1 Introduction

Coal seams are naturally fractured reservoirs with coalbed methane (CBM) consisting of matrix blocks, where most gas is adsorbed onto coal inner surfaces. A network of cleats, usually saturated with water in an in situ state, provide the major flow paths for gas and water in the coal seams (Clarkson and McGovern 2005; Pan and Connell 2012; Chen et al. 2014; Zhao et al. 2014). The magnitude and orientation of in situ stress can substantially influence coal deformation and destruction (including both cleat and matrix), and with increase of that stress, its role becomes more prominent (Ryan 2003; Bell 2006; Liu et al. 2014). Clearly, in situ stress measurement in the downhole is indispensable for coal and CBM development (Iannacchione et al. 2007; Karacan et al. 2008; Gentzis 2009, 2011a, b; Chatterjee and Pal 2010; Meng et al. 2010; Liu 2011; Talebi et al. 2014).

In situ stress is a type of internal stress in the earth crust, and its formation is closely related to various dynamic actions during historical geologic periods, including gravitational and tectonic stresses (Cai et al. 2010; Kang et al. 2009, 2010). The gravitational stress field caused by rock mass gravity is relatively simple and can be estimated by the specific weight of overburden and buried depth. On the contrary, the tectonic stress field is much more complicated, and is greatly influenced by tectonic movements (especially in the horizontal) and rock geological structures. In addition, the tectonic stress field is extremely irregular in spatial distribution and almost impossible to describe by precise analytical solutions, because it is continually changing with geologic age (Kang et al. 2010). However, at the human scale these stresses (tectonic and gravitational) can be assumed constant. Only local changes in the in situ stress occur caused by the reservoir exploitation (Segall and Fitzgerald 1998; Jeanne et al. 2015).

During CBM development, accurate measurement of in situ stress is particularly important for coal reservoir permeability evaluation and CBM recoverability assessment (Haimson 2005; Paul and Chatterjee 2011; Li et al. 2014). Coal reservoir permeability is determined by the fracture system resulting from the ancient and current tectonic stress fields, which immediately affect fracture aperture, morphology and propagation (direction and dip) (Meng et al. 2011). Moreover, coal reservoir pressure or effective stress, coal matrix swell associated with gas adsorption (e.g., CBM or other gases injected in the coal reservoir), and shrinkage effects during CBM desorption and production also affect fracture aperture (White et al. 2005; Bustin et al. 2008; Kumar et al. 2010, 2012; Wang et al. 2011; Pan and Connell 2012; Singh et al. 2015). A series of permeability dynamic prediction models have been established by considering the aforementioned coal effective stress and matrix shrinkage effects (Palmer and Mansoori 1998; Shi and Durucan 2004; Connell et al. 2010; Pan and Connell 2012; Zhao et al. 2013). Nevertheless, these models are based on stress or strain that can be simulated in the laboratory and cannot be used to analyze the effect of in situ stress on CBM reservoir permeability, because of the contrasting time scales between CBM development and in situ stress variability. Recently, after revelations of successful CBM development in the United States (Johnson and Flores 1998; Nelson 2003; Tonnsen and Miskimin 2010), the government of China has explored and developed CBM with great resource potential in the Ordos Basin. However, the ubiquitous characteristics of underpressured, CBM-undersaturated, low-porosity and low permeability in Chinese CBM reservoirs may cause coal reservoir permeability to be sensitive to stress change (Su et al. 2005; Yao et al. 2008; Keim et al. 2011; Xu et al. 2014). Moreover, both the complex stress conditions and poor reservoir properties could bring little economic benefit and inefficient development (Zhao et al. 2014). Chinese CBM development is mainly aimed at shallow coal seams, usually with depths 700–1000 m (Zhao et al. 2015a; Xu et al. 2015a, b; Lv et al. 2012; Li et al. 2014, 2015; Chen et al. 2014; Liu et al. 2014). Therefore, it is of major and practical significance to analyze the relationship between coal reservoir permeability, in situ stress and burial depth, which is beneficial to determine reasonable operating practice, reduce reservoir damage, improve CBM well productivity as well as guide deep CBM (>1000 m depths) exploration and development.

