Introduction

It is imperative to efficiently harvest coal bed methane (CBM) within coal mining regions, denoted as Coal Mine Methane (CMM), to ensure the sustainable exploitation of coal reserves, maintain a reliable energy supply, and realize pivotal objectives of achieving “carbon peak and carbon neutrality.” China has an estimated 36.8 trillion m3 geological CBM reserves; those within coal mining areas (specifically CMM) surpass 16 trillion m3 and thus constitute 43.5% of the nation’s total CBM endowment1. However, in contrast to the commercial development of CBM, the extraction of CMM cannot be confined to areas with high yield and permeability, and a more holistic approach is needed to satisfy the exigencies of coal mining operations2,3 that considers effective strategies for extracting CMM in areas with fragmented, soft, and low coal permeability4,5, and which deviates from the profiles sought in commercial CBM ventures. The geological features of CMM in China are notably complex, and fragmented soft coal seams account for 82% of China’s coalfield area and contain more than 50% of the potential CBM resources in China6. Such coal seams are problematic; they exhibit low reservoir pressure, poor permeability, and an elevated risk of coal protrusion. In addition, the mechanical strength of the coal mass is diminished, rendering the drilling and hole formation processes arduous. These seams are also characterized by strong adsorption, making the extraction CMM a complex endeavour, and they are particularly prone to outbursts. It is therefore imperative to address these issues when developing and implementing novel methodologies for CMM extraction from fragmented soft coal seams. Resolving such extraction challenges is paramount for the advancement of safe and efficient coal resource mining and CMM recovery practices7.

Cavitation completion is an important CMM completion technology, and its application in fragmented soft coal seams has yielded good results. Using the soft coal seam mechanical strength, broken structure, and natural characteristics of deformation, the local coal boundary conditions can be changed by removing part of the coal manufacturing cave; such changes include boundary stress unloading, boundary constraint removal, local coal deformation, swelling, the promotion of stress redistribution, increased permeability, and the promotion of CMM extraction. Compared with hydraulic fracturing completion, cavitation completion is more applicable for broken soft coal seams and shorter gas production cycles, and a good complementary effect with hydraulic fracturing coal seam gas wells can be formed. In this respect, mechanical8,9, water jet10,11, and aerodynamic cavitation12,13 completion technology types can be employed. The use of cavitation completion technologies varies depending on the type of construction well employed, and under coalmine directional drilling and ground directional and multi-branch wells can be employed. These technologies have been widely used in the Huainan mining area of China, and the application results are listed in the following Table 1.

Table 1 Current application status of cavitation completion technology in the Huainan mining area.
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Ground cavitation technology provides more advantages than coalmine cavitation technology because the site employed is unrestricted and more equipment can be employed. Several large-diameter caves can be formed via a directional well and multi-branch well cavitation; however, the overall efficiency requires improvement. Therefore, horizontal-well cavitation methods have been proposed. For example, Wang et al.14 proposed a method for inducing and controlling collapse cavitation formation after mechanically expanding holes in horizontal wells. They considered that mechanical drilling, hydraulic injection, and fluid loading and unloading could be sequentially used to induce controllable coal collapse and cave formation, thereby achieving enhanced permeability and drainage in fragmented soft coal seams. Furthermore, Huang et al.15 and Yang et al.16 proposed horizontal well hydraulic cavitation technology based on the use of a water jet.

However, horizontal wells in fragmented soft coal seams must be completed using casing or screen tubes, and the key issue in horizontal well drilling technology is how to overcome the influence of casing or screen tubes. Water jet technology is more commonly used in open hole wells, and the concept of hydraulic loading and unloading evolved from aerodynamic cavitation. However, due to the almost incompressible nature of water, its effectiveness is far inferior to that of aerodynamic cavitation. Aerodynamic cavitation, due to its uncontrollable cavitation process, is less feasible in horizontal wells. Therefore, it is necessary to propose a hydraulic drilling method suitable for cased horizontal wells.

