Experimental and modeling study of the stress-dependent permeability of a single fracture in shale under high effective stress

Fuel 257 (2019) 116078

Contents lists available at ScienceDirect

Fuel journal homepage: www.elsevier.com/locate/fuel

Full Length Article

Experimental and modeling study of the stress-dependent permeability of a single fracture in shale under high effective stress ⁎

Jian Zhoua,b,c, Luqing Zhanga,b, Xiao Lia,b,d, , Zhejun Panc,

T

⁎

a

Key Laboratory of Shale Gas and Geoengineering, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China Institutions of Earth Science, Chinese Academy of Sciences, Beijing 100029, China c CSIRO Energy, Private Bag 10, Clayton South, VIC 3169, Australia d College of Earth Science, University of Chinese Academy of Sciences, Beijing 100049, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Permeability model Shale fractures Fracture compressibility Apparent permeability High stress

It is of great importance to investigate the change in the permeability of stimulated shale gas reservoirs under high effective stress because the depths of producing shale gas reservoirs in China have gradually exceeded 3500 m. In this study, a series of permeability measurements were performed on two fractured Longmaxi shale samples under confining pressures from 5 to 80 MPa. Experimental results show that the measured apparent permeability of the fractured samples sharply decreases with effective stress. The intrinsic permeability of fractured shale decreases with increasing effective stress, and the Klinkenberg constant is almost invariable. Then, a stress-dependent fracture permeability model was derived based on the fracture compressibility models. The modeling results indicate that the permeability model derived in this study can accurately describe the experimental data, with errors of 5.63% and 2.24% for samples I and II, respectively. A comparison of the modeling results using this model to those using permeability models available in the literature indicates that the model derived in this study is preferable, especially for a broad effective stress range. Moreover, the average compressibility of fractured Longmaxi shale decreases from 0.077 to 0.014 MPa−1 with effective stress increasing from 4.13 to 79.65 MPa, showing that fracture compressibility is strongly stress-dependent. The results in this paper will be important for the prediction of permeability change and gas production behavior for the development of deep shale gas.

1. Introduction Shale gas production in North America has achieved great success due to the development of horizontal drilling and multistaged hydraulic fracturing technologies. The induced fracture network may enhance existing natural fracture networks in the reservoir [1], further shortening the distance from the matrix to the producing well. The production performance of shale gas highly relies on the stimulated reservoir volume, where the permeability is highly improved by the opening of existing fractures and newly induced hydraulic fractures. As pore pressure is drawdown during the production of gas, the effective stress acting on the shale reservoir significantly increases, which causes deformation of the internal pore and fracture geometry and results in changes in reservoir permeability [2–7]. Therefore, knowledge of stress-dependent permeability is of great significance for gas production behavior from unconventional gas reservoirs. Investigating the

permeability of shale fractures under varying stress is a significant step towards a more realistic evaluation of long-term shale gas production behavior. Existing studies in this area mainly focus on conducting experimental tests on core samples and characterizing the relationship between the effective stress acting on solid grains [8,9] and fracture permeability by different empirical functions, such as logarithmic functions [10,11], cubic functions [12,13] and exponential functions [4,14–16]. Considering the effects of gas slippage and effective stress, Klinkenberg’s model was expanded to a stress-dependent permeability model of gas shale [4,17]. However, most of the experiment and modeling were conducted under low confining pressure; thus, the results are not suitable for the current development of deeply buried shale gas. Taking the Barnett shale and Marcellus shale as examples, with average burial depths of approximately 2200 m and 2600 m, respectively, Yu and Sepehrnoori [18] predicted that the maximum closure

⁎ Corresponding authors at: Key Laboratory of Shale Gas and Geoengineering, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China (X. Li). E-mail addresses: [email protected] (X. Li), [email protected] (Z. Pan).

https://doi.org/10.1016/j.fuel.2019.116078 Received 10 May 2019; Received in revised form 14 August 2019; Accepted 21 August 2019 Available online 26 August 2019 0016-2361/ © 2019 Elsevier Ltd. All rights reserved.

Fuel 257 (2019) 116078

J. Zhou, et al.

pressure of these two reservoirs would reach 3379 psi (~23.3 MPa) and 5480 psi (~37.8 MPa), respectively, due to gas depletion. As reported by Gutierrez et al. [3], a reservoir burial depth greater than 4 km causes a considerable overburden stress of 100 MPa. In China, there are many gas shale reservoirs buried at depths greater than 3.5 km [19]. For example, the Weiyuan reservoir has a depth of 1530–3500 m, the FushunYongchuan reservoir has a depth of 3000–4500 m, and the Changning (-Zhaotong) reservoir has a depth of 2300–4000 m. Thus, as the gas is extracted from these reservoirs, a high effective stress (> 60 MPa) will be generated. Although the influence of stress on shale matrix permeability and fracture conductivity has been investigated in more detail [2,4,7,11,16,20–27], as summarized by Chen et al. [28] via the experimental data from the literature, in most of the current studies on shale permeability, the effective stress is less than 30 MPa. Thus, it is still challenging to precisely predict the stress-dependent permeability of shale formation due to the limited data on stress-induced permeability under high-level confining pressure. Therefore, it is essential to investigate the stress sensitivity of shale fracture permeability under higher confining and effective stresses. In this work, a series of permeability measurements in the laboratory within a wide range of confining pressures (from 4 to 80 MPa) is performed on two fractured shale samples obtained from an outcrop of the Longmaxi formation in Chongqing, China. Then, we derive a prediction equation for shale fracture permeability with a function of effective stress; this equation is based on the characteristic of fracture geometry. The newly derived model is applied to match the experimental data, and the coefficients in the model are studied. Finally, a comparison and discussion on the reasonability and application of our model and previous models are provided.

