Introduction

Hydraulic fracturing is a process of injecting high-viscosity, high-volume fracturing fluids into rock formations with high pressure to increase the connectivity ability between wellbore and formation (Guo et al. 2020). This technique is widely used in the oil and gas extraction industry to enhance formation permeability (Fang et al. 2022). The recently proposed drilling and fracturing in the roof of coal seam has obtained good results in the extraction of coalbed methane (CBM) in the soft coal seam with low permeability. However, there are significant difference of mechanical properties of coal seam and its sandstone roof, and the size and shape of hydraulic fractures are crucial factors in determining the effectiveness of gas production enhancement (Wu et al. 2022). If the fracture width is too small, it can impede the transportation and placement of proppants into the coal layer. And the hydraulic fractures may close due to the influence of ground stress without efficient support of the proppants after the fracturing process is completed. This closure reduces fracture flow conductivity and hampers the achievement of desired reservoir improvement objectives.

The hydraulic fracturing process involves a complex array of steps, including the flow and potential loss of fracturing fluid, rock deformation, fracture propagation, and various mechanical factors to consider (Zhang et al. 2023a, b). Since the 1950s, scholars have extensively studied hydraulic fracturing and have proposed various analytical or semi-analytical models to describe and predict the geometry of hydraulic fractures. The PKN model, developed by Nordgren (1972) and Perkins and Kern (1961), is based on the classical Sneddon plane-strain crack solution. It assumes a constant seam height in the direction of the seam length and an elliptical crack section. The KGD model, developed by Geertsma and Klerk (1969), assumes a rectangular fracture section and is applicable when the seam height is much larger than the seam length. Spence (1985) proposed a semi-analytical solution for the KGD model, which takes into account some energy dissipation but ignored consider the filtration loss of fracturing fluid. Bunger et al. (2005) developed a more comprehensive system of semi-analytical solutions for the KGD model by considering energy dissipation ratios and fracturing fluid fugacity patterns in the rock formation. Some researchers have proposed three-dimensional models of hydraulic fracturing, which take into account the complexities of fracture propagation in three dimensions. Recently, Shel and Paderin (2019) presented a semi-analytical solution for a proposed 3D model that considers filtration loss of fracturing fluid. These models and solutions contribute to our understanding of hydraulic fracturing and help in predicting fracture behavior and optimizing fracture design in various geological conditions.

Classical models can be used to calculate the relationship between pressure and fracture geometry. However, these models do not consider the effects of natural fractures and rock non-homogeneity on fracture expansion. To address this, scholars have used numerical simulations and laboratory experiments to comprehend the expansion pattern of hydraulic fractures. The flow rate of the fracturing fluid stands as a critical parameter in the hydraulic fracturing process. Typically, a higher fluid flow rate leads to a broader hydraulic fracture width. (Guo et al. 2017; Cao et al. 2022). Increasing the viscosity of the fracturing fluid reduces fluid loss and shows a positive correlation with the width of the hydraulic fracture (Cong et al. 2022b). When carbon dioxide foam is used as a fracturing fluid, increasing its quality from 0.5 to 0.8 more than doubles the hydraulic fracture width (Cong et al. 2022a). The spacing between clusters significantly affects fracture width in segmented cluster hydraulic fracturing, making the design of cluster spacing crucial (Tang et al. 2018). Apart from hydraulic fracturing process parameters, formation parameters also play a vital role in determining fracture width. Natural fractures and laminar surfaces also have significant impacts on hydraulic fracture morphology (Guo et al. 2021). The interaction between hydraulic fractures and these surfaces can cause a sudden drop in hydraulic fracture width (Shi et al. 2022). Moreover, the smaller the angle of intersection between hydraulic and natural fractures, the larger the width of natural cracks (Zheng et al. 2020). The reduction in tensile strength of natural weak surfaces increases the width of natural cracks (Sun et al. 2020). Additionally, the hindering effect of crack expansion becomes more pronounced with an increase in the number of natural cracks in the layer, resulting in a smaller width of hydraulic cracks (Ren et al. 2022). The physical and mechanical properties of the formation significantly influence hydraulic fracture width. Smaller permeability and loss coefficients result in larger hydraulic fracture length and width (Guo et al. 2022; Zhang et al. 2023c). The direction of hydraulic fracture crack expansion is usually parallel to the maximum horizontal ground stress. Hence, ground stress has a vital influence on hydraulic fracture morphology, with greater differences in horizontal ground stress and bias stress leading to increased hydraulic fracture width (Saberhosseini and Keshavarzi 2019). When the hydraulic fracture encounters formations with higher modulus of elasticity, the hydraulic fracture width is constrained to be narrower (Zhang et al. 2022). Acoustic emission is commonly used for hydraulic fracture detection in hydraulic fracturing experiments. This method captures the acoustic emission signals released during the generation of hydraulic fractures in rocks, enabling the detection of changes in fracture length and height throughout the process. However, it has limited effectiveness in detecting changes in fracture width (Chitrala et al. 2013; Stoeckhert et al. 2015; Patel et al. 2017; Sakhaee-Pour and Agrawal 2018). CT scanning provides an effective way for comprehensive characterization of complex pores and fractures (Gou et al. 2019). CT scanning of rock specimens after hydraulic fracturing has also been employed by some researchers to obtain hydraulic fracture morphology. Nevertheless, this method fails to capture the dynamic changes in fracture morphology during the entire process (He et al. 2017; Zhang et al. 2021). In the use of hydraulic fracturing to control coal seam gas, due to the weak mechanical properties of the coal seam, the possibility of collapsing the hole in the coal seam is greater, and it is more difficult to form a hole. Therefore, hydraulic fracturing in the rock layer above the coal seam is an effective method to solve this problem. In hydraulic fracturing operations targeting the roof of coal seams, there is a limited amount of research available specifically focusing on the changes in hydraulic fracture widths within both the roof and coal seam after the hydraulic fracture enters the coal seam. This area requires further investigation and study to better understand the behavior and characteristics of hydraulic fractures in this specific context.

