# Research on Seepage Characteristics of Y-Shaped Fractures under Different Fracture Roughness

Authors: Ruifeng Tang; Jing Xie; Shouning Xue; Zhilong Zhang; Junjun Liu; Bengao Yang
Journal: Geofluids
 (2022)

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## Abstract

Fractures are widely distributed in coal, and studying the seepage characteristics of fluids in fractures is of great significance for unconventional natural gas extraction and prevention of gas disasters. In this work, based on the W-M fractal function, a Y-shaped fracture model with different roughness is established. The seepage characteristics of Y-shaped fracture with different roughness are interpreted from two perspectives. Firstly, the seepage law of the Y-shaped fracture with the same roughness is studied. Secondly, the fluid competitive diversion capacity caused by the difference of branch fracture roughness is discussed. The results show that the fracture roughness is an important factor affecting the seepage characteristics of the fracture. When the roughness of the fracture is identical, the outlet flow rate, velocity, and Re are all positively correlated with pressure, and the flow regime is unchanged. The increase in the fracture roughness will lead to a significant flow rate and momentum loss, resulting in maximum loss up to 45.44%, besides, enhance the flow resistance of the fracture,Re increasing by 771, 713, 489, and 355, respectively, at four patterns. And there is a threshold DM between 1.1 and 1.2 so that the major influencing factor on hydraulic conductivity changes. In addition, the roughness difference between the branch fractures of the Y-shaped fracture is the key factor to control the fluid competitive diversion capacity. The larger the roughness difference, the greater difference in flow velocity, and the more significant change in flow rate proportion, which proportion differs by 4%, 44%, and 54%, respectively, and the stronger smooth branch fracture competitive diversion capacity, and the lower the rough branch fracture hydraulic conductivity. As the inlet pressure increases, the two branch fractures Re gradually increases but Eu decreases, which Eu in smooth branch fracture is smaller is 15, 19, 20, and 21 smaller than that of rough branch, respectively, and the rough branch fracture competitive diversion capacity is weakened. Both roughness difference and inlet pressure will affect the competitive diversion capacity of the Y-shaped fracture. These results are expected to provide new insights for the exploitation of underground fluid resources.


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## Body

## 1. Introduction

Due to the exhaustion of shallow mineral resources, resource development continues to move towards the deep part of the earth, deep mineral resource mining has become normal [1–3], and the expansion and development of fractures is closely related to the stress evolution caused by resource mining [4, 5]. Fractures can provide channels for the migration of underground fluids [6–9], and can also provide space for fluid storage [10]. As a clean energy with abundant reserves and high combustion calorific value, coalbed methane is widely stored in the fractures of coal [11–13]. How to make efficient use of coalbed methane resources is an important link in tackling key energy problems. However, deep resources usually have typical characteristics such as complex occurrence environment and low initial permeability. The migration law of coalbed methane is ambiguous, and there are many interference factors. A slight carelessness in the development process will induce major coal mine safety accidents [14, 15]. Therefore, how to scientifically evaluate fluid flow behavior is very important.Single fractures and cross-fractures are the basic components of complex fracture networks [16], widely distributing in outcrop rocks [17] and coal as shown in Figure 1. At present, academia has achieved certain research results in single-fracture seepage, cross-fracture seepage, and fracture network seepage. Experts and scholars at home and abroad have achieved rich results in the seepage characteristics of single fractures. Stress state and fracture roughness are two important factors that affect the seepage state of single fractures. The fracture hydraulic conductivity and the normal stress acting on the upper and lower sides of the fracture usually meet an exponential or power function relationship [18, 19], and the relationship between the normal stress and the normal deformation of the fracture is generally considered to be an exponential curve and a hyperbolic type [20]. Chang et al. [21] and Yang et al. [22] established a physical model under three-dimensional stress of a single fracture and deduced the calculation formula of the hydraulic conductivity. They believed that the influence of the lateral stress of a fractured rock sample on its hydraulic conductivity is a negative exponential relationship. Ju et al. [23] explained the flow resistance composition of water seepage and proposed a rough single-fracture hydraulic conductivity calculation formula based on fractal theory. Wang et al. [24] used JRC curves and 3D technology to prepare fractured samples with different roughness. The results showed that when there is no normal pressure, the hydraulic conductivity decreases in a negative exponential function with the increase of roughness. When the normal pressure is constant, the hydraulic conductivity of the fractured sample decreases linearly with the increase of roughness. Hydraulic resistance has a huge effect on the fluid behavior at the intersection of cross-fractures [25], which is related to factors such as fracture aperture and roughness [26, 27]. Therefore, some scholars have carried out research on the influence of the fracture intersection on seepage characteristics. Jannes et al. [28] combined numerical simulations and laboratory experiments to study the effect of droplet and rivulet flow modes on the fluid distribution at the fracture intersection, revealing the behavior of fluid diversion combined with multiple factors such as fracture aperture, droplet size, and fracture inclination. Zhu et al. [29] deduced the accurate flow calculation formula of the cross-fracture and found that the flow shrinkage or expansion effect at the intersection of the fracture is the real mechanism of excessive pressure drop loss. Xue et al. [30] proposed a set of droplet splitting theory and calculation model based on the transient static balance method, which explained the dynamic splitting behavior from a mesolevel and revealed the influence of factors such as the droplet length, fracture inclination, and the liquid accumulation in the channel on the interfacial velocity and the diffluence volume proportion. In the research of fracture network seepage, due to the complexity and randomness of the fracture networks, the research on the seepage of the fracture network mainly stays in theoretical models, numerical calculations, and a few on-site monitoring. Zhu et al. [31] used the concept of topology to evaluate the connectivity of fractures. The pore size distribution has the strongest influence on the overall efficiency of the fracture network. The connectivity of the fracture network decreases with the increase of the number of small fractures. Alireza and Jing [32] used numerical analysis methods to analyze the equivalent permeability and anisotropy characteristics of the fracture network under two-dimensional conditions for the discrete fracture network. Gao et al. [12] determined the fractal dimension and connectivity rate of the coal mining fracture network through field monitoring and theoretical algorithms and divided the evolution of the fracture network into three stages, which provided a theoretical basis for the study of fracture network seepage characteristics.Figure 1
Schematic diagram of extraction and simplification of the Y-shaped fracture. The red and blue parts represent two different kinds of fluids, respectively, such as gas, water, or oil, and the yellow arrow represents the fluids flow direction.Because the seepage law of the fracture network is difficult to capture, it is extremely important to explore the seepage characteristics of basic fracture units such as single fractures and cross-fractures. The fractures in natural coal have different shapes and messy distribution, and the geometric characteristics of the fractures are difficult to determine. There is some difference in the roughness of natural cross-fractures. Under multiple working conditions, the mechanism of fluid seepage path selection in cross-fractures is unclear, which needs to be deeply studied.This work takes the Y-shaped fracture under the cross-fracture as the main research object and conducts a systematic study on the different seepage characteristics caused by rough morphology characteristics and roughness difference of the Y-shaped fracture. Combined with numerical simulation software, seven Y-shaped patterns were established, the seepage characteristics of fractures under different working conditions were explored, and the competitive diversion behavior of the fracture was quantitatively discussed. The research results can provide new cognition for fracture seepage and are the prerequisite for scientific description of coalbed methane migration. However, this research only discusses one of the many influencing factors, and more research will be carried out in the future.


