Introduction

Hydraulic fracturing is a majority method to improve the oil production in reservoir development with low permeability and low porosity formation. The research on hydraulic fracturing includes experimental research (Liao et al. 2019) and numerical simulation research. Numerical simulation has developed rapidly because of its advantages of low cost and no site restrictions. At present, the commonly used numerical simulation methods include CZM (Elices et al. 2002), DDA (Hu et al. 2021), XFEM (Zhao et al. 2020), DEM (Zhang et al. 2017, 2019; Hou et al. 2022a, b; Hou et al. 2022a, b), FEM (Liao et al. 2020; Cordero et al. 2019) and phase-field method (Ni et al. 2020), etc., and a series of progress has been made in the application of these methods. CZM (cohesive zone model) is one of the better alternative methods for fracture propagation. In this model, it is assumed that there is a process zone at the fracture tip, and the fracture in the process zone is described by traction–separation (T–S) constitutive model, which avoids the stress singularity at the fracture tip in linear elastic fracture mechanics. Dehghan (2020) pointed out that the expansion of hydraulic fractures (HF) in the presence of natural fractures (NF) is fundamentally different from that in reservoirs without natural fractures. For this reason, a large number of scholars have carried out the research on the intersection behavior of hydraulic fracture and natural fracture (Yao et al. 2018; Shakib et al. 2012; Zou et al. 2021; Luo et al. 2019; Chen et al. 2016; Rahman 2009; Dahi and Olson 2014). These studies divide the intersection behave of hydraulic fractures and natural fractures into three categories: Hydraulic cracks pass through natural cracks, and hydraulic cracks are arrested by natural cracks and turn along natural crack. Further research shows that the intersection behavior of hydraulic fractures and natural fractures is mainly controlled by the difference of in situ stress, the cementation strength and angle of natural fractures (Wang et al. 2021; Liu et al. 2022; Zhao et al. 2021; Zhao et al. 2022a, b). Larger stress difference, bigger approach angle and higher cementation strength of natural fractures will facilitate the passing through natural fracture of hydraulic fractures and thus resulting in a simpler fracture network (Liu et al. 2019). In fractured reservoirs, natural fractures are staggered, and the fracture intersection behavior in the process of hydraulic fracturing is more complex (Dou et al. 2021; Liu et al. 2021a, b; Hou et al. 2014; Wu and Olson 2016). Liu et al. found that the distribution of natural fractures is the main factor affecting the fracture network through true 3D fracturing test and CT scanning (Liu et al. 2021a, b). Zhao et al. pointed out that when the development degree of natural fractures is small, the fracture propagation mode is mainly affected by the minimum horizontal principal stress. On the contrary, when the development of natural fractures is large, the fracture propagation mode is affected by the distribution of natural fractures (Zhao et al. 2022a, b). Zou et al. found that the more complex the natural fracture, the higher the fracture pressure of the formation, and the complex bifurcation caused by the natural fracture will also shorten the fracture length (Zou et al. 2021). Further, Li et al. and Dong et al. studied the influence of different natural fracture orientations on the scale of fracture network and considered that the random fracture trend angle is more conducive to the formation of fracture network than the regular distribution (Li et al. 2020; Dong et al. 2021). However, there is a lack of targeted research on the factors restricting the formation of fracture network in the strata with the regular distribution of natural fractures.

In this paper, four groups of “checkerboard-liked” staggered natural fractures are generated. The propagation of hydraulic fractures in fractured formations is simulated by CZM method. The scale and connectivity of fracture network after fracturing are studied, and the influencing factors are clarified, which has good application value for guiding the fracturing of such reservoirs.

Influence of fracture types and failure modes on fracture network conductivity

Fractures types after fracturing and its performance

Volume fracturing refers to the continuous expansion of natural fractures and shear slip of brittle rock in the process of hydraulic fracturing, forming a fracture network of natural and artificial fractures interleaved with each other, so as to increase the reconstructed volume and improve the initial production and final recovery. Generally, three kinds of fractures formed in reservoirs after volume fracturing: main fractures, branch fractures and self-supported fractures (Fig. 1). Main fractures are the primary fracture formed by artificial fracturing and the backbone of the fracture network formed by volume fracturing. Main fractures’ characteristic is that the fracture width is large, but the shape is simple, and proppant is needed to keep the fracture open. Branch fractures are mainly formed by the connection of hydraulic fracture and natural fracture. Because of random distribution of the natural fractures in the reservoir, the morphology of branch fractures is complex, and distribution range is large. Self-supported fractures mainly formed by slip of natural fractures, but these fractures generally not connected with main fracture.

