Introduction

Reservoir evaluation is an important work related to deepening our understanding of reservoirs, predicting sweet spots, and guiding scientific decision-making throughout the whole process of oil and gas exploration and development (Rostirolla et al. 2003). Reservoir evaluations are mostly deterministic, focusing on an individual aspect of the reservoir, such as the evaluation of physical properties using log data, the evaluation of reservoir thickness using seismic data, and the evaluation of reservoir pore structure using lab analysis (Ye et al. 2011; Hou and Liu 2012; Lai et al. 2015; Zhu et al. 2015). However, in the case of more complex reservoirs, such as low-permeability reservoirs, it is necessary to consider the influence of various factors in order to obtain a more comprehensive and accurate evaluation (Lai et al. 2015). Moreover, it is necessary to take into account any uncertainty related to the ambiguity and grayness caused by subjective and objective conditions, such as imperfect reservoir understanding, inconsistent classification standards, and incomplete data. As the complexity of a system increases, our ability to form an accurate and meaningful description of it decreases. As a result, the accuracy of description and complexity of a system are almost mutually exclusive after a certain threshold is reached (Rui et al. 2017). In recent years, evaluation methods such as fuzzy mathematics and gray theory have been partially applied to the complexity and uncertainty presented by low-permeability reservoir evaluations. Fuzzy mathematics and gray theory are two comprehensive evaluation methods that have been widely used in recent years and provide the comprehensive quantitative evaluation of complex matters (Olatunji et al. 2014; Zhu et al. 2017). Blockley et al. (1997) and Wu and Yang (2012) systematically summarized the uncertainties in reservoir evaluation. Ghadami et al. (2015) used fuzzy mathematics and artificial intelligence to effectively characterize and classify a large carbonate reservoir. Ranjbar-Karami and Shiri (2014) improved the reliability of elastic performance evaluation of a consistent tight gas reservoir in the southern Persian Gulf using a fuzzy system method. Liang et al. (2013) used the gray correlation method in order to quantitatively analyze the factors affecting the single well productivity of the Baken tight oil reservoir in the Williston Basin and the relationships between them. Tu et al. (2012) applied the gray correlation method to the reservoir evaluation of Qingdong sag and solved the problem of quantitative evaluation under the condition in which there was insufficient data. Ye et al. (2012) established the corresponding comprehensive classification evaluation criteria for reservoirs based on the characteristics of low-permeability sandstones using fuzzy analysis and gray correlation analysis.

Although current uncertainty evaluation methods based on fuzzy mathematics and gray theory have been applied in reservoir evaluation, the majority of the evaluations carried out still focus on individual aspects of these methods. Moreover, the fact that differences in evaluation results may be attributed to fuzzification processing and a reduced degree of discrimination is often overlooked or ignored. These factors affect final evaluation performance (Zadeh 1973; Shi et al. 2002). Thus, in order to further improve reservoir evaluation, the comprehensiveness and uncertainty of the evaluation process must be determined and the differences between evaluation results must be examined in detail. Taking the fuzzy mathematics and gray theory models as a base, a new fuzzy-gray comprehensive evaluation method is established through a combination and improvement of the models described above. This method is then used in the evaluation of the low-permeability gas reservoir of upper interval of xiaganchaigou group (E32) in the Nanbaxian oil and gas field on the northern margin of the Qaidam Basin. The proposed method is expected to provide a geological basis for the rapid and effective development of gas fields in the early stages, which may be a useful reference for use with similar low-permeability reservoir evaluation methods.

Method description

The proposed method involves the following tasks: comprehensively considering the methodological principles and advantages of fuzzy comprehensive evaluation and gray relational evaluation; combining these methods using membership as the link for information superposition; improving the internal structures and algorithms; adding feature information to the model, maximizing its comprehensive, quantitative and differential features; and reducing inherent uncertainties and ensuring the improved application of this method to low-permeability reservoir evaluation. The work flow of method is shown in Fig. 1.

