Introduction

Multiple and detailed observations of attributes related to principal stress directions, fracture orientation, and AOFP of gas well probed the community of geoscientists and petroleum engineers alike to ask the following question. How does the angle between maximum principal stress direction and natural fracture strike influence gas well productivity and reservoir modification? Geostress state, distribution, and orientation of natural fractures play an important role in controlling the performance of fractured reservoirs (Chatterjee et al., 2017). A lot of methods pertain to geostress and natural fracture survey were proposed in studies. Magnitude of maximum horizontal principal stress was estimated by using borehole wall collapse analysis and strength tests on core (Vernik et al., 1992). A new approach was applied to determine the orientations of natural fractures in subsurface siliciclastic rocks (Laubach 1997). Fracture attitude of ten wells was surveyed with resistivity images data, and maximum horizontal principal stress direction was determined through borehole wall collapse analysis (Tezuka et al., 2002). True direction of maximum horizontal compressive stress was determined through a method based on four seismologically measurable parameters (Lund et al., 2007). Effectiveness of natural fractures in Xujiahe formation was investigated with outcrop, core, thin slice, imaging logging, and production data (Li et al., 2019). Effect of natural fractures on gas productivity depends on seepage ability of them. Permeability of natural fracture in subsurface depends on mineral filling state (Laubach et al., 2004). Investigations indicate that natural gas flows through high permeability fractures and into high permeability beds (Cook et al., 2008). In fact, gas well productivity was influenced by many factors. Hydraulic fracturing treatment is one of the impact factors to influence the gas well productivity in tight gas reservoirs. Furthermore, fracturing effect is affected by the distribution of natural fractures and various fracture treatment parameters such as fracture fluids and proppant size (Abaa et al., 2013). A comprehensive revision on the interactions between hydraulic fractures and natural fractures was conducted (Kolawole et al., 2019). Relationship between individual factor and gas reservoir fracturing production was analyzed with factor analysis method (Jiang et al. 2014). The importance and effect of affecting factors on productivity of oil and gas construction projects were analyzed through analytical hierarchical process and paired comparison (Barati et al. 2015). Grey correlation analysis was applied to study the influences of geological, engineering, and other factors on coalbed methane wells productivity (Zhao et al. 2017). Bedrock type, geographic setting, proximity to fracture-correlated lineaments, and structural position are the factors influencing well productivity, which was obtained by using one-way analysis of variance technique (Mabee 1999). The sensitivity factors affecting productivity of low-permeability condensate gas well were analyzed in different ranges of bottom hole pressure based on orthogonal experiment design method (Yan et al. 2019). The primary and secondary relationships of different factors influencing gas productivity were determined through grey correlation analysis method (Han et al. 2020). In this work, 14 natural gas wells in gas reservoir T of Kuqa Depression in Tarim Basin were taken to explore the influencing factors of gas well productivity. This work will optimize well deployment and improve natural gas productivity in fractured tight sandstone reservoir.

Geologic setting

Gas reservoir T is located in Kuqa Depression of Tarim Basin. Under the action of strong north–south tectonic compression, it has high geostress and develops high dip angle fractures. Gas reservoir T is a fault anticline structure sandwiched by two north dipping thrust faults, with a length of 26 km from east to west and a width of 3 km from south to north. The trap area is 53.25 km2, with an amplitude of 425 m. Elevation of the high point is − 5075 m, with three local high points, as shown in Fig. 1.