This study proposes a model for horizontal-well hydraulic cavitation technology based on seepage erosion, with the aim of destroying fragmented soft coal seams and forming caves. A mathematical model was established and its feasibility is verified using numerical simulation methods. This new technological approach enhances permeability and stress release in fragmented soft coal seams.

Introduction of hydraulic cavitation technology in horizontal wells of fragmented soft coal seams and its differences from traditional water jet cavitition methods

Conventional water jet open-hole cavitation technology

Conventional hydraulic open-hole cavitation technology is a type of open-hole water jet cavitation technology that is used both in under-coal mines and on the ground. The technical principle involves using a water jet to break coal in an open drilling environment, as shown in Fig. 1.

Fig. 1
figure 1

Schematic diagram of conventional hydraulic open-hole cavitation technology using ground directional well water jet cavitation technology as an example.

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Coal is broken via the water jet using conventional hydraulic open-hole cavitation technology, and the non-coal seam section (the borehole casing annulus) is used as the hydraulic slag discharge channel to form the hydraulic cycle of the column-coal seam-cavity-borehole-casing annulus and realize the cavitation operation. However, this concept cannot be applied with hydraulic hole technology in the horizontal well of a broken soft coal seam because broken soft coal seams have poor drillability. In addition, the casing must be supported after drilling or the hole will collapse in the short term, destroying the gas extraction channel, and for the same reason, cased horizontal wells cannot form open holes at multiple points. Therefore, hydraulic drilling technology for horizontal wells can only be used in the perforation section, which increases the difficulty of using hydraulic cavitition technology in horizontal wells.

Hydraulic cavitation technology for horizontal cased wells in fragmented soft coal seams

Currently, almost all cavity completion technologies involve open-hole completion, which implies that there is no casing in the coal seam section. However, it is necessary to construct the hydraulic hole in a horizontal well of a broken soft coal seam under the condition of full-hole casing. Therefore, we propose hydraulic cavitation technology for horizontal-cased wells in fragmented soft coal seams, as described in Fig. 2.

Fig. 2
figure 2

Schematic diagram of hydraulic cavitation technology for horizontal-cased wells in fragmented soft coal seams.

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Horizontal wells in fragmented soft coal seams are constructed with a casing not fixed in the horizontal section, and a set of perforations is made simultaneously at regular intervals on the casing. One set of packers (including two packers, referred to as the front packer in the bottom hole direction, and the rear packer in the other direction) is used to isolate a section of space inside the casing. There is a space between at least one set of perforations at the front and at least one set of perforations at the back of the rear packer. There is a restrictor between the two packers, and water flows out of the restrictor. After the space separated by the two packers is filled with water, the water flows into the formation along the perforation hole of the casing and then returns to the annulus between the casing and the column through the perforation hole at the back of the rear packer, forming a hydraulic cycle, as shown in Fig. 2.

The differences between the hydraulic cavitition technology of horizontal wells in fragmented soft coal seams and conventional water jet cavitition technology are summarized as follows:

  1. (1)

    The construction environment: the conventional water jet cavitation environment is an open hole, and that of horizontal-well hydraulic cavitation is a cased environment, but there is communication with the coal seam through the perforating holes.

  2. (2)

    Coverage range of the water jets: the conventional water jet cavitation method involves rotating the water jet nozzle and moving it up and down with the pipe column within striking range of a cylinder. When horizontal well hydraulic holes are made, the water jet flows out through the perforation hole of the casing. Although multiple perforation holes exist as outlets, they cannot move, and the range of impact of the water jet is only within a few straight lines.

  3. (3)

    The hydraulic circulation backflow paths: in conventional water jet cavitation, water carries coal powder directly into the casing (non-coal seam section) and column annulus under hydraulic lifting, and it then returns to the ground. In horizontal well hydraulic cavitation, water carries coal powder outside the casing where it enters the annulus between the casing and pipe column through the perforation hole, and it then flows back to the surface (Fig. 2).

