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Experimental study of the effective stress coefficient for coal permeability with different water saturations
Journal of Petroleum Science and Engineering 182 (2019) 106282
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Experimental study of the effective stress coefficient for coal permeability with different water saturations
T
Xiaoyang Zhanga,*, Caifang Wub,c, Ziwei Wangb,c a
College of Earth Sciences & Engineering, Shandong University of Science and Technology, Qingdao 266590, China Key Laboratory of Coalbed Methane Resource and Reservoir Formation Process, Ministry of Education, Xuzhou 221008, China c School of Resources & Geosciences, China University of Mining and Technology, Xuzhou 221008, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Effective stress coefficient for permeability Water saturation Response surface method Stress sensitivity Dynamic change of permeability
The effect of effective stress on coal reservoir permeability is inevitable during coalbed methane (CBM) production. Studying the effective stress coefficient for permeability (ESCK) is crucial because it determines the effective stress and stress sensitivity of a coal reservoir. Considering the actual drainage process of a CBM well, a series of stress sensitivity experiments with different water saturations were designed and implemented. The experimental results show that the gas measured permeability of the coal sample decreased with increasing water saturation. The change in permeability with different water saturations had a significant power relationship with increasing net pressure. Using the response surface method (RSM), ESCK values with different water saturations were obtained. The ESCK value, regardless of the stress conditions, is a constant at a certain water saturation, and it shows an increasing trend with increasing water saturation. The ESCK values with different water saturations indicate that the effective stress decreases with increasing water saturation. The stress sensitivity and dynamic change in coal reservoir permeability were also analyzed using ESCK values with different water saturations. The stress sensitivity coefficient decreased with increased water saturation, and the reservoir permeability decreased more obviously with an ESCK value of 0.7675 than that with a value of 1.0. During CBM well production, water saturation decreases, and the stress sensitivity of coal reservoir permeability increases.
1. Introduction The concept of effective stress was first introduced by Terzaghi (1936) for the use in soil analyses, and it is commonly known as Terzagi's effective stress law (Dassanayake and Fujii, 2014). The effective stress law transforms external stress (σ) and pore pressure (p) into a single equivalent variable (σeffective). Biot and Wills (1957) suggested using an effective stress coefficient and modified the effective stress principle into a more widely applicable form, expressed as σeffective = σ – αp, where α is the effective stress coefficient (Zhao et al., 2011; Dassanayake et al., 2015). For porous media, every property such as permeability, deformation, strength, and storage capacity, has its own particular effective stress coefficient (Sang et al., 2017). The effective stress coefficient reflects the relative sensitivity of a particular property to σ and p. It is an important parameter in the geotechnical mechanics of porous media, particularly in determining the effective stress of the rock skeleton (Shen et al., 2017; Siggins et al., 2004). For oil and gas production operations, one of the most important
*
reservoir properties is permeability (Li et al., 2009; Liu et al., 2015; Zhang et al., 2017; Kudasik, 2019). Effective stress within the porefracture system increases during oil and gas production because of the reservoir depressurization (Shen et al., 2017; Salmachi et al., 2019). Thus, the pore-fracture system in the reservoir is compressed, and the permeability decreases (Moore, 2012; Wang et al., 2015). Many scholars have studied the effective stress coefficient for permeability (ESCK) of different rock types, such as granite, synthetic samples, shale and sandstone. The experimental results can be divided into three categories of ESCK values: 1.0, less than 1.0 and greater than 1.0 (Li et al., 2009; Zhao et al., 2011; Qiao et al., 2012; Moghadam et al., 2016; Glubokovskikh and Gurevich, 2015). In general, the ESCK value is less than 1.0 (Li et al., 2009; Dassanayake et al., 2015; Vasquez et al., 2015; Shen et al., 2017). Wang et al. (2015) also found that the reservoir ESCK values of fine sandstone are affected by fluid media. The ESCK value was measured to be the smallest by injecting nitrogen and the largest by injecting water, with brine solution falling in between. The experimental analysis methods for calculating ESCK values
Corresponding author. 579 Qianwangang Road, Huangdao District, Qingdao, Shandong Province, 266590, China. E-mail address: [email protected] (X. Zhang).
https://doi.org/10.1016/j.petrol.2019.106282 Received 10 January 2019; Received in revised form 2 July 2019; Accepted 15 July 2019 Available online 16 July 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.
Journal of Petroleum Science and Engineering 182 (2019) 106282
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mainly include differentiation, cross-plot, and the response surface method. Kranz et al. (1979) found that permeability was not a simple function of the difference between external confining pressure and internal fluid pressure. Changes in permeability were found to be proportional to (b dσ – a dp), where a and b are constants. Walsh (1981) obtained ESCK values by plotting the permeability isogram under different confining pressures and pore pressures. Bernabé (1987) derived the differential equation between permeability and confining pressure and pore pressure. The ESCK value can be calculated by changing the confining pressure and pore pressure or maintaining the equivalent permeability during stress sensitivity tests. Warpinski and Teufel (1992) obtained the ESCK values of sandstones and chalk by applying the response surface technique. This approach is particularly useful for tight rocks with relaxation microcracks. Xiao et al. (2013) developed a new method for calculating the nonlinear effective pressure (the secant effective pressure). It was observed that the secant effective pressure better fit the results than those fit using the tangent effective pressure. Based on the experimental analysis of the effective stress for sandstone, it was found that the ESCK value is usually not fixed and that the effective stress equation is nonlinear. The ESCK value varies with confining pressure and pore pressure as well as with loading or unloading cycles (Li et al., 2009; Xiao et al., 2013; Moghadam et al., 2016; Shen et al., 2017). Many scholars have studied the stress sensitivity of coal permeability and found that the permeability decreases exponentially as the net stress increases (McKee et al., 1987; Shi and Durucan, 2004; Pan et al., 2010; Salmachi et al., 2019). However, accurate determinations of coal ESCK values have not been established. Chen et al. (2011) demonstrated that the effective stress coefficient for coal is not equal to unity based on the experimental observations of a series of gas flowthrough experiments. This study obtained the effective coefficients of 0.945 for core No. 01, and 0.842 and 0.855 for No. 02 at pore pressures of 2.0 and 3.0 MPa, respectively. Coal reservoir permeability varies significantly during coalbed methane (CBM) production. The permeability decreases with increasing effective stress during the depressurization stage (saturated water single-phase flow). While the gas desorbs, the reservoir phase state becomes two phase – gas and water. The permeability will be affected by the effective stress, matrix shrinkage and gas slippage (Gray, 1987; Palmer and Mansoori, 1998; George and Barakat, 2001; Shi and Durucan, 2005; Cui and Bustin, 2005; Tao et al., 2012; Pan and Connell, 2012; Zhao et al., 2014; Chen et al., 2015; Connell, 2016). Many researchers have investigated the dynamic change of permeability and its controlling mechanisms through physical simulation experiments and theoretical derivation (Li et al., 2014; Chen et al., 2015). A number of models (strain-based simulation, stress-based simulation and production data-based inversion) have been developed to predict changes in coal permeability during CBM production. The strain-based simulation models are built by converting the effects of stress into volumetric strain of the coal matrix (Levine, 1996; Palmer and Mansoori, 1998; Xu et al., 2014). The stress-based simulation models are established by introducing the coal matrix shrinkage into the change of effective stress (Shi and Durucan, 2005; Cui and Bustin, 2005; Robertson and Christiansen, 2006). The production data-based inversion models are proposed by combining the relationship between water production, gas production and reservoir permeability (Xu et al., 2014; Zhao et al., 2014; Chen et al., 2015). Extensive and in-depth studies indicate that coal reservoir permeability exhibits an asymmetric U type variation with descending firstly and then ascending during production. The effective stress is the only negative effect among the influence factors (effective stress, matrix shrinkage and gas slippage) of coal reservoir permeability during CBM production. It can significantly decrease permeability during the entire production process. Thus, it is important to recognize the dynamic variation of ESCK values for coal reservoirs during CBM production. In this paper, stress sensitivity experiments with different water
Fig. 1. The experimental instruments for the permeability tests.
saturations were designed based on actual variations in reservoir pressure and water saturation during CBM production. The effective stress effect was analyzed, and ESCK values were calculated using the RSM with different water saturations. Subsequently, the stress sensitivity and dynamic variation of reservoir permeability were analyzed using ESCK values with different water saturations. 2. Experimental measurements Gas permeability with different water saturations was measured by instrumentation designed to determine the physical properties of a rock sample (Fig. 1). The experimental instrumentation is composed of three parts: the gas supply part, the core control part (complete with a core holding unit and confining pressure pump) and the back-pressure control part. The experimental sample was collected from the Qudi mine, Jincheng City in Shanxi Province. The underlying parameters of the coal sample are shown in Table 1. A coal core was drilled parallel to the direction of the bedding plane. The ends of the core were polished to yield flat surfaces, and no obvious fractures were observed in the core (Fig. 2). All experiments were performed on one coal sample to ensure that the test results were comparable. A total of 18 sets of experiments were performed in this study. In each set of experiments, the pore pressure (Pp) remained constant. The change in gas permeability with different water saturations (0%, 20%, 40%, 55%, 70%, 85%) was measured with increasing confining pressure (Pc). The permeability measurements with different water saturations were performed according to the experimental scheme in Fig. 3 and the pressure setting were consistent. The water saturation of the coal sample was established by the core displacement method and the low temperature drying method. Water saturation was established by the following steps: 1. The coal sample was dried and then saturated with water of 8% salinity in a vacuum. The dry weight (m0) and saturated weight (ms) were recorded. 2. The saturated coal sample was placed in the core holding unit, and the gas supply (helium) was turned on to displace the water in the sample. The displacement pressure and confining pressure were set at 1 MPa and 3.5 MPa, respectively. During the displacement process, the weight of the coal sample (mc) was weighed continuously. Water saturation (Sw) was calculated by Eq. (1), and precise water saturation was obtained by controlling the coal sample weight (mc).
