Research on Horizontal Well Multi-Fracture Propagation Law under the Synergistic Effect of Complex Natural Fracture and Reservoir Heterogeneity

Abstract

Conventional fracturing design generally only considers the propagation of a single fracture in homogeneous storage, but the heterogeneity characteristics of the reservoir and the development of complex natural fractures will affect the fracturing effect. In order to study the horizontal well fracture propagation law under the synergistic effect of complex natural fracture reservoir heterogeneity, the globally embedded cohesive element finite model is established, and the heterogeneous reservoirs are built by Python programming. The influencing factors of multi-fracture propagation with heterogeneous reservoirs are analyzed, including the in situ stress difference, the injection rate of fracturing fluid, the perforation spacing, and the perforating clusters number. The results show that the heterogeneity and the natural fracture of the reservoir will affect the in situ stress distribution and propagation direction. When the in situ stress difference is low, the reservoir heterogeneity and natural fractures have a great influence on the fracture morphology. With the increase in injection rate, the fracture length and fracture width increase. With the in situ stress difference increase, the fracture deflection angle decreases. With the increase in fracturing fluid injection rate, the fracture length increases. The number of hydraulic fractures communicating with natural fractures increases with the natural fracture number and the number of perforation clusters under the synergistic effect of complex natural fracture and reservoir heterogeneity.

1. Introduction

Oil and gas are very important resources, but the main oil and gas reserves are concentrated in mountainous areas, most of which have experienced complex geological structures and diagenetic processes; therefore, the reservoirs exhibit significant heterogeneity [1,2]. Hydraulic fracturing technology is a key method to improve the recovery of oil and gas resources in complex environmental reservoirs. The reservoir heterogeneity and the distribution of complex NF (natural fractures) have a great influence on the HF (hydraulic fractures) propagation, which has attracted the attention of many scholars.

Shen et al. [3] established a fluid-structure coupling model considering porous media flow and analyzed the heterogeneous effects on fracture propagation. Chen et al. [4] obtained the finite element characterization of heterogeneous coal by combining geometric information with heterogeneous physical property parameters. Yang et al. [5] combined digital core technology with the construction of a heterogeneous sandstone model to analyze the influence of mineral particles on the spatial distribution morphology and propagation law of HF in heterogeneous sandstone. Li et al. [6] set a built-in unit of 0 thickness in the finite element lattice to establish a new method for the instantaneous expansion of HF. The correctness and effectiveness of the method were verified by model and experiment. Wang et al. [7] used the extended finite element method combined with the cohesive element method to simulate and study the simultaneous expansion of multiple fractures in hydraulic fracturing. It was found that the joint length and width of multiple cracks are mainly affected by fracture spacing, in situ stress difference, and the stress shadow effect. Shimizu et al. [8] analyzed the influence of rock particle size on hydraulic fracturing under heterogeneous conditions through numerical simulation. Duan [9] analyzed the influence of reservoir heterogeneity on HF propagation by studying the law of multiple HF propagation. By studying the propagation path of HF, Renard et al. [10] found that mineral cleavage and intergranular space in heterogeneous reservoirs affect the propagation behavior of HF.

The above scholars studied the heterogeneity of reservoirs but did not explore the random propagation process of HF in fractured reservoirs. Rueda et al. [11] proposed a new mesh crushing technique to analyze the cross-random propagation process of HF-NF under different influencing factors. Qiu and Yu [12,13] conducted a secondary development of the random propagation process of HF. To address the presence of NF in orebodies, Wang et al. [14] embedded cohesive elements in the established finite element model and found that the horizontal distance and length of NF had a significant influence on fracture propagation. Li et al. [15] considered the Mohr–Coulomb failure criterion for NF, improved the relationship between double linear adhesion structures, and proposed a two-dimensional pore pressure adhesion structure model. Wang [16] proposed a fully coupled HF propagation model to study the interference and coalescence of fluid-driven HF induced by horizontal wells, taking into account rock brittleness and toughness. Yang et al. [17] used an algorithm embedding cohesive elements in a solid finite element mesh to improve the prediction of real and complex fracture processes using heterogeneous models. Hou and Liu et al. [18,19] established a microstructure model through secondary development to explore the relationship between mineral interfaces and the initiation and HF propagation. Wang [20] studied the stress shadow of the HF process by secondary development.

In general, the current model force on the single influence factor of fracture propagation; there are deficiencies in understanding fracture propagation under the influence of the heterogeneity and complex fracture. In this paper, the HF propagation model was established that comprehensively considers the synergistic effect of reservoir heterogeneity and complex NF development. The influencing factors of multi-fracture propagation in heterogeneous reservoirs are analyzed, including the in situ stress difference, the injection rate of fracturing fluid, the perforation spacing, and the number of perforating clusters. This provides theoretical guidance for HF design in fractured and heterogeneous reservoirs.