  4. (4)

    The hydraulic coal-breaking methods: in conventional water jet cavitation, the water jet directly impacts the coal body to break the coal. In horizontal well hydraulic cavitation, the water jet is unable to directly hit the hole-making area. Through water flow erosion, the coal body is peeled off layer-by-layer to achieve the cavitation effect.

It is evident that although there are similarities between hydraulic cavitation technology for horizontal wells and conventional water jet cavitation technology, the working environments and mechanisms differ. Therefore, as a new application technology, it is necessary to analyse the mechanism involved in hydraulic cavitation technology for horizontal-cased wells in fragmented soft coal seams.

Principle of hydraulic cavitation technology used in horizontal cased wells within fragmented soft coal seams

The hydraulic cavitation of horizontal wells in fragmented soft coal seams can be divided into stages in which the water-rock interaction modes differ, as shown in Fig. 3.

Fig. 3
figure 3

Hydraulic cavitation technological process.

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(1) Stage 1: Local cavity formation.

In the initial stage of hydraulic drilling in a horizontal well, high-pressure water forms in the middle of the two packers. This high-pressure water forms a jet al.ong the perforation hole; it hits the coal body near the hole, damaging the coal structure within a small area and expanding the contact area between water and coal. As the moisture content of the coal increases, its mechanical strength decreases17, which is conducive to further hydraulic cavitation formation.

(2) Stage 2: Seepage and erosion.

A high water pressure is formed between the two packers, creating a water pressure difference between the front and rear perforation holes. Under the action of this water pressure difference, a water seepage path is formed, which increases the local coal moisture content and reduces its mechanical strength. In addition, under erosion, a small cavity is gradually formed between the two sets of perforation holes, providing a channel for hydraulic circulation.

(3) Stage 3: Hydraulic circulation cavitation.

When the small cavity is formed, water flow within it causes erosion of the coal body on the cavity wall, and a larger cavity is gradually formed.

(4) Stage 4: Instability of cavity structure.

The formation of the cavity changes the original three-dimensional stress state of the coal body. The coal body loses support in the radial direction (perpendicular to the cavity wall), and plastic and loose bands are formed under the action of tangential stress. The collapse and deformation of the coal body lead to shrinkage of the cavity, which continues to erode under the action of hydraulic circulation, repeating the process of cavity formation.

Numerical model of hydraulic cavitation and seepage process in horizontal wells

Thermoelastic seepage constitutive equation for middling coal in hydraulic seepage process

The thermoelastic constitutive equation for a porous medium is as follows

$$\:{\epsilon}_{ij}=\frac{1}{2G}{\sigma}_{ij}-\left(\frac{1}{6G}-\frac{1}{9K}\right){\sigma}_{kk}{\delta}_{ij}+\frac{\alpha}{3K}p{\delta}_{ij},$$
(1)

where \(\:{\epsilon\:}_{ij}\) is the strain tensor; \(\:{\sigma\:}_{ij}\) is the stress tensor; and \(\:G\) is the shear modulus (\(\:G=E/2\left(1+\nu\:\right)\), in which \(\:E\) is the elastic modulus and \(\:{\upnu\:}\) is Poisson’s ratio; \(\:K\) is the Bulk modulus (\(\:K=E/3\left(1-2\nu\:\right)\); the Biot coefficient \(\:\alpha\:=1-K/{K}_{s}\) (where \(\:{K}_{s}\) is the bulk modulus of the coal matrix); and \(\:{\delta\:}_{ij}\) is the Kronecker symbol.

The volumetric strain of coal, \(\:{{\upepsilon\:}}_{\text{v}}\), can be calculated using the following,

$$\:{\epsilon\:}_{v}={\epsilon\:}_{11}+{\epsilon\:}_{22}+{\epsilon\:}_{33}.$$
(2)

From Eqs. (1) and (2), we can conclude that

$$\:{\epsilon\:}_{v}=\frac{1}{K}\left(\frac{1}{3}{\sigma\:}_{kk}+\alpha\:p\right),$$
(3)

and

$$\:{\epsilon\:}_{v}=-\frac{1}{K}\left(\stackrel{-}{\sigma\:}-\alpha\:p\right),$$
(4)

where \(\:\stackrel{-}{\sigma\:}=-\frac{1}{3}{\sigma\:}_{kk}\) represents the average compressive stress.

