1 Introduction

Coalbed methane (CBM) stands as a crucial unconventional natural gas resource in China, known for its abundant reserves and wide distribution (Ding et al. 2012; Huang et al. 2023). Deep CBM (with a depth over 1000 m) is about 22.53 trillion m3, which accounts for 61.22% of China’s known reserves (Wei et al. 2020; Li et al. 2018; Sun et al. 2023). Thus, the development of deep CBM in China holds significance. Due to the extremely low permeability and the complex mechanical behaviour of the deep CBM reservoirs (Zhang et al. 2023a), efficient extraction of the deep CBM remains a challenge (Liu et al. 2016; Wang et al. 2015; Zhao et al. 2023). Therefore, reservoir stimulation is essential to enhance reservoir permeability. Hydraulic fracturing (HF) stands as the predominant stimulation technology employed in China for enhancing CBM recovery (Zhang et al. 2020; Ge et al. 2019; Jin and Gu 2015). However, in the Linxing CBM field (with a depth of approximately 2000 m), situated in the eastern part of the Erdos Basin in China, the application of this method was unsuccessful. According to historical production data, the average production rate is approximately 500–1000 m3/d (Meng et al. 2018). This failure to enhance CBM recovery in the deep coal seam might be attributed to the high-stress difference and the ductile behavior of the coal, which hinder the effective generation of fractures. In addition, HF is a pressurizing process (Fig. 1a), which can significantly hinder the desorption of the CBM.

Fig. 1
figure 1

Distribution of average stress change induced by a HF and b HS

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Hydraulic slotting (HS) is another stimulation technology that is widely used in the extraction of CBM in underground coal mines to prevent gas outbursts (Lu et al. 2010; Yan et al. 2015; Lin et al. 2015a; Li et al. 2021). The void space generated by HS can cause both stress and pressure relief around the slots (Liu et al. 2022), enhancing the reservoir permeability meanwhile promoting gas desorption (Cui et al. 2023; Jiang et al. 2018; Lin et al. 2012; Shen et al. 2015). Nevertheless, the stimulation extent of a single hydraulic slot is limited. To overcome this constraint, Lu et al. (2021) introduced a novel concept (Fig. 2) that combines directional wells and hydraulic slots. Thus, a large extent of the reservoir can be stimulated. This paper aims to investigate the effectiveness of the new simulation approach in enhancing gas recovery in deep coal seams.

Fig. 2
figure 2

Schematic diagram of directional wells with multiple hydraulic slots in the coal seam

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The spatial permeability enhancement and gas desorption caused by stress relief around a hydraulic slot must be evaluated to investigate the gas extraction capability of directional wells with multiple hydraulic slots. There are amounts of publications related to this topic, in which the stress relief around a hydraulic slot is determined using different numerical methods, e.g., finite difference method (FDMLu et al. 2011; Shen et al. 2015; Si et al. 2019), finite element method (FEM, Wong et al. 2002; Yan et al. 2015; Yang et al. 2015; Liu et al. 2022), block-based distinct element method (Zhao et al. 2018; Gui et al. 2017) and particle-based distinct element method (Liu et al. 2015; Lin et al. 2015b). According to these studies, the characteristics of the stress relief zones caused by a hydraulic slot were quantitatively investigated, in which a damaged or cracked zone with significant stress reduction and permeability enhancement was induced around the hydraulic slot. Meanwhile, the stress gradually decreased outer the damage zone.

Then relevant scholars have conducted numerous field tests on gas extraction capability by HS (Zhang et al. 2023b; Lin et al. 2012, 2015b; Zou and Lin 2018; Yan et al. 2015; Shen et al. 2012). Most of these studies have shown that the hydraulic slot gives gas concentration and volume higher and improves CBM extraction efficiency significantly compared to conventional techniques. Furthermore, several investigations have been conducted to explore the impact of slotting layout on CBM extraction. Zhao et al. (2020) examined the impact of the Langmuir volumetric strain constant, the homogeneity coefficient, slot distance, and overburden stress on gas extraction. The results demonstrated that the most suitable slot distance was 8 m. Liu et al. (2015) investigated the impact of slot inclination angle on mechanical properties and quantified its effect on CBM extraction at various gas-drainage depths. Fan et al. (2021) analyzed the relationship between slot parameters (e.g., the spacing and width) and the pressure relief of the coal seam, observing that the hydraulic slots with greater width and narrower spacing resulted in more significant pressure relief.

These studies gave a basic understanding of how hydraulic slots promote gas recovery. Nevertheless, new challenges have emerged in studying stress relief in deep CBM reservoirs (e.g., Linxing gas field) due to the relatively low strength, high in-situ stress, and ductile behavior of deep coal rock. These conditions can result in large deformation and displacement around the hydraulic slot (Gu et al.2023), ultimately leading to self-contact of the hydraulic slot boundary (Fig. 3). However, the previous used numerical methods are problematic to handle these mechanical issues. The simulation based on FEM and FDM can cause grid distortion and embedding under large deformation, resulting in computational errors. In addition, the contact algorithm is difficult to implement for solving the contact problem of the large deformed hydraulic slot boundary. The particle-based DEM is easy to handle the contact problem. Nevertheless, the particle is rigid, which is naturally inadequate for treating the large deformation problem. Secondly, the stress distribution is difficult to evaluate by this method. Similar to the FEM, the hybrid DEM-FEM, and the block-based DEM also share the same problem. Therefore, according to the author’s knowledge, there is limited study on quantitative evaluation of the stress relief caused by large deformed hydraulic slots in deep CBM. However, the distribution of the permeability enhancement is the precondition to quantitatively study the gas recovery by the directional well with multiple hydraulic slots.

Fig. 3
figure 3

Illustration of large deformation and boundary self-contact around the hydraulic slot

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This study developed two numerical tools to study the effectiveness of directional well with multiple hydraulic slots for enhanced gas recovery in the deep coal seam. The first is based on the material point method and continuum damage mechanics, which can solve the large deformation (Zhou et al. 2023) and complex contact problems associated with hydraulic slots in deep coal seams. This computational program is coded in the C++ programming language and performed through Visual studio software. The second one is a gas production simulator, which considers both porous flow and directional-well flow and flow exchange through the perforations connecting the porous and the directional wells. The finite volume method (FVM) and FDM are utilized to address the sequential coupled process. In addition, the effects of stress-dependent permeability are considered. The computational code was developed using the C++ programming language and incorporated into FLAC3D through its secondary development feature. This ability to incorporate the user-defined code into FLAC3D offers the advantage of tailoring simulations more closely to specific research requirements, ensuring greater accuracy in our modeling approach. Both numerical tools are verified by theoretical and experimental results, respectively. We conduct a numerical case study in the Linxing gas field using these tools. The extraction failure reason is firstly studied based on the history matching of the filed production by HF. And then, the influences of stress relief, damage zone, gas desorption, and permeability enhancement induced by a large deformed hydraulic slot are revealed. Finally, enhanced gas recovery by directional well with multiple hydraulic slots is simulated, and sensitivities of the slot spacing, length, and width were also investigated in detail. The findings may potentially contribute to an innovative comprehension of the advancement of deep CBM exploitation.

