Advances in Fuzzy Systems

Volume 2012 (2012), Article ID 134068, 8 pages

http://dx.doi.org/10.1155/2012/134068

## Production Performance Appraisal Rating for Reservoir Management Units Based on Fuzzy Clustering

^{1}Changqing Oilfield Branch, PetroChina Company Limited, Shanxi, Xi’an 710018, China^{2}Department of Graduate, Southwest Petroleum University, Sichuan, Chengdu 610500, China

Received 13 July 2012; Accepted 2 September 2012

Academic Editor: Salvatore Sessa

Copyright © 2012 An-Qi Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In view of the existing situation of oilfield development, one kind of method to evaluate the production performance of reservoir management units (RMUs) was presented in this paper. Among the commonly used indicators of oilfield development, select 12 indicators from the three aspects of production task, production technology, and reservoir development. According to the principle of fuzzy analytic hierarchy process (FAHP), this paper introduced one kind of new method to get the weights of indicators. By means of the method of TOPSIS, it is easy to obtain the rankings for all the RMUs through calculating the weighted Euclidean distance between each RMU and the positive or negative ideal RMU. Considering the gap between the differences in RMUs, the production performance appraisal ratings of RMUs are determined by fuzzy clustering. This evaluation method could constantly improve the management level of reservoir units and deepen the delicacy management of oilfield development.

#### 1. Introduction

The oilfield companies mostly take the management concept of “Benchmarking" during the process of oilfield development [1]. According to the dynamic analysis of reservoir development, the technical section provides a kind of development scheme and sets some feasible goals that should be achieved. And the reservoir management units (RMUs) must achieve the production goals in accordance with the development requirements [2]. Currently, the development department in the process of oilfield development evaluates the production performance of the reservoir management units (RMUs) based on their own statistics data and the assessment results calculated by themselves [3]. It means that the evaluation accuracy is not high enough and the crosswise contrast is not enough. In order to make the development department accurately and timely, grasp the current situation of development and management of RMUs, it needs to establish a relatively perfect evaluation system to really respond to the management level, efficiency, and development effect of RMUs, promoting the delicacy management of oilfield development. In this paper, we first present the evaluation indicators and their computing method. In order to reasonably decide the weight of each indicator, a kind of fuzzy AHP is introduced in Section 3. Next, we introduce the method of TOPSIS to decide the comprehensive ranking of RMUs and use the Euclidean distance to describe the proximity between two RMUs. However, the proximity among the RMUs is different, so we adopt the fuzzy clustering to classify the RMUs. In order to obtain the optimal classification, the -statistics is mentioned in Section 4.

#### 2. Evaluation Indicators and Their Computing Method

Through the analysis, the production performance evaluation indicators of RMU are divided into three aspects of production task, reservoir development, and production technology [1–5] (see Figure 1), including 12 indicators in the following.

##### 2.1. Production Task

The production task [4, 5], denoted as , contains the task completion rate of crude oil () and the task completion rate of water injection ().

* Task Completion Rate of Crude Oil*

, where is the actual production of crude oil and is the planned output of crude oil, unit: “tons”.

* Task Completion Rate of Water Injection*

, where is the actual volume of water injection and is the geology-required volume of water injection, unit: m^{3}.

##### 2.2. Reservoir Development

The Reservoir development [6, 7], denoted as , contains the controlled degree of natural decline (), the controlled degree of composite decline (), the controlled degree of the rising of composite water cut (), and the formation pressure maintenance level ().

* Controlled Degree of Natural Decline*

, where is the actual natural decline rate and is the control target of natural decline rate, unit: %.

* Controlled Degree of Composite Decline*

, where is the actual composite decline rate and is the control target of composite decline rate, unit: %.

* Controlled Degree of the Rising of Composite Water Cut*

, where is the actual rising rate of composite water cut and is the control target of the raising rate of composite water cut, unit: %.

* Formation Pressure Maintenance Level*

, where is the current formation pressure and is the original formation pressure, unit: “MPa”.

##### 2.3. Production Technology

The production technology [8], denoted as , contains the utilization rate of oil-water wells (), the hour utilization rate of oil production, and water injection (), the qualified rate of injection allocation (), the qualified rate of injected water quality (), the dynamic monitoring completion rate (), and the measure effective rate of old wells ().