(a) Fractured sample I

(b) Fractured sample II Fig. 1. The rejoined cylindrical shale sample from the two separated parts, for permeability measurements.

approximately 28 μm for the sample used in this work. The original CT data were processed by using Avizo software, and a resulting 3D rendered image is presented in Fig. 2, which clearly shows that only one macroscale fracture cut across the shale sample. Using the same shale block, Zhou et al. [29] tested the mechanical properties of other intact samples 50 mm in diameter and 100 mm in length and showed that the average Young’s modulus and Poisson’s ratio are 27.3 GPa and 0.20, respectively, and that the mean uniaxial compressive strength is 159.8 MPa.

2. Experimental procedure 2.1. Samples

2.2. Experimental setup and data processing method A shale block with dimensions of 500 × 500 × 500 mm3 was obtained from an outcrop of the Lower Silurian Longmaxi formation in Shizhu County; this outcrop is a natural extension of the Pengshui shale gas block in the Sichuan Basin, Southwest China. Several short cylindrical samples with a diameter of 50 mm and a height of 50 or 25 mm were drilled from the shale block, along the bedding planes. The short cylinders were made with the high-precision standards suggested by the ISRM, with end surface flatness of 0.03 mm, parallelism of 0.05 mm, and verticality of 0.25°. To eliminate the influence of moisture, before permeability measurements were taken, the two samples were dried in a vacuum oven at 105 °C for 24 h and were weighed every few hours until the weight remained unchanged. After drying, a macroscale fracture appeared along the bedding plane in the center of each cylindrical sample, possibly due to the drying-induced contraction in the vacuum oven. Each of the fractured samples was broken into two halves by hand (Fig. 1) and then they were put together carefully and placed in a heat-shrinkable flexible polyolefin sleeve. The contact surfaces in each rejoined cylindrical sample were coupled perfectly, as they were in the intact samples. The samples in Fig. 1 will be utilized to mimic a fracture transversely crossing the sample’s axis for permeability measurements. The mineral composition influences the mechanical properties of shale samples, and the surface topography of a fracture significantly influences its permeability. Thus, the mineral composition analysis and X-ray μ-CT analysis of the shale samples were conducted at the Multiscale Imaging and Characterization Laboratory at the Institute of Geology and Geophysics, Chinese Academy of Sciences. Table 1 shows the area percentage of each mineral by the AmicScan system, in which the content of fragile minerals reaches 56.9%, and that of clay minerals is approximately 18.5%. X-ray μ-CT by Xradia 520 Versa 3D was used to detect the internal surface of sample I, with a resolution of

The permeability measurement apparatus utilized in the experiment is a coupled multifield testing system developed by the authors. The schematic of the permeability test is shown in Fig. 3 and is based on the transient method developed by Brace et al. [30]. The apparatus can provide a confining pressure and bias stress acting on the samples during the measurement of permeability. A servo-hydraulic booster with a pressure control accuracy of 0.01 MPa was employed to control the confining pressure up to 150 MPa, and the ISCO 260D pump was utilized to control the axial force, which can provide a maximum force of 2400 kN. Another ISCO 65D double pump with a maximum pressure of 20,000 psi (~137.9 MPa) and minimum flow rate of 10 nL/min was applied to regulate the gas pressure and flow. Two high-resolution gas pressure transducers with a range from 0 to 10 MPa are used to monitor the changes in gas pressure between the downstream and upstream of the sample, respectively, and a differential pressure sensor is adopted to detect the pressure difference between the downstream and upstream of the sample in the gas line. The Longmaxi formation shale matrix is a tight assembly of very fine particles with very low porosity (2%), which results in extremely low permeability, ranging from μD to nD. The permeability measurement is very time-consuming. Thus, a set of pipelines and valves from Swagelok Company were utilized in the apparatus to ensure a very good gas tightness of the system. Two cylindrical gas tanks were assembled downstream and upstream of the sample container to adjust the volume of the gas line. After the examination of gas tightness, a calibration shows that both the void volumes of the downstream and upstream of the sample are 22.0 ml. When installing a sample, the upper and lower platens holding the fractured shale sample were wrapped by a thermal-shrinkable sleeve. The inner wall of the thermal-shrinkable tube contains a layer of 2

Fuel 257 (2019) 116078

J. Zhou, et al.

Table 1 Mineral composition of the shale samples. Minerals Area percentage (%) Minerals Area percentage (%) Minerals Area percentage (%)

Quartz 30.32 Albite 4.74 Rutile 0.13

Ankerite 12.81 Oligoclase 4.5 Apatite 0.14

Illite 13.49 Orthoclase 3.47 Plagioclase 0.13

Else 9.76 Biotite 2.33 Benitoite 0.03

Dolomite 7.77 Calcite 2.37 Zircon 0.02

Chlorite 5.01 Pyrite 0.96 Pores 2

respectively. Pu and Pd represent the pressures of the upstream and downstream of the gas systems at time t, respectively. α is the pressure decay coefficient, which can be written as [7,30]:

α=

⎜

⎟

(2)

where ka represents the apparent permeability, A is the cross-sectional area of the sample, L is the length of the sample, μ is the gas fluid viscosity, and Vu and Vd represent the volume of the upstream and downstream of the gas systems, respectively. As mentioned above, the fractured shale samples were rejoined cylinders. To avoid permeability measurement error induced by the unrecoverable deformation of the contact surfaces, four cycles of loading and unloading were initially conducted as the confining pressure was increased from 5 to 80 MPa and then decreased back to 5 MPa. Then, to estimate the influence of effective stress on the permeability of a shale fracture, a differential gas pressure of approximately 0.5 MPa was applied on the sample’s ends for transient pressure pulse testing under hydraulic confining pressures varying from the initial pressure to 80 MPa. Different average gas pressures (0.35 MPa, 0.87 MPa, 1.34 MPa, and 4.33 MPa) were considered in the experimental measurements to study the gas slippage effect. All the tests were conducted at room temperature (25 °C).

Fig. 2. The CT images of rejoined cylindrical shale sample I show a transversely crossing fracture along the bedding plane.