According to Darcy’s law,

$$\:v=-\frac{k}{\mu\:}\nabla\:p,$$
(5)

where \(\:\nabla\:=\left(\frac{\partial\:}{\partial\:\text{x}},\frac{\partial\:}{\partial\:\text{y}}\right),\:\text{v}\) is the seepage velocity, \(\:k\) is permeability, and \(\:{\upmu\:}\) is the viscosity of water. Of these, \(\:k\) can be obtained by combining the cubic law,

$$\:\frac{k}{{k}_{0}}={\left(\frac{\varphi\:}{{\varphi\:}_{0}}\right)}^{3},$$
(6)
$$\:k={k}_{0}{\left(1+\frac{{\epsilon\:}_{v}}{{\varphi\:}_{0}}\right)}^{3},$$
(7)

where \(\:{k}_{0}\) is the initial permeability, \(\:\varphi\:\) is porosity, and \(\:{\varphi\:}_{0}\) is initial porosity.

In the above, Eq. (4) describes the volume change of coal under the action of pore pressure, which leads to an increase in porosity; Eq. (7) shows the impact of the above changes on permeability; and the relationship between velocity and pore pressure is established through Darcy’s law in Eq. (5). Combining Eqs. (4), (5), and (7), we obtain

$$\:v=-\frac{{k}_{0}}{\mu\:}{\left(\frac{{{\varnothing}}_{0}K-\stackrel{-}{\sigma\:}+\alpha\:p}{{{\varnothing}}_{0}K}\right)}^{3}\nabla\:\:p,$$
(8)

which is used to calculate the relationship between velocity and pore pressure.

Geometric models and boundary conditions

The hydraulic cavitation range of the horizontal well can be approximated as a cylinder, where a half section of the cylinder is taken as the research object. A geometric model is established in Fig. 4.

Fig. 4
figure 4

Geometrical model of the hydraulic cavitation seepage stage in horizontal wells.

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As shown in Figs. 4 and 5, the boundary conditions are as follows:

Fig. 5
figure 5

Boundary conditions of the numerical hydraulic hole seepage model.

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(1) The upper and lower boundaries of the geometric model are fixed,

$$\:\frac{\partial\:u}{\partial\:x}=0,\frac{\partial\:u}{\partial\:y}=0,$$
(9)

where \(\:\text{u}\) is the displacement.

(2) The left and right boundaries of the geometric model represent stress loads that remain constant over time,

$$\:\sigma\:=\stackrel{\sim}{\sigma\:}\left(t\right),$$
(10)

where \(\:\sigma\:\) represents the stress load and \(\:\stackrel{\sim}{\sigma\:}\left(t\right)\) represents the function of the stress load at the boundary over time.

(3) The inlet and outlet of the model are under pressure boundary conditions that remain constant over time,

$$\:{P}_{in}=\stackrel{\sim}{{P}_{in}}\left(t\right);\:{P}_{out}=\stackrel{\sim}{{P}_{out}}\left(t\right),$$
(11)

where \(\:{P}_{in}\), \(\:{P}_{out}\) represent the water pressures at the inlet and outlet, respectively, and \(\:\stackrel{\sim}{{P}_{in}}\left(t\right)\), \(\:\stackrel{\sim}{{\:P}_{out}}\left(t\right)\) represent the function of the stress load over time, at the inlet and outlet, respectively.

Discriminant conditions of coal destruction under hydraulic hole seepage in horizontal wells

For one microelement of the coal body, the infiltration and erosion processes by the water flow are shown in Fig. 6.

Fig. 6
figure 6

Mechanical analysis of coal microelement eroded by hydraulic seepage.

