# Simulation Analysis on Water’s Micro Seepage Laws under Different Pressure Gradients Using Computed Tomography Method

Authors: Gang Zhou; Lei Qiu; Wenzheng Zhang; Jiao Xue
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

The aim of this paper was to develop a model that can characterize the actual micropore structures in coal and gain an in-depth insight into water’s seepage rules in coal pores under different pressure gradients from a microscopic perspective. To achieve this goal, long-flame coals were first scanned by an X-ray 3D microscope; then, through a representative elementary volume (REV) analysis, the optimal side length was determined to be 60μm; subsequently, by using Avizo software, the coal’s micropore structures were acquired. Considering that the porosity varies in the same coal sample, this study selected four regions in the sample for an in-depth analysis. Moreover, numerical simulations on water’s seepage behaviors in coal under 30 different pressure gradients were performed. The results show that (1) the variation of the simulated seepage velocity and pressure gradient accorded with Forchheimer’s high-velocity nonlinear seepage rules; (2) the permeability did not necessarily increase with the increase of the effective porosity; (3) in the same model, under different pressure gradients, the average seepage pressure decreased gradually, while the average seepage velocity and average mass flow varied greatly with the increase of the seepage length; and (4) under the same pressure gradient, the increase of the average mass flow from the inlet to the outlet became more significant under a higher inlet pressure.


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## Body

## 1. Introduction

In recent years, rock bursts, coal-gas outbursts, and dust explosion accidents have occurred frequently in underground coal mines [1–3]. Coal seam water injections have proved to be an effective solution to these problems and consequently are widely applied in coal mines [4–7]. However, coal is a type of porous medium with a loose structure, whose density, specific surface area, and porosity all show a wide range of variations. These microstructural properties can significantly affect the characteristics of the flow of water in coal pores [8–10]. The pressure water seepage in coal pores refers to the flow of water into coal that is driven by the water injection pressure. Therefore, an investigation into the flow characteristics of water pressure in coal is important and of practical significance to ascertaining the essentials of water injections at a microlevel and improving their efficiency [11–13].As coal is usually stored underground, it is fairly difficult to explore its structural properties through observation. Various experimental methods, which mainly include the mercury intrusion method [14], low-temperature nitrogen adsorption method [15], microtomy method [16], and computed tomography (CT) [17], have been adopted by many scholars for investigating coal’s microstructure. Micro-CT, which is a high-resolution three-dimensional (3D) imaging method based on the principles of X-ray imaging, can obtain a coal sample’s high-resolution 3D images without any damage to the sample’s structure and can further examine coal’s micropore structure. Accordingly, CT has some major advantages compared with the other methods [18–20] and researchers have begun to employ micro-CT in the digitalization of cores and 3D spatial characterizations of coal’s pore structures at a microscopic scale. For example, Gong et al. proposed an image-description-based multiscale precise description method of core rock’s CT images, which was developed based on the multiscale structural characteristics of coal rock’s pore system [21]. Moreover, on the basis of nondestructive, precise, and quantitative characterization, Yao et al. proposed a novel characterization method in combination with the microfocus CT scanning technique, which can provide more precise quantitative characterizations [22]. However, in these studies, CT was only proposed as a characterization method and has not yet been broadly applied in engineering practices. Wildenchild and Song et al. conducted a precise and quantitative characterization of the seepage pores in coal using the micro-CT technique [23, 24]. Cai et al. studied the material distribution, heterogeneity, pore development, porosity, and permeability of coal at multiscales through the use of multiscale X-ray CT, scanning electron microscope (SEM), and mercury intrusion porosimetry (MIP) [25]. Feng et al. analyzed the microstructure and deformation rules after methane adsorption on coal by scanning microscopic coal samples using SEM and CT [26]. Simons et al. achieved a quantitative characterization of coal based on a CT reconstruction, with a characterization accuracy of approximately 53 µm [27]. Golab et al. also applied micro-CT to investigate the occurrence states of minerals in coals [28]. Through the use of CT, Mathews et al. explored coal’s thermal fracture laws and analyzed the variation of pore space in subbituminous coal during the thermal drying process [29]. Finally, Bird et al. also employed the CT technique to analyze the pore structure in rocks and conducted numerical simulations using Avizo and COMSOL for investigating the effects of permeability, formation factors, and sampling on the numerical results [30]. However, as the accuracy of CT scanning has been continuously improving, the errors in the aforementioned research have become more prominent, and the acquired results do not reflect the actual structures accurately.In addition, in terms of seepage simulations, Coles and Karacan et al. focused on the gas adsorption and migration characteristics in coal pores and fractures [31, 32]; Busch et al. investigated the methane and carbon dioxide’s adsorption and diffusion behaviors in coal and developed a simple dual-modeling method that was more suitable for simulation [33]; and Zhou et al., using a method that combined SEM-EDS and a micro-CT scan, observed the mesostructural deformations of coal during three methane adsorption-desorption cycles [34]. Furthermore, Watanabe et al. performed simulations on the characteristics of water’s flow in different fracture surfaces under different confining pressures [35, 36]; Teng et al. carried out simulations on the seepage and migration rules of gas in fractured coal using the lattice Boltzmann method (LBM) [37]; and Luo et al. conducted numerical simulations on the coal seam water injection process in the excavation face of the test mine and ascertained the variation of the coal’s fracture degree in the fracturing process using the water injection pressure [38]. However, these seepage models that are used during the water injection process are ideal models that are constructed through 3D modeling and differ to a certain degree from the actual pore models based on CT scanning. Thus, CT images should first be acquired through high-precision micro-CT technology, and a digital 3D model that can characterize coal’s actual pore and skeleton structures with the use of FDK 3D reconstruction should be constructed. On that basis, the numerical simulations on seepage behaviors during the water injection process can more accurately reflect the conditions in actual engineering practices.Accordingly in this study, a domestic advanced X-ray 3D microscope (nanoVoxel-2000 series, Sanying Precision Instrument Ltd., China) was employed for conducting scanning and microstructural reconstruction on long-flame coals from the Daliuta Coal Mine; then, the coal’s micropore structural model was developed through denoising, rendering, and data segmentation, which were performed on the Avizo platform; finally, appropriate pressure inlet and outlet were set, and the seepage behaviors during the water injection process under different pressure gradients were simulated using finite element software. The seepage characteristics were then analyzed from a microscopic perspective, based on the simulation results. The present study could provide a new way of gaining insights into the nature of water injections in coal seams and contribute to the improvement of water injection efficiency [39].


## 2. Methodology

### 2.1. CT Scanning and the 3D Reconstruction of Coals

In this study, long-flame coals from the Daliuta Coal Mine, Shanxi, China, were tested. The coal samples show low metamorphism degrees and well-developed pores and thus are suitable for investigating the characteristics of pore structures. For each coal sample, an ore core with a diameter of approximately 2 mm and a length of approximately 5 mm was drilled, which was then sealed by wax. This was done to prevent the evaporation of water in coal during scanning, which would then affect the experimental results due to the changes in shape. Through measurements, it was determined that the coal core after sealing was 2.29 mm in diameter, as shown in Figure1. The sealed core was fixed at the tip of a toothpick, placed on the sample stage (as shown in Figure 2, the scanning accuracy of the device is as high as 0.5 m, which can achieve a 360-degree rotation) and fixed in place by screws. Next, the main control computer (MCC) was switched on so that it could operate the platform. The sample stage was adjusted to the center from four dimensions, x, y, z, and r; then, the X-ray source was opened and a mathematical model with a pixel size of 1024 ∗ 1024 ∗ 1024 was acquired through a super-wide field scanning, whose actual size was 1.024 mm ∗ 1.024 mm ∗ 1.024 mm. Table 1 lists the specific CT scanning parameters, in which SOD denotes the distance between the ray source and the detector and ODD denotes the distance between the sample stage and the detector. During the scanning process, a 20X lens was used as the detector, and the number of scanning frames was set as 900; that is, the sample was scanned once after the sample stage was rotated by 0.4°. The exposure time, that is, the time required for a rotation of the sample stage, was set as 65 s.Figure 1
Picture of a coal sample.Figure 2
Picture of the nanoVoxel-2000 series X-ray 3D microscope.Table 1
Setting of scanning parameters of the X-ray 3D microscope.
Detector
SOD/ODD (mm/mm)
Voltage (kV)
Current (µA)
Number of scanning frames
Consumed time (h)
Exposure time (s)
Penetration rate (%)
20X
35/13
41
240
900
19
65
57A micro-CT system generally adopts a 3D cone-beam reconstruction algorithm. Among various 3D cone-beam reconstruction algorithms, the approximation algorithm has a number of advantages, including simple mathematical expression and easy implementation, which means it can achieve favorable reconstruction results at relatively small cone angles, and thus, it is widely applied in practice. FDK algorithm is a mainstream algorithm in various approximation algorithms based on a filtered back projection. The FDK algorithm, proposed by Feldkamp et al., is a kind of approximate reconstruction algorithm based on circular orbit scanning. In this study, this algorithm was used to conduct a 3D reconstruction on the scanned images. It is essentially an expansion of the 2D fan-beam filtered back projection algorithm in 3D space and includes the steps of the preweighting of projection data, one-dimensional filtering, and back projection. Specifically, the FDK algorithm includes the following steps:(1)
First, weighted processing was conducted on the projection data, using a function similar to the cosine function, in order to modify the distance between the voxel and source point and the angle difference appropriately.(2)
Then, one-dimensional filtering was conducted on the projection data from different projection angles along the horizontal direction.(3)
Finally, a 3D back projection was performed along the direction of the X-ray. The reconstructed voxel values were the sum of the rays passing the voxel at all projection angles.The corresponding formulas are written as follows:(1)p˜β,a,b=wa,b⋅pβ,a,b∗ga,fFDKx,y,z=∫02πR2Ux,y,β2p˜β,a,bdβ,where(2)wa,b=RR2+a2+b2=cosγcosκ,Ux,y,β=R+xcosβ+ysinβ,where p˜β,a,b denotes the projection data; β denotes the position of the projection source, that is, the projection angle °; a,b denotes the coordinates of the detector array; ga denotes a frequently used ramp filter function; R denotes the radius of the circular orbit; γ denotes the sector angle; and κ denotes the cone angle [40].


