019-09-02.xml*