2 Numerical model for simulating large deformation and complex contact

2.1 Governing equations

2.1.1 Continuum damage model

The linear elasticity theory guides the mechanical deformation patterns of coal formations, encompassing the equilibrium equation (Eq. (1)), the geometrical equation (Eq. (2)), and the constitutive equation (Eq. (3)). This theory elucidates the intricate relationships among stress, strain, and displacement.

$$\nabla \sigma + \rho \left( {b - \frac{\partial v}{{\partial t}}} \right) = 0$$
(1)
$$\Delta \varepsilon = \frac{1}{2}\left[ {\nabla \left( {\Delta u} \right) + \nabla^{T} \left( {\Delta u} \right)} \right]$$
(2)
$$\Delta \sigma = \left( {K - \frac{2}{3}G} \right)\Delta \varepsilon_{V} + 2G\Delta \varepsilon_{dev}$$
(3)

where ρ represents the rock density (kg/m3); σ stands for the stress tensor presented in matrix form (Pa); G and K refer to the shear modulus and the bulk modulus of rock, respectively (Pa); v is the velocity vector (m/s); the terms εv represents the volumetric strain (−) while the terms εdev is used for the deviatoric strain (−); the tensor Δε corresponds to the strain increment, depicted in vector form (−); u indicates the displacement vector (m); t symbolizes the time (s); the vector b signifies the body force per unit volume (m/s2).

Mohr–Coulomb’ criterion (Eq. (4)) and plastic yield criteria of tensile failure (Eq. (5)) are preconditions of the plastic deformation from the shear slip behavior and tensile behavior, respectively.

$$f_{s} = \sigma_{1} - N_{\varphi } \sigma_{3} + 2c\sqrt {N_{\varphi } }$$
(4)
$$f_{t} = \sigma_{3} - \sigma_{t}$$
(5)

where fs and ft are the yielding shear strength and yielding tensile strength, respectively (MPa); σ3 donates the minimum principle stress (Pa), while σ1 refers to the maximum principle stress (Pa); σt indicates the tensile strength (Pa). The term c symbolizes the cohesion (Pa); the formula Nφ = (1 + sinφ)/(1 − sinφ) relates to the internal friction angle, represented by φ (°).

Both the magnitude and the orientation of the plastic strain derive from the plastic flow rule. Using the principles of plastic potential theory, the increment in plastic strain is computed, detailed as follows:

$$\Delta \varepsilon = \lambda \frac{\partial g}{{\partial \sigma }}$$
(6)

where the term g is representative of the plastic potential function, which for shear and tensile yielding are given by Eq. (4) and Eq. (5), respectively adhering to the principles of the associated flow rule; λ denotes the nonnegative scalar (-) with its value dictated by the consistency condition, as proposed by Drucker and Prager (2013).

The shear damage is captured by the second invariant, formulated from deviatoric stresses in relation to the plastic strain. On the other hand, the principal strain from the plastic tensile behavior delineates the tensile damage sustained by the coal rock (Irwin 1956). The shear and tensile damage are respectively illustrated in Eqs. (7) and (8).

$$\varepsilon_{d,s} = \sqrt {\frac{{\left( {\varepsilon_{1}^{p} - \varepsilon_{2}^{p} } \right)^{2} + \left( {\varepsilon_{2}^{p} - \varepsilon_{3}^{p} } \right)^{2} + \left( {\varepsilon_{3}^{p} - \varepsilon_{1}^{p} } \right)^{2} }}{6}}$$
(7)
$$\varepsilon_{d,t} = \varepsilon_{3}^{p}$$
(8)

where εd,s and εd,t are plastic damage of shear and tensile, respectively(−); εp represents the plastic strain (−); the subscripts 1, 2, and 3 indicates the principal direction within the principal stress space; the superscript p serves as an abbreviation for ‘plastic’;

To represent the link between material strength and plastic damage, a linear model with a plastic damage threshold was utilized. When the plastic damage accords with this threshold, the material strength no longer weakens, maintaining its residual strength. The area containing the damaged particles was set as the damaged area where the residual cohesion and tensile strength were set to zero.

2.1.2 Models on permeability enhancement and gas desorption

Because the stiffness of the coal skeleton is 2–3 orders of magnitude more than that of the porous medium, the deformation volume of the coal skeleton can be viewed as the change in porous volume (Eq. (9)). Equation (10) expresses the link between coal permeability and the cubic of coal porosity, being proportional (Mckee et al. 1988). The proportion between the coal seam’s enhanced permeability following slotting and its natural permeability is what we refer to as the coal permeability change ratio, which is given by Eq. (11) by substituting Eq. (9) into Eq. (10).

$$\Delta \phi = \Delta e_{v}^{e} + \Delta e_{v}^{p}$$
(9)
$$k = \frac{{r^{2} }}{c}\phi^{3}$$
(10)
$$k_{1} /k_{0} = \left( {\phi_{0} + \Delta \phi } \right)^{3} /\phi_{0}^{3}$$
(11)

where ϕ is the porosity (−); Δeve and Δevp are elastic and plastic volume strains, respectively (−); k is the permeability (m2); c is the dimensionless constant (−); r is the hydraulic radius of the pore throat (m); k0 is the initial permeability(m2); ϕ0 is the initial porosity (−).

With an increase in pore volume, the gas pressure in coal pores decreases, causing the gas in the adsorption state to become free gas. According to the principle of mass conservation (Eq. (12-1)), ease the incrin free gas in the pores is equal to the gas quantity desorbed from the coal matrix as a result of pressure relief. Then, the Eq. (12-2) can be used to calculate the volume of desorbed gas per unit volume.

$$\frac{{MP_{1} V\left( {\phi_{0} + \Delta \varepsilon_{V}^{e} + \Delta \varepsilon_{V}^{p} } \right)}}{RT} - \frac{{MP_{0} V\phi_{0} }}{RT} = \frac{{V_{L} P_{0} }}{{P_{L} + P_{0} }}\rho_{c} V\rho_{g} - \frac{{V_{L} P_{1} }}{{P_{L} + P_{1} }}\rho_{c} V\rho_{g}$$
(12-1)
$$V_{d} = \frac{{V_{L} P_{0} }}{{P_{L} + P_{0} }}\rho - \frac{{V_{L} P_{1} }}{{P_{L} + P_{1} }}\rho$$
(12-2)

where M represents the molar mass of CBM (kg/mol); V stands for a unit volume (m3); the initial pore pressure is denoted by P0 (MPa), while P1 is the pore pressure after pressure relief (MPa); CBM density and coal density are symbolized by ρg (kg/m3) and ρc (kg/m3), respectively; T refers to temperature, measured in Kelvin (K); Vd signifies the desorbed gas volume per unit volume (m3/m3); R is a constant (J/(K mol)); Langmuir pressure and Langmuir volume are denoted by PL (MPa) and VL (m3/kg), respectively.

The proposed Eq. (11) and Eq. (12–2) are respectively used to demonstrate the permeability enhancement effect and evaluate the gas desorption effect related to mechanical deformation in the following Sect. 4.2.2.