### 2.2. The Representative Elementary Volume Analysis

The pores should be connected in simulating seepage behaviors during the water injection process. Generally, a larger 3D digital core can provide a more accurate characterization of a core’s micropore structure and more accurate simulation results of its macroscopic physical properties; however, it would place higher requirements on the computer’s storage and calculation capabilities. Since a computer generally has limited storage and calculation capacities, the size of a 3D core cannot be too large. Accordingly, a representative elementary volume (REV) analysis [41] was conducted on the constructed 3D digital core before the construction of a pore network model; that is, the most appropriate 3D digital core size should be selected so that the constructed model can accurately reflect the rock’s micropore structural and physical characteristics. A REV analysis of a digital core is primarily aimed at selecting the REV. We first randomly selected a point in the 3D digital core model and centered a cube on this point; then, we gradually increased the side length of this cube until the model’s porosity reached a stable value. The side length of the cubic model under this stable state was thus regarded as the optimal size. As shown in Figure 3, four different voxels were selected in the long-flame coal model, denoted as P1, P2, P3, and P4, respectively; then, using these four points as the centers, the different cubic regions were set. By gradually increasing the side length (a) of these cubes, four variation curves between the coal rock’s porosity and the cube’s side length were acquired. These curves represent the calculated porosities of the coal rocks using the four different voxels of the 3D digital rock as the cube centers. Table 2 lists the calculated porosity values of the long-flame coal sample when the cube’s side length was set at different values, and Figure 4 displays the variation of the calculated porosity and the side length of the cube. As shown in Figure 4, when the side length of the cube increased to 62 μm, the calculated porosity gradually tended to be stable, suggesting that this size of digital core can adequately characterize the core’s physical properties. For the convenience of calculation, the size of the coal model was finally determined to be 60 pixel ∗ 60 pixel ∗ 60 pixel, with an actual size of 0.06 mm × 0.06 mm × 0.06 mm.Figure 3
Distribution of the selected models using different points as the voxel centers.Table 2
Calculated results of coal rock porosity using different points (P1, P2, P3, and P4) as the voxel centers.
Size of the volume element (μm)
30
40
50
55
60
65
70
80
90
Calculated porosity using P1 as the voxel center (%)
34.56
30.95
28.18
26.43
26.32
26.24
26.31
26.07
26.30
Calculated porosity using P2 as the voxel center (%)
32.40
30.76
29.72
28.75
28.77
28.80
28.74
28.73
28.81
Calculated porosity using P3 as the voxel center (%)
35.74
33.19
32.01
30.27
30.10
30.13
30.09
30.15
30.20
Calculated porosity using P4 as the voxel center (%)
37.43
36.91
35.28
34.53
34.57
34.49
34.53
34.52
34.50Figure 4
Variations of the calculated porosity with the increase of volume element size for different models.It can also be observed from Figure4 that the calculated porosities are different for the same coal sample [42, 43]. Through a REV analysis, this study adopted four coal regions, whose centers were P1, P2, P3, and P4, respectively, as the models for simulating the seepage behaviors during water injections under different pressure gradients. These four models are hereafter referred to as the P1 model, P2 model, P3 model, and P4 model, respectively. The absolute porosities of these four regions were 26.32%, 28.77%, 30.10%, and 34.57%, respectively.


### 2.3. The Construction of the Geometric Model

Various kinds of system noises exist in digital cores after CT scanning, which would not only affect the image quality but also affect the subsequent quantitative analysis. Before image segmentation, the images should be filtered so as to reduce noises, improve the image quality, and thus make it easier for image segmentation and display. The common filtering algorithms include Gaussian filtering, variance filtering, median filtering, recursive exponential filtering, and low-pass linear filtering [44]. Through comparison, median filtering was selected in this study, which is a classical noise smoothing method and generally used for protecting the edge information. Figure 5 shows the model before and after filtering.Figure 5
Comparison of the P1 model before and after denoising: (a) before denoising and (b) after denoising.
(a)
(b)In order to better distinguish pores from skeletons and make quantitative descriptions, an appropriate binarization should be conducted on the grayscale images. On an Avizo platform, the coal model was first segmented by pressing the “Image Segmentation” button, and then the segmented image was displayed by pressing the “Surface View” button. Thus, the microscopic models of coal pores based on actual coal pore structures were acquired, as shown in Figures6–9.Figure 6
Model of coal’s micropore structure in model P1. (a) Coal sample. (b) Pores and skeletons. (c) Pores (with an absolute porosity of 26.30%).
(a)
(b)
(c)Figure 7
Model of coal’s micropore structure in model P2. (a) Coal sample (b) Pores and skeletons. (c) Pores (with an absolute porosity of 28.77%).
(a)
(b)
(c)Figure 8
Model of coal’s micropore structure in model P3. (a) Coal sample. (b) Pores and skeletons. (c) Pores (with an absolute porosity of 30.10%).
(a)
(b)
(c)Figure 9
Model of coal’s micropore structure in model P4. (a) Coal sample. (b) Pores and skeletons. (c) Pores (with an absolute porosity of 34.57%).
(a)
(b)
(c)


### 2.4. Grid Optimization and Simulation Parameter Setting

Next, the acquired pore model was imported into ICEM CFD software for grid generation and the establishment of the finite element model [45–47]. According to the requirements of the numerical simulations on seepage behaviors, the effective pores in the model should be interconnected. To ensure the successful implementation of simulations and a quick convergence, the pores that are disconnected with each other should be eliminated, that is, the islands should be eliminated. After island elimination, the effective porosities of the four models were 21.15%, 18.26%, 17.57%, and 23.43%, respectively. The pressure inlet and outlet were then set. To reduce the calculation burden, the pressure inlet and outlet were set as −X plane and +X plane, respectively, and the other four surfaces were set as free slippage walls, as shown in Figure 10. After the setting of the pressure inlet and outlet, the grids were automatically generated, as shown in Figure 11. We also checked the generated grids, eliminated the low-quality grids, and gradually improved the grid quality.Figure 10
Settings of boundary conditions.
(a)
(b)Figure 11
Grid generation.After the mesh has been divided effectively, the seepage behaviors in the four different coal microstructure models during the water injection process were simulated under different pressure gradients using CFX15.0 software. In simulations, the nonstationary Navier–Stokes equation, the most basic equation that can accurately describe the fluid’s actual flow characteristics, was used as the control equation. The standard k-epsilon turbulence equation was also used [48, 49]. The flow was set as downstream at normal temperature (298 K), the pressure at the outlet was kept at 0.1 MPa, and the inlet pressure was varied for different values, as shown in Table 3. Specifically, the inlet pressures of 0.3–4.3 MPa, 4.9–8.1 MPa, 8.9–13.1 MPa, and 14.1–18.1 MPa correspond to water injections at low pressure, medium pressure, high pressure, and ultrahigh pressure, respectively.Table 3
Average seepage velocities in coal seams under different pressure gradients.
Pressure increase (MPa)
Inlet pressure (MPa)
Pressure gradient (109 MPa·m−1)
Average seepage velocities in coal’s different regions (m·s−1)
P1
P2
P3
P4
0.2
0.3
3.93
1.01
1.24
1.18
0.95
0.5
7.97
1.77
2.22
2.11
1.70
0.7
12.14
2.40
3.07
2.91
2.30
0.9
16.52
2.95
3.82
3.65
2.83
1.1
21.16
3.44
4.53
4.39
3.32
1.3
26.77
3.98
5.19
5.35
3.79
0.4
1.7
35.36
4.92
6.12
6.13
4.70
2.1
45.31
5.72
7.15
7.19
5.48
2.5
53.26
6.46
8.14
7.55
6.01
2.9
62.47
7.14
9.04
8.42
6.31
3.3
71.88
7.79
9.68
9.21
6.91
3.7
82.88
8.36
10.71
9.99
7.49
0.6
4.3
99.06
9.21
11.95
11.12
8.32
4.9
113.87
10.02
12.63
12.20
9.11
5.5
130.78
10.8
13.75
13.25
9.85
6.1
148.37
11.52
14.85
14.31
10.59
6.7
161.51
12.24
16.27
15.37
11.31
7.3
174.18
13.05
17.14
16.24
12.01
0.8
8.1
196.07
13.98
18.01
17.35
12.92
8.9
230.39
14.95
18.94
18.42
13.82
9.7
259.86
15.84
20.84
19.45
14.66
10.5
285.29
16.69
21.98
20.45
15.52
11.3
310.44
17.54
23.10
21.41
16.08
12.1
339.06
18.39
24.20
22.35
17.37
1.0
13.1
373.47
19.39
25.55
23.48
18.39
14.1
407.36
20.44
26.88
24.58
19.06
15.1
443.90
21.48
28.20
25.64
19.90
16.1
480.27
22.40
29.50
26.68
20.72
17.1
518.03
23.38
30.80
27.67
21.52
18.1
556.90
24.35
32.11
28.66
22.30