2. Hydraulic Fracturing Fluid-Solid Coupling Governing Equation

2.1. Momentum Equilibrium Equation for Solid Deformation in Porous Media

During HF, mechanical deformation is caused by pressure changes in fractures. The equivalent continuum theory is adopted to describe rock mechanical behavior, including the equilibrium differential equation, geometric equation, and constitutive equation [1].

$$\nabla \sigma +\rho \left(B-{\displaystyle \frac{dv}{dt}}\right)=0$$

(1)

$$d\epsilon ={\displaystyle \frac{1}{2}}[\nabla du+{(\nabla du)}^{\mathrm{T}}]$$

(2)

$$d\epsilon =\mathrm{Dd}\sigma$$

(3)

where $\sigma$ is the stress tensor, MPa; $\rho$ is rock density, kg/m³; B is a physical force vector per unit volume, m/S²; v is a velocity vector, m/s; u is a displacement vector, m; $\epsilon$ is a strain tensor; D is the constitutive matrix.

In saturated porous media, according to Biot’s law:

$$\sigma =\overline{\sigma }+\alpha {\mathrm{p}}_{\mathrm{w}}$$

(4)

where $\sigma$ is the total stress in a porous medium, MPa; $\overline{\sigma }$ is the effective stress in porous media, MPa; $\alpha$ is the porous elastic coefficient related to rock properties, the Biot coefficient is dimensionless.

Assuming that HF propagation is a quasi-static process, the deformation equation in porous rock media can be derived from Equation (1) as follows

$${{\displaystyle \int }}_v\left(\overline{\sigma }+p_{w}I\right){\delta }_{\epsilon }dV={{\displaystyle \int }}_{s}t\cdot {\delta }_{V}dS+{{\displaystyle \int }}_{V}b\cdot {\delta }_{V}dV$$

(5)

where I is the identity matrix; ${\delta }_{\epsilon }$ is a virtual strain rate matrix, S⁻¹; dV is the cell volume, m³; t is the force vector per unit area, N/m²; ${\delta }_v$ is a virtual velocity matrix, m/s; dS is the unit area of the surface force applied, m².

2.2. The Continuous Equation of Flow of Fluid during Fracture

It is assumed that the depletion fluid is an incompressible Newtonian fluid. During the fracturing process, the width of HF is generally in the range of millimeters, much smaller than the length of the fracture. Therefore, the flow of fluid inside the wavefront can be simplified to a one-dimensional fluid. The continuity equation is [21]

$${\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial }{\partial t}}(J{\rho }_w{n}_w)+{\displaystyle \frac{\partial }{\partial x}}({\rho }_w{n}_w{v}_w)=0$$

(6)

where J is the volume change of fluid in porous media, dimensionless; ${\rho }_w$ is fluid density, kg/m³; $n_w$ is the porosity ratio related to the porosity of porous media without dimension; $v_w$ is the flow velocity of the fluid, m/s; x is a spatial vector related to the spatial step size, m.

2.3. The Continuity Equation of Fluid Flow in Fractures

Let us assume it to be an incompressible Newtonian fluid. In the fracturing process, the width of HF is usually in the range of millimeters and much smaller than the length of the fracture. Thus, the flow of fluid in the fracture can be simplified as one-dimensional flow. Then its continuity equation is [1]

$${\displaystyle \frac{d{q}_f}{dx}}-{\displaystyle \frac{dw}{dt}}+q_l=0$$

(7)

where $q_f$ is the local velocity per unit height, m²/s; w is the fracture width, m.

As shown in Figure 1, the flow modes in HF are tangential flow and normal flow. Tangential flow is the driving force of fracture propagation, and normal flow represents the leakage of fracturing fluid into the formation [1]:

$$q_{f}w=-{\displaystyle \frac{w^3}{12\mu }}\nabla p_f=-k_t\nabla p_f$$

(8)

where $q_1$ can be obtained from the normal flow equation of the upper and lower fracture surfaces [22]:

$$q_l=q_t+q_b$$

(9)

where the normal flow equation at the top of the fracture is

$$q_t=c_t(p_f-p_w)$$

(10)

The normal flow equation of the fracture surface at the bottom of the fracture is

$$q_b=c_b(p_f-p_w)$$

(11)

where $k_t$ is the tangential flow permeability coefficient of the fluid in the fracture, m/s; $c_t$, ${\mathrm{c}}_{\mathrm{b}}$ are the filtration coefficients of the top and bottom fracture surfaces, respectively, m/(Pa·s). $p_w$ is the pore pressure, MPa.

