Permeability Evolution of Intact and Fractured Coals Subject to Varying Pore Pressures: Laboratory Tests and Analytical Modeling

Abstract

Due to the application of hydraulic fracturing, a proper understanding of the permeability evolution of multiscale fractures in coal is essential for predicting coalbed methane production and formulating development strategies. In this paper, we adopted laboratory seepage experiments (conducted under controlled pore pressure conditions) and analytical modeling techniques (including permeability characterization models for different fracture types) to investigate the permeability evolution patterns of multiscale fractures in coal under varying pore pressures. Experimental results reveal that when the pore pressure increases from 0.5 to 3 MPa, the permeability of intact coal presents a trend of first decreasing and then increasing, whereas the permeability of self-propped coal samples exhibits a monotonic increasing trend. Additionally, the permeability of the propped fracture is significantly higher than that of intact coal and self-propped samples, showing an approximately linear increase with increasing pore pressure. Notably, under the same pore pressure conditions, a higher proppant concentration and larger proppant size correspond to greater propped fracture permeability. The experimental results also indicate that the permeability of fractures propped with larger proppants is more sensitive to pore pressure changes. At low proppant concentrations, propped fracture permeability increases significantly with increasing concentration; however, this enhancement diminishes as the concentration further increases. Analytical modeling results demonstrate that the proposed permeability model for coal can effectively characterize both intact coal and self-propped samples, exhibiting stronger applicability than existing models, particularly for self-propped samples. Moreover, the proposed permeability characterization model for a propped fracture shows excellent consistency with experimental data and outperforms existing models in capturing permeability variations under different pore pressures, proppant concentrations, and sizes.

1. Introduction

Within the broader context of greenhouse gas mitigation and efforts to advance carbon capture and storage technologies, the sustainable production of coalbed methane (CBM) has made significant contributions to global energy supply while also holding relevance for balancing energy security with climate goals. , However, due to the compact structure of coal, most CBM is stored in low-permeability coal seams. − To achieve commercial natural gas production, a series of technologies for coalbed permeability enhancement have been proposed, including thermal shock, cyclic freezing-thawing, and hydraulic fracturing. − Among these, hydraulic fracturing has established a core position in CBM development due to its prominent advantages such as high operability and broad applicability. , Nevertheless, hydraulic fracturing not only creates hydraulic fractures kept open by proppants but also forms an unmodified blind zone and a self-propped zone that completely rely on the own structure of coal. The coexistence of such multiscale fractures has greatly increased the complexity of studying CBM seepage laws. Furthermore, during CBM production operations, the gas pressure in coal reservoirs continuously decreases, leading to changes in the seepage capacity of such multiscale fractures. − The variations in permeability of multiscale fractures, in turn, further affect the feasibility, efficiency, and economic benefits of CBM extraction. − Thus, a proper understanding of the permeability evolution of multiscale fractures in coal under the influence of pore pressure is essential for predicting CBM production and formulating development strategies.

Numerous scholars have conducted a series of studies on the influence of the pore pressure on the permeability of coal. Somerton et al. systematically tested the permeability evolution of three bituminous coals with distinct hardness and fracture degrees under varying pore pressures, reporting an exponential decline in coal permeability with decreasing pressure and revealing that the dominant mechanism behind this phenomenon is the fracture closure effect. In contrast, Harpalani et al. found that coal permeability conversely decreases with increasing pore pressure, arguing that adsorption swelling strain induced by rising gas pressure serves as the dominant factor governing changes in coal permeability. This finding has broken through the traditional cognitive boundary that permeability monotonically decreases with a decrease in pore pressure. Meanwhile, Robertson found that with the decrease of pore pressure, the permeability of coal presents a biphasic trend of initial decrease followed by increase and attributed this phenomenon to the coupled influence of adsorption swelling effect and stress sensitivity effect on seepage channels. Moreover, Xiao and Pan verified the existence of gas slip effect in low-permeability coal samples through slip effect tests. Feng et al. conducted seepage tests under different confining pressures and pore pressures, finding that the influence of the slip effect on coal permeability cannot be ignored at low pore pressures and proposing the view that the permeability of coal at low pore pressures is primarily governed by the slip effect. In recent years, with the large-scale application of hydraulic fracturing technology in CBM extraction, the influence of pore pressure on the seepage capacity of propped fractures has become an area of increasing research interest in the international academic community. For instance, Tan et al. took sand-gravel as the research object and selected methane as the test fluid, obtaining the dynamic evolution law of propped fracture permeability under different pore pressures based on the pulse decay method and finding that propped fracture permeability exhibits a linear increasing trend with the gradual increase of pore pressure. Bandara et al. reported on the variations in propped fracture permeability in response to pore pressure and proppant layers, finding that increasing the number of proppant layers is an effective means to improve the permeability of propped fractures under the same pore pressure. Wu et al. conducted a series of laboratory experiments on cylindrical sandstone cores with proppant layers under static pressure conditions, finding that apart from the number of proppant layers, proppant embedment and deformation induced by decreasing pore pressure are among the main factors causing changes in propped fracture permeability. In summary, existing studies of the effect of pore pressure on coal permeability primarily focus on intact coal samples. The dynamic variation law of permeability for self-propped coal samples under different pore pressure conditions remains unclear. In addition, while some scholars have attempted to clarify the permeability characteristics of propped fractures, the influence of pore pressure on the permeability of propped fractures in coal still requires further investigation, specifically taking the proppant concentration and size into account.