## 2.1. CT Scanning and the 3D Reconstruction of Coals

In this study, long-flame coals from the Daliuta Coal Mine, Shanxi, China, were tested. The coal samples show low metamorphism degrees and well-developed pores and thus are suitable for investigating the characteristics of pore structures. For each coal sample, an ore core with a diameter of approximately 2 mm and a length of approximately 5 mm was drilled, which was then sealed by wax. This was done to prevent the evaporation of water in coal during scanning, which would then affect the experimental results due to the changes in shape. Through measurements, it was determined that the coal core after sealing was 2.29 mm in diameter, as shown in Figure1. The sealed core was fixed at the tip of a toothpick, placed on the sample stage (as shown in Figure 2, the scanning accuracy of the device is as high as 0.5 m, which can achieve a 360-degree rotation) and fixed in place by screws. Next, the main control computer (MCC) was switched on so that it could operate the platform. The sample stage was adjusted to the center from four dimensions, x, y, z, and r; then, the X-ray source was opened and a mathematical model with a pixel size of 1024 ∗ 1024 ∗ 1024 was acquired through a super-wide field scanning, whose actual size was 1.024 mm ∗ 1.024 mm ∗ 1.024 mm. Table 1 lists the specific CT scanning parameters, in which SOD denotes the distance between the ray source and the detector and ODD denotes the distance between the sample stage and the detector. During the scanning process, a 20X lens was used as the detector, and the number of scanning frames was set as 900; that is, the sample was scanned once after the sample stage was rotated by 0.4°. The exposure time, that is, the time required for a rotation of the sample stage, was set as 65 s.Figure 1
Picture of a coal sample.Figure 2
Picture of the nanoVoxel-2000 series X-ray 3D microscope.Table 1
Setting of scanning parameters of the X-ray 3D microscope.
Detector
SOD/ODD (mm/mm)
Voltage (kV)
Current (µA)
Number of scanning frames
Consumed time (h)
Exposure time (s)
Penetration rate (%)
20X
35/13
41
240
900
19
65
57A micro-CT system generally adopts a 3D cone-beam reconstruction algorithm. Among various 3D cone-beam reconstruction algorithms, the approximation algorithm has a number of advantages, including simple mathematical expression and easy implementation, which means it can achieve favorable reconstruction results at relatively small cone angles, and thus, it is widely applied in practice. FDK algorithm is a mainstream algorithm in various approximation algorithms based on a filtered back projection. The FDK algorithm, proposed by Feldkamp et al., is a kind of approximate reconstruction algorithm based on circular orbit scanning. In this study, this algorithm was used to conduct a 3D reconstruction on the scanned images. It is essentially an expansion of the 2D fan-beam filtered back projection algorithm in 3D space and includes the steps of the preweighting of projection data, one-dimensional filtering, and back projection. Specifically, the FDK algorithm includes the following steps:(1)
First, weighted processing was conducted on the projection data, using a function similar to the cosine function, in order to modify the distance between the voxel and source point and the angle difference appropriately.(2)
Then, one-dimensional filtering was conducted on the projection data from different projection angles along the horizontal direction.(3)
Finally, a 3D back projection was performed along the direction of the X-ray. The reconstructed voxel values were the sum of the rays passing the voxel at all projection angles.The corresponding formulas are written as follows:(1)p˜β,a,b=wa,b⋅pβ,a,b∗ga,fFDKx,y,z=∫02πR2Ux,y,β2p˜β,a,bdβ,where(2)wa,b=RR2+a2+b2=cosγcosκ,Ux,y,β=R+xcosβ+ysinβ,where p˜β,a,b denotes the projection data; β denotes the position of the projection source, that is, the projection angle °; a,b denotes the coordinates of the detector array; ga denotes a frequently used ramp filter function; R denotes the radius of the circular orbit; γ denotes the sector angle; and κ denotes the cone angle [40].


## 2.2. The Representative Elementary Volume Analysis

The pores should be connected in simulating seepage behaviors during the water injection process. Generally, a larger 3D digital core can provide a more accurate characterization of a core’s micropore structure and more accurate simulation results of its macroscopic physical properties; however, it would place higher requirements on the computer’s storage and calculation capabilities. Since a computer generally has limited storage and calculation capacities, the size of a 3D core cannot be too large. Accordingly, a representative elementary volume (REV) analysis [41] was conducted on the constructed 3D digital core before the construction of a pore network model; that is, the most appropriate 3D digital core size should be selected so that the constructed model can accurately reflect the rock’s micropore structural and physical characteristics. A REV analysis of a digital core is primarily aimed at selecting the REV. We first randomly selected a point in the 3D digital core model and centered a cube on this point; then, we gradually increased the side length of this cube until the model’s porosity reached a stable value. The side length of the cubic model under this stable state was thus regarded as the optimal size. As shown in Figure 3, four different voxels were selected in the long-flame coal model, denoted as P1, P2, P3, and P4, respectively; then, using these four points as the centers, the different cubic regions were set. By gradually increasing the side length (a) of these cubes, four variation curves between the coal rock’s porosity and the cube’s side length were acquired. These curves represent the calculated porosities of the coal rocks using the four different voxels of the 3D digital rock as the cube centers. Table 2 lists the calculated porosity values of the long-flame coal sample when the cube’s side length was set at different values, and Figure 4 displays the variation of the calculated porosity and the side length of the cube. As shown in Figure 4, when the side length of the cube increased to 62 μm, the calculated porosity gradually tended to be stable, suggesting that this size of digital core can adequately characterize the core’s physical properties. For the convenience of calculation, the size of the coal model was finally determined to be 60 pixel ∗ 60 pixel ∗ 60 pixel, with an actual size of 0.06 mm × 0.06 mm × 0.06 mm.Figure 3
Distribution of the selected models using different points as the voxel centers.Table 2
Calculated results of coal rock porosity using different points (P1, P2, P3, and P4) as the voxel centers.
Size of the volume element (μm)
30
40
50
55
60
65
70
80
90
Calculated porosity using P1 as the voxel center (%)
34.56
30.95
28.18
26.43
26.32
26.24
26.31
26.07
26.30
Calculated porosity using P2 as the voxel center (%)
32.40
30.76
29.72
28.75
28.77
28.80
28.74
28.73
28.81
Calculated porosity using P3 as the voxel center (%)
35.74
33.19
32.01
30.27
30.10
30.13
30.09
30.15
30.20
Calculated porosity using P4 as the voxel center (%)
37.43
36.91
35.28
34.53
34.57
34.49
34.53
34.52
34.50Figure 4
Variations of the calculated porosity with the increase of volume element size for different models.It can also be observed from Figure4 that the calculated porosities are different for the same coal sample [42, 43]. Through a REV analysis, this study adopted four coal regions, whose centers were P1, P2, P3, and P4, respectively, as the models for simulating the seepage behaviors during water injections under different pressure gradients. These four models are hereafter referred to as the P1 model, P2 model, P3 model, and P4 model, respectively. The absolute porosities of these four regions were 26.32%, 28.77%, 30.10%, and 34.57%, respectively.


## 2.3. The Construction of the Geometric Model

Various kinds of system noises exist in digital cores after CT scanning, which would not only affect the image quality but also affect the subsequent quantitative analysis. Before image segmentation, the images should be filtered so as to reduce noises, improve the image quality, and thus make it easier for image segmentation and display. The common filtering algorithms include Gaussian filtering, variance filtering, median filtering, recursive exponential filtering, and low-pass linear filtering [44]. Through comparison, median filtering was selected in this study, which is a classical noise smoothing method and generally used for protecting the edge information. Figure 5 shows the model before and after filtering.Figure 5
Comparison of the P1 model before and after denoising: (a) before denoising and (b) after denoising.
(a)
(b)In order to better distinguish pores from skeletons and make quantitative descriptions, an appropriate binarization should be conducted on the grayscale images. On an Avizo platform, the coal model was first segmented by pressing the “Image Segmentation” button, and then the segmented image was displayed by pressing the “Surface View” button. Thus, the microscopic models of coal pores based on actual coal pore structures were acquired, as shown in Figures6–9.Figure 6
Model of coal’s micropore structure in model P1. (a) Coal sample. (b) Pores and skeletons. (c) Pores (with an absolute porosity of 26.30%).
(a)
(b)
(c)Figure 7
Model of coal’s micropore structure in model P2. (a) Coal sample (b) Pores and skeletons. (c) Pores (with an absolute porosity of 28.77%).
(a)
(b)
(c)Figure 8
Model of coal’s micropore structure in model P3. (a) Coal sample. (b) Pores and skeletons. (c) Pores (with an absolute porosity of 30.10%).
(a)
(b)
(c)Figure 9
Model of coal’s micropore structure in model P4. (a) Coal sample. (b) Pores and skeletons. (c) Pores (with an absolute porosity of 34.57%).
(a)
(b)
(c)


## 2.4. Grid Optimization and Simulation Parameter Setting

Next, the acquired pore model was imported into ICEM CFD software for grid generation and the establishment of the finite element model [45–47]. According to the requirements of the numerical simulations on seepage behaviors, the effective pores in the model should be interconnected. To ensure the successful implementation of simulations and a quick convergence, the pores that are disconnected with each other should be eliminated, that is, the islands should be eliminated. After island elimination, the effective porosities of the four models were 21.15%, 18.26%, 17.57%, and 23.43%, respectively. The pressure inlet and outlet were then set. To reduce the calculation burden, the pressure inlet and outlet were set as −X plane and +X plane, respectively, and the other four surfaces were set as free slippage walls, as shown in Figure 10. After the setting of the pressure inlet and outlet, the grids were automatically generated, as shown in Figure 11. We also checked the generated grids, eliminated the low-quality grids, and gradually improved the grid quality.Figure 10
Settings of boundary conditions.
(a)
(b)Figure 11
Grid generation.After the mesh has been divided effectively, the seepage behaviors in the four different coal microstructure models during the water injection process were simulated under different pressure gradients using CFX15.0 software. In simulations, the nonstationary Navier–Stokes equation, the most basic equation that can accurately describe the fluid’s actual flow characteristics, was used as the control equation. The standard k-epsilon turbulence equation was also used [48, 49]. The flow was set as downstream at normal temperature (298 K), the pressure at the outlet was kept at 0.1 MPa, and the inlet pressure was varied for different values, as shown in Table 3. Specifically, the inlet pressures of 0.3–4.3 MPa, 4.9–8.1 MPa, 8.9–13.1 MPa, and 14.1–18.1 MPa correspond to water injections at low pressure, medium pressure, high pressure, and ultrahigh pressure, respectively.Table 3
Average seepage velocities in coal seams under different pressure gradients.
Pressure increase (MPa)
Inlet pressure (MPa)
Pressure gradient (109 MPa·m−1)
Average seepage velocities in coal’s different regions (m·s−1)
P1
P2
P3
P4
0.2
0.3
3.93
1.01
1.24
1.18
0.95
0.5
7.97
1.77
2.22
2.11
1.70
0.7
12.14
2.40
3.07
2.91
2.30
0.9
16.52
2.95
3.82
3.65
2.83
1.1
21.16
3.44
4.53
4.39
3.32
1.3
26.77
3.98
5.19
5.35
3.79
0.4
1.7
35.36
4.92
6.12
6.13
4.70
2.1
45.31
5.72
7.15
7.19
5.48
2.5
53.26
6.46
8.14
7.55
6.01
2.9
62.47
7.14
9.04
8.42
6.31
3.3
71.88
7.79
9.68
9.21
6.91
3.7
82.88
8.36
10.71
9.99
7.49
0.6
4.3
99.06
9.21
11.95
11.12
8.32
4.9
113.87
10.02
12.63
12.20
9.11
5.5
130.78
10.8
13.75
13.25
9.85
6.1
148.37
11.52
14.85
14.31
10.59
6.7
161.51
12.24
16.27
15.37
11.31
7.3
174.18
13.05
17.14
16.24
12.01
0.8
8.1
196.07
13.98
18.01
17.35
12.92
8.9
230.39
14.95
18.94
18.42
13.82
9.7
259.86
15.84
20.84
19.45
14.66
10.5
285.29
16.69
21.98
20.45
15.52
11.3
310.44
17.54
23.10
21.41
16.08
12.1
339.06
18.39
24.20
22.35
17.37
1.0
13.1
373.47
19.39
25.55
23.48
18.39
14.1
407.36
20.44
26.88
24.58
19.06
15.1
443.90
21.48
28.20
25.64
19.90
16.1
480.27
22.40
29.50
26.68
20.72
17.1
518.03
23.38
30.80
27.67
21.52
18.1
556.90
24.35
32.11
28.66
22.30


## 3. Results and Discussion

The seepage parameters under different conditions are now analyzed based on the simulation results of the water injection seepage under different pressure gradients.