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Set any loose coal microelement, its shape is a cube, the area in either direction is \(\:\text{d}\text{s}\), and the pressure difference between the two ends is \(\:\varDelta\:\text{p}\). The seepage pressure \(\:{F}_{s}\), can then be determined,

$$\:{F}_{s}=\varDelta\:pds.$$
(12)

The limit equation for the balance is as follows,

$$\:\left(2{\tau\:}_{21}+2{\tau\:}_{31}+{\sigma\:}_{t}\right)ds={F}_{s}-{\sigma\:}_{11}ds,$$
(13)

where\(\:\:{\tau\:}_{y}\), \(\:{\tau\:}_{z}\) is the shear strength in the y and z directions, respectively; \(\:{\sigma\:}_{t}\) is the tensile strength of the coal body; \(\:{\sigma\:}_{x}\) is the stress in the x direction, positive for tension and negative for compression; and \(\:{\uptau\:}\) is the shear strength, which can be calculated using the Mohr Coulomb criterion (Eq. 14),

$$\:\tau\:=c+{\sigma\:}_{n}tan\theta\:,$$
(14)

where \(\:\text{c}\) is the cohesion of coal, and \(\:{{\upsigma\:}}_{\text{n}}\) is the stress perpendicular to the shear plane, which corresponds here to \(\:{{\upsigma\:}}_{22}\), \(\:{{\upsigma\:}}_{33}\), respectively.

From Eqs. (12), (13), and (14), we can conclude that

$$\:4c+2\left({\sigma\:}_{22}+{\sigma\:}_{33}\right)tan\theta\:+{\sigma\:}_{t}=\varDelta\:p+{\sigma\:}_{11},$$
(15)

where \(\:\varDelta\:\text{p}\) is a function of the horizontal distance, x.

Equation (15) can be organised as a function of \(\:\varDelta\:\text{p}\),

$$\:\varDelta\:p=4c+2\left({\sigma\:}_{22}+{\sigma\:}_{33}\right)tan\theta\:+{\sigma\:}_{t}-{\sigma\:}_{11}.$$
(16)

Using Eq. (16), it is possible to determine whether the coal body is damaged based on the pressure gradient.

On this basis, integrating in the x direction there is,

$$\:P={\int\:}_{0}^{L}\varDelta\:pdx=\left(4c+{\sigma\:}_{t}\right)L+2tan\theta\:{\int\:}_{0}^{L}\left({\sigma\:}_{22}+{\sigma\:}_{33}\right)dx-{\int\:}_{0}^{L}{\sigma\:}_{11}dx,$$
(17)

where \(\:\text{P}\) is the difference between the inlet and outlet water pressures of the two perforation sections, and \(\:\text{L}\) is the distance between the two perforations.

As the research area in the x direction is much longer than that in the other two directions during the hydraulic cavitation process, the hydraulic cavitation process can be regarded as a plane strain problem,

$$\:{\sigma\:}_{11}=-\nu\:\left({\sigma\:}_{22}+{\sigma\:}_{33}\right),$$
(18)

where \(\:{\upnu\:}\) is the Poisson’s ratio.

Combining Eqs. (17) and (18), we obtain

$$\:P=\left(4c+{\sigma\:}_{t}\right)L+\left(2tan\theta\:+\nu\:\right){\int\:}_{0}^{L}\left({\sigma\:}_{22}+{\sigma\:}_{33}\right)dx.$$
(19)

For the original occurrence state of the coal body, \(\:{\sigma\:}_{22}\) and \(\:{\sigma\:}_{33}\) have a fixed value, but due to the influence of hydraulic injection, they are functions of the horizontal x. By the effective stress principle,

$$\:{\sigma\:}^{*}=\sigma\:-p,$$
(20)

where \(\:\sigma\:\) is the geostress, and the geostress of y and z directions are \(\:{\sigma\:}_{yy}\) and \(\:{\sigma\:}_{zz}\), respectively, \(\:{\sigma\:}^{*}\) is effective stress, and \(\:\text{p}\) is pore water pressure.