## 2. Model Building

At present, a lot of useful understanding have been obtained in the academic circles about the influence mechanism of Y-shaped fracture seepage. However, there are few related studies that consider the influence of the difference in roughness of two branch fractures on the seepage characteristics of Y-shaped fracture. Therefore, this work considers seven patterns, in which four patterns are set for the same roughness of two branch fractures of the Y-shaped fracture, and three patterns are set to highlight the influence of the roughness difference between the two branch fractures, as shown in Figure2. The fracture roughness is controlled by fractal dimension D.Figure 2
Establishing a Y-shaped fracture model based on the fractal curve.Assuming that the fluid in the fracture is an isothermal and stable incompressible Newtonian viscous fluid, the seepage flow is controlled by the Navier-Stokes [33–35] equation:
(1)∇∙u=0,(2)ρu∙∇u−μ∇2u=−∇P,where∇2 is the Laplacian; u is the flow velocity, m/s; ρ is the density of the fluid, kg/m3; P is pressure, Pa; and μ is the fluid viscosity coefficient, Pa·s.In this work, the Weierstrass-Mandelbrot fractal function is used to simulate the two-dimensional rough surface curve, in which the fractal dimensionD is an important parameter representing the degree of roughness, reflecting the complexity and irregularity of the contour of the fractal characteristic curve. The Weierstrass-Mandelbrot function [16, 36] expression is as follows:
(3)Wt=∑n=−∞∞1−eibnteiφnb2−Dn,where b is the frequency density factor, a real number greater than 1, which reflects the degree of deviation between the curve and the straight line. φn is a random phase between 0 and approximately 2π on the rough surface, fractal dimension DD∈1∼2.In order to simulate the contour curve of a two-dimensional rough surface, the real part of the functionWt is regarded as a continuous and nondifferentiable fractal control function Ct:
(4)Ct=ReWt=∑n=−∞∞1−cosbntb2−Dn.The fractal dimension satisfies the following relationship:(5)DHB−Bb≤D≤DHB,where B is a constant and DHB stands for the Hausdorff-Besicovitch dimension.Set the parameterb to 1.4 [23] to generate fractal curves with different fractal dimensions. The fractal curve and the Y-shaped fracture model are shown in Figure 2.In this work, COMSOL Multiphysics 5.4 software is used as a numerical simulation tool to establish a Y-shaped fracture model based on the fractal curve. The fluid density and viscosity coefficient was set to 1000 kg/m3 and 1.01×10−3 Pa·s, and set four inlet pressures of 50, 100, 150, and 200 Pa. All the related parameters of the fracture are listed in Table 1.Table 1
Parameters of the Y-shaped fracture.
ParametersFracture AFracture BFracture CFractal dimension1.01.0/1.1/1.2/1.31.0/1.1/1.2/1.3 1.5Average aperture (mm)2Length (mm)200Intersection angleθ (°)120