### 3.1. An Analysis of the Relationship between the Seepage Velocity and Pressure Gradient

Using CFD-POST software, the average seepage velocities at different water injection pressures were extracted. Table3 lists the calculated seepage velocities in four coal models under 30 different pressure gradients.As shown in Table3, the seepage velocities of the coal samples from different regions all increased gradually with the increase of the pressure gradient. For the same coal sample, the porosity differed slightly in the different regions. Therefore, under the same pressure gradient, the seepage velocity was mainly subjected to the connectivity of the pores. Obviously, the water seepage velocity is greater in the coal with more favorably interconnected pores. For example, under a pressure gradient of 556.90 × 109 MPa·m−1, the seepage velocities in the P2 and P4 models were 32.11 m·s−1 and 22.30 m·s−1, respectively; that is, the pores in P2 were better interconnected. In order to analyze the relationship between the pressure gradient and seepage velocity more clearly, the variations of the seepage velocity with the pressure gradient of the four different regions in the coal sample were plotted based on the data in Table 3, as shown in Figure 12.Figure 12
Variation of average seepage velocities and pressure gradient.More evidently, it can be observed that the water seepage velocities in the four different regions increased gradually with the increase of the pressure gradient. Both factors show an obvious nonlinear relationship. Wang et al. found that, under the conditions with quite large pores and fractures or large hydraulic slopes, the Reynolds number of water flow was significant, and the underground water seepage velocity exhibited a complex nonlinear relationship with the hydraulic slope [50]. Considering the complexity of nonlinearity, there is still a lack of a uniform formula for describing the flow to date. Currently, two formulas are commonly used. One of them describes the non-Darcy seepage relationship as proposed by Forchheimer in 1901, also referred to as the Forchheimer equation:(3)J=Av+Bv2,where(4)A=μk,B=βρ.The other is the exponential formula:(5)J=cv∧d,where J denotes the hydraulic gradient, with a unit of MPa·m−1; V denotes the seepage velocity, with a unit of m·s−1; A, B, c, and d are the constants connected with the seepage medium and fluid; μ denotes the coefficient of the dynamic viscosity, with a unit of Pa·s; k denotes the permeability, with a unit of mD; β denotes the non-Darcy coefficient, with a unit of m−1; and ρ denotes the density of the fluid, with a unit of kg/m3.Compared with the latter formula, the Forchheimer formula possesses more favorable theoretical foundations, which can be derived from the Navier–Stokers equation in fluid mechanics. Therefore, the Forchheimer formula was used in this study for fitting the relationship between the pressure gradientJ and seepage velocity v. The black curve in Figure 12 displays the fitting results. The fitting coefficient was as high as 0.99, suggesting an excellent fitting.Through fitting, the values of A and B of different curves were acquired; moreover, both the coefficient of the dynamic viscosity and seepage velocity of water at a normal temperature are constants. Then, according to equation (4), the coal’s permeability and non-Darcy coefficients were calculated, with the results listed in Table 4.Table 4
Coal’s permeability and non-Darcy coefficients after fitting.
Coal region
Effective porosity
Correlation coefficient
Fitted value ofA
Fitted value ofB
Permeability (k/10−4mD)
Non-Darcy coefficient (β/m−1)
P1
18.15%
0.99967
3.64826
0.94244
2.30
942
P2
15.26%
0.9995
3.44116
0.80657
2.44
807
P3
14.57%
0.99912
3.40512
0.65068
2.46
651
P4
20.43%
0.99926
3.35691
0.44063
2.50
441As shown in Table4, the permeability showed no monotonic increase with the increase of porosity. As for the relationship between porosity and permeability, some scholars concluded that a monotonic increasing relationship exists [51–53], while other scholars stated that the positive correlation between the two factors is not absolute and not applicable under all conditions. A low porosity is a sufficient but not necessary condition for low permeability; similarly, a high permeability is also a sufficient but not necessary condition for high porosity. For some high-porosity coal samples, the permeability may be low and the non-Darcy coefficient increases gradually with the increase of permeability [54, 55].


### 3.2. An Analysis of the Simulated Seepage Pressure and Velocity Fields in the Same Model under Different Pressure Gradients

For analyzing the effects of the water injection pressure on seepage in coal, the model P1 was selected in this study for seepage simulations during water injections under different pressure gradients, in which the injection pressure was set as 0.3 MPa, 1.7 MPa, 4.3 MPa, 8.1 MPa, 13.1 MPa, and 18.1 MPa. Figures 13–15 show the acquired pressure fields, velocity fields, and mass flow distributions under different pressure gradients, respectively.Figure 13
Distributions of pressure fields in model P1 under different water injection pressures. (a) 0.3 MPa. (b) 1.7 MPa. (c) 4.3 MPa. (d) 8.1 MPa. (e) 13.1 MPa. (f) 18.1 MPa.
(a)
(b)
(c)
(d)
(e)
(f)Figure 14
Distributions of velocity fields in model P1 under different water injection pressures. (a) 0.3 MPa. (b) 1.7 MPa. (c) 4.3 MPa. (d) 8.1 MPa. (e) 13.1 MPa. (f) 18.1 MPa.
(a)
(b)
(c)
(d)
(e)
(f)Figure 15
Distributions of mass flows in model P1 on different sections under different water injection pressures. (a) 0.3 MPa. (b) 1.7 MPa. (c) 4.3 MPa. (d) 8.1 MPa. (e) 13.1 MPa. (f) 18.1 MPa.
(a)
(b)
(c)
(d)
(e)
(f)As shown in Figure13, along the seepage direction (i.e., along the positive direction of X), the seepage pressure decreased gradually; however, when water passed through the small pores, the pressure rapidly decreased at first and then increased gradually, overall showing a decreasing trend. As the pressure gradient increased, the pressure field showed more significant variations. Due to the existence of some pore channels with a good connectivity and large diameters, the pressure fields changed more gently, suggesting that water passed more easily through these connected pores.It can be observed from Figure14 that the seepage velocity overall increased with the increase of the pressure gradient. The maximum seepage velocity changed most significantly, increasing from 11.49 m·s−1 at 0.3 MPa to 297.3 m·s−1 at 18.1 MPa. The maximum seepage velocity generally appeared at the small pores in the intermediate region with a well-developed flow. With the increase of the water injection pressure and the prolongation of injection time, some originally closed pores collapsed and became connected with the others to form the seepage channels for the passage of flowing water; while some originally interconnected pores were destroyed by large water pressures and some large fractures were formed, contributing to more regular seepage behaviors. One can also observe that at a water injection pressure of 8.1 MPa, the maximum seepage velocity, which was approximately 145 m·s−1, appeared in the black block region in Figure 14. As the water injection pressure increased to 13.1 MPa, the maximum seepage velocity increased to 191.2 m·s−1 and was also found at the same position; as the injection pressure further increased to 18.1 MPa, the maximum seepage velocity increased to 297.3 m·s−1, while the seepage velocity in the black block region was 200 m·s−1.As shown in Figure15, with the increase of the pressure gradient, the maximum mass flow on the same section increased gradually and appeared at the same position with the maximum seepage velocity. Under the same pressure gradient (e.g., 18.1 MPa), the maximum mass flow was largest (3.24 × 10−8 kg·s−1) at the inlet (X = 0 μm) and relatively smaller at the outlet (X = 60 μm), being approximately 2.32 × 10−8 kg·s−1. This is mainly due to the fact that the pore structures in coal are quite complex and some water cannot flow out through the outlet and still remains in the coal.In order to gain an in-depth insight into the water injection seepage rules of the same model under different pressure gradients, 13 sections that were equally spaced were selected, and the seepage parameters, which mainly included the average seepage pressure, average seepage velocity and average mass flow on each section, were simulated, as shown in Table5. Based on the data listed in Table 5, the variations of the average seepage pressure, average seepage velocity, and average mass flow rate with the seepage length were plotted, as shown in Figures 16–18.Table 5
Simulated results of average seepage pressure, average seepage velocity, and average mass flow on each section under different inlet pressures.
Inlet pressure (MPa)
Seepage parameters
Distance from the inlet section (μm)
0
5
10
15
20
25
30
35
40
45
50
55
60
0.3
P1
0.30
0..29
0.29
0.26
0.23
0.21
0.18
0.16
0.15
0.13
0.12
0.11
0.10
V2
0.82
0.92
1.19
1.63
1.54
1.69
1.51
1.26
1.31
1.00
1.15
1.10
1.17
M3
13.34
3.96
4.45
5.61
5.59
6.10
5.35
4.93
4.31
4.12
4.19
4.61
5.12
1.7
P
1.70
1.69
1.62
1.38
1.05
0.87
0.71
0.50
0.44
0.30
0.25
0.15
0.10
V
4.20
4.58
5.96
7.93
7.77
7.99
7.43
6.73
6.70
4.61
5.82
5.27
5.31
M
37.15
18.98
21.34
26.92
26.82
29.27
25.69
23.62
20.65
19.76
20.09
22.13
27.33
4.3
P
4.30
4.27
4.11
3.49
2.60
2.13
1.71
1.13
1.00
0.62
0.52
0.23
0.10
V
7.86
8.77
10.99
14.49
14.10
14.77
14.08
13.25
12.83
8.87
11.10
9.71
9.72
M
61.01
34.00
38.23
48.22
48.04
52.43
46.01
42.31
36.99
35.40
36.00
39.65
45.55
8.1
P
8.10
8.06
7.76
6.57
4.83
3.96
3.17
2.04
1.77
1.08
0.92
0.36
0.10
V
12.75
13.67
16.06
20.72
20.70
21.97
21.03
20.35
19.52
13.39
16.67
14.83
14.56
M
84.14
49.22
55.34
69.80
69.53
75.90
66.60
61.25
53.55
51.24
52.10
57.39
64.06
13.1
P
13.25
13.08
12.48
10.54
7.75
6.39
5.13
3.26
2.68
1.67
1.47
0.55
0.10
V
19.55
19.46
20.88
26.97
27.91
29.31
28.25
28.26
26.69
18.29
22.77
20.15
19.47
M
107.91
64.66
72.70
91.70
91.35
99.70
87.50
80.47
70.34
67.26
68.43
75.39
85.92
18.1
P
18.10
18.38
17.34
14.67
10.63
8.74
7.00
4.44
3.64
2.28
2.01
0.74
0.10
V
26.54
26.07
25.11
32.72
34.09
35.39
33.81
34.76
32.6
22.31
27.75
24.52
23.33
M
126.26
76.67
86.21
108.74
108.33
118.24
103.77
95.43
83.41
79.75
81.15
89.4
101.49
Note:1P denotes the average seepage pressure, with a unit of MPa, 2V denotes the average seepage velocity, with a unit of m·s−1, and 3M denotes the average mass flow, with a unit of 10−11 Kg·s−1.Figure 16
Variations of average seepage pressure with seepage length under different inlet pressures.Figure 17
Variations of average velocity with seepage length under different inlet pressures.Figure 18
Variations of average seepage mass flow with seepage length under different inlet pressures.By analyzing Table5 and Figures 16–18, the following conclusions were reached:(1)
As shown in Figure16, at a water injection pressure of 0.3 MPa, the seepage pressure varied slowly; as the water injection pressure increased, the seepage pressure varied more significantly. Overall, the average seepage pressure showed almost identical variation tendencies under different inlet pressure conditions, all decreasing gradually with the increase of the seepage length.(2)
As shown in Figure17, at a water injection pressure of 0.3 MPa, the seepage velocity presented a mild variation; as the injection pressure increased, the average seepage velocities overall showed a variation tendency of increase-decrease-increase-decrease. On the basis of the variation tendency, we divided the whole seepage length into the following four segments for an in-depth analysis: 0∼10 μm, 10∼25 μm, 25∼45 μm, and 45∼60 μm. In the first segment (0∼10 μm), at an inlet pressure below 13.1 MPa, the average seepage velocities increased with the increase of the seepage length, and at an inlet pressure of 18.1 MPa, the average seepage velocity decreased gradually with the increase of the seepage length. This is because with an inlet pressure of 18.1 MPa, part of the water injection pressure contributed to the expansion of the cross section. In the second segment (10∼25 μm), at an inlet pressure below 8.1 MPa, the average seepage velocities increased rapidly at first and then decreased slowly and eventually increased, and at the inlet pressure above 13.1 MPa, the average seepage velocities increased rapidly at first, then slowly, and finally reached the maximum at a seepage length of 25 μm. In the third segment (25∼45 μm), at an inlet pressure below 8.1 MPa, the average seepage velocities decreased, and at the inlet pressure above 13.1 MPa, the average seepage velocities underwent similar variations of decrease-increase-decrease. In the fourth segment (45∼60 μm), the average seepage velocities first increased and then decreased. Although the seepage velocity varied greatly in the entire water injection process, the seepage velocities at the inlet and outlet only differed slightly. At the same seepage length, the average seepage velocities changed significantly under different pressure gradients; overall, the greater the pressure gradient, the larger the average seepage velocity. For example, when the seepage length was 25 μm, the seepage velocity increased from 1.69 m·s−1 to 35.39 m·s−1 as the inlet pressure increased from 0.3 MPa to 18.1 MPa.(3)
As shown in Figure18, at an injection pressure of 0.3 MPa, the average mass flows at various sections varied slightly; however, with the increase of injection pressure, the average mass flows of various sections along the direction of the seepage length showed significant changes, which were all greatest at the inlet (i.e., at a seepage length of 0 μm). At a seepage length of 0∼5 μm, the average mass flows decreased dramatically; as the seepage length increased to 5∼25 μm, the average mass flows first increased and then decreased and finally increased, showing overall increasing trends; as the seepage length further increased to 25∼45 μm, the average mass flows decreased gradually; finally, as the seepage length increased to 45∼60 μm, the average mass flows increased gradually. By analyzing Figures 17 and 18, it can be concluded that the parameters showed obvious changes at the seepage lengths of 25 μm and 45 μm. To be specific, the average seepage velocities and average mass flow rates reached their maximum levels at 25 μm (the results at the inlet were not taken into account); at 45 μm, both the average seepage velocities and average mass flow rates began to increase. Due to the complexity of pores, the effective cross-sectional area of pore distribution differs among different analysis nodes and the channel showed many twists and turns; accordingly, some back flows passed some pores, while no flow of water passed others, and the average mass flow exhibited a variation trend of decrease-increase-decrease-increase. This was concluded from the present pore structure model, and the average mass flow shows different variations in different pore structure models; however, both the inlet and outlet mass flows imposed slight effects and the mass flow shows almost identical variation tendencies under different hydraulic pressures. This can further verify the existence of the effects of the pore structure.