3. Numerical Model

Assuming the mechanical model is 2D, the geometric size of the model is 50 m × 50 m, and the global size of the grid is 0.1 m. There are two groups of NF, as shown in Figure 2a. Reservoir heterogeneity is characterized by elastic modulus heterogeneity, and the elastic modulus distribution follows a Weibull distribution through secondary development [23], as shown in Figure 2b. The rock grid type is CPE4P (a 4-node plane strain quadrilateral, bilinear displacement, bilinear pore pressure). Through Python secondary opening, the cohesive elements are embedded globally, and the grid type is COH2D4P (a 6-node two-dimensional pore pressure cohesive element), as shown in Figure 2c [24]. The fracture cluster spacing is 8 m. The horizontal well multi-fracture propagation model under the synergistic effect of complex natural fractures and reservoir heterogeneity was established. The parameters of the model are shown in

Table 1. The numerical simulation parameters of fractured reservoirs in HF.

The Parameter (The Unit)	The Numerical Value	The Parameter (The Unit)	The Numerical Value
Group 1 Natural Fracture Angle (°)	45	Group 2 Natural Fracture Angle (°)	150
Elasticity Modulus (GPa)	15.9	Injection rate (m3/min)	0.6
Tensile strength of rock (MPa)	6	Initial perforation length (m)	2
Rock density (kg/m3)	2300	Hydraulic fracture filtration coefficient (m/Pas)	1 × 10−14
Poisson’s ratio	0.25	The in situ stress difference (MPa)	1
Viscosity (Pa.s)	0.001	Fracturing fluid specific gravity (kN/m3)	9800

.

The fracture propagation model of homogeneous reservoir and heterogeneous reservoir are established. The fracture morphology are plotted in Figure 3.

It can be seen from Figure 3 that the fracture on the left side mainly propagates downward in the homogeneous reservoir, with little upward propagation, while the fracture on the right side mainly propagates upward, with little downward propagation, showing a certain symmetry. However, the heterogeneity model exhibits relatively few symmetries and produces a large angle deflection due to the reservoir’s heterogeneity. The stress distribution in the reservoir is more complicated [24]. Under the influence of heterogeneity, the fracture on the left side flexes when encountering NF due to the difference in elastic modulus. In contrast, the homogeneous reservoir is not affected by the elastic modulus, with fractures propagating along the NF. The right fracture in the heterogeneous reservoir does not intersect with the NF, while the right fracture in the homogeneous reservoir connects two fractures. Therefore, the heterogeneity of the reservoir influences stress distribution and fracture propagation.

4. Result Analysis

In order to further study the mechanism of HF propagation in heterogeneous reservoirs, the influence of factors such as the in situ stress difference, fracturing fluid injection rate, number of fracture clusters, and cluster spacing were analyzed [25,26].

4.1. The Influence of the In Situ Stress Difference

During the actual fracturing operation of a heterogeneous reservoir, the in situ stress difference affects the fracture propagation morphology. To investigate the effect of the in situ stress difference on the morphology of fracture growth in heterogeneous reservoirs, a fracture growth model with in situ stress differences of 0.5 MPa, 1 MPa, 2 MPa, and 3 MPa was established to analyze the distribution of the stress field and fracture width during HF growth.

As can be seen from Figure 4, the HF on the left side of the figure propagates downward, while the HF on the right side propagates upward. This is due to the stress disturbance between fractures, forming mutually exclusive fractures. This is consistent with the fracture propagation law in homogeneous reservoirs. The fracture propagation pattern is more complex under the influence of heterogeneity, including turns, deflections, and captures. When the in situ stress difference is low, the fracture is captured by the NF at 45°. When the in situ stress difference is high, the fracture is captured by the NF at 150°. There are many stress concentration points in the reservoir. As the stress difference increases, the stress in the heterogeneous reservoir increases. Compared with in situ stress difference of 0.5 MPa, 1 MPa, 2 MPa, and 3 MPa, it can be seen that the deflection angle of the HF decreases gradually, and the propagation trend of the fracture becomes closer to the direction of the maximum principal stress. The fracture deflection angle decreases gradually, and the fracture propagation tends to be closer to the direction of the maximum principal stress. The fracture propagation tends to open the NF under the combined influence of heterogeneity and NF. The heterogeneity of stress distribution increases.