Sw =
ms − mc × 100% ms − m 0
(1)
3. Coal is strongly hydrophilic, and irreducible water saturation is usually high (Zhang et al., 2017). When the gas could not discharge the water in the coal sample (approximately 70% of the water saturation), water saturation was established by the low temperature (25 °C) drying method (the coal sample was put into a vacuum 2
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Table 1 Underlying parameters of the cylindrical sample. Sample
Length (cm)
QD-9
4.702
Diameter (cm)
2.466
Ro,
max
(%)
3.31
Industrial analysis Mad (%)
Ad (%)
Vdaf (%)
St,
0.82
15.09
7.66
0.37
d
(%)
Abbreviations: Ro, max = maximum reflectance of vitrinite; Mad = moisture content at air-dried basis; Ad = ash yield at dried basis; Vdaf = volatile matter yield at dried and ash free basis; St, d = total sulphur content at dried basis.
3. Experimental results The results of stress sensitivity experiments are shown in Table 2. The gas measured permeability shows a negative power relationship with the increasing of net pressure (confining pressure – pore pressure). In the initial stage of increasing net pressure, the permeability decreased substantially, and the stress sensitivity was strong. When the net pressure was greater than 6 MPa, the decrease permeability was reduced with increased net pressure (Fig. 4). Coal seam is a naturally fractured reservoir that consists of the matrix pore and fracture network (Laubach et al., 1998; Zhou et al., 2016; Wang et al., 2017; Du et al., 2018). Fluid flow was mainly affected by the fractures and pores with diameters greater than 100 nm (defined as seepage pores in coal) (Shi and Durucan, 2005; Zhang et al., 2017; Li et al., 2019). The fractures and pores were closed during the initial stage of increasing net pressure. Thus, the flow path was narrowed, substantially impacting on permeability. As shown in Fig. 4, the permeability was inconsistent under different cycles (1, 2 and 3 MPa of the pore pressure), indicating that the ESCK value of 1.0 is insufficient for calculating effective stress according to Terzagi's effective stress law. With the increase in water saturation, the gas measured permeability of the coal sample decreased under different pore pressures. The space of the pore-fracture system was occupied by water; thus, the higher the water saturation, the greater the impact on permeability. The permeability under different water saturations showed a significant power relationship with increasing net pressure (Fig. 5).
Fig. 2. Cylindrical sample for the permeability experiments.
4. Determination of the effective stress coefficient for permeability
Fig. 3. Experimental scheme of gas measured permeability at one water saturation.
The ESCK is essential for analyzing reservoir stress sensitivity, and many scholars have studied the variation of ESCK values of low permeability sandstone using RSM. It has been observed that the ESCK value of low permeability sandstone decreases with increasing confining pressure and decreases with decreasing pore pressure (Warpinski and Teufel, 1992; Li et al., 2009; Zhao et al., 2011; Xiao et al., 2013; Wang et al., 2015; Moghadam et al., 2016). This paper mainly used RSM to analyze the variability of the ESCK values of coal under different confining pressures, pore pressures and water saturations. The calculation steps of ESCK are as follows:
drying oven and weighed continuously, and then the precise water saturation was obtained by controlling the coal sample weight). The steady state method was used to measure the gas permeability of coal with different water saturation and the test gas was helium. The coal sample established by different water saturation was placed in the core holding unit, and a specified confining pressure (Pc) was applied on the coal sample by the confining pressure pump. The outlet was connected to the outside, and the gas supply was used to control the inlet pressure (Pi) of the specimen. A soap bubble flowmeter was used to measure gas flow. After forming steady state flow between the upper and lower surfaces of the coal sample, the gas measured permeability was obtained by Eq. (2) (Somerton et al., 1975; Jasinge et al., 2011; Xue et al., 2015).
Kg =
2QμLP0 × 103 A (Pi2 − P02)
1. Transform the permeability data to a simpler form so as to minimize the sum of squared residuals. Box and Draper (1987) described a technique (Box-Cox method) to optimize the transformation for a given set of data by means of an empirical, maximum-likelihood approach. Generally, transformations of the form k' = kλ were applied to the data, where k is experimental data, k' is transformed data, and λ is a power generally between −3 and +3, with λ = 0 being a log transformation (Warpinski and Teufel, 1992). The transformation formula of Box-Cox method is expressed in Eq. (3) and the determination was established by the maximum-likelihood approach.
(2)
where Kg is the gas measured permeability, mD; Q is the gas flow, cm3/ s; μ is the helium viscosity, mPa·s; L is the length of coal sample, cm; A is the sectional area of the coal sample, cm2; and P0 is the barometric pressure, 101.325 × 10−3 MPa.
λ
y (λ) =
3
⎧ y − 1, λ ≠ 0 ⎫ λ ⎨ log (y ), λ = 0 ⎬ ⎩ ⎭
(3)
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Table 2 Results of gas measured permeability with different water saturations. Water saturation (%)
Confining pressure (MPa)
Pore pressure (MPa)
Permeability (mD)
Confining pressure (MPa)
Pore pressure (MPa)
Permeability (mD)
Confining pressure (MPa)
Pore pressure (MPa)
Permeability (mD)
0
3.53 5.04 6.55 8.05 9.50 11.02 13.01 3.52 5.02 6.55 8.03 9.53 11.04 13.03 3.54 5.03 6.50 8.02 9.00 11.01 13.00 3.52 5.01 6.5 8 9.02 11.04 13.03 3.51 5 6.5 8 9.53 11.06 13.03 3.55 5.05 6.53 8.03 9.55 11 13.02
0.99 0.98 0.99 1.01 1.06 1.09 1.11 1.01 0.97 0.97 0.96 0.99 1.02 1.11 0.99 1.07 1.08 1.11 1.13 1.15 1.16 1 0.98 1.01 1.06 1.06 1.09 1.09 0.99 1 1.05 1.09 1.11 1.14 1.15 1 1.09 1.15 1.18 1.2 1.21 1.23
0.40498 0.11668 0.04273 0.02078 0.01242 0.00839 0.00533 0.26840 0.08153 0.03307 0.01735 0.01001 0.00634 0.00376 0.14609 0.05051 0.02360 0.01384 0.01009 0.00610 0.00410 0.09123 0.02924 0.01572 0.00962 0.00744 0.00465 0.00320 0.07377 0.02764 0.01190 0.00610 0.00348 0.00226 0.00140 0.02371 0.00739 0.00329 0.00161 0.00088 0.00049 0.00027
4.5 6.03 7.5 9.01 10.52 12.01
1.99 2 2.03 2.09 2.09 2.15
0.19559 0.06561 0.02982 0.01656 0.01008 0.00686
5.52 7.03 8.53 10.00 11.51 13.01
3.08 3.07 3.08 3.1 3.13 3.05
0.16937 0.06357 0.03024 0.01719 0.01091 0.00714
4.52 6.03 7.52 9.01 10.52 12.05
1.98 1.99 1.99 2.04 2.04 2.07
0.18347 0.06408 0.02785 0.01555 0.00981 0.00672
5.58 7.02 8.52 10.03 11.54 13
2.98 2.97 2.96 2.98 2.98 2.96
0.12642 0.05123 0.02535 0.01449 0.00910 0.00581
4.52 6.03 7.5 9.03 10.52 12.05
1.99 1.98 1.97 1.97 2.04 2.06
0.10977 0.04156 0.01900 0.01038 0.00671 0.00434
5.54 7.00 8.51 10.00 11.50 13.04
3.1 3.1 3.11 3.18 3.22 3.23
0.10809 0.04254 0.02164 0.01290 0.00872 0.00619
4.52 6.02 7.52 9.01 10.53 12.02
1.99 2.02 2.06 2.1 2.13 2.12
0.10239 0.03964 0.01954 0.01191 0.00811 0.00579
5.53 7.03 8.53 10.01 11.51 13.05
3.03 3.01 3.01 3 3.02 3.01
0.06666 0.02664 0.01340 0.00809 0.00523 0.00357
4.55 6 7.51 9.03 10.52 12.04
2.04 1.98 1.97 1.96 1.94 1.94
0.05944 0.02587 0.01419 0.00887 0.00626 0.00467
5.57 7.06 8.54 10.06 11.55 13.04
3.1 3.1 3.14 3.13 3.13 3.13
0.04738 0.02291 0.01330 0.00857 0.00602 0.00442
4.55 6.05 7.54 9.08 10.51 12.01
1.99 1.94 1.95 1.94 1.94 1.94
0.02815 0.01186 0.00644 0.00377 0.00247 0.00160
5.5 7.02 8.53 10.01 11.5 12.99
2.97 2.92 3.04 2.98 2.99 3
0.02726 0.01197 0.00664 0.00416 0.00275 0.00191
20
40
55
70
85
data of coal showed a binary linear regression with Pc and Pp. The transformed permeability can thus be fit by Eq. (6), and the ESCK value can be calculated by Eq. (7).