2.2 Numerical solution based on MPM

Material points and the background grid are the two independent spatial discretizations utilized by MPM. The latter is only used to solve the equations of motion, after which it is reset, either maintaining its original position or moving to a new location (Wang et al. 2016). Therefore MPM avoids the issue of mesh distortion that could be induced because of large deformations using FEM. The MPM was utilized in this section to solve the continuous damage model proposed in Sect. 2.1.1. Figure 4 depicts the mapping and remapping process between the background computational mesh and the material points. MPM solution consists of the following steps (Wang et al. 2016; Sulsky et al. 1994):

Fig. 4
figure 4

Illustration of computational principles of MPM

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Step 1 A two-dimensional continuum body is segmented into a collection of material points, each embodying all necessary state variables. Simultaneously, the background grid is discretized, with the nodal displacement initially at zero.

Step 2 The state variables from the material points are projected onto the nodes of the discretized background mesh. This mapping process results in an updated set of nodal information that is formulated as follows:

$$m_{I\left( n \right)} = \sum\limits_{p \in I} {N_{Ip} m_{p} }$$
(13)
$$p_{I\left( n \right)} = \sum\limits_{p \in I} {N_{Ip} m_{p} v_{p\left( n \right)} }$$
(14)
$$f_{I,int\left( n \right)} = \sum\limits_{p \in I} {\frac{{\partial N_{Ip} }}{\partial j}\sigma_{ij\left( n \right)} V_{p} }$$
(15)

where the symbols i, j are the spatial coordinates of x, y; the subscripts p and I indicate the material points and the background grid nodes, respectively; fI,int stands for the nodal force (N); V signifies the volume (m3); n is the time step; v represents the velocity (m/s); p is the momentum (kg⋅m/s); NIp is the shape function, taking the quadratic B-splines in this solution; m represents the mass (kg).

Step 3: The equation of motion, indicated as Eq. (16), is resolved throughout the background mesh to compute the prevailing velocity and acceleration, represented by Eq. (17) and Eq. (18), respectively. Note that the integral computation of the elements is grounded on the material points;

$$f_{{I\left( {n + 1} \right)}} = f_{I,load\left( n \right)} + m_{I\left( n \right)} g - f_{I,int\left( n \right)} - \alpha f_{I\left( n \right)}$$
(16)

where α represents the damping coefficient to dissipate the system’s kinetic energy; g signifies the gravity acceleration (m/s2); fI,load(n) symbolizes the external force (N);

$$v_{{I\left( {n + 1} \right)}} = \frac{{p_{I\left( n \right)} + f_{{I\left( {n + 1} \right)}} \Delta t}}{{m_{I\left( n \right)} }}$$
(17)
$$a_{{I\left( {n + 1} \right)}} = \frac{{f_{{I\left( {n + 1} \right)}} }}{{m_{I\left( n \right)} }}$$
(18)

where Δt is the time (s); a represents the acceleration (m/s2).

Step 4 The state variables on the material points are refreshed through remapping from the deformed background mesh. These variables constitute displacement, velocity, and strain increment, respectively detailed as:

$$u_{{p\left( {n + 1} \right)}} = u_{p\left( n \right)} + \Delta t\sum\limits_{I \in p} {N_{Ip} } v_{{I\left( {n + 1} \right)}}$$
(19)
$$v_{{p\left( {n + 1} \right)}} = v_{p\left( n \right)} + \Delta t\sum\limits_{I \in p} {N_{Ip} } a_{{I\left( {n + 1} \right)}}$$
(20)
$$\Delta \varepsilon_{{p,ij\left( {n + 1} \right)}} = \Delta t\sum\limits_{I \in p} {\frac{1}{2}\left( {\frac{{\partial N_{Ip} }}{\partial j}v_{{I,i\left( {n + 1} \right)}} + \frac{{\partial N_{Ip} }}{\partial i}v_{{_{I,j} \left( {n + 1} \right)}} } \right)}$$
(21)

where Δє represents the increment in strain (−); u signifies the displacement, measured in meters (m).

Stage 5 Elasticity testing is conducted, and if the stress under testing surpasses the yield strength, plastic strain is determined via the plastic flow rule.

Step 6 After deformation, the background mesh is removed, placing the material points at their updated location, followed by resetting of the new mesh.

This cycle, encompassing Procedures 2–6, is continually reiterated until the completion of the specified loading steps/ time.

2.3 Verification

The effectiveness of the continuum damage model in a two-dimensional scope was validated by testing the MPM code against uniaxial compression in this section. The utilized specimen, as depicted in Fig. 5, had dimensions of 0.2 × 0.4 m2, encompassing a total of 3200 material points for numerical modeling. The bottom of the specimen was subjected to a fixed constraint boundary. Throughout the testing duration, a downward loading velocity (Vz = 1 × 10–8 m/s) was enforced on the specimen’s upper surface until failure of the specimen was observed. Table 1 lists the mechanical parameters for this test.

Table 1 Parameters for the tested specimen
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Fig. 5
figure 5

Diagram of the 2D specimen

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In general, ideal elastoplastic material under uniaxial compression test goes through two deformation stages: compressive elastic stage and yield deformation stage (that is plastic stage). During the elastic phase, the relationship between stress and strain is defined by Hooke’s law (Eq. (22-1)). Conversely, for the plastic stage, the Mohr–Coulomb yield criterion (Eq. (22-2)) is employed to ascertain the yielding strength. Therefore, Eq. (22) is utilized to procure the analytical stress–strain curve, denoted by the green dot in Fig. 6.

$$\sigma = \left\{ {\begin{array}{*{20}l} {E\varepsilon ,} \hfill & {Elastic\;stage} \hfill \\ { - 2c\sqrt {N_{\varphi } } ,} \hfill & {Plastic\;stage} \hfill \\ \end{array} } \right.$$
(22)
Fig. 6
figure 6

The stress–strain curves observed during the uniaxial compression test

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Figure 6 shows the specimen’s stress–strain curve (the solid blue line) obtained from numerical simulation. Upon the external load being applied, the specimen first experiences elastic deformation. An abrupt loss of stability occurred in the specimen when the peak strength is reached, leading to its failure. Figure 7 reveals a typical brittle failure mode in the specimen, demonstrated by ‘X’ shaped conjugate cracks. This numerical observation aligns well with rock shear failure encountered in lab tests. On the other hand, for the mechanical modeling without rock damage by MPM, the numerical stress–strain curve (the solid red line in Fig. 6) is consistent with the analytical result (the green dot in Fig. 6) of ideal elastoplastic material. In sum, the proposed model was valid, and its numerical code could simulate large deformations on the 2D scale well.

Fig. 7
figure 7

Specimen failure mode obtained from the uniaxial compression test through numerical modeling

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3 Numerical model for gas extraction by a directional well with multiple perforations

As shown in Fig. 8, the gas extraction progress contains both porous and directional-well flows, with multiple perforations connecting the porous and directional-well flows. The gas extraction model is formulated on the basis of the following assumptions:

  1. (1)

    Neither the capillary pressure nor the moisture content do not noticeably affect coal’s permeability.

  2. (2)

    Given the isothermal conditions in the coal seam, the influences of temperature on gas adsorption or desorption is considered negligible.