This study investigates the evolution of hydraulic fracture width during the hydraulic fracturing of coal seam roof layers in coalbed methane mining. Understanding the temporal changes in hydraulic fracture width is crucial for optimizing coalbed methane extraction and enhancing reservoir performance.

True triaxial coal seam roof hydraulic fracturing experiments

Specimen properties and preparation

Due to the poor mechanical properties of the interface between the coal seam and the roof rock, it is not possible to obtain the continuous coal seam and roof rock from the field, this paper adopts different ratios of concrete to replace the coal seam and the rock layer. The mechanical properties can be used to characterize the coal and rock seams through rock mechanical tests, and their mechanical parameters are shown in Table 1. The size of the specimen was 300 mm*300 mm*300 mm, and the thickness of the specimen was 120 mm sandstone, 80 mm mudstone and 100 mm coal seam from top to bottom. Strain gauges were arranged in the mudstone and coal bed to measure the deformation of the specimens during the hydraulic fracturing process. After the specimens were poured and left to rest for 72 h, holes were pre-fabricated in the center of the specimens and epoxy resin was used to fasten the fracturing tubes to the holes. The end of the fracturing tube will be arranged at the center of the mudstone since the scenario simulated in this experiment is hydraulic fracturing in the top plate of the coal seam. The schematic diagram of the specimen is shown in Fig. 1, and the strain gauges are located on the upper surface of the strain brick in the mudstone and the lower surface of the strain brick in the coal, respectively.

Table 1 Mechanical parameters of similar materials
Full size table
Fig. 1
figure 1

Schematic diagram of specimen

Full size image

Experimental apparatus

The hydraulic fracturing test system used in this study comprises various components, as illustrated in Fig. 2 These components include: a true triaxial fracturing tank, a confining pressure loading control box, a constant pressure and constant current pump, a digital hydraulic displacement and strain collector (DHDAS). Where the True triaxial fracturing tank is responsible for subjecting the hydraulic fracturing specimens to stress in a manner that simulates real formation conditions. It is designed to apply loading independently in the X, Y, and Z directions, allowing for comprehensive stress simulation. The maximum loading pressure of the tank is 25 MPa with a control accuracy of 0.25%. The Confining pressure loading control box works in conjunction with the true triaxial fracturing tank to regulate the applied confining pressure on the specimens. The confining pressure can be adjusted to achieve the desired stress conditions during the hydraulic fracturing process. The pumping system consists of a fracturing pump and a water tank and sensor. The fracturing pump is powered by an air compressor and can inject fracturing fluid into the fractured specimen in two ways: constant flow and constant pressure. The constant flow rate is up to 100 ml/min with an accuracy of 0.001 ml, and the constant pressure is up to 40 MPa with an accuracy of 0.1%. The DHDAS connect with the stain transduces pre-placed in the specimen. This strain collector detects and records the strain changes occurring inside the specimen during the hydraulic fracturing process. It has a strain range of ± 60,000 με with a minimum strain resolution of 0.1 με. The combination of these components allows for accurate and controlled testing of hydraulic fracturing behavior. The experimental system is shown in Fig. 2.