Fig. 1
figure 1

Work flow of method

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Step 1: establishing an evaluation set

The establishment of an evaluation set is the basis of the evaluation method. The evaluation set generally consists of a set of factors, a set of comments, and a set of weights. The factor set is the evaluation index system. There are many indicators for reservoir evaluation. These indicators must follow certain principles, such as integrity, scientificity, comparability, and practicability (Ou et al. 2016). Moreover, reservoir evaluation must conform to objective reality and ensure the availability and reliability of data. The set of comments, also called the evaluation level, can be a set of qualitative and orderly descriptions. For example, (excellent, good, moderate, poor, very poor) or (one, two, three) can also be a set of quantitative ordered values (such as 0.8, 0.6, 0.4, 0.2). The evaluation level is generally divided into between three and five grades. A weighted set is used to reflect the evaluation factors, that is, the mutual relationship and importance of each reservoir index. Common weight determination methods include the Delphi method, the principal component analysis method, the analytic hierarchy process, the entropy weight method, and the eigenvector method (Chen and Wei 2015; Zeng et al. 2015, 2017; Liu et al. 2004, 2015). Among these methods, the entropy weight method relies on the concept of information entropy (Huang 2008). It is generally believed that if an indicator differs greatly in the comprehensive evaluation and the degree of variation is higher, the more information it provides, the greater the effect will be, and the smaller the information entropy will be. Therefore, the weight of the indicator can be reflected according to the size of the information entropy, and the arbitrariness caused by the human judgment can be avoided, which is more objective, as shown in formulas (3)–(5) (Huang 2008).

The final set of factors is given by

$$ U=\left\{{u}_1,\kern0.5em {u}_2,\kern0.5em ...,\kern0.5em {u}_{\mathrm{n}}\right\} $$
(1)

where \( {u}_1,\kern0.5em {u}_2,\kern0.5em ...,\kern0.5em {u}_n \) represent n evaluation factors.

The final set of comments is described by

$$ V=\left\{{v}_1,\kern0.5em {v}_2,\kern0.5em ...,\kern0.5em {v}_{\mathrm{m}}\right\} $$
(2)

where \( {v}_1,\kern0.5em {v}_2,\kern0.5em ...,\kern0.5em {v}_m \) represent m comment levels.

The formula for calculating the entropy weight method is

$$ {P}_{ki}=\frac{x_{ki}}{\sum \limits_{k=1}^K{x}_{ki}},i=1,2,...,n;k=1,2,...,K $$
(3)
$$ {E}_i=-\frac{1}{\ln (K)}\sum \limits_{k=1}^K{P}_{ki}\ln {P}_{ki} $$
(4)
$$ {a}_i=\frac{1-{E}_i}{\sum \limits_{i=1}^n\left(1-{E}_i\right)} $$
(5)

In the above formulas, K is the number of evaluation sample objects; xki is the sample data of the ith indicator of the kth evaluation object; Pki is the proportion of the kth evaluation sample data in the ith indicator; Ei is the information entropy of the ith indicator, and is dimensionless; and ai is the weight of the ith indicator, which is dimensionless.

The final set of weights is given by

$$ A=\left\{{a}_1,\kern0.5em {a}_2,\kern0.5em ...,\kern0.5em {a}_n\right\} $$
(6)

where \( {a}_1,\kern0.5em {a}_2,\kern0.5em ...,\kern0.5em {a}_n \) represents n weight values.

Step 2: determination of the membership function

The membership function provides a quantitative method for the processing of uncertain matter description. In the past, reservoir evaluation has been based mainly on grading standards for reservoir classification and scoring; i.e., each attribute can only be fixed to a particular class according to the median rate, with no other intermediate state available. However, the boundaries between different categories of low-permeability reservoirs, which are uncertain in reality, are not very strict and accurate. At present, there is no complete and effective method for establishing the membership function. Common methods for determining the membership degree include the statistical test method, the expert experience method, the binary comparison sorting method, and the distribution function method. Among these approaches, the distribution function method is a commonly used method. This method is based on different distributions such as the rectangular, trapezoidal, Cauchy, and ridge distributions (Zhao 2004). Note that although different studies have established different forms of membership functions, consistent results in solving practical problems can still be achieved with each of these as long as they reflect the same fuzzy concept. The obtained membership degree is only an approximation. In practice, the membership degree must be adjusted according to information feedback in order to gradually be improved. Thus, the membership function must be confirmed to be suitable for the actual and applied effects related to a particular system. Finally, the membership evaluation degree is established by the membership function in order to determine the following basic evaluation matrix:

$$ R={\left({r}_{ij}\right)}_{n\times m}=\left[\begin{array}{cccc}{r}_{11}& {r}_{12}& ...& {r}_{1m}\\ {}{r}_{21}& {r}_{22}& ...& {r}_{2m}\\ {}\vdots & \vdots & \vdots & \vdots \\ {}{r}_{n1}& {r}_{n2}& ...& {r}_{nm}\end{array}\right] $$
(7)

where rij represents the degree of membership of the jth factor of the evaluation object.

Step 3: using the improved fuzzy comprehensive evaluation method

Traditional fuzzy comprehensive evaluation mainly relies on obtaining the comprehensive membership degree of the evaluation object through a fuzzy synthesis operation of the weight set and the membership basic matrix, before normalizing the comprehensive membership degree as the weighted sum of the weight and numerical comment sets in order to obtain the fuzzy evaluation value (Anifowose and Abdulraheem 2011; Wang et al. 2017). This method is represented by the following formula:

$$ B=\left({b}_j\right)=A\circ R=\sum \limits_{i=1}^n{a}_i{r}_{ij},j=1,2,...,m $$
(8)
$$ C=\left({c}_k\right)=\frac{\sum \limits_{j=1}^m{b}_j^k{v}_j}{\sum \limits_{j=1}^m{b}_j^k},k=1,2,...,K $$
(9)

In the above expressions, bj represents the comprehensive membership of the evaluation object to the jth comment; ck represents the fuzzy evaluation value of the kth evaluation object, which is dimensionless; and \( {b}_j^k \) represents the comprehensive membership of the jth evaluation object to the jth comment.

However, there are several problems related to the traditional fuzzy comprehensive evaluation method: (1) the range of data obtained from the evaluation results is within the scope of the evaluation set. Depending on the set of reviews, it may not be convenient to unify and compare the results obtained by using other evaluation methods, and (2) when using the fuzzy evaluation process of membership degree conversion and the weighted average processing of comprehensive membership degree, evaluation results can be easily overlapped and redundant, leading to a low degree of discrimination and thereby negatively affecting reservoir evaluation.

Based on an original improvement to the method, the traditional fuzzy evaluation results are compared with the best and the worst comments; thus, the evaluation results are between 0 and 1, and result dispersion is expanded in order to improve the feature extraction ability of the evaluation model. The final fuzzy evaluation value is expressed as

$$ M=\left({m}_k\right)=\frac{1}{1+{\left[\frac{\underset{j}{\max }{v}_j-{c}_k}{c_k-\underset{j}{\min }{v}_j}\right]}^p},k=1,2,...,K $$
(10)

where mk represents the improved fuzzy evaluation value for the kth evaluation object, which is dimensionless, and p is the resolution coefficient, also dimensionless. In order to obtain a better distinguishing effect while ensuring the superior nature of the results, the resolution coefficient is usually set to be 2 in term of the experience of value and the general cognitive habits of people. This can also be adjusted according to the evaluation effects, the needs, and preferences of the evaluator.

