The analysis of the numerical simulation results for frequency conversion seepage flow in fracture focuses primarily on the qualitative analysis of flow rate distribution characteristics. In this study, the entry flow line is set to 30. Based on the observed streamline distribution characteristics, the seepage characteristics of the fracture can be understood to a certain extent. The expression of the streamline is presented as follows:

$$u=\sqrt {u_{x}^{2}+u_{y}^{2}}$$

17

In Eq. (17), *u**x* and *u**y* represents the flow rate in the x direction and the y direction, respectively.

## 4.1 Constant water injection scheme

Constant water injection refers to the relative seepage pressure (0 Pa) and fluid rate (100 ml/s) remain all constant. As shown in Fig. 5, upon observing the distribution of streamlines, we observed that the narrow watershed appears darker, indicating a higher flow rate, the fluid does not seem to follow the shortest path through the fracture due to the joint effect of side wall obstruction and a relative larger fluid rate, resulting in a unique flow trajectory. The upper fracture was observed to have a high flow rate resulting in a relatively smooth streamline, whereas the flow rate at the bottom was slower, measuring less than 0.06 m/s. Near the low flow rate area’s edge wall, a clear eddy was observed, the emergence of eddy current will lead to the loss of kinetic energy and reduce the permeability of fractures. Further examination of the flow line showed that the fluid flowed clockwise in this eddy area, with a notable dividing line between it and the main flow channel.

## 4.2 Variable amplitude water injection scheme

We select the last cycle, that is, the fifth cycle, for analysis to avoid interference caused by aperiodic factors. The initial flow value, flow peak, and flow valley values are the three characteristic time points of t = 4 s, t = 4.25 s, and t = 4.75 s of the fifth cycle, as shown in Fig. 6.

Figure 6 shows the variations in fluid flow rates within the fractures. The fluid appears to flow more rapidly in the narrow area of the fracture and the outlet, while slower flow rates are observed at the edge of the flow channel due to the boundary effects. While observing the flow rate across various time points, the flow rate inside the fracture increases from its initial value (t = 4 s) to its peak value (t = 4.25 s), allowing for the expansion of the dark high flow rate area. However, when the boundary flow decreases from its peak value (t = 4.25 s) to its valley (t = 4.75 s), the flow rate in the fracture decreases as a whole. Furthermore, the eddy region expands, and small eddies are separated from the larger eddies.

The comparisons between different water injection amplitudes indicates that, as the amplitude increases from 10 ml/s to 30 ml/s, there is an overall increase in flow rate within the fracture corresponding to the flow peak, and a decrease in the flow rate corresponding to the flow valley. Furthermore, an increase in amplitude also causes a change in flow pattern within the eddy region. The evolution of flow rate over time is found to be closely related to changes in boundary flow rates, while the influence of amplitude changes on flow rate distribution is mostly reflected in the eddy region.

Using Darcy’s law and Cube’s law, the law of fracture permeability over time is calculated based on Eq. (11), and the results are shown in Fig. 7. We observe that, starting from the second cycle, when the inlet boundary is constant water injection, permeability remains relatively constant. However, when the inlet boundary adopts variable frequency water injection, the permeability begins to fluctuate over time, showing stable and regular fluctuation from the second cycle. Observing the permeability curve at various water injection amplitudes, it manifests that the peak permeability tends to increase as the water injection amplitude increases while the valley permeability decreases as the water injection amplitude increases. Notably, the increase in water injection amplitude has a greater impact on peak permeability than on the valley permeability. As a result, the permeability amplitude increases with increasing water injection amplitude. The phenomenon is reasonable because larger amplitude, generating larger dilation and aperture volume, resulting in the increment of permeability.

Figure 8 displays a clear nonlinear relationship between inlet boundary flow and fracture permeability. The abscissa of the figure represents the inlet boundary flow, while the ordinate represents the fracture permeability. The circular pattern in the figure reflects the process of boundary flow rising and falling, and each anticlockwise cycle, in the direction of the arrow, forms a complete water injection cycle. Using the curve with an amplitude of 100 ± 30 ml/s in Fig. 9 as an example, the relationship between inlet boundary flow and fracture permeability is analyzed. Starting from a flow rate of 70 ml/s, the inlet boundary flow increases in the direction indicated by the red arrow, resulting in a decrease in permeability within the red area. As the flow rate increases to 126 ml/s, the permeability starts to increase again. When the flow rate reaches its peak at 130 ml/s and enters the blue area, the flow rate begins to decrease, while the permeability continues to increase. Ultimately, when the flow rate drops to 74 ml/s, the permeability reaches a peak of 12×10− 8 m2 before moving back into the red zone to complete a water injection cycle.

## 4.3 Variable frequency injection scheme

Under the conditions of six sets of periodic cyclic water injection at different frequencies, the evolution of flow rate with time is illustrated in Fig. 10.

The flow pattern of the fluid changes periodically over time, with the analysis progressing from left to right in time. As the flow rate increases from its initial value to its peak value, the flow rate inside the fracture increases concurrently with larger boundary flow. This is manifested through the expansion of the dark high flow rate area. Conversely, when the boundary flow decreases from the peak to the valley, the flow rate in the fracture decreases as a whole, resulting in the expansion of the eddy area. This phenomenon becomes more pronounced at higher frequencies and follows a similar pattern to that of variable amplitude water injection.

