Seepage Characteristics of Shale Fracture and the Effect of Filling Sand under Normal Stress

Abstract

As a new type of unconventional natural gas resource, shale gas plays a vital role in energy supply. In order to deeply understand the shale fracture seepage characteristics, filled and unfilled fracture seepage tests were carried out on shale samples with different fracture surface topography with the use of self-developed shale fracture seepage testing equipment. The fitting formula that the seepage discharge decreased as a negative exponential function with the increase in normal stress was obtained in the unfilled fracture seepage test; the fitting coefficient had a good logarithmic and exponential function relationship with joint roughness coefficient JRC and surface development interface area ratio S_dr. Meanwhile, the modified cubic law was obtained by adding the correction coefficient. The correction coefficient had an exponential relationship with the anisotropy value K_a. Compared with the unfilled one, the fracture surface topography had little effect on the seepage discharge when it was filled. The experimental results show the effects of water head difference, fracture surface topography, particle size, and thickness of filling sand on shale fracture seepage characteristics under different normal stress conditions, which have a certain significance for improving the efficiency of shale gas production.

1. Introduction

With the increase in natural gas demand, as a new type of unconventional natural gas resource, shale gas, which is a kind of natural gas mainly stored in organic-rich shale interlayers [1], has attracted more and more attention. The shale gas reserves are very rich in China. By 2021, the sustainable yield of shale gas had reached 22 trillion m³, and the identified resource reserves of shale gas had reached 365.97 billion m³ [2]. In addition, it is projected to achieve 80–100 billion m³ of shale gas production by 2030. Developing shale gas can increase the supply of clean energy, optimize the energy structure, and promote the revolution of energy supply in China [3]. However, due to the low permeability of shale reservoirs, when shale gas is mined, hydraulic fracturing must be carried out to form fractures, and these fractures are usually filled with proppant, which plays a role in the rapid release and recovery of shale gas [1]. Therefore, it is very important to deeply understand the flow conductivity law in shale fractures and to study the fracture seepage characteristics under stress and filling conditions for the hydraulic fracturing process design, reservoir stimulation technology research, and the growth of gas recovery ratio.

In 1951, based on the parallel plate model, the cubic law was established by Lomize [4]. The cubic law states that the seepage flow per unit width is proportional to the cube of the crack aperture. According to the cubic law, the seepage discharge changes with the change in the fracture aperture, and the fracture aperture is controlled by the stress acting on the fracture surface. According to the in-depth study on the coupling characteristics of fracture seepage–stress over the years, researchers believed that there might be exponential (Louis, C., 1972; Chen, D., 2015; Cardona, A., 2020) [5,6,7], power function (Kranz, R., 1979; Zhang, Y., 1997; Li, B., 2021) [8,9,10], logarithmic function (Jones, F., 1975) [11], and composite function(Yang, J., 2023) [12] relationships between permeability coefficient and normal stress.

In addition, since the actual rock fracture surface is not smooth and straight, the uneven fracture surface has a significant impact on the fluid flow, which makes the cubic law not applicable to the fracture seepage [13,14]. Therefore, many scholars proposed corresponding correction formulas after studying the seepage characteristics in rough fractures. Due to the influence of the roughness of the fracture surface and channel curvature, the calculation result of the cubic law on seepage in rough fractures was three or more orders of magnitude larger than the actual value (Tsang, Y., 1984) [15]. Via the seepage test of different tooth-shaped fractures, the power exponent relationship between the fracture seepage discharge and the fracture aperture was analyzed, and the cubic law and the sub-cubic law suitable for describing the seepage characteristics of rough fractures were proposed by Xu, G. in 2003 [16]. Considering the local curvature and roughness of the fracture, an improved local cubic law was proposed by Wang, L. in 2015 [17], which could calculate the fracture seepage discharge more accurately, with an effective error ranging from 13.4% to 23.4%.