In this work, the data of reservoir pressure, in situ stress and well test permeability from 55 CBM wells were collected in the eastern margin of the Ordos Basin. Through statistics and regression analysis, correlations between stress, permeability and burial depth were determined to reveal the in situ stress distribution and address the impact of that stress on coal reservoir permeability, which is expected to provide theoretical support and guidance to CBM exploration and development.

2 Geological Setting

The eastern margin of the Ordos Basin crosses the Inner Mongolia Autonomous Region, Shanxi and Shaanxi provinces, and is a long and narrow zone along the Yellow River. This zone covers about 2.5 × 104 km2. It is nearly 560 km long in the north–south direction and 50–200 km wide in the east–west direction (Fig. 1). The zone has become the second most successful CBM development base after the Qinshui Basin since the 1980s (Xu et al. 2012a, b; Yang et al. 2013; Zhao et al. 2015b; Feng et al. 2015), and contains an estimated 9.0 × 1012 m3 of total CBM reserves that are buried less than 1500 m deep (Jie 2010). By the end of 2013, nearly 2000 CBM wells had been completed and vertical well productivity exceeded 6000 m3/day; that of horizontal wells reached 16,000 m3/day (Chen et al. 2015).

Fig. 1
figure 1

Location of the eastern margin of the Ordos Basin, China. a Locations of the Qinshui and Ordos Basins in the China Map; b general location of the eastern margin of the Ordos Basin in the tectonic division map; c specific locations of the CBM blocks and wells in the eastern margin of the Ordos Basin; AB is the cross section from south to north

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The eastern margin of the Ordos Basin is composed of three tectonic units, the Jinxi fold, eastern Yimeng uplift, and eastern Weibei uplift (Xue et al. 2011; Tang et al. 2012) (Fig. 1). The area is a large west-dipping monocline with dip angle 3°–10°, where the structures are relatively simple and stable. Only some slight northeast and north–northeast trending folds and small-scale faults are developed, which are beneficial to CBM preservation and exploration (Yao et al. 2009; Jie 2010).

The main coal-bearing sequences in the area are in the Carboniferous Taiyuan and Permian Shanxi Formations. The coal-bearing strata are largely preserved, providing a good basis for CBM generation and accumulation (Chen et al. 2015). Figure 2 shows a geologic cross section of the two formations from south to north. There are more than 20 coal seams of various thicknesses, but their distribution is stable. The Taiyuan Formation is predominantly deposited in a tidal flat and delta system. The number of principal mineable coal seams ranges from 2 to 4 and the total number is between 4 and 8, with net coal thickness 3–40 m. The Shanxi Formation is mainly deposited in a shallow water delta, lagoon-gulf system consisting of three to nine coal seams. There are one to three major mineable coal seams that have general coal thickness approximately 4–15 m in total. Overall, coal thickness in the northern portion is the greatest (20–35 m), followed by 10–20 m in the central portion and <5 m in the south (Jie 2010). Average vitrinite reflectance of coal is from 0.44 to 2.35 % and the roof and floor of the major coal seams consist primarily of mudstone, sandy mudstone and sandstone (Jiang et al. 2012). These generally have low permeability and favorable CBM sealing effects, although the limestone roofs are usually aquifers that can dissolve gases and transport them out of the CBM reservoir in the Liulin block (Xu et al. 2015a).

Fig. 2
figure 2

A geologic cross section (AB) from south to north showing the Permian Shanxi and Carboniferous Taiyuan Formations (the section line could be found in Fig. 1c; GR gamma ray, LLD deep lateral resistivity, HC Hancheng block, YCN Yanchuannan block, DN-JX Daning–Jixian block, SL Shilou block, LL Liulin block, SJ Sanjiao block, LX Linxing block, BD Baode block, HQ Hequ block; the 4 + 5# and 8 + 9# are the coal seam numbers of Shanxi and Taiyuan Formations, respectively)

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3 Methodology

After a CBM well is completed and before it is put into production, the coal seam is usually tested to obtain parameters such as reservoir pressure, fracturing pressure, closing pressure, permeability and others, which can provide a reliable basis for regional CBM assessment. Generally, injection/falloff well testing accompanied by in situ stress measurement is done to obtain the above parameters (Li et al. 2014).