As can be seen from Figure 5, the influence of heterogeneity and NF on HF propagation leads to an increase in both the length and width of fracture propagation over time However, the growth of fracture fluctuation is caused by the existence of reservoir non-uniformity and natural destruction [27,28]. In addition, when the in situ stress difference increases, the total length of the fracture shows a certain downward trend, while the maximum width of the fracture shows an increasing trend. This indicates that the increase in the in situ stress difference leads to a certain pressure suppression and, at the same time, causes an increase in normal stress during the fracture propagation. As a result, more energy is directed toward the propagation of the two walls of the fracture, leading to an increase in fracture width.

4.2. The Influence of the Fracturing Fluid Injection Rate

In order to investigate the effect of reservoir modification and the influence of different injection rates under the same injection volume, fracture propagation models were established with a total injection volume of 0.102 m³ and injection rates of 0.002 m³/s, 0.004 m³/s, 0.006 m³/s, and 0.008 m³/s, respectively. The stress field and fracture width distribution of the reservoir during HF propagation for these four injection velocities were plotted separately, as shown in Figure 6.

As shown in Figure 6, the fracture propagation morphology is similar at different injection rates. It can be observed that when the injection velocity is 0.002 m³/s, there are four HF communicating with NF, including three on the left side. As the injection rate increases, the fracture length increases while the injection volume remains the same. Due to the heterogeneity of the right HF, the resulting angular deviation is the greatest. The left fracture propagation is mainly influenced by the heterogeneity of the reservoir, resulting in different stress responses of the particles under certain stress. When the injection rate is 0.002 m³/s, the maximum and minimum stresses appear near the fracture particles due to the influence of heterogeneity. For example, when the rate increases to 0.004 m³/s, the maximum stress increases from top to bottom, and similarly, the minimum stress increases from bottom to top. At injection rates of 0.006 m³/s and 0.008 m³/s, the maximum and minimum stresses will appear near the fracture particles due to the influence of heterogeneity. These results indicate that velocity and heterogeneity play a significant role.

To analyze the threshold conditions for fracture growth, the fracture growth rate with the injection rate of fracturing fluid is plotted as shown in Figure 7.

As shown in Figure 7, the fracture growth rate per unit injection volume increases from the lowest to the highest when the injection rate increases from 0.002 m³/s to 0.006 m³/s. The fracture growth rate per unit injection volume decreases when the injection rate increases from 0.006 m³/s to 0.008 m³/s. This indicates that the fracture growth rate does not increase linearly with the injection rate increasing. At a low injection rate (0.002 m³/s), most of the fracturing fluid only fills the existing fracture and does not significantly propagate the fracture. When the threshold is approached (0.006 m³/s), it indicates that more fracturing fluid is being used for fracture propagation rather than just filling the existing fracture space. When the threshold is exceeded (0.006 m³/s), the fracturing fluid is used almost exclusively for fracture propagation.

As shown in Figure 8, the propagation direction of the hydraulic fracture will change due to differences in the elastic modulus. As the injection rate increases, the tangential stress increases, and the fracture will propagate forward. However, the stress interference generated will cause the normal stress to be locally compressed, leading to a decrease in fracture width. This is because the reduction in, the injection time decreases the filtration loss of fracturing fluid in the formation, resulting in more energy being used for fracture tip propagation, leading to an increase in the fracture length and fracture propagation rate.

4.3. The Influence of the Perforation Spacing

In actual fracturing operations, the perforation spacing is closely related to the effectiveness of the fracturing process. When the perforation spacing is too small, it can cause stress interference during fracturing, which can suppress the propagation of fractures. When the perforation spacing is too large, it will prevent the formation of a fracture network between adjacent clusters, negatively impacting oil and gas extraction. Therefore, the study of perforation spacing is very important. Fracture propagation models were established with perforation spacings of 8 m, 10 m, 12 m, and 14 m, respectively, and a distribution map of the reservoir stress field and fracture width during HF propagation was created, as shown in Figure 9.

As shown in Figure 9, it can be observed that when the cluster spacing is 8 m, there are two connections between the left HF and the NF. The upper part propagates along the 150° direction of the NF, and the propagation trend connects to the right HF. The lower HF intersects with an NF that also runs through 150 degrees. There are three connections between the right HF and the NF. The upper part runs through the NF, while the lower part propagates along the direction of the NF. When the perforation spacing increases, the left fracture tends to propagate upward, and the direction turns. This is due to the influence of the NF network distribution on the propagation direction. The fractures near each perforation may become more independent, forming multiple single fractures that are not connected to each other.