Fig. 1
figure 1

Three types of fractures in reservoir after fracturing

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These three types of fractures have obviously improved the conductivity of the formation, but the degree of improvement is different. The main fracture is the most important oil flow channel due to its wide width and large remaining flow space after proppant injection and backflow. However, its morphology is usually relatively simple, and the actual area affected in the stratum is not large. The width of branch fracture is slightly smaller than the main fracture, but it continuously forks from the main fracture, greatly improving the size of the fracture network, and because it is connected to the main fracture, it also has good conductivity in the formation. Self-supported fractures are special because they are not connected to the main fractures and exist in isolation in the formation, so their role as oil flow channels is not obvious. However, due to the unique structure (as shown in Fig. 2) of the self-supported fracture that allows it to remain open without proppant, these small and discrete flow spaces effectively increase the permeability of the formation as a whole and thus improve fracture performance.

Fig. 2
figure 2

Natural fractures slip and self-supporting after opened. Subfigure (a) is a closed natural fracture; Subfigure (b) is an opened natural fracture, which slides under the displacement caused by the opening of other fractures, resulting in self-supporting. The blue part is the additional seepage space caused by self-supporting phenomenon

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Relations between fractures type and failure modes

What kind of fractures formed after fracturing is related to the failure modes. The failure modes can be divided into three basic types: tensile failure, shear failure and tear failure, in fracturing process, it is mainly governed by the stress state and, the mechanical property and interactive conditions of hydraulic and natural fractures. Research shows that under the anisotropic in situ stress field, natural fractures tend to slip, when the pore pressure increasing caused by fracturing fluids injection, so some micro-crack slip and form a self-supported fracture. Hydraulic fracture mainly formed by tensile stress, when hydraulic fracture intersects with a natural fracture, branch fracture formed, and it usually caused by the combination of tensile stress and shear stress.

The comparison of the three types of fractures is shown in Table 1.

Table 1 Comparison of the three types fractures
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Therefore, it is not comprehensive to evaluate the fracturing effect only from the perspective of fracture width and fracture length. In this paper, a method is adopted to comprehensively evaluate the actual effect of fracture network through the change of reservoir pressure and seepage velocity. In fact, proppant transport and bridge location also affect the fracturing effect, but generally only affect the fracture width. In this paper, the distribution and transport of proppant during fracturing are not considered.

Numerical simulation and governing equation

Numerical simulation model and parameter setting

The distribution of natural fractures is varied. Some are dominated by these in a certain direction, some are dominated by intersecting fractures, and some are stochastic fractures. In this paper, the number and angle of natural fractures are the key research factors. Therefore, the form of intersecting fractures is selected. Moreover, such fractures are real. For example, this kind of natural fracture network usually appears in Sichuan province of China (Zhao et al. 1980).

The distribution characteristics of natural fractures in the information can be described by characteristic parameters such as trend, dip angle and length. On the two-dimensional plane perpendicular to the depth direction, it can be simplified to describe natural fractures by two parameters: trend and length. In this paper, a series of oblique natural cracks similar to checkerboard shape are randomly generated on a 10 m * 10 m square thin plate through script, as shown in Fig. 3.

Fig. 3
figure 3

Four groups natural crack with various trend and density. Group (a) contains 54 cracks with a angle of − 20°and 80°from horizontal. Group (b) contains 54 cracks with an angle of ± 40°from horizontal. Group (c) contains 100 cracks with an angle of − 20°and 80°from horizontal. Group (d) contains 100 cracks with an angle of ± 40°from horizontal

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The centers of the above four groups of random cracks are uniformly distributed on the plane, and the crack length is subject to normal distribution, so as to avoid the influence of local natural crack concentrated on the simulation results. The simulation is carried out in a uniform in situ stress field, and the parameters used in the model are shown in Table 2.

Table 2 Parameters used in simulation
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Fracture initiation and propagation

Traction and separation are two majority variables to describe deformation and failure of cohesive element. Traction is the result of simplifying six stress components into three principal stresses, and separation is the distance between element top surface and bottom surface, which are assumed to zero at beginning.