Fig. 2
figure 2

True triaxial hydraulic fracturing experimental system

Full size image

Experimental program

The morphology of hydraulic fractures plays a crucial role in the success of hydraulic fracturing operations. The height and length of hydraulic fractures determine the extent of the fracturing, while the width of hydraulic fractures affects the transportation and placement of proppants within the fractures. In this study, a true triaxial hydraulic fracturing experimental system is utilized to simulate hydraulic fracturing in the top plate of a coal seam. The experiment considers the effects of ground stress and flow rate on the expansion pattern of hydraulic fractures within the seam. The fracturing fluid used is an active water fracturing fluid commonly employed in coalbed methane development. Three different flow rates of fracturing fluid, namely 40 ml/min, 50 ml/min, and 60 ml/min, are injected into the specimen at a constant flow rate. Specific experimental parameters can be found in Table 2. By conducting these experiments and analyzing tracer traces on each face of the fractured specimen, the initial judgment of the height and length of the hydraulic fracture can be made. Additionally, the strain within the specimen provides an initial characterization of the width of the hydraulic fracture. These experiments aim to provide insights into the expansion patterns of hydraulic fractures in the coal seam’s roof under different ground stresses and flow rates.

Table 2 Parameter settings for hydraulic fracturing experiments
Full size table

Experimental results and analysis

Morphological analysis of hydraulic cracks

The crack expansion patterns under different horizontal ground stress differences were examined in this study using specimens labeled as 1#, 2#, and 3#. The horizontal stress differences applied to these specimens were 3 MPa, 2 MPa, and 1 MPa, respectively. The experimental results, as shown in Fig. 3, indicate that when the horizontal stress difference is 3 MPa and 2 MPa, hydraulic fractures initiate and propagate from the mudstone layer. Subsequently, they propagate through the sandstone-mudstone interface and the mudstone-coal interface, extending into the sandstone and coal layers. Furthermore, the traces of fracturing fluid observed at the sandstone-mudstone interface and the mudstone-coal interface suggest that the damage area at the sandstone-mudstone interface is relatively larger. This can be attributed to the fact that the tensile strength of the sandstone is significantly greater than that of the coal. Consequently, under the same geo-stress conditions, the rupture pressure of the sandstone surpasses that of the coal. As a result, the fracturing fluid exerts relatively higher pressure at the sandstone-mudstone interface. Under a stress difference of 1 MPa, hydraulic fractures are only activated at the sandstone-mudstone interface and the mudstone-coal interface, without penetrating into the coal and sandstone layers. This is attributable to the increasing fracture pressure of both the sandstone and coal resulting from the rising minimum horizontal ground stress. As a consequence, it requires higher fracturing fluid pressure to establish hydraulic fractures at the interface. However, due to the significantly weaker nature of the interface compared to the sandstone and coal, the interface is fully activated, preventing the hydraulic fractures from propagating into the coal and sandstone layers.

Fig. 3
figure 3

Hydraulic crack patterns under different horizontal stress differences

Full size image

In order to investigate the effect of fracturing fluid flow rate on the hydraulic fracture through the layer, 4#, 2# and 5# specimens were selected, and the fracturing fluid flow rate corresponded to 40 ml/min, 50 ml/min and 60 ml/min. Figure 4 shows the hydraulic fracture extension under different fracturing fluid flow rates, and it can be seen that hydraulic fracture crosses the interface and enters into the coal seam when the flow rate is 40 ml/min, but does not enter into the sandstone. This is due to the fact that when the flow rate of fracturing fluid is small, the pressure of fracturing fluid at the interface has not reached the rupture pressure of sandstone, but the hydraulic fracture has already extended to the boundary of sandstone-mudstone interface, so the pressure of fracturing fluid at the interface cannot reach the rupture pressure of sandstone. The hydraulic fracture pattern is similar when the flow rate is 50 ml/min and 60 ml/min, and the hydraulic fracture crosses the interface into the sandstone and coal bed. The higher flow rate of fracturing fluid helps the hydraulic fracture to penetrate the layer.