0.2 mm thick hot melt adhesive that can completely couple the sample surface and thermal-shrinkable sleeve. Nitrogen with a purity of 99.999% was used in the permeability tests. During the permeability measurement, the hydraulic confining pressure was initially applied at a rate of 0.1 MPa/s to the target pressure. Then, the fractured shale sample and gas injection system were vacuumed for at least 24 h. The pressures in the upstream and downstream cylinders were recorded over time and used in the permeability calculation. The permeability measurements were described in detail by Pan et al. [7], and a short description is provided for the calculation method. The gas pressure difference between the two ends of the sample was measured to calculate the permeability using the following equation [7,30]:

(Pu − Pd ) = e−αt (Pu,0 − Pd,0 )

ka A (Pu,0 − Pd,0 ) ⎛ 1 1⎞ − 2μL Vd ⎠ ⎝ Vu

3. Results Fig. 4 shows an example of the gas pressure evolution in the upstream and downstream of samples under different confining pressures. Based on the recorded gas pressure evolution data, the apparent gas permeability ka results determined directly from Brace’s formula (Eq. (2)) are listed in Tables 2–4. Tables 2 and 3 present the apparent permeabilities of the fractured shale samples I and II at different effective

(1)

where Pu,0 and Pd,0 are the pressures of the upstream and downstream of the gas system at the start of the permeability measurement,

Fig. 3. The schematic diagram of the permeability measurement system used in this study (modified from Ref. [7]). 3

Fuel 257 (2019) 116078

J. Zhou, et al.

Table 4 The apparent permeability of the intact sample under various effective stresses. Pp = 0.35 MPa

Pp = 1.34 MPa

Pp = 4.90 MPa

σe (MPa)

ka (μD)

σe (MPa)

ka (μD)

σe (MPa)

ka (μD)

4.65 6.65 9.65 12.65 15.65 19.65 24.65 29.65 34.65 39.65 49.65 59.65 65.65 79.65

96.38 94.17 89.58 83.65 81.8 76.54 70.02 67.52 63.41 59.27 53.45 47.45 44.55 38.32

3.66 5.66 8.66 12.66 16.66 20.66 24.66 28.66 38.66 48.66 58.66 68.66 78.66 –

26.33 25.57 23.4 21.82 20.55 19.24 17.84 16.89 14.67 13.14 11.48 10.52 9.41 –

5.1 10.1 13.1 16.1 20.1 25.1 30.1 40.1 50.1 65.1 75.1 – – –

8.21 7.62 7.2 7.03 6.4 5.77 5.1 4.43 3.96 3.21 2.71 – – –

Fig. 4. The history records of the evolution of upstream gas pressure Pu and downstream gas pressure Pd under different confining pressures (10 MPa, 20 MPa, 40 MPa, and 80 MPa). Table 2 The apparent permeability of fractured sample I under various effective stresses. Pp = 0.35 MPa

Pp = 0.87 MPa

Pp = 1.39 MPa

Pp = 4.33 MPa

σe (MPa)

ka (μD)

σe (MPa)

ka (μD)

σe (MPa)

ka (μD)

σe (MPa)

ka (μD)

4.65 6.65 7.65 9.65 12.65 14.65 17.65 19.65 24.65 29.65 34.65 39.65 44.65 49.65 59.65 69.65 79.65

421.66 353.07 304.65 256.23 207.81 181.98 154.14 139.82 109.75 89.17 75.25 64.16 56.09 49.23 39.75 33.09 28.45

4.13 7.13 9.13 12.13 14.13 17.13 19.13 24.13 29.13 34.13 39.13 44.13 49.13 59.13 69.13 – –

338.00 250.23 207.65 168.35 147.38 123.15 112.67 85.81 72.71 61.97 51.42 45.26 38.91 30.85 25.61 – –

4.61 6.61 8.61 11.61 13.61 16.61 18.61 23.61 28.61 33.61 38.61 43.61 48.61 58.61 68.61 78.61 –

232.64 207.23 170.08 136.07 117.30 98.14 87.97 69.99 57.48 48.87 44.96 39.49 34.84 28.54 23.89 20.37 –

5.67 7.67 10.67 13.67 15.67 20.67 25.67 30.67 35.67 40.67 45.67 50.67 55.67 60.67 65.67 75.67 –

179.39 170.07 147.94 111.13 99.25 76.07 63.60 53.70 42.87 36.46 32.97 29.00 25.98 23.76 21.90 18.64 –

(a) Fractured shale sample I

Table 3 The apparent permeability of fractured sample II under various effective stresses. Pp = 0.35 MPa

Pp = 0.87 MPa

Pp = 1.39 MPa

Pp = 4.33 MPa

σe (MPa)

ka (μD)

σe (MPa)

ka (μD)

σe (MPa)

ka (μD)

σe (MPa)

ka (μD)

4.65 6.65 9.65 12.65 14.65 19.65 24.65 29.65 34.65 39.65 44.65 49.65 59.65 69.65 79.65 –

601.23 466.05 340.96 274.38 240.09 180.37 144.25 119.03 100.67 87.56 77.07 68.8 56.69 48.02 41.76 –

4.13 6.13 9.13 12.13 14.13 17.13 19.13 24.13 29.13 34.13 39.13 44.13 49.13 59.13 69.13 79.13

450.67 363.55 258.74 203.06 176.21 146.73 130.35 103.5 85.81 72.71 62.43 55.29 49.65 40.87 34.98 30.79

4.61 6.61 8.61 11.61 13.61 16.61 18.61 23.61 28.61 33.61 38.61 43.61 48.61 58.61 68.61 78.61

461.37 326.48 262.75 197.45 171.26 140.37 124.73 96.58 79.37 67.25 59.04 51.22 47.31 37.93 32.34 28.31

5.57 7.57 10.57 13.57 15.57 20.57 25.57 30.57 35.57 40.57 45.57 55.57 65.57 75.57 – –

264.42 232.97 178.22 142.11 123.47 95.87 78.51 65.93 57.08 49.51 44.15 35.53 30.52 26.79 – –

(b) Fractured shale sample II Fig. 5. Variation in apparent permeability of fractured shale samples with effective stress and pore pressure.