As a key scientific issue connecting laboratory-scale phenomena with engineering-scale production prediction, constructing mathematical models that accurately characterize the dynamic evolution of permeability based on laboratory research has always been a focus of academic research. In 1987, Gray proposed a dynamic permeability evolution model for coal considering stress sensitivity and adsorption swelling effects, based on the assumption of a matchstick model and uniaxial strain conditions. The core limitation of this model lies in its assumption that the adsorption swelling effect is quantitatively proportional to the equivalent adsorption pressure, while the permeability of natural fractures in coal reservoirs is highly sensitive to changes in effective stress. , During CBM extraction, the reservoir pressure decline simultaneously induces fracture compression via increased effective stress and enhances fracture opening via gas desorption-induced matrix shrinkage. This dynamic coupling of positive and negative effects transforms permeability theoretical simulation into a complex nonlinear problem that requires breaking through the traditional linear framework, necessitating the inclusion of dynamic evolution characteristics of multimechanism interactions in models to approximate actual reservoir responses. Against this backdrop, Palmer and Mansoori proposed the renowned P–M model under uniaxial strain conditions, by coupling coal adsorption swelling (as a function of pressure) with stress-sensitive effects. However, when coal porosity changes exceed 30%, the model’s predictive accuracy is questionable. In view of this, Shi and Duruncan proposed the SD model considering pore pressure changes, arguing that the strain induced by the adsorption swelling effect is proportional to the volume of adsorbed gas. Subsequently, a series of theoretical models describing coal permeability variation with pore pressure emerged, from the pioneering CB model considering linear elastic deformation constitutive equations, to the RC model based on cubic models and triaxial stress assumptions, and the MA model based on constant volume assumptions. − These models have gradually constructed a sophisticated academic system for coal rock seepage theory. Regarding the permeability of propped fractures, in the early stage of research, scholars generally believed that the influence of the adsorption swelling effect on propped fracture permeability could be ignored, and a series of empirical models based on traditional stress sensitivity effect have been proposed to characterize the permeability of propped fracture. , Unfortunately, empirical models fail to reveal the internal mechanisms influencing propped fracture permeability, and the physical meanings of empirical constants remain unclear. On this basis, Cheng et al. and Li et al. derived mechanism-based models for the variation of propped fracture permeability, but these models insufficiently considered the proppant concentration and size, and meanwhile equivalently treated the effective stress on propped fractures as that on individual proppants.

The primary target of this study is to attain a better knowledge of the variation in permeability of multiscale fracture in coal. To achieve this research target, the variation in the permeability of multiscale fractures under different pore pressures was investigated based on the multiscale fracture seepage experimental system, and more compatible characterization models for multiscale fracture permeability were proposed.

2. Method

2.1. Sample Preparation

The experimental coal sample, with dimensions of 30 × 15 × 15 cm³, was collected from the Sihe Coal Mine in the southern Qinshui Basin. The proppants used for the experiments were purchased from Hebei Huayuan Mining Co., Ltd., China. Proppant particles were sieved with different mesh sieves (20–30, 40–60, 60–80, and 80–100). The basic parameters’ test results of the coal sample and proppant are shown in Tables and , respectively.

1. Basic Properties of Coal .

proximate analysis				elemental analysis
M ad (%)	A d (%)	V d (%)	FCad (%)	C (%)	H (%)	N (%)	O (%)	S (%)	others (%)
1.16	12.95	9.84	78.48	91.27	3.44	1.35	3.38	0.31	0.25

^aNotes: M _ad and FC_ad represent moisture and fixed carbon on air-dried basis; A _d and V _d are ash and volatile matter on dry basis.

2. XRF Analysis of the Proppant.

CaO (%)	SiO2 (%)	MgO (%)	Al2O3 (%)	SO3 (%)	Na2O (%)	others (%)
50.233	22.576	24.519	1.731	0.285	0.449	0.207

The experiments investigated test samples containing fractures at different scales, with the detailed preparation procedure shown in Figure . As shown in Figure , the preparation process of test samples can be divided into the following steps: (1) standard cylindrical coal samples measuring 50 mm in length and 25 mm in diameter were drilled from the large bulk coal specimen to represent the intact coal material. (2) Standard cylindrical coal samples without obvious external fractures were selected and then axially cut to create artificial fractures. Subsequently, the test coal samples containing artificial fractures were wrapped with heat-shrink tubing to form self-propped coal samples. (3) For the proppant-propped fracture, this study includes two experimental types, namely, the influence of proppant concentrations (the first experimental type in Figure ) and proppant sizes (the second experimental type in Figure ) on the test results. The preparation process of proppant-propped test samples can be further divided into two steps. First, under the condition of consistent proppant particle size (proppants with particle sizes between 60 and 80 mesh were selected in this study), proppants of varying masses were measured and then evenly placed across the artificial fracture plane. Subsequently, the proppant-loaded cores were tightly encapsulated with heat-shrink tubing to form proppant-propped coal samples, which were used to examine the effects of varying proppant placement concentrations on the permeability of the propped fracture. Second, under the condition of consistent proppant placement concentration (proppant placement concentration of 0.525 kg·m^–2 was selected in this study), proppants of different particle sizes were measured and then evenly placed across the artificial fracture plane. Subsequently, the proppant-containing cores were tightly wrapped with heat-shrink tubing to form proppant-propped coal samples, which were used to examine the effects of varying proppant sizes on the permeability of the propped fracture.

1.

Preparation process of experimental coal samples.

2.2. Experimental Apparatus

The permeability of coal samples was measured on a multiscale fracture seepage experimental system, as shown in Figure . It can be seen from Figure that the multiscale fracture seepage experimental system consists of a stainless-steel core holder designed to house cylindrical coal samples, two high-pressure methane gas cylinders (99.99% purity) supplying the test fluid at precisely regulated pressures, an ISCO pump with a pressure range of 0.007–70 MPa for confining pressure application, two six-way valves enabling precise flow path switching, a vacuum pump operating at −100 to −300 kPa for system evacuation to eliminate residual gas effects on experimental results, two manometers with an accuracy of 0.1 kPa for monitoring inlet and outlet pressures of the core holder, a gas flowmeter with a precision of 0.01 mL/min for measuring outlet gas flow from the core holder, and a temperature control system maintaining the core holder at 40 °C throughout the experiments.

2.

Multiscale fracture seepage experimental system.