The seepage process in the model can be described by Darcy’s law,

$$\:\varDelta\:p=v\frac{\mu\:}{k}dx,$$
(21)

where \(\:\text{v}\) is the flow velocity, \(\:\text{k}\) is permeability, \(\:{\upmu\:}\) is the viscosity of water, and \(\:\text{d}\text{x}\) is the length of the microelements.

The pore pressure at the horizontal position x is

$$\:p={\int\:}_{x}^{0}v\frac{\mu\:}{k}dx=-\frac{v\mu\:x}{k}.$$
(22)

According to Eqs. (19), (20), and (22),

$$\:P=\left(4c+{\sigma\:}_{t}\right)L+\left(2tan\theta\:+\nu\:\right)\left({\sigma\:}_{yy}L+{\sigma\:}_{zz}L+\frac{v\mu\:}{k}{L}^{2}\right),$$
(23)

which can be used to obtain the relationship between the range of the hydraulic fracturing action (i.e., the range of coal damage under seepage erosion) \(\:L\) and the ultimate fracture pore pressure \(\:{p}_{min.}\)

Equation (23) can be rewritten in a quadratic polynomial form,

$$\:\left\{\begin{array}{c}y=intercept+B1x+B2{x}^{2}\\\:B1=\left(2tan\theta\:+\nu\:\right)\frac{v\mu\:}{k}\\\:B2=\left(2tan\theta\:+\nu\:\right)\left({\sigma\:}_{yy}+{\sigma\:}_{zz}\right)+\left(4c+{\sigma\:}_{t}\right)\\\:intercept=0,\end{array}\right.$$
(24)

where x represents the distance from a certain point to the outlet, and y represents the ultimate fracture pore pressure, \(\:{\text{p}}_{\text{m}\text{i}\text{n}}\). In practical applications, a quadratic polynomial curve can be drawn using Eq. (24) in combination with actual parameters as a discrimination curve to determine whether the coal body is damaged during the hydraulic coal cavitation process, as shown in Fig. 7.

Fig. 7
figure 7

Relationship between the perforation spacing calculated using Eq. (24) and the minimum inlet pressure that can cause coal damage (using the parameters in Table 2 as an example).

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Numerical simulation

To construct a numerical model, COMSOL Multiphysics software was employed to building the geometric model, mathematical model, and the boundary conditions established in Sect. 3.1 and 3.2, respectively. The constructed geometric model is shown in Fig. 8, the mesh division is shown in Fig. 9, and the parameters are listed in Table 2.

Fig. 8
figure 8

Structured grid division.

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Fig. 9
figure 9

Geometric model.

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Table 2 Numerical simulation parameter settings18,19.
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The inlet and outlet were set as pressure boundary conditions, and a numerical simulation was conducted. Taking the simulation results (Fig. 10) for the high-pressure water seepage in coal at an inlet pressure of \(\:50\times\:{10}^{6}\:\text{Pa}\) as an example, we derived the overall distribution pattern of the pore pressure. The horizontal and vertical coordinates in the figure represent the length of the simulation area. The highest pore pressure at the inlet was 50 MPa, and the pore pressure at the outlet was the lowest (close to 0 MPa). The colour changes in the cloud map reflect the distribution of pore pressure, and the legend indicates the corresponding pore pressure values for different colours. The direction and density of the streamlines in the Fig. 10 provide information about the direction and velocity of Darcy flow. The streamline direction indicates the direction of the fluid flow, and the streamline density reflects the fluid speed. In this figure, the streamline clearly shows the flow path of the fluid in the porous media from the high-pressure region (inlet) to the low-pressure region (outlet).

Fig. 10
figure 10

Distribution of void pressure and Darcy velocity field streamline in the study area when the inlet pressure was \(\:50\times\:{10}^{6}\:\text{Pa}\)

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The numerical simulation results were analysed and a discussion about these results is provided in the following section.