* Utilization Rate of Oil-Water Wells*

, where is the utilization rate of oil wells, is the active number of oil wells, and is the total number of oil wells; and , where is the utilization rate of water injection wells, is the active number of water injection wells, and is the total number of water injection wells. Therefore, the utilization rate of oil-water wells can be defined by

* Time Utilization Rate of Oil Production and Water Injection*

, where is the time utilization rate of oil production, is the actual time of oil production and is the calendar time of oil production; , where is the time utilization rate of water injection, is the actual time of water injection, and is the calendar time of water injection, unit: “day”. Therefore, the time utilization rate of oil production and water injection can be defined by

* Qualified Rate of Injection Allocation*

, where is the sum of the number of wells and the number of layers for water injection allocation and is the sum of the qualified wells and the qualified layers.

* Qualified Rate of Injected Water Quality*

, where is the total number of water quality monitoring and is the number of qualified water sample.

* Dynamic Monitoring Completion Rate*

is the dynamic monitoring completion rate of the th project, and the comprehensive dynamic monitoring completion rate is defined as
where is the number of monitoring project, is the total number of planned monitoring, is the number of planned monitoring for the th project, and is the actual number of monitoring for the th project, unit: times.

* Measure Effective Rate of Old Wells*

, where is the total number of measures and is the number of effective measures.

#### 3. The Method to Determine the Weights of Evaluation Indicators

At present, with regard to determining the weights of evaluation indicators, the analytic hierarchy process (AHP) is a kind of relatively ideal method. While the traditional AHP needs to do the consistency test and constantly adjust the judgment matrix, some scholars put forward the fuzzy analytic hierarchy process (FAHP) [9–18]. We introduce a kind of fuzzy AHP to determine the weights of indicators in this section. The principle is as follows.

*Definition 1 (see [9–11]). *Assume that is an *n*-order matrix, denoted as ,(a)is called a fuzzy matrix if for all , satisfies ;(b) is called a fuzzy complementary matrix if for all , also satisfies ;(c) is called a fuzzy consistent judgment matrix if for any , further satisfies .

*Property 1. * is a fuzzy consistent judgment matrix if and only if for any and all , there exists a constant of satisfying .

*Proof. * Suppose that is a fuzzy consistent judgment matrix such that
for any , and all . Therefore, we have

Suppose that for any given , , there exists a constant of for each such that
and when , we have
Consequently, we get
Since , so we can get . Finally, we obtain

*Property 2. *Assume is a fuzzy complementary judgment matrix, we define a kind of fuzzy transform:
where , . Then, is a fuzzy consistent judgment matrix.

*Proof. *Firstly, we prove that is a fuzzy matrix. Since , we know that
Therefore, we have . When , we get .

Secondly, we prove that is a fuzzy complementary matrix.

Finally, we prove that is a fuzzy consistent judgment matrix

*Property 3. *Assume that is a fuzzy complementary judgment matrix, and is the weight vector or ordering vector, then(1)when is consistent, by using the normalizing rank aggregation method, the weight is given by
(2)when is not entirely consistent, first carry out the fuzzy consistent transformation by (11), and then through the normalizing rank aggregation method, the weight is given by

*Proof. * When is completely consistent,

When is not entirely consistent,

From the above analysis, the steps for determining the weights can be summarized in the following.

*Step 1. *The expert gives out the fuzzy complementary judgment matrix by using the pairwise comparison method based on the quantity scale of [9] (see Table 1).

*Step 2. *Check whether is consistent or not. If consistent, calculate the weights by (15); if not, calculate the weight by (16).

#### 4. Production Performance Appraisal Rating for RMUs

In this section, we introduce the method of TOPSIS [19] to decide the comprehensive ranking of RMUs and use the Euclidean distance to describe the proximity between two RMUs. However, the proximity among the RMUs is different, for this reason, we adopt the fuzzy clustering to classify the RMUs. In order to obtain the optimal classification, the -statistics is mentioned in this paper.

##### 4.1. Comprehensive Ranking

Assume that there are RMUs and evaluation indicators, the decision data matrix is denoted by . According to the method of TOPSIS, the comprehensive ranking procedure for RMUs consists of the following steps.