2.3. Experimental Procedure

The following experiments were conducted using traditional gas measurement permeability methods for intact coal and self-propped coal samples, as shown in eq . The specific experimental procedures are as follows:

• (1)Place the experimental core into the core holder, open valves ②, ③, and ④, while closing valves ①, ⑤, ⑥, ⑦, and ⑧. Apply a confining pressure of 6 MPa and then turn on the vacuum pump to evacuate the experimental system for more than 12 h.
• (2)After the vacuum process is complete, open valve ① and the methane gas cylinder. Apply a methane pressure of 0.5 MPa and maintain it for more than 48 h to allow the core to fully adsorb and reach equilibrium.
• (3)Open valves ⑤, ⑥, and ⑧, keeping the outlet pressure of the methane gas cylinder constant (0.5 MPa). Record the data from the gas flow meter and calculate the permeability of the test sample under different pore pressures by eq .
• (4)Repeat the above steps, gradually increasing the pressure of the methane gas cylinder in the following sequence: 0.5 MPa → 1 MPa → 1.5 MPa → 2 MPa → 2.5 MPa → 3 MPa.

$$k=\frac{2p_{\mathrm{a}}{q}_{\mathrm{g}}{\mu }_{\mathrm{g}}L}{A(p_1^2-p_2^2)}$$ 1

where k is the permeability of the test coal sample; p _a is the atmospheric pressure, which can be regarded as a constant of 0.1 MPa; μ_g is the dynamic viscosity of gas; L is the length of the experimental core; A is the cross-sectional area of the test core; p ₁ is the inlet pressure of the core holder; p ₂ is the outlet pressure of the core holder.

For proppant-containing samples, it is difficult to maintain stable methane pressure at the inlet during experiments due to the presence of the propped fracture. Given this, the permeability of proppant-propped samples was measured using the pulse-decay method, as shown in eq . , The specific experimental procedure is as follows:

• (1)Place the experimental core into the core holder, open valves ②, ③, and ④, while closing valves ①, ⑤, ⑥, ⑦, and ⑧. Apply a confining pressure of 6 MPa and then activate the vacuum pump to evacuate the experimental system for over 12 h.
• (2)After evacuation, open valve ① and the methane gas cylinder. Apply a methane pressure of 0.5 MPa and maintain it for at least 48 h to ensure full adsorption equilibrium of the core.
• (3)Open valve ⑦ between the methane gas cylinder and the core sample, allowing gas to flow from the upstream methane source through the core to the downstream side. Record the upstream-downstream pressure difference in real-time using pressure gauges until equilibrium is reached. Then, calculate the permeability of the sample under different pore pressures using eqs and .
• (4)Repeat the above steps, incrementally increasing the methane gas cylinder pressure in the sequence: 0.5 MPa → 1 MPa → 1.5 MPa → 2 MPa → 2.5 MPa → 3 MPa.

$$\frac{p_{\mathrm{u}}-p_{\mathrm{d}}}{p_{\mathrm{u}0}-p_{\mathrm{d}0}}={\mathrm{e}}^{-\alpha t}$$ 2

$$\alpha =\frac{k_{\mathrm{F}}{V}_{\mathrm{R}}}{{\mu }_{\mathrm{g}}{C}_{\mathrm{g}}{L}^2}(\frac{1}{V_{\mathrm{u}}}-\frac{1}{V_{\mathrm{d}}})$$ 3

where k _F is the permeability of the propped fracture; p _u – p _d is the pressure difference between the upstream and downstream sides; p _u0 – p _d0 is the initial pressure difference between the upstream and downstream sides, MPa; t is the equilibration time; α is the slope of the decay curve; μ_g is the dynamic viscosity of gas, mpa·s; C _g is the gas compressibility; L is the length of the propped fracture; V _R is the volume of the propped fracture; V _u and V _d are the volumes of the upstream and downstream gas cylinders.

3. Experimental Results and Discussion

3.1. Effect of Pore Pressure on Intact Coal Permeability

Figure presents the test results of the dynamic permeability changes in intact coal under varying pore pressures. As shown in Figure , when the pore pressure is low, the permeability of intact coal gradually decreases with an increase in pore pressure. As the pore pressure continues to rise, the permeability of intact coal begins to recover gradually. Within the experimental range, the permeability of intact coal exhibits an overall “U”-shaped trend with increasing pore pressure. For intact coal, the change in permeability is influenced by two factors: the stress sensitivity effect and the matrix swelling effect. The stress sensitivity effect dominates under high pore pressure conditions, while the matrix swelling effect prevails under low pore pressure conditions. As the pore pressure increases, both the methane adsorption capacity and surface free energy of the coal matrix increase accordingly. Since the surface free energy of the coal matrix is directly proportional to its swelling deformation, the increase in pore pressure inevitably leads to swelling deformation of the coal matrix. At low pore pressures, the swelling deformation of the coal matrix caused by increasing pore pressure results in the closure of seepage channels. Although the increase in pore pressure partially mitigates the stress-induced closure effect of seepage channels, it cannot fully offset the impact of matrix swelling on permeability. When the pore pressure exceeds a certain critical value, the adsorption of methane molecules by the coal matrix approaches saturation. Further increases in pore pressure gradually diminish the influence of matrix swelling deformation on seepage channels. At this stage, the increase in pore pressure weakens the stress-induced closure effect of seepage channels, leading to a gradual recovery of coal permeability.

3.

Effect of pore pressure on the permeability of intact coal.

3.2. Effect of Pore Pressure on Self-Propped Coal Permeability

Figure presents the test results of the dynamic permeability changes in self-propped coal samples under varying pore pressures. It can be seen from Figure that unlike intact coal, the permeability of self-propped coal samples exhibits a monotonically increasing trend with rising pore pressure, and the permeability of self-propped coal samples is higher than that of intact coal. This phenomenon indicates that artificial fracture not only enhances the permeability of coal samples but also significantly amplifies the influence of the stress sensitivity effect on permeability. The primary reason for this difference lies in the fact that an artificial fracture increases the compressibility of self-propped coal samples. As demonstrated in the research by Chen et al., the compressibility coefficient of self-propped samples increases by an order of magnitude compared to that of raw samples. In this case, the stress sensitivity effect has a far greater impact on permeability than the adsorption-induced swelling effect. Additionally, observations from Figure (taking sample 1 as an example) reveal that when the pore pressure increases from 0.5 to 3 MPa, the permeability of the self-propped coal sample increases by approximately 5-fold. This is mainly because an artificial fracture provides the primary seepage channels for self-propped coal samples, and the aperture of artificial fracture expands with increasing pore pressure.