4

Fuel 257 (2019) 116078

J. Zhou, et al.

Fig. 8. Sheet-type fracture network. a is the fracture spacing, and s is the matrix width. Fig. 6. Variation in apparent permeability of intact shale samples with effective stress and pore pressure.

Fig. 7 shows that the intrinsic permeability of both samples decreases with increasing effective stress; the Klinkenberg constant of the intact sample increases sharply, while that of the fractured sample II is almost invariable. The gas slippage effect for fractured shale is not very significant, which may be attributed to the fact that the fracture pore size is much larger than the mean free path size required for nitrogen gas.

stresses, respectively. As a comparison, Table 4 lists the apparent gas permeabilities of an intact shale sample with varying effective stresses and pore pressures. In this study, the effective stress is defined as the confining pressure minus the mean pore pressure (σc-Pp). All of the measured permeability data also presented in Figs. 5 and 6. Fig. 5 indicate that the apparent permeability of the fractured samples sharply decreases with increasing effective stress. For example, with effective stress changing from 4.65 to 79.65 MPa at an average pore pressure of 0.35 MPa, the apparent permeability of samples I and II decrease from 421.66 to 28.45 μD and from 601.23 to 41.76 μD, respectively. The drop in the fractured shale permeability results from the reduction in the effective fracture porosity due to the compression of the fracture surfaces. Moreover, Fig. 6 shows that the logarithm of the apparent permeability of intact shale has an approximately linear negative correlation with effective stress, while Fig. 5 presents that the logarithm of the apparent permeability of fractured samples has a nonlinear negative correlation with effective stress. Compared with intact shale, the permeability of fractured cores is at least three orders of magnitude higher than that of an intact core under a certain pressure state, which is in line with the experimental results from Chen et al. [28]. Table 4 and Fig. 6 demonstrate that due to the gas slippage effect, the permeability of intact shale drops sharply as the average pore pressure slightly increases. Based on the results in Tables 3 and 4, we obtained the interpolation of permeability at the effective stress of 10 to 70 MPa with a step of 10 MPa and then calculated the Klinkenberg constant and intrinsic permeability [31] of fractured sample II and an intact sample.

4. Model development Fractures commonly exist in shale. Experimental tests have revealed that the permeability of fractures is highly dependent on the effective confining pressure. The stress-dependent permeability of a fracture mainly results from changes in the porosity of fractured rock, which is generally defined by the fracture pore compressibility [32]. Under a certain assumption, the complex fracture system could be simplified into an idealized fracture reservoir with a sheet-type fracture structure (Fig. 8). An analytical model initially given by Reiss [33] presents the permeability of a sheet-type fracture network as:

kf =

a2ϕf3 (3)

12

where kf is fracture permeability and ϕf represents the fracture porosity. During the study of stress-dependent cleat permeability in coals, Seidle [14] derived a negative exponential relationship between the permeability and the multiplication of fracture compressibility and effective stress σ:

(a) Fractured shale sample II

(b) Intact shale sample

Fig. 7. Intrinsic permeability and Klinkenberg constant of shale samples at different effective stresses. 5

Fuel 257 (2019) 116078

J. Zhou, et al.

kf = k f0 e−3Cf (σ − σ0)

(4)

where kf0 is the fracture permeability at reference average confining pressure σ0 and represents the fracture compressibility. In the appendixes, the fracture compressibility models are derived, each based on a conceptual assumption or a physical assumption. The fracture compressibility models in Eq. (A11) and Eq. (B12) have the same form. Compared with the compressibility of a fracture, the compressibility of a rock matrix Cs can be neglected. Thus, the coefficients of ac in Eq. (A11) and ap in Eq. (B12) can be simplified to zero. Then, the compressibility of a shale fracture Cf can be expressed as:

Cf =

c σ+b

(5)

Eq. (5) is in line with the fracture compressibility given by Li et al. [15]. Considering that the experimental results are averaged fracture pore compressibility values, the average pore compressibility C¯f can be calculated as:

C¯f =

1 σ − σ0

∫σ

σ

0

Cf dσ =

c σ+b⎞ ln ⎛ σ − σ0 ⎝ σ0 + b ⎠ ⎜

⎟

(6)

(a) Fractured shale sample I

The work of Klinkenberg [31] has shown that if the pore size in a fractured shale approaches the mean free path size of the gas, gas slippage will significantly influence the fracture permeability. Klinkenberg’s model is often employed to evaluate the gas slippage phenomenon, which is approximately a linear function of the inverse of the mean pore pressure (1/Pm). To extend Klinkenberg’s model to incorporate both the effective stress and gas slippage effects, we propose a new permeability model for fractured rock, expressed as:

B ⎞ −3c¯f (σ − σ0) kf = kini ⎛⎜1 + ⎟e p m⎠ ⎝

(7)

where kf is the permeability of the fractured rock subjected to pore pressure and confining pressure, kini is the initial permeability of the fractured rock, and B is similar to the Klinkenberg constant. As mentioned above, the Klinkenberg constant of fractured sample II is almost invariable at different effective stresses. Thus, the coefficient B in Eq. (7) is also assumed to be a constant, independent of effective stress. 5. Discussion

(b) Fractured shale sample II

5.1. Modeling results of experimental data

Fig. 9. Permeability of fractured shales under different effective stresses and pore pressures, and modeling results from the model derived in this study.