### 3.3. An Analysis of the Seepage Pressure and Velocity Fields in Different Models under Different Water Injection Pressure Gradients

The porosities of different regions are different even in the same coal sample. Thus, a coal sample may exhibit different water injection processes under the same injection gradient. In this study, the seepage rules of four different models were analyzed with four types of pressure gradients, low pressure (4.3 MPa), medium pressure (8.1 MPa), high pressure (13.1 MPa), and ultrahigh pressure (18.1 MPa). Figures19–22 show the simulated pressure and velocity fields of the four models under these different inlet pressures, respectively.Figure 19
Distribution of pressure and velocity fields in four different models under a low inlet pressure of 4.3 MPa. (a) Results in model P1. (b) Results in model P2. (c) Results in model P3. (d) Results in model P4.
(a)
(b)
(c)
(d)Figure 20
Distribution of pressure and velocity fields in four different models under a medium inlet pressure of 8.1 MPa. (a) Results in model P1. (b) Results in model P2. (c) Results in model P3. (d) Results in model P4.
(a)
(b)
(c)
(d)Figure 21
Distribution of pressure and velocity fields in four different models under a high inlet pressure of 13.1 MPa. (a) Results in model P1. (b) Results in model P2. (c) Results in model P3. (d) Results in model P4.
(a)
(b)
(c)
(d)Figure 22
Distribution of pressure and velocity fields in four different models under an ultrahigh inlet pressure of 18.1 MPa. (a) Results in model P1. (b) Results in model P2. (c) Results in model P3. (d) Results in model P4.
(a)
(b)
(c)
(d)Table6 lists the simulated seepage parameters in the four models under a low inlet pressure of 4.3 MPa. When this table is analyzed in combination with Figure 19, the following can be concluded:(1)
Under an injection pressure of 4.3 MPa, the pressure fields of the four models were different. The maximum pressures all fluctuated around 4.3 MPa and appeared around the inlet; similarly, the minimum pressures fluctuated around 0.1 MPa and appeared around the outlet.(2)
Under an injection pressure of 4.3 MPa, the velocity fields of the four models also showed significant differences. The maximum seepage velocities appeared in the intermediate region where the pore diameter was suddenly reduced. In the four models, the maximum seepage velocities differed slightly; specifically, the maximum seepage velocity in the P4 model was greatest, followed by those in the P1 and P2 models, while the value in P3 was smallest. This is mainly because at the center of the small pore channel of the section where the water flowed, the water seepage velocity increased as the pore radius and bending degree decreased.(3)
Under the same injection pressure, the total mass flows at different inlet areas were different. At the inlet area of 12.10 × 10−10 m2, the total inlet mass flow was 5.61 × 10−6 kg·s−1; at the inlet area of 9.29 × 10−10 m2, the total inlet mass flow was 10 × 10−6 kg·s−1; at the inlet area of 13.02 × 10−10 m2, the total inlet mass flow was 9.90 × 10−6 kg·s−1; and at the inlet area of 9.60 × 10−10m2, the total inlet mass flow was 7.13 × 10−6 kg·s−1. When these are ranked in a descending order, the total mass flow in the P2 model is greatest, followed by the values in P3 and P4, and finally by that in P1. Since the present seepage simulations were conducted based on an actual coal pore model, water would flow into the pores that were not connected with the outlet due to the seepage effect and the total mass flows at the outlet were smaller than those at the inlet.Table 6
Simulated results of seepage parameters in four models under a low inlet pressure of 4.3 MPa.
Model
P
max (MPa)
P
min (MPa)
V
max (m·s−1)
V
min (10−5 m·s−1)
S
IN (10−10 m2)
M
IN (10−6 kg·s−1)
S
OUT (10−10 m2)
M
OUT (10−6 kg·s−1)
P1
4.13
0.08
97.60
10.16
12.10
5.61
8.37
5.61
P2
4.33
0.11
87.08
6.38
9.29
10
8.30
9.98
P3
4.38
0.11
81.55
4.21
13.02
9.90
8.60
9.89
P4
4.32
0.11
98.70
3.28
9.60
7.13
8.53
7.13Table7 lists the seepage parameters of the four models under a medium inlet pressure of 8.1 MPa. When this table is analyzed in combination with Figure 20, the following conclusions regarding the medium inlet pressure of 8.1 MPa can be obtained:(1)
The pressure and velocity fields under a medium inlet pressure were similar to those under a low pressure. With the increase of the water injection pressure, the maximum seepage velocity increased to be approximately 125 m·s−1, with basically the same increasing amplitudes in the four models. The maximum seepage velocity in the P4 model was greatest, followed by those in P1 and P3, and finally by the value in P2.(2)
Compared with the results at a low inlet pressure (4.3 MPa), the total mass flows at an 8.1 MPa at the inlet and outlet increased. The total mass flow at the inlet in the P2 model was greatest (14.95 × 10−6 kg·s−1), followed by the values in the P3 and P4 models (13.78 × 10−6 kg·s−1 and 10.54 × 10−6 kg·s−1, resp.); finally, the value in the P1 model was smallest (7.40 × 10−6 kg·s−1). Since the inlet area remained unchanged, the total mass flow increased gradually with the increase of pressure.Table 7
Simulated results of seepage parameters in four models under a medium inlet pressure of 8.1 MPa.
Model
P
max (MPa)
P
min (MPa)
V
max (m·s−1)
V
min (10−5 m·s−1)
S
IN (10−10 m2)
M
IN (10−6 kg·s−1)
S
OUT (10−10 m2)
M
OUT (10−6 kg·s−1)
P1
8.17
0.062
145.4
17.37
12.10
7.40
8.37
7.32
P2
8.08
0.13
117.3
2.213
9.29
14.95
8.30
14.90
P3
8.12
0.11
121.8
0.9
13.02
13.78
8.60
13.78
P4
8.11
0.15
150.8
6.042
9.60
10.54
8.53
10.54Table8 lists the seepage parameters of the four models under a high inlet pressure of 13.1 MPa. When this table is analyzed in combination with Figure 21, the following conclusions regarding the high inlet pressure of 13.1 MPa can be obtained:(1)
The seepage behaviors in the four models showed almost the same variation tendencies. The pressures decreased along the seepage direction, which were greatest at the inlet and smallest at the outlet. Although these four models had different porosities, the seepage pressures at both the inlet and outlet fluctuated at around 13.1 MPa and showed slight variations. The average seepage velocities varied significantly in the intermediate regions, and the maximum seepage velocities reached up to 202.4 m·s−1.(2)
Compared with the results under low and medium inlet pressures, the total mass flows at the inlet and outlet under a high pressure of 13.1 MPa increased more significantly. Owing to the increase of the water injection pressure, some pores which were originally closed were opened and connected with each other to form seepage channels; thus, more water flowed out compared with the conditions under low and medium pressures. Under a high inlet pressure, the mass flow differences between the inlet and outlet were small.(3)
The total mass flow in the P2 model was greatest, followed by the values in the P3 and P4 models, and finally by the value in the P1 model. Similarly, with the increase of the pressure gradient, the total mass flows increased, with almost the same increasing amplitudes in the four models.Table 8
Simulated results of seepage parameters in four models under a high inlet pressure of 13.1 MPa.
Model
P
max (MPa)
P
min (MPa)
V
max (m·s−1)
V
min (10−5 m·s−1)
S
IN (10−10 m2)
M
IN (10−6 kg·s−1)
S
OUT (10−10 m2)
M
OUT (10−6 kg·s−1)
P1
13.2
0.11
191.2
14
12.10
9.56
8.37
9.49
P2
13.1
0.14
256
1.37
9.29
20.26
8.30
20.1
P3
13.07
0.092
161.8
3.43
13.02
19.83
8.60
19.76
P4
13.1
0.1
202.4
3.80
9.60
13.94
8.53
13.93Table9 lists the seepage parameters of the four models under an ultrahigh inlet pressure of 18.1 MPa. When this table is analyzed in combination with Figure 22, the following conclusions regarding the ultrahigh inlet pressure of 18.1 MPa can be reached:(1)
The seepage behaviors showed same variation tendencies in the four models. With the increase of the pressure gradient, the maximum seepage velocity increased steadily. The maximum seepage velocity in the P2 model was as high as 469.9 m·s−1, while the minimum seepage velocity remained at approximately 2 × 10−5 m·s−1. The total mass flow increased steadily with the increase of the pressure gradient. The total mass flows in the four models differed slightly; specifically, the total mass flow in the P2 model was greatest, followed by the values in the P3 and P4 models, and finally by the value in the P1 model.(2)
At 18.1 MPa, some pores that originally showed a favorable connectivity might have broken under the impact of the hydraulic pressure, and some large fractures were formed. Thus, fluids flowed out easily. Taking the P1 model as the example, water was more inclined to flow out through the left-center pores and less inclined to flow out through the upper right pores.Table 9
Simulated results of seepage parameters in four models under an ultrahigh inlet pressure of 18.1 MPa.
Model
P
max (MPa)
P
min (MPa)
V
max (m·s−1)
V
min (10−5 m·s−1)
S
IN (10−10 m2)
M
IN (10−6 kg·s−1)
S
OUT (10−10 m2)
M
OUT (10−6 kg·s−1)
P1
17.99
0.093
297.3
3.43
12.10
11.50
8.37
11.23
P2
18.02
0.13
469.9
1.97
9.29
24.96
8.30
24.42
P3
18.1
0.1
198.1
3.49
13.02
23.86
8.60
23.76
P4
18.01
0.11
243.2
2.83
9.60
16.64
8.53
16.63