Injection/falloff is a type of transient well test that is usually conducted before in situ stress measurement basing on Chinese national standard GB/T 24504-2009 (AQSIO and SAC 2009). Throughout the test, because the formation pressure is always higher than the CBM critical desorption pressure, the fluid just maintains single-phase water flow in the cleats. Permeability of the coal matrix is much lower than that of the cleats, which demonstrates that the measured permeability is mainly from the cleats rather than the coal matrix. Here, two methods are available to examine well test data, including characteristic straight line analysis and chart board matching analysis. Usually, the selected analysis model is determined by the log–log curve characteristic of test data. If the log–log pressure derivative curve appears as a horizontal straight line segment in the middle period, the semi-log method can be used to calculate reservoir permeability. Other cases could be investigated by typical curve fitting (Hopkins et al. 1998). In the present work, to ensure that the parameters are accurate and reliable, pressure history matching curves were compared and verified with measured curves. Figure 3 shows the workflow for the permeability calculation with the injection/falloff data. Figure 4 portrays the injection/falloff curves of an actual CBM well.

Fig. 3
figure 3

Workflow for the permeability calculation with the injection/falloff well test data

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Fig. 4
figure 4

Injection/falloff curves of an actual CBM well. a Relationship between the pressure/temperature and time; b relationship between the injection rate at the surface and time; V i represents the total injection volume; v i,ave represents the average injection rate at the surface; p represents the pressure; p i represents the injection pressure; t represents the time; t i represents the injection time; t s represents the well shut-in time

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It should be noted, however, that the maximum injection pressure at the surface before an injection/falloff well test is usually unknown for a new area or areas with less data. Consequently, a minifrac is typically required to determine that pressure (Meng et al. 2010, 2011; Li et al. 2014). Figure 5 shows minifrac curves of an actual CBM well. During an injection/falloff well test, to reduce damage from stress on reservoir permeability, injection pressure at the surface is initiated at a relatively low value, and the maximum injection pressure at the injection rate is kept lower than fracturing pressure during the designed injection period. The injection period should be longer than the end time of the wellbore storage effect, and the well shut-in time should not be shorter than twice the injection period. In the present study, flow-static injection was continued for 12 h and the well was then shut in for at least 24 h to obtain the curves between temperature, pressure and time. Those curves were used to calculate reservoir permeability and initial reservoir pressure (Fig. 4).

Fig. 5
figure 5

Minifrac curves of an actual CBM well. a Relationship between the pressure and time; b relationship between the injection rate at the surface and time; p i,max represents the maximum injection pressure at the surface; V r represents the backflow volume; t lp represents the last pumping time

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In situ stress is often measured by the multi-cycle hydraulic fracturing method (Hubbert and Willis 1957; Kang et al. 2010; Liu et al. 2014; Li et al. 2014, 2015), according to Chinese earthquake industry standard DB/T 14-2000 (CEA 2000), which is also appropriate for the in situ stress measurement of coal (Kang et al. 2010; Liu et al. 2014), following the injection/falloff well test (Li et al. 2014, 2015). During the in situ stress measurement, the objective of pumping a thin fluid (water) at a certain rate into the target coal seam is to create a small fracture. Initially, pressure in the fracture is less than the fracture closing pressure and the fracture is closed. When the pressure in the fracture is greater than the fracture closing pressure, the fracture opens. Once this occurs, the pumps are shut down and the pressure falloff with time is recorded. When the fracture closes, the fracture closing pressure is equal to the minimum horizontal stress. This is why stresses can be deduced from the fluid injection test. Therefore, through the injection curve, reservoir fracturing pressure can be obtained. With the falloff curve, reservoir fracture closing pressure can be calculated (Zuber et al. 1990). During in situ stress tests, two to three cycles with clear fracturing and closing effects were chosen to calculate in situ stress parameters with the time square root method (Li et al. 2014), and were further verified by the log–log method. Figure 6 reveals the in situ stress measurement curves of an actual CBM well.