2. Match the transformed permeability data by quadric surface:
k′ = a1 + a2Pc + a3 Pp + a 4 P 2c + a5 Pc Pp + a 6 P 2p
(4)
k′ = a1 + a2 Pc + a3 Pp α=−
where a1, a2, a3, a4, a5, and a6 are coefficients; Pc is confining pressure, MPa; and Pp is pore pressure, MPa.
∂k / ∂Pp ∂k / ∂Pc
=−
a3 + a5 Pc + 2a6 Pp a2 + 2a4 Pc + a5 Pp
(7)
The ESCK value with different water saturations is a constant according to Eq. (7), and the results are shown in Table 3. The water in the coal was displaced by gas at 85% water saturation, and the test result was substantially affected; thus, the ESCK value at 85% water saturation was eliminated. The effective stress with different water saturations was calculated by σeffective = Pc – αpPp, where αp is the ESCK value. It was found that the effective stress obtained by the RSM was more coincident with the definition of effective stress. When the effective stress was constant, the permeability remained unchanged regardless of the variation in Pc and Pp (Fig. 6). With the increase of water saturation, the ESCK value showed an increasing trend (Fig. 7). This result indicates that the effective stress decreases with increasing water saturation under the same Pc and Pp. During the drainage process of the CBM well, when the reservoir pressure dropped below the critical gas production pressure, the gas began to desorb from the coal reservoir, and water saturation decreased. Thus, the effective stress in the coal reservoir increases during
3. Calculating the ESCK value under different Pc and Pp according to the definition of ESCK:
α=−
a3 a2
(6)
(5)
If a4, a5, and a6 are negligible, the ESCK value is a constant; if a2 equals a3 additionally, the ESCK value equals 1.0. The experimental results of permeability were first transformed and then matched using the RSM. The Box and Draper method suggested that an F value equal to 10 times the F-distribution percentage point is needed to ensure that the surface adequately fits the data (Warpinski and Teufel, 1992). Unlike the quadric relationship between transformed permeability and Pc and Pp of sandstone, the transformed permeability 4
Journal of Petroleum Science and Engineering 182 (2019) 106282
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Fig. 4. Test results of gas measured permeability with different water saturations.
Permeability and effective stress values obtained by the RSM were plotted on a scatter diagram, and the stress sensitivity coefficient was determined by a linear fit using the opposite of the slope value (Fig. 8). The stress sensitivity coefficients of coal with different water saturations were all greater than 1.0 (1.1464–1.4508), and decreased gradually with increasing water saturation. The results show that the stress sensitivity of coal increases with decreasing water saturation, and that coal has quite strong stress sensitivity (Table 4).
gas production, and the trend grows nonlinearly. 5. Discussion 5.1. Stress sensitivity evaluation of coal Stress sensitivity was mainly evaluated by the permeability damage rate and the value of the stress sensitivity coefficient (Xiao et al., 2016). In this paper, the stress sensitivity coefficient was used to evaluate the stress sensitivity of coal because the value obtained by this method has uniqueness.
Ss =
1 − (k / k 0)1/3 lg (σeff / σeff 0)
5.2. Dynamic change of coal reservoir permeability The fluid phase of the coal reservoir changes continuously during the drainage process of CBM well. The permeability also varies with different factors (effective stress, matrix shrinkage and gas slippage). At present, the effective stress coefficient is usually taken as 1.0 to study the dynamic change of permeability during the drainage process. The conclusions obtained in Section 3 and 4 have shown this to be insufficient to explain the gas-water two-phase flow stage when the gas
(8)
where Ss is the stress sensitivity coefficient of permeability, dimensionless; σeff is the effective stress, MPa; k is the permeability under σeff , mD; σeff 0 is the minimum of effective stress, MPa; and k 0 is the initial permeability under σeff 0 , mD. 5
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Fig. 5. Relationship between permeability and net pressure under different pore pressures.
production. Thus, the S&D model (Eq. (9)) was chose to analyze the influence of different ESCK values on dynamic change law of coal reservoir permeability.
Table 3 Conversion coefficients and ESCK values with different water saturations. Water saturation (%) 0 20 40 55 70
λ
−0.24 −0.21 −0.25 −0.34 −0.14
a1
−0.244 −0.587 −1.324 −1.3662 −2.2718
a2
−0.4351 −0.4287 −0.3852 1.5386 −0.456
a3
0.1964 0.2565 0.2956 −1.1127 0.4334
ESCK
0.4513 0.5982 0.7675 0.7233 0.9503
F
140.47 214.56 139.38 262.18 128.49
Ft
⎧ ⎪ ki = ⎨ 3Cf ⎪k0 e ⎩
2.67 2.67 2.67 2.67 2.67
3Cf ν
k 0 e 1−ν
(P − P0)
P ≥ Pj
⎛ ν (αi P − α 0 P0) + Eεmax ⎛ Pj − Pj ⎞ ⎟⎞ ⎜ 1−ν 3(1 − ν ) Pj + Pε P + Pε ⎝
⎜
⎟
⎝
⎠⎠
P < Pj
(9)
where P is the reservoir pressure, MPa; P0 is the initial reservoir pressure, MPa; ki is the dynamic permeability of the coal reservoir, 10−3μm2; k0 is the initial permeability of the coal reservoir, 10−3μm2; Cf is the cleat compressibility, MPa−1; ν is Poisson's ratio; αi is the ESCK value corresponding to P; α 0 is the ESCK value corresponding to P0; E is the elasticity modulus of coal, MPa; εmax is the maximum bulk swelling strain; Pε is the Langmuir pressure for the swelling isotherm, MPa; and Pj is the critical gas production pressure, MPa. In the stable period (mid and late periods) of CBM well production, gas production rises steadily, and water production is negligible (very low or none). The water saturation of the coal reservoir in this period can be regarded as irreducible water saturation. A CBM well (well Z8) near the Qudi mine was selected to analyze the dynamic change of coal reservoir permeability. Well Z8 is one of the typical high-yielding wells in the Zhengzhuang Block (Fig. 10), and it has an average gas and water production rate of 2127.71 m3/d and 1.20 m3/d, respectively (Fig. 11). The reservoir parameters of well Z8 are shown in Table 5. Gas cannot drive the water in the coal sample when the water saturation reaches 70%; thus, the irreducible water saturation is regarded
desorbs from the reservoir and water saturation decreases. The permeability of coal and effective stress show a significant power relationship (Fig. 6), as well as a negative exponential relationship (Fig. 9). The existence of slippage is good for increasing coal reservoir permeability. However, the gas slippage only indirectly increases the gas permeability and does not cause any change in reservoir properties. The effect of increasing permeability is relatively limited and the impact stage is difficult to quantify in actual production. Many researchers neglect the gas slippage in the modeling of dynamic change for coal reservoir permeability (Niu et al., 2018). Shi and Durucan (2005) developed a permeability model (S&D model) from the constitutive equations for isotropic linear poroelasticity. Using the exponential expression of the S&D model, the confining pressure can be eliminated in the computation of dynamic permeability analysis during the CBM 6
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Fig. 6. Effective stress law of coal with different water saturations.
permeability (such as the sharp decrease of reservoir pressure and water saturation).
as 70% in the dynamic permeability calculation of well Z8. Dynamic permeability with the ESCK values of 1.0 and 0.7675 was calculated using Eq. (7) (Fig. 11). The results show that the reservoir permeability decreases more with an ESCK value of 0.7675 than that with a value of 1.0 when the reservoir pressure decreases. The results indicate that the decrease of water saturation affects the seepage capability of the coal reservoir during the CBM production. However, the matrix shrinkage effect will increase the permeability when the reservoir pressure decreases to a certain extent. When the matrix shrinkage effect plays a leading role, the permeability of the coal reservoir starts to increase continuously, and the permeability difference between ESCK values of 0.7675 and 1.0 is reduced (Fig. 12). During the drainage process, the drainage rate should be controlled to avoid serious damage to reservoir
6. Conclusions 1. Gas measured permeability shows a negative power relationship with increasing net pressure. With the increase of water saturation, the gas measured permeability of the coal sample decreases under different pore pressures. The permeability with different water saturations also has a significant power relationship with increasing net pressure. 2. The effective stress coefficient for permeability obtained by the response surface method with different water saturations is a constant, 7
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Fig. 7. The relationship between ESCK and water saturation.