Step 4: using the improved gray correlation analysis method

The traditional gray evaluation model has been improved and adapted as follows. First, the dimensionless gray comparison sequence in the previous gray correlation is directly processed using operators such as the initial value, mean value, and normalization based on the input data for evaluation (the model data) (Wong et al. 2001; Tamiloli et al. 2016), without considering any ambiguity related to the reservoir. Using the membership degree as the link, the fuzzification process is conducted, and the dimensionless comparison sequence of the evaluation object is obtained by synthesizing the membership degree matrix for the weight and the comment set in order to achieve an organizing combination with the fuzzy comprehensive evaluation according to the following formula:

$$ D=\left({d}_i\right)=V\circ R=\frac{\sum \limits_{j=1}^m{v}_j{r}_{ij}}{\sum \limits_{j=1}^m{r}_{ij}},i=1,2,...,n $$
(11)

where di represents the gray sequence value of the ith factor of the evaluation object, which is dimensionless.

Second, previous gray correlation analysis methods mainly relied on determining a reference sequence as the parent sequence for comparison in the model structure. Based on the evaluation idea contained in the Topsis method (Huang 2008; Wood 2016), comparative information is fully utilized in order to form an optimal reference sequence and a worst reference sequence according to the best reviews and the worst comments in the reservoir evaluation set, written as

$$ {D}_g={\left({d}_i\right)}_n={\left(\underset{j}{\max }{v}_j\right)}_n $$
(12)
$$ {D}_b={\left({d}_i\right)}_n={\left(\underset{j}{\min }{v}_j\right)}_n $$
(13)

In the above expressions, Dg is the optimal reference sequence and Db is the worst reference sequence.

Deng’s correlation method (Deng 1993) is used in order to calculate the correlation coefficient between each factor and two reference series, and the optimal and worst gray correlation degree is obtained by weighting the correlation coefficient; that is, the relative positive and negative ideal solutions are obtained as follows:

$$ {E}_{gk}=E\left({D}_g,{D}_k\right)=\left({e}_k^i\right)=\frac{\underset{k}{\min}\underset{i}{\min}\mid {d}_g^i-{d}_k^i\mid +\xi \underset{k}{\max}\underset{i}{\max}\mid {d}_g^i-{d}_k^i\mid }{\mid {d}_g^i-{d}_k^i\mid +\xi \underset{k}{\max}\underset{i}{\max}\mid {d}_g^i-{d}_k^i\mid },i=1,2,...,n;k=1,2,...,K $$
(14)
$$ {F}_{gk}=F\left({D}_g,{D}_k\right)=\left({f}_k\right)=\sum \limits_{i=1}^n{a}_i{e}_k^i $$
(15)
$$ {E}_{bk}=E\left({D}_b,{D}_k\right)=\left({e}_k^i\right)=\frac{\underset{k}{\min}\underset{i}{\min}\mid {d}_b^i-{d}_k^i\mid +\xi \underset{k}{\max}\underset{i}{\max}\mid {d}_b^i-{d}_k^i\mid }{\mid {d}_b^i-{d}_k^i\mid +\xi \underset{k}{\max}\underset{i}{\max}\mid {d}_b^i-{d}_k^i\mid },i=1,2,...,n;k=1,2,...,K $$
(16)
$$ {F}_{bk}=F\left({D}_g,{D}_k\right)=\left({f}_k\right)=\sum \limits_{i=1}^n{a}_i{e}_k^i $$
(17)

In the above expressions, Egk and Fgk represent the optimal correlation coefficient and the optimal correlation degree, respectively, which are dimensionless and are also the results given by the traditional gray correlation analysis; Ebk and Fbkrepresent the worst correlation coefficient and the worst correlation degree, respectively, and are also dimensionless; \( {e}_k^i \) represents the gray correlation coefficient of the ith factor of the kth object, also dimensionless; fk represents the gray degree of association of the kth object, also dimensionless; and ξ is the dimensionless resolution coefficient, ξ ∈ (0, 1). Note that in order to weaken the influence of data distortion that is due to the large absolute difference value, a resolution coefficient of 0.5 is generally adopted (Ghadami et al. 2015) according to the principle of minimum information.