## 3.1. An Analysis of the Relationship between the Seepage Velocity and Pressure Gradient

Using CFD-POST software, the average seepage velocities at different water injection pressures were extracted. Table3 lists the calculated seepage velocities in four coal models under 30 different pressure gradients.As shown in Table3, the seepage velocities of the coal samples from different regions all increased gradually with the increase of the pressure gradient. For the same coal sample, the porosity differed slightly in the different regions. Therefore, under the same pressure gradient, the seepage velocity was mainly subjected to the connectivity of the pores. Obviously, the water seepage velocity is greater in the coal with more favorably interconnected pores. For example, under a pressure gradient of 556.90 × 109 MPa·m−1, the seepage velocities in the P2 and P4 models were 32.11 m·s−1 and 22.30 m·s−1, respectively; that is, the pores in P2 were better interconnected. In order to analyze the relationship between the pressure gradient and seepage velocity more clearly, the variations of the seepage velocity with the pressure gradient of the four different regions in the coal sample were plotted based on the data in Table 3, as shown in Figure 12.Figure 12
Variation of average seepage velocities and pressure gradient.More evidently, it can be observed that the water seepage velocities in the four different regions increased gradually with the increase of the pressure gradient. Both factors show an obvious nonlinear relationship. Wang et al. found that, under the conditions with quite large pores and fractures or large hydraulic slopes, the Reynolds number of water flow was significant, and the underground water seepage velocity exhibited a complex nonlinear relationship with the hydraulic slope [50]. Considering the complexity of nonlinearity, there is still a lack of a uniform formula for describing the flow to date. Currently, two formulas are commonly used. One of them describes the non-Darcy seepage relationship as proposed by Forchheimer in 1901, also referred to as the Forchheimer equation:(3)J=Av+Bv2,where(4)A=μk,B=βρ.The other is the exponential formula:(5)J=cv∧d,where J denotes the hydraulic gradient, with a unit of MPa·m−1; V denotes the seepage velocity, with a unit of m·s−1; A, B, c, and d are the constants connected with the seepage medium and fluid; μ denotes the coefficient of the dynamic viscosity, with a unit of Pa·s; k denotes the permeability, with a unit of mD; β denotes the non-Darcy coefficient, with a unit of m−1; and ρ denotes the density of the fluid, with a unit of kg/m3.Compared with the latter formula, the Forchheimer formula possesses more favorable theoretical foundations, which can be derived from the Navier–Stokers equation in fluid mechanics. Therefore, the Forchheimer formula was used in this study for fitting the relationship between the pressure gradientJ and seepage velocity v. The black curve in Figure 12 displays the fitting results. The fitting coefficient was as high as 0.99, suggesting an excellent fitting.Through fitting, the values of A and B of different curves were acquired; moreover, both the coefficient of the dynamic viscosity and seepage velocity of water at a normal temperature are constants. Then, according to equation (4), the coal’s permeability and non-Darcy coefficients were calculated, with the results listed in Table 4.Table 4
Coal’s permeability and non-Darcy coefficients after fitting.
Coal region
Effective porosity
Correlation coefficient
Fitted value ofA
Fitted value ofB
Permeability (k/10−4mD)
Non-Darcy coefficient (β/m−1)
P1
18.15%
0.99967
3.64826
0.94244
2.30
942
P2
15.26%
0.9995
3.44116
0.80657
2.44
807
P3
14.57%
0.99912
3.40512
0.65068
2.46
651
P4
20.43%
0.99926
3.35691
0.44063
2.50
441As shown in Table4, the permeability showed no monotonic increase with the increase of porosity. As for the relationship between porosity and permeability, some scholars concluded that a monotonic increasing relationship exists [51–53], while other scholars stated that the positive correlation between the two factors is not absolute and not applicable under all conditions. A low porosity is a sufficient but not necessary condition for low permeability; similarly, a high permeability is also a sufficient but not necessary condition for high porosity. For some high-porosity coal samples, the permeability may be low and the non-Darcy coefficient increases gradually with the increase of permeability [54, 55].


## 3.2. An Analysis of the Simulated Seepage Pressure and Velocity Fields in the Same Model under Different Pressure Gradients