Fig. 8. Stress sensitive coefficients of coal permeability with different water saturations.
Fig. 10. Location of well Z8.
Table 4 The evaluation criterion of stress sensitivity using stress sensitivity coefficient (Xiao et al., 2016). Stress sensitivity coefficient
Ss < 0.3
0.3≤Ss ≤ 0.7
0.7≤Ss ≤ 1.0
Ss > 1.0
Sensitivity degree
Weak
Medium
Strong
Quite strong
Fig. 11. Production curve of well Z8.
and it shows an increasing trend with the increase of water saturation. 3. The stress sensitivity coefficient decreases with increasing water saturation, and the coal has strong stress sensitivity. The dynamic change of coal reservoir permeability can be more accurately analyzed using the effective stress coefficient for permeability value obtained by the response surface method. The reservoir permeability decreases more with an effective stress coefficient of 0.7675 than that with a value of 1.0 when the reservoir pressure decreases. The change of water saturation of the coal reservoir has a significant influence on permeability during the coalbed methane well production. The stress sensitivity of coal reservoir permeability becomes stronger with decreasing water saturation.
Fig. 9. The negative exponential relationship between permeability and effective stress.
Acknowledgements This work was supported by the Natural Science Foundation of 8
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Table 5 The reservoir parameters of well Z8. Thickness (m)
Burial depth (m)
Reservoir pressure (MPa)
Permeability (mD)
Pj (MPa)
Cf (MPa−1)
ν
E (MPa)
εmax
Pε (MPa)
5.28
702.21
3.96
1.75
3.13
0.143
0.32
1220
0.008
1.36
Gray, I., 1987. Reservoir engineering in coal seams: Part I-The physical process of gas storage and movement in coal seams. SPE Reserv. Eng. 2 (1), 28–34. https://doi.org/ 10.2118/12514-PA. Jasinge, D., Ranjith, P.G., Choi, S.K., 2011. Effects of effective stress changes on permeability of latrobe valley brown coal. Fuel 90, 1292–1300. http://doi.org/10.1016/j. fuel.2010.10.053. Kranz, R.L., Frankel, A.D., Engelder, T., Scholz, C.H., 1979. The permeability of whole and jointed Barre granite. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 16, 225–234. http://doi.org/10.1016/0148-9062(79)91197-5. Kudasik, M., 2019. Investigating permeability of coal samples of various porosities under stress conditions. Energies 12 (4), 762. http://doi.org/10.3390/en12040762. Laubach, S.E., Marrett, R.A., Olson, J.E., Scott, A.R., 1998. Characteristics and origins of coal cleat: a review. Int. J. Coal Geol. 35, 175–207. http://doi.org/10.1016/S01665162(97)00012-8. Levine, J.R., 1996. Model study of the influence of matrix shrinkage on absolute permeability of coalbed reservoirs. Geol. Soc. 109, 197–212. http://doi.org/10.1144/ GSL.SP.1996.109.01.14. Li, M., Bernabé, Y., Xiao, W.I., Chen, Z.Y., Liu, Z.Q., 2009. Effective pressure law for permeability of E-bei sandstones. J. Geophys. Res. 114, B07205. https://doi.org/10. 1029/2009JB006373. Li, X., Fu, X.H., Ranjith, P.G., Xu, J., 2019. Stress sensitivity of medium- and high volatile bituminous coal: an experimental study based on nuclear magnetic resonance and permeability-porosity tests. J. Pet. Sci. Eng. 172, 889–910. https://doi.org/10.1016/ j.petrol.2018.08.081. Li, Y., Tang, D.Z., Xu, H., Meng, Y.J., Li, J.Q., 2014. Experimental research on coal permeability: the roles of effective stress and gas slippage. J. Nat. Gas Sci. Eng. 21, 481–488. https://doi.org/10.1016/j.jngse.2014.09.004. Liu, B., Du, J.Y., Qi, Y.G., Zhang, F.N., Yu, Y.Q., Zhao, H.H., 2015. A new coal particles cleanout technology in coalbed methane wells in China. J. Pet. Sci. Eng. 127, 445–451. http://doi.org/10.1016/j.petrol.2015.01.042. McKee, C.R., Bumb, A.C., Koenig, R.A., 1987. Stress-dependent permeability and porosity of coal and other geologic formations. SPE Form. Eval. 3, 183–193. https://doi.org/ 10.2118/12858-PA. Moghadam, J.N., Mondol, N.H., Aagaard, P., Hellevang, H., 2016. Effective stress law for the permeability of clay-bearing sandstones by the Modified Clay Shell model. Greenh. Gases: Sci. Technol. 0, 1–23. https://doi.org/10.1002/ghg. Moore, T.A., 2012. Coalbed methane: a review. Int. J. Coal Geol. 101, 36–81. http://doi. org/10.1016/j.coal.2012.05.011. Niu, Y.F., Mostaghimi, P., Shikhov, I., Chen, Z.X., Armstrong, R.T., 2018. Coal permeability: gas slippage linked to permeability rebound. Fuel 215, 844–852. http://doi. org/10.1016/j.fuel.2017.11.082. Palmer, I., Mansoori, J., 1998. How permeability depends on stress and pore pressure in coalbeds: a new model. SPE Reserv. Eval. Eng. 1 (6), 557–563. https://doi.org/10. 2118/52607-PA. Pan, Z., Connell, L.D., Camilleri, M., 2010. Laboratory characterisation of coal reservoir permeability for primary and enhanced coalbed methane recovery. Int. J. Coal Geol. 82, 252–261. https://doi.org/10.1016/j.coal.2009.10.019. Pan, Z.J., Connell, L.D., 2012. Modelling permeability for coal reservoirs: a review of analytical models and testing data. Int. J. Coal Geol. 92, 1–44. https://doi.org/10. 1016/j.coal.2011.12.009. Qiao, L.P., Wong, R.C.K., Aguilera, R., Kantzas, A., 2012. Determination of biot's effectivestress coefficient for permeability of Nikanassin Sandstone. J. Can. Pet. Technol. 51 (03), 193–197. http://doi.org/10.2118/150820-PA. Robertson, E.P., Christiansen, R.L., 2006. A permeability model for coal and other fractured sorptive-elastic media. In: SPE Eastern Regional Meeting, 1-13 October, Canton, Ohio, USA, . http://doi.org/10.2118/104380-MS. Salmachi, A., Dunlop, E., Rajabi, M., Yarmohammadtooski, Z., Begg, S., 2019. Investigation of permeability change in ultra deep coal seams using time-lapse pressure transient analysis: a pilot project in the Cooper Basin, Australia. AAPG Bull. 103 (1), 91–107. http://doi.org/10.1306/05111817277. Sang, G.J., Elsworth, D., Liu, S.M., Harpalani, S., 2017. Characterization of swelling modulus and effective stress coefficient accommodating sorption-induced swelling in coal. Energy Fuels 31, 8843–8851. http://doi.org/10.1021/acs.energyfuels.7b00462. Shen, Y.H., Luan, G.H., Zhang, H.Y., Liu, Q., Zhang, J.J., Ge, H.K., 2017. Novel method for calculating the effective stress coefficient in a tight sandstone reservoir. KSCE J. Civ. Eng. 21 (6), 2467–2475. http://doi.org/10.1007/s12205-016-0514-5. Shi, J.Q., Durucan, S., 2004. Drawdown induced changes in permeability of coalbeds: a new interpretation of the reservoir response to primary recovery. Transp. Porous Media 56, 1–16. https://doi.org/10.1023/B:TIPM.0000018398.19928.5a. Shi, J.Q., Durucan, S.A., 2005. A model for changes in coalbed permeability during primary and enhanced methane recovery. SPE Reserv. Eval. Eng. 8 (4), 291–299. https://doi.org/10.2118/87230-PA. Siggins, A.F., Dewhurst, D.N., Dodds, K., 2004. Effective stress and the Biot-Willis coefficient for reservoir sandstones. In: SEG Annual Meeting, 10-15 October, Denver, Colorado, . http://doi.org/10.1190/1.1845168. Somerton, W.H., Söylemezoḡlu, I.M., Dudley, R.C., 1975. Effect of stress on permeability of coal. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 12 (5–6), 129–145. http://doi.
Fig. 12. Dynamic permeability change during coalbed methane production.
