Exploring the propagation of hydraulic fracture with nonuniform stress based on the block distinct element method

The stress field is a critical factor for fracture propagation in hydraulic fracturing. Attention should be paid to fracture propagation with the nonuniform stress field. A stress field with gradient was established to investigate the influence of the nononuniform stress on fracture propagation. Based on the three‐dimensional block distinct element model the fracture propagation law with nononuniform stress was obtained. The mechanism of stress gradient and magnitude of uneven fracture propagation was revealed. Aiming at the uneven propagation, the influence of elastic modulus and construction parameters was analyzed. The results show that the influence of the elastic modulus follows the χ2 distribution equation and the influence of construction parameters can be expressed by the product of the injection rate and fluid viscosity. This conclusion is of theoretical significance for fracturing with nonuniform in situ stress.

parameters, and stratum structure. 6,7 The in situ stress is critical to fracture propagation. Previous studies conducted a serious conclusion about hydraulic fracturing based on the assumption of uniform stress. They mainly focused on the influence of stress magnitude and direction. The fractures propagated perpendicular to the direction of minimum principal stress. And the stress magnitude affected the aperture and length of fractures. However, the in situ stress is always not uniform because of stratum structure and fracture interference. Therefore, the propagation under nonuniform stress gradually attracted attention. Zhang et al. 8 pointed out that in situ stress affects the propagation of hydraulic fractures. So, attention should be paid to the stress variation during the planning and construction stages of fracturing. Tang et al. 9 used the three-dimensional displacement discontinuity method (3D-DDM) to simulate the propagation of multiple fractures. The fracture geometry was affected by in situ stress and stress shadow between fractures. Liu et al. 10 pointed out that stress anisotropy controlled the propagation length of multiple fractures. In conclusion, in situ, stress is an essential factor in controlling fracture propagation. Attention should be paid to the evolution law of in situ stress during fracturing. And the influence of stress on fracture propagation should be comprehensively considered in all processes. The stress evolution in different fracturing objects is different. Ren et al. 11 studied the propagation of hydraulic fractures near faults under a nonuniform stress field. The propagation and steering of hydraulic fractures were different with fault types and fault distances. Liu et al. 12 considered the stress waves caused by microseismic events and engineering disturbances during fluid injection in deep reservoirs and investigated the influence of stress waves on the fracture propagation path. Chen et al. 13 investigated stress change and fracture propagation during excavation. Results showed that the fracture propagation was perpendicular to the direction of minimum principal stress. The excavation led to nonuniform distribution of in situ stress. For gas extraction, fracturing is generally carried out in the excavation roadway. According to previous studies, nonuniform stress has a significant impact on fracture propagation. Therefore, fracture propagation with nonuniform in situ stress will be discussed in this study.
There are some factors inducing the nonuniform stress distribution. First, the fluid injection can cause the nonuniform stress. There are two reasons. One case is that the pore pressure variation induced by fluid injection changes the effective stress of formation. Another case is the change of total stress of formation. 14 Besides fluid injection, the fracture surface deformation by fluid extrusion can also cause stress disturbance to the surrounding stratum. Guo et al. 15 showed that the normal stress variation on the fracture surface could reach 10 MPa. Hu et al. 16 carried out a field monitoring on the stress of coal seam fracturing. The results showed that the maximum stress disturbance distance could reach 70 m, and the in situ stress was different during fracturing and after fracturing. Wang et al. 17 studied the stress variation of coal seam through nine stress monitoring wells during fracking. Chen et al. 18 found that extraction in coal seam led to stress release through 59 sets of test data from 32 wells. In addition to field monitoring, numerical simulation was also widely used to simulate stress variation. Pouriabeh 19 used the boundary element method to simulate the change of in situ stress during the liquid injection. Tang et al. 9 used the 3D-DDM method to calculate the induced stress during multifracture propagation and obtained the potential stress redistribution region. In addition to the injection itself, the geological structure also affects the stress distribution. Liu et al. 20 studied the stress variation around and along the fault area during hydraulic fracturing. Ren et al. 11 calculated the stress disturbance around the fault and pointed out that the fault would cause nonuniform distribution of stress around the stratum. In addition to faults, the factors such as local density, strength differences, and lithospheric flexure also cause nonuniform distribution of in situ stress. 21 And the nonuniform stress field caused by the temperature difference of fluid injection should also be concerned. 22 It is worth noting that stress changes throughout the entire process. An et al. 23 have employed a dislocation-based analytical model to calculate the stress changes in the entire process of multistage fracturing. It showed that the stress disturbance led to the nonuniform distribution of in situ stress. During coal seam gas extraction, not only the stress caused by early fracturing but also the change caused by excavation should be considered. Yang et al. 24 analyzed the stress evolution at different positions during the entire mining process, and the results showed that stratum excavation could lead to the reduction of stress. Chen et al. 13 analyzed the influence of excavation on stress distribution, and the result showed that the overall change of stress from the excavation face to the deep rock layer was monotonic. In this paper, the formation stress is nonuniform considering the influence of excavation and early fracturing. To simplify the nonuniform stress, the stress gradient is used to characterize the nonuniform stress field.
In conclusion, stress is an important factor for fracture propagation. There are many results about the influence of stress on fracture propagation. However, most of them are based on the assumption of uniform stress. Actually, the in situ stress is nonuniform. Therefore, fracture propagation with a nonuniform stress field was studied. The influence of stress condition (stress gradient and stress magnitude), elastic modulus, and construction parameters (injection rate and viscosity) on facture propagation under a nonuniform stress field were analyzed.
The block distinct element method has an advantage in simulating the hydro-mechanic coupling behavior in the fracture. In this paper, the block distinct element method was used to simulate hydraulic fracturing. The block distinct element describes the discontinuity by a set of distinct blocks. So, it has significant advantages in modeling discontinuities. Each block is subdivided into finite difference elements consisting of tetrahedral regions and nodes. The velocities, displacements, and joint forces of all nodes at different time steps follow Newton's laws of motion. The discontinuity is expressed by the boundary between blocks. 25 The 3D DEM has been used by Zheng. See literature by Zhang and colleagues 26,27 for details about the validation of this method.