*Step 1. **Standardize the decision data matrix.* The standardized decision data matrix is denoted by , and the transformation formula are given in the following;(a)when the th indicator is the benefit type,
(b)when the th indicator is the cost type,
(c)when the th is the target type,
where is the target, and .

*Step 2. **Determine the weights of indicators.* The weight vector can be obtained by FAHP. Furthermore, we could calculate the weighted decision matrix , where .

*Step 3. **Determine the positive ideal vector and the negative ideal vector.* Respectively, denoted by , where , .

*Step 4. **Calculate the Euclidean distance from the positive ideal vector and the negative ideal vector*. The Euclidean distance between the th RMU and the positive ideal vector is denoted by
The Euclidean distance between the th RMU and the negative ideal vector is denoted by

*Step 5. **Calculate the relative closeness to the positive ideal vector*. The relative closeness can be defined as

*Step 6. **Decide the ranking according to the value of *. The bigger the closeness shows the better the ranking.

##### 4.2. Rating of RMUs

Considering the gap between the differences in RMUs, the comprehensive ranking still is not enough. It is necessary to classify the RMUs with fuzzy clustering. Therefore, further we work out the distance matrix of RMUs, denoted by , where From (25), we can get the dynamic fuzzy clustering and the dynamic clustering figure.

In order to reasonably determine the number of classification, we introduce a kind of statistics [20]: where is the number of classification and is the number of elements in the th classification. is the average value of the th indicator in the th classification. is the average value of the th indicator of all the RMUs.

We can calculate the values of for each classification scheme by (26), under a given reliability , and find out the critical values of . If , the corresponding classification is feasible. Generally, take the classification number corresponding directly with as the optimal classification number and finally get the best classification rating.

#### 5. Example Analysis

The statistical data of 12 reservoir management units (RMUs) of an oilfield in the year of 2011 are listed in Table 2.

According to the basic data in Table 2, we could obtain the evaluation results.

*Step 1. *Standardize the above decision data. The 12 indicators are all the benefit type; their standardized decision data are shown in Table 3.

*Step 2. *Get the judgment matrixes of all the hierarchies through expert scoring Table 4.

Therefore, we can calculate the weights for the evaluation indicators shown in Table 5.

*Step 3. *Calculate the relative closeness of every RMU (see Table 6).

From Table 6, we know the comprehensive ranking as follows:

*Step 4. *Determine the best classification rating. First of all, by (25), calculate the distance between any two RMUs (see Table 7).

Next, we can draw the dynamic fuzzy clustering figure (see Figure 2).

Lastly, calculate the values of for every kind of classification by -statistics. The values of are listed in Table 8.

From Table 8, it can be seen that the best classification number is “six”, namely,

#### 6. Conclusions

Through analyzing the actual situation in the process of oilfield development, we first present some practically feasible evaluation indicators and their computing method in the second section. In order to reasonably decide the weight of each indicator, we introduce a kind of fuzzy AHP in Section 3. Next, by means of the method of TOPSIS, it is easy to decide the comprehensive rankings of RMUs through calculating the weighted Euclidean distance between every RMU and the positive or negative ideal RMU. And we use the Euclidean distance to describe the proximity between two RMUs. Considering the proximity among the RMUs is different, the fuzzy clustering is introduced to classify the RMUs, and the production performance appraisal ratings of RMUs are determined by fuzzy clustering. In order to obtain the optimal classification, the -statistics is mentioned in Section 4. Finally, a practical example is illustrated to explain the feasibility of this method.

In order to make the development department of oilfield companies accurately and timely grasp the current situation of oilfield development and management of RMUs, it needs to establish a relatively perfect evaluation method to really respond to the management level, efficiency, and development effect of RMUs, promoting the delicacy management of oilfield development. As we know, by means of the relatively effective evaluation method to ascertain the appraisal rating of the RMUs, we cannot only know their production performance, but also it is helpful to motivate the enthusiasm of practical production for all the RMUs. By using the evaluation method proposed in this paper, the management level could be constantly improved, and the delicacy management of oilfield development can be deepened. And the oilfield companies continuously strengthen the digital construction, so it will support the accuracy and objectivity of the evaluation method.

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