  3. (3)

    Coal embodies an elastic material of heterogeneous and isotropic nature, encompassing the pores, matrix, and coal skeleton.

  4. (4)

    In the coal seam, the gas adheres to ideal gas properties, and gas movement through pores complies with Darcy’s law.

Fig. 8
figure 8

Illustration of the wellbore-pore flow system in the gas extraction

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3.1 Governing equations

3.1.1 Porous flow and directional-well flow

Concerning the fluid dynamics in a porous medium, the velocity of flow is captured by the Darcy equation (Eq. (23)). To complete the calculation of the pore pressure field, the mass conservation equation (Eq. (24)) must be invoked alongside the flow model.

$$\upsilon_{p} = \frac{{K_{e} }}{\mu }\frac{{\partial P_{p} }}{\partial L}$$
(23)
$$\frac{{V\phi_{m} }}{{R_{g} T}}\frac{{\partial P_{p} }}{\partial t} = Q_{s,p} + \nabla (\upsilon_{p} \cdot \rho_{g} )$$
(24)

where Ke is the element permeability (m2); Qper denotes the gas production rate through the perforation (kg/s); Qde is the gas desorption rate in a pore medium (kg/s); Qs,p = Qde + Qper, corresponds to the source term in a pore medium (kg/s); Pp is the gas pressure in a pore medium (Pa); ϕm is the porosity (dimensionless). vp stands for the gas flow rate in a pore medium (m3/s);

The Langmuir isotherm model was most commonly used in gas adsorption on coal rock (Langmuir 1917). Based on this model, the gas desorption rate in a porous medium can be expressed by Eq. (25).

$$Q_{de} = \frac{\partial }{\partial t}\frac{{\rho_{g} \rho_{c} V_{L} VP_{p} }}{{P_{p} + P_{L} }}$$
(25)

For a CBM reservoir, the gas within the pore-matrix system is a combination of adsorbed and free gas. Consequently, the total gas mass can be expressed as (Zhang et al. 2018):

$$m_{tot} = \rho_{g} V\phi_{m} + \frac{{\rho_{g} \rho_{c} V_{L} VP_{p} }}{{P_{p} + P_{L} }}$$
(26)

Modeling of flow rates from the pore into the wellbore employs the concept of radial flow so that the production rate through a perforation can be expressed by Eq. (27).

$$Q_{perf} = \frac{{2\pi K_{m} h\rho_{g} \left( {P_{w} - P_{p} } \right)}}{{\mu \ln \frac{{L_{e} }}{{r_{w} }}}}$$
(27)

where Le is the characteristic length of the production element (m); rw denotes the wellbore radius (m); h represents the height of the open well section (m); Pw refers to the production pressure (Pa).

To characterize the flow dynamics within the wellbore, we utilize the Hagen–Poiseuille equation (found in Eq. (28)), a derivative of the Navier–Stokes equation (Valko and Economides 1995). According to the mass conservation equation (Eq. (29)), the total production rate should equal the sum of all the production rates of each perforation in the wellbore.

$$\upsilon_{w} = \frac{{\pi D_{w}^{4} }}{128\mu }\frac{{\partial P_{w} }}{\partial L}$$
(28)
$$Q_{pro} { = }\sum\limits_{j = 1}^{n} {Q_{perf,j} }$$
(29)

where Dw represents the wellbore diameter (m); vw is the gas flow rate in the wellbore (m3/s); n indicates the total amount of perforations; μ is the gas viscosity (Pa s); Qperf,i is the gas flow rate through perforation i (kg/s); Qpro corresponds to the total gas production rate (kg/s).

3.1.2 Stress-dependent permeability models

To accurately describe methane flow behavior in a coal seam, the stress-dependent permeability of coal rock should be adopted considering the hydro-mechanical coupled effects.

The relationship between coal permeability and effective stress in porous media is frequently modeled using an exponential function (Eq. (30), Seidle et al. 1992).

$$k_{m} = k_{m,0} {\text{e}}^{{ - 3c_{f} (\sigma_{eff} - \sigma_{eff,0} )}}$$
(30)

where cf characterizes compression coefficient (MPa−1), describing the stress sensitivity of the matrix based on experiment data; km,0 and km represent the initial permeability and the matrix permeability and (m2), respectively.

HS is one method to enhance coal rock permeability. In response to the geo-stress conditions in the deep coal seam, gradual compaction occurred within the slot leading to stress release. As a result, the slot permeability varies with the changing stress. Equation (30) can be used to model slot permeability but with different values of km,0, and cf.

HF is another approach to improve the coalbed conductivity. The fractures propped by the proppant during the stimulation are named propped fractures; otherwise, they are unpropped.

The calculation of aperture and permeability for the propped fractures utilized Eqs. (31) and (32) considering the impact of the loading path on proppant deformation (Zhou et al. 2019a). We calculate propped fracture conductivity by combining both equations in the modeling.

$$w_{p} = w_{p,0} - \frac{{a_{up} \sigma_{n,eff} }}{{b_{up} + \sigma_{n,eff} }}(c_{up} + D^{{ - d_{up} }} )w_{p,0} - \frac{{\sigma_{n,eff} }}{{K_{E} }}$$
(31)
$$K_{f,p} = a_{kp} e^{{ - b_{kp} \sigma }} (c_{kp} + D^{{ - d_{kp} }} )$$
(32)

where wp is the aperture of propped fracture (mm); wp,o is the initial aperture of propped fracture (mm); D indicates the thickness of the proppant (m); akp, bkp, ckp, dkp, bup, cup, dup are the fitting parameters; σ is the stress (Pa); σn,eff is the effective stress (MPa); Kf,p is the propped fracture permeability (m2); KE is the embedment modulus (MPa/mm).

The hydraulic conductivity of the unpropped fractures aligns with the “cubic law” (Snow and David 1969). Therefore, the permeability of the unpropped fractures can be determined through Eq. (33).

$$\left\{ \begin{gathered} w_{up} = a_{eh} + \frac{{b_{eh} \sigma_{n,eff} }}{{c_{eh} + \sigma_{n,eff} }} \hfill \\ K_{f,up} = \frac{{w_{up}^{2} }}{12} \hfill \\ \end{gathered} \right.$$
(33)

where w is the equivalent hydraulic aperture (µm); aeh, beh, ceh are the fitting parameters from Shen et al. (2020); Kf,up is the unpropped fracture permeability (m2).

These permeability models mentioned above are adopted for the numerical simulation of gas production in Sect. 4. As for a rock element containing a fracture, which we refer to as the ‘fracture element.’ Its conductivity is calculated using the equivalent permeability of matrix permeability and fracture permeability. On the contrary, only matrix permeability is set for an intact element without fractures.