4.

Effect of pore pressure on the permeability of self-propped coal.

3.3. Effect of Pore Pressure on Propped Fracture Permeability

Figure presents the test results of the dynamic changes in the propped fracture permeability under varying proppant placement concentrations and proppant sizes at different pore pressures. As shown in Figure , the permeability of the propped fracture is significantly higher than that of intact coal and self-propped coal samples, and it exhibits a nearly linear increasing trend with rising pore pressure. This phenomenon indicates that the introduction of proppants not only substantially enhances the permeability of hydraulic fractures but also greatly suppresses the impact of matrix swelling on the permeability. The underlying mechanism is that the proppant provides rigid structural support, which effectively mitigates permeability impairment caused by coal matrix swelling deformation, thereby maintaining a high fracture permeability. It should be noted that under the same pore pressure conditions the permeability of the propped fracture further increases with higher proppant placement concentrations and larger proppant particle sizes. This trend can be attributed to two main mechanisms: (1) an increase in proppant concentration leads to a denser distribution of proppant particles within the fracture, enabling better resistance to fracture wall closure under the same stress conditions and maintaining a larger effective aperture. (2) Larger proppant particle sizes directly enhance the propping height of individual proppant grains against the fracture walls, resulting in the formation of wider dominant flow channels on the macroscopic scale.

5.

Effect of pore pressure on the permeability of the propped fracture under (a) different proppant concentrations with proppant sizes between 60 and 80 mesh and (b) different proppant sizes with a proppant concentration of 0.525 kg/m².

Moreover, it can also be seen from Figure a that, at a certain pore pressure, when the proppant concentration is low, the permeability of propped fractures increases significantly with an increasing proppant concentration. However, this enhancement effect attenuates progressively with further increases in proppant concentration. Taking the pore pressure of 0.5 MPa as an example, when the proppant concentration increases from 0.116 kg/m² to 0.525 kg/m², the propped fracture permeability increases from 149.2 to 195.4 mD, with a percentage increase of 30.97%. Nonetheless, when the proppant concentration increases from 0.525 kg/m² to 1.684 kg/m², the propped fracture permeability increases from 195.4 to 220.7 mD, with a percentage increase of 12.95%. This conclusion is consistent with the finding of Meng et al., where they found that the permeability of propped fractures does not increase indefinitely with additional proppant layers, and they attributed this phenomenon to the misalignment of proppants. Specifically, under the action of effective pressure, proppants will be compressed and gradually undergo elastic embedment, leading to the multidirectional movement of proppants (as shown in Figure a). The displacement and rearrangement of proppants within the fracture will alter the original number of layers. As a result, the number of proppant layers under effective stress is no longer strictly an integer, consequently leading to a nonstrictly linear relationship between fracture conductivity and proppant concentration. Figure b presents that, at a certain proppant concentration, when the pore pressure increases from 0.5 to 3 MPa, there is a significant divergence in propped fracture permeability variation across distinct proppant sizes. Taking the proppant size of 20–30 mesh and 80–100 mesh as an example, when the pore pressure increases from 0.5 to 3 MPa, the permeability of the fracture propped with a smaller-sized proppant increases from 185.8 mD to 194.2 mD (with a percentage increase of 4.41%), while the permeability of the fracture propped with a larger-sized proppant increases from 253.1 mD to 267.9 mD (with a percentage increase of 5.89%). This result indicates that the permeability of the fracture propped with a larger-sized proppant exhibits greater sensitivity to pore pressure variations. One reasonable explanation is that at a certain proppant concentration the number of proppants in the fracture propped by larger-sized proppants is significantly less than that in the fracture propped by smaller-sized proppants, as shown in Figure b. In our previous study, we have found that a fewer number of proppants corresponds to a greater effective stress acting on a single proppant, leading to more severe embedment and deformation of the proppants. Therefore, at a certain proppant concentration, the permeability of the fracture propped with a larger proppant exhibits greater sensitivity to pore pressure.

6.

Schematic of (a) the displacement and rearrangement of proppants and (b) the distribution of proppants under different proppant sizes.

4. Establishment and Validation of Permeability Characterization Models

As previously mentioned, current research on characterization models for the dynamic evolution of coal permeability primarily focuses on intact coal, leaving a cognitive gap in characterization models for the permeability evolution of self-propped coal samples under varying pore pressures. Moreover, although existing studies have characterized the dynamic response of propped fracture permeability to pore pressure changes, the effects of proppant size and concentration on this relationship remain unquantified. Therefore, the primary target of this section is to establish analytical models characterizing the permeability evolution of intact coal, self-propped coal samples, and propped fracture under varying pore pressure conditions. On this basis, the proposed permeability characterization models will be further validated through comparative analysis with experimental data and existing models.

4.1. Permeability Characterization Model for Raw and Self-Propped Coal Samples

The adsorption/desorption process of methane in coal will induce expansion-/shrinkage-induced volumetric strain, which can be described by eq .

$$\epsilon =\frac{{\epsilon }_{\mathrm{L}}p}{p_{\mathrm{L}}+p}$$ 4

where ε is the expansion-/shrinkage-induced volumetric strain of coal; ε_L is the maximum expansion/shrinkage volumetric strain constant of coal; p _L is the gas pressure corresponding to half of the maximum expansion/shrinkage volumetric strain constant of coal; p is the gas pressure.

When the gas pressure changes from p ₀ to p ₁, eq can be further expressed by eq .

$${\epsilon }_1=\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)}$$ 5

Additionally, as the gas pressure changes, the effective stress of coal is bound to change, and when the gas pressure changes from p ₀ to p ₁, the porosity in coal can be expressed by eq .

$$\phi ={\phi }_0\exp [c_{\mathrm{f}}(p_1-p_0)]$$ 6

where φ is the porosity of coal; φ₀ is the initial porosity of coal; c _f is the compressibility coefficient of coal; p ₁ is the gas pressure subsequent to the change; p ₀ is the gas pressure prior to the change.