| Deformation and failure of joints
The fractures are represented by the preset joints in the block distinct element method (BDEM). The failure of the joint means the propagation of fracture. And the joints are described by contacts in BDEM. The basic model for joints is the Columb slip joint model. This model considers the shear failure, tensile failure, and dilatancy of joints. During the elastic stage, the contacts are described by normal and tangential stiffness. And the normal force is expressed as 14 The tangential behavior is defined as where A c is the contact area. F Δ n is the increment of the normal force. F Δ i s is the increment of shear force. K n is the normal stiffness. K s is the tangential stiffness. U Δ n is the increment of normal displacement. U Δ i s is the increment of tangential displacement.
The maximum normal tensile force for an intact joint, with no slip or open, is where T is the tensile strength.
The maximum tangential force of the joint is where c is the cohesive force. φ is the friction angle.
Once the force on the joint exceeds the tensile or shear strength, the contacts fail. Then, the tensile strength and cohesion of the joint are set as 0. The maximum tensile and tangential force on joints are expressed as After failure, the contact force between blocks will be updated. And the compression force is positive in this article. The updated tensile failure is as follows: The tangential displacement may cause normal displacement. It can be described by dilatancy angle ψ. The relationship is Considering the dilatancy, the normal force becomes where u = u(xi) is the distance from a point x i to the impermeable boundary. H is the hydraulic head. ρ is the fluid density. μ is the fluid viscosity. The fluid velocity can be derived from Equation (10) as follows: where the permeability for a single fracture is u 12 2 . The hydraulic conductivity is k = H u ρg μ 12 3 . It is worth noting that this formula is suitable for fluid flow in fractures or in very thin planar flow channels.