It can be seen that when the perforation spacing is 8 m, the maximum stress is close to the minimum stress, which is because the maximum stress changes under the influence of heterogeneity and perforation distance. When the perforation spacing is 10 m, the maximum stress and minimum stress appear on the fracture, and the maximum stress is opposite to the minimum stress. When the perforation spacing is 12 m, the maximum stress appears in a different position and does not appear at the tip, which is due to the inhomogeneity, natural fractures, weak interface, and other inhomogeneity. When the perforation spacing is 14 m, the maximum stress propagates from the bottom up, and the minimum stress is the same.

As shown in Figure 10, the stress shadow between the HF and the NF leads to changes in fracture propagation direction and HF propagation length. When the perforation spacing is 10 m, the fracture propagation rate is slow in the early stage, but when the HF propagation reaches the intersection with the NF and the stress shadow exists, the fracture propagation length increases rapidly [29]. When the perforation spacing is 8 m, the fracture propagation length is the shortest and the width is the largest. This is mainly due to the HF propagation turning when it reaches the intersection of two connected NF groups, causing a part of the fracturing fluid to continue propagating forward under pressure, resulting in an increase in the connectivity at the intersection. As the cluster spacing increases, the number of connected NF increases.

4.4. The Influence of the Number of Perforating Clusters

The number of perforation clusters is a very important factor in hydraulic fracturing operations, directly affecting the initial initiation point and the effectiveness of fracturing. To explore the effectiveness of HF with different numbers of perforation clusters, HF propagation models were established with two, three, four, and five clusters. The stress field and fracture width distribution of the reservoir during HF propagation are plotted as shown in Figure 11.

As shown in Figure 11, when there are two perforation clusters, it can be seen that there are a total of seven branches of HF, connecting five NF, and the propagation direction follows the direction of the NF. The results show that NF has a significant effect on HF direction under certain stress due to the heterogeneity of the reservoir. The intermediate perforation propagates less in the direction of the maximum principal stress, and the particles show different stress responses. When there are four perforation clusters, the perforation positions are set to −12 m, −4 m, 4 m, and 12 m, respectively, connecting a total of five NF, and the propagation direction mainly follows the NF direction. The lower ends of the fractures at 12 m and −4 m are connected, and there is almost no propagation at the lower end of the HF at 4 m. Moreover, the HF at −4 m and 4 m in the middle are shorter compared to the propagation of the HF on both sides. This shows that HF will experience a certain degree of stress interference during the propagation process due to the heterogeneity of the reservoir. The particles show different stress responses, and there is a certain competitive relationship in the propagation process. As the number of perforations increases, the fracture network may become more complex, which is generally beneficial for improving reservoir permeability but also causes the fractures to become shorter, resulting in multiple branching fractures. The propagation length of the middle fracture is shorter and tends to propagate along the direction of the NF.

The maximum stress in two, four, and six perforations occurs outside the fracture because the fracture intersects or is adjacent to the NF, and this interaction may cause stress concentration to occur near the NF rather than within the fracture. Additionally, because some areas of the formation may be more prone to stress concentration than others, especially if the material has significant weak surfaces, inclusions, or sources of stress concentration, heterogeneity can cause stress to propagate outside the fracture.

As shown in Figure 12, the propagation length and width of fractures increase as the number of perforation clusters increases. This indicates that the number of perforation clusters plays a direct role in the transformation effect of reservoirs [30,31]. When the injection rate is constant, the number of perforation clusters is directly proportional to the net pressure inside the fracture. As the number of perforation clusters increases, the net pressure inside the fracture also increases, leading to an increase in the number of fracture initiation points. This facilitates the formation of longer fractures and complex fracture networks.

5. Conclusions

(1) An HF propagation model considering heterogeneity and complex NF propagation is established, and the law of multi-fracture propagation in horizontal wells with the synergistic effect of heterogeneity and NF in reservoirs is analyzed through secondary development. The heterogeneity of the reservoir leads to different stress responses at different locations and changes in fracture propagation length under certain stress, and the difference in elastic modulus also leads to changes in the direction of hydraulic fracture propagation.

(2) The fracture deflection angle decreases gradually as the in situ stress difference increases. As the fracturing fluid injection rate increases, the fracture length first increases and then decreases. The gain effect of natural fractures near the perforation on the length of fracture propagation varies.

(3) The increase in the NF near the perforation increases the HF in contact with the NF. At the same time, it will increase the filtration loss of fracturing fluid in the formation, leading to a certain impact on the length of fractures. As the number of perforation clusters increases, the connected NF increase and the stress interference generated by each cluster will suppress the fracturing effect of intermediate perforations.

(4) It is recommended that under on-site fracturing conditions, the injection rate, the perforation location and the number of perforation clusters should not be blindly selected based on the number of NF. Instead, the reservoir properties of the fractured formation should be fully understood before optimizing the fracturing parameters.