Finally, in order to fully exploit the information related to the differences between objects in the model results, the optimal and worst correlations are compared and contrasted to increase the discriminating interval of the results as much as possible and improve the recognition and adaptability of the model. The final gray evaluation information is expressed as

$$ H=\left({h}_k\right)=\frac{1}{1+{\frac{\left[\left(1-{F}_{gk}\right){F}_{bk}\right]}{{F_{gk}}^{2p}}}^p},k=1,2,...,K $$
(18)

where hk is the improved gray evaluation value for the kth evaluation object, which is dimensionless; p is the resolution coefficient, also dimensionless, and its value affects the distribution and discrete trend of the gray evaluation results. In order to obtain a suitable value interval, a resolution coefficient of 1 is generally adopted, although this can be adjusted according to the particular evaluation process.

Step 5: integration of the two evaluation results

In a sense, the essence of evaluation is a process of overlapping various pieces of information. The fuzzy and gray evaluation results that are obtained are used as the new reservoir information (Li 2007) and superimposed, and then the fuzzy comprehensive evaluation and gray correlation analysis of the organic connection are the evaluation outputs. In order to further improve the resolution of the superimposed information, the image enhancement algorithm based on image blur inverse processing differentiates the results through nonlinear transformation in order to enhance the final evaluation effect. This algorithm is given by

$$ X=\left({x}_k\right)=\alpha {m}_k+\beta {h}_k $$
(19)

where xk is the dimensionless evaluation value of the kth evaluation object; α + β = 1, 0 ≤ α ≤ 1, and 0 ≤ β ≤ 1, for which each value can be set to 0.5 or be adjusted according to expert experience or the effect on evaluation.

$$ Z=\left({z}_k\right)=\left(\sigma +\tau -2\right){X}^3+\left(3-2\sigma -\tau \right){X}^2+\sigma X $$
(20)

where zk is the dimensionless superimposed evaluation value of the kth evaluation object; 0 ≤ σ ≤ 1 and 0 ≤ τ ≤ 1, where σ and τ control the degree and style of the transformation and the value of each can be set to 0.5 or be adjusted according to the actual effect on evaluation. When both values are equal to 1, it is the original transformation.

Case study

Geological setting and basic reservoir characteristics

The Qaidam Basin is located in the north-eastern part of the Qinghai-Tibet Plateau and belongs to the southern block of the Tarim-China-DPRK plate. The Qaidam Basin is the third largest inland basin in China. The northern margin of the Qaidam Basin consists of a fault block to the west and the Delingha sag to the east. It consists of a number of secondary structural belts, depressions, and swellings. The Nanbaxian oil and gas field is located in the north-eastern part of the Qaidam Basin in the Qinghai Province and is a tertiary structure on the fault block in the northern part of the Qaidam Basin (Fig. 2). There are eight sets of strata in the study area, namely, Shangyoushashan group (N22), Xiayoushashan group (N21), Shanggancaigou group (N1), upper interval of Xiaganchaigou group (E32), lower interval of Xiaganchaigou group (E31), Lulehe group (E1 + 2), Xiaomeigou group (J2), and basement (O3), of which E31 and E1 + 2, E1 + 2 and J2, J2, and basement are contacted by unconformities. The E32 formation is the target layer in this study (Fig. 3). The main depositional system of the target layer is the delta front. The distributary channel of the delta front, river mouth bar, and distal bar mainly consists of sandstone. In general, the reservoir exhibits the characteristics of low porosity, low permeability, and poor connectivity. Oil and gas production declines rapidly is difficulty to remain stable. The reservoir is a typical heterogeneous lithologic reservoir with ultra-low permeability. The microscopic pore types of the reservoir are diverse, the structure is complex, and the reservoir performance differs from that of other reservoirs. The E32 formation is mainly comprised of lithic feldspar sandstone, with feldspar lithic sandstone and feldspar sandstone also observed, mainly in a silt-fine grain structure. The grain size is mainly distributed within 0.05–0.25-mm diameter, and the grains are medium-good sorted. However, the roundness is poor, and the roundness is sub-edge to sub-circular. The reservoir has poor physical properties and is highly heterogeneous. A core lab analysis showed a porosity of between 3 and 20% with an average of 13.1% as well as a permeability of between 0.003 and 105.1 mD, with an average of 12.784 mD. Thus, the reservoir may be described as a typical low-permeability reservoir.