## 3. Result and Discussion

### 3.1. Analysis of the Influence of Fracture Roughness on Seepage Law

In order to explore the influence of fracture roughness on the seepage characteristics, the relationship between the outlet flow rate and velocity and inlet pressures was determined. The test results of the flow rate and velocity under theDB=DC condition are shown in Table 2, and 1.0-1.0-1.0 represents the fractal dimensions of the three branch fractures A, B, and C. Outlet A is the outlet of the branch fracture B, and outlet B is the outlet of the branch fracture C. The pressure at the inlet is as mentioned above.Table 2
Fluid seepage results under different inlet pressures (DB=DC).
Fracture roughnessInlet pressure (Pa)Flow rate of outlet AQA (cm3/s)Flow velocity of outlet AVA (cm/s)Flow rate of outlet BQB (cm3/s)Flow velocity of outlet BVB (cm/s)1.0-1.0-1.0501.165.801.155.751002.4912.462.4512.251504.0420.193.9219.592005.8829.405.6928.431.0-1.1-1.1501.165.781.155.761002.4212.122.4112.051503.8419.193.8119.052005.5227.605.4727.341.0-1.2-1.2501.115.561.115.541002.1710.862.1610.821503.1715.843.1515.762004.1020.514.0820.391.0-1.3-1.3501.035.171.035.161001.839.161.839.161502.4712.332.4712.342003.2116.043.2116.05According to the evolution of flow rate in Figure3 and the flow rate proportion in Figure 4, it is not difficult to find that with the increase of fracture roughness, the outlet flow rate under the four patterns gradually decreases. The greater the roughness, the more obvious flow rate loss, the faster the flow rate decline speed, and the smaller the proportion. Moreover, the inlet pressure also significantly affects its proportion. When the fracture roughness is certain, the flow rate proportion decreases with the increase of inlet pressure, which can be understood as that the inlet pressure affects the fluid seepage velocity in the fracture, but due to the existence of the fracture roughness, the fluid velocity flowing through the fracture surface attenuates, resulting in the reduce of proportion with a maximum loss of 45.44%. All these indicate that the greater the fracture roughness, the greater the flow resistance of the fracture. The flow velocity of smooth fracture is much higher than that of rough fracture, in which the main reason is that the increase in roughness makes the undulation of the fracture surface more complicated, the resistance of the fluid passing through the fracture surface increases, and the flow energy decreases [24]. However, with the increase of inlet pressure, the flow rate increases linearly, and at high pressure level, the greater the velocity difference under each roughness, e. g., V1.3 is 13.36 cm/s smaller than V1.0 when pressure is 200 Pa. The fracture fractal dimension has a negative correlation with the velocity growth rate, indicating that the fracture roughness is an important factor affecting the seepage capacity of the whole Y-shaped fracture. When the pressure at the inlet continues to increase, the flow rates at the two outlets rise at the same time, and the flow rate ratio is close to 1 : 1. This phenomenon shows that when branch fracture roughness is identical, the roughness does not affect the fluid flow path, and the fluid flows into two branch fractures on average, as shown in Figure 5, which reflects that the main factor that affects fluid seepage characteristics is fluid inflow pressure, and the influence of roughness on seepage is mainly reflected in two aspects of flow velocity and flow rate instead of fluid path selection.Figure 3
Evolution of flow and velocity with inlet pressure.
(a)
Outlet A flow rate and velocity(b)
Outlet B flow rate and velocityFigure 4
Evolution of outlet flow rate proportion with inlet pressure.
(a)
Outlet A(b)
Outlet BFigure 5
Fluid flow behavior in fracture (velocity streamline) whenDB=DC.In addition, the difference value in outlet flow rate is an important feature of seepage characteristics. Overall, when the pressure is less than 150 Pa, the flow rate loss shows a gradual increasing trend, which maximum loss is up to 0.70 cm3/s, indicating that within a certain pressure threshold, the greater the fracture roughness, the more fluid will be lost through the fracture. However, when the pressure exceeds this threshold, such as 200 Pa in Figure 6, it is found that the flow rate loss is abnormal. At this time, the maximum loss at ΔD1.1−1.2 is 1.42 cm3/s, which the fractal dimension continues to increase, the loss decreases. The reason for this phenomenon is that the pressure has a nonlinear effect on the seepage. There is a threshold that makes the seepage characteristics before and after the threshold deviate. The flow rate loss of the fracture at low pressure levels is mainly controlled by the roughness of the fracture. As the inlet pressure increases, the pressure affects the role is gradually becoming more prominent, based on the seepage characteristics of low flow rate under high roughness or high pressure conditions, so the flow rate loss under high pressure and high fractal dimensions is lower than the loss under high pressure and low fractal dimensions. From the perspective of actual engineering, the greater the initial roughness of natural fractures in coal, the greater the resistance of fluids passing through the fractures, and the more difficult it is for fluids to flow through the fractures, which makes it hard for the water or gas stored in the fractures to gush out [37]. The mining disturbance causes the fluid pressure to rise, and disasters such as water inrush or gas outburst occur frequently. Therefore, when exploiting underground fluid resources, we should try to avoid the fluid under high pressure or use other methods to reduce fracture roughness and ensure safe mining. The greater the pressure, the greater the flow rate loss, and the degree of flow loss will have different levels of influence on the fracture hydraulic conductivity. Hydraulic conductivity is an important index to characterize the permeability of coal body, and the expression [38] is
(6)k=QμLA∆P,(7)K=kρgμ=QLρgA∆P,where K is the fracture hydraulic conductivity, m/s; k is the fracture permeability, m2; μ is the viscosity coefficient, Pa·s; Q is the outlet rate flow, m3/s; L is the fracture length, m; ρ is the fluid density, kg/m3; A is the seepage cross-sectional area, m2; and ΔP is the seepage pressure difference, Pa.Figure 6
The relationship between flow rate loss and fractal dimension.The evolution of hydraulic conductivity is shown in Figure7. The greater the fracture roughness, the lower the fracture hydraulic conductivity, indicating that the fracture flow resistance is increased, the higher the fractal dimension value, the change of hydraulic conductivity is gradually increased, and the influence of roughness on hydraulic conductivity is gradually enhanced. When the inlet pressure is constant, the change rate of the hydraulic conductivity increases with the increase of the fractal dimension, which drops rapidly at 200 Pa and the maximum and minimum values are 2.88 and 1.57 m/s. However, there is a threshold DM for the fracture fractal dimension between 1.1 and 1.2, so that under four pressure conditions, the fracture hydraulic conductivity is a fixed value. When D<DM, the hydraulic conductivity is positively correlated with the inlet pressure, and when D>DM, the hydraulic conductivity is negatively correlated with the inlet pressure. The reason for this phenomenon is as follows: Q is mainly controlled by D and P. When D<DM, P is the dominant factor, and its influence on Q is greater than the influence of D on Q. Therefore, the hydraulic conductivity and P are positively correlated, but when D approaches DM, the influence of P gradually decreases, and the influence of D strengthens. When D>DM, the effect of D on Q gradually increases and dominates, and its influence is greater than that of P on Q. The hydraulic conductivity and P are negatively correlated. When D is farther away from DM, D is more dominant. In summary, when D<DM, the inlet pressure is the main influencing factor of the hydraulic conductivity, when D>DM, the fractal dimension replaces pressure and becomes the new dominant factor. Moreover, the farther away from DM, the stronger their control ability.Figure 7
Evolution of hydraulic conductivity with fractal dimension.
(a)
Fracture B(b)
Fracture C


