Fuzzy Comprehensive Evaluation of Oil Field Waterflooding Effect

Oil field waterflooding is a complex man-controlled systematic behavior, and the related evaluation methods vary greatly. This paper put forward a fuzzy comprehensive method of evaluating controlled development level by analysis of the macroscopic evaluation to oil field waterflooding effect with com-bination of original reservoir geological state. This fuzzy evaluation technique bears unique advantages because there is little difference among evaluation indexes which represent the dynamic and static state of regional neighbor-hood of development units (blocks, Production Company, etc.). Not only the mathematical method for evaluating oil field waterflooding effect is set up, but also the method is applied in three blocks of D oil field. The calculated results show the effectiveness and practicability of the method.


Introduction
Waterflooding as the dominant technology in oil field development is facing a universal problem of associated high water cut and how to improve the waterflooding efficiency during high water cut stage has become a highlight for all the researchers in the world. Many of the oil fields in China have entered extra-high water cut stage after several decades of development, which are now encountered by the conjuncture of intractable remaining oil exploitation, deficient reserve replacements and continuous production declination. Reasonable and proper development effect evaluation would help to determine the direction for remaining oil exploitation and adjustment and thus guiding more effective and scientific reservoir development, as well as providing reference to the composi-International Journal of Geosciences tion of development plans and long-term decisions. The essence of evaluating the operational performance of any system is to judge the scientism, rationality and optimality of the system. Comprehensive evaluation of the operational performance of a system should set off from the perspective of systematology. Judging from the above point of view, during development effect evaluation, the first step is to find out evaluation indexes [1] [2]. This paper just applies these evaluation indexes, and focuses on comprehensive techniques.
Many of the development effect evaluation methods usually lean on oil production mechanism [3]- [14], however development effect of water flooding is a cumulative, macroscopic, integral and systematical issue [15] [16] that requires evaluation from the perspective of systematology [17] [18]. Although considering the development effect from the perspective of oil production mechanism could seize the key point and provide detailed operational ways for production adjustment, this method is restricted by the short coming of partial evaluation.
On the other side, many single indexe [6] [14] evaluation or correlation evaluation of practical curves and theoretical curves (usually 2D and two indexes) [7] [13] are confined. The theory of systematology [3] [8] should be adopted in the analysis of injection and production correlations in partial well clusters which have undergone water flooding, in order to ascertain the efficiency of water flooding and the utility of water flooding adjustment and plan. The limitation of this method is that it is only fit for partial well clusters and the range of the evaluation unit is too small which is not advisable to apply to the whole block or even an oil field. Evaluating water flooding effect from the perspective of systematology is discussed in this paper. Because of the slight differential of evaluation attributes caused by geographic proximity of the evaluated units, we would choose fuzzy mathematics methods [19] to handle it. Fuzzy remarks range is The universe of ambiguity factors could be defined as two aspects, first, factors that decide the original geologic property of the reservoir, and second, factors that dominate the development level of the oil field. To determine whether the water flooding effect is good or not mainly depends on the amount of oil production, which if explained from systematology is mainly controlled by two attributes, one is original reservoir geologic property and the other is controlled development level. Original reservoir geologic property refers to the natural oil productivity of the reservoir, while the controlled development level refers to treatments taken during water flooding process, including water injection pattern, well array, perforation, fracturing and acid treating, etc., some reservoirs with predominance natural oil productivity probably would not obtain satisfactory crude oil production after exploration, only by scientific and rational development control could the maximum crude oil recovery rate be reached. This Contexture of this paper follows the contents of regular comprehensive evaluation, first building an evaluation index system, and then calculating the evaluation index weight, successively, setting up unified standard function with fuzzy membership degree for each evaluation index, and finally the comprehensive evaluation.

Comprehensive Evaluation Index Set
Through system theory and control theory analysis, the structure of systematic behavior of oil field waterflooding can be found out. Water well system is defined as the group composed of all water wells, oil well system is defined as the group composed of all oil wells.
The index system of original reservoir property is the information indicator of reservoir system, while the index system of controlled development level is the information indicator of oil and water well system. Based on comprehensive evaluation of the oil field reality (D oil field), through oil field development engineering mechanism, the index system of the two oil field development attributes can be defined as follows by using quantitative and qualitative analysis (multiple correlation analysis).
All these indexes can be screen out as follows, the details are referenced to [1] [2].
Factors for evaluating original reservoir geological property include: 1) channel sand proportion hd R ; 2) effective permeability e K ; 3) net thickness e h ; 4) coefficient of permeability variation K V ; 5) initial oil saturation o S ; 6) oil viscosity o µ ; 7) difference between reservoir pressure and saturation pressure b P ; 8) reserves abundance q Factors for evaluating controlled development level include: 1) total pressure difference P ∆ ; 2) oil and water well ratio j R ; 3) well spacing density j A ; 4) cumulative water storage rate w γ ; 5) cumulative injection production ratio rc R ; 6) waterflooding controlled degree h C ; 7) reserves recovery degree h D .