In the study of seepage characteristics in filled fractures. On the basis of a large number of experimental and theoretical studies, the filled fracture seepage could still be described by the cubic law, but it needed to be modified, and the modified coefficient depended on the porosity of the filling medium (Chen, J., 2011) [18]. Combining with the seepage test results of artificial prefabricated filled fractures, the permeability coefficient of filled fracture was modified by utilizing fracture aperture, filling medium particle composition, and filling medium porosity by Su, B. in 1994 [19]. In the seepage test of the naturally filled fracture, the thin filling medium contributed less to the discharge capacity, and most of the water flowed out through the contact gap between the fracture walls and the filling medium (Zhang, Y., 2005) [20]. Via the seepage test of a complex fracture network, the influences of the closure pressure, the density of filling sand, the filling medium, and the fracture network structure on the discharge capacity of the fracture network were studied by Wen, Q. in 2016 [21]. The influences of HPG fracturing fluid residue and filling medium on fracture conductivity were explored by Ni, X. in 2016 [22]. The research results showed that with the increase in the HPG fracturing fluid residue and the filling medium crushing rate, the flow conductivity showed the exponential and linear decrease trend, respectively. The seepage characteristics experiments of turning fractures and branched fractures were carried out by using ceramsite and coated sand by Wang, L. in 2017 [23]. The test results showed that the equivalent flow conductivity of the turning fracture and multiple branched fractures was lower than that of the single fracture. In addition, some scholars studied the transport and migration of fillings in fractures [24,25].

In summary, as the basis for the study, the seepage characteristics of single fracture in rock mass had been studied, but it was not very clear for the influence of normal stress and fracture surface topography in the seepage characteristics of rock fracture. There were some differences in the seepage characteristics between filled and unfilled rock fractures [26], and there were few studies on seepage in filled rock fractures. The seepage tests of filled and unfilled shale fractures were conducted by using self-developed fracture seepage–stress coupling testing equipment. At the same time, by using self-developed rock fracture surface roughness scanning equipment, 3D coordinates were obtained, and 3D images were drawn. During normal stress loading, the influences of water head difference, fracture surface topographic characteristics, particle size and thickness of filling sand, and other factors on the seepage law of shale fractures were studied.

2. Materials and Methods

2.1. Testing Equipment

By using computer-controlled electro-hydraulic servo fracture stress–seepage coupling testing equipment developed by Nanjing Hydraulic Research Institute, the shale fracture seepage tests were carried out. The testing equipment, which is shown in Figure 1, is made up of the loading system, the water supply system, and the weighing system. The loading system includes a hydraulic source, computer, base, normal load loading system, normal load sensor, normal displacement sensor, normal load loader, upper shear box, lower shear box, etc. The water supply system includes a water supply tank, water tank lifting device, water supply pipe, water supply valve, and pressure head sensor; the weighing system includes outlet pipe, measuring cylinder, precision balance, etc.

The computer was controlled to apply the normal load to the upper surface of the sample by the vertical loading header. The height of the water supply tank was adjusted to change the pressure head. The pressure water entered the lower shear box through the supply pipe and then flowed from right to left in the fracture sample into the measuring cylinder. The normal load was recorded and stored in the computer by the normal load sensor (with an accuracy of 0.01 kN), and the normal displacement was recorded and stored in the computer by the normal displacement sensor (with an accuracy of 0.001 mm). The mass change in the water in the measuring cylinder was recorded by the balance (with an accuracy of 0.01 g), and then the corresponding seepage discharge was obtained. The water pressure was obtained by the pressure head sensor on the water supply pipe.

2.2. Sample Preparation

Shale material was used in the test, whose main mineral components were kaolinite, hydromica, and so on. The sample preparation steps were as follows: (1) the shale rock material was processed into a cube sample of 200 mm × 200 mm × 150 mm (L × W × H) via casting, polishing, and other procedures; (2) the pre-fracture was made at the axial midpoint of the sample to facilitate splitting; and (3) tensile fractures were carried out by applying pressure at the pre-fracture location using the self-made sample splitting fixture (Figure 2a). The fracture surface obtained is shown in Figure 2b.