### 3.2. Analysis of the Influence of Different Roughness on the Competitive Diversion Capacity of the Fracture

In the previous part, in order to study the influence of fracture roughness on the seepage law, two branch fractures have the same roughness. However, in actual engineering, it is almost difficult to find two fractures with the identical roughness in the natural fracture networks. Therefore, when the roughness of the two fractures is different to a certain extent, the seepage characteristics and the competitive diversion capacity of the Y-shaped fracture need to be reconsidered.As shown in Figure8, the difference in roughness between the two branch fractures is an important factor in controlling the competitive diversion capacity of the branch fracture. Table 3 lists the simulation results of each outlet flow rate and velocity under the condition of DB≠DC. When the inlet pressure is between 0and approximately 200 Pa, the flow rate and velocity at each outlet show a typical linear evolution. The flow rate of smooth and rough branch fractures with different fractal dimension combination increases linearly with the inlet pressure, but the total outlet flow rate decreases with the increase of fractal dimension of rough branch fracture. After the bifurcation of the Y-shaped fracture, the rough branch fracture flow rate decreases sharply with the increase of the fractal dimension, which is reduced by up to 66%, while the smooth branch fracture flow rate gradually increases. At the same time, the smooth branch fracture flow velocity is much higher than that of rough branch fracture. As the inlet pressure increases, the difference of VA and VB becomes larger and larger that is up to 20.73 cm/s. With the increase of the roughness difference between two branches, the maximum VA:VB ratio also increases from 1.1 to 3.3. However, in the process of increasing the velocity ratio, it is found that when the fractal dimension increases from 1.1 to 1.3, the velocity descent rate is greater than that when the fractal dimension increases from 1.3 to 1.5. It is speculated that the reason for this phenomenon is that the fractal dimension determines the fluctuation degree of the fracture surface. The smaller the fractal dimension is, the smoother the fracture is. When fractal dimension increases from 1.1 to 1.3, the number of concave convex on the fracture surface is more than that increased by fractal dimension from 1.3 to 1.5, resulting in the nonlinear reduction of the velocity with the evolution of fractal dimension. This indicates that the roughness not only affects the seepage capacity of rough branch fracture but also affects the seepage capacity of smooth branch fracture to a large extent and is an important factor influencing the seepage capacity of the entire Y-shaped fracture.Figure 8
Evolution of flow and velocity with inlet pressure.
(a)
1.0-1.0-1.1(b)
1.0-1.0-1.3(c)
1.0-1.0-1.5Table 3
Fluid seepage results under different inlet pressures.
Fracture roughnessInlet pressure (Pa)Flow rate of outlet AQA (cm3/s)Flow velocity of outlet AVA (cm/s)Flow rate of outlet BQB (cm3/s)Flow velocity of outlet BVB (cm/s)1.0-1.0-1.1501.165.791.155.741002.5112.572.3811.881504.0820.423.7418.722005.9329.645.3826.901.0-1.0-1.3501.236.160.974.841002.7413.721.608.001504.3521.742.0110.072005.9329.642.3211.621.0-1.0-1.5501.346.700.854.371002.9114.561.316.561504.4822.381.608.022005.9829.891.839.16Figure9 shows the evolution of the flow rate proportion of the rough and smooth branch fractures with the inlet pressure. The results show that when the pressure is 0 ~ 200 Pa, the proportion of rough branch fracture shows a downward trend, and under high pressure, the downward trend gradually slows down and then tends to be stable. The smooth branch fracture proportion shows an opposite trend, and the proportion eventually tends to be flat. When DC=1.1, the rough branch fracture proportion is about 48%, while that of smooth branch fracture is about 52%. Due to the small difference between the fractal dimensions of DB and DC, the influence of pressure on the proportion is not obvious, and they ended up changing by only 2%. When DC=1.3, the rough branch fracture proportion is mainly between 28% and 44%. If the fluid flow regime remains unchanged, it will eventually stabilize at 25%, which reduced by 16% in total. And the smooth branch fracture proportion of mainly between 56% and 72%. When DC=1.5, the flow rate proportion of rough branch fracture is mainly between 23% and 40% (finally stabilized at about 20%), and the flow rate proportion of smooth branch fracture is mainly between 60% and 77% (finally stabilized at about 80%), which changes significantly with the increase of inlet pressure, speculating that it could be reduced by up to 20%. It can be found that when the branch fracture roughness changes, the flow rate proportion of smooth branch fracture is significantly greater than that of rough branch fracture, which the proportion in the two branches differs by 4%, 44%, and 54% for the three fractal dimensions, respectively. However, with the increase of fractal dimension of rough branch fracture, the increased flow rate proportion of smooth branch fracture is the same as that decreased of rough branch fracture under the same inlet pressure. With the increase of fractal dimension of rough branch fracture, the proportional difference between smooth branch fracture and rough branch fracture is greater. In other words, the greater the roughness of the rough branch fracture of the Y-shaped fracture, the stronger the competitive diversion capacity of the smooth branch fracture. When the roughness is constant, as the inlet pressure increases, the flow rate loss of rough branch fracture is greater. In the process of competitive diversion, the smooth branch fracture become stronger and stronger, forcing the fluid to flow into the smooth branch fracture, resulting in the decrease of rough branch fracture flow rate, as shown in Figure 10, changing the flow path of fluid in the fracture, and the path selection mechanism of fluid changes fundamentally.Figure 9
Evolution of outlet flow rate proportion with inlet pressure.
(a)
1.0-1.0-1.1(b)
1.0-1.0-1.3(c)
1.0-1.0-1.5Figure 10
Fluid flow behavior in fracture (velocity streamline) whenDB≠DC.From the perspective of hydraulic conductivity, the competitive diversion capacity of branch fractures is analyzed. As shown in Figure11, the larger the fractal dimension of rough branch fracture, the lower the hydraulic conductivity, but the smooth branch fracture hydraulic conductivity increases gradually, indicating that the flow resistance of rough branch fracture enhances. The fluctuation degree of fracture surface becomes further complex, the number of concave convex becomes more, the resistance of fluid passing through the fracture surface increases, and it is more difficult for fluid to enter the rough branch fracture. Due to the existence of fracture resistance, fluid flows to the smooth branch fracture, so smooth branch fracture occupies an advantage in fluid competitive diversion. It shows that the change of roughness of one branch fracture in the Y-shaped fracture will not only affect the seepage law of the branch fracture but also affect the seepage characteristics of adjacent branch. According to Figure 11(a), it can be found that with the increase of rough branch fracture fractal dimension, the hydraulic conductivity of smooth branch fracture increases significantly, while the slope gradually slows down under four pressure conditions. However, Figure 11(b) shows the opposite trend, that is, the greater the pressure, the more significant the variety. The smooth branch maximum and rough branch minimum all appear under 200 Pa and DC=1.5, which is 2.9 m/s and 0.9 m/s, and the hydraulic conductivity difference is largest under this condition. Combined with the above flow rate proportion and Figure 11, it can be found that on the premise that the fluid flow regime is unchanged, the influence of adjacent fracture roughness on smooth fracture hydraulic conductivity is in a specific range, which can be considered that under high pressure, the pressure is the main reason affecting the smooth fracture hydraulic conductivity. However, for the rough fracture, like Figure 7, the pressure and hydraulic conductivity are positively correlated at first, and then positively correlated, which is closely related to the fracture fractal dimension, and large fractal dimension is unfavorable to the passage of fluid. In practical engineering, when gas or other fluids flow in fractures, it can be controlled artificially only when the fluid flow regime does not change; then, the engineering disaster at this time can be controlled to the minimum by some specific means. Therefore, it is urgent for engineers to use specific methods to prevent the fluid from being high pressure in practical engineering, so as to reduce the frequency of accidents. In summary, the larger the fractal dimension of rough branch fracture, the weaker the competitive diversion capacity is, the greater the flow resistance is, and the fluid path selection mechanism changes to a certain extent.Figure 11
Evolution of hydraulic conductivity with fractal dimension.
(a)
Fracture B(b)
Fracture C