Fig. 2
figure 2

Structural divisions and position of research area (Dai et al. 2003; Shao et al. 2014))

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Fig. 3
figure 3

Stratigraphic column and lithology profile of Nanbaxian gas field

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Data analysis: evaluation set and membership function

According to the evaluation parameters selection principle, a comprehensive set of evaluation indicators with relatively clear attribute meanings and wide availability has been determined. These indicators include sand layer thickness, porosity, permeability, gas saturation, shale content, and a heterogeneity coefficient of permeability. Among these indicators, sand layer thickness, porosity, permeability, and gas saturation are directly proportional to reservoir quality; i.e., these are positive correlation indices. The shale content and heterogeneity coefficient of permeability are the opposite, representing a negative correlation index. The corresponding set of factors is expressed as

$$ U=\left\{H,\kern0.5em \varPhi, \kern0.5em K,\kern0.5em \mathrm{Sg},\kern0.5em \mathrm{Sh},\kern0.5em \mathrm{Tk}\right\} $$
(21)

where H is the thickness of the sand layer, m; Φ is the porosity, %; K is the permeability, ×10−3 μm2; Sg is the gas saturation, %; Sh is the shale content, %; and Tk is the heterogeneity coefficient of permeability, which is dimensionless.

According to the general reservoir classification habits combined with the reservoir development and index characteristics of E32 formation, the comments are divided into four levels, and a review set is created:

$$ V=\left\{\mathrm{I}\kern0.5em \mathrm{I}\mathrm{I}\kern0.5em \mathrm{I}\mathrm{I}\mathrm{I}\kern0.5em \mathrm{I}\mathrm{V}\right\}=\left\{\begin{array}{cccc}0.8& 0.6& 0.4& 0.2\end{array}\right\} $$
(22)

According to the principle of entropy weight method, by substituting the sample data of 37 wells in the work area into equations (3)–(5), the weight of each indicator may be established in the following weight set:

$$ A=\left\{0.193\kern0.5em 0.184\kern0.5em 0.191\kern0.5em 0.189\kern0.5em 0.122\kern0.5em 0.121\right\} $$
(23)

Quantifying the membership degree and establishing the affiliation of different categories is the main purpose and task of establishing the membership function. The ridge series distribution curve in the fuzzy distribution has the characteristics of a wide main value interval, gentle transition zone, and a strong anti-interference capacity, which is close to human cognition characteristics and is widely used in evaluation (Li 2007). The ridge-shaped distribution may be described by three specific expressions (Table 1) in the same manner as other fuzzy distributions, according to the description relationship: small (descending-half-ridge-shaped distribution), large (rising-half-ridge-shaped distribution), and intermediate (middle-ridge-shaped distribution). According to the curve type and formula of the ridge-shaped distribution and the evaluation index of the reservoir index in the study area, the distribution function of each evaluation parameter to the different evaluation levels is constructed. Using the established membership function curve, a single reservoir sample can be transformed into a multi-value membership relationship; then uncertainty relationship mapping from the factor set to the comment set can be realized. Taking the sand body sample of the XZ58 well as an example, the thickness of the sand body is H = 12 m. Substituting the membership function curve formula of the sand body thickness parameter by membership function (Table 2), the degree of membership for the four types of reservoir reviews are found to be 0.345, 1, 0.655, and 0. Performing a similar process, the membership degrees of all the parameter data for the sample are obtained. The basic evaluation matrix R that constitutes the reservoir evaluation of the sand body sample of the XZ58 well is as follows:

$$ {\displaystyle \begin{array}{l}\kern1.32em \mathrm{I}\kern2.75em \mathrm{I}\mathrm{I}\kern2.5em \mathrm{I}\mathrm{I}\mathrm{I}\kern2em \mathrm{I}\mathrm{V}\kern1.32em \\ {}R=\left[\begin{array}{cccc}0.345& 1& 0.655& 0\\ {}1& 0.873& 0& 0\\ {}1& 0.014& 0& 0\\ {}0& 0.875& 1& 0.125\\ {}0.309& 1& 0.691& 0\\ {}0.454& 1& 0.546& 0\end{array}\right]\begin{array}{l}\mathrm{H}\\ {}\varPhi \\ {}\mathrm{K}\\ {}\mathrm{Sg}\\ {}\mathrm{Sh}\\ {}\mathrm{Tk}\end{array}\end{array}} $$
(24)
Table 1 Curve and formula of ridge-shaped distribution membership function
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Table 2 Membership function of sand body thickness index
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Evaluation results and discussion

Taking the sand layer sample of the XZ58 well in the study area as an example, the basic membership degree and weight vector are substituted into formula (10) in order to calculate the comprehensive membership degree B = (0.534, 0.765, 0.466, 0.024), the fuzzy evaluation value C = 0.602, and the improved fuzzy evaluation value M = 0.806. The obtained membership base matrix, R, is substituted into the gray correlation evaluation formula (11) in order to obtain the gray comparison sequence of the reservoir D = (0.569, 0.707, 0.797, 0.475, 0.562, 0.591). The traditional gray evaluation result of Fg = 0.669 was used to calculate the optimal gray correlation degree using formulas (14) and (15). The worst gray correlation degree Fb = 0.425 was calculated using formulas (16) and (17). The final fuzzy-gray superimposed evaluation results of X = 0.783 and Z = 0.831 were obtained using formulas (16) and (17).

In the same manner, all the reservoir evaluation results in the whole district were obtained, and the data characteristics of the evaluation results before and after improvement were compared (Fig. 4). The traditional fuzzy evaluation results before improvement give a C-value interval of 0.257–0.770, with a dispersion standard deviation of 0.096, and the gray-related evaluation results have a relatively compact numerical range of 0.352–0.955, and a standard deviation of 0.106. The data interval of the superposition evaluation results is 0.304–0.862 with a standard deviation of 0.099; the improved fuzzy evaluation result M value interval is 0.011–0.997, with a standard deviation of 0.253. The improved gray evaluation value H distribution interval is 0.178–0.984, with a standard deviation of 0.170. The numerical interval of the improved superimposed evaluation results is extended to 0.060–0.995, with a standard deviation of 0.239, a data range which is close to 0–1. The resolution is also improved similarly to the image enhancement processing. A comparison of the frequency distribution of the evaluation results also indicates that the improved results greatly enhance the differences within and discrimination of the data structure; thus, the reservoir may be portrayed in more detail and with increased accuracy.

Fig. 4
figure 4

Comparison of reservoir evaluation results before and after improvement

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According to the obtained results and the actual data based on the improved fuzzy-gray method, the reservoirs are classified into type I (combined scores of 0.75–1.0), type II (combined scores of 0.5–0.75), type III (combined scores of 0.25–0.50), and type IV (combined scores of 0–0.25). The comprehensive classification combines multiple parameters in order to avoid the uniqueness and inconsistency of single reservoir parameter classification (Table 3). The coincidence rate with the actual capacity classification result can be above 80%, which indicates a great improvement in accuracy. The evaluation classification results are approximately consistent with the existing development results (Fig. 5a), and the comprehensive evaluation value is in good agreement with the production data (Fig. 5b), which are well tested to a certain extent and can be used as geological guidance (Lai et al. 2016). For the specific evaluation example discussed here, the comprehensive score of the sand body sample of XZ58 is 0.831, and the evaluation grade belongs to the class I reservoir. After testing, the daily gas volume of the single well is approximately 1.35 × 104 m3, which indicates high productivity, and the evaluation results are consistent with the actual production data.