Fig. 4
figure 4

Hydraulic fracture morphology under different fracturing fluid flow rates

Full size image

Strain curve analysis

The strain data of specimen 3# is shown in Fig. 5, which can be divided into three stages. The first stage is the stress loading stage, where the X direction represents the minimum horizontal ground stress of 2 MPa and the Z direction represents the vertical ground stress of 5 MPa. Since the vertical stress is much larger than the minimum horizontal ground stress, the rock generally shows compressive but not tensile resistance. Therefore, the strain brick is stretched in the X direction, and the strain shows a rising trend. The second stage is the hydraulic crack expansion stage, where the crack pattern of specimen 3# is a penetrating crack in YZ direction. As a result, the hydraulic crack expansion in mudstone and coal leads to the strain brick being compressed in the X direction, resulting in a decreasing trend in the strain data. However, the decreasing value is smaller than the increasing value due to the loading of the ground stress. The third stage is the hydraulic crack closure stage, where the strength of the interface is lower than that of the mudstone and coal. Consequently, the interface will take priority over the hydraulic crack in the mudstone and coal to expand to the boundary. Thus, when the pressure in the hydraulic crack drops, the crack width decreases, resulting in the strain brick being stretched in the X-direction. The strain data shows a rising trend in this stage. It demonstrates the different response of rock materials to the ground stress and the hydraulic fracture process. The analysis of the strain data can provide insights into the deformation and failure mechanisms of rock materials under complex stress conditions.

Fig. 5
figure 5

Strain curve of reference specimen

Full size image

Numerical model

Guidelines for crack initiation and expansion

Crack initiation guidelines

In this paper, the tensile-separation criterion with degraded elements stiffness is used to simulate the crack initiation and extension in ABAQUS. As the stress on the cohesive elements increases, the stiffness of the elements degrades gradually, and when the element stiffness is 0, the cohesive element starts to crack and expand. In the finite element method, the following criteria are commonly used to determine the crack initiation: maximum positive stress criterion, maximum positive strain criterion, quadratic stress criterion and quadratic strain criterion. In this paper, the maximum positive stress criterion is used, i.e., as long as the unit reaches its critical stress in any direction, the unit will start to crack.

$$\max \left\{ {\frac{{\sigma_{n} }}{{\sigma_{n}^{\max } }},\frac{{\tau_{s} }}{{\tau_{s}^{\max } }},\frac{{\tau_{t} }}{{\tau_{t}^{\max } }}} \right\} = 1$$
(1)

where \(\sigma_{n}^{\max }\) refers to the maximum tensile stress that the element can withstand in the vertical direction, i.e. the tensile strength of the coal rock, and \(\tau_{s}^{\max }\) and \(\tau_{t}^{\max }\) refer to the maximum shear stress that the unit can withstand in both directions, i.e. the shear strength of the coal rock, respectively. Considering that coal rock is anisotropic material, the shear strength in different directions is different.

Crack expansion criteria

Introduce the dimensionless damage factor D, the value range is between 0 and 1, when D = 0, the material is not damaged, when D = 1, the material is completely damaged, the formation of cracks and continue to expand forward, 0 < D > 1, when the material is being damaged, the expression is as follows:

$$\sigma _{{\text{n}}} = \left\{ \begin{gathered} (1 - D)\sigma _{n}^{'} ,\sigma _{n}^{'} \ge 0 \hfill \\ \sigma _{n}^{'} ,\sigma _{n}^{'} < 0 \hfill \\ \end{gathered} \right\}$$
(2)
$$\sigma _{{\text{s}}} = (1 - D)\sigma _{s} '$$
(3)
$$\sigma _{{\text{t}}} = (1 - D)\sigma _{t} '$$
(4)

where \(\sigma _{n} '\) ≥ 0 means that the cohesive is subjected to tensile stress and \(\sigma_{n}{\prime}\) < 0 means that the cohesive unit is subjected to compressive stress. \(\sigma_{n}\), \(\sigma_{s}\), \(\sigma_{t}\) represent the normal stress component and two tangential stress components of the cohesive unit in the actual situation, while \(\sigma _{n} '\), \(\sigma _{s} '\), \(\sigma _{t} '\) represent the normal stress component and two tangential stress components of the cohesive element calculated under linear elasticity conditions before the material damage occurs.