From the comparison between different frequencies, the water injection frequency increases from 0.05 Hz to 1 Hz, the flow rate of water injection changes more severely, the perturbation of the fracture flow is greater, the overall flow rate inside the fracture corresponding to the flow peak is larger, the flow rate corresponding to the flow valley value is smaller, and the increase of frequency causes the change of the flow pattern in the eddy area, the higher the frequency, the greater the flow rate, but the influence of frequency on the flow pattern in the fracture is smaller than the amplitude.

Four typical variable frequency water injection schemes (*f* = 0.05 Hz, *f* = 0.5 Hz, *f* = 0.75 Hz and *f* = 1 Hz) are selected to plot the permeability curve with time, the fracture permeability changes periodically. Comparing Fig. 11 with Fig. 4, the fluctuation of permeability and boundary flow does not rise or fall synchronously. Taking *f* = 1 Hz as an example, as shown in Fig. 4f, the flow curve rises from 0 s to 0.25 s, but the permeability curve in Fig. 10d gradually decreases. Nevertheless, when the flow rate decreased from a peak of 0.25 s to a valley of 0.75 s, the permeability increased from a valley of 8.9×10− 8 m2 to 11.1×10− 8 m2.

Figure 11 displays the circulation curve of permeability versus boundary flow, which expands longitudinally with the increase of frequency. For instance, the curve of 1 Hz, the inlet boundary flow increases along the direction indicated by the red arrow, starting from a flow rate of 80 ml/s. As the flow rate increases to 118 ml/s, permeability decreases within the red area. Nevertheless, when the flow rate reaches its peak at 120 ml/s and enters the blue area, the permeability continues to increase even as the flow rate decreases. Ultimately, when the flow rate drops to 84 ml/s, the permeability reaches a peak of 11.2×10− 8 m2, before moving back into the red zone to complete a water injection cycle.

Combining the analysis of Fig. 10 and Fig. 11, we observeed that the flow rate near the boundary does not change under constant water injection, resulting in a corresponding constant permeability shown as a point on the figure. However, under variable frequency water injection, the permeability of rock fractures changes periodically with the inlet flow, resulting in a permeability-flow curve with time that resembles a closed elliptical shape. The lateral width of the elliptical curve represents the variation range of flow, while the longitudinal height represents the varied range of permeability. Accordingly, an increase in the inlet flow amplitude results in the elliptical curve expanding outward from the same center. This expansion reflects that the amplitude of permeability increases with the increase in flow amplitude, with the impact on peak permeability being more pronounced than on the valley. With the increase of inlet flow frequency, the elliptic curve expands along the longitudinal height, reflecting that the permeability amplitude increases with larger flow frequency, and the influence on the peak permeability is more obvious than the valley.

## 4.4 Mainstream domain of rock seepage

We propose a method of identifying the mainstream domain and the eddy area which quantitatively calculate the proportion of the mainstream in the flow region, and explore the relationship between permeability and the domain area.

(1) Mainstream domain identification method

Affected by the geometry boundary, the fluid in the fracture does not flow uniformly, but is concentrated in a certain area. The mainstream domain can be separated from the eddy area with low velocities. Zhou et al. (2019) designed an automatic identification method for the eddy zone in fracture flow, which is used to detect the boundary between the recirculation and the mainstream region.

This paper proposes a simpler and effective method for identifying the mainstream region. Specifically, the area within the flow channel where flow rate is greater than the average flow rate is defined as the mainstream region, while the area where flow rate is less than the average flow rate is defined as the eddy region. This method can divide the mainstream in real-time within COMSOL software and extract it directly during post-processing

Figure 12 illustrates the flow area before and after the identification. The boundary between the two kinds of regions is clear, with the blue area representing the mainstream and the white area representing the eddy. Within the mainstream region, the streamlines parallel each other, extending from the inlet to the outlet, with minimal resistances resulting in faster flow rate. In contrast, the eddy region’s streamline forms closed loops, circulating within the white area and not participating in the mainstream area’s flow. Consequently, the eddy has negative effect on the fracture permeability.

(2) Calculation and analysis of the proportion of mainstream region

Figure 13 illustrates the evolution of permeability and the mainstream proportion over time at various water injection frequencies. As demonstrated in Fig. 13a, the permeability is closely linked to the area of the mainstream domain, with an increase in permeability resulting in a corresponding increase in the mainstream domain area. Similarly, a decrease in permeability leads to a decrease in the mainstream domain area. As a result, both variables fluctuate periodically within a stable cycle.

Zhou et al. (2019) proposed that an increase in the eddy region during steady-state flow results in additional kinetic energy dissipation, leading to a reduction in permeability. this study adopted the transient flow analysis, moreover, we conclude that an increase in the eddy area leads to a decrease in the mainstream domain area, resulting in additional kinetic energy dissipation and a reduction in fluid passing capacity. Consequently, a decrease in permeability occurs, perhaps explaining the periodic fluctuations of permeability over time. Essentially, this process represents transient, nonlinear flow.

Figure 14a illustrates the periodic fluctuations of permeability over time, with the fifth period emphasized in Fig. 14b. A phase difference exists between the two curves within this stable period. This phase difference between permeability and the main watershed gradually increases as the water injection frequency increase, which appears the similar phenomenon between dynamic normal stress and shear stress in previous rock fracture shearing experiments (Dang et al. 2016; Dang et al. 2022). This phenomenon arises due to the intensification of nonlinear flow within the fracture as the water injection frequency rises. Specifically, changes in the eddy zone will lag behind changes in the permeability due to this intensification, resulting in a time difference between the two variables. Such deviation increases accordingly with the injection frequency.