China (41272178, 41572140), the National Major Special Project of Science and Technology of China (2016ZX05044001), the Fundamental Research Funds for the Central Universities (2014ZDPY26, 2015XKZD07), and the Qing Lan Project. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.petrol.2019.106282. References Bernabé, Y., 1987. The effective pressure law for permeability in Chelmsford granite and Barre granite. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 23 (3), 267–275. http:// doi.org/10.1016/0148-9062(86)90972-1. Biot, M.A., Wills, D.G., 1957. The elastic coefficients of the theory of consolidation. ASME J. Appl. Mech. 24, 594–601. Box, G.P., Draper, N.R., 1987. Empirical Model-Building and Response Surfaces. John Wiley & Sons Inc., New York City. Chen, Y.X., Liu, D.M., Yao, Y.B., Cai, Y.D., Chen, L.W., 2015. Dynamic permeability change during coalbed methane production and its controlling factors. J. Nat. Gas Sci. Eng. 25, 335–346. https://doi.org/10.1016/j.jngse.2015.05.018. Chen, Z.W., Pan, Z.J., Liu, J.S., Connell, L.D., Elsworth, D., 2011. Effect of the effective stress coefficient and sorption-induced strain on the evolution of coal permeability: experimental observations. Int. J. Greenh. Gas Control 5, 1284–1293. https://doi. org/10.1016/j.ijggc.2011.07.005. Connell, L.D., 2016. A new interpretation of the response of coal permeability to changes in pore pressure, stress and matrix shrinkage. Int. J. Coal Geol. 162, 169–182. http:// doi.org/10.1016/j.coal.2016.06.012. Cui, X., Bustin, R.M., 2005. Volumetric strain associated with methane desorption and its impact on coalbed gas production from deep coal seams. Am. Assoc. Pet. Geol. Bull. 89 (9), 1181–1202. https://doi.org/10.1306/05110504114. Dassanayake, N., Fujii, Y., 2014. Biot's effective stress coefficient of rocks for peak and residual strengths by modified failure envelope method. Rock Eng. Rock Mech. Struct. Rock Masses 5, 155–160. Dassanayake, N., Fujii, Y., Fukuda, D., Kodama, J., 2015. A new approach to evaluate effective stress coefficient for strength in Kimachi sandstone. J. Pet. Sci. Eng. 131, 70–79. https://dx.doi.org/j.petrol.2015.04.015. Du, W., Zhang, Y., Meng, X., Zhang, X., Li, W., 2018. Deformation and seepage characteristics of gas-containing coal under true triaxial stress. Arab. J. Geosci. 11, 190. http://doi.org/10.1007/s12517-018-3543-1. George, J.D.S., Barakat, M.A., 2001. The change in effective stress associated with shrinkage from gas desorption in coal. Int. J. Coal Geol. 45, 105–113. http://doi.org/ 10.1016/S0166-5162(00)00026-4. Glubokovskikh, S., Gurevich, B., 2015. Effect of micro-inhomogeneity on the effective stress coefficients and undrained bulk modulus of a poroelastic medium: a double spherical shell model. Geophys. Prospect. 63 (3), 656–668. http://doi.org/10.1111/ 1365-2478.12222.
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Xiao, W.L., Li, M., Zhao, J.Z., Zhao, S.X., 2013. Calculation of non-linear effective pressure. Chin. J. Geophys. 56 (8), 2808–2817. http://doi.org/10.6038/cjg20130829. Xiao, W.L., Li, T., Li, M., Zhao, J.Z., Zheng, L.L., Li, L., 2016. Evaluation of the stress sensitivity in tight reservoirs. Pet. Explor. Dev. 43 (1), 115–123. https://doi.org/10. 1016/S1876-3804(16)30013-1. Xu, H., Tang, D.Z., Tang, S.H., Zhao, J.L., Meng, Y.J., Tao, S., 2014. A dynamic prediction model for gas–water effective permeability based on coalbed methane production data. Int. J. Coal Geol. 121, 44–52. https://doi.org/10.1016/j.coal.2013.11.008. Xue, Y., Gao, F., Liu, X.G., 2015. Effect of damage evolution of coal on permeability variation and analysis of gas outburst hazard with coal mining. Nat. Hazards 79 (2), 999–1013. http://doi.org/10.1007/s11069-015-1888-2. Zhang, X.Y., Wu, C.F., Liu, S.X., 2017. Characteristic analysis and fractal model of the gaswater relative permeability of coal under different confining pressures. J. Pet. Sci. Eng. 159, 488–496. http://doi.org/10.1016/j.petrol.2017.09.057. Zhao, J.L., Tang, D.Z., Xu, H., Meng, Y.J., Lv, Y.M., Tao, S., 2014. A dynamic prediction model for gas-water effective permeability in unsaturated coalbed methane reservoirs based on production data. J. Nat. Gas Sci. Eng. 21, 496–506. https://doi.org/10. 1016/j.jngse.2014.09.014. Zhao, J.Z., Xiao, W.L., Li, M., Xiang, Z.P., Li, L.J., Wang, J., 2011. The effective pressure law for permeability of clay-rich sandstones. Pet. Sci. 8, 194–199. http://doi.org/10. 1007/s12182-011-0134-0. Zhou, G., Zhang, Q., Bai, R., Ni, G., 2016. Characterization of coal micro-pore structure and simulation on the seepage rules of low-pressure water based on CT Scanning data. Minerals 6 (3), 1–16. http://doi.org/10.3390/min6030078.
org/10.1016/0148-9062(75)91244-9. Tao, S., Wang, Y.B., Tang, D.Z., Xu, H., Lv, Y.M., He, W., Li, Y., 2012. Dynamic variation effects of coal permeability during the coalbed methane development process in the Qinshui Basin, China. Int. J. Coal Geol. 93, 16–22. https://doi.org/10.1016/j.coal. 2012.01.006. Terzaghi, K., 1936. The shearing resistance of saturated soils and the angle between planes of shear. In: First International Conference of Soil Mechanics and Foundation Engineering. Harvard University Press, Cambridge, Massachusetts. Vasquez, G.F., Morschbaher, M.J., Justen, J., Silveira, A.J., 2015. New experimental results on dynamic biot coefficient on Brazilian reservoir rocks. In: the 14th International Congress of the Brazilian Geophysical Society, Rio de Janeiro, Brazil, August 3-6, . http://doi.org/10.1190/sbgf2015-170. Walsh, J.B., 1981. Effect of pore pressure and confining pressure on fracture permeability. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 18, 429–435. http://doi.org/10.1016/ 10.1016/0148-9062(81)90006-1. Wang, C., Zhai, P., Chen, Z., Liu, J., Wang, L., Xie, J., 2017. Experimental study of coal matrix-cleat interaction under constant volume boundary condition. Int. J. Coal Geol. 181, 124–132. http://doi.org/10.1016/j.coal.2017.08.014. Wang, Y., Li, G.Z., Li, M., Zhang, J., 2015. The applicability of different fluid media to measure effective stress coefficient for rock permeability. J. Chem. 11, 391851. http://doi.org/10.1155/2015/391851. Warpinski, N.R., Teufel, L.W., 1992. Determination of the effective stress law for permeability and deformation in low-permeability rocks. SPE Form. Eval. 7 (2), 123–131. http://doi.org/10.2118/20572-PA.
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Experimental study of the effective stress coefficient for coal permeability with different water saturations
T
Xiaoyang Zhanga,*, Caifang Wub,c, Ziwei Wangb,c a
College of Earth Sciences & Engineering, Shandong University of Science and Technology, Qingdao 266590, China Key Laboratory of Coalbed Methane Resource and Reservoir Formation Process, Ministry of Education, Xuzhou 221008, China c School of Resources & Geosciences, China University of Mining and Technology, Xuzhou 221008, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Effective stress coefficient for permeability Water saturation Response surface method Stress sensitivity Dynamic change of permeability
The effect of effective stress on coal reservoir permeability is inevitable during coalbed methane (CBM) production. Studying the effective stress coefficient for permeability (ESCK) is crucial because it determines the effective stress and stress sensitivity of a coal reservoir. Considering the actual drainage process of a CBM well, a series of stress sensitivity experiments with different water saturations were designed and implemented. The experimental results show that the gas measured permeability of the coal sample decreased with increasing water saturation. The change in permeability with different water saturations had a significant power relationship with increasing net pressure. Using the response surface method (RSM), ESCK values with different water saturations were obtained. The ESCK value, regardless of the stress conditions, is a constant at a certain water saturation, and it shows an increasing trend with increasing water saturation. The ESCK values with different water saturations indicate that the effective stress decreases with increasing water saturation. The stress sensitivity and dynamic change in coal reservoir permeability were also analyzed using ESCK values with different water saturations. The stress sensitivity coefficient decreased with increased water saturation, and the reservoir permeability decreased more obviously with an ESCK value of 0.7675 than that with a value of 1.0. During CBM well production, water saturation decreases, and the stress sensitivity of coal reservoir permeability increases.