Modeling fluid flow within a crack

The fluid pressure in the crack is the source of energy for crack expansion. The direction of fluid flow in the cohesive element is divided into tangential flow along the cohesive cell and normal flow perpendicular to the cohesive element.

Tangential flow equations in cracks

The fluid is assumed to be continuous and incompressible. Tangential flow follows the Newtonian flow equation

$${\text{q}} = - \frac{{d^{2} }}{12\mu }\nabla p$$
(5)

where \(q\) is the fracturing fluid displacement, \({\text{m}}^{3} /\min\); \(d\) is the fracture opening width, \(m\); \(\mu\) is the fracturing fluid viscosity, \({\text{mP}}a * s\); \(p\) is the flow pressure, \({\text{MPa}}\).

Normal fluid equations for the upper and lower surfaces of a crack

Fluid flow in a fracture is mainly tangential, while a small amount of fluid will be leached out through normal flow in the fracture to the formation above and below the surface of the cohesive unit. The equation for the normal seepage of fracturing fluid in the cohesive unit is:

$$\left\{ \begin{gathered} q_{{\text{t}}} = c_{t} (p_{i} - p_{t} ) \hfill \\ q_{{\text{b}}} = c_{b} (p_{i} - p_{b} ) \hfill \\ \end{gathered} \right.$$
(6)

\(q_{t}\), \(q_{b}\) for the normal-phase fluid flow rate into the upper and lower surfaces of the crack, \({\text{m}}^{3} /{\text{s}}\); \(c_{t}\), \(c_{b}\) are the fluid loss coefficients at the upper and lower surfaces of the crack, respectively, \({\text{m}}/{\text{s}}^{1/2}\). \(p_{i}\) for the cohesive fluid pressure in the middle of the surface,\(p_{t}\), \(p_{b}\) for the crack of the upper and lower surfaces of the fluid pressure, respectively, \({\text{MPa}}\).

Model parameter setting

Figure 6 shows the three-dimensional flow-solid coupling numerical model established in this paper, \(\sigma_{H}\) is the maximum horizontal principal stress, \(\sigma_{{\text{h}}}\) is the minimum horizontal principal stress, \(\sigma_{V}\) is the vertical principal stress. In order to study the influence of various factors on the hydraulic crack width after the hydraulic crack enters into the coal seam from the top plate, the three-dimensional flow-solid coupling hydraulic crack extension model is simplified into a three-layer entity according to the actual situation of the stratum, in which the computational parameters of the top layer, the intermediate layer and the bottom layer are consistent with the pre-setting of the Cohesive elements surfaces in the horizontal and vertical directions, which are used to simulate the lithological interfaces and the expansion path of the hydraulic crack, respectively. The depth of the coal seams in the site is mainly between 753 and 768 m, the thickness of the coal seams varies from 4.5 and 7 m, the average thickness of the mudstone on the top of the coal seams is 4.2 m, and the average thickness of the sandstone is 5.1 m. The thicknesses of overburden rock, top plate and coal seam in the model are 5 m, 4 m and 6 m, respectively, and the injection point of fracturing fluid is located in the middle of the mudstone, and the dimensions of the 3D model are 15 m × 15 m × 20 m. The solid elements, C3D8P, characterizes the deformation and damage of the reservoir rock and the seepage of the pore fluids in the three-dimensional space.

Fig. 6
figure 6

Application of cohesive elements hydraulic fracturing simulation model

Full size image

The mechanical properties of each rock layer in the model and the ground stress to which it is subjected are shown in Table 3.

Table 3 Numerical simulation parameter settings
Full size table

In order to verify the correctness of the numerical model, the numerical model is established with reference to the benchmark specimen of the physical experiment. Figure 7 shows the morphology of hydraulic cracks, comparing the numerical simulation with the results of physical experiments, it can be found that the hydraulic cracks all penetrate the interface to reach the coal and sandstone, and the crack morphology is consistent, which verifies the correctness of this paper.

