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

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 𝐹-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.

Figure 1 

Hierarchical relationship of the evaluation indicators.

2.1. Production Task

The production task [4, 5], denoted as 𝑢₁, contains the task completion rate of crude oil (𝑢₁₁) and the task completion rate of water injection (𝑢₁₂).

(1) Task Completion Rate of Crude Oil
𝑢₁₁=𝑞_𝑜/𝑞_𝑜×100%, where 𝑞_𝑜 is the actual production of crude oil and 𝑞_𝑜 is the planned output of crude oil, unit: “tons”.

(2) Task Completion Rate of Water Injection
𝑢₁₂=𝑞_𝑤/𝑞_𝑤×100%, where 𝑞_𝑤 is the actual volume of water injection and 𝑞_𝑤 is the geology-required volume of water injection, unit: m³.

2.2. Reservoir Development

The Reservoir development [6, 7], denoted as 𝑢₂, contains the controlled degree of natural decline (𝑢₂₁), the controlled degree of composite decline (𝑢₂₂), the controlled degree of the rising of composite water cut (𝑢₂₃), and the formation pressure maintenance level (𝑢₂₄).

(1) Controlled Degree of Natural Decline
𝑢₂₁=(𝐷_𝑟−𝐷_𝑟)/𝐷_𝑟×100%, where 𝐷_𝑟 is the actual natural decline rate and 𝐷_𝑟 is the control target of natural decline rate, unit: %.

(2) Controlled Degree of Composite Decline
𝑢₂₂=(𝐷_𝑡−𝐷_𝑡)/𝐷_𝑡×100%, where 𝐷_𝑡 is the actual composite decline rate and 𝐷_𝑡 is the control target of composite decline rate, unit: %.

(3) Controlled Degree of the Rising of Composite Water Cut
𝑢₂₃=(Δ𝑓_𝑤−Δ𝑓_𝑤)/Δ𝑓_𝑤×100%, where Δ𝑓_𝑤 is the actual rising rate of composite water cut and Δ𝑓_𝑤 is the control target of the raising rate of composite water cut, unit: %.

(4) Formation Pressure Maintenance Level
𝑢₂₄=𝑝/𝑝₀×100%, where 𝑝 is the current formation pressure and 𝑝₀ is the original formation pressure, unit: “MPa”.

2.3. Production Technology

The production technology [8], denoted as 𝑢₃, contains the utilization rate of oil-water wells (𝑢₃₁), the hour utilization rate of oil production, and water injection (𝑢₃₂), the qualified rate of injection allocation (𝑢₃₃), the qualified rate of injected water quality (𝑢₃₄), the dynamic monitoring completion rate (𝑢₃₅), and the measure effective rate of old wells (𝑢₃₆).

(1) Utilization Rate of Oil-Water Wells
𝑢^𝑜₃₁=𝑛_𝑜/𝑛_𝑜×100%, where 𝑢^𝑜₃₁ is the utilization rate of oil wells, 𝑛_𝑜 is the active number of oil wells, and 𝑛_𝑜 is the total number of oil wells; and 𝑢^𝑤₃₁=𝑛_𝑤/𝑛_𝑤×100%, where 𝑢^𝑤₃₁ is the utilization rate of water injection wells, 𝑛_𝑤 is the active number of water injection wells, and 𝑛_𝑤 is the total number of water injection wells. Therefore, the utilization rate of oil-water wells can be defined by 𝑢₃₁=𝑛_𝑜𝑛_𝑜+𝑛_𝑤×𝑢^𝑜₃₁+𝑛_𝑤𝑛_𝑜+𝑛_𝑤×𝑢^𝑤₃₁. (1)

(2) Time Utilization Rate of Oil Production and Water Injection
𝑢^𝑜₃₂=𝑇_𝑜/𝑇_𝑜×100%, where 𝑢^𝑜₃₂ is the time utilization rate of oil production, 𝑇_𝑜 is the actual time of oil production and 𝑇_𝑜 is the calendar time of oil production; 𝑢^𝑤₃₂=𝑇_𝑤/𝑇_𝑤×100%, where 𝑢^𝑤₃₂ is the time utilization rate of water injection, 𝑇_𝑤 is the actual time of water injection, and 𝑇_𝑤 is the calendar time of water injection, unit: “day”. Therefore, the time utilization rate of oil production and water injection can be defined by 𝑢₃₂=𝑛_𝑜𝑛_𝑜+𝑛_𝑤×𝑢^𝑜₃₂+𝑛_𝑤𝑛_𝑜+𝑛_𝑤×𝑢^𝑤₃₂. (2)

(3) Qualified Rate of Injection Allocation
𝑢₃₃=𝑐_𝑤/𝑐_𝑤̃×100%, where 𝑐_𝑤̃ is the sum of the number of wells and the number of layers for water injection allocation and 𝑐_𝑤 is the sum of the qualified wells and the qualified layers.