| Block model
Focused on fracture propagation under nonuniform stress, a single fracture model with nonuniform stress field is established, as shown in Figure 1A. The block model is 60 m × 50 m × 10 m. There is a preset joint in the center of the model setting as the hypothetical hydraulic fracture. It is joined by contacts before failure. The joints fail when the stress on the joints meets the strength. Then, the fractures open, and the fluid can flow through the joints. The injection point is at the center of the model, as shown in Figure 1B. Set this point as the reference point and set its axis as 0. The initial reference stress and stress gradient are based on this point. The nonuniform stress field is expressed by stress gradient. Figure 1C demonstrates the effective normal stress to fracture surface, which is the initial stress of 15 MPa, the stress gradient of 0.05 MPa/m, and the pore pressure of 5 MPa. The stress increases from left to right in the figure. Based on this model, the fracture propagation with the nonuniform stress field is analyzed.

| Base case
The fracture propagation of the base case was simulated based on the parameters in Table 1. The propagation results are shown in Figure 2. During the early stage of fracking (100 s), the hydraulic fracture only evenly opened and propagated near the injection point. Then, the fracture became longer. However, the length of the left fracture was shorter than the right fracture because of the stress gradient. In other words, the fracture trended to propagate to the left, the lower stress zone. And the position of the maximum aperture moved to the left.
To describe the offset of fracture, the geometrical center of fracture was calculated by the location of the where x 0 is the position of the injection point. x L is the position of the left end of the fracture. x R is the position of the right end of the fracture.

| Results of the base case
The two fracture tips changing with time are shown in Figure 3A. To facilitate the description of uneven propagation, the hydraulic fracture to the left of the injection point was named the left hydraulic fracture. The hydraulic fracture to the right of the injection point was named the right hydraulic fracture. As shown in the figure, the right fracture propagated slowly during 100s-300s and stopped to extend after 7.5 m. While the left fracture extended during the whole fracturing. When the right fracture stopped expanding, the length of the left fracture was approximately linear with time. The center of the hydraulic fracture shifted to the left with time, as shown in Figure 3B.
In conclusion, the low-stress zone had an attractive effect on the hydraulic fracture due to the influence of nonuniform stress. During the early fracturing, the stress difference between the left and right tips of the fracture was small. Therefore, the hydraulic fracture propagates evenly. Then, the stress difference between the two fracture tips was gradually significant with propagation. At this time, the normal stress of the left fracture gradually decreases, so the required fracturing pressure of the left side was lower than that of the right side. Therefore, the hydraulic fracture was more likely to propagate along the left side. While the required fracturing pressure of the right tip increased gradually as the stress on the right tip increased. When the right fracture extends to a certain location, the pressure in the fracture could not open the fracture. Then, the propagation of the right fracture stopped.
The pressure variation in the fracture can reveal the internal reasons for fracture propagation under the nonuniform stress field. The curves of injection pressure are shown in Figure 3C. After fracture initiation, the pressure in the fracture maintained the opening of the existing fracture and the extension of the fracture tip. During the 100-600 s, there was a slight initial decrease (from 19.4 to 18.6 MPa) before the pressure in the fracture stabilized at 18.6 MPa. Despite the stress gradient, the curve remained stable. It is noteworthy that according to the fracture initiation criterion, the required pressure for fracking decreases with the stress reduction at the fracture tip. But why does the injection pressure remain constant in the simulation?