Calculation about Index Weight
For comprehensive evaluation to any complex system, it is not easy to determine relies greatly on the samples need to be evaluated. There are three blocks (samples) in this D oil field. According to oil field waterflooding system attributes, the multiple correlation coefficient and variation coefficient are used to calculate index weight in this paper. Then, the final weight is got by geometric averaging the two weights.

Index Weight of Reservoir Geological Status
Assume that the multiple correlation coefficients with m indexes are 1 2 , , , m ρ ρ ρ  , then the number k coefficient reflects the ability of all the other indexes except k index to replace k index. Hence, the larger the k index, the smaller its function, so the reciprocal value of multiple correlation coefficient can be taken as weight.
Then the weight formula for multiple correlation coefficients is developed, Table 1 (all the subscription number is in accordance with the index sequence in factor range).
The weight is calculated to be: , , , 0.172, 0.105, 0.088, 0.123, 0.101, 0.098, 0.088, 0.118 In addition, from statistical theory we can know that, the larger the variance of each index, the stronger the corresponding attributes of each index for evaluating the targets, the more attention should be paid to this index, thus the larger of this index weight. This is the so called theory of determining index weight by variation coefficient.
As above mentioned, by using data in Table 4, weight can be obtained by variation coefficient calculating method [15].
In order to avoid the unilateralism of different methods for calculating weights, comprehensive weight is obtained by geometric averaging the weights obtained from the two above methods. Table 1. Multiple correlation coefficients of geological statue index system. Formula for geometric average is: In order to show the relationship between the indexes weight with the index itself, these data is tabularized in Table 2.

Controlling Development Index Weight
Based on the same principle, the weight of control index can be obtained. The These data is tabularized in Table 3.

Establishment of Fuzzy Membership Function
A unified evaluation standard is needed in order to rationally evaluate each index; this is the function with fuzzy membership degree of each index. Based on characteristic analysis of 9 geological indexes and 7 controlling indexes, it is found that the comprehensive evaluation index can be classified into two types: one is tendency index, the other is moderate index. x x x x x x respectively, among them, superscript +1 stands for points larger than 1, −1 stands for points smaller than 1.
Based on above analysis, the broken line function with fuzzy membership degree of three types of index can be set up.
 Function with fuzzy membership degree of the larger the better index x  Function with fuzzy membership degree of the smaller the better index x x x x

Comprehensive Evaluation of Waterflooding Effect
There are two index system for oil field waterflooding effect evaluation: Original reservoir state evaluation; controlled development level evaluation. Comprehensive evaluation result can be got be combining the two results. Here formula (5.7) is used.  Table 4 and Table 5, the average value for 3 samples of each index is n x , the average value for samples larger or smaller than n x is g x or b

Fuzzy Evaluation of Vectors by Single Index
x , etc. Others will not be listed here for sake of limited space.

Fuzzy Evaluation Vectors for Blocks
The fuzzy evaluation result of each block is obtained by weighting the single in- , , , is the evaluation result of k block, synthetic fuzzy operator "  " is ( ) ( ) 9  T   1 2  9  11 21  91  1  1   , , , , , , inf ,1 when calculating the fuzzy evaluation result of reservoir geological property of three blocks using formula (5.1), since the weight ( ) T 1 2 9 , , , 3)) and each factor in fuzzy matrix ( ) 9  , , , The fuzzy evaluation result of three blocks calculated by using formula (5.3) and (5.4) is listed in Table 6.

Convert Fuzzy Evaluation of Vector to a Number
For the convenience of application and comparison, we need assign a real number for these fuzzy vectors. Fuzzy levels are assigned as "good, normal, poor", and the corresponding numerical values are "100, 50, 0" respectively for better and direct understanding, so the numerical value for fuzzy comprehensive evaluation vector ( ) ( ) Fuzzy evaluation of vectors of three blocks is determined by using formula (5.5) and listed in Table 6.

Comprehensive Evaluation and Calculation
The scores for original reservoir geological property and controlled development level of three blocks in D oil field are calculated by using formula (5.3)-(5.5), the result is listed in attached C. The oil field waterflooding effect is mainly determined by controlled development level. Yet controlling development level depends greatly on original reservoir geological property. Therefore, the waterflooding effect can be defined as score for controlled development level divided by score for reservoir geological property, i.e. score for comprehensive development effect score for controlled development level score for reservoir geological property = (5.6) In ideal condition, the formula bears "normal geological property (score 50) and normal control level (score 50)". Hence, in order to rationally evaluate the comprehensive effect, the above formula should be adapted to the following to cal-