### 3.3. Evaluation of Flow Regime Based on Reynolds Number and Euler Number

Reynolds number and Euler number can quantitatively describe the flow regime of fluid, in which Reynolds number reflects the ratio of viscous force to inertial force, and can determine the resistance of fluid in flow [39]. The Euler number is the relationship between fluid pressure and inertial force [40]. The expressions of Reynolds number Re and Euler number Eu [16, 39, 40] are as follows:
(8)Re=ρvRμ,(9)Eu=∆Pρv2,where ρ is the fluid density, kg/m3; v is the flow velocity, m/s; R is the characteristic length, m; μ is the fluid viscosity coefficient, Pa·s; ΔP is the pressure difference between the start point and end point of the relevant section, Pa; and the hydraulic diameter is R.As shown in Figure12, for the case of DB=DC, Re of branch fractures is positively correlated with pressure, and the maximum value of the four patterns shall not exceed 1000. Referring to the fluid flow in a circular tube, it is generally considered that when Re is less than the critical Reynolds number of 2000, the control effect of viscous force in the flow process is substantially stronger than that of inertial force, and the flow in rough branch fracture is laminar flow [16]. Taking pattern 1 as an example, with the increase of inlet pressure, the fluid flow regime in the two branch fractures always maintains laminar flow, in which Re is between 168 and 960, and the viscous force is the main control force in the flow process. Since DB=DC, the change trend of Re in the two branch fractures is the same and the values are similar. And the other three patterns always maintain laminar flow, and Re increases with the increase of inlet pressure, which increases by 713, 489 and 355, respectively. Although the fluid in the two branch fractures is always laminar flow, the control effect of viscous force is gradually weakened and the control of inertial force is enhanced with the increase of inlet pressure. Compared with the four patterns, due to the increase of fracture fractal dimension, the Re of each branch fracture decreases significantly. And the larger the fractal dimension, the slower the upward trend with the pressure, reflecting that the fracture roughness does not affect the path selection mechanism of fluid when the roughness of the two branch fractures are identical, and it mainly affects its fluid regime. Under certain pressure conditions, the greater the fractal dimension, the more difficult it is for the fluid to transition from laminar flow to eddy flow. At this time, the fluid flow in the fracture becomes more and more stable. In addition, with the increase of inlet pressure, Eu decreases significantly, which the maximum and minimum values are 19 and 2, indicating that the loss rate of momentum decreases gradually. However, Eu increases significantly when the fracture fractal dimension increases, indicating that the flow resistance increases gradually. When the fluid is at low inlet pressure, its momentum loss is much higher than that at high pressure. Fracture roughness mainly affects the flow resistance, so the lower the inlet pressure and the greater the fractal dimension of the fracture, the fluid momentum loss reaches the peak. The fluid momentum loss determines its flow regime. The greater the loss, the more stable the fluid is, and the fluid can maintain a stable laminar flow in the fracture. Therefore, in engineering practice, because it is impossible to directly observe the fracture roughness in coal, in order to ensure engineering safety, it is necessary to ensure that the fluid pressure always maintain a low level, so as to make sure that the fluid is always in laminar flow and reduce the risk of accidents.Figure 12
Evolution of Re and Eu (DB=DC). Re-S and Re-R represent the Reynolds number of the smooth and rough branch fracture. Eu-S and Eu-R represent the Euler number of the smooth and rough branch fracture.
(a)
Pattern 1: 1.0-1.0-1.0(b)
Pattern 2: 1.0-1.1-1.1(c)
Pattern 3: 1.0-1.2-1.2(d)
Pattern 4: 1.0-1.3-1.3As shown in Figure13, for the case of DB≠DC, with the increase of inlet pressure, the Re of rough and smooth branch fractures gradually increases, the growth rate of smooth branch fracture Re is greater than that of rough branch fracture, and the larger the fractal dimension of rough branch fracture, the larger the Re gap between the two branch fractures. Although the roughness of the branch fracture is different, the fluid maintains the laminar flow. Due to the increase of the roughness difference, Re in the rough branch fracture decreases continuously, Re in smooth branch fracture is larger is 76, 261, 469 and 678 larger than that of rough branch, respectively, the action of viscous force in rough branch continues to enhance and the flow resistance increases. While Re in the smooth branch fracture increases gradually, and the action of inertial force increases. The change of the opposite Re trend of the two branch fractures leads to that part of the fluid that should flow to the rough branch fracture flows to the smooth branch fracture, so as to improve the competitive diversion capacity of the smooth branch fracture. The change trend of Eu is opposite, which Eu decreases with the increase of inlet pressure, and the larger the fractal dimension, the slower the change trend, and the greater the Eu difference between the two branch fractures. By comparing the three patterns, it can be found that with the increase of the roughness difference, the smooth branch fracture Eu reduces, Eu in smooth branch fracture is smaller is 15, 19, 20, and 21 smaller than that of rough branch, respectively, the flow resistance and fluid momentum loss in smooth branch decreases, while the flow resistance and the fluid momentum loss of the rough branch fracture increases and smooth branch competitive diversion capacity is strengthened. It can be interpreted as due to the large momentum loss and slow flow velocity of fluid in the rough branch fracture, it is more difficult for the fluid to flow out of the branch fracture, which makes it hard for the subsequent fluid to enter the rough branch fracture and flow to the smooth branch fracture. The smooth branch fracture has an advantage in the fracture fluid competition.Figure 13
Evolution of Re and Eu (DB≠DC).
(a)
Pattern 1: 1.0-1.0-1.1(b)
Pattern 2: 1.0-1.0-1.3(c)
Pattern 3: 1.0-1.0-1.5