The model derived in Section 4 is verified by matching it with the experimental permeability data on the fractured shale samples listed in Section 3. Assuming that the initial effective stress (σ0) is zero, the model parameters in Eqs. (6) and (7) are solved based on the programming solver in Excel. Fig. 9 presents the fitted curves based on the original experimental permeability data of fractured shale samples I and II. The permeability model fits the measured data well. The average relative errors between the experimental data and model prediction are 5.63% and 2.24% for sample I and II, respectively. The permeability model coefficients of kini, B, b and c are 399.52 μD, 0.35 MPa, 7.53 MPa and 0.45 for sample I, respectively, and they are 919.07 μD, 0.30 MPa, 3.16 MPa and 0.38 for sample II, respectively. The major difference between these two samples is the initial permeability value. Fig. 10 shows the absolute value of the difference between measured permeability and model prediction, indicating that major errors arise under low effective stress (< 15 MPa) and that the errors decrease with increasing effective stress. Fig. 11 also presents the modeling results using the derived model in this study for the experimental permeability data of fractured Longmaxi shale and Niutitang shale from Chen et al. [28] with the confining pressure varying from 2 to 60 MPa. Fig. 11 shows that the derived permeability model fits well with the experimental data from Chen et al. [28]. The permeability model coefficients of kini, B, b and c are 1193.89 μD, 0.59 MPa, 7.50 MPa and 0.83 for the

fractured Longmaxi shale, respectively, and they are 1298.81 μD, 1.95 MPa, 11.44 MPa and 1.35 for the fractured Niutitang shale, respectively. 5.2. Model comparison The permeability model in Eq. (4) indicates that it is highly dependent on the fracture compressibility, which is analogous to the pore volume compressibility of conventional reservoirs. Seidle et al. [14] investigated the permeability model of a coal cleat, assuming that the coal cleat compressibility was constant. Under confining pressure, the contact surfaces of a rock fracture approach each other, and the fracture compressibility can be considered a negative linear function of effective stress (Cf = Cf0 − ασ). Moreover, a negative exponential expression between fracture compressibility and stress initially proposed by Shi and Durucan [34], abbreviated as the S&D model, was widely used in recent works [6,35] and is written as

C¯f = 6

Cf 0 σ − σ0

[1 − e−α (σ − σ0)]

(8)

Fuel 257 (2019) 116078

J. Zhou, et al.

(a )Fractured shale sample I

(b )Fractured shale sample II

Fig. 10. The absolute value of the difference between the measured permeability and model prediction.

(a) Fractured shale sample I

Fig. 11. Experimental permeability data from Chen et al. [28], and modeling results using the derived model in this study.

where Cf0 is the initial fracture compressibility and α is the declining rate of the fracture compressibility with increasing stress. A comparison of the fit of permeability models for fractured shale to experimental data was performed, considering the different fracture compressibility models. The permeability model considering the negative linear fracture compressibility model is named Model I, while the S &D model considering the permeability model is called Model II. Fig. 12 shows the fitted curves based on Model I for the permeability data of shale samples I and II, respectively. The test data agree well with the fitted curves, with a mean relative error of approximately 6.00% when the effective stress is less than 20 MPa. However, when the effective stress exceeds 20 MPa, the fitted curves deviate entirely from the test data for both samples. The experimental data indicate that the fracture permeability decreases monotonically as the effective stress increases, while the fitted curves in Fig. 12 are not monotonically changing, which implies that permeability testing at a high effective stress is essential for a model match. Taking the S&D model (Eq. (8)) into account, the permeability model curves based on Model II and fit to the measured permeability data are presented in Fig. 13. The fitted results are obviously better than those in Fig. 12, and the average relative errors between the experimental data and the fitted model are 6.85% and 7.12% for sample I and II, respectively. However, Fig. 13 also shows that the model prediction gradually deviates from the experimental results when the effective stress exceeds 50 MPa. The model parameters and mean relative errors from the model fitting for samples I and II are summarized in Table 5, respectively. By comparing the fitted curves in Figs. 9, 12 and 13 and the mean relative error listed in Table 5, we are confident that the stress-dependent permeability model derived in this

(b) Fractured shale sample II Fig. 12. The fitted curves for the shale samples when the negative linear fracture compressibility model (Model I) was used.

study for fractured shale is more reasonable, especially at a high effective stress. Notably, our newly derived permeability model is validated with data from shale samples with a nearly flat fracture only. Whether this model can be applied to complexly fractured shale requires further experimental study.

7

Fuel 257 (2019) 116078

J. Zhou, et al.

and completion of gas wells. Tan et al. [16] summarized the recent research results on the fracture compressibility of shale and stated that the fracture compressibility ranges from 0.001 MPa−1 to 0.5 MPa−1. Most of the shale fracture compressibility results vary from 0.01 to 0.1 MPa−1. Table 6 lists a comparison of shale (with or without fractures) compressibility calculated from this study based on Eq. (6) and those from the literature. The intact shale compressibility is lower than that of fractured shale under the same effective stress, and the fractured shale compressibility is more sensitive to the change in effective stress. The shale fracture compressibility obtained in this study is highly consistent with the results from the literature [6,25–28]. Table 6 also shows that the gas and fluid are different during the permeability test. Thus, the effect of the gas and fluid used in permeability measurements on the back analysis of shale fracture compressibility should be considered in future studies. 6. Conclusions Currently, the depths of shale gas reservoirs being developed in China have gradually exceed 3500 m. Thus, it is of great significance to investigate the change in the permeability of stimulated shale reservoirs under high effective stress because the permeability evolution is important for predicting gas production behavior and designing the drilling and completion of gas wells. In this study, the relationship between shale fracture permeability and effective stress was investigated. A series of permeability measurements were performed on intact and fractured Longmaxi shale samples under confining pressure varying from 4 to 80 MPa. A stress-dependent permeability model was derived for fractured shale. The main results are summarized as follows.