Fig. 1
figure 1

Geological structure of gas reservoir T, maximum principal stress direction, and natural fracture strike of each gas well in gas reservoir T. The maximum principal stress direction is marked with pointer needle in rose diagrams of fracture strike

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Influencing factors and approach

Influencing factors

Absolute open flow potential (AOFP) is the output of natural gas under the condition of unit differential production pressure. It is the most direct parameter to measure gas well productivity. Maximum principal stress direction, natural fracture strike, and AOFP of 14 wells located in gas reservoir T before and after reservoir modification are listed in Table 1. Natural fracture strike of six wells, T1, T3, T4, T5, T13, and T14, is relatively consistent. Angles between maximum principal stress direction and natural fracture strike are less than 20 °. Corresponding AOFP of each gas well is quite different. No positive or negative correlation exists between the angles and AOFPs. Well T1 located in the east high point of gas reservoir T has the strongest AOFP. Natural fractures of wells T2, T6, T7, T8, T9, T11, T12 can roughly be divided into two groups. Angles between maximum principal stress direction and strike of main fracture group are in the range 5 °–77.5 °. Similarly, no positive or negative correlation exists between the angle and AOFP.

Table 1 Maximum principal stress direction, natural fracture strike, and AOFP before and after reservoir modification
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Data of ten factors influencing AOFP are listed in Table 2. No positive or negative correlation relationships exist between influencing factors and AOFP of gas well through intuitive quantitative analysis. Comparing AOFPs before and after reservoir modification, a remarkable feature could be revealed that AOFP of each gas well was increased several times, even hundreds of times, except T10 well, after reservoir modification. This indicates that reservoir modification plays a significant role in increasing gas well productivity. Connection between natural fractures near wellbore and natural fractures in deep reservoir was greatly improved by reservoir modification.

Table 2 Factors influencing AOFP of gas well
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Gray correlation analysis

The concept of grey correlation degree was put forward in grey system theory (Deng 2005). It is a measure of the relevance of two factors in system. The size of correlation degree directly reflects the influence of various factors in the system on target value. General steps of factor analysis by grey correlation analysis are as follows.

Step 1 Determining analysis sequences. Reference sequence reflects the system behavior characteristics, which is also called parent sequence. It is assumed that there is a reference sequence as follows.

$$x_{j} = \left( {x_{j} \left( 1 \right),x_{j} \left( 2 \right),x_{j} \left( 3 \right), \cdots ,x_{j} \left( k \right), \cdots ,x_{j} \left( n \right)} \right),j = 1,2,3, \cdots ,s$$
(1)

Comparison sequence affects the system behavior, which is also called subsequence. To suppose that there is a comparison sequence as follows:

$$x_{i} = \left( {x_{i} \left( 1 \right),x_{i} \left( 2 \right),x_{i} \left( 3 \right), \cdots ,x_{i} \left( k \right), \cdots ,x_{i} \left( n \right)} \right),i = 1,2,3, \cdots ,t$$
(2)

Step 2 Nondimensionalizing variables. Data in each factor column of system may be different in dimension, as shown in Table 2. It is difficult to compare or to get correct conclusion. So the data are generally nondimensionalized in gray correlation analysis in order to ensure the reliability of results.

Step 3 Calculating correlation coefficient. Correlation coefficient can be expressed as

$$\xi_{ji} \left( k \right) = \frac{{\min \min \left| {x_{j} \left( k \right) - x_{i} \left( k \right)} \right| + \rho \max \max \left| {x_{j} \left( k \right) - x_{i} \left( k \right)} \right|}}{{\left| {x_{j} \left( k \right) - x_{i} \left( k \right)} \right| + \rho \max \max \left| {x_{j} \left( k \right) - x_{i} \left( k \right)} \right|}}$$
(3)

where \(\xi_{ji} \left( k \right)\) represents correlation coefficient of the k-th sample in the i-th comparison sequence and in the j-th reference sequence.\(\min \min \left| {x_{j} \left( k \right) - x_{i} \left( k \right)} \right|\) and \(\max \max \left| {x_{j} \left( k \right) - x_{i} \left( k \right)} \right|\) express the maximum and minimum values that are obtained by subtracting the elements of comparison sequence from the ones of reference sequence. \(\left| {x_{j} \left( k \right) - x_{i} \left( k \right)} \right|\) is called Hamming distance, and reciprocal of Hamming distance is called inverse reciprocal distance. \(\rho\) is resolution, \(\rho \in \left( {0,\infty } \right)\). It is generally taken the values in interval [0, 1]. The smaller the \(\rho\), the greater the resolution. \(\rho\) is usually taken 0.5 in practical analysis.