1. Introduction The concept of effective stress was first introduced by Terzaghi (1936) for the use in soil analyses, and it is commonly known as Terzagi's effective stress law (Dassanayake and Fujii, 2014). The effective stress law transforms external stress (σ) and pore pressure (p) into a single equivalent variable (σeffective). Biot and Wills (1957) suggested using an effective stress coefficient and modified the effective stress principle into a more widely applicable form, expressed as σeffective = σ – αp, where α is the effective stress coefficient (Zhao et al., 2011; Dassanayake et al., 2015). For porous media, every property such as permeability, deformation, strength, and storage capacity, has its own particular effective stress coefficient (Sang et al., 2017). The effective stress coefficient reflects the relative sensitivity of a particular property to σ and p. It is an important parameter in the geotechnical mechanics of porous media, particularly in determining the effective stress of the rock skeleton (Shen et al., 2017; Siggins et al., 2004). For oil and gas production operations, one of the most important
*
reservoir properties is permeability (Li et al., 2009; Liu et al., 2015; Zhang et al., 2017; Kudasik, 2019). Effective stress within the porefracture system increases during oil and gas production because of the reservoir depressurization (Shen et al., 2017; Salmachi et al., 2019). Thus, the pore-fracture system in the reservoir is compressed, and the permeability decreases (Moore, 2012; Wang et al., 2015). Many scholars have studied the effective stress coefficient for permeability (ESCK) of different rock types, such as granite, synthetic samples, shale and sandstone. The experimental results can be divided into three categories of ESCK values: 1.0, less than 1.0 and greater than 1.0 (Li et al., 2009; Zhao et al., 2011; Qiao et al., 2012; Moghadam et al., 2016; Glubokovskikh and Gurevich, 2015). In general, the ESCK value is less than 1.0 (Li et al., 2009; Dassanayake et al., 2015; Vasquez et al., 2015; Shen et al., 2017). Wang et al. (2015) also found that the reservoir ESCK values of fine sandstone are affected by fluid media. The ESCK value was measured to be the smallest by injecting nitrogen and the largest by injecting water, with brine solution falling in between. The experimental analysis methods for calculating ESCK values
Corresponding author. 579 Qianwangang Road, Huangdao District, Qingdao, Shandong Province, 266590, China. E-mail address: [email protected] (X. Zhang).
https://doi.org/10.1016/j.petrol.2019.106282 Received 10 January 2019; Received in revised form 2 July 2019; Accepted 15 July 2019 Available online 16 July 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.
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mainly include differentiation, cross-plot, and the response surface method. Kranz et al. (1979) found that permeability was not a simple function of the difference between external confining pressure and internal fluid pressure. Changes in permeability were found to be proportional to (b dσ – a dp), where a and b are constants. Walsh (1981) obtained ESCK values by plotting the permeability isogram under different confining pressures and pore pressures. Bernabé (1987) derived the differential equation between permeability and confining pressure and pore pressure. The ESCK value can be calculated by changing the confining pressure and pore pressure or maintaining the equivalent permeability during stress sensitivity tests. Warpinski and Teufel (1992) obtained the ESCK values of sandstones and chalk by applying the response surface technique. This approach is particularly useful for tight rocks with relaxation microcracks. Xiao et al. (2013) developed a new method for calculating the nonlinear effective pressure (the secant effective pressure). It was observed that the secant effective pressure better fit the results than those fit using the tangent effective pressure. Based on the experimental analysis of the effective stress for sandstone, it was found that the ESCK value is usually not fixed and that the effective stress equation is nonlinear. The ESCK value varies with confining pressure and pore pressure as well as with loading or unloading cycles (Li et al., 2009; Xiao et al., 2013; Moghadam et al., 2016; Shen et al., 2017). Many scholars have studied the stress sensitivity of coal permeability and found that the permeability decreases exponentially as the net stress increases (McKee et al., 1987; Shi and Durucan, 2004; Pan et al., 2010; Salmachi et al., 2019). However, accurate determinations of coal ESCK values have not been established. Chen et al. (2011) demonstrated that the effective stress coefficient for coal is not equal to unity based on the experimental observations of a series of gas flowthrough experiments. This study obtained the effective coefficients of 0.945 for core No. 01, and 0.842 and 0.855 for No. 02 at pore pressures of 2.0 and 3.0 MPa, respectively. Coal reservoir permeability varies significantly during coalbed methane (CBM) production. The permeability decreases with increasing effective stress during the depressurization stage (saturated water single-phase flow). While the gas desorbs, the reservoir phase state becomes two phase – gas and water. The permeability will be affected by the effective stress, matrix shrinkage and gas slippage (Gray, 1987; Palmer and Mansoori, 1998; George and Barakat, 2001; Shi and Durucan, 2005; Cui and Bustin, 2005; Tao et al., 2012; Pan and Connell, 2012; Zhao et al., 2014; Chen et al., 2015; Connell, 2016). Many researchers have investigated the dynamic change of permeability and its controlling mechanisms through physical simulation experiments and theoretical derivation (Li et al., 2014; Chen et al., 2015). A number of models (strain-based simulation, stress-based simulation and production data-based inversion) have been developed to predict changes in coal permeability during CBM production. The strain-based simulation models are built by converting the effects of stress into volumetric strain of the coal matrix (Levine, 1996; Palmer and Mansoori, 1998; Xu et al., 2014). The stress-based simulation models are established by introducing the coal matrix shrinkage into the change of effective stress (Shi and Durucan, 2005; Cui and Bustin, 2005; Robertson and Christiansen, 2006). The production data-based inversion models are proposed by combining the relationship between water production, gas production and reservoir permeability (Xu et al., 2014; Zhao et al., 2014; Chen et al., 2015). Extensive and in-depth studies indicate that coal reservoir permeability exhibits an asymmetric U type variation with descending firstly and then ascending during production. The effective stress is the only negative effect among the influence factors (effective stress, matrix shrinkage and gas slippage) of coal reservoir permeability during CBM production. It can significantly decrease permeability during the entire production process. Thus, it is important to recognize the dynamic variation of ESCK values for coal reservoirs during CBM production. In this paper, stress sensitivity experiments with different water
Fig. 1. The experimental instruments for the permeability tests.
saturations were designed based on actual variations in reservoir pressure and water saturation during CBM production. The effective stress effect was analyzed, and ESCK values were calculated using the RSM with different water saturations. Subsequently, the stress sensitivity and dynamic variation of reservoir permeability were analyzed using ESCK values with different water saturations. 2. Experimental measurements Gas permeability with different water saturations was measured by instrumentation designed to determine the physical properties of a rock sample (Fig. 1). The experimental instrumentation is composed of three parts: the gas supply part, the core control part (complete with a core holding unit and confining pressure pump) and the back-pressure control part. The experimental sample was collected from the Qudi mine, Jincheng City in Shanxi Province. The underlying parameters of the coal sample are shown in Table 1. A coal core was drilled parallel to the direction of the bedding plane. The ends of the core were polished to yield flat surfaces, and no obvious fractures were observed in the core (Fig. 2). All experiments were performed on one coal sample to ensure that the test results were comparable. A total of 18 sets of experiments were performed in this study. In each set of experiments, the pore pressure (Pp) remained constant. The change in gas permeability with different water saturations (0%, 20%, 40%, 55%, 70%, 85%) was measured with increasing confining pressure (Pc). The permeability measurements with different water saturations were performed according to the experimental scheme in Fig. 3 and the pressure setting were consistent. The water saturation of the coal sample was established by the core displacement method and the low temperature drying method. Water saturation was established by the following steps: 1. The coal sample was dried and then saturated with water of 8% salinity in a vacuum. The dry weight (m0) and saturated weight (ms) were recorded. 2. The saturated coal sample was placed in the core holding unit, and the gas supply (helium) was turned on to displace the water in the sample. The displacement pressure and confining pressure were set at 1 MPa and 3.5 MPa, respectively. During the displacement process, the weight of the coal sample (mc) was weighed continuously. Water saturation (Sw) was calculated by Eq. (1), and precise water saturation was obtained by controlling the coal sample weight (mc).
Sw =
ms − mc × 100% ms − m 0
(1)
3. Coal is strongly hydrophilic, and irreducible water saturation is usually high (Zhang et al., 2017). When the gas could not discharge the water in the coal sample (approximately 70% of the water saturation), water saturation was established by the low temperature (25 °C) drying method (the coal sample was put into a vacuum 2
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Table 1 Underlying parameters of the cylindrical sample. Sample
Length (cm)
QD-9
4.702
Diameter (cm)
2.466
Ro,
max
(%)
3.31
Industrial analysis Mad (%)
Ad (%)
Vdaf (%)
St,
0.82
15.09
7.66
0.37
d
(%)
Abbreviations: Ro, max = maximum reflectance of vitrinite; Mad = moisture content at air-dried basis; Ad = ash yield at dried basis; Vdaf = volatile matter yield at dried and ash free basis; St, d = total sulphur content at dried basis.