Fig. 7
figure 7

Hydraulic fracture morphology of the validated model

Full size image

Numerical simulation results and analysis

Figure 8 illustrates the progressive propagation of the hydraulic fracture through different layers and highlights the importance of studying factors that affect the width and success of hydraulic fracture propagation at the interface between mudstone and coal. In Fig. 8a, the hydraulic fracture initiates by opening in the mudstone layer and then propagates towards the interface between the mudstone and coal layers. Due to the weaker mechanical properties of the interface compared to coal, the hydraulic fracture continues to extend into the interface. The fracturing fluid pressure within the fracture keeps increasing until the hydraulic fracture penetrates the coal seam, as shown in Fig. 8c. Since coal seams have weaker mechanical properties compared to mudstone, the fracturing fluid pressure required for hydraulic fracture expansion in coal is lower than that needed for expansion in mudstone. As the hydraulic fracture crosses the interface and enters the coal seam, the pressure within the fracture decreases. This decrease eventually leads to a reduction in the width of the hydraulic fracture in the mudstone. It is worth noting that there may be a phenomenon called “excess fluid without excess sand” where the fracturing fluid can pass from the mudstone into the coal seam but the proppants (sand or other materials) are unable to do so. This indicates the need to study the influence of various factors on the width of the hydraulic fracture at the interface. Understanding these factors can help ensure that the proppant is able to enter and remain within the coal seam for an extended period of time.

Fig. 8
figure 8

Hydraulic crack patterns at different times

Full size image

Influence of horizontal ground stress difference on the width of hydraulic cracks at the interface

The width of hydraulic fractures is significantly influenced by the horizontal geostress difference. To examine the impact of interlayer horizontal ground stress difference on hydraulic crack width at the coal-rock interface, the minimum horizontal ground stress of the coal seam was altered. The minimum horizontal ground stress difference between the coal seam and the roof plate was set to 5 MPa, 4 MPa, 3 MPa, and 2 MPa respectively. Figure 9 illustrates the variation in hydraulic crack width in mudstone at the interface under different horizontal ground stress differences. It also displays the hydraulic crack width of the coal when the width of the hydraulic cracks in the coal exceeds 0.6 mm. Proppant size in CBM development is typically 20–80 mesh to ensure effective transport within the fractures. The fracture width needs to be at least three times larger than the proppant particle size. Therefore, only statistical data on hydraulic fracture width changes when the width exceeds 0.6 mm are considered. From Fig. 9, it is apparent that the time taken for the proppant to enter the coal seam is 1.46 s, 1.27 s, 1.17 s, and 2.95 s for ground stress differences of 5 MPa, 4 MPa, 3 MPa, and 2 MPa respectively. Hence, when the ground stress difference is larger, the time for proppant entry into the coal seam remains relatively consistent. Consequently, the influence of local stress on the change in crack width at the interface is not significant during these scenarios. However, when the ground stress difference is 1 MPa, the time required for proppant entry into the coal seam noticeably increases compared to scenarios with higher ground stress differences (4 MPa, 3 MPa, and 2 MPa). Additionally, Fig. 9 indicates that the maximum width of the hydraulic fracture at the coal-rock interface exhibits an increasing trend as the ground stress difference decreases. Upon analysis, it is concluded that a smaller horizontal stress difference corresponds to a larger minimum horizontal stress in the coal seam. As a result, more hydraulic energy needs to be accumulated in the roof plate to break through the coal-rock interface(Li et al. 2021). This is consistent with the time at which proppant starts entering the coal seam through the interface in Fig. 9. The delayed entry of proppant indicates higher hydraulic energy accumulation as the ground stress difference diminishes. Furthermore, Fig. 10 presents a cloud diagram depicting the pore pressure at the beginning of hydraulic fracturing into the coal seam under different levels of geostress difference. It is evident that smaller geostress differences yield greater hydraulic energy accumulation at the interface. Consequently, narrower stress differences result in wider hydraulic fracture widths at the interface and longer durations for proppant entry into the coal seam.