(a) Fractured shale sample I

(1) Experimental results indicate that the apparent permeability of the fractured samples sharply decreases with effective stress, and the logarithm of the apparent permeability of the fractured shale samples has a nonlinear negative correlation with effective stress. The decrease in fractured shale permeability may result from the reduction in effective fractured shale porosity due to the compression of fracture surfaces. The intrinsic permeability of fractured shale decreases with increasing effective stress, and the Klinkenberg constant is almost invariable. (2) A unified fracture compressibility model, which is vital in fracture permeability modeling, is derived. This model not only reflects the compressibility of macroscopic fractures at different scales but also captures the characteristic of the physical contact of fracture walls. Moreover, a stress-dependent fracture permeability model was then derived based on the unified fracture compressibility model. This permeability model can accurately describe the experimental data and is preferable for a broad effective stress range. (3) The permeability of the fractured rock is highly dependent on the effective stress due to the variation in fracture compressibility with effective stress. The average compressibility of the Longmaxi shale fractures in this study decreases from 0.077 to 0.014 MPa−1 as the effective stress increases from 4.13 to 79.65 MPa. Although the two surfaces of a fracture approach each other under high stress, there is still open space remaining, supported by the asperities. It can be deduced that the conductivity of fractured shale reservoirs is always superior to that of the shale matrix. The results in this paper are useful for predicting permeability change and gas production behavior for the development of deep shale gas reservoirs.

(b) Fractured shale sample II Fig. 13. The fitted curves for the shale samples when the S&D fracture compressibility model (Model II) was used. Table 5 The comparison of model parameters and mean relative errors from the modeling results. Permeability Models

Sample I

Sample I

Parameters kini

B

b or Cf0

c or α

Mean relative error

Model derived in this study Model I Model II

399.52

0.35

7.53

0.45

5.63%

276.29 476.66

0.396 0.306

0.028 0.033

2.17E−04 2.75E−04

10.47% 6.85%

Model derived in this study Model I Model II

919.07

0.30

3.16

0.38

2.24%

476.66 588.86

0.306 0.301

0.033 0.048

2.75E−04 0.047

6.05% 7.12%

Acknowledgments

5.3. Shale fracture compressibility

This research is funded by the National Natural Science Foundation of China (Grants 41572312, 41672321), the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB10050202), National key research and development program (2018YFB1501801), China Postdoctoral Science Foundation (2018M630204, 2019T120133), and

As mentioned above, the permeability of fractured shale is highly dependent on its compressibility. The accuracy of fracture compressibility is significant not only for the prediction of the permeability change and gas production behavior but also for designing the drilling 8

Fuel 257 (2019) 116078

J. Zhou, et al.

Table 6 Comparison of shale (with or without fractures) compressibility between the results of this study and data from the literature. Source

Lithologic

fluid

Effective stress (MPa)

Compressibility (MPa−1)

This study

Fractured Longmaxi shale sample I Fractured Longmaxi shale sample II

N2

4.13–79.65

0.014–0.047 0.016–0.077

Chen et al. [28]

Intact Longmaxi shale Intact Niutitang shale Longmaxi shale with visible fracture Niutitang shale with visible fracture

He

1.50–59.50

0.010–0.083 0.007–0.035 0.018–0.079 0.041–0.182

Tan et al. [26,27]

Longmaxi shale – Horizontal 1 Longmaxi shale – Horizontal 2

CH4, He

2.61–8.50 2.56–8.49

0.055–0.069 0.052–0.062

Zhang et al. [25]

Longmaxi shale

N2

0–25

0.021–0.050

Chen et al. [7]

Devonian shale Chinshui shale

N2

12–41 2–119

0.016–0.06 0.060–0.425

China Scholarship Council (201804910293) to the first author.

Appendix A. Conceptual model of fracture compressibility Due to the complexity of fracture surfaces, a rock fracture is usually simplified as two flat surfaces in contact. They are rough with a complex geometry. As the two surfaces of a rock fracture approach one another, contact occurs at only high asperities, and there is a large amount of open space remaining, which could be storage and migration space for the formation fluid. The asperities in contact change with increasing normal force acting on the rock fracture. The fracture compressibility was adopted to describe the deformation feature of a fracture with changing confining pressure, which significantly impacts the transport properties of rock. Experiments have shown that the fracture compressibility of shale is highly dependent on the confining pressure. The pores in rock fractures have a variety of geometric shapes. A sketch of the evolution of macroscale fracture geometry is presented in Fig. A1, which shows that with an increasing normal stress, the two surfaces of the fracture approach each other, and additional parts of the fracture come into contact; however, those parts never fully closed due to the roughness of the contact surface. In the study of the strain-stress behavior of fractured rock by Liu et al. [4], a rock body was divided into two distinct parts, a solid part and soft part. According to the ideas of Liu et al., we assumed that under a certain confining pressure, the porosity of a macroscale fracture could be separated into two parts. One part is like a wide pore, and the other is like a microscale fracture in the rock matrix. Based on the above assumption, the pore volume of a macroscale fracture Vp can be regarded as the summation of two parts of the compressible pore volume V1 and V2. V1 is the volume of the large pores, and V2 is the volume of the micropores.

Vp = V1 + V2

(A1)

Therefore, the fracture pore compressibility can be written as:

Fig. A1. The evolution of macroscale fracture geometry with increasing normal stress. 9

Fuel 257 (2019) 116078

J. Zhou, et al.

Cp = −

1 dVp 1 dV1 1 dV2 ⎞ V1 ⎛ 1 dV1 ⎞ V2 ⎛ 1 dV2 ⎞ = −⎛ + = − + − Vp dσ V1 + V2 dσ ⎠ V1 + V2 ⎝ V1 dσ ⎠ V1 + V2 ⎝ V2 dσ ⎠ ⎝ V1 + V2 dσ ⎜

⎟

⎜

⎟

⎜

⎟

(A2)

The compressibility of these two types of pores are assumed to be constant and are represented by C1 and C2, respectively.