Therefore, the induced stress–strain of the coal sample caused by the change in gas pressure can be expressed by eq .

$${\epsilon }_2=\phi -{\phi }_0={\phi }_0[\exp (c_{\mathrm{f}}\mathrm{\Delta }p)-1]$$ 7

where Δp is the difference of gas pressure.

Since the pore pressure is gradually increased during the experiment, the expansion strain exhibits a negative effect. Thus, the total strain of coal (ε_v) can be expressed by eq .

$${\epsilon }_{\mathrm{v}}={\phi }_0[\exp (c_{\mathrm{f}}\mathrm{\Delta }p)-1]-\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)}$$ 8

where ε_v is the total strain of coal.

According to the definition of porosity, when the coal porosity changes, the coal porosity can be expressed by eq .

$$\phi =1-\frac{V_{\mathrm{s}0}(1+\mathrm{\Delta }{V}_{\mathrm{s}}/V_{\mathrm{s}0})}{V_0(1+\mathrm{\Delta }V/V_0)}=1-\frac{1-{\phi }_0}{1+{\epsilon }_{\mathrm{v}}}(1+\frac{\mathrm{\Delta }{V}_{\mathrm{s}}}{V_{\mathrm{s}0}})$$ 9

where ΔV is the change in the overall volume of coal; V ₀ is the initial volume of coal; ΔV _s is the change in the volume of the coal skeleton; and V _s0 is the initial volume of the coal skeleton.

Moreover, according to the definition of porosity, ΔV/V ₀ can be further expressed by eq .

$$\frac{\mathrm{\Delta }V}{V_0}=\frac{\mathrm{\Delta }{V}_{\mathrm{S}}}{V_{\mathrm{S}0}}+\frac{\mathrm{\Delta }\phi }{1-\phi }$$ 10

Substituting eqs and into eq , the porosity of coal can be further expressed by eq .

$$\phi =\frac{{\phi }_0+{\phi }_0[\exp (c_{\mathrm{f}}\mathrm{\Delta }p)-1]-\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)}}{1+{\phi }_0[\exp (c_{\mathrm{f}}\mathrm{\Delta }p)-1]-\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)}}$$ 11

Additionally, according to the previous study, there is a fixed relationship between the porosity and permeability of coal, which can be expressed by eq .

$$\frac{k}{k_0}={\left(\frac{\phi }{{\phi }_0}\right)}^3$$ 12

where k is the permeability of coal and k ₀ is the initial permeability of coal.

Substituting eq into eq , the permeability of coal can be further expressed by eq .

$$\frac{k}{k_0}={\left\{\frac{{\phi }_0+{\phi }_0[\exp (c_{\mathrm{f}}\mathrm{\Delta }p)-1]-\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)}}{{\phi }_0(1+{\phi }_0[\exp (c_{\mathrm{f}}\mathrm{\Delta }p)-1]-\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)})}\right\}}^3$$ 13

When estimating coal permeability, it is generally considered necessary to account for the influence of the slip effect, as shown in eq .

$$k=k_{\mathrm{a}}(1+\frac{B}{p})$$ 14

where k _a is the absolute permeability of coal and B is a constant factor.

Therefore, by substituting eqs into , the permeability of coal can be further expressed by eq .

$$k=k_0(1+\frac{B}{p}){\left\{\frac{{\phi }_0+{\phi }_0[\exp (c_{\mathrm{f}}\mathrm{\Delta }p)-1]-\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)}}{{\phi }_0(1+{\phi }_0[\exp (c_{\mathrm{f}}\mathrm{\Delta }p)-1]-\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)})}\right\}}^3$$ 15

As shown in eq , in the proposed model for characterizing the permeability of coal, aside from the parameters required to describe the slip effect and adsorption strain, there are only two uncertain parameters in the model, that is, the initial porosity and compressibility coefficient of coal. Therefore, if the artificial fracture and the original porosity of the self-propped coal sample are equivalent to the initial porosity and the compressibility of the artificial fracture and coal is equivalent to the overall compressibility of the self-propped coal sample, eq can be used to characterize not only the permeability of intact coal but also that of the self-propped coal sample.

4.2. Permeability Characterization Model for the Propped Fracture

When the gas pressure changes from p ₀ to p ₁, under the action of stress, the pore space of the propped fracture will be changed, and the resulting variation of propped fracture permeability in this process can be expressed by eq .

$$k_{\mathrm{F}}=k_{\mathrm{F}0}\exp [-3c_{\mathrm{F}}({\sigma }_1-{\sigma }_0)]=k_{\mathrm{F}0}\exp (-3c_{\mathrm{F}}\mathrm{\Delta }{\sigma }_{\mathrm{e}})$$ 16

where k _F is the initial permeability of the propped fracture; c _F is the compressibility coefficient of the propped fracture; and σ₁ and σ₀ are the effective stress of propped fracture corresponding to p ₀ and p ₁.

In our previous study, we have proposed the model to describe the variation in the compressibility coefficient of the propped fracture, taking effective stress, the embedment and deformation of proppant, as well as the proppant concentration and size into account, as shown in eqs and . Among them, eq is used to describe the compressibility of the fracture propped by a single layer of proppants, as shown in Figure a. Meanwhile, eq is used to describe the compressibility of the fracture propped by multilayers of proppants, as shown in Figure b. Moreover, the proppant layer is determined by the proppant concentration. When the proppant concentration is less than C _min (C _min = πρD/6, note: ρ and D are the density and diameter of proppant, respectively), the fracture is propped by a single layer of proppants. When the proppant concentration is greater than C _min, the fracture is propped by multilayers of proppants (the detailed deduced process of propped fracture compressibility is supplied in the Supporting Information).