(4) Qualified Rate of Injected Water Quality
𝑢₃₄=𝑚_𝑤/𝑚_𝑤×100%, where 𝑚_𝑤 is the total number of water quality monitoring and 𝑚_𝑤 is the number of qualified water sample.

(5) Dynamic Monitoring Completion Rate
𝑢^𝑗₃₅=𝑘_𝑗/𝑘_𝑗×100% is the dynamic monitoring completion rate of the 𝑗th project, and the comprehensive dynamic monitoring completion rate is defined as 𝑢₃₅=^𝑛_𝑗=1𝑘_𝑗𝐾×𝑢^𝑗₃₅, (3) where 𝑛 is the number of monitoring project, 𝐾 is the total number of planned monitoring, 𝑘_𝑗 is the number of planned monitoring for the 𝑗th project, and 𝑘_𝑗 is the actual number of monitoring for the 𝑗th project, unit: times.

(6) Measure Effective Rate of Old Wells
𝑢₃₆=𝑛_eff/𝑛×100%, where 𝑛 is the total number of measures and 𝑛_eff 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 𝐴 is an n-order matrix, denoted as 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛,(a)𝐴is called a fuzzy matrix if for all 𝑖,𝑗∈{1,2,…,𝑛}, 𝐴 satisfies 0≤𝑎_𝑖𝑗≤1;(b)𝐴 is called a fuzzy complementary matrix if for all 𝑖,𝑗∈{1,2,…,𝑛}, 𝐴 also satisfies 𝑎_𝑖𝑗+𝑎_𝑗𝑖=1;(c)𝐴 is called a fuzzy consistent judgment matrix if for any 𝑖,𝑗,𝑘∈{1,2,…,𝑛}, 𝐴 further satisfies 𝑎_𝑖𝑗=𝑎_𝑖𝑘−𝑎_𝑗𝑘+0.5.

Property 1. 𝐴 is a fuzzy consistent judgment matrix if and only if for any 𝑖,𝑗∈{1,2,…,𝑛} and all 𝑘∈{1,2,…,𝑛}, there exists a constant of 𝑎 satisfying 𝑎_𝑖𝑘−𝑎_𝑗𝑘=𝑎.

Proof. (1) Suppose that 𝐴 is a fuzzy consistent judgment matrix such that 𝑎_𝑖𝑗=𝑎_𝑖𝑘−𝑎_𝑗𝑘+0.5, (4) for any 𝑖, 𝑗 and all 𝑘. Therefore, we have 𝑎_𝑖𝑘−𝑎_𝑗𝑘=𝑎_𝑖𝑗−0.5=𝑎. (5)
(2) Suppose that for any given 𝑖, 𝑗, there exists a constant of 𝑎 for each 𝑘 such that 𝑎_𝑖𝑘−𝑎_𝑗𝑘=𝑎, (6) and when 𝑗=𝑘, we have 𝑎_𝑖𝑗−𝑎_𝑗𝑗=𝑎. (7) Consequently, we get 𝑎_𝑖𝑘−𝑎_𝑗𝑘=𝑎_𝑖𝑗−𝑎_𝑗𝑗. (8) Since 𝑎_𝑖𝑗+𝑎_𝑗𝑖=1, so we can get 𝑎_𝑗𝑗=0.5. Finally, we obtain 𝑎_𝑖𝑗=𝑎_𝑖𝑘−𝑎_𝑗𝑘+0.5. (9)

Property 2. Assume 𝐴 is a fuzzy complementary judgment matrix, we define a kind of fuzzy transform: 𝑎𝐹∶𝐴→𝑅,_𝑖𝑗→𝑟_𝑖𝑗=𝑟_𝑖−𝑟_𝑗𝛼+0.5,𝛼≥2(𝑛−1), (10)(11) where 𝑟_𝑖=∑^𝑛_𝑗=1𝑎_𝑖𝑗, 𝑖=1,2,…,𝑛. Then, 𝑅 is a fuzzy consistent judgment matrix.