3.2 Numerical solution based on FVM and FDM

In the proposed gas extraction model, the rock matrix is discretized using hexahedral elements, while the FVM and the FDM are adopted to solve pore flow and directional-well flow, respectively. We utilized the sequential coupling solution, whose flowchart can be viewed in Fig. 9. With the calculation initiating from a presumption of gas desorption rates in the coal rock being zero (Eq. (34)), a dynamic radial flow is utilized to outline the gas transition from pores into the wellbore within a time step, illustrated implicitly in Eq. (35). To mitigate the effect of the element size on the wellbore size, an empirical fiction factor is incorporated in Eq. (35).

$$Q_{de,o}^{{({\text{t}} + 1)}} = 0.0$$
(34)
$$Q_{perf,o}^{{({\text{t}} + 1)}} = C_{1} \frac{{2\pi hK_{m} (P_{w}^{{({\text{t}})}} P_{w}^{{({\text{t}})}} - P_{p,0}^{{({\text{t}})}} P_{pold,0}^{{({\text{t}} + 1)}} )}}{{\mu ln\frac{{L_{e} }}{{r_{w} }}}}\frac{1}{{2R_{g} T}}$$
(35)

where o is the index of the central element; t + 1 represents the current time step, while t represents the preceding one.

Fig. 9
figure 9

Flow diagram illustrating the sequential coupling approach

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Following the determination of source terms, we performed the hydraulic analysis in the pores. The transport equation (Eq. (24)) is linearized in an implicit schema through the use of the FVM, as depicted in Eq. (36).

$$\sum\limits_{Ni} {\rho_{io} } \frac{{K_{m} A_{i} }}{\mu \Delta L}\left( {P_{pold,i \, }^{{({\text{t}} + 1)}} - P_{pold,o \, }^{{({\text{t}} + 1)}} } \right) + Q_{de,o \, }^{{({\text{t}} + 1)}} - Q_{perf,o \, }^{{({\text{t}} + 1)}} = \frac{{V\phi_{m} }}{{R_{g} T\Delta t}}\left( {P_{pold,o \, }^{{({\text{t}} + 1)}} - P_{p,o}^{{({\text{t}})}} } \right)$$
(36)

where Ni is the total number of neighbor elements; i denotes the index of the adjacent elements; ΔL represents the distance between two neighboring elements (m).

After computation of the pressure field in pores, the gas flow rate from the adjacent rock elements into the center rock element can be estimated by Eq. (37). Moreover, the production rate through a perforation can be estimated using Eq. (35), implying that the production rate through a perforation is not calculated in the wellbore hydraulic computation.

$$Q_{inter,o \, }^{{({\text{t}} + 1)}} = \sum\limits_{Ni} {\rho_{io} } \frac{{K_{m} A_{i} }}{\mu \Delta L}\left( {P_{p,i}^{{({\text{t}} + 1)}} - P_{pold,o \, }^{{({\text{t}} + 1)}} } \right)$$
(37)

Given the assumption that the CBM flow rate within the wellbore remains constant, Eq. (38) is employed to estimate the total production rate at the outlet. Subsequently, in order to compute the gas pressure within the wellbore, the transport equation, drawn from Eq. (27), is linearized via the FDM and represented in an implicit form, as illustrated in Eq. (39).

$$Q_{pro}^{{({\text{t}} + 1)}} = \sum\limits_{Nj} {Q_{perf,j}^{{({\text{t}} + 1)}} }$$
(38)
$$\frac{{\pi D_{w}^{4} }}{{128\mu \left( {L_{j + 1} - L_{j} } \right)}}\rho \left( {P_{w,j}^{{({\text{t}} + 1)}} - P_{w,j + 1}^{{({\text{t}} + 1)}} } \right) = Q_{j\sim j + 1,w}^{{({\text{t}} + 1)}} = \sum\limits_{k = 1}^{j} {Q_{perf,k}^{{({\text{t}} + 1)}} }$$
(39)

where j is the index of the perforations; (Lj+1 − Lj) presents the distance from the perforation j to the adjacent perforation j + 1 (m); Qj~j+1,w is the flow rate between the perforation j and the adjacent perforation j + 1; Nj is the number of all the perforations.

After the hydraulic calculations in pores and wellbore, the pressure should be updated considering the CBM desorption induced by pore pressure drop. When the rock matrix is full of absorbed gas, and the absorption volume can be described by the pressure-dependant Langmuir model in the isothermal condition, the CBM desorption rate derivated from Eq. (25) in the coal rock is described in an implicit form as Eq. (40).

The solution to this equation can be obtained using the Newton iteration method. Consequently, the Eq. (26), combined with the porous flow equation (Eq. (28)), is linearized and converted into an implicit form (Eq. (41)). According to this, the pore pressure correction is done.

$$Q_{de,o}^{{({\text{t}} + 1)}} = \frac{1}{\Delta t}\left( {\rho_{g}^{{({\text{t}})}} \rho_{c} \frac{{V_{L} P_{p,o}^{{({\text{t}})}} }}{{P_{p,o}^{{({\text{t}})}} + P_{L} }}V - \rho_{c} \rho_{g}^{{({\text{t}} + 1)}} \frac{{V_{L} P_{pnew,o}^{{({\text{t}} + 1)}} }}{{P_{pnew,o \, }^{(t + 1)} + P_{L} }}V} \right)$$
(40)
$$\begin{aligned} & \rho_{g}^{{({\text{t}} + 1)}} \rho \frac{{V_{L} P_{ \, }^{{({\text{t}} + 1)}} }}{{P_{pnew,o}^{{({\text{t}} + 1)}} + P_{L} }}V + \frac{{V\phi_{m} }}{{R_{g} T}}P_{pnew,o \, }^{{({\text{t}} + 1)}} = Q_{s,p} + Q_{inter,o}^{{({\text{t}} + 1)}} - Q_{perf,o \, }^{{({\text{t}} + 1)}} \\ & + \rho_{g}^{{({\text{t}})}} \rho \frac{{V_{L} P^{{({\text{t}})}} }}{{P^{{({\text{t}})}} + P_{L} }}V + \frac{{V\phi_{m} }}{{R_{g} T}}P_{p,o}^{{({\text{t}})}} \\ \end{aligned}$$
(41)

The beneficial aspect of this numerical strategy is its capability to separately resolve the pore and wellbore flow processes, demanding only one iteration. As a result, the derived solution is more efficient and maintains higher stability than those generated from a monolithic coupling. Furthermore, this implicit formulation enables the selection of a comparatively larger time step.

3.3 Verification

The proposed model was verified by simulation of a laboratory test for CBM coproduction and comparison of the experimental and numerical results. Xu et al. (2021) performed a laboratory CBM coproduction test at a large specimen box (400 × 400 × 1050 mm3) using the apparatus for the commingled drainage of multiple layers superposed CBM system. Figure 10a shows the physical picture of the used specimen, which contains four coal layers, with the remainder being impermeable gas barriers. Moreover, the effective dimension of each coal layer is around 380 × 390 × 200 mm3, and the in-site stress loading is shown in Fig. 10. The gas drainage pipe was centralized in the specimen box and placed along the height of the block. The parameters are listed in Table 2.

Fig. 10
figure 10

Geometric model consisting of impermeable gas barriers in yellow and four coal reservoirs in grey, along with a depiction of the stress and pressure boundary condition: a laboratory specimen (Xu et al. 2021); b simplified specimen for the simulation verification

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Table 2 Calculation parameters in the verification example (Xu et al. 2021)
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For the numerical verification in this work, the large specimen is simplified to a specimen with dimensions of 380 mm × 390 mm × 1050 mm. Figure 10b shows the geometrical model of this simplified specimen, consisting of four coal layers and five impermeable interlayers, and the four lateral boundaries of the model are set to be impermeable. The production wellbore is located in the middle of the geometric model, and a total production time of 360 min at an outlet pressure of 0.3 MPa was simulated. Figure 11 indicates an increase in cumulative gas production and a decrease in instantaneous gas flow rate over the operation time. We found that numerical simulation results align well with laboratory tests. In general, this confirms the suitability of the developed model in FLAC3D to simulate CBM production processes.