$${C_F}_1=\frac{4D^2{\left(\frac{E^{*}}{C{A}_{\mathrm{f}}}\right)}^{\frac{2}{3}}{\sigma }_{\mathrm{e}}^{-\frac{1}{3}}+4.16D^2{\left(\frac{E_0}{C{A}_{\mathrm{f}}}\right)}^{\frac{2}{3}}{\sigma }_{\mathrm{e}}^{-\frac{1}{3}}}{3-6D^2{\left(\frac{E^{*}{\sigma }_{\mathrm{e}}}{C{A}_{\mathrm{f}}}\right)}^{\frac{2}{3}}-6.24D^2{\left(\frac{E_0{\sigma }_{\mathrm{e}}}{C{A}_{\mathrm{f}}}\right)}^{\frac{2}{3}}}+\frac{4}{{\sigma }_{\mathrm{e}}}\frac{{(E_3{\sigma }_{\mathrm{e}})}^{\frac{2}{3}}}{1-2{(E_3{\sigma }_{\mathrm{e}})}^{\frac{2}{3}}}$$ 17

where C is the proppant concentration; A _f is the area of the fracture wall; ΔV _s is the change in the volume of the coal skeleton; E ₀ and E* are the equivalent moduli that describe the proppant and the mutual influence of the coal and proppant, respectively; E ₃ is related to the Young’s modulus (E ₁) and Poisson’s ratio of proppant (v ₁), and E ₃ can be expressed as E ₃ = 3­(1 −v ₁ ²)/E ₁.

$$\begin{array}{l}{C_F}_2=\frac{4D^2{\left(\frac{E^{*}}{C{A}_{\mathrm{f}}}\right)}^{2/3}{\sigma }_{\mathrm{e}}^{-1/3}+4.16D^2[1+\sqrt{3}(\frac{CS}{D}-1)]{\left(\frac{E_0}{C{A}_{\mathrm{f}}}\right)}^{2/3}{\sigma }_{\mathrm{e}}^{-1/3}}{3+\sqrt{6}(\frac{CS}{D}-1)-6D^2{\left(\frac{E^{*}{\sigma }_{\mathrm{e}}}{C{A}_{\mathrm{f}}}\right)}^{2/3}-6.24D^2[1+\sqrt{3}(\frac{CS}{D}-1)]{\left(\frac{E_0{\sigma }_{\mathrm{e}}}{C{A}_{\mathrm{f}}}\right)}^{2/3}}\\ +\frac{4D+\frac{2\sqrt{6}}{3}(\frac{CS}{D}-1)D}{{\sigma }_{\mathrm{e}}[D+\frac{\sqrt{6}}{3}(\frac{CS}{D}-1)D]}\frac{{(E_3{\sigma }_{\mathrm{e}})}^{2/3}}{1-\frac{2D+\frac{\sqrt{6}}{3}(\frac{CS}{D}-1)D}{D+\frac{\sqrt{6}}{3}(\frac{CS}{D}-1)D}{(E_3{\sigma }_{\mathrm{e}})}^{2/3}}\end{array}$$ 18

where S is related to the density of the proppant (v ₁), which can be expressed as 6/πρ.

7.

Schematic of (a) proppants state for a single-layer propped fracture and (b) proppant’s state for a multilayer propped fracture.

Therefore, incorporating eqs and into eq , the permeability of a propped fracture under the action of stress can be further expressed by eq .

$$\{\begin{array}{l}{k}_{\mathrm{F}}=k_{\mathrm{F}0}\exp (-3C_{\mathrm{F}1}\mathrm{\Delta }{\sigma }_{\mathrm{e}}),C\le C_{\min }\\ k_{\mathrm{F}}=k_{\mathrm{F}0}\exp (-3C_{\mathrm{F}2}\mathrm{\Delta }{\sigma }_{\mathrm{e}}),C>C_{\min }\end{array}$$ 19

It should be noted that eq presents the model of propped fracture permeability under the action of stress, while the impact of the adsorption swelling effect on the propped fracture permeability should not be neglected. According to a previous study, the adsorption swelling effect will reduce the opening of a propped fracture to a certain extent (as shown by the black line in Figure ). When the gas pressure changes from p ₀ to p ₁, the changes in the opening of the propped fracture can be expressed by eq .

$$\mathrm{\Delta }b=-\beta \frac{3b_0}{{\phi }_{\mathrm{f}}}\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)}$$ 20

where β is the adsorption strain coefficient, with a value range of 0–1; b ₀ is the initial opening of the propped fracture; and φ_f is the initial porosity of the propped fracture.

8.

Schematic of the variation in the propped fracture opening under the action adsorption swelling effect.

Moreover, there is a fixed relationship between the opening and permeability of the propped fracture, which can be expressed by eq .

$$\frac{k_{\mathrm{F}}}{k_{\mathrm{F}0}}={\left(\frac{b}{b_0}\right)}^3={(1+\frac{\mathrm{\Delta }b}{b_0})}^3$$ 21

Substituting eq into eq , the changes in propped fracture permeability induced by the adsorption swelling effect can be expressed by eq .

$$\frac{k_{\mathrm{F}}}{k_{\mathrm{F}0}}={(1-\beta \frac{3}{{\phi }_{\mathrm{f}}}\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)})}^3$$ 22

Based on the assumption that the proppant is spherical, the mass of a single proppant (m ₀) can be expressed by eq .

$$m_0=\frac{\rho \pi D^3}{6}$$ 23

Therefore, the number of proppants and the total volume of proppants can be expressed by eqs and , respectively.

$$N=\frac{6C{A}_{\mathrm{f}}}{\rho \pi D^3}$$ 24

$$V_{\mathrm{T}}=C{A}_{\mathrm{f}}/\rho$$ 25

For a fracture propped by a single layer of proppants, the volume of the propped fracture is related to the diameter of the proppant and the area of the fracture and can be expressed by eq .

$$V_{\mathrm{F}\mathrm{S}}=D{A}_{\mathrm{f}}$$ 26

Therefore, the initial porosity of the fracture propped by a single layer of proppants can be expressed by eq .

$${\phi }_{\mathrm{S}}={(V_{\mathrm{F}\mathrm{S}}-V_{\mathrm{T}})/V}_{\mathrm{F}\mathrm{S}}=1-C/\rho D$$ 27

For a fracture propped by multilayers of proppants, assuming the proppants are placed in a rhombic pattern within the fracture and only the elastic behavior of the proppants is considered, the volume of propped fracture can be expressed by eq .

$$V_{F\mathrm{M}}=A_{\mathrm{f}}(D+\frac{\sqrt{6}}{3}(n-1)D)$$ 28

where n is the average number of proppant layers.