Fig. 9
figure 9

Effect of horizontal stress difference on hydraulic fracture width (MHFW mudstone hydraulic fracture width, CHFW coal hydraulic fracture width)

Full size image
Fig. 10
figure 10

Pore pressure cloud during hydraulic fracture penetration under different ground stress differences

Full size image

Influence of coal seam elastic modulus on the width of hydraulic cracks at the interface

The higher the elastic modulus of coal, the greater the stress required to create hydraulic fractures in the coal. Therefore, it is important to study the influence of coal elastic modulus on the variation of hydraulic crack width at the interface between coal and mudstone. Keeping other conditions constant, we changed the elastic modulus of the coal seam to 1 GPa, 1.5 GPa, 3 GPa, 4 GPa, and 5 GPa. Figure 11 shows that as the elastic modulus of the coal seam decreases, the time for proppant to enter the coal seam also decreases. This can be attributed to the fact that a lower elastic modulus of coal requires less hydraulic energy for hydraulic cracks to penetrate the coal-rock interface and propagate within the coal bed (Jiang et al. 2016). The pore pressure cloud plot in Fig. 12 indicates that hydraulic fractures expand more rapidly in coal seams with lower elastic modulus. Consequently, the width of hydraulic cracks in the roof decreases at a faster rate, ultimately preventing the proppant from passing through the roof into the coal seam. Hence, a lower elastic modulus of the coal seam results in a shorter period for proppant entry. Additionally, Fig. 11 demonstrates that a lower elastic modulus of the coal bed corresponds to a larger maximum hydraulic fracture width at the coal-rock interface. This is due to the greater deformation capacity of coal beds with lower elastic modulus. Therefore, when considering the width of hydraulic cracks at the interface and the timing of proppant entry, it can be concluded that a smaller elastic modulus of the coal bed does not necessarily facilitate more proppant infiltration. A comprehensive assessment is necessary in such cases.

Fig. 11
figure 11

Effect of coal elastic modulus on hydraulic fracture width (MHFW mudstone hydraulic fracture width, CHFW coal hydraulic fracture width)

Full size image
Fig. 12
figure 12

Pore pressure cloud under coal seam with different elastic modulus at the same time

Full size image

Influence of permeability on the width of hydraulic cracks at the interface

In order to explore the influence of coal seam permeability on the width of hydraulic cracks at the interface, the permeability of the coal seams is set to 0.1 md, 0.5 md, 1 md, and 5 md. Figure 13 shows the width of hydraulic cracks at the interface at different coal seams. It can be seen from Fig. 13 that the time when the coal seam penetration rate increases from 0.1 to 5 md, the time when the support agent can enter the coal seams is 3.43 s, 1.71 s, 1.17 s, 1.26 s, respectively. This is because the lower permeability of the coal seam, the more difficult it is to flow in the coal seam. Under the same fracture fluid flow, this will help maintain the pressure of the internal fracture fluid of hydraulic cracks (Lan and Gong 2020). Figure 14 shows the pore pressure cloud at the same time, from which it can be seen that the lower the permeability of the coal seam the higher the pressure of the fracturing fluid in the hydraulic fracture. Therefore, the faster the time when the width of the hydraulic crack reaches the support agent and the longer the crack width is maintained. In addition, it can be seen in Fig. 13 that the permeability of the coal seam is the most width at the time of hydraulic fractures at the time of 5 md. According to analysis, the greater permeability of the coal seam, the more difficult the hydraulic cracking layer is. Therefore, the greater the hydraulic energy of the top plate accumulated when the hydraulic crack can be worn, the larger the maximum value of the hydraulic crack width. Comprehensive consideration, the smaller the permeability, the width of the hydraulic cracking can enter the overall view of the coal seam, the more helpful the support agent to enter the coal seam.

Fig. 13
figure 13

Effect of coal permeability on hydraulic fracture width (MHFW mudstone hydraulic fracture width, CHFW coal hydraulic fracture width)

Full size image
Fig. 14
figure 14

Pore pressure cloud at the same time with different coal seam permeability

Full size image

Interface strength on the width of hydraulic fractures at the interface

The strength of the interface plays a crucial role in determining the hydraulic crack pattern. Therefore, it is important to study the effect of laminated surface strength on the width of hydraulic fractures at the interface. To investigate this influence, we set the interface strength values as follows: 0.5 MPa, 1 MPa, 1.5 MPa, and 2 MPa. Figure 15 illustrates the variation in hydraulic crack width at the interface under different laminated surface strengths. It can be observed that as the laminated surface strength increases, the time required for the proppant to pass through the interface also increases. However, the change in hydraulic crack width at the interface is not significant. This can be analyzed by considering that an increase in laminated surface strength leads to a smaller area of the laminated surface being damaged. Figure 16 displays the area of the laminated surface destroyed within the same time, which supports the aforementioned analysis. Consequently, more fracturing fluid enters the hydraulic fracture of the coal seam and the top plate at the same fluid flow rate, allowing the proppant to pass through the interface for a longer duration. Compared with factors such as ground stress difference, coal bed elastic modulus, and coal bed permeability, it can be concluded that the strength of the laminated surface has a more pronounced effect on changes in hydraulic crack width at the interface.