Quadratic nominal stress criterion is used to describe fracture initiation (Li 2017), as shown in Eq. (1):

$$\left\{ {\frac{{\left\langle {t_{n} } \right\rangle }}{{t_{n}^{0} }}} \right\}^{2} + \left\{ {\frac{{t_{s} }}{{t_{s}^{0} }}} \right\}^{2} + \left\{ {\frac{{t_{t} }}{{t_{t}^{0} }}} \right\}^{2} = 1$$
(1)

where \(t_{n}\), \(t_{s}\), \(t_{t}\) are the traction in three principle directions of cohesive element; \(t_{n}^{0}\), \(t_{s}^{0}\), \(t_{t}^{0}\) are the traction when cohesive element damaged; specifically, \(\left\langle {t_{n} } \right\rangle\) means cohesive element only damaged in tensile, when element suffer from pressure, \(\left\langle {t_{n} } \right\rangle\) treated as zero.

Before cohesive elements damaged, coupled linear elastic model was used to modeling the relation between traction and separation, as shown in Eq. (2):

$$\left\{ {\begin{array}{*{20}c} {t_{n} } \\ {t_{s} } \\ {t_{t} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {k_{nn} } & {k_{ns} } & {k_{nt} } \\ {k_{sn} } & {k_{ss} } & {k_{st} } \\ {k_{tn} } & {k_{ts} } & {k_{tt} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\delta_{n} } \\ {\delta_{s} } \\ {\delta_{t} } \\ \end{array} } \right\}$$
(2)

where \(\delta_{n}\), \(\delta_{s}\), \(\delta_{t}\) are the separation in three principle directions of cohesive element; \(k_{ij}\) is stiffness tensor.

After cohesive elements damaged, damage variable \(da\) was used to describe progressive damage and strength reduction of cohesive elements, which can be calculated by Eq. (3):

$$da = \frac{{\delta_{m}^{f} \left( {\delta^{\max } - \delta_{m}^{0} } \right)}}{{\delta^{\max } \left( {\delta_{m}^{f} - \delta_{m}^{0} } \right)}}$$
(3)

where \(\delta_{m}^{f}\) is the current separation; \(\delta_{m}^{0}\) is the separation when cohesive element begins to damaged; and \(\delta^{\max }\) is the separation when cohesive element fully damaged. Influenced by reduction of strength, traction after damaged also decrease, it can be calculated by Eq. (4):

$$\left\{ \begin{gathered} t_{n} = \left\{ \begin{gathered} (1 - da)\overline{{t_{n} }} {\text{ (tension)}} \hfill \\ \overline{{t_{n} }} {\text{ (compression) }} \hfill \\ \end{gathered} \right. \hfill \\ t_{s} = (1 - da)\overline{{t_{s} }} \hfill \\ t_{t} = (1 - da)\overline{{t_{t} }} \hfill \\ \end{gathered} \right.$$
(4)

where \(\overline{{t_{n} }}\), \(\overline{{t_{s} }}\), \(\overline{{t_{t} }}\) are tractions assumed no damage happened and calculated by Eq. (2). The whole process from deformation to failure of cohesive elements can be represented by bilinear traction–separation diagram, as shown in Fig. 4.

Fig. 4
figure 4

A typically bilinear traction–separation diagram

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Fracturing fluids flow in porous media and crack

The flow of fracturing fluid in the formation is mainly the gap flow in the fracture, Darcy flow in the porous medium and the leakage of the fracture facing to the porous medium. Assuming that the formation is isotropic and homogeneous, the Darcy flow of fluid in the matrix can be expressed by Eq. (5):

$${\mathbf{q}} = - \frac{k}{{\mu_{w} }}\nabla p$$
(5)

where \({\mathbf{q}}\) is flow rate vector, \(k\) is permeability of porous media, \(\mu_{w}\) is viscosity of fracturing fluids, \(\nabla p\) is gradient of pore pressure in porous media.

Since the crack width is very small relative to the size of the whole model, the normal pressure gradient in the crack is ignored and only the tangential flow is considered. At this time, the velocity can be calculated by Eq. (6):

$$q_{h} = - \frac{{d^{3} }}{{12\mu_{w} }}\nabla p_{h}$$
(6)

where \(q_{h}\) is tangential flow velocity in fractures, \(d\) is fractures opening, and \(\nabla p_{h}\) is tangential pressure gradient in fractures.