For analyzing the effects of the water injection pressure on seepage in coal, the model P1 was selected in this study for seepage simulations during water injections under different pressure gradients, in which the injection pressure was set as 0.3 MPa, 1.7 MPa, 4.3 MPa, 8.1 MPa, 13.1 MPa, and 18.1 MPa. Figures 13–15 show the acquired pressure fields, velocity fields, and mass flow distributions under different pressure gradients, respectively.Figure 13
Distributions of pressure fields in model P1 under different water injection pressures. (a) 0.3 MPa. (b) 1.7 MPa. (c) 4.3 MPa. (d) 8.1 MPa. (e) 13.1 MPa. (f) 18.1 MPa.
(a)
(b)
(c)
(d)
(e)
(f)Figure 14
Distributions of velocity fields in model P1 under different water injection pressures. (a) 0.3 MPa. (b) 1.7 MPa. (c) 4.3 MPa. (d) 8.1 MPa. (e) 13.1 MPa. (f) 18.1 MPa.
(a)
(b)
(c)
(d)
(e)
(f)Figure 15
Distributions of mass flows in model P1 on different sections under different water injection pressures. (a) 0.3 MPa. (b) 1.7 MPa. (c) 4.3 MPa. (d) 8.1 MPa. (e) 13.1 MPa. (f) 18.1 MPa.
(a)
(b)
(c)
(d)
(e)
(f)As shown in Figure13, along the seepage direction (i.e., along the positive direction of X), the seepage pressure decreased gradually; however, when water passed through the small pores, the pressure rapidly decreased at first and then increased gradually, overall showing a decreasing trend. As the pressure gradient increased, the pressure field showed more significant variations. Due to the existence of some pore channels with a good connectivity and large diameters, the pressure fields changed more gently, suggesting that water passed more easily through these connected pores.It can be observed from Figure14 that the seepage velocity overall increased with the increase of the pressure gradient. The maximum seepage velocity changed most significantly, increasing from 11.49 m·s−1 at 0.3 MPa to 297.3 m·s−1 at 18.1 MPa. The maximum seepage velocity generally appeared at the small pores in the intermediate region with a well-developed flow. With the increase of the water injection pressure and the prolongation of injection time, some originally closed pores collapsed and became connected with the others to form the seepage channels for the passage of flowing water; while some originally interconnected pores were destroyed by large water pressures and some large fractures were formed, contributing to more regular seepage behaviors. One can also observe that at a water injection pressure of 8.1 MPa, the maximum seepage velocity, which was approximately 145 m·s−1, appeared in the black block region in Figure 14. As the water injection pressure increased to 13.1 MPa, the maximum seepage velocity increased to 191.2 m·s−1 and was also found at the same position; as the injection pressure further increased to 18.1 MPa, the maximum seepage velocity increased to 297.3 m·s−1, while the seepage velocity in the black block region was 200 m·s−1.As shown in Figure15, with the increase of the pressure gradient, the maximum mass flow on the same section increased gradually and appeared at the same position with the maximum seepage velocity. Under the same pressure gradient (e.g., 18.1 MPa), the maximum mass flow was largest (3.24 × 10−8 kg·s−1) at the inlet (X = 0 μm) and relatively smaller at the outlet (X = 60 μm), being approximately 2.32 × 10−8 kg·s−1. This is mainly due to the fact that the pore structures in coal are quite complex and some water cannot flow out through the outlet and still remains in the coal.In order to gain an in-depth insight into the water injection seepage rules of the same model under different pressure gradients, 13 sections that were equally spaced were selected, and the seepage parameters, which mainly included the average seepage pressure, average seepage velocity and average mass flow on each section, were simulated, as shown in Table5. Based on the data listed in Table 5, the variations of the average seepage pressure, average seepage velocity, and average mass flow rate with the seepage length were plotted, as shown in Figures 16–18.Table 5
Simulated results of average seepage pressure, average seepage velocity, and average mass flow on each section under different inlet pressures.
Inlet pressure (MPa)
Seepage parameters
Distance from the inlet section (μm)
0
5
10
15
20
25
30
35
40
45
50
55
60
0.3
P1
0.30
0..29
0.29
0.26
0.23
0.21
0.18
0.16
0.15
0.13
0.12
0.11
0.10
V2
0.82
0.92
1.19
1.63
1.54
1.69
1.51
1.26
1.31
1.00
1.15
1.10
1.17
M3
13.34
3.96
4.45
5.61
5.59
6.10
5.35
4.93
4.31
4.12
4.19
4.61
5.12
1.7
P
1.70
1.69
1.62
1.38
1.05
0.87
0.71
0.50
0.44
0.30
0.25
0.15
0.10
V
4.20
4.58
5.96
7.93
7.77
7.99
7.43
6.73
6.70
4.61
5.82
5.27
5.31
M
37.15
18.98
21.34
26.92
26.82
29.27
25.69
23.62
20.65
19.76
20.09
22.13
27.33
4.3
P
4.30
4.27
4.11
3.49
2.60
2.13
1.71
1.13
1.00
0.62
0.52
0.23
0.10
V
7.86
8.77
10.99
14.49
14.10
14.77
14.08
13.25
12.83
8.87
11.10
9.71
9.72
M
61.01
34.00
38.23
48.22
48.04
52.43
46.01
42.31
36.99
35.40
36.00
39.65
45.55
8.1
P
8.10
8.06
7.76
6.57
4.83
3.96
3.17
2.04
1.77
1.08
0.92
0.36
0.10
V
12.75
13.67
16.06
20.72
20.70
21.97
21.03
20.35
19.52
13.39
16.67
14.83
14.56
M
84.14
49.22
55.34
69.80
69.53
75.90
66.60
61.25
53.55
51.24
52.10
57.39
64.06
13.1
P
13.25
13.08
12.48
10.54
7.75
6.39
5.13
3.26
2.68
1.67
1.47
0.55
0.10
V
19.55
19.46
20.88
26.97
27.91
29.31
28.25
28.26
26.69
18.29
22.77
20.15
19.47
M
107.91
64.66
72.70
91.70
91.35
99.70
87.50
80.47
70.34
67.26
68.43
75.39
85.92
18.1
P
18.10
18.38
17.34
14.67
10.63
8.74
7.00
4.44
3.64
2.28
2.01
0.74
0.10
V
26.54
26.07
25.11
32.72
34.09
35.39
33.81
34.76
32.6
22.31
27.75
24.52
23.33
M
126.26
76.67
86.21
108.74
108.33
118.24
103.77
95.43
83.41
79.75
81.15
89.4
101.49
Note:1P denotes the average seepage pressure, with a unit of MPa, 2V denotes the average seepage velocity, with a unit of m·s−1, and 3M denotes the average mass flow, with a unit of 10−11 Kg·s−1.Figure 16
Variations of average seepage pressure with seepage length under different inlet pressures.Figure 17
Variations of average velocity with seepage length under different inlet pressures.Figure 18
Variations of average seepage mass flow with seepage length under different inlet pressures.By analyzing Table5 and Figures 16–18, the following conclusions were reached:(1)
As shown in Figure16, at a water injection pressure of 0.3 MPa, the seepage pressure varied slowly; as the water injection pressure increased, the seepage pressure varied more significantly. Overall, the average seepage pressure showed almost identical variation tendencies under different inlet pressure conditions, all decreasing gradually with the increase of the seepage length.(2)
As shown in Figure17, at a water injection pressure of 0.3 MPa, the seepage velocity presented a mild variation; as the injection pressure increased, the average seepage velocities overall showed a variation tendency of increase-decrease-increase-decrease. On the basis of the variation tendency, we divided the whole seepage length into the following four segments for an in-depth analysis: 0∼10 μm, 10∼25 μm, 25∼45 μm, and 45∼60 μm. In the first segment (0∼10 μm), at an inlet pressure below 13.1 MPa, the average seepage velocities increased with the increase of the seepage length, and at an inlet pressure of 18.1 MPa, the average seepage velocity decreased gradually with the increase of the seepage length. This is because with an inlet pressure of 18.1 MPa, part of the water injection pressure contributed to the expansion of the cross section. In the second segment (10∼25 μm), at an inlet pressure below 8.1 MPa, the average seepage velocities increased rapidly at first and then decreased slowly and eventually increased, and at the inlet pressure above 13.1 MPa, the average seepage velocities increased rapidly at first, then slowly, and finally reached the maximum at a seepage length of 25 μm. In the third segment (25∼45 μm), at an inlet pressure below 8.1 MPa, the average seepage velocities decreased, and at the inlet pressure above 13.1 MPa, the average seepage velocities underwent similar variations of decrease-increase-decrease. In the fourth segment (45∼60 μm), the average seepage velocities first increased and then decreased. Although the seepage velocity varied greatly in the entire water injection process, the seepage velocities at the inlet and outlet only differed slightly. At the same seepage length, the average seepage velocities changed significantly under different pressure gradients; overall, the greater the pressure gradient, the larger the average seepage velocity. For example, when the seepage length was 25 μm, the seepage velocity increased from 1.69 m·s−1 to 35.39 m·s−1 as the inlet pressure increased from 0.3 MPa to 18.1 MPa.(3)
As shown in Figure18, at an injection pressure of 0.3 MPa, the average mass flows at various sections varied slightly; however, with the increase of injection pressure, the average mass flows of various sections along the direction of the seepage length showed significant changes, which were all greatest at the inlet (i.e., at a seepage length of 0 μm). At a seepage length of 0∼5 μm, the average mass flows decreased dramatically; as the seepage length increased to 5∼25 μm, the average mass flows first increased and then decreased and finally increased, showing overall increasing trends; as the seepage length further increased to 25∼45 μm, the average mass flows decreased gradually; finally, as the seepage length increased to 45∼60 μm, the average mass flows increased gradually. By analyzing Figures 17 and 18, it can be concluded that the parameters showed obvious changes at the seepage lengths of 25 μm and 45 μm. To be specific, the average seepage velocities and average mass flow rates reached their maximum levels at 25 μm (the results at the inlet were not taken into account); at 45 μm, both the average seepage velocities and average mass flow rates began to increase. Due to the complexity of pores, the effective cross-sectional area of pore distribution differs among different analysis nodes and the channel showed many twists and turns; accordingly, some back flows passed some pores, while no flow of water passed others, and the average mass flow exhibited a variation trend of decrease-increase-decrease-increase. This was concluded from the present pore structure model, and the average mass flow shows different variations in different pore structure models; however, both the inlet and outlet mass flows imposed slight effects and the mass flow shows almost identical variation tendencies under different hydraulic pressures. This can further verify the existence of the effects of the pore structure.