To answer the above question, the pressure distribution curves along the x-axis in the fracture were extracted, as shown in Figure 3D. In the figure, the injection point was at 0 m where the pressure in the fracture was the highest. The pressure decreased gradually from the injection point to both ends. Take 600 s as an example, the pressure curve showed a rapid downward trend near the injection point and then decreased linearly outward. The pressure dropped sharply to 0 MPa at the fracture tip. It indicated that the fluid pressure in the fracture was lost along the fracture, and the loss was approximately linear. It can be explained by the head loss formula of laminar flow (Equation 13). The pressure loss is proportional to the square of the flow velocity v and the flow path length l. The flow velocity of the fluid was high near the injection point, so the pressure loss was significant, which caused a rapid decrease in the pressure near the injection point. As the fluid flowed outward, the flow velocity in the fracture was the same, and v at this time can be regarded as a constant. Therefore, the head loss h f was proportional to l, and the pressure in the fracture decreased linearly.
where h f is the head loss of fluid along the flow path. λ is the loss coefficient along the flow, which is related to the flow characteristics and the roughness of the fracture. l is the flow length. R is the hydraulic radius of the crosssection; v is the average velocity of the cross-section. g is the gravitational acceleration. Note that the pressure curves for the cracking zone coincide from 200 to 600 s and the gradient of pressure loss for the linear reduction stage stays the same. This paper assumes that the fracture height is the same. When the right fracture propagation stopped, the hydraulic fracture only propagates to the left. It means that fluid flows only to the left, so the flow rate inside the fracture is constant. Since the other parameters are the same and v is regarded as a constant, h f can remain constant at different times. This is the reason why the hydraulic fracture pressure loss is consistent. Here, the reason why the pressure required to fail at the fracture tip decreases but the injection pressure remains basically stable is inferred.
A constant injection rate determines the propagation rate of the left fracture (this can be confirmed in Figure 3A). At this point, there is a critical value of the fluid velocity in the fracture. When this critical value is exceeded, the pressure loss in the fracture is too fast. Then, the pressure at the fracture tip is too low to propagate. When the front of the fracture impedes the fluid flow, the flow velocity is reduced to the critical value. By contraries, if the flow velocity is lower than the critical value, the fluid pressure loss is reduced. Then, as the pressure at the fracture tip increases, the fracture expands. And the new fracture space can increase the fluid flow velocity to the critical value. Therefore, this critical value of fluid velocity in the fracture determines the stability of h f , which ensures that the fluid injection pressure remains stable in the later stage of fracking. Since this paper focuses on fracture propagation under nonuniform in situ stress conditions, the stability of fluid injection pressure is not the main point of attention. Therefore, the exact value and verification of the critical velocity are not discussed. However, the critical velocity value can be used as a potential hypothesis to explain the stability of injection pressure.
According to the calculation of the base case, the fracture propagation shows differences with the nonuniform stress. The low-stress area has an attractive effect on the propagation of hydraulic fractures. At the early stage of fluid injection, the rate of fracture propagation to the low stress is higher than that to the high stress. After a specific time of injection, the hydraulic fracture stops propagating along the high-stress region, and the propagation rate along the low-stress region tends to be stable. In the base case, the propagation of hydraulic fractures will be offset with the nonuniform stress field. However, the influence of different factors on the offset of hydraulic fractures (hereinafter referred to as uneven propagation) is still unclear. By clarifying the influencing mechanism of each factor on the hydraulic fracture offset, the fracturing plan can be designed based on the formation condition. Besides, the hydraulic fracture offset can be artificially controlled by changing the relevant parameters. Therefore, the influence mechanism of hydraulic fracture offset under nonuniform stress conditions will be analyzed from three aspects, namely, in situ stress, rock mechanics parameters, and fracturing construction parameters.