Fig. 6
figure 6

In situ stress measurement and injection rate curves of an actual CBM well. a Relationship between the pressure and time; b relationship between the injection rate and time at the surface; ISIP represents the instantaneous shut-in pressure; p f represents the fracturing pressure; p c represents the closing pressure

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For the in situ stress measurement of a vertical well, when the formation pressure increases to the fracturing pressure by injecting water after the perforation is packed, the maximum horizontal principal stress can be expressed as (Bredehoeft et al. 1976):

$$\sigma_{\text{H}} = 3p_{\text{c}} - p_{\text{f}} - p_{0} + T,$$
(1)

where, σ H is the maximum horizontal principal stress, MPa; p c is the closing pressure, MPa; p f is the fracturing pressure, MPa; p 0 is the rock pore pressure (initial reservoir pressure); T is tensile strength of the rock around the borehole, MPa.

If the fluid is injected continuously to pressurize the reservoir, the fracture will extend to the deep site. Otherwise, if fluid injection is halted and the fracturing loop is kept closed, the fracture will immediately stop expanding and tend to close. Here, the balance pressure that can just keep the fracture open is called closing pressure and is equivalent to the minimum horizontal principal stress perpendicular to the fracture surface (Haimson and Fairhurst 1970; Haimson and Cornet 2003), i.e.,

$$\sigma_{\text{h}} = p_{\text{c}} ,$$
(2)

where σ h is the minimum horizontal principal stress, MPa.

Similarly, if the injection process is repeated, the fracture will reopen and the refracturing pressure will be attained. Because the rock has been fractured, the rock tensile strength equals 0 and the fracturing pressure can be written as

$$p_{\text{f}} (T = 0) = p_{\text{r}} = 3\sigma_{\text{h}} - \sigma_{\text{H}} - p_{0} ,$$
(3)

where p r is the refracturing pressure, MPa.

Substituting Eqs. (2) and (3) into Eq. (1), rock tensile strength is expressed as

$$T = p_{\text{f}} - p_{\text{r}}$$
(4)

The vertical principal stress can be computed according to the weight of overlying rock from 116 in situ stress test results worldwide (Hoek and Brown 1980), yielding the prediction equation of that stress used herein:

$$\sigma_{\text{v}} = 3\gamma H \approx 0.027H$$
(5)

where, σ v is the vertical principal stress, MPa; γ is rock bulk density, kN/m3 (1 kN/m3 = 0.001 MPa/m); H is burial depth, m.

During the in situ stress tests, tubing strings are used to lower the bottom-hole pressure gauge, packer and shut-in tools into the well. The packer is installed on an immediate roof that can be identified by logging data. Through the pipelines, underground equipment is connected with the injection system at the surface. In a short time, the injection fluid (formation water or filtered clean water) is pumped into the wellbore at a high rate, which makes the flowing bottom-hole pressure higher than the fracturing pressure of the target strata. Once the target strata fracture, the tested well is shut in and then the falloff curve may be acquired from the falloff data (Fig. 7).

Fig. 7
figure 7

Schematic diagram of the in situ stress measurement

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To accurately estimate the in situ stress and eliminate uncertainties in the measurements, four cycles were performed in the hydraulic fracturing and all in situ stress data were calculated on the basis of the hydraulic fracturing test. The general measurement steps are as follows: (1) starting the injection pump at the surface and injecting water into the well at a certain rate with the ladder displacement method; (2) observing change of wellhead pressure until fracture occurrence after pumping 2–6 min continuously, shutting in the well for 20 min and measuring the reservoir pressure drop; (3) opening the well again and recording backflow volume; (4) keeping the injection rate as in the first cycle and running another 2–4 cycles per the designed injection and shut-in time; (5) comparing the measured and pressure gauge data and if the former is qualified, ending the in situ stress test (Fig. 6). Table 1 shows in situ stress measurement data from an actual CBM well in the eastern margin of the Ordos Basin.