Fig. 11
figure 11

Evolution of a cumulative production volume and b instantaneous flow rate under a production pressure of 0.3 MPa. ER and SR represent the experimental result and simulation result, respectively. The experimental data here was collected from Xu et al. (2021)

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4 Case study in Linxing CBM field

4.1 Simulation of the field CBM production in well LX-XX

In the Linxing region, the technology of vertical well and HF is commonly used to extract CBM from the deep Benxi group coal seam (Lu et al. 2021). The fracturing fluid was injected with proppant in the Benxi group coal seam of well LX-XX. The CBM extraction was performed in the permeability-enhanced reservoir with propped hydraulic fractures. The gas production lasted for 270 days. The recorded daily production rate fluctuated throughout the production period, with a daily production rate of 617 m3/d on average. This average rate was lower than 2000 m3/d, the minimum CBM production rate for effective production in a single well (Meng et al. 2018). Furthermore, there was no sign of stable production in the fieldwork. We found that the deep CBM extraction from the deep Benxi group coal seam in the Linxing region did not succeed.

To find the reason for the failure of deep CBM mining in the Linxing region, we performed numerical modeling of CBM extraction according to the field condition in this section. Since the studied deep coal seam at the depth of around 2000 m has a temperature of around 56 degrees Celsius, the effects of the temperature on the deep coal seam are ignored in this work.

4.1.1 Model generation

Based on the data from the Benxi group coal seam in well LX-XX, in the Linxing region of Shanxi Province, China, a 120 m × 80 m × 5 m coal seam model was created in this section. Figure 12 illustrates the 3D geometrical model, with a model center at 2003.3 m. In previous work, including HF experiments (Huang et al. 2017; Sun et al. 2019; Wu et al. 2018; Jiang and Zhang 2016) and numerical studies (Zou et al. 2016; Zhou et al. 2019b; Chen et al. 2015;), a typical single hydraulic fracture was generated by HF operation because of high-stress difference in the deep coal seam. A single 2D vertical hydraulic fracture plane within 3D elements was assumed parallel to the maximum horizontal stress traversing the model’s center, with a length of 100 m. In the middle of the fracture, a vertical borehole was placed, and a production element with a production pressure of 0.1 MPa was inserted. The total gas extraction time was 270 days. The model’s upper, lower, and lateral boundaries are impermeable and subject to fixed constraint conditions. Table 3 lists additional parameters, in which the geo-stress and rock physical properties are based on real formation data.

Fig. 12
figure 12

Geometrical model based on HF

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Table 3 Parameters used in the field simulation
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4.1.2 Contrast between numerical modeling and field monitoring

Figure 13 illustrates the convergence of simulated predictions and on-field statistics of gas production over a period of 270 days. It can be observed that when the 100-m fracture is fully propped by proppant (the solid purple line), the gas production with the highest total volume of 3.59 × 106 m3 after 270-day production is 21.5 times the field production, while the scenario considering a fully 100-m unpropped fracture (the solid green line) is ranked second, with a total volume being twice the field production volume. These significant differences in gas production indicate that the assumed hydraulic fractures above cannot match the actual HF result in the fieldwork. In the Linxing field fracturing operation, solid proppants were incorporated into the injected fluid to hinder complete fracture closure. As a result, further testing was undertaken with a series of simulations considering the hydraulic fractures at different propped lengths. Results reveal that the total gas production decreases as the propped length of a single main hydraulic fracture reduces. As the propped length reduces to 3 m, there is a notable correlation between the numerical (the solid red line) and field production. This history matching thus implies that a 3-m propped fracture is comparable to the field stimulation outcome. Furthermore, the effective aperture of the fracture following this production modeling aligns closely with that observed in the experiment by Shen et al. 2020, measuring at the micrometer scale. Therefore, the field CBM extraction failure in the Linxing region could be because of the inefficient reservoir stimulation, for which fractures cannot be effectively generated after HF leading to poor extraction performance. Li et al. (2020) similarly inferred this through field monitoring and observation of the fracturing fissures in underground CBM wells. Factors such as the ductile characteristics of the deep coal seam, in conjunction with the considerable stress differential, may play a role in the failure of HF, as suggested by Zhou et al. (2020). Therefore, other stimulation technology instead of HF should be implemented for efficient extraction of the deep CBM in the Linxing region.

Fig. 13
figure 13

Results comparison between field production and numerical production: a cumulative production volume; b instantaneous production rate

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4.2 Simulation of enhanced CBM recovery by a directional well with multiple hydraulic slots.

One effective measure instead of the HF is needed to improve CBM extraction, as CBM resources are abundant and widely distributed in the Linxing region. In this section, we used the technology of directional wells and HS proposed by (Lu et al. 2021) and tested its efficiency of enhanced CBM recovery on deep coal seams in the Lingxin area.

4.2.1 Geometric model

Based on the 3D reservoir model described in Sect. 4.1.1, a 100-m horizontal well was established at the center (Fig. 14b), aligning parallel to the maximum horizontal principal stress. Along this horizontal well, several slots were uniformly arranged at a spacing of Δd, with a perforation positioned at the center of every slot. For the gas production over a 5-year period, the outlet pressure was configured to be 0.1 MPa. This 3D model is utilized to study the enhanced CBM performance in the deep coal seam based on the technology of directional wells and HS.

Fig. 14
figure 14

Geometric models for modeling: a mechanical behavior of a single slot after HS; b deep CBM extraction considering multiple slots

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The effects of multiple slots in permeability enhancement and CBM desorption can be represented by superimposing every slot’s effect. Briefly, the stimulated effect of multiple hydraulic slots on the reservoir is simplified to the superposition of reservoir deformation caused by a single slot. Due to the irreversibility of reservoir plastic deformation, the superposition effect among the region of plastic deformation is ignored and only the deformation of the region where elastic deformation occurs within the range of influence of the two single slots is superimposed. Following this, we obtain the change of porosity. Based on the formula relationship between porosity and permeability, the enhanced permeability on the reservoir is estimated. So that we can determine the stimulation effect of the multiple slots on the reservoir.

As such, the effect of a single hydraulic slot on stress release, CBM desorption and permeability improvement within the coal seam should be initially elucidated prior to the gas production simulation. For simplification purposes, we are treating the slot-induced deformation problem as a plane strain scenario. A 2D coal seam model is employed to investigate this problem in coal seam based on the MPM algorithm. This model measures 30 m in length and 24 m in width, with a 4 m long and 0.15 m wide slot fixed in its center (Fig. 14a). Four-node square elements with dimensions of 0.05 m × 0.05 m are used to build the computational grid and within each element, there are four material points initially placed. And thus, the model is divided into 288,000 background grids and contains over 1 million material points.