| FRACTURE PROPAGATION WITH THE NONUNIFORM STRESS
There are two parameters that determine the in situ stress distribution. The first is the nonuniformity of the in situ stress. The stress gradient is one of the key factors leading to nonuniform stress. The second is the magnitude of the total stress. First, we analyzed the fracture propagation under different in situ stress gradients. Then, we analyzed the influence of the total stress magnitude on fracture propagation with the same gradient.

| Effect of stress gradient
The inhomogeneity of the in situ stress that changes gradually along a certain direction can be described by the concept of stress gradient. To analyze the influence of the in situ stress gradient on hydraulic fractures, the stress gradient was changed based on the base case. The stress gradients were set as 0 MPa/m (uniform stress), 0.025, 0.050, 0.075, and 0.100 MPa/m. The results of five cases are shown in Figure 4.
The stress gradient has a significant effect on the propagation. When the stress was uniform, the hydraulic fracture propagates evenly to both sides. As the stress gradient increases, the fracture moved to the left. The length of the left fracture increased gradually, while the length of the right fracture reduced gradually. Figure 4 also represents the aperture of the hydraulic fracture. The location of the maximum aperture moved to the left with the increase of the stress gradient. To analyze the influence of stress gradient on hydraulic fracture propagation in detail, the coordinate positions of the two fracture ends of the five cases were extracted. And the FOD according to Equation (12) is shown in Figure 5. Figure 5A shows the coordination of the hydraulic fracture tip. With the uniform in situ stress, the lengths of the left and right fractures were the same, both of which were 22.5 m. With the increase of the in situ stress gradient, the length of the right fracture reduced, and the left increased. With the increase of the in situ stress gradient, the in situ stress on the left side of the injection point decreased greatly, which means that the in situ stress encountered by the left fracture was smaller. Similarly, the in situ stress value on the right side was larger. The in situ stress determined the required failure F I G U R E 4 Fracture contours with different stress gradients. JIA ET AL. | 2799 pressure in the fracture tip. Therefore, the large in situ stress gradient increased the propagation difficulty of the right fracture and reduced the propagation difficulty of the left fracture. The fracture length can be calculated according to the coordinates of the left and right tips. The fracture length increased with the stress gradient according to the figure. Besides, the FOD increased with the stress gradient, as shown in Figure 5B. Figure 5C shows the injection pressure with different stress gradients. The stress gradient has little effect on injection pressure. Figure 5D shows the pressure distribution in the fracture. According to the figure, the stress gradient affected the pressure loss in the fracture. The greater the stress gradient, the greater the pressure loss along the fracture. In short, the stress gradient has little effect on the injection pressure but led to pressure loss along the fracture. When Equations (5) and (13) are combined, the larger the pressure gradient, the faster the fracture propagation on the left side and the faster the fluid velocity in the fracture are, so the higher the pressure head loss h f is.

| Effect of initial stress magnitude
The above results described the effect of stress gradient on the propagation of hydraulic fractures. In addition to the stress gradient, the stress magnitude is another characteristic of in situ stress. The in situ stress magnitude is related to the depth of the formation and determined by the initial stress at the injection point in numerical modeling. Therefore, we named the stress at the injection point as the initial stress in this article. Based on the base case, different initial stress schemes (10, 15, 20, 25, and 30 MPa) were designed to analyze their effects on fracture propagation. The results are shown in Figure 6.
The stress magnitude has a significant effect on propagation. When the initial stress was 10 MPa, the length of the left fracture was 56.4 m, and the length of the right fracture was 7.5 m. In this case, the FOD was the largest. With the increase of initial stress, the length of the left fracture gradually reduced. When the initial stress was 30 MPa, the length of the left fracture was 21.6 m. In addition, the total length of the fracture was larger with lower stress. As the initial stress increased, the length of the fracture decreased. It is consistent with the understanding that the fracture propagated easier in low-stress strata. The initial stress magnitude represents the depth of the stratum. The greater the buried depth, the higher the in situ stress. Therefore, the shallower the buried depth of the stratum was, the greater the influence of nonuniform stress on the hydraulic fracture under the same in situ stress gradient, and the longer the fracture length. In other words, the effect of stress gradient in deep strata is reduced. Figure 7A shows the pressure distribution in the fracture with different initial stresses. The pressure in the fracture increased with the increase of initial stress. Figure 7B shows the injection pressure corresponding to different initial stress values. The injection pressure was linearly related to the initial stress, and the fitting effect of the linear equation was well. Although the pressure in the fracture caused by the initial stress was different, the pressure distribution curve on the left side of the fracture was approximately parallel, indicating that the head loss along the fracture was the same.