Table 1 In situ stress measurement data of an actual CBM well in the eastern margin of the Ordos Basin
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4 Results and Discussions

4.1 Results of In Situ Stress and Injection/Falloff Well Tests

The in situ stress data measured by four-cycle hydraulic fracturing of 55 CBM wells in the eastern margin of the Ordos Basin indicates that at coal seam burial depths (H) 455.60–1323.10 m (average 832.46 m), initial reservoir pressure (p 0) is 2.58–12.22 MPa (average 6.73 MPa) and the pressure gradient (δ 0) 0.40–1.42 MPa/100 m (average 0.82 MPa/100 m), which represents an predominant underpressured reservoir. The closing pressure (p c) is 6.98–26.88 MPa (average 15.04 MPa) and the pressure gradient (δ c) 1.22–2.94 MPa/100 m (average 1.83 MPa/100 m). The coal seam fracturing pressure (p f) is 8.17–31.08 MPa (average 16.39 MPa) and the pressure gradient (δ f) 1.32–3.35 MPa/100 m (average 1.99 MPa/100 m) (Fig. 8; Table 2). Overall, the burial depth in this area is greater than that of the Qinshui Basin which is accompanied by lower initial reservoir pressure, closing pressure and fracturing pressure (Fig. 9; Table 2, Meng et al. 2010). Coal reservoir temperature is 16.08–45.37 °C with average 32.01 °C (Fig. 8; Table 3).

Fig. 8
figure 8

Stock chart of injection/falloff well test and in situ stress measurement parameters in the Ordos Basin (H represents the burial depth; p 0 represents the initial reservoir pressure; σ H represents the maximum horizontal principal stress; σ h represents the minimum horizontal principal stress; σ v represents the vertical stress; δ 0 represents the initial reservoir pressure gradient; δ c represents the closing pressure gradient; δ f represents the fracturing pressure gradient; ε h represents the minimum horizontal principal stress gradient; ε H represents the maximum horizontal principal stress gradient; Temp represents the reservoir temperature; k represents the permeability)

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Table 2 CBM reservoir parameters associated with in situ stress in the Qinshui and Ordos Basins
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Fig. 9
figure 9

Stock chart of injection/falloff well test and in situ stress measurement parameters in the Qinshui Basin (based on Meng et al. 2010)

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Table 3 Injection/falloff and in situ stress test parameters in the eastern margin of the Ordos Basin
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With Eqs. (1), (2) and (5), magnitudes of the maximum and minimum horizontal principal stresses and the vertical principal stress were computed. This shows that the maximum horizontal principal stress (σ H) is 10.13–37.84 MPa with average 22.50 MPa [its stress gradient (ε H) is 1.43–4.13 MPa/100 m with average 2.75 MPa/100 m]. The minimum horizontal principal stress (σ h) is 6.98–26.88 MPa with average 15.04 MPa, for which the stress gradient (ε h) is 1.22–2.94 MPa/100 m with average 1.83 MPa/100 m. The vertical principal stress (σ v) is 12.30–35.72 MPa with average 22.48 MPa (Fig. 8; Table 3). Compared with the Black Warrior Basin in the United States (where minimum horizontal principal stress is generally 1–6 MPa, McKee et al. 1988) and the Sydney and Bowen Basins in Australia (minimum horizontal principal stress generally 1–10 MPa, with a few values of 14 MPa, Enever and Henning 1997), in situ stress is greater in the coal seams in the eastern margin of the Ordos Basin. Overall, that stress is moderate to strong, because 85 % of the stresses are 10–30 MPa according to the assessment standard (Kang et al. 2009).

Similar to the Qinshui Basin, the initial reservoir pressure, fracturing pressure, and closing pressure all have a linearly increasing trend with increase of coal burial depth (Fig. 10a–c, Meng et al. 2010). The relevant relationships are as follows.

Fig. 10
figure 10

Scatter diagrams of coal reservoir pressure, fracturing pressure, closing pressure and burial depth

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1. Initial reservoir pressure:

$$p_{0} = 0.0064H + 1.3849$$
(6)

2. Fracturing pressure:

$$p_{\text{f}} = 0.0168H + 2.368$$
(7)

3. Closing pressure:

$$p_{\text{c}} = 0.0147H + 2.7772$$
(8)

Moreover, the fracturing pressure has a clear positive correlation with closing pressure (Fig. 10d), expressed as

$$p_{\text{c}} = 0.8324p_{\text{f}} + 1.396$$
(9)

The results of the injection/falloff well tests indicate that coal reservoir permeability is 0.01–3.33 mD with average 0.65 mD, and 75 % of values are <1 mD. This reveals that the coal seam is a typical low permeability reservoir (Fig. 8; Table 3).