2.3. Test Scheme Design

Ten samples were selected for the seepage test. Samples S1–S5 were carried out via an unfilled fracture seepage test; S6–S10 were carried out via a filled fracture seepage test; the filling medium was sand. The filling sand of samples S6–S8 had the same particle size and different thicknesses, and the sand mass was used to control the sand thickness; samples S6, S9, and S10 were carried out via filled tests with different particle sizes and single-layer sand thicknesses. In the process of the test, normal stress σ_F and water head difference ΔH were adjusted. The loading sequence of normal stress were 0.5, 1.0, 1.5, 2.0, and 2.5 MPa. The water head difference was changed for each level of normal stress. The adjustment sequence of the water head difference was 1, 3, and 5 m. When the sands filled the fractures, the sands were spread evenly on the lower fracture surface. Sand might flow out with the water flow and block the water outlet during seepage, so fine gauze was used to block the water outlet of the shear box. The specific test conditions are shown in

Table 1. Shale fractures seepage test conditions.

Sample Number	Loading of Normal Stress /MPa	Loading of Water Head Difference /m	Filled Fracture Conditions
			Particle Size of Filling Sand/mm	Mass of Filling Sand/g
S1–S5	0.5, 1.0, 1.5, 2.0, 2.5	1, 3, 5	Unfilled
S6			1.0–1.6	72.2
S7				144.4
S8				216.6
S9			2.5–3.2	109.3
S10			3.2–4.0	153.5

.

3. Fracture Surface Topographic Characteristics

In order to obtain the fracture surface topography and quantitatively analyze the influence of the fracture surface topographic characteristics on the seepage characteristics, a fracture surface roughness scanning equipment was designed. The scanning device includes the computer, the gathering device, the control device, and the scanner, as shown in Figure 3. The scanner used a Panasonic CMOS laser sensor with an average accuracy of 0.01~0.03 mm and a maximum coverage scanning area of 300 mm × 300 mm. The equipment could be used to scan the 3D coordinates of each point in the fracture surface of the test sample. At the same time, 3D images were generated by graphics processing software. The 3D mesh model of the fracture surface after processing is shown in Figure 4.

The joint roughness coefficient JRC was introduced to study the influence of the fracture surface roughness on the seepage characteristics. The surface-developed interfacial area ratio S_dr and anisotropy degree K_a were used to analyze the influence of the relief degree and anisotropy on the seepage characteristics. The calculation formulas of each parameter were listed as follows [27,28,29]:

$$JRC=32.2+32.47\lg Z_{2S}$$

(1)

$$Z_{2S}=\left.\left\{\frac{1}{\left(N_x-1\right)\left(N_y-1\right)}\right.·\left[{\displaystyle \sum _{i=1}^{N_x-1}{\displaystyle \sum _{j=1}^{N_y-1}\frac{{\left(Z_{i+1,j+1}-Z_{i,j+1}\right)}^2+{\left(Z_{i+1,j}-Z_{i,j}\right)}^2}{2\Delta x^2}}}+{\displaystyle \sum _{i=1}^{N_x-1}{\displaystyle \sum _{j=1}^{N_y-1}\frac{{\left(Z_{i+1,j+1}-Z_{i+1,j}\right)}^2+{\left(Z_{i,j+1}-Z_{i,j}\right)}^2}{2\Delta y^2}}}\right]\right\}$$

(2)

In which Z_2S is the average gradient mode of the surface, N_x and N_y are the number of coordinate points in the x and y coordinates, and Z_i,j are the Z coordinate values of the coordinate points.

$$S_{dr}=\frac{A_t-A}{A}\times 100\%$$

(3)

$$A_t=\left(\Delta x\Delta y\right)·{\displaystyle \sum _{i=1}^{N_x-1}{\displaystyle \sum _{j=1}^{N_y-1}\sqrt{1+{\left(\frac{z_{i+1,j}-z_{i,j}}{\Delta x}\right)}^2+{\left(\frac{z_{i,j+1}-z_{i,j}}{\Delta y}\right)}^2}}}$$