Leakage from fractures to porous media can be calculated by Eq. (7):

$$\left\{ \begin{gathered} q_{t} = c\left( {p - p_{t} } \right) \hfill \\ q_{b} = c\left( {p - p_{b} } \right) \hfill \\ \end{gathered} \right.$$
(7)

where \(q_{t}\), \(q_{b}\) are leakage velocity, \(c\) is leakage coefficient, and \(p\), \(p_{t}\), \(p_{b}\) are pressure of inside of fractures, outside of crack upper surface and outside of crack lower surface, respectively.

Results and discussion

Pore pressure distribution characteristic and fractures scale

The distribution of formation pore pressure at 30 min after fracturing of four groups of natural fractures is shown in Fig. 5. The results show that the angle and position of natural fractures in the formation have a significant effect in distribution of pore pressure in the formation. Comparing the pore pressure and its spread range of groups (c), (d) and (a), (b) in the figure, it is found that when the number of natural fractures is small (Figure (a) and (b)), the spread range of fracturing fluid is narrower, and the overall level of pore pressure in the whole region is relatively small, but the phenomenon of a large increase of pore pressure is easy to occur in local fractures, regardless of the size of the fracture angle, which is mainly due to the small number of fractures are difficult to pore pressure transfer, resulting in local pressure suppression. When the number of natural fractures is large, the spread range of fracturing fluid is wider, and the overall level of pore pressure in the whole region is greatly increased (Figure (c) and (d)). However, when the natural fractures angle is different, the pore pressure within fractures is slightly different. When the angle is larger, the pore pressure within fractures is lower (Figure (c)). However, a larger increase in pore pressure in local fractures occurs when the angle is smaller (Figure (d)).

Fig. 5
figure 5

Pore pressure distribution in formation at 30 min during fracturing. Each subfigure represents a type of natural fracture group which randomly generated previously. The positions of injection points are all near the midpoint of the figure

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The fracturing operation lasts for 1 h in total. The distribution of formation pressure after completion is shown in Fig. 6. The distribution is roughly the same as that in Fig. 5. The greater the number of natural fractures, the greater the range of pore pressure elevation at the end of fracturing, and the greater the overall level of pore pressure in that range.

Fig. 6
figure 6

Pore pressure distribution in formation at end of fracturing

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To clarify the effect of natural fracture location and angle on pore pressure distribution, the forms of natural fractures opened after the end of fracturing were distinguished, as shown in Fig. 7. The black ones in the figure are tensile fractures, and the red ones are shear fractures. The number of natural cracks opened in groups (a) and (b) is small, while the number of natural cracks opened in groups (c) and (d) is large and dominated by shear cracks. A significant difference between shear-induced fractures and tensile-induced fractures is the tensile fractures will gradually close as the net pressure within the fractures decreases before proppant injection, but the shear fractures will remain open due to the shear self-support phenomenon, resulting in a significant improvement in the overall permeability of the formation, as shown in Fig. 2. Therefore, the distribution of natural fractures will affect the development of hydraulic fractures, then affect the failure form of fractures, and finally lead to the difference of pore pressure distribution in the formation.

Fig. 7
figure 7

Fracturing induced natural fractures mode

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Impact of natural fractures trend and density on hydraulic fracture connectivity

The four groups of natural fracture networks shown in Fig. 3 include two types of natural fracture angle and natural fracture density, which will be used in this section to reveal the impact of natural fracture trend and density on the scale and connectivity of the natural fracture network formed by hydraulic fracturing.

On this basis, in order to quantitatively evaluate the impact of trend and density of natural fractures on the scale and connectivity of fracture network after fracturing, the natural fractures after fracturing are divided into three categories: connected with artificial fractures, not connected with artificial fractures and not opened, as shown in Fig. 8. The quantities of each type are counted, respectively, to calculate their proportion.

Fig. 8
figure 8

Three types of natural fractures after hydraulic fracturing. Red parts represent main fractures, while blue is natural fractures. Subfigure (a) means natural fractures connected with main fracture and branch fractures formed (blue); Subfigure (b) means natural fractures opened but not connected with artificial fractures and self-supported fractures formed (blue); Subfigure (c) means natural fractures not opened ignore whether connected with artificial fractures

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The simulation results corresponding to the four groups of natural fracture networks are shown in Fig. 9. In the uniform in situ stress field, the four groups of simulations have formed a certain scale of fracture network. In the figure, the fracture network of group (a) is the simplest, while the group (c) is the most complex; the overall shape of the fracture network is basically consistent with the angle of the natural fractures. The main fractures are mainly connected with the natural fractures in the propagation process. When the natural fractures in the formation are almost orthogonal to each other, the fractures will produce branches and gradually turn to the natural fractures in advance (as shown in subfigure (b)) or expand along one of the natural fractures and directly pass through the natural fractures in the other direction (as shown in subfigure (d)). Comparing group (a) with group (c) and comparing group (b) with group (d), it is found that the more natural fractures in the formation, the more natural fractures will open after fracturing, and the proportion of opened fractures is also increasing (Fig. 10). But a considerable part of the open natural fractures is not connected with hydraulic fractures and remains isolated from each other in the formation.