4.2 Principal Stress Variation with Depth

Figure 11 shows that the maximum and minimum horizontal principal stresses and the vertical principal stress are all positively correlated with buried depth. Because the minimum horizontal principal stress has a strong and even decisive impact on the fracturing pressure and its gradient and limits the expansion of hydraulic fractures, this stress becomes one of the key parameters in CBM well drilling design and coal seam reinforcement (Tang et al. 2011; Hallam and Last 1991; Hubbert and Willis 1957). In a common sedimentary basin, this stress is typically ~70 % of the vertical stress (Meng et al. 2013). However, 70 % of the coal reservoir vertical stress cannot accurately describe the magnitude and variation of the minimum horizontal principal stress in the eastern margin of the Ordos Basin. For example, at depths >1000 m, 70 % of the vertical stress is larger than the minimum horizontal principal stress. Overall, the minimum horizontal principal stress in the coal reservoir is slightly weaker than that of sedimentary rock strata in a common sedimentary basin. This is mainly attributed to coal as the organic matter and its weaker mechanical strength.

Fig. 11
figure 11

Scatter diagrams of maximum horizontal principle stress, minimum horizontal principle stress, vertical principle stress and burial depth (I represents σ H > σ v > σ h where the horizontal principal stress is dominant revealing a strike slip regime; II represents σ H ≈ σ v > σ h showing a stress transition zone; III represents σ v > σ H > σ h which indicates that the vertical principal stress is dominant demonstrating a normal stress regime)

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Based on the actual data, a general trend of the in situ stress change can be observed and in the vertical direction, stress field types in the study area are as follows. At depths <700 m, the coal reservoir in situ stress state is such that σ H > σ v > σ h, the minimum horizontal principal stress is <19 MPa, and the current in situ stress field type is a strike slip regime. At depths 700–1000 m, the coal reservoir in situ stress state changes to σ H ≈ σ v > σ h, indicating a stress transition zone where the average minimum horizontal principal stress is ~15 MPa. In the deep coal seam with burial depths >1000 m, vertical stress increases conspicuously and is greater than the maximum principal stress, i.e., σ v > σ H > σ h, with the minimum principal stress increasing from 14.80 to 26.88 MPa and averaging 19.88 MPa. These strata show a normal stress regime type, which is advantageous for normal fault activity and indicates an extension zone (Anderson 1951, Fig. 11).

4.3 Principal Stress Ratio Variation with Depth

Generally, when researching the stress field change behavior, the lateral stress coefficient is used to represent the in situ stress state at some point (Brown and Hoek 1978; Hoek and Brown 1980; Han et al. 2012). The lateral stress coefficient is defined as the ratio of average horizontal principal stress to vertical stress, expressed as

$$\lambda = \frac{{\sigma_{\text{H}} + \sigma_{\text{h}} }}{{2\sigma_{\text{v}} }},$$
(10)

where, λ is the lateral stress coefficient.

Measurements of lateral stress coefficient in the eastern margin of the Ordos Basin indicate that this parameter ranges from 0.51 to 1.21 with average 0.84. These values are between the Hoek–Brown inner and outer in situ stress envelopes representing the relationship between the ratio and burial depth worldwide (Brown and Hoek 1978) and those in China (Zhao et al. 2007) (Fig. 12). At depths <700 m, the lateral stress coefficient variation is 0.61–1.21 with average 0.94. The lateral stress coefficient is 0.51–1.12 with average 0.82 for 700–1000 m depths. For depths >1000 m, that coefficient is 0.62–0.98 with average 0.76. The lower limit of the in situ stress transformation depth corresponding to the depth at which the lateral stress coefficient equals 1 appears around 900 m, just within the range 700–1000 m. In other words, at depths <900 m, the horizontal principal stress is dominant and the lateral stress coefficient is large and disperse. The vertical principal stress is primary at depths >900 m where the lateral stress coefficient converges. This result is consistent with the analysis of Brown and Hoek (1978) and Zhao et al. (2007), who discussed lateral stress coefficient variation with depth worldwide and in China, respectively.