## 3.3. An Analysis of the Seepage Pressure and Velocity Fields in Different Models under Different Water Injection Pressure Gradients

The porosities of different regions are different even in the same coal sample. Thus, a coal sample may exhibit different water injection processes under the same injection gradient. In this study, the seepage rules of four different models were analyzed with four types of pressure gradients, low pressure (4.3 MPa), medium pressure (8.1 MPa), high pressure (13.1 MPa), and ultrahigh pressure (18.1 MPa). Figures19–22 show the simulated pressure and velocity fields of the four models under these different inlet pressures, respectively.Figure 19
Distribution of pressure and velocity fields in four different models under a low inlet pressure of 4.3 MPa. (a) Results in model P1. (b) Results in model P2. (c) Results in model P3. (d) Results in model P4.
(a)
(b)
(c)
(d)Figure 20
Distribution of pressure and velocity fields in four different models under a medium inlet pressure of 8.1 MPa. (a) Results in model P1. (b) Results in model P2. (c) Results in model P3. (d) Results in model P4.
(a)
(b)
(c)
(d)Figure 21
Distribution of pressure and velocity fields in four different models under a high inlet pressure of 13.1 MPa. (a) Results in model P1. (b) Results in model P2. (c) Results in model P3. (d) Results in model P4.
(a)
(b)
(c)
(d)Figure 22
Distribution of pressure and velocity fields in four different models under an ultrahigh inlet pressure of 18.1 MPa. (a) Results in model P1. (b) Results in model P2. (c) Results in model P3. (d) Results in model P4.
(a)
(b)
(c)
(d)Table6 lists the simulated seepage parameters in the four models under a low inlet pressure of 4.3 MPa. When this table is analyzed in combination with Figure 19, the following can be concluded:(1)
Under an injection pressure of 4.3 MPa, the pressure fields of the four models were different. The maximum pressures all fluctuated around 4.3 MPa and appeared around the inlet; similarly, the minimum pressures fluctuated around 0.1 MPa and appeared around the outlet.(2)
Under an injection pressure of 4.3 MPa, the velocity fields of the four models also showed significant differences. The maximum seepage velocities appeared in the intermediate region where the pore diameter was suddenly reduced. In the four models, the maximum seepage velocities differed slightly; specifically, the maximum seepage velocity in the P4 model was greatest, followed by those in the P1 and P2 models, while the value in P3 was smallest. This is mainly because at the center of the small pore channel of the section where the water flowed, the water seepage velocity increased as the pore radius and bending degree decreased.(3)
Under the same injection pressure, the total mass flows at different inlet areas were different. At the inlet area of 12.10 × 10−10 m2, the total inlet mass flow was 5.61 × 10−6 kg·s−1; at the inlet area of 9.29 × 10−10 m2, the total inlet mass flow was 10 × 10−6 kg·s−1; at the inlet area of 13.02 × 10−10 m2, the total inlet mass flow was 9.90 × 10−6 kg·s−1; and at the inlet area of 9.60 × 10−10m2, the total inlet mass flow was 7.13 × 10−6 kg·s−1. When these are ranked in a descending order, the total mass flow in the P2 model is greatest, followed by the values in P3 and P4, and finally by that in P1. Since the present seepage simulations were conducted based on an actual coal pore model, water would flow into the pores that were not connected with the outlet due to the seepage effect and the total mass flows at the outlet were smaller than those at the inlet.Table 6
Simulated results of seepage parameters in four models under a low inlet pressure of 4.3 MPa.
Model
P
max (MPa)
P
min (MPa)
V
max (m·s−1)
V
min (10−5 m·s−1)
S
IN (10−10 m2)
M
IN (10−6 kg·s−1)
S
OUT (10−10 m2)
M
OUT (10−6 kg·s−1)
P1
4.13
0.08
97.60
10.16
12.10
5.61
8.37
5.61
P2
4.33
0.11
87.08
6.38
9.29
10
8.30
9.98
P3
4.38
0.11
81.55
4.21
13.02
9.90
8.60
9.89
P4
4.32
0.11
98.70
3.28
9.60
7.13
8.53
7.13Table7 lists the seepage parameters of the four models under a medium inlet pressure of 8.1 MPa. When this table is analyzed in combination with Figure 20, the following conclusions regarding the medium inlet pressure of 8.1 MPa can be obtained:(1)
The pressure and velocity fields under a medium inlet pressure were similar to those under a low pressure. With the increase of the water injection pressure, the maximum seepage velocity increased to be approximately 125 m·s−1, with basically the same increasing amplitudes in the four models. The maximum seepage velocity in the P4 model was greatest, followed by those in P1 and P3, and finally by the value in P2.(2)
Compared with the results at a low inlet pressure (4.3 MPa), the total mass flows at an 8.1 MPa at the inlet and outlet increased. The total mass flow at the inlet in the P2 model was greatest (14.95 × 10−6 kg·s−1), followed by the values in the P3 and P4 models (13.78 × 10−6 kg·s−1 and 10.54 × 10−6 kg·s−1, resp.); finally, the value in the P1 model was smallest (7.40 × 10−6 kg·s−1). Since the inlet area remained unchanged, the total mass flow increased gradually with the increase of pressure.Table 7
Simulated results of seepage parameters in four models under a medium inlet pressure of 8.1 MPa.
Model
P
max (MPa)
P
min (MPa)
V
max (m·s−1)
V
min (10−5 m·s−1)
S
IN (10−10 m2)
M
IN (10−6 kg·s−1)
S
OUT (10−10 m2)
M
OUT (10−6 kg·s−1)
P1
8.17
0.062
145.4
17.37
12.10
7.40
8.37
7.32
P2
8.08
0.13
117.3
2.213
9.29
14.95
8.30
14.90
P3
8.12
0.11
121.8
0.9
13.02
13.78
8.60
13.78
P4
8.11
0.15
150.8
6.042
9.60
10.54
8.53
10.54Table8 lists the seepage parameters of the four models under a high inlet pressure of 13.1 MPa. When this table is analyzed in combination with Figure 21, the following conclusions regarding the high inlet pressure of 13.1 MPa can be obtained:(1)
The seepage behaviors in the four models showed almost the same variation tendencies. The pressures decreased along the seepage direction, which were greatest at the inlet and smallest at the outlet. Although these four models had different porosities, the seepage pressures at both the inlet and outlet fluctuated at around 13.1 MPa and showed slight variations. The average seepage velocities varied significantly in the intermediate regions, and the maximum seepage velocities reached up to 202.4 m·s−1.(2)
Compared with the results under low and medium inlet pressures, the total mass flows at the inlet and outlet under a high pressure of 13.1 MPa increased more significantly. Owing to the increase of the water injection pressure, some pores which were originally closed were opened and connected with each other to form seepage channels; thus, more water flowed out compared with the conditions under low and medium pressures. Under a high inlet pressure, the mass flow differences between the inlet and outlet were small.(3)
The total mass flow in the P2 model was greatest, followed by the values in the P3 and P4 models, and finally by the value in the P1 model. Similarly, with the increase of the pressure gradient, the total mass flows increased, with almost the same increasing amplitudes in the four models.Table 8
Simulated results of seepage parameters in four models under a high inlet pressure of 13.1 MPa.
Model
P
max (MPa)
P
min (MPa)
V
max (m·s−1)
V
min (10−5 m·s−1)
S
IN (10−10 m2)
M
IN (10−6 kg·s−1)
S
OUT (10−10 m2)
M
OUT (10−6 kg·s−1)
P1
13.2
0.11
191.2
14
12.10
9.56
8.37
9.49
P2
13.1
0.14
256
1.37
9.29
20.26
8.30
20.1
P3
13.07
0.092
161.8
3.43
13.02
19.83
8.60
19.76
P4
13.1
0.1
202.4
3.80
9.60
13.94
8.53
13.93Table9 lists the seepage parameters of the four models under an ultrahigh inlet pressure of 18.1 MPa. When this table is analyzed in combination with Figure 22, the following conclusions regarding the ultrahigh inlet pressure of 18.1 MPa can be reached:(1)
The seepage behaviors showed same variation tendencies in the four models. With the increase of the pressure gradient, the maximum seepage velocity increased steadily. The maximum seepage velocity in the P2 model was as high as 469.9 m·s−1, while the minimum seepage velocity remained at approximately 2 × 10−5 m·s−1. The total mass flow increased steadily with the increase of the pressure gradient. The total mass flows in the four models differed slightly; specifically, the total mass flow in the P2 model was greatest, followed by the values in the P3 and P4 models, and finally by the value in the P1 model.(2)
At 18.1 MPa, some pores that originally showed a favorable connectivity might have broken under the impact of the hydraulic pressure, and some large fractures were formed. Thus, fluids flowed out easily. Taking the P1 model as the example, water was more inclined to flow out through the left-center pores and less inclined to flow out through the upper right pores.Table 9
Simulated results of seepage parameters in four models under an ultrahigh inlet pressure of 18.1 MPa.
Model
P
max (MPa)
P
min (MPa)
V
max (m·s−1)
V
min (10−5 m·s−1)
S
IN (10−10 m2)
M
IN (10−6 kg·s−1)
S
OUT (10−10 m2)
M
OUT (10−6 kg·s−1)
P1
17.99
0.093
297.3
3.43
12.10
11.50
8.37
11.23
P2
18.02
0.13
469.9
1.97
9.29
24.96
8.30
24.42
P3
18.1
0.1
198.1
3.49
13.02
23.86
8.60
23.76
P4
18.01
0.11
243.2
2.83
9.60
16.64
8.53
16.63


## 4. Conclusions

In this paper, through a REV analysis, the optimum border length of the finite element analysis data was determined to be 0.06 mm, and the water seepage numerical simulation under 30 different pressure gradients was carried out using CFX software. The following results were ascertained:(1)
At a microscopic scale, the simulated results of the seepage velocity based on actual pore models exhibited complex nonlinear relationships with the pressure gradient rather than simple linear relationships. The seepage behaviors accorded with Forchheimer’s high-velocity nonlinear seepage rules. The permeability did not necessarily increase with the increase of effective porosity; that is, the positive correlation between them was not absolute. The non-Darcy flow efficient increased gradually with the increase of permeability.(2)
In the same model (P1), under different water injection pressure gradients, the following can be concluded:(a)
Along the seepage direction (i.e., the positive direction ofX), the seepage pressures decreased overall; however, when passing through small pores, the seepage pressures rapidly decreased at first and then gradually increased, showing overall decreasing tendencies. With the increase of the pressure gradient, the pressure fields showed more significant variations. In some pore channels with a favorable connectivity and large diameters, the seepage pressure fields showed mild variations, suggesting that water easily passed through these connected pores.(b)
At a low pressure water injection, the seepage velocities fluctuated evenly; with the increase of the water injection pressure, the average seepage velocities overall showed the variation tendency of increase-decrease-increase-decrease. Although the seepage velocities varied significantly during the entire water injection process, the seepage velocities at the inlet and outlet showed slight variations. At the same seepage length, the average seepage velocities varied significantly under different pressure gradients; the larger the pressure gradient, the greater the average seepage velocity. The average mass flows on the various sections fluctuated evenly. However, as the water injection pressure increased, the average mass flows at various sections fluctuated, notably along the direction of the seepage length; the average mass flow at the inlet (with a seepage length of 0μm) was the largest.(3)
The overall direction of the seepage is insignificantly influenced by the injection pressure. The seepage pressures decreased gradually along the seepage direction, which were greatest at the inlet and smallest at the outlet. The maximum seepage velocities all appeared in the intermediate regions with suddenly decreased pores. Under the same injection pressure, the total mass flows from the different inlets with different areas also varied. The increase of the total mass flow from the inlet to outlet was more significant under a higher injection pressure.



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*Source: 3401341-2018-09-06.xml*