(4)

In which A_t is the actual area of the rock fracture surface, and A is the projected area of the rock fracture surface on the xy plane.

$$Z_2={\left[\frac{1}{M{\left(\Delta x\right)}^2}{{\displaystyle \sum _{\mathrm{i}=1}^M\left(z_{i+1}-z_i\right)}}^2\right]}^{\frac{1}{2}}$$

(5)

$$K_a=\frac{\min \left\{\left.JR{C}^{2D}\right\}\right.}{\max \left\{\left.JR{C}^{2D}\right\}\right.}$$

(6)

In which profile lines are obtained at intervals of 5 mm on the fracture surface along the water flow direction, and Z₂ is the slope root-mean-square of the profile line.

The average values of the upper and lower fracture surface topographic characteristic parameters are shown in

Table 2. Fracture surfaces topographic characteristic parameters.

Sample Number	Z2S	JRC	Sdr	K a
S1	0.170	7.203	1.377	0.195
S2	0.128	3.078	0.785	0.156
S3	0.114	1.584	0.632	0.059
S4	0.151	5.543	1.101	0.188
S5	0.118	2.002	0.674	0.017
S6	0.109	0.992	0.590	0.132
S7	0.118	2.085	0.686	0.285
S8	0.139	4.346	0.947	0.003
S9	0.130	3.465	0.820	0.076
S10	0.158	6.182	1.156	0.382

.

4. Results and Discussion

4.1. Unfilled Shale Fracture Seepage

4.1.1. Seepage State Analysis

The Reynolds number Re can reflect the influence of parameters such as flow velocity, viscosity coefficient, and seepage channel shape in fracture. By calculating the Reynolds number, the flow state of fracture flow can be decided. The calculation formula was as follows:

$$Re=\frac{v{L}_F}{\upsilon }=\frac{v{L}_F\rho }{\mu }$$

(7)

$$v=\frac{q}{A}=\frac{q}{e_h{L}_w}$$

(8)

where Re is the Reynolds number; v is the characteristic flow velocity with the unit of m/s; υ is the kinematic viscosity coefficient of flow with the unit of m²/s; L_F is the characteristic length with the unit of meter, and it would be taken 2 times of the fracture equivalent hydraulic aperture [30]; ρ is the density of water with the unit of kg/m³; and μ is the dynamic viscosity coefficient of flow. During the test, the water temperature was about 20 °C, the value of μ was 0.001 Pa∙s; q was the seepage discharge with the unit of m³/s; A was the characteristic area of fracture outlet with the unit of m²; e_h was the equivalent hydraulic aperture with the unit of meter; and L_w was the width of samples with the unit of meter.

According to the test schemes, when the normal stress was 0.5 MPa, and the water head difference was 5 m, the seepage velocity was the largest, and the calculated Reynolds number was also the largest. The calculated Reynolds number of the samples at this time is shown in

Table 3. Initial Reynolds number.

Sample Number	Normal Stress/MPa	Water Head Difference/m	Re
S1	0.5	5	12.869
S2			8.141
S3			5.542
S4			10.989
S5			6.470

.

As can be seen from Table 3, the maximum Reynolds number of the seepage test is 12.869, which is far less than the critical Reynolds number of 500 [31]. Therefore, it could be considered that the flow state of fracture water in the seepage test is laminar flow. In addition, the permeability of intact rock was generally small, and its permeability coefficient was generally less than 10⁻¹⁰ m/s. After calculations, the minimum permeability coefficient of shale fracture samples in this paper was 3.724 × 10⁻⁶ m/s, which was four orders of magnitude higher than that of intact rock, so the influence of shale permeability on fracture seepage test results could be ignored.