Fig. 12
figure 12

Scatter diagrams of coefficient of lateral stress and burial depth. a Relationship between the lateral stress coefficient and the Hoek–Brown stress envelopes; b the relationship between the lateral stress coefficient and the stress envelopes in China

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Referencing the expression of the Hoek–Brown inner and outer envelopes, the lateral stress coefficient is rewritten in the following form:

$$\lambda = \frac{a}{H} + b,$$
(11)

where, a and b are undetermined coefficients, respectively.

Equation (11) shows a linear relationship between the lateral stress coefficient and reciprocal of the buried depth. Thus, by regression of the measured lateral stress coefficients after linearizing the data, the trend line and inner and outer in situ stress envelopes of the eastern margin of the Ordos Basin were obtained. From Fig. 12, the outer envelope in the area is basically coincident with the Hoek–Brown outer envelope. The inner envelope is within the Hoek–Brown inner envelope and essentially coincident with the Chinese in situ stress inner envelope, indicating that the in situ stress measurements from the hydraulic fracturing method are credible.

Furthermore, by similar linear regression of the stress ratios and burial depth, trend lines of the ratios and their general ranges at various depths were acquired (Fig. 13). It is seen that the ratios of horizontal principal stresses in the study area are 1.08–1.70 with average 1.48. The ratios of maximum principal stress to vertical principal stress are 0.53–1.53 with average 1.01, and ratios of minimum principal stress to vertical principal stress are 0.45–0.90 with average 0.67. Overall, the three in situ stress ratios slowly decreased with buried depth. At <700 m depths, variation of the three ratios is great and their decrease with burial depth is rapid. For 700–1000 m depths, that decrease slows substantially, and is near zero at depths >1000 m. This general change behavior is consistent with the analysis of Liu et al. (2014), who addressed lateral stress coefficient variation with depth in the Huainan coalfield.

Fig. 13
figure 13

Scatter diagrams of in situ stress ratios and burial depth. a Relationship between the ratio of the horizontal principal stresses and the burial depth; b relationship between the ratio of the maximum principal stress to the vertical principal stress and the burial depth; c relationship between the ratio of the minimum principal stress to the vertical principal stress and the burial depth

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4.4 Permeability Variation Under the Principal Stress

Permeability is one of the key determinants of CBM productivity (Wang et al. 2011, 2013). Coal reservoir permeability is extremely sensitive to in situ stress and with increase of that stress, the permeability decreases exponentially (Somerton et al. 1975). This has been verified by field measurements, e.g., in coal of the Australian Sydney and Bowen Basins (Enever and Henning 1997; Enever et al. 1998), the American San Juan, Piceance and Black Warrior Basins (McKee et al. 1988; Sparks et al. 1995), and the Chinese Qinshui Basin (Meng et al. 2011). Generally, in situ stress has clear effects on coal reservoir permeability. When other factors (e.g., coal pore-fracture structures, petrographic constituents, and engineering operations) are equal, the greater stress could bring the lower the permeability and the poorer the CBM well productivity. In such a case, permeability may be roughly determined based on in situ stress magnitude (Li et al. 2014).

In our study, a negative exponent was used to analyze the relationship between coal reservoir well test permeability and in situ stress (Seidle et al. 1992; Zhao et al. 2015b), as

$$k = k_{\text{0}} {\text{e}}^{{- \alpha\Delta \sigma }} ,$$
(12)

where, k is permeability of a given stress condition, mD; k 0 is permeability of the initial stress condition, mD; ∆σ is the effective stress difference between an initial and a given stressed state, MPa; α is a coefficient dependent on the principal stress type.