4.2.2 Effects of stress relief, gas desorption, and permeability enhancement induced by HS

Figure 15 shows the deformation process of a single hydraulic slot calculated by the MPM. It can be observed that the slot gets compacted as time goes on until it is fully closed. This deformation behavior of coal rock after HS is actually a mechanical problem involving large deformation, large displacement, and self-contact of slot internal boundaries. The same calculation was also carried out using FLAC3d software, and Fig. 16 shows the numerical results of the plastic zone calculated by MPM and FEM. When the computational mesh retains its initial place, the hydraulic slot, characterized as a vacant zone with non-contact slot boundaries unable to withstand any stress, results in significant damage to the slot-disturbed coal seam (Fig. 16b). On the other hand, upon adjustment of the computational mesh in response to deformation after each time step, severe mesh distortion transpires due to the substantial deformation, subsequently causing computational errors (Fig. 16c). Therefore, FEM is problematic in handling these issues in terms of slot-induced coal rock deformation; by contrast, Fig. 16a shows that MPM can achieve an accurate and efficient solution.

Fig. 15
figure 15

Slot patterns at different numerical time steps by MPM

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Fig. 16
figure 16

Comparison of the plastic zones calculated by MPM and FEM: a Scenario 1 calculated by MPM b Scenario 2 where the node location is not updated with the displacement by FEM; c Scenario 3 where the node location is updated with the displacement by FEM

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The variation in stress distribution resulting from mechanical deformation post-slotting in the coal seam is demonstrated in Fig. 17a. These are the stress relief region (Δσ < 0), the stress concentration region (Δσ > 0), and the initial stress region (Δσ = 0). Lu et al. (2016) suggest an effective depressurization occurs with a stress reduction of more than 5%. Therefore, we use a threshold of 2 MPa to identify areas with effective stress drop in our study. The effective region of stress relief can be categorized into two primary subsections: Zone 1 and Zone 2. Zone 1 is the approximately elliptical zone around the slot, with a rapid stress drop over 7 MPa, whereas Zone 2, which is shown in a broader scope surrounding Zone 1, exhibits a moderate stress decline between 2 and 7 MPa, resembling the numeral ‘8’ in shape. The stress redistribution in HS-disturbed coal seam leads to local stress concentrations where the stress could easily reach the yielding strength and thus induces plastic damage. The zone of damage propagates from the slot, extending to the neighboring area, and ultimately exhibits an elliptical layout. The elliptical region as illustrated in Fig. 17b represents the plastic zone where plastic strain exists while the rest region without plastic strain is classified as the elastic zone. It can be found that the extent of the plastic deformation zone matches that of the stress rapid relief zone (Zone 1), which means the rapid drop in stress in Zone 1 is caused by large plastic damage; on the contrary, the small elastic deformation occurs in the stress slow drop zone (Zone 2).

Fig. 17
figure 17

The effects of mechanical deformation on Deep CBM reservoir after HS: a stress variation distribution; b plastic deformation zone; c permeability change ratio; d desorbed gas volume per unit

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The spatial distribution of the coal seam permeability ratio following and preceding HS is displayed in Fig. 17c. Within the elastic deformation zone, the coal seam’s permeability enhanced by the HS technique is between 1 and 5 times the original permeability. This numerical result is comparable to the physical experimental result (Xue et al. 2015). Whereas the coal seam permeability after HS increases over 1000% in the plastic deformation zone. It is noticeable that there is an increasing ratio of over 200 around both ends of the slot. This occurrence can be attributed to the high stress concentration at the slot ends, as illustrated in Fig. 17a, leading to the facilitated crushing of coal rock and the generation of cracks in these areas. Under the influence of a slot, the overall permeability enhancement effect in the plastic zone is greater compared to that in the elastic zone. Due to the compaction deformation of the slot and the slot-induced failure in the coal seam, the gas is converted from the adsorption state into the free state, which subsequently increases the quantity of desorbed gas. The distribution of the gas desorption amount per unit volume is demonstrated in Fig. 17d, corresponding to the distribution pattern of the permeability alteration. By statistical computing, the influence of a single hydraulic slot results in the enhanced CBM desorption of 556 m3. These alterations in free gas content and the permeability within the coal seam, as a consequence of HS, are likely to facilitate a notable enhancement in CBM extraction efficiency. In summary, the slot deformation process after HS was accurately simulated by MPM, and the effects of a single hydraulic slot on stress relief, increased permeability, and enhanced desorption were analyzed.

4.2.3 Direction of the horizontal well

In order to determine the drilling direction of the horizontal well, the effect of the slot inclination angle on mechanical deformation and gas desorption disturbed by the slot is studied. The inclination angle of the slot (α) is characterized as the angle between the direction of the minimum horizontal principal stress (σh) and the slot direction. The inclination angles were respectively set at 0°, 30°, 45°, 60°, and 90° and Fig. 18 illustrates the stress variation in the coal seam under these angles, as obtained through the MPM calculation. Figure 19 represents the association between the gas desorption amount, the area of the stress relief zone, and the slot inclination angle. Notably, the area of the plastic stress relief zone exhibits minimal variation under all slot inclination angles, about 20 m2; on the contrary larger α leads to a more remarkable decline of the elastic stress relief zone area. The elastic zone induced by slot deformation is the largest when α = 0°, with an area of 95 m2, which is about five times the area of the plastic zone. However, the gas amount desorbed in the smaller plastic stress relief zone is the primary source of the total CBM desorption amount, at the difference over 400 m3 with the gas amount desorbed in the large elastic zone for each slot inclination angle. It is because the significant stress drop over 7 MPa in the plastic stress release zone caused the high gas desorption amount.

Fig. 18
figure 18

Calculated stress variation around the slot under different inclination angles. The gray picture shows the slot inclination angles. The grayscale image denotes the slot inclination angle (α), identified as the angle between the direction of the minimum horizontal principal stress (σh) and the slot direction

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Fig. 19
figure 19

Relationship between a gas desorption amount, b areas of the elastic–plastic stress relief zones, and slot deviation angles

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Overall, both the total amount of gas desorption and the total area of the stress release zone show a decreasing trend with the rising slot inclination angle. This suggests that an increase in the inclination angle lessens the stress relief effect induced by slot deformation and amplifies the counteraction to gas desorption in the coal matrix. We conclude that a slot at the inclination angle of 0° favors a better enhanced CBM recovery. Therefore, a horizontal well, outfitted with multiple slots, should be drilled in parallel alignment with the maximum horizontal principal stress.

4.2.4 Results of enhanced CBM recovery

Figure 20 shows gas production amounts and gas flow rates in four simulated reservoirs over five years. They are Case 1 by a directional well with multiple hydraulic slots, Case 2 by a 100-m fully propped fracture, Case 3 by a 100-m unpropped fracture, and Case 4 by matching with field production. A directional well with multiple slots demonstrated a maximum gas production amount after five years, at 9600 × 103 m3 being 13 times the field gas production amount. Figure 21 initially presents a 2D horizontal slice at a depth of 2003.3 m within the 3D model, followed by illustrations of the CBM pressure distributions in four simulated reservoirs at the end of a five-year production at this depth. It can be found the largest range of pressure relief at a higher magnitude is presented in Case 1 by a directional well with multiple hydraulic slots, contributing to the most effective gas extraction among these cases, while the pressure relief in Case 4 by matching with field production occurs in a pretty small zone just around a 3-m propped fracture. This indicates that the larger the range and the magnitude of the pressure drop, the better the enhanced efficiency of CBM recovery is. Moreover, Case 2, with the 100-m fully propped fracture, is ranked second; however, forming this ideal entirely propped fracture in the deep coal seam is quite difficult according to field fracturing simulation (Zhou et al. 2020). Therefore, it is recommended to adopt the more controllable and effective reservoir stimulation technology, that is, HS combined with directional wells.