Discussion

In Fig. 10, two-dimensional segments were drawn at 5 levels, including, \(\:\text{y}=0.4\), \(\:\text{y}=0.2\), \(\:\text{y}=0\), \(\:\text{y}=-0.2\), \(\:\text{y}=-0.4\). A line graph was drawn with the arc length of the segment on the x-axis and the pore pressure on the y-axis, as shown in Fig. 11.

Fig. 11
figure 11

Line graph with x-axis arc length and y-axis pore pressure.

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As shown in Fig. 12, the overall pore pressure variation pattern at different levels was consistent near the inlet and outlet of the Darcy flow; that is, it decreased linearly from the Darcy inlet to the outlet. The variation law of pore pressure varied in different y-axis horizontal segments. The segment closest to the casing exhibited obvious nonlinear characteristics in its variation curve, and a significant pressure drop was observed near the inlet and outlet. This mainly occurred due to the open boundaries at the inlet and outlet, which were in an unloaded state, while the other boundaries were all subjected to geostress loading; significant stress changes occurred near the inlet and outlet, leading to changes in porosity and permeability, thereby affecting the pore pressure gradient.

Fig. 12
figure 12

(a) Relationship between pore pressure and minimum failure pore pressure with a perforation step of x = 2 m. (b) Relationship between pore pressure and minimum failure pore pressure with a perforation step of x = 3 m. (c) Relationship between pore pressure and minimum failure pore pressure with a perforation step of x = 4 m.

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Taking a segment at position y = − 0.4, the pore pressure curves for inlet pressures \(\:{\text{P}}_{\text{in}}\) of \(\:10\times\:{10}^{6}\:\text{Pa}\), \(\:20\times\:{10}^{6}\:\text{Pa}\),\(\:30\times\:{10}^{6}\:\text{Pa}\), \(\:40\times\:{10}^{6}\:\text{Pa}\), and \(\:50\times\:{10}^{6}\:\text{Pa}\), respectively, were plotted and are shown in Fig. 13.

Fig. 13
figure 13

A line graph at \(\:\text{y}=1\), with a cross-sectional arc length on the x-axis and pore pressure on the y-axis.

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As shown in the figure, the slope of the pressure change curve increased significantly from x = 0.5 to x = 1. The increase in the pressure drop was conducive to the destruction of the coal structure, and the coal was therefore stripped and caves formed. As the inlet pressure \(\:{\text{P}}_{\text{in}}\) increased, the downward trend of the pressure near the outlet became more pronounced.

Taking an inlet pressure of \(\:{\text{P}}_{\text{in}}=50\times\:{10}^{6}\:\text{Pa}\), the total lengths of the model in the x direction were 2 m, 4 m, 6 m, 8 m, and 10 m. Taking a segment at \(\:\text{y}=-0.4\) and the outlet end in the x direction from 0 to 2 m, a line graph was compiled with the x-axis as the segment length (0 to 2 m) and the y-axis as the pore pressure, as shown in Fig. 14.

Fig. 14
figure 14

Pore pressure distribution curve: x-axis represents the length of the cross-Sects. (0–2 m), and the y-axis represents pore pressure.

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As shown in Fig. 14, as the total length of the model in the x-direction decreased, the slope of the pore pressure descent increased. In this model, the x-axis length represents the interval between the perforated sections of the casing. Based on the above analysis, it can be concluded that during cased horizontal well cavitition operation, a smaller perforation spacing and higher injection pressure are more conducive it increasing the pore pressure gradient at the outlet, thereby damaging the coal structure, promoting the collapse of the coal body at the outlet, and completing the coal cavitation process.

According to the derivation in Sect. 3.3, Eq. (24) can be used as a standard to determine whether a coal body is damaged. Using the numerical examples provided in Sect. 3.4 as examples, the construction parameters can be optimized.