| Comprehensive effect of nonuniform stress on fracture propagation
As mentioned above, the fractures propagate differently in two ends (named uneven propagation) with nonuniform F I G U R E 6 Fracture contour with different initial stress magnitudes.
F I G U R E 7 Pressure distribution with different initial stresses. (A) Pressure distribution in fracture and (B) injection pressure with initial stress. stress that is described by the stress gradient and initial stress magnitude. The simulation results of the stress gradient showed that the increase of the stress gradient resulted in an increase of FOD. The simulation results of initial stress showed that the increase of initial stress resulted in the decrease of FOD with the same stress gradient. Therefore, the difference in the propagation of hydraulic fracture was the combined effect of the stress gradient and initial stress magnitude. It is not comprehensive to describe the uneven propagation of hydraulic fracture by a single factor. Therefore, it is necessary to establish an evaluation parameter considering the comprehensive effect of the two factors. In this paper, the stress difference coefficient shown in Equation (14) was defined to study the relationship between FOD and the stress field.
where σ 1 is the stress at the rightmost side of the model (the stress at 60 m). σ 2 is the stress at the leftmost side (the stress at −60 m), and σ 0 is the initial stress (the in situ stress at 0 m, the injection point).
Based on the results in Sections 3.1 and 3.2, a set of simulations of different initial stress magnitudes under the 0.025 MPa/m stress gradient was added. The results of Sections 3.1, 3.2, and 3.3 are summarized in Figure 8. The linear fitting was used to fit FOD and the stress difference coefficient. The fitting accuracy R 2 = 0.9892, which was high. Therefore, linear relations can be used to describe the relationship between FOD and the stress difference coefficient. Their relationship is In Equation (14), the σ 1 and σ 2 are the stress at the rightmost side and leftmost side. Their values are influenced by their location. Their distances for different cases are the same. Therefore, the difference can be expressed by the stress gradient in Equation (14). To eliminate the impact of location, Equation (15) In conclusion, the influence of the nonuniform stress on the propagation of hydraulic fractures can be determined by the stress difference coefficient, comprehensively considering stress gradient and initial stress magnitude. According to Figure 8, the FOD and stress difference coefficient follows the linear relationship and can be described by the direct proportional function.

| Uneven propagation with the elastic modulus
The elastic modulus reflects the deformation ability of a rock. The smaller the elastic modulus, the larger the deformation with the same stress field. The normal deformation of the fracture surface has an important influence on the fracture aperture distribution. To explore the influence of elastic modulus on the propagation under the nonuniform stress, the influence of different elastic moduli (5,10,15,20, and 25 GPa) on hydraulic fracture propagation based on the parameters of the base case was analyzed. The calculation results are shown in Figure 9.
When the elastic modulus is 5 GPa, the hydraulic fracture was short and the FOD was small. The aperture of this condition was the largest compared with others. When the elastic modulus was 10 GPa, the hydraulic fracture length increased and the fracture aperture reduced greatly. Then, the uneven propagation regularity of the hydraulic fracture was strong with the elastic modulus ranging from 10 to 25 GPa. The length of the left fracture reduced, the length of the right fracture increased slowly, and the FOD decreased. In conclusion, we can speculate that when the elastic modulus was low (<10 GPa), the FOD induced by the nonuniform stress increases with the increase of the elastic modulus. And F I G U R E 8 Fracture offset distance (FOD) with the stress difference coefficient. when the elastic modulus was high (>10 GPa), the FOD induced by the nonuniform stress decreases with the increase of the elastic modulus.

| Uneven propagation with construction parameters
The influence of elastic modulus on the uneven propagation due to the stress field was described above. In general, the stress field and elastic modulus are the existing characteristics of the stratum and are difficult to be controlled artificially. The parameters that can be directly changed are the construction parameters, including the injection rate and the fluid viscosity. According to previous studies, the injection rate and the viscosity have important effects on fracture propagation. However, its effect on nonuniform expansion is not clear. Therefore, we analyzed the influence of the injection rate and viscosity.

| Effect of the injection rate
Based on the base case, we added some new cases with different injection rates (0.005, 0.010, 0.015, and 0.020 m 3 /s). To keep the same total injection volume, different injection times (1200, 600, 400, and 300 s) were set. The simulation results are shown in Figure 10. According to the figure, with the increase in the injection rate, the fracture length was reduced. The left fractures became shorter and the right fractures increased slightly. The FOD decreased and the aperture increased.
The uneven propagation was closely related to the pressure in the fracture. Figure 11 shows the pressure distribution in the fracture with different rates. According to the figure, when the injection rate was 0.005 m 3 /s, the overall pressure in the fracture was low, and the injection pressure was lower than other rates. With the increase of the injection rate, the overall pressure in the fracture gradually increases, and the injection pressure increases.
According to the pressure increase amplitude, the pressure increased greatly when the injection rate was low. Besides, the pressure gradient in the fracture was small with a low rate. It is because the pressure in the fractures can be effectively transferred to the fracture tip with a low flow rate. The injection pressure is F I G U R E 9 Fracture propagation with different elastic moduli.
F I G U R E 10 Fracture propagation with different injection rates.
F I G U R E 11 Pressure character of fracture with different injection rates. determined by the failure pressure at the fracture tip and the stress gradient. Therefore, a low-stress gradient means a low injection pressure. Then, the low injection pressure determines the overall low pressure in the fracture. At this time, the difficulty of propagation on the right increased, and the fracture propagation on the left side was easier. This was ultimately characterized by an increase in FOD. On the contrary, if the injection rate was large, the FOD was small. Therefore, the low injection rate enhances the degree of uneven propagation of hydraulic fracture, while the high injection rate attenuates the uneven propagation.