Therefore, the porosity of the fracture propped by multilayers of proppants can be expressed by eq .

$${\phi }_{\mathrm{M}}=(V_{\mathrm{F}\mathrm{M}}-V_{\mathrm{T}})/V_{\mathrm{F}\mathrm{M}}=1-\frac{C}{\rho }\frac{1}{D+\frac{\sqrt{6}}{3}(n-1)D}$$ 29

In our previous study, we have proved that the average number of proppant layers is related to the diameter, density, and concentration of the proppant, as shown in eq .

$$n=\frac{6C}{\pi D\rho }$$ 30

Substituting eq into eq , the initial porosity of fracture propped by multilayers of proppants can be further expressed by eq .

$${\phi }_{\mathrm{M}}=(V_{\mathrm{F}\mathrm{M}}-V_{\mathrm{T}})/V_{\mathrm{F}\mathrm{M}}=1-\frac{C}{\rho D}\frac{1}{1+\frac{2\sqrt{6}C}{\pi D\rho }-\frac{\sqrt{6}}{3}}$$ 31

Substituting eqs and into eq , the changes in propped fracture permeability induced by the adsorption swelling effect can be further expressed by eq .

$$\{\begin{array}{l}\frac{k_{\mathrm{F}}}{k_{\mathrm{F}0}}={(1-\beta \frac{3}{{\phi }_{\mathrm{S}}}\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)})}^3,C\le C_{\min }\\ \frac{k_{\mathrm{F}}}{k_{\mathrm{F}0}}={(1-\beta \frac{3}{{\phi }_{\mathrm{M}}}\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)})}^3,C>C_{\min }\end{array}$$ 32

Therefore, by combining eqs and , the permeability of the propped fracture considering the comprehensive effects of stress sensitivity and adsorption swelling can be expressed by eq .

$$\{\begin{array}{l}{k}_{\mathrm{F}}=k_{\mathrm{F}0}\exp (-3C_{\mathrm{F}1}\mathrm{\Delta }{\sigma }_{\mathrm{e}}){(1-\beta \frac{3}{{\phi }_{\mathrm{S}}}\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)})}^3,C\le C_{\min }\\ k_{\mathrm{F}}=k_{\mathrm{F}0}\exp (-3C_{\mathrm{F}2}\mathrm{\Delta }{\sigma }_{\mathrm{e}}){(1-\beta \frac{3}{{\phi }_{\mathrm{M}}}\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)})}^3,C>C_{\min }\end{array}$$ 33

Moreover, the impact of the slip effect on the propped fracture permeability cannot be ignored. Substituting eq into eq , the permeability of the propped fracture considering the comprehensive effects of stress sensitivity, adsorption swelling, and slip can be further expressed by eq .

$$\{\begin{array}{l}{k}_{\mathrm{F}}=k_{\mathrm{F}0}\exp (-3C_{\mathrm{F}1}\mathrm{\Delta }{\sigma }_{\mathrm{e}})(1+\frac{B}{p}){(1-\beta \frac{3}{{\phi }_{\mathrm{S}}}\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)})}^3,C\le C_{\min }\\ k_{\mathrm{F}}=k_{\mathrm{F}0}\exp (-3C_{\mathrm{F}2}\mathrm{\Delta }{\sigma }_{\mathrm{e}})(1+\frac{B}{p}){(1-\beta \frac{3}{{\phi }_{\mathrm{M}}}\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p_1-p_0)}{(p_{\mathrm{L}}+p_1)(p_{\mathrm{L}}+P_0)})}^3,C>C_{\min }\end{array}$$ 34

4.3. Verification of the Permeability Characterization Model

Figure shows the fitting results of experimental data and the permeability characterization model for intact coal and self-propped coal samples (the adopted parameters are presented in Table ). The compressibility ranges of intact coal and self-propped coal are referred to the study by Hu et al., whose adopted coal samples are from the same region as those in this study. It can be seen from Figure that the proposed model generally aligns well with the experimental data. In addition, we adopted Tan’s model and the SD model, , and further validated the accuracy of our model in characterizing the permeability variation of self-propped coal samples by comparing with the experimental data of the first self-propped coal sample. As shown in Figure , our model fits experimental data better than both SD and Tan’s models. Moreover, when comparing the SD model and Tan’s model, Tan’s model outperforms the SD model in fitting experimental data. This is mainly because the SD model characterizes the permeability by incorporating Young’s modulus and Poisson’s ratio of coal, but relying solely on mechanical parameters of coal cannot accurately reflect the influence of artificial fractures on the permeability of self-propped coal. It can also be seen from Figure that the difference in fitting experimental data between our model and Tan’s model mainly appears in the stage of a larger pore pressure. Moreover, as pore pressure increases, the deviation of Tan’s model from experimental data gradually grows. The main reason for this phenomenon is that Tan’s model mainly focuses on the influence of the slippage effect and stress sensitivity on coal permeability, while ignoring the effect of adsorption swelling.

9.

Comparison between the proposed model and (a) experimental data of intact coal and (b) experimental data of self-propped coal.

3. Parameters of the Permeability Model for Raw and Self-Propped Coal.

parameters	value	parameters	value
the maximum expansion/shrinkage volumetric strain constant of coal (εL)	0.032	porosity of self-propped coal (sample-3)	0.14
the gas pressure corresponding to half of the εL (pL)	0.67	compressibility of intact coal (sample-1)	0.069
compressibility of self-propped coal (sample-1)	0.34	compressibility of intact coal (sample-2)	0.048
self-propped coal compressibility (sample-2)	0.23	compressibility of intact coal (sample-3)	0.027
compressibility of self-propped coal (sample-3)	0.18	porosity of intact coal (sample-1)	0.073
porosity of self-propped coal (sample-1)	0.25	porosity of intact coal (sample-2)	0.052
porosity of self-propped coal (sample-2)	0.16	porosity of intact coal (sample-3)	0.039

10.

Comparison between the proposed permeability model and existing models with experimental data.