## 3.1. Analysis of the Influence of Fracture Roughness on Seepage Law

In order to explore the influence of fracture roughness on the seepage characteristics, the relationship between the outlet flow rate and velocity and inlet pressures was determined. The test results of the flow rate and velocity under theDB=DC condition are shown in Table 2, and 1.0-1.0-1.0 represents the fractal dimensions of the three branch fractures A, B, and C. Outlet A is the outlet of the branch fracture B, and outlet B is the outlet of the branch fracture C. The pressure at the inlet is as mentioned above.Table 2
Fluid seepage results under different inlet pressures (DB=DC).
Fracture roughnessInlet pressure (Pa)Flow rate of outlet AQA (cm3/s)Flow velocity of outlet AVA (cm/s)Flow rate of outlet BQB (cm3/s)Flow velocity of outlet BVB (cm/s)1.0-1.0-1.0501.165.801.155.751002.4912.462.4512.251504.0420.193.9219.592005.8829.405.6928.431.0-1.1-1.1501.165.781.155.761002.4212.122.4112.051503.8419.193.8119.052005.5227.605.4727.341.0-1.2-1.2501.115.561.115.541002.1710.862.1610.821503.1715.843.1515.762004.1020.514.0820.391.0-1.3-1.3501.035.171.035.161001.839.161.839.161502.4712.332.4712.342003.2116.043.2116.05According to the evolution of flow rate in Figure3 and the flow rate proportion in Figure 4, it is not difficult to find that with the increase of fracture roughness, the outlet flow rate under the four patterns gradually decreases. The greater the roughness, the more obvious flow rate loss, the faster the flow rate decline speed, and the smaller the proportion. Moreover, the inlet pressure also significantly affects its proportion. When the fracture roughness is certain, the flow rate proportion decreases with the increase of inlet pressure, which can be understood as that the inlet pressure affects the fluid seepage velocity in the fracture, but due to the existence of the fracture roughness, the fluid velocity flowing through the fracture surface attenuates, resulting in the reduce of proportion with a maximum loss of 45.44%. All these indicate that the greater the fracture roughness, the greater the flow resistance of the fracture. The flow velocity of smooth fracture is much higher than that of rough fracture, in which the main reason is that the increase in roughness makes the undulation of the fracture surface more complicated, the resistance of the fluid passing through the fracture surface increases, and the flow energy decreases [24]. However, with the increase of inlet pressure, the flow rate increases linearly, and at high pressure level, the greater the velocity difference under each roughness, e. g., V1.3 is 13.36 cm/s smaller than V1.0 when pressure is 200 Pa. The fracture fractal dimension has a negative correlation with the velocity growth rate, indicating that the fracture roughness is an important factor affecting the seepage capacity of the whole Y-shaped fracture. When the pressure at the inlet continues to increase, the flow rates at the two outlets rise at the same time, and the flow rate ratio is close to 1 : 1. This phenomenon shows that when branch fracture roughness is identical, the roughness does not affect the fluid flow path, and the fluid flows into two branch fractures on average, as shown in Figure 5, which reflects that the main factor that affects fluid seepage characteristics is fluid inflow pressure, and the influence of roughness on seepage is mainly reflected in two aspects of flow velocity and flow rate instead of fluid path selection.Figure 3
Evolution of flow and velocity with inlet pressure.
(a)
Outlet A flow rate and velocity(b)
Outlet B flow rate and velocityFigure 4
Evolution of outlet flow rate proportion with inlet pressure.
(a)
Outlet A(b)
Outlet BFigure 5
Fluid flow behavior in fracture (velocity streamline) whenDB=DC.In addition, the difference value in outlet flow rate is an important feature of seepage characteristics. Overall, when the pressure is less than 150 Pa, the flow rate loss shows a gradual increasing trend, which maximum loss is up to 0.70 cm3/s, indicating that within a certain pressure threshold, the greater the fracture roughness, the more fluid will be lost through the fracture. However, when the pressure exceeds this threshold, such as 200 Pa in Figure 6, it is found that the flow rate loss is abnormal. At this time, the maximum loss at ΔD1.1−1.2 is 1.42 cm3/s, which the fractal dimension continues to increase, the loss decreases. The reason for this phenomenon is that the pressure has a nonlinear effect on the seepage. There is a threshold that makes the seepage characteristics before and after the threshold deviate. The flow rate loss of the fracture at low pressure levels is mainly controlled by the roughness of the fracture. As the inlet pressure increases, the pressure affects the role is gradually becoming more prominent, based on the seepage characteristics of low flow rate under high roughness or high pressure conditions, so the flow rate loss under high pressure and high fractal dimensions is lower than the loss under high pressure and low fractal dimensions. From the perspective of actual engineering, the greater the initial roughness of natural fractures in coal, the greater the resistance of fluids passing through the fractures, and the more difficult it is for fluids to flow through the fractures, which makes it hard for the water or gas stored in the fractures to gush out [37]. The mining disturbance causes the fluid pressure to rise, and disasters such as water inrush or gas outburst occur frequently. Therefore, when exploiting underground fluid resources, we should try to avoid the fluid under high pressure or use other methods to reduce fracture roughness and ensure safe mining. The greater the pressure, the greater the flow rate loss, and the degree of flow loss will have different levels of influence on the fracture hydraulic conductivity. Hydraulic conductivity is an important index to characterize the permeability of coal body, and the expression [38] is
(6)k=QμLA∆P,(7)K=kρgμ=QLρgA∆P,where K is the fracture hydraulic conductivity, m/s; k is the fracture permeability, m2; μ is the viscosity coefficient, Pa·s; Q is the outlet rate flow, m3/s; L is the fracture length, m; ρ is the fluid density, kg/m3; A is the seepage cross-sectional area, m2; and ΔP is the seepage pressure difference, Pa.Figure 6
The relationship between flow rate loss and fractal dimension.The evolution of hydraulic conductivity is shown in Figure7. The greater the fracture roughness, the lower the fracture hydraulic conductivity, indicating that the fracture flow resistance is increased, the higher the fractal dimension value, the change of hydraulic conductivity is gradually increased, and the influence of roughness on hydraulic conductivity is gradually enhanced. When the inlet pressure is constant, the change rate of the hydraulic conductivity increases with the increase of the fractal dimension, which drops rapidly at 200 Pa and the maximum and minimum values are 2.88 and 1.57 m/s. However, there is a threshold DM for the fracture fractal dimension between 1.1 and 1.2, so that under four pressure conditions, the fracture hydraulic conductivity is a fixed value. When D<DM, the hydraulic conductivity is positively correlated with the inlet pressure, and when D>DM, the hydraulic conductivity is negatively correlated with the inlet pressure. The reason for this phenomenon is as follows: Q is mainly controlled by D and P. When D<DM, P is the dominant factor, and its influence on Q is greater than the influence of D on Q. Therefore, the hydraulic conductivity and P are positively correlated, but when D approaches DM, the influence of P gradually decreases, and the influence of D strengthens. When D>DM, the effect of D on Q gradually increases and dominates, and its influence is greater than that of P on Q. The hydraulic conductivity and P are negatively correlated. When D is farther away from DM, D is more dominant. In summary, when D<DM, the inlet pressure is the main influencing factor of the hydraulic conductivity, when D>DM, the fractal dimension replaces pressure and becomes the new dominant factor. Moreover, the farther away from DM, the stronger their control ability.Figure 7
Evolution of hydraulic conductivity with fractal dimension.
(a)
Fracture B(b)
Fracture C