| Effect of fluid viscosity
The fluid viscosity is another effective construction parameter. Based on the base case, the fluid viscosity was set as 100, 200, 300, and 400 cp. The uneven propagation of hydraulic fractures with different viscosities was analyzed. The simulation results are shown in Figure 12. According to the figure, the total fracture length reduced with the increase in viscosity. The left fractures became shorter and the right fracture slightly increased. The FOD decreased and the aperture increased.
Similarly, the pressure distribution in the fracture under different viscosities was analyzed. The results of pressure distribution with different viscosities are plotted in Figure 13. The injection pressure and the pressure gradient increased with the increase in viscosity. Similar to the injection rate, the viscosity determined the transfer of fluid pressure. If the fluid viscosity was low, the fluid pressure was easy to transfer to the fracture tip, that is, the pressure loss along the fracture was small. For a certain point (same coordinate point), the failure stress was the same. But the required injection pressure was low due to the small pressure loss. It resulted in a reduction in the overall pressure, which in turn makes it more difficult to propagate for the right fracture. The final result was that the fracture extends to the left, and the FOD was large. On the contrary, if the fluid viscosity was high, the FOD was small. Therefore, the lowviscosity fluid enhances the degree of uneven propagation, while the high-viscosity fluid attenuates the uneven propagation.

| Mechanism of the elastic modulus on uneven propagation
The results about the elastic modulus showed that the FOD increased first and then decreased with the increase of elastic modulus. The elastic modulus indicates the deformation ability of rock, so it should be analyzed from the deformation of fractures (i.e., the aperture). Figure 14A shows the aperture characteristics with different elastic moduli. According to the figure, when the elastic modulus was 5 GPa, the aperture was much larger than in the other four cases because the elastic modulus was small. Figure 14B shows the relationship between the maximum aperture and the elastic modulus. The maximum aperture was negatively correlated with the elastic modulus. When the elastic modulus was small, the aperture decreases rapidly with the increase of elastic modulus. When the elastic modulus was large, the reduction rate of aperture became smaller. The power function can be used to fit the relationship between the maximum aperture and the elastic modulus. And the F I G U R E 12 Fracture propagation with different viscosities. fitting accuracy is 0.9910, which is high. In Dontsov's 28 approximate solution of penny-shaped fractures based on different fracture regimes, the relationship between the maximum aperture and the elastic modulus was considered. And its formula without considering formation filtration can be referred to Zheng et al. 26 The relation between aperture and elastic modulus in the formula was expressed by power function relation. The consistency of the two results (simulation results and equation) indicates the rationality of the simulation and the correctness of using power function to describe the relationship between them.
According to Figure 14A, the deformation of the fracture was large when the elastic modulus was small. Then, the deformation of the fracture surface reduced the fluid pressure in the fracture, which resulted in the short length of the hydraulic fracture. There was little stress difference between the two fractures ends when the fracture length was short. Therefore, the FOD was small at low elastic modulus. According to Figure 14B, with the increase of elastic modulus, the aperture decreased rapidly with a small elastic modulus and decreased slowly with a large elastic modulus. Therefore, when the elastic modulus was large, the aperture had little influence on the fracture length because the difference of aperture was small. Due to the small aperture, the fluid pressure in the fracture can be effectively increased. The increase of pressure in the fracture contributed to the propagation of the right fracture, so the FOD decreased.
In conclusion, the essential mechanism of elastic modulus on fracture propagation is rock deformation.
The rock deformation determines the pressure in the fracture. For a small elastic modulus, the pressure in the fracture was too low to affect the fracture propagation distance. Because the fracture propagation distance was the main factor controlling the fracture offset, the FOD was not large. With the increase of elastic modulus, the difference of aperture with elastic modulus decreased, and the pressure was sufficient to support fracture propagation. Currently, the pressure in the fracture becomes the main factor controlling the fracture offset. The aperture decreased with the increase of elastic modulus. Then, the pressure in the fracture increased and the FOD decreased.
There are few data about small elastic modulus in the above scheme. To obtain the FOD under a small elastic modulus, the calculation schemes with elastic moduli of 3, 4, 6, and 7 GPa were supplemented. The simulated FODs are summarized in Figure 15. The FOD curve showed a trend of increasing first and then decreasing. This trend was similar to the χ 2 distribution in the probability distribution. Therefore, this basic form was used for data fitting, and the fitting result is shown in Figure 11. The fitting accuracy was 0.96, which was high. Therefore, this equation can be used to describe the relation between FOD and E. Their relationship is expressed by E e FOD = 2.285 .
To determine the rationality of the curve, the curve was extended outward to 60 GPa. The results showed that with the increase of elastic modulus, the FOD decreased. The effect of elastic modulus on FOD was inapparent at high modulus. In conclusion, the influence of elastic modulus on uneven propagation was mainly in the middle-and low-magnitude ranges. The FOD first increases and then decreases with the increase of elastic modulus.