Based on the parameters presented in Tables and , the comparison between the calculation results of the propped fracture permeability by the proposed model and the experimental results is given in Figure . It can be seen from Figure that the calculated results from the proposed model are in good agreement with the experimental data. Moreover, to further validate the proposed model, we took the experimental results with proppant concentration and sizes of 0.525 kg·m^–2 and 20–30 mesh as an example to compare the permeability calculation results of a propped fracture by Cheng’s model and Li’s model. , It can be seen from Figure that compared with Cheng’s model and Li’s model, the proposed model presents a higher degree of fit with the experimental data. In addition, it can also be seen from Figure that with the increase of pore pressure the modeling curves of the other two models are both above the modeling curve of the proposed model. It means that the calculation results of propped fracture permeability obtained from Cheng’s model and Li’s model present a stronger sensitivity to changes in pore pressure. The main reason for this difference is the discrepancy in the way our model, compared with Cheng’s model and Li’s model, handles the effective stress acting on the proppant. As stated in our previous study, Cheng’s model and Li’s model simply equate the effective stress acting on the propped fracture to that acting on individual proppants, while our model avoids this issue. Figure also indicates the modeling curve of Li’s model lies above that of Cheng’s model, and the gap between the two curves gradually widens as pore pressure increases. This discrepancy is primarily attributed to the fact that Li’s model does not account for the impact of adsorption swelling on the opening of the propped fracture.

4. Parameters of the Permeability Model for the Propped Fracture .

parameters	value
Young’s modulus of coal (GPa)	2.09
Poisson’s ratio of coal	0.36
Young’s modulus of proppant (GPa)	21.08
Poisson’s ratio of the proppant	0.32
the distance coefficient	1
density of the proppant (g/cm3)	2.65
fracture wall area (cm2)	12.5
average proppant size (mm)	0.75 (20–30 mesh)
	0.38 (40–60 mesh)
	0.25 (60–80 mesh)
	0.18 (80–100 mesh)

11.

Comparison between the proposed model and experimental data (a) under different proppant concentrations with proppant sizes between 60 and 80 mesh and (b) under different proppant sizes with a proppant concentration of 0.525 kg/m².

12.

Comparison between the proposed propped fracture permeability model and existing models with experimental data.

5. Potential Research Direction

In this study, the variation in the permeability of multiscale fractures in coal under different pore pressures was investigated based on the multiscale fracture seepage experimental system, and more compatible characterization models for multiscale fracture permeability were proposed based on geometric effects. However, previous studies have shown that the permeability of multiscale fracture is also influenced by several additional factors. For example, the impact of real gas effects on the permeability of deep coal seams cannot be ignored. Specifically, when the pore diameter is close to the molecular diameter, the intrinsic volume of molecules occupies the effective flow space, reducing the actual pore cross-sectional area available for flow, increasing flow resistance, and decreasing permeability. Sharma and Guria reported that for deep coal seams, due to the influence of real gas effects, the traditional slippage effect, which describes a linear relationship between permeability and the reciprocal of pore pressure, is no longer applicable. Apart from methane, CBM also contains small amounts of other gases, such as carbon dioxide and nitrogen. Tian et al. reported that coal exhibits a stronger adsorption capacity for carbon dioxide than for methane, and a weaker adsorption capacity for nitrogen than for methane. The coexistence of such multiple gases during CBM extraction will affect the volumetric strain of coal and further influence the permeability variation trend of multiscale fractures in coal. In summary, while this study has conducted a preliminary investigation into the impact of pore pressure on the permeability of multiscale fractures in coal, the effect of pore pressure on such permeability, specifically considering real gas effects and gas adsorption properties, remains to be further studied.

In addition, numerical simulation has become a key technical means to connect laboratory research and on-site development. Up to now, as an efficient numerical simulation method, the embedded discrete fracture model (EDFM) characterizes fracture elements using structured grid embedding technology and achieves dynamic coupling of the fracture-matrix system through cross-scale connection factors, which has been widely applied in the field of unconventional oil and gas reservoir development. However, the reliability of numerical simulation results is highly dependent on the accuracy of mathematical models. Therefore, further clarifying the variation trend of permeability of multiscale fractures in coal under the influence of real gas effects and gas adsorption properties, and establishing corresponding permeability characterization models, followed by deeply integrating these models with EDFM, will be the key focus of subsequent research.

6. Conclusions

Understanding the permeability evolution characteristics of multiscale fractures under different pore pressures is crucial for ensuring the efficient development of coalbed methane. In this paper, a combination of laboratory experiments and analytical models was adopted to investigate the permeability evolution patterns of multiscale fractures in coal under varying pore pressures, and corresponding characterization models were constructed. Conclusions show that

• 1.As the pore pressure increases from 0.5 to 3 MPa, the permeability of intact coal presents a trend of first decreasing and then increasing, whereas the permeability of self-propped coal samples exhibits a monotonic increasing trend. Additionally, the permeability of a propped fracture is significantly higher than that of intact coal and self-propped samples, showing an approximately linear increase with increasing pore pressure.
• 2.Under the same conditions of pore pressure and proppant concentration, the larger proppant size corresponds to greater propped fracture permeability. Moreover, at a certain proppant concentration, the permeability of the fracture propped with larger-sized proppants is more sensitive to pore pressure changes. With the proppant concentration of 0.525 kg/m², as pore pressure increases from 0.5 to 3 MPa, the permeability growth rates of the propped fracture with 20–30 mesh and 80–100 mesh proppants are 5.89% and 4.41%, respectively.
• 3.At low proppant concentrations, propped fracture permeability increases significantly with increasing concentration; however, this enhancement diminishes as the concentration further increases. When the pore pressure is 0.5 MPa and the proppant sizes are between 60 and 80 mesh, as the proppant concentration increases from 0.116 kg/m² to 0.525 kg/m² and further to 1.684 kg/m², the percentage increases of the propped fracture permeability are 30.97% and 12.95%, respectively.
• 4.The proposed permeability model for coal can effectively characterize both intact coal and self-propped samples, exhibiting stronger applicability than existing models, particularly for self-propped samples. Moreover, the proposed permeability characterization model for propped fracture outperforms existing models in capturing permeability variations under different pore pressures, proppant concentrations, and sizes.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.5c06980.

• Derivation process of propped fracture compressibility and permeability data; (PDF)

The authors declare no competing financial interest.