## 3.2. Analysis of the Influence of Different Roughness on the Competitive Diversion Capacity of the Fracture

In the previous part, in order to study the influence of fracture roughness on the seepage law, two branch fractures have the same roughness. However, in actual engineering, it is almost difficult to find two fractures with the identical roughness in the natural fracture networks. Therefore, when the roughness of the two fractures is different to a certain extent, the seepage characteristics and the competitive diversion capacity of the Y-shaped fracture need to be reconsidered.As shown in Figure8, the difference in roughness between the two branch fractures is an important factor in controlling the competitive diversion capacity of the branch fracture. Table 3 lists the simulation results of each outlet flow rate and velocity under the condition of DB≠DC. When the inlet pressure is between 0and approximately 200 Pa, the flow rate and velocity at each outlet show a typical linear evolution. The flow rate of smooth and rough branch fractures with different fractal dimension combination increases linearly with the inlet pressure, but the total outlet flow rate decreases with the increase of fractal dimension of rough branch fracture. After the bifurcation of the Y-shaped fracture, the rough branch fracture flow rate decreases sharply with the increase of the fractal dimension, which is reduced by up to 66%, while the smooth branch fracture flow rate gradually increases. At the same time, the smooth branch fracture flow velocity is much higher than that of rough branch fracture. As the inlet pressure increases, the difference of VA and VB becomes larger and larger that is up to 20.73 cm/s. With the increase of the roughness difference between two branches, the maximum VA:VB ratio also increases from 1.1 to 3.3. However, in the process of increasing the velocity ratio, it is found that when the fractal dimension increases from 1.1 to 1.3, the velocity descent rate is greater than that when the fractal dimension increases from 1.3 to 1.5. It is speculated that the reason for this phenomenon is that the fractal dimension determines the fluctuation degree of the fracture surface. The smaller the fractal dimension is, the smoother the fracture is. When fractal dimension increases from 1.1 to 1.3, the number of concave convex on the fracture surface is more than that increased by fractal dimension from 1.3 to 1.5, resulting in the nonlinear reduction of the velocity with the evolution of fractal dimension. This indicates that the roughness not only affects the seepage capacity of rough branch fracture but also affects the seepage capacity of smooth branch fracture to a large extent and is an important factor influencing the seepage capacity of the entire Y-shaped fracture.Figure 8
Evolution of flow and velocity with inlet pressure.
(a)
1.0-1.0-1.1(b)
1.0-1.0-1.3(c)
1.0-1.0-1.5Table 3
Fluid seepage results under different inlet pressures.
Fracture roughnessInlet pressure (Pa)Flow rate of outlet AQA (cm3/s)Flow velocity of outlet AVA (cm/s)Flow rate of outlet BQB (cm3/s)Flow velocity of outlet BVB (cm/s)1.0-1.0-1.1501.165.791.155.741002.5112.572.3811.881504.0820.423.7418.722005.9329.645.3826.901.0-1.0-1.3501.236.160.974.841002.7413.721.608.001504.3521.742.0110.072005.9329.642.3211.621.0-1.0-1.5501.346.700.854.371002.9114.561.316.561504.4822.381.608.022005.9829.891.839.16Figure9 shows the evolution of the flow rate proportion of the rough and smooth branch fractures with the inlet pressure. The results show that when the pressure is 0 ~ 200 Pa, the proportion of rough branch fracture shows a downward trend, and under high pressure, the downward trend gradually slows down and then tends to be stable. The smooth branch fracture proportion shows an opposite trend, and the proportion eventually tends to be flat. When DC=1.1, the rough branch fracture proportion is about 48%, while that of smooth branch fracture is about 52%. Due to the small difference between the fractal dimensions of DB and DC, the influence of pressure on the proportion is not obvious, and they ended up changing by only 2%. When DC=1.3, the rough branch fracture proportion is mainly between 28% and 44%. If the fluid flow regime remains unchanged, it will eventually stabilize at 25%, which reduced by 16% in total. And the smooth branch fracture proportion of mainly between 56% and 72%. When DC=1.5, the flow rate proportion of rough branch fracture is mainly between 23% and 40% (finally stabilized at about 20%), and the flow rate proportion of smooth branch fracture is mainly between 60% and 77% (finally stabilized at about 80%), which changes significantly with the increase of inlet pressure, speculating that it could be reduced by up to 20%. It can be found that when the branch fracture roughness changes, the flow rate proportion of smooth branch fracture is significantly greater than that of rough branch fracture, which the proportion in the two branches differs by 4%, 44%, and 54% for the three fractal dimensions, respectively. However, with the increase of fractal dimension of rough branch fracture, the increased flow rate proportion of smooth branch fracture is the same as that decreased of rough branch fracture under the same inlet pressure. With the increase of fractal dimension of rough branch fracture, the proportional difference between smooth branch fracture and rough branch fracture is greater. In other words, the greater the roughness of the rough branch fracture of the Y-shaped fracture, the stronger the competitive diversion capacity of the smooth branch fracture. When the roughness is constant, as the inlet pressure increases, the flow rate loss of rough branch fracture is greater. In the process of competitive diversion, the smooth branch fracture become stronger and stronger, forcing the fluid to flow into the smooth branch fracture, resulting in the decrease of rough branch fracture flow rate, as shown in Figure 10, changing the flow path of fluid in the fracture, and the path selection mechanism of fluid changes fundamentally.Figure 9
Evolution of outlet flow rate proportion with inlet pressure.
(a)
1.0-1.0-1.1(b)
1.0-1.0-1.3(c)
1.0-1.0-1.5Figure 10
Fluid flow behavior in fracture (velocity streamline) whenDB≠DC.From the perspective of hydraulic conductivity, the competitive diversion capacity of branch fractures is analyzed. As shown in Figure11, the larger the fractal dimension of rough branch fracture, the lower the hydraulic conductivity, but the smooth branch fracture hydraulic conductivity increases gradually, indicating that the flow resistance of rough branch fracture enhances. The fluctuation degree of fracture surface becomes further complex, the number of concave convex becomes more, the resistance of fluid passing through the fracture surface increases, and it is more difficult for fluid to enter the rough branch fracture. Due to the existence of fracture resistance, fluid flows to the smooth branch fracture, so smooth branch fracture occupies an advantage in fluid competitive diversion. It shows that the change of roughness of one branch fracture in the Y-shaped fracture will not only affect the seepage law of the branch fracture but also affect the seepage characteristics of adjacent branch. According to Figure 11(a), it can be found that with the increase of rough branch fracture fractal dimension, the hydraulic conductivity of smooth branch fracture increases significantly, while the slope gradually slows down under four pressure conditions. However, Figure 11(b) shows the opposite trend, that is, the greater the pressure, the more significant the variety. The smooth branch maximum and rough branch minimum all appear under 200 Pa and DC=1.5, which is 2.9 m/s and 0.9 m/s, and the hydraulic conductivity difference is largest under this condition. Combined with the above flow rate proportion and Figure 11, it can be found that on the premise that the fluid flow regime is unchanged, the influence of adjacent fracture roughness on smooth fracture hydraulic conductivity is in a specific range, which can be considered that under high pressure, the pressure is the main reason affecting the smooth fracture hydraulic conductivity. However, for the rough fracture, like Figure 7, the pressure and hydraulic conductivity are positively correlated at first, and then positively correlated, which is closely related to the fracture fractal dimension, and large fractal dimension is unfavorable to the passage of fluid. In practical engineering, when gas or other fluids flow in fractures, it can be controlled artificially only when the fluid flow regime does not change; then, the engineering disaster at this time can be controlled to the minimum by some specific means. Therefore, it is urgent for engineers to use specific methods to prevent the fluid from being high pressure in practical engineering, so as to reduce the frequency of accidents. In summary, the larger the fractal dimension of rough branch fracture, the weaker the competitive diversion capacity is, the greater the flow resistance is, and the fluid path selection mechanism changes to a certain extent.Figure 11
Evolution of hydraulic conductivity with fractal dimension.
(a)
Fracture B(b)
Fracture C