Fig. 20
figure 20

Comparison of CBM extraction over five years in deep CBM reservoirs stimulated by different approaches: a Cumulative production volume; b Instantaneous production rate

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Fig. 21
figure 21

a Depiction of the 2D slice’s positioning within the 3D model and b–e illustrations of fluid pressure distributions after 5-year CBM drainage in deep CBM reservoirs stimulated by different approaches at this slice

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4.3 Sensitivity analysis

Evaluating the influence of slot parameters on enhanced CBM) recovery by a directional well with multiple hydraulic slots is vital for the optimization of the HS design. Enhanced CBM recovery simulation in Sect. 4.2.4 by a 100-m directional well with ten hydraulic slots (Δd = 10 m), with each slot being 4 m in length and 0.15 m in width, was used as the base case for the parameter sensitivity study in this section.

These factors were individually examined while maintaining the other parameters unchanged in the numerical simulations. In addition to the Base case (Fig. 22a), the length is set as 3 m and 4 m; likewise, the width is set as 0.1 m and 0.2 m. Figure 22a presents the slot layout for the base case, while the designed cases featuring multiple slots at intervals of 5 m, 15 m, and 20 m, respectively, are portrayed in Fig. 22b–d. Given the Base case, the length is set to 3 m and 4 m; similarly, the width is set to 0.1 m and 0.2 m. The findings from this investigation are anticipated to deliver critical insights for the implementation of HS in the process of deep CBM recovery.

Fig. 22
figure 22

Profile of the slot layout in a directional well with multiple slots at various spacing

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4.3.1 Slot spacing

Figure 23 shows the comparison of CBM extraction performance under slot spacings of 5 m, 10 m, 15 m, and 20 m over five years. As production time progresses, all production rates rise initially to the peak, show a gradual decline after the peak, and maintain constant eventually. This is because, at the initial extraction stage, abundant free CBM, which contains desorbed gas enhanced by hydraulic slots, contributes to a high extraction rate. As the extraction continued, both the enhanced permeability and the CBM desorption induced by a slot exerted a weakened influence, resulting in a gradually declined extraction capacity.

Fig. 23
figure 23

Comparison of CBM extraction performance under different slot spacings: a Instantaneous production rate; b Cumulative production volume

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During the CBM extraction process, the CBM cumulative production volumes over five years are shown in Fig. 23b. As the slot spacing increases, the cumulative production volume decreases, while the average contributions of each slot towards the total gas production amount exhibit an opposing trend, being approximately 576.9 × 103 m3 (Δd = 5 m), 960 × 103 m3 (Δd = 10 m), 1141.1 × 103 m3 (Δd = 15 m), and 1263 × 103 m3 (Δd = 20 m), respectively. As slotting spacing decreases, the production of CBM increases, but with a rising operation cost budget at a growing slot number for the same stimulation range. Consequently, the appropriate slot spacing and number should be arranged regarding economic production.

4.3.2 Slot length

Figure 24 presents the rates of gas production and the cumulative production volumes for slot lengths of 3, 4, and 5 m over a five-year period of CBM extraction. As illustrated in Fig. 24, the highest peak in gas production rate and the largest cumulative production volume are attributed to the scenario at the shortest slot length, with the smallest pressure relief range in Fig. 25. This finding contradicts some perspectives (Lu et al. 2011; Shen et al. 2015) that propose better production is associated with a wider pressure relief range. This is because of a sudden increase of the slot permeability at both ends of each slot (Fig. 25) that is closer to the production unit in the middle, in which the CBM is rapidly conducted, improving production performance. Thus, the proper short slotting operation can increase CBM production volume efficiently, as well as lower engineering costs.

Fig. 24
figure 24

Comparison of CBM extraction performance under different slot lengths: a Instantaneous production rate; b Cumulative production volume

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Fig. 25
figure 25

Permeability change ratio after HS considering different slot lengths: a 5 m; b 4 m; c 3 m

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4.3.3 Slot width

Figure 26 shows the cumulative production volume and the instantaneous production rate considering the slot width of 0.1, 0.15, and 0.2 m over five years. There is a marked rise in the instantaneous production rate in the early extraction stage as the width increases, characterized by the growing peak values being 5082 m3/d (w = 0.1 m), 8000 m3/d (w = 0.15 m), and 12,236 m3/d (w = 0.2 m), respectively (Fig. 26a). With an increase in the slot width, the CBM production volume appears an increasing trend, as shown in Fig. 26b. It is noticed that the amount of 5-year gas production increases by 2.5 × 106 m3 from 0.1-m width to 0.15-m width and by 1.5 × 106 m3 from 0.15-m width to 0.2-m width. Therefore, an increase in slot width in filed applications can contribute to enhanced CBM recovery. In addition, the enhancement contribution by the slot width change to the gas production is more significant than that by the slot length change.

Fig. 26
figure 26

Comparison of CBM extraction performance under different slot widths: a Instantaneous production rate; b Cumulative production volume

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5 Conclusions

This paper has presented a valid model developed using MPM and continuum damage mechanics to simulate the large deformation and complex contact process associated with hydraulic slots in deep coal seams. Furthermore, we established a gas production simulator accounting for both porous and directional-well flow with multiple perforations. Through numerical modeling, the enhanced gas recovery in the Linxing gas field was simulated using both a vertical well with a single main fracture and directional wells with multiple hydraulic slots. Based on this analysis, we have drawn the following conclusions:

  1. (1)

    The field CBM extraction failure in the Linxing region may be a result of unsuccessful reservoir stimulation. Here, hydraulic propped fractures are not effectively generated due to ductile behavior and the high-stress difference in the deep coal seam, leading to poor extraction performance.

  2. (2)

    Reservoir stimulation using directional wells with multiple hydraulic slots improves enhanced CBM recovery at deep reservoir levels in the Linxing region. This approach yields a total gas production of 13 times the field production. After HS in the deep coal seam, a slot gradually compacts until it is fully closed. The large plastic damage induced causes a rapid stress drop of over 7 MPa in the elliptical zone around the hydraulic slot. Elastic deformation and plastic deformation zones see a permeability increase of up to five times and over ten times the instinct coal seam permeability, respectively. Moreover, a total gas desorption of 556 m3 is observed from these zones.

  3. (3)

    Proper short and wide slotting operations can increase CBM production volume efficiently and reduce engineering costs. Additionally, as slotting spacing decreases, the production of CBM increases, but the economic benefit does not proportionately follow. In addition, the design of slot spacing should appropriately overlap within the range of the stimulated area of a slot.