1 dV1 − = C1 V1 dσ

(A3)

1 dV2 = C2 V2 dσ

(A4)

−

Thus, V1 and V2 can be given by

V1 = V1,0 e−C1 (σ − σ0)

(A5)

V2 = V2,0 e−C2 (σ − σ0)

(A6)

where σ0 is the initial stress, and V1,0 and V2,0 represent the volumes of the two types of pores at the initial stress, respectively. Thus, Eq. (A2) can be rewritten as:

Cp =

V1 C1 + V2 C2 V1 + V2

(A7)

V1 and V2 are approximated by the first-order Taylor expression as follows:

V1 =

V1,0 1 + C1 (σ − σ0)

(A8)

V2 =

V2,0 1 + C2 (σ − σ0)

(A9)

Substituting Eqs. (A8) and (A9) into Eq. (A7) yields:

Cp =

V1,0 C1 + V2,0 C2 + (V1,0 C1 C2 + V2,0 C2 C1)(σ − σ0) V1,0 + V2,0 + (V1,0 C2 + V2,0 C1)(σ − σ0)

(A10)

Defining(V1,0 C1 C2 + V2,0 C2 C1)/(V1,0 C2 + V2,0 C1) = ac , (V1,0 + V2,0)/(V1,0 C2 + V2,0 C1) − σ0 = bc and [2V1,0 V2,0 C1 (C1 − C2)]/(V1,0 C2 + can be calculated as:

Cp = ac +

cc bc + σ

V2,0 C1)2

= cc , Cp

(A11)

Although the above derivation is for pore compressibility Cp, not directly for fracture compressibility Cf, the difference between Cp and Cf is negligible according to previous research. Appendix B:. Physical model of fracture compressibility The physical model of fracture compressibility was first discussed by Walsh [31], Brace [36] and Walsh and Grosenbaugh [37]. Fig. B1 shows a fracture model in which the surfaces are propped open by the contacting asperities. With an increasing confining pressure dσ, an increased contact force dF is generated at an asperity. The change in the pressure dσ at the model boundary could be separated into two parts, as shown in Fig. B1(b) and (c) [37]. One part is the increase in dσc with the average contact pressure, which is defined as the pressure caused by the sum of the increased forces at all of the asperities.

Adσc =

∑ dF

(B1)

Fig. B1. (a) A confining pressure of dp applied to a fracture model causing contact force dF at asperities can be separated into two parts: (b) the average contact pressure dσc caused by the sum of increased forces at all of the asperities; (c) the remaining pressure sustained by the fracture without contacting asperities. 10

Fuel 257 (2019) 116078

J. Zhou, et al.

where A is the nominal contact area of the fracture surface. The effective compressibility Ceff of a macroscale fracture is defined as:

Ceff = −

1 dV V dσ

(B2)

where V is the volume of the fracture model. The change volume dV of the body in Fig. B1 (a) includes the changes caused by the two stress components in Fig. B1(b) and (c) [37].

dV = Cs dσc + C0 (dσ − dσc ) V

(B3)

where Cs is the compressibility of the solid matrix and C0 is the compressibility of the fracture without contacting asperities. According to the reciprocal theorem by Walsh [31], the volume change enclosed by the reference surfaces (in Fig. B1(a)) caused by pressure increase dσ can be given as: (B4)

dVi = Vi Cs dσc + V (C0 − Cs )(dσ − dσc )

where Vi is the volume enclosed by the reference surfaces. On the other hand, the deformation enclosed by the reference surface in Fig. B1(a) is due to two sources, the deformation of asperities and that of the solid matrix supporting them, and can be given as:

dVi = Vi Cs dσc + Ca ∑ dF

(B5)

where Ca is the compliance of the asperities. Combination Eqs. (B1), (B3) and (B5), we obtain:

V (C0 − Cs )(dσ − dσc ) = Ca Adσc

(B6)

Greenwood and Williamson [38] performed a mechanical analysis of the elastic deformation of two rough contact surfaces. In their analysis, it is assumed that all the tips of the asperities have the same radius of curvature and that the probability density function of the height of the composite topography approximately followed an exponential distribution ϕ(z). Thus, the relationship between the applied normal pressure σ and the deformation of the contacting asperities is presented as:

σ=

4 E 1 ηA r2 3 (1 − ν 2)

∞

∫D

3

(z − D) 2 ϕ (z )dz

(B7)

1 e− δ z ,

E and ν are the elastic modulus and Poisson’s ratio of the solid matrix, respectively, η represents the number of asperities per unit where ϕ (z ) = area of surface, z is the height of an asperity, δ is the standard deviation of the height of the asperities, and D is the distance between the reference surface and contact surface. The integral part in Eq. (B7) can be written as:

σ=

π (ηrδ )

D E 1 (δ / r ) 2 Ae− δ (1 − υ2)

(B8)

Eq. (B8) can be rewritten as: −1

E 1 ⎛ ⎞ ⎤ D = −δ ln ⎡ ⎢ π (ηrδ ) (1 − υ2) (δ / r ) 2 A σ⎥ ⎠ ⎦ ⎣⎝ ⎜

⎟

(B9)

Thus, the compliance of the asperities Ca is given as:

Ca = −

∂D δ = ∂σ σ

(B10)

The combination of Eqs. (B2), (B3), (B6) and (B10) yields [37]:

V 1 1 − = σ Ceff − Cs C0 − Cs δA

(B11)

Defining ap = Cs , bp = δA/ V (C0 − Cs ) , and cp = δA/ V , Eq. (B11) can be rewritten as:

Ceff = ap +

cp σ + bp

(B12)

[7] Pan Z, Ma Y, Connell LD, Down DI, Camilleri M. Measuring anisotropic permeability using a cubic shale sample in a triaxial cell. J Nat Gas Sci Eng 2015;26:336–44. [8] Terzaghi K. Theoretical soil mechanics. New York: John Wiley & Sons; 1943. [9] Biot MA, Willis DG. The elastic coefficients of the theory of consolidation. J Appl Mech 1957;24:594–601. [10] Kranzz RL, Frankel AD, Engelder T, Scholz CH. The permeability of whole and jointed Barre granite. Int J Rock Mech Min Sci Geomech Abs 1979;16(4):225–34. [11] Walsh JB. Effect of pore pressure and confining pressure on fracture permeability. Int J Rock Mech Min Sci Geomech Abs 1981;18(5):429–35. [12] Gangi AF. Variation of whole and fractured porous rock permeability with confining pressure. Int J Rock Mech Min Sci Geomech Abs 1978;15(5):249–57. [13] Kwon O, Kronenberg AK, Gangi AF, Johnson B. Permeability of Wilcox shale and its effective stress law. J Geophys Res Solid Earth 2001;106(B9):19339–53. [14] Seidle JP, Jeansonne MW, Erickson DJ. Application of matchstick geometry to stress dependent permeability in coals. SPE rocky mountain regional meeting. Society of Petroleum Engineers; 1992. [15] Li S, Tang D, Pan Z, Xu H, Huang W. Characterization of the stress sensitivity of