4.1.2. Relationship between Seepage Discharge and Water Head Difference

In the process of normal loading, the relationship between seepage discharge via sample S1 and water head difference is shown in Figure 5. For different samples, the change law of seepage discharge with water head difference was similar, and sample S1 was taken as an example in this paper. It can be seen from Figure 5 that there is a linear increasing relationship between seepage discharge and water head difference:

$$q=K\Delta H$$

(9)

where q is the seepage discharge with the unit of m³/s; ΔH is the water head difference with the unit of meter; the fitting coefficient K represents the permeability of the fracture, and it is related to the normal stress with the unit of m²/s.

4.1.3. Relationship between K and σ_F

The relationships between K and σ_F are shown in Figure 6. K showed a decreasing trend of negative exponential function with the increase in normal stress; meanwhile, the fracture closure Δb showed an increasing trend of logarithmic function. The specific was the following:

$$K=\frac{q}{\Delta H}={\lambda }_1{e}^{{\lambda }_2{\sigma }_F}$$

(10)

where σ_F is the normal stress with the unit of MPa; λ₁ and λ₂ are the fitting coefficients with λ₁ > 0 and λ₂ < 0.

$$\Delta b=a+b\ln \left({\sigma }_F+c\right)$$

(11)

where, Δb is the fracture closure with the unit of mm.

According to Formula (10), when the water head difference is constant, the seepage discharge via the fracture decreases with the increase in the normal stress, showing a negative exponential function relationship. The exploratory experimental research on the relationship between seepage and stress of a single fracture was carried out, and the exponential empirical formula was obtained (Yin, L., 2013) [32]. This showed that the rate of seepage discharge decreased continuously, and the influence of normal stress on seepage discharge decreased continuously with the increase in normal stress.

The reason was that at the initial stage of normal stress loading, the contact area of the upper and lower fracture surfaces was small. Under the action of stress, the fracture closure rapidly increased, the fracture surface rapidly closed, and the seepage discharge rapidly decreased. With the increase in normal stress, the protrusions on the contact surface were crushed, the contact area of the upper and lower fracture surfaces became larger and larger, and the normal stiffness of the fracture surface became larger and larger. According to Formula (11), the fracture closure increases logarithmically with the increase in the normal stress, and the growth rate continues to decrease; that is, the fracture closure becomes smaller and smaller under the same normal stress increment, and the reduction rate of the seepage discharge of the fracture also decreases, which also well explains the changing trend of seepage discharge with the change in normal stress.

In terms of the relationship between fracture closure and normal stress. The exponential function was used to characterize the normal closure deformation characteristics of structural surfaces by Sun G in 1983 [33] (Formula (12)). The double-curve deformation formulas (Formulas (13) and (14)) of normal deformation of fracture surface changing with normal stress were proposed by Goodman, R. in 1980 [34] and Bandis, S. in 1983 [35].

$$\Delta b=b_{m0}\left(1-e^{-\frac{{\sigma }_F}{k_n}}\right)$$

(12)

$$\Delta b=b_{m0}\left(1-\frac{\xi }{{\sigma }_F}\right)$$

(13)

$$\Delta b=\frac{{\sigma }_F}{k_{n0}+\frac{{\sigma }_F}{b_{m0}}}$$

(14)

where b_m0 is the initial mechanical aperture with the unit of mm; k_n is the normal stiffness coefficient; and ξ is the constant.

Although the expression forms of Formulas (12)–(14) were different from that in this paper, but, $\frac{d\Delta b}{d{\sigma }_F}>0$ and $\frac{d^2\Delta b}{d{\sigma }_F{}^2}<0$, it could be seen that as σ_F increased, Δb increased, its increase rate was slower and slower, and the changing trend was the same as that of Formula (11), which also verified the change law of seepage discharge under the change in normal stress.

For different samples, the fitting parameters λ₁ and λ₂ in Formula (10) were different, and this was related to the roughness and relief degree of the fracture surface, among which λ₁ increased logarithmically with JRC (Figure 7a) and λ₂ decreased exponentially with S_dr