Figure 14 shows that a decreasing behavior can describe the relationship between the maximum horizontal principal stress and coal reservoir permeability. The same applies to the minimum horizontal principal stress. This result is in accord with correlations between the measured permeability of coal seam and in situ horizontal stresses in the Hancheng and Liulin blocks in the eastern margin of the Ordos Basin, found by Tang (2001) and Li et al. (2014), respectively. After ~20 MPa in Figs. 14a and 15 MPa in Fig. 14b, high permeability data disappears. In contrast, the relationship between permeability and vertical principal stress is relatively complicated. With increasing stress, the permeability shows a tendency of decrease (<20 MPa), increase (20–25 MPa) and decrease (>25 MPa). When that stress is >30 MPa, the permeability is extremely low. This permeability change is in accord with Li et al. (2014, 2015).

Fig. 14
figure 14

Scatter diagrams of in situ stresses and well test permeability. a Relationship between the maximum horizontal principal stress and the permeability; b relationship between the minimum principal stress and the permeability; c relationship between the vertical principal stress and the permeability

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Fig. 15
figure 15

Scatter diagrams of well test permeability and burial depth

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In fact, the above permeability variation with vertical principal stress represents the relationship between permeability and burial depth, as that stress is calculated by Eq. (5). Figure 15 indicates that the permeability has three distinct vertical bands with increase of buried depth. At depths <700 m, the permeability decreases with buried depth and is 0.01–3.33 mD with average 0.89 mD. For 700–1000 m depths, permeability increases with depth and is 0.01–3.26 mD with average 0.73 mD. At depths >1000 m, permeability is very low, at 0.01–0.42 mD with average 0.11 mD.

Taking the stress variation into consideration, coal reservoir permeability inflection points at various depths are basically consistent with transition points of the stress field types and states at corresponding depths (Meng et al. 2011). In this work, for coal seams at burial depths <700 m, the horizontal principal stress is dominant. There, coal fractures gradually close and permeability decreases gradually with burial depth, corresponding to increase of the in situ stresses. During in situ stress state transitions (700–1000 m), with further increase of those stresses, both elastic and some inelastic deformation may occur. This results in part of the pores and fractures opening, because of friction and slide of the fractures or even coal destruction. Therefore, in this in situ stress transition zone, the stress-dependent permeability somewhat increases. Nonetheless, permeability of the coal seams buried more than 1000 m deep decreases rapidly instead of continuously increasing, although it is an extensional stress regime where the rock is in a compressive state but the vertical compressive stress is higher than the horizontal compressive stress. The main reason for this is that under the stronger vertical principal stress in a deep coal seam, the coal reservoir pore and fracture system tends to close again and cannot be effectively connected which results a rapid permeability reduction. Obviously, the essential influence of deep burial on permeability is its control by in situ stress control, i.e., the deformation and destruction of coal pore structures under in situ stress. This is consistent with Li et al. (2014, 2015), who discovered the same situation.

Finally, in the eastern margin of the Ordos Basin, optimizing the relationships among buried depth, in situ stress and permeability is very important for improvement of CBM productivity. Moreover, deep-level (>1000 m) CBM development faces the extreme challenge of “extremely low permeability and extremely high in situ stress,” which is the current problem in need of urgent solution for CBM development in China.

5 Conclusions

Through the regression of measured injection/falloff and in situ stress well test data for 55 CBM wells in the eastern margin of the Ordos Basin, the distribution of in situ stress and its effect on permeability were analyzed systematically.

  1. 1.

    The maximum and minimum horizontal principal stress and vertical principal stress are all positively correlated with buried depth. Stress ratios and the lateral stress coefficient have a slow attenuation trend with burial depth.

  2. 2.

    Overall, there are three in situ stress states in the eastern margin of the Ordos Basin. At burial depths <700 m, the in situ stress state is σ H > σ v > σ h, revealing a strike slip regime. From 700 to 1000 m depths, σ H ≈ σ v > σ h, showing a stress transition regime. At depths >1000 m, the in situ stress state changes to σ v > σ H > σ h demonstrating a normal stress regime.

  3. 3.

    Well test permeability decreases with horizontal principal stress but has a decrease–increase–decrease trend with increased vertical stress. This is basically consistent with the change of stress state at a certain burial depth, whose essence is the deformation and destruction of coal pore structures under the action of stresses.

  4. 4.

    With increase of buried depth, the vertical principal stress gradually dominates. A state of “extremely low permeability and extremely high in situ stress” means that it is very difficult to develop the deep-level CBM resource (>1000 m depths).