3. Experimental results The results of stress sensitivity experiments are shown in Table 2. The gas measured permeability shows a negative power relationship with the increasing of net pressure (confining pressure – pore pressure). In the initial stage of increasing net pressure, the permeability decreased substantially, and the stress sensitivity was strong. When the net pressure was greater than 6 MPa, the decrease permeability was reduced with increased net pressure (Fig. 4). Coal seam is a naturally fractured reservoir that consists of the matrix pore and fracture network (Laubach et al., 1998; Zhou et al., 2016; Wang et al., 2017; Du et al., 2018). Fluid flow was mainly affected by the fractures and pores with diameters greater than 100 nm (defined as seepage pores in coal) (Shi and Durucan, 2005; Zhang et al., 2017; Li et al., 2019). The fractures and pores were closed during the initial stage of increasing net pressure. Thus, the flow path was narrowed, substantially impacting on permeability. As shown in Fig. 4, the permeability was inconsistent under different cycles (1, 2 and 3 MPa of the pore pressure), indicating that the ESCK value of 1.0 is insufficient for calculating effective stress according to Terzagi's effective stress law. With the increase in water saturation, the gas measured permeability of the coal sample decreased under different pore pressures. The space of the pore-fracture system was occupied by water; thus, the higher the water saturation, the greater the impact on permeability. The permeability under different water saturations showed a significant power relationship with increasing net pressure (Fig. 5).
Fig. 2. Cylindrical sample for the permeability experiments.
4. Determination of the effective stress coefficient for permeability
Fig. 3. Experimental scheme of gas measured permeability at one water saturation.
The ESCK is essential for analyzing reservoir stress sensitivity, and many scholars have studied the variation of ESCK values of low permeability sandstone using RSM. It has been observed that the ESCK value of low permeability sandstone decreases with increasing confining pressure and decreases with decreasing pore pressure (Warpinski and Teufel, 1992; Li et al., 2009; Zhao et al., 2011; Xiao et al., 2013; Wang et al., 2015; Moghadam et al., 2016). This paper mainly used RSM to analyze the variability of the ESCK values of coal under different confining pressures, pore pressures and water saturations. The calculation steps of ESCK are as follows:
drying oven and weighed continuously, and then the precise water saturation was obtained by controlling the coal sample weight). The steady state method was used to measure the gas permeability of coal with different water saturation and the test gas was helium. The coal sample established by different water saturation was placed in the core holding unit, and a specified confining pressure (Pc) was applied on the coal sample by the confining pressure pump. The outlet was connected to the outside, and the gas supply was used to control the inlet pressure (Pi) of the specimen. A soap bubble flowmeter was used to measure gas flow. After forming steady state flow between the upper and lower surfaces of the coal sample, the gas measured permeability was obtained by Eq. (2) (Somerton et al., 1975; Jasinge et al., 2011; Xue et al., 2015).
Kg =
2QμLP0 × 103 A (Pi2 − P02)
1. Transform the permeability data to a simpler form so as to minimize the sum of squared residuals. Box and Draper (1987) described a technique (Box-Cox method) to optimize the transformation for a given set of data by means of an empirical, maximum-likelihood approach. Generally, transformations of the form k' = kλ were applied to the data, where k is experimental data, k' is transformed data, and λ is a power generally between −3 and +3, with λ = 0 being a log transformation (Warpinski and Teufel, 1992). The transformation formula of Box-Cox method is expressed in Eq. (3) and the determination was established by the maximum-likelihood approach.
(2)
where Kg is the gas measured permeability, mD; Q is the gas flow, cm3/ s; μ is the helium viscosity, mPa·s; L is the length of coal sample, cm; A is the sectional area of the coal sample, cm2; and P0 is the barometric pressure, 101.325 × 10−3 MPa.
λ
y (λ) =
3
⎧ y − 1, λ ≠ 0 ⎫ λ ⎨ log (y ), λ = 0 ⎬ ⎩ ⎭
(3)
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Table 2 Results of gas measured permeability with different water saturations. Water saturation (%)
Confining pressure (MPa)
Pore pressure (MPa)
Permeability (mD)
Confining pressure (MPa)
Pore pressure (MPa)
Permeability (mD)
Confining pressure (MPa)
Pore pressure (MPa)
Permeability (mD)
0
3.53 5.04 6.55 8.05 9.50 11.02 13.01 3.52 5.02 6.55 8.03 9.53 11.04 13.03 3.54 5.03 6.50 8.02 9.00 11.01 13.00 3.52 5.01 6.5 8 9.02 11.04 13.03 3.51 5 6.5 8 9.53 11.06 13.03 3.55 5.05 6.53 8.03 9.55 11 13.02
0.99 0.98 0.99 1.01 1.06 1.09 1.11 1.01 0.97 0.97 0.96 0.99 1.02 1.11 0.99 1.07 1.08 1.11 1.13 1.15 1.16 1 0.98 1.01 1.06 1.06 1.09 1.09 0.99 1 1.05 1.09 1.11 1.14 1.15 1 1.09 1.15 1.18 1.2 1.21 1.23
0.40498 0.11668 0.04273 0.02078 0.01242 0.00839 0.00533 0.26840 0.08153 0.03307 0.01735 0.01001 0.00634 0.00376 0.14609 0.05051 0.02360 0.01384 0.01009 0.00610 0.00410 0.09123 0.02924 0.01572 0.00962 0.00744 0.00465 0.00320 0.07377 0.02764 0.01190 0.00610 0.00348 0.00226 0.00140 0.02371 0.00739 0.00329 0.00161 0.00088 0.00049 0.00027
4.5 6.03 7.5 9.01 10.52 12.01
1.99 2 2.03 2.09 2.09 2.15
0.19559 0.06561 0.02982 0.01656 0.01008 0.00686
5.52 7.03 8.53 10.00 11.51 13.01
3.08 3.07 3.08 3.1 3.13 3.05
0.16937 0.06357 0.03024 0.01719 0.01091 0.00714
4.52 6.03 7.52 9.01 10.52 12.05
1.98 1.99 1.99 2.04 2.04 2.07
0.18347 0.06408 0.02785 0.01555 0.00981 0.00672
5.58 7.02 8.52 10.03 11.54 13
2.98 2.97 2.96 2.98 2.98 2.96
0.12642 0.05123 0.02535 0.01449 0.00910 0.00581
4.52 6.03 7.5 9.03 10.52 12.05
1.99 1.98 1.97 1.97 2.04 2.06
0.10977 0.04156 0.01900 0.01038 0.00671 0.00434
5.54 7.00 8.51 10.00 11.50 13.04
3.1 3.1 3.11 3.18 3.22 3.23
0.10809 0.04254 0.02164 0.01290 0.00872 0.00619
4.52 6.02 7.52 9.01 10.53 12.02
1.99 2.02 2.06 2.1 2.13 2.12
0.10239 0.03964 0.01954 0.01191 0.00811 0.00579
5.53 7.03 8.53 10.01 11.51 13.05
3.03 3.01 3.01 3 3.02 3.01
0.06666 0.02664 0.01340 0.00809 0.00523 0.00357
4.55 6 7.51 9.03 10.52 12.04
2.04 1.98 1.97 1.96 1.94 1.94
0.05944 0.02587 0.01419 0.00887 0.00626 0.00467
5.57 7.06 8.54 10.06 11.55 13.04
3.1 3.1 3.14 3.13 3.13 3.13
0.04738 0.02291 0.01330 0.00857 0.00602 0.00442
4.55 6.05 7.54 9.08 10.51 12.01
1.99 1.94 1.95 1.94 1.94 1.94
0.02815 0.01186 0.00644 0.00377 0.00247 0.00160
5.5 7.02 8.53 10.01 11.5 12.99
2.97 2.92 3.04 2.98 2.99 3
0.02726 0.01197 0.00664 0.00416 0.00275 0.00191
20
40
55
70
85
data of coal showed a binary linear regression with Pc and Pp. The transformed permeability can thus be fit by Eq. (6), and the ESCK value can be calculated by Eq. (7).
2. Match the transformed permeability data by quadric surface:
k′ = a1 + a2Pc + a3 Pp + a 4 P 2c + a5 Pc Pp + a 6 P 2p
(4)
k′ = a1 + a2 Pc + a3 Pp α=−
where a1, a2, a3, a4, a5, and a6 are coefficients; Pc is confining pressure, MPa; and Pp is pore pressure, MPa.