Step 4 Calculating correlation degree. The computing formula is as follows.

$$r_{ji} = {{\sum\limits_{k = 1}^{n} {\xi_{ji} \left( k \right)} } \mathord{\left/ {\vphantom {{\sum\limits_{k = 1}^{n} {\xi_{ji} \left( k \right)} } n}} \right. \kern-\nulldelimiterspace} n}$$
(4)

Combining with actual background, positive effect is called positive correlation, and vice versa. It is called strong correlation when \(\left| {r_{ji} } \right|\) is greater than 0.7. And it is called weak correlation when \(\left| {r_{ji} } \right|\) is less than 0.3.

Results and discussion

Factors influencing AOFP before reservoir modification

In order to clarify which factors were relatively important, correlation degree of each factor to gas well productivity was computed. Results indicated that five factors had relatively high correlation degree. These factors are natural fracture density, porosity, permeability, elevation of geological top surface, and gas saturation, as shown in Fig. 2. Their correlation degrees are 0.7778, 0.7403, 0.7403, 0.7380, and 0.7375, respectively. Natural fracture density is the most important factor influencing gas well productivity before reservoir modification. This result is consistent with the fact that gas reservoir T is a typical fractured tight sandstone reservoir. In fractured tight sandstone reservoir, natural fracture is the major seepage channels of natural gas.

Fig. 2
figure 2

Factors influencing gas well productivity before reservoir modification and corresponding correlation degrees. This chart shows the influential intensity of each influencing factor on well productivity before reservoir modification

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Factors influencing AOFP after reservoir modification

Connection between natural fractures near wellbore and natural fractures in deep reservoir was greatly improved by reservoir modification. Porosity, gas saturation, elevation of geological top surface, minimum principal stress, and permeability became the central factors influencing the AOFP after reservoir construction, as shown in Fig. 3. The correlation degrees of them are 0.7857, 0.7853, 0.7836, 0.7829, and 0.7806, respectively. Minimum principal stress became one of the central factors influencing the AOFP after reservoir construction. This conforms to the general viewpoint that fracturing resistance decreases with reducing geostress. The correlation degree of natural fracture density was decreased to 0.7433, which indicated that natural fracture density was no longer the main factor for gas well productivity after reservoir modification. Natural fractures on wellbore wall that could be employed in fracturing or acidizing are extremely restricted.

Fig. 3
figure 3

Factors influencing gas well productivity after reservoir modification and corresponding correlation degrees. This chart shows the influential intensity of each influencing factor on well productivity after reservoir modification

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Conclusion

In this study, factors influencing AOFP of gas well in fractured tight sandstone reservoir T were investigated. Ten factors influencing AOFP were summarized. Gray correlation analysis was utilized to distinguish the influential intensity of each influencing factor. Following conclusions were obtained.

  1. (1)

    Angle between maximum principal stress direction and natural fractures strike is not a central factor influencing gas well productivity. No positive or negative correlation relationship exists between the angle and AOFP of gas well. Reservoir modification plays a vitally important role in raising productivity of gas well in fractured tight sandstone reservoir. Obviously, it effectively improves the connection between natural fractures near wellbore and natural fractures in deep reservoir.

  2. (2)

    Elevation of geological top surface, porosity, permeability, gas saturation are the common central factors influencing the AOFP of gas well before and after reservoir modification. Natural fracture density is the most important factor influencing the AOFP of gas well before reservoir modification. Development degree of natural fracture reflects directly the quality of reservoir.

  3. (3)

    Minimum principal stress is one of the central factors influencing the AOFP of gas well after reservoir modification. Effect of reservoir modification depends partly on the magnitude of minimum horizontal principal stress. Minimum principal stress reflects the size of fracturing resistance.