Fig. 9
figure 9

Fracturing results under different natural fracture distributions. The colors represent the damage degree of the rock. From blue to red, the damage degree of the rock gradually increases, blue means the rock is not damaged, and red means the fracture has opened

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Fig. 10
figure 10

The proportion of three kinds of natural fractures after hydraulic fracturing

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The proportion of three kinds of natural fractures is further counted, and the statistical results are shown in Fig. 10.

The statistical results in the figure are compared and analyzed. Comparing the results of group (a) and group (b), the proportion of branch fractures after fracturing is the same, which is 11%; the proportion of self-supported fractures increased significantly from 6 to 22%, which shows that when the number of natural fractures is small, its angle has a great influence on the opening of natural fractures, but has a smaller influence on the connectivity of fracture network; the results of comparing groups (c) and (d) also reflect the phenomenon that the greater the angle of natural fractures, the more the number of opened fractures after fracturing. Comparing the results of group (b) and (d), the proportion of branch fractures after fracturing is slightly increased, but not obvious, only 2%; the proportion of self-supported fractures has increased by 8%, which indicates that the more natural fractures, both of the branch fractures and self-supported fractures in the fracture network will increase; the comparison of group (a) and (c) also reflects this phenomenon, and the phenomenon is even more dramatic. Group (c) has the best fracturing effect, and its unopened natural fractures are the least, accounting for only 47%, which is due to the significant increase in the proportion of branch fractures compared with the other three groups; the proportion of branch fractures in group (d) increases fewer under the same natural fracture density, indicating that the natural fractures with larger angles in uniform stress field are easier to be connected by hydraulic fractures.

Fracturing performance evaluation

In order to evaluate the conductivity of various fractures and the overall effect of fracture network more comprehensively and accurately, the formation seepage field has been studied.

In fact, the process of fracturing fluid flow is basically the same with oil production, and the hydrodynamic and percolation mechanics equations are the same to describe them. Thus, the flow of fracturing fluid in the formation at the end of fracturing can be used to reflect the flow of oil during production. The flow trajectory of the fluid under the four groups of fracture nets is shown in Fig. 11.

Fig. 11
figure 11

The flow trajectory of the fluid under the four groups of crack nets. White vectors represent the streamline

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The white arrow in the figure is the fluid flow vector. The larger the arrow, the faster the fluid flow rate. It can be seen from the subfigures (a) and (b) that the flow velocity near the main fracture is the maximum and slows down rapidly along the direction away from the main fracture. The flow velocity in the region without fracture distribution is minimal. Although the area circled in black in subfigure (c) is far away from the main fracture, the flow rate drops slightly, but it still maintains a higher level compared with figure (a) and (b), because there are a large number of self-supporting fractures in the black area, which improves the overall conductivity of the nearby area. Branch fractures exist in the area circled in yellow in subfigures (c) and (d). It can be seen that branch fractures extend the influence range of the main fractures to a large extent.

In general, the fracturing effect of group (c) is the most ideal among the four fracture networks. Moreover, the overall flow rate of the fluid in the formation is significantly faster, and the effective flow area is larger. Compared with the other three groups, group (c) obviously has more self-supporting fractures and branch fractures, so the overall flow conductivity of the final fracture network is also stronger.

Conclusions

  1. (1)

    The shear or tensile-generated branch fracture and self-supported fracture can improve the SRV and the overall conductivity of the reservoir, thus improving the fracturing effect.

  2. (2)

    The angle and number of natural fractures have a significant effect on fracture network scale and fracturing effect. With the increasing of number of natural fractures, branching fractures and self-supported fractures activated. When the angle of natural fractures is larger, the connectivity of fracture network improved.

  3. (3)

    To obtain a great fracturing effect, both tensile-induced main fracture and shear/tensile-induced branch fracture or self-supported fracture should be considered. The former provides flow channel to oil, and the latter refines the conductivity and connectivity of overall fracture networks and finally contributes to fracturing performance.