References [1] Gale JFW, Reed RM, Holder J. Natural fractures in the Barnett shale and their importance for hydraulic fracture treatment. AAPG Bull 2007;91(4):603–22. [2] Gutierrez M, Øino LE, Nygaard R. Stress-dependent permeability of a de-mineralised fracture in shale. Mar Pet Geol 2000;17(8):895–907. [3] Gutierrez M, Katsuki D, Tutuncu A. Determination of the continuous stress-dependent permeability, compressibility and poroelasticity of shale. Mar Pet Geol 2015;68:614–28. [4] Liu HH, Rutqvist J, Berryman JG. On the relationship between stress and elastic strain for porous and fractured rock. Int J Rock Mech Min Sci 2009;46(2):289–96. [5] Tao S, et al. Material composition, pore structure and adsorption capacity of lowrank coals around the first coalification jump: a case of eastern Junggar Basin, China. Fuel 2018;211:804–15. [6] Chen D, Pan Z, Ye Z. Dependence of gas shale fracture permeability on effective stress and reservoir pressure: model match and insights. Fuel 2015;139:383–92.

11

Fuel 257 (2019) 116078

J. Zhou, et al.

[16]

[17]

[18] [19] [20] [21]

[22]

[23]

[24]

[25]

[26] Tan Y, Pan Z, Liu J, Wu Y, Haque A, Connell LD. Experimental study of permeability and its anisotropy for shale fracture supported with proppant. J Nat Gas Sci Eng 2017;44:250–64. [27] Tan Y, Pan Z, Liu J, Feng XT, Connell LD. Laboratory study of proppant on shale fracture permeability and compressibility. Fuel 2018;222:83–97. [28] Chen T, Feng XT, Cui G, Tan Y, Pan Z. Experimental study of permeability change of organic-rich gas shales under high effective stress. J Nat Gas Sci Eng 2019;64:1–14. [29] Zhou J, Zhang L, Pan Z, Han Z. Numerical investigation of fluid-driven near-borehole fracture propagation in laminated reservoir rock using PFC2D. J Nat Gas Sci Eng 2016;36:719–33. [30] Brace W, Walsh JB, Frangos WT. Permeability of granite under high pressure. J Geophys Res 1968;73(6):2225–36. [31] Klinkenberg LJ. The permeability of porous media to liquids and gases. Drilling and production practice. American Petroleum Institute; 1941. [32] Walsh JB. The effect of cracks on the compressibility of rock. J Geophys Res 1965;70(2):381–9. [33] Reiss LH. The reservoir engineering aspects of fractured formations. Gulf Publishing Company; 1980. [34] Shi JQ, Durucan S. Exponential growth in San Juan Basin Fruitland coalbed permeability with reservoir drawdown: model match and new insights. SPE Reservoir Eval Eng 2010;13(06):914–25. [35] Li M, Yin G, Xu J, Cao J, Song Z. Permeability evolution of shale under anisotropic true triaxial stress conditions. Int J Coal Geol 2016;165:142–8. [36] Brace WF. Relation of elastic properties of rocks to fabric. J Geophys Res 1965;70(22):5657–67. [37] Walsh JB, Grosenbaugh MA. A new model for analyzing the effect of fractures on compressibility. J Geophys Res Solid Earth 1979;84(B7):3532–6. [38] Greenwood JA, Williamson JP. Contact of nominally flat surfaces. Proc R Soc Lond Ser A Math Phys Sci 1966;295(1442):300–19.

pores for different rank coals by nuclear magnetic resonance. Fuel 2013;111:746–54. Tan Y, Pan Z, Feng XT, Zhang D, Connell LD, Li S. Laboratory characterisation of fracture compressibility for coal and shale gas reservoir rocks: a review. Int J Coal Geol 2019. Yang D, Wang W, Chen W, Wang S, Wang X. Experimental investigation on the coupled effect of effective stress and gas slippage on the permeability of shale. Sci Rep 2017;7:44696. Yu W, Sepehrnoori K. Simulation of gas desorption and geomechanics effects for unconventional gas reservoirs. Fuel 2014;116:455–64. Caineng Zou, et al. Shale gas in China: characteristics, challenges and prospects (I). Pet Explor Dev 2015;42(6):753–67. Pedrosa Jr. OA. Pressure transient response in stress-sensitive formations. SPE California regional meeting. Society of Petroleum Engineers; 1986. Kwon O, Kronenberg AK, Gangi AF, Johnson B, Herbert BE. Permeability of illitebearing shale: 1. Anisotropy and effects of clay content and loading. J Geophys Res Solid Earth 2004;109(B10). Dong JJ, Hsu JY, Wu WJ, Shimamoto T, Hung JH, Yeh EC, et al. Stress-dependence of the permeability and porosity of sandstone and shale from TCDP Hole-A. Int J Rock Mech Min Sci 2010;47(7):1141–57. Cho Y, Ozkan E, Apaydin OG. Pressure-dependent natural-fracture permeability in shale and its effect on shale-gas well production. SPE Reservoir Eval Eng 2013;16(02):216–28. Zheng J, Zheng L, Liu HH, Ju Y. Relationships between permeability, porosity and effective stress for low-permeability sedimentary rock. Int J Rock Mech Min Sci 2015;78:304–18. Zhang R, Ning Z, Yang F, Wang X, Zhao H, Wang Q. Impacts of nanopore structure and elastic properties on stress-dependent permeability of gas shales. J Nat Gas Sci Eng 2015;26:1663–72.

12