∂k / ∂Pp ∂k / ∂Pc
=−
a3 + a5 Pc + 2a6 Pp a2 + 2a4 Pc + a5 Pp
(7)
The ESCK value with different water saturations is a constant according to Eq. (7), and the results are shown in Table 3. The water in the coal was displaced by gas at 85% water saturation, and the test result was substantially affected; thus, the ESCK value at 85% water saturation was eliminated. The effective stress with different water saturations was calculated by σeffective = Pc – αpPp, where αp is the ESCK value. It was found that the effective stress obtained by the RSM was more coincident with the definition of effective stress. When the effective stress was constant, the permeability remained unchanged regardless of the variation in Pc and Pp (Fig. 6). With the increase of water saturation, the ESCK value showed an increasing trend (Fig. 7). This result indicates that the effective stress decreases with increasing water saturation under the same Pc and Pp. During the drainage process of the CBM well, when the reservoir pressure dropped below the critical gas production pressure, the gas began to desorb from the coal reservoir, and water saturation decreased. Thus, the effective stress in the coal reservoir increases during
3. Calculating the ESCK value under different Pc and Pp according to the definition of ESCK:
α=−
a3 a2
(6)
(5)
If a4, a5, and a6 are negligible, the ESCK value is a constant; if a2 equals a3 additionally, the ESCK value equals 1.0. The experimental results of permeability were first transformed and then matched using the RSM. The Box and Draper method suggested that an F value equal to 10 times the F-distribution percentage point is needed to ensure that the surface adequately fits the data (Warpinski and Teufel, 1992). Unlike the quadric relationship between transformed permeability and Pc and Pp of sandstone, the transformed permeability 4
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Fig. 4. Test results of gas measured permeability with different water saturations.
Permeability and effective stress values obtained by the RSM were plotted on a scatter diagram, and the stress sensitivity coefficient was determined by a linear fit using the opposite of the slope value (Fig. 8). The stress sensitivity coefficients of coal with different water saturations were all greater than 1.0 (1.1464–1.4508), and decreased gradually with increasing water saturation. The results show that the stress sensitivity of coal increases with decreasing water saturation, and that coal has quite strong stress sensitivity (Table 4).
gas production, and the trend grows nonlinearly. 5. Discussion 5.1. Stress sensitivity evaluation of coal Stress sensitivity was mainly evaluated by the permeability damage rate and the value of the stress sensitivity coefficient (Xiao et al., 2016). In this paper, the stress sensitivity coefficient was used to evaluate the stress sensitivity of coal because the value obtained by this method has uniqueness.
Ss =
1 − (k / k 0)1/3 lg (σeff / σeff 0)
5.2. Dynamic change of coal reservoir permeability The fluid phase of the coal reservoir changes continuously during the drainage process of CBM well. The permeability also varies with different factors (effective stress, matrix shrinkage and gas slippage). At present, the effective stress coefficient is usually taken as 1.0 to study the dynamic change of permeability during the drainage process. The conclusions obtained in Section 3 and 4 have shown this to be insufficient to explain the gas-water two-phase flow stage when the gas
(8)
where Ss is the stress sensitivity coefficient of permeability, dimensionless; σeff is the effective stress, MPa; k is the permeability under σeff , mD; σeff 0 is the minimum of effective stress, MPa; and k 0 is the initial permeability under σeff 0 , mD. 5
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Fig. 5. Relationship between permeability and net pressure under different pore pressures.
production. Thus, the S&D model (Eq. (9)) was chose to analyze the influence of different ESCK values on dynamic change law of coal reservoir permeability.
Table 3 Conversion coefficients and ESCK values with different water saturations. Water saturation (%) 0 20 40 55 70
λ
−0.24 −0.21 −0.25 −0.34 −0.14
a1
−0.244 −0.587 −1.324 −1.3662 −2.2718
a2
−0.4351 −0.4287 −0.3852 1.5386 −0.456
a3
0.1964 0.2565 0.2956 −1.1127 0.4334
ESCK
0.4513 0.5982 0.7675 0.7233 0.9503
F
140.47 214.56 139.38 262.18 128.49
Ft
⎧ ⎪ ki = ⎨ 3Cf ⎪k0 e ⎩
2.67 2.67 2.67 2.67 2.67
3Cf ν
k 0 e 1−ν
(P − P0)
P ≥ Pj
⎛ ν (αi P − α 0 P0) + Eεmax ⎛ Pj − Pj ⎞ ⎟⎞ ⎜ 1−ν 3(1 − ν ) Pj + Pε P + Pε ⎝
⎜
⎟
⎝
⎠⎠
P < Pj
(9)
where P is the reservoir pressure, MPa; P0 is the initial reservoir pressure, MPa; ki is the dynamic permeability of the coal reservoir, 10−3μm2; k0 is the initial permeability of the coal reservoir, 10−3μm2; Cf is the cleat compressibility, MPa−1; ν is Poisson's ratio; αi is the ESCK value corresponding to P; α 0 is the ESCK value corresponding to P0; E is the elasticity modulus of coal, MPa; εmax is the maximum bulk swelling strain; Pε is the Langmuir pressure for the swelling isotherm, MPa; and Pj is the critical gas production pressure, MPa. In the stable period (mid and late periods) of CBM well production, gas production rises steadily, and water production is negligible (very low or none). The water saturation of the coal reservoir in this period can be regarded as irreducible water saturation. A CBM well (well Z8) near the Qudi mine was selected to analyze the dynamic change of coal reservoir permeability. Well Z8 is one of the typical high-yielding wells in the Zhengzhuang Block (Fig. 10), and it has an average gas and water production rate of 2127.71 m3/d and 1.20 m3/d, respectively (Fig. 11). The reservoir parameters of well Z8 are shown in Table 5. Gas cannot drive the water in the coal sample when the water saturation reaches 70%; thus, the irreducible water saturation is regarded
desorbs from the reservoir and water saturation decreases. The permeability of coal and effective stress show a significant power relationship (Fig. 6), as well as a negative exponential relationship (Fig. 9). The existence of slippage is good for increasing coal reservoir permeability. However, the gas slippage only indirectly increases the gas permeability and does not cause any change in reservoir properties. The effect of increasing permeability is relatively limited and the impact stage is difficult to quantify in actual production. Many researchers neglect the gas slippage in the modeling of dynamic change for coal reservoir permeability (Niu et al., 2018). Shi and Durucan (2005) developed a permeability model (S&D model) from the constitutive equations for isotropic linear poroelasticity. Using the exponential expression of the S&D model, the confining pressure can be eliminated in the computation of dynamic permeability analysis during the CBM 6
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Fig. 6. Effective stress law of coal with different water saturations.
permeability (such as the sharp decrease of reservoir pressure and water saturation).
as 70% in the dynamic permeability calculation of well Z8. Dynamic permeability with the ESCK values of 1.0 and 0.7675 was calculated using Eq. (7) (Fig. 11). The results show that the reservoir permeability decreases more with an ESCK value of 0.7675 than that with a value of 1.0 when the reservoir pressure decreases. The results indicate that the decrease of water saturation affects the seepage capability of the coal reservoir during the CBM production. However, the matrix shrinkage effect will increase the permeability when the reservoir pressure decreases to a certain extent. When the matrix shrinkage effect plays a leading role, the permeability of the coal reservoir starts to increase continuously, and the permeability difference between ESCK values of 0.7675 and 1.0 is reduced (Fig. 12). During the drainage process, the drainage rate should be controlled to avoid serious damage to reservoir
6. Conclusions 1. Gas measured permeability shows a negative power relationship with increasing net pressure. With the increase of water saturation, the gas measured permeability of the coal sample decreases under different pore pressures. The permeability with different water saturations also has a significant power relationship with increasing net pressure. 2. The effective stress coefficient for permeability obtained by the response surface method with different water saturations is a constant, 7
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Fig. 7. The relationship between ESCK and water saturation.
Fig. 8. Stress sensitive coefficients of coal permeability with different water saturations.
Fig. 10. Location of well Z8.
Table 4 The evaluation criterion of stress sensitivity using stress sensitivity coefficient (Xiao et al., 2016). Stress sensitivity coefficient
Ss < 0.3
0.3≤Ss ≤ 0.7
0.7≤Ss ≤ 1.0
Ss > 1.0
Sensitivity degree
Weak
Medium
Strong
Quite strong
Fig. 11. Production curve of well Z8.
and it shows an increasing trend with the increase of water saturation. 3. The stress sensitivity coefficient decreases with increasing water saturation, and the coal has strong stress sensitivity. The dynamic change of coal reservoir permeability can be more accurately analyzed using the effective stress coefficient for permeability value obtained by the response surface method. The reservoir permeability decreases more with an effective stress coefficient of 0.7675 than that with a value of 1.0 when the reservoir pressure decreases. The change of water saturation of the coal reservoir has a significant influence on permeability during the coalbed methane well production. The stress sensitivity of coal reservoir permeability becomes stronger with decreasing water saturation.
Fig. 9. The negative exponential relationship between permeability and effective stress.
Acknowledgements This work was supported by the Natural Science Foundation of 8
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Table 5 The reservoir parameters of well Z8. Thickness (m)
Burial depth (m)
Reservoir pressure (MPa)
Permeability (mD)
Pj (MPa)
Cf (MPa−1)
ν
E (MPa)
εmax
Pε (MPa)
5.28
702.21
3.96
1.75
3.13
0.143
0.32
1220
0.008
1.36
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