Table 3 Comprehensive evaluation table of the reservoir (portion)
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Fig. 5
figure 5

a Classification results of reservoirs and b comprehensive evaluation value versus production data

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In general, according to geological evaluation, type I and II reservoirs are considered to be good reservoirs, with corresponding mercury intrusion curves that show good sorting and rough slanting degrees. The curve has a relatively flat platform in its lower right part. The displacement pressure and median pressure are lower, the maximum mercury saturation is higher, and the average pore throat radius is larger; the pore structure is dominated by large pores and coarse throat, and the reservoir performance is better (Fig. 6a, b), when compared to type III and IV reservoirs. The average production capacity is above 1.0 × 104 m3/day, which constitutes the main production layer.

Fig. 6
figure 6

Characteristics of four types of reservoirs and corresponding mercury intrusion curves

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The mercury intrusion curves corresponding to the type III reservoir are characterized by medium sorting and slightly rough slanting degree. The curve has a slightly steeper platform in the middle, the displacement pressure and median pressure are at medium levels, the maximum mercury saturation is higher, and the pore throat range is wider (Fig. 6c). The pore structure is complex (characterized by micropores and middle throats, with medium storage performance), the gas production of the partial reservoir is relatively low, and the development effects are biased.

The mercury intrusion curves corresponding to the type IV reservoir are characterized by poor sorting, smaller slanting degrees, and a large slope. The displacement pressure and median pressure are higher, the maximum mercury saturation is smaller, the pore throat range is narrower, and the average pore throat radius is smaller, when compared to the other reservoir types. The structure of the pore throat is complex, primarily belonging to the small hole or the extremely thin throat type, and the storage performance is poor (Fig. 6d). The gas production is extremely low, mostly dry and water layers, and the reservoir does not have any industrial development value.

According to the improved reservoir comprehensive evaluation plane distribution map, it can be seen that there is a better distinction among the different reservoir types when using the new method, compared to the results of standard methods. The obtained results are also closer to the statistical laws and knowledge base related to the delta front sediments and approximately match existing practice data and single-well geological knowledge (Fig. 7). Type I and II reservoirs are mostly strip-shaped and basically correspond to the geological development scale of the underwater distributary channel in the studied area. The single-well sedimentary facies types are mainly underwater distributary channels and mouth bars. The sand layer is thick, and the grain size is coarse. It is mainly composed of gas-bearing gravel and middle-fine sandstone and has good physical properties. Type III and IV reservoirs are mostly flat and scattered on the outer sides of the type I and II reservoirs, and the regularity of the distribution on the plane is poor (mainly corresponding to low-energy channels, subsea natural barrier, and flood fans), with a fine grain size (mainly comprised of fine sandstone and siltstone), with an increased muddy quality, and poor physical properties. In general, type I and type II reservoirs are well developed, exhibit high gas production, and can be used as a geological desert; thus, type I and type II reservoirs are key areas for natural gas enrichment and further development.

Fig. 7
figure 7

Lateral distribution of four reservoir types

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Conclusion

  1. (1)

    A new fuzzy-gray comprehensive and systematic evaluation method based on a combination of the fuzzy comprehensive and gray relational evaluation methods has been proposed. This combined method was used in order to evaluate low-permeability reservoirs in the E32 formation of the Nanbaxian oil and gas field. Compared to current methods, the proposed approach gives a better indication of the uncertainty related to the low-permeability reservoir evaluation process and the differences between evaluation results by integrating the fuzzy, gray, and differential treatments of reservoir evaluation. This method provides a new approach that may be adopted for use in similar examples of low-permeability reservoir evaluation.

  2. (2)

    The improved fuzzy-gray comprehensive evaluation method was applied to the classification and characterization of a reservoir in a particular study area. The reservoirs in the study area were divided into four categories, where type I and II reservoirs were found to be favorable targets. The degree of discrimination of the evaluation results showed great improvement and the reservoirs were characterized in more lateral detail compared to the results of previous evaluations. The observations resulting from this approach were found to be approximately consistent with the current geological and production-based understanding of the E32 formation in the Nanbaxian oil and gas field. The results obtained here can be used to better guide exploration and development in the studied area.