Fig. 15
figure 15

Effect of interface strength on hydraulic fracture width (MHFW mudstone hydraulic fracture width, CHFW coal hydraulic fracture width)

Full size image
Fig. 16
figure 16

Expansion of crack shape at different strength interfaces at the same time

Full size image

Effect of flow rate on the width of hydraulic fractures at the interface

Among the various construction parameters, the flow rate of the fracturing fluid has a significant impact on the width of hydraulic fractures. Therefore, this study investigates the effect of different fracturing fluid flow rates: 0.033 m3/s, 0.042 m3/s, 0.050 m3/s, and 0.058 m3/s on the hydraulic fracture width. Figure 17 demonstrates that as the fracturing fluid flow rate decreases, the time taken for the proppant to enter the coal seam through the interface increases. Once the hydraulic fracture breaks through the laminated surface and enters the coal seam, the accumulated hydraulic energy in the roof plate rapidly acts on the coal seam. As a result, the hydraulic fracture is more likely to expand in the coal seam than in the roof plate. Consequently, the pressure of the fracturing fluid decreases after the hydraulic fracture breaks through the laminated surface and enters the coal seam. This leads to a rapid reduction in the width of the hydraulic fracture in the roof plate, ultimately preventing the proppant from passing through the hydraulic fracture in the roof plate. Increasing the fracturing fluid flow rate under the same stratigraphic parameters accelerates this process. Figure 18 presents a cloud diagram illustrating the expansion of fractures at different fracturing fluid flow rates at the same time. It is evident that higher fracturing fluid flow rates result in faster hydraulic fracture expansion. Therefore, as the fracturing fluid flow rate increases, the time for the proppant to enter the coal seam decreases. The maximum width of the hydraulic fracture at the interface depicted in Fig. 17 decreases with increasing fracturing fluid flow rates, although the trend of width reduction is not significant. Based on comprehensive analysis, it is crucial to ensure that the fracturing fluid flow rate enables maximum penetration through the laminated surface and entry into the coal seam.

Fig. 17
figure 17

Effect of fracturing fluid flow rate on hydraulic fracture width (MHFW mudstone hydraulic fracture width, CHFW coal hydraulic fracture width)

Full size image
Fig. 18
figure 18

Fracture expansion pattern under different fracturing fluid flow rates at the same time

Full size image

Conclusions

In this paper, the effects of horizontal stress difference and flow rate on hydraulic fracture morphology were analyzed using true triaxial hydraulic fracturing experiments, and the strain in the direction of hydraulic fracture width was analyzed. Using cohesion model to analyze the effect of stress difference, coal bed elastic modulus, coal bed permeability, interface strength, and fracturing fluid flow rate on the width of the hydraulic fracture through the seam in hydraulic fracturing of the coal seam top plate was done and the following conclusions were obtained.

  1. 1.

    Physical experiments and numerical simulations have revealed that during the hydraulic fracturing process in the roof plate, the width of the hydraulic fracture exhibits a non-monotonic increase, followed by a decrease. This indicates that the relationship between the width of the hydraulic fracture and other parameters is not simply linear.

  2. 2.

    Hydraulic fracture more likely to expanded along the coal-rock interface instead of break through it with the small horizontal stress difference and low flow rate injection condition. Compared with the fracturing fluid flow rate, the horizontal stress difference has a greater influence on whether the hydraulic fracture can penetrate the seam.

  3. 3.

    Interface strength and modulus of elasticity had the greatest influence on the hydraulic crack width at the interface, where the hydraulic crack width was able to pass through the proppant for the longest time of 15.07 s when the Interface Strength was 2 MPa. The elastic modulus of the coal seam, permeability of the coal seam, and interface Strength are directly proportional to the time allowed for the proppant to pass through the interface by the hydraulic fracture width, and inversely proportional to the horizontal stress difference and the fracturing fluid flow rate.