| Interaction effect of the injection rate and fluid viscosity
The effects of the fluid injection rate and viscosity on the uneven propagation were discussed above. Both showed that the low value enhanced the uneven propagation, while the high value attenuated the uneven propagation. Both had the same mechanism of uneven propagation. Therefore, we attempted to establish the law of uneven propagation under the interaction of the injection rate and viscosity.
Based on the above study, more 12 cases with different viscosities (100, 200, and 300 cp) and injection rates (0.005, 0.010, 0.015, and 0.020 m 3 /s) were designed. The simulation results are shown in Figure 16. Figure 16A shows the relationship between fracture tip position and the product. The left fracture became shorter, and the right fracture slightly increased with the increase of the product. Figure 16B shows the relationship between FOD and the product. According to Figure 16B, the FOD was basically the same for different schemes with the same product. Therefore, the product of rate and viscosity can be considered as the combined independent variable to describe the fracture propagation. The FOD decreased with the increase of the product. To quantitatively describe the relationship, the R 2 fitted by the power function was 0.9981, and the fitting effect was good. Their relationship can be expressed by μq FOD = 4.52 + 27.55( ) , 0.65 (18) where μ is the fluid viscosity and q is the injection rate.
Therefore, through the product of the injection rate and viscosity, the influence of the two factors on the uneven fracture propagation can be comprehensively considered. The curve results showed that when the product was low (corresponding to low viscosity and low rate), the uneven propagation of hydraulic fracture was obvious. The large product, which corresponds to the high viscosity and high rate, can effectively attenuate the uneven propagation. Therefore, in fracturing construction, the high-viscosity and high-rate injection can be used to eliminate the difference of fracture propagation caused by nonuniform stress. This conclusion has important guiding significance for fracturing design.

| CONCLUSION
In this paper, the propagation of hydraulic fractures with the nonuniform stress was simulated by the block distinct element method. The effects of elastic modulus and construction parameters on hydraulic fracture offset due to nonuniform stress were analyzed. The main conclusions are as follows: (1) The low-stress zone has an attractive effect on hydraulic fracture due to the influence of nonuniform stress. The stress gradient affects the uneven propagation through the stress differences between two fracture tips. And the stress magnitude affects the uneven propagation through the difficulty of propagation. The fracture offset distance is related with the stress difference coefficient considering stress gradient and magnitude. Their relationship is linear and can be described by direct proportional function. (2) The relationship between the FOD and elastic modulus can be described by χ 2 distribution. The influence of elastic modulus on uneven propagation is mainly in the middle-and low-magnitude range. The FOD first increases and then decreases with the increase of elastic modulus. (3) The effect of construction parameters on uneven propagation can be described by the product of the injection rate and viscosity. A higher product can attenuate uneven propagation during fracturing.