## 3.3. Evaluation of Flow Regime Based on Reynolds Number and Euler Number

Reynolds number and Euler number can quantitatively describe the flow regime of fluid, in which Reynolds number reflects the ratio of viscous force to inertial force, and can determine the resistance of fluid in flow [39]. The Euler number is the relationship between fluid pressure and inertial force [40]. The expressions of Reynolds number Re and Euler number Eu [16, 39, 40] are as follows:
(8)Re=ρvRμ,(9)Eu=∆Pρv2,where ρ is the fluid density, kg/m3; v is the flow velocity, m/s; R is the characteristic length, m; μ is the fluid viscosity coefficient, Pa·s; ΔP is the pressure difference between the start point and end point of the relevant section, Pa; and the hydraulic diameter is R.As shown in Figure12, for the case of DB=DC, Re of branch fractures is positively correlated with pressure, and the maximum value of the four patterns shall not exceed 1000. Referring to the fluid flow in a circular tube, it is generally considered that when Re is less than the critical Reynolds number of 2000, the control effect of viscous force in the flow process is substantially stronger than that of inertial force, and the flow in rough branch fracture is laminar flow [16]. Taking pattern 1 as an example, with the increase of inlet pressure, the fluid flow regime in the two branch fractures always maintains laminar flow, in which Re is between 168 and 960, and the viscous force is the main control force in the flow process. Since DB=DC, the change trend of Re in the two branch fractures is the same and the values are similar. And the other three patterns always maintain laminar flow, and Re increases with the increase of inlet pressure, which increases by 713, 489 and 355, respectively. Although the fluid in the two branch fractures is always laminar flow, the control effect of viscous force is gradually weakened and the control of inertial force is enhanced with the increase of inlet pressure. Compared with the four patterns, due to the increase of fracture fractal dimension, the Re of each branch fracture decreases significantly. And the larger the fractal dimension, the slower the upward trend with the pressure, reflecting that the fracture roughness does not affect the path selection mechanism of fluid when the roughness of the two branch fractures are identical, and it mainly affects its fluid regime. Under certain pressure conditions, the greater the fractal dimension, the more difficult it is for the fluid to transition from laminar flow to eddy flow. At this time, the fluid flow in the fracture becomes more and more stable. In addition, with the increase of inlet pressure, Eu decreases significantly, which the maximum and minimum values are 19 and 2, indicating that the loss rate of momentum decreases gradually. However, Eu increases significantly when the fracture fractal dimension increases, indicating that the flow resistance increases gradually. When the fluid is at low inlet pressure, its momentum loss is much higher than that at high pressure. Fracture roughness mainly affects the flow resistance, so the lower the inlet pressure and the greater the fractal dimension of the fracture, the fluid momentum loss reaches the peak. The fluid momentum loss determines its flow regime. The greater the loss, the more stable the fluid is, and the fluid can maintain a stable laminar flow in the fracture. Therefore, in engineering practice, because it is impossible to directly observe the fracture roughness in coal, in order to ensure engineering safety, it is necessary to ensure that the fluid pressure always maintain a low level, so as to make sure that the fluid is always in laminar flow and reduce the risk of accidents.Figure 12
Evolution of Re and Eu (DB=DC). Re-S and Re-R represent the Reynolds number of the smooth and rough branch fracture. Eu-S and Eu-R represent the Euler number of the smooth and rough branch fracture.
(a)
Pattern 1: 1.0-1.0-1.0(b)
Pattern 2: 1.0-1.1-1.1(c)
Pattern 3: 1.0-1.2-1.2(d)
Pattern 4: 1.0-1.3-1.3As shown in Figure13, for the case of DB≠DC, with the increase of inlet pressure, the Re of rough and smooth branch fractures gradually increases, the growth rate of smooth branch fracture Re is greater than that of rough branch fracture, and the larger the fractal dimension of rough branch fracture, the larger the Re gap between the two branch fractures. Although the roughness of the branch fracture is different, the fluid maintains the laminar flow. Due to the increase of the roughness difference, Re in the rough branch fracture decreases continuously, Re in smooth branch fracture is larger is 76, 261, 469 and 678 larger than that of rough branch, respectively, the action of viscous force in rough branch continues to enhance and the flow resistance increases. While Re in the smooth branch fracture increases gradually, and the action of inertial force increases. The change of the opposite Re trend of the two branch fractures leads to that part of the fluid that should flow to the rough branch fracture flows to the smooth branch fracture, so as to improve the competitive diversion capacity of the smooth branch fracture. The change trend of Eu is opposite, which Eu decreases with the increase of inlet pressure, and the larger the fractal dimension, the slower the change trend, and the greater the Eu difference between the two branch fractures. By comparing the three patterns, it can be found that with the increase of the roughness difference, the smooth branch fracture Eu reduces, Eu in smooth branch fracture is smaller is 15, 19, 20, and 21 smaller than that of rough branch, respectively, the flow resistance and fluid momentum loss in smooth branch decreases, while the flow resistance and the fluid momentum loss of the rough branch fracture increases and smooth branch competitive diversion capacity is strengthened. It can be interpreted as due to the large momentum loss and slow flow velocity of fluid in the rough branch fracture, it is more difficult for the fluid to flow out of the branch fracture, which makes it hard for the subsequent fluid to enter the rough branch fracture and flow to the smooth branch fracture. The smooth branch fracture has an advantage in the fracture fluid competition.Figure 13
Evolution of Re and Eu (DB≠DC).
(a)
Pattern 1: 1.0-1.0-1.1(b)
Pattern 2: 1.0-1.0-1.3(c)
Pattern 3: 1.0-1.0-1.5


## 4. Conclusions

Taking the Y-shaped fracture as the entry point, this work systematically explores the influence of different roughness on fracture seepage characteristics, expounds the seepage law of the Y-shaped fracture from two perspectives, and explores the competitive diversion capacity affected by roughness difference. The main conclusions are as follows:(1)
WhenDB=DC, the ratio of the two branch fractures outlet flow rate is close to 1 : 1. The flow rate and velocity are linear with pressure, which vary among 1.16~5.88 cm3/s and 5.80~29.40 cm/s. The greater the fracture roughness, the smaller the outlet flow rate, resulting in maximum loss up to 45.44%. There is a threshold DM between 1.1 and 1.2, so that the major influencing factor on hydraulic conductivity changes. The greater the fracture roughness is, the lower the fracture hydraulic conductivity is, and the fracture flow resistance is enhanced(2)
WhenDB≠DC, the flow rate and velocity of the two branch fractures have a linear relationship with pressure, but the total outlet flow rate decreases with the rough branch fracture roughness increasing. The rough branch flow rate decreases sharply with the increase of roughness, and the flow velocity of smooth branch is much higher than that of rough branch. With the increase of inlet pressure, the flow rate proportion of rough branch fracture shows a downward trend, while that of smooth branch fracture shows an increasing trend, which eventually tends to be stable and differs by 4%, 44%, and 54%, respectively. The greater the roughness difference, the stronger smooth branch fracture competitive diversion capacity, and the lower the rough branch fracture hydraulic conductivity, and the hydraulic conductivity difference is largest under 200 Pa and DC=1.5(3)
The calculated Reynolds number and Euler number show that for the case ofDB=DC, with the increase of inlet pressure, Re of two branch fractures gradually increases, increasing by 771, 713, 489, and 355, respectively, at four patterns, Eu gradually decreases, the maximum and minimum values are 19 and 2, and the momentum loss rate gradually decreases. The larger the roughness, the obvious increase of Eu and flow resistance(4)
For the case ofDB≠DC, the smooth branch fracture growth rate of Re is greater than that of rough branch fracture. Due to the increase of roughness difference, Re in smooth branch fracture is larger is 76, 261, 469 and 678 larger than that of rough branch, respectively, and the flow resistance increases. Eu in smooth branch fracture is smaller is 15, 19, 20, and 21 smaller than that of rough branch, respectively, fluid momentum loss in smooth branch decreases, and competitive diversion capacity is strengthened



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*Source: 7521955-2022-04-01.xml*