Proof. Firstly, we prove that 𝑅 is a fuzzy matrix. Since 0≤𝑎_𝑖𝑗≤1, we know that 0.5≤^𝑛_𝑘=1𝑎_𝑖𝑘≤𝑛−0.5,0.5−𝑛≤−^𝑛_𝑘=1𝑎_𝑗𝑘≤−0.5. (12) Therefore, we have |𝑟_𝑖−𝑟_𝑗|≤𝑛−1. When 𝛼≥2(𝑛−1), we get 0≤𝑟_𝑖𝑗≤1.
Secondly, we prove that 𝑅 is a fuzzy complementary matrix. 𝑟_𝑖𝑗+𝑟_𝑖𝑗=𝑟_𝑖−𝑟_𝑗𝛼+𝑟+0.5^𝑗−𝑟_𝑖𝛼+0.5=1. (13)
Finally, we prove that 𝑅 is a fuzzy consistent judgment matrix 𝑟_𝑖𝑘−𝑟_𝑗𝑘𝑟+0.5=^𝑖−𝑟_𝑘𝛼−𝑟+0.5^𝑗−𝑟_𝑘𝛼=𝑟+0.5+0.5_𝑖−𝑟_𝑗𝛼+0.5=𝑟_𝑖𝑗. (14)

Property 3. Assume that 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 is a fuzzy complementary judgment matrix, and 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) is the weight vector or ordering vector, then(1)when 𝐴 is consistent, by using the normalizing rank aggregation method, the weight is given by 𝑤_𝑖=2𝑛^2𝑛_𝑗=1𝑎_𝑖𝑗,𝑖=1,2,…,𝑛, (15)(2)when 𝐴 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 𝑤_𝑖=1𝑛−1𝛼+2𝑛𝛼^𝑛_𝑗=1𝑎_𝑖𝑗,𝛼≥2(𝑛−1). (16)

Proof. (1) When 𝐴 is completely consistent, 𝑤_𝑖=∑^𝑛_𝑗=1𝑎_𝑖𝑗∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝑎_𝑖𝑗=∑^𝑛_𝑗=1𝑎_𝑖𝑗∑^𝑛_{1≤𝑖<𝑗≤𝑛}𝑎_𝑖𝑗+𝑎_𝑗𝑖=∑+0.5𝑛_𝑛𝑗=1𝑎_𝑖𝑗𝑛²=2/2𝑛^2𝑛_𝑗=1𝑎_𝑖𝑗. (17)
(2) When 𝐴 is not entirely consistent, 𝑤_𝑖=∑^𝑛_𝑗=1𝑟_𝑖𝑗∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝑟_𝑖𝑗=∑^𝑛_𝑗=1𝑟_𝑖𝑗𝑛²=∑/2_𝑛𝑗=1𝑟_𝑖−𝑟_𝑗/𝛼+0.5𝑛²=1/2𝑛−1𝛼+2𝑛𝛼^𝑛_𝑗=1𝑎_𝑖𝑗. (18)
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 𝐴 by using the pairwise comparison method based on the quantity scale of 0.1∼0.9 [9] (see Table 1).

Table 1 

The quantity scale of 0.1∼0.9.

Step 2. Check whether 𝐴 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 𝐹-statistics is mentioned in this paper.

4.1. Comprehensive Ranking

Assume that there are 𝑚 RMUs and 𝑛 evaluation indicators, the decision data matrix is denoted by 𝑋=(𝑥_𝑖𝑗)_𝑚×𝑛. 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 𝑌=(𝑦_𝑖𝑗)_𝑚×𝑛, and the transformation formula are given in the following;(a)when the 𝑗th indicator is the benefit type, 𝑦_𝑖𝑗=𝑥_𝑖𝑗−min_𝑖𝑥_𝑖𝑗max_𝑖𝑥_𝑖𝑗−min_𝑖𝑥_𝑖𝑗. (19)(b)when the 𝑗th indicator is the cost type, 𝑦_𝑖𝑗=max_𝑖𝑥_𝑖𝑗−𝑥_𝑖𝑗max_𝑖𝑥_𝑖𝑗−min_𝑖𝑥_𝑖𝑗. (20)(c)when the 𝑗th is the target type, 𝑦_𝑖𝑗𝑥=1−^𝑖𝑗−𝑥₀maxmax_𝑖𝑥_𝑖𝑗−𝑥₀,𝑥₀−min_𝑖𝑥_𝑖𝑗, (21) where 𝑥₀ is the target, and min_𝑖{𝑥_𝑖𝑗}≤𝑥₀≤max_𝑖{𝑥_𝑖𝑗}.