The perforation step x was set to three levels: 1 m, 2 m, and 3 m, and the injection pressure, Pin, was set to six levels: \(\:50\times\:{10}^{6}\:\text{Pa}\), \(\:60\times\:{10}^{6}\:\text{Pa}\), \(\:70\times\:{10}^{6}\:\text{Pa}\), \(\:80\times\:{10}^{6}\:\text{Pa}\), \(\:90\times\:{10}^{6}\:\text{Pa}\) and \(\:100\times\:{10}^{6}\:\text{Pa}\), respectively. Line diagrams were drawn and use the calculation results of Eq. (24) (Fig. 10) were employed to determine whether the coal body was damaged. The results are shown in Fig. 12a–c.

Figure 12 shows the pore pressure and minimum failure pore pressure on the cutoff line at different injection pressures. The y-axis on the left side of the figure represents the pore pressure, corresponding to the pore pressure under different injection pressures (black curve) in the figure; the y-axis on the right represents the minimum failure pore pressure pmin, corresponding to the minimum failure pore pressure curve (red) in the figure. In the figure, coal is destroyed when the pore pressure is greater than pmin, otherwise it is not destroyed. The calculation example in this article shows that hydraulic cavitation can be completed under the condition of a perforation step of x = 2 m and an injection pressure of \(\:{\text{P}}_{\text{in}}\ge\:60\times\:{10}^{6}\:\text{Pa}\). Under a condition of x = 3 m perforation step and \(\:{\text{P}}_{\text{in}}\ge\:60\times\:{10}^{6}\:\text{Pa}\:\)injection pressure, hydraulic cavitation can be completed. However, when the perforation step distance x > 3 m, hydraulic cavitation cannot be completed with an injection pressure of \(\:{\text{P}}_{\text{in}}\le\:100\times\:{10}^{6}\:\text{Pa}\).

Conclusion

This study proposes an innovative horizontal-well hydraulic cavitition technology for the particular geological conditions of fragmented and soft coal seams in coal mining areas, with the aim of improving the efficiency of CMM extraction and addressing the technical challenges in current CMM development. The main highlights of this study are as follows:

  1. 1.

    Development of innovate technology and associated principle. The proposed horizontal-well hydraulic hole-making technology is based on the principle of hydraulic seepage. By perforating through the casing in a horizontal well, a local high-pressure hydraulic environment is formed, causing the destruction of the coal body and the formation of caves. This technology is particularly suitable for fractured soft coal seams with low permeabilities and easy outburst characteristics.

  2. 2.

    Construction of mathematical model and theoretical contribution. We established a mathematical model to describe the hydraulic cavitation process in horizontal wells based on the thermal elastic constitutive equation of porous media and Darcy’s law. This model comprehensively considers the elasticity and seepage characteristics of coal seams, and provides a theoretical basis for the response of coal bodies to hydraulic action in the field of coalbed methane development.

  3. 3.

    Verification using numerical simulation. The accuracy of the mathematical model was verified through numerical simulation experiments conducted using COMSOL Multiphysics software. The simulation results showed that under the set construction parameters, the distribution pattern of the pore pressure in the coal body was consistent with the theoretical prediction, confirming the practicality and reliability of the model.

  4. 4.

    Analysis of key construction parameters. In this study, we focused on two key construction parameters, perforation step distance and injection pressure. The results indicated that a smaller perforation step and higher injection pressure improved the pore pressure gradient of the coal body, thereby effectively destroying the coal structure and achieving coal cavitation.

  5. 5.

    Optimization of construction parameters and application guidance. Through numerical simulations, we conducted an optimization analysis of the construction parameters and determined the critical conditions for coal body failure under different perforation step spacings and injection pressure conditions. These findings provide an important reference for practical engineering applications and can assist in guiding onsite construction.

Positive results were achieved in this study; however, there is room for further exploration. To achieve a broader technological application and deeper theoretical development, future research should focus on determining the adaptability of the technology in on-site applications, conducting an economic analysis, and analyzing any synergistic effects with other CMM extraction technologies.

In summary, this study provides new perspectives and solutions at both the theoretical and practical levels, laying a solid foundation for the effective development and utilization of fragmented soft coal seam coalbed methane. With further development and technological improvements, the model is expected to make a great contribution to safe coal mine production and clean energy development.