Step 2. Determine the weights of indicators. The weight vector 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛) can be obtained by FAHP. Furthermore, we could calculate the weighted decision matrix 𝑍=(𝑧_𝑖𝑗)_𝑚×𝑛, where 𝑧_𝑖𝑗=𝑤_𝑗⋅𝑦_𝑖𝑗.

Step 3. Determine the positive ideal vector and the negative ideal vector. Respectively, denoted by 𝑧⁺=(𝑧⁺₁,𝑧⁺₂,…,𝑧⁺_𝑛)𝑧⁻=(𝑧⁻₁,𝑧⁻₂,…,𝑧⁻_𝑛), where 𝑧⁺_𝑗=max_𝑖{𝑧_𝑖𝑗}, 𝑧⁻_𝑗=min_𝑖{𝑧_𝑖𝑗}.

Step 4. Calculate the Euclidean distance from the positive ideal vector and the negative ideal vector. The Euclidean distance between the 𝑖th RMU and the positive ideal vector is denoted by 𝑑⁺_𝑖=^𝑛_𝑘=1𝑧_𝑖𝑘−𝑧⁺_𝑘². (22) The Euclidean distance between the 𝑖th RMU and the negative ideal vector is denoted by 𝑑⁻_𝑖=^𝑛_𝑘=1𝑧_𝑖𝑘−𝑧⁻_𝑘². (23)

Step 5. Calculate the relative closeness to the positive ideal vector. The relative closeness can be defined as 𝑓^∗_𝑖=𝑑⁻_𝑖𝑑⁻_𝑖+𝑑⁺_𝑖,𝑖=1,2,…,𝑚. (24)

Step 6. Decide the ranking according to the value of 𝑓^∗_𝑖. 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 𝐷=(𝑑_𝑖𝑗)_𝑚×𝑚, where 𝑑_𝑖𝑗=1𝑛^𝑛_𝑘=1𝑧_𝑖𝑘−𝑧_𝑗𝑘. (25) 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 𝐹 statistics [20]: ∑𝐹=^𝑟𝑖=1𝑛_𝑖∑^𝑚_𝑘=1𝑥_𝑖𝑘−𝑥_𝑘²/(𝑟−1)∑^𝑟_𝑖=1∑^𝑛𝑖_𝑗=1∑^𝑚_𝑘=1𝑥_𝑖𝑘−𝑥_𝑗𝑘²/()(),𝑛−𝑟∼𝐹𝑟−1,𝑛−𝑟 (26) where 𝑟 is the number of classification and 𝑛_𝑖 is the number of elements in the 𝑖th classification. 𝑥_𝑖𝑘=1𝑛_𝑖^𝑛𝑖_𝑗=1𝑥_𝑗𝑘,𝑘=1,2,…,𝑚 (27) is the average value of the 𝑘th indicator in the 𝑖th classification. 𝑥_𝑘=1𝑛^𝑛_𝑗=1𝑥_𝑗𝑘,𝑘=1,2,…,𝑚 (28) is the average value of the 𝑘th indicator of all the RMUs.

We can calculate the values of 𝐹 for each classification scheme by (26), under a given reliability 𝛼=0.05, and find out the critical values of 𝐹_0.05. If 𝐹>𝐹_0.05, the corresponding classification is feasible. Generally, take the classification number corresponding directly with max{𝐹−𝐹_0.05} 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.

Table 2 

The statistical data of 12 RMUs in 2011.

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.

Table 3 

The standardized decision data.

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.

Table 4 

Table 5 

The weights of all the indicators.

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

Table 6 

The relative closeness of 12 RMUs.

From Table 6, we know the comprehensive ranking as follows: 𝑓^∗₄≻𝑓^∗₂≻𝑓^∗₈≻𝑓^∗₉≻𝑓^∗₇≻𝑓^∗₁₀≻𝑓^∗₁₁≻𝑓^∗₅≻𝑓^∗₁₂≻𝑓^∗₃≻𝑓^∗₁≻𝑓^∗₆. (29)

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

Table 7 

The distance between any two RMUs.

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

Figure 2 

The dynamic clustering figure.

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

Table 8 

The values of 𝐹 and the critical values of 𝐹_0.05.

From Table 8, it can be seen that the best classification number is “six”, namely, {4}≻{2}≻{8,9}≻{7}≻{3,5,10,11,12}≻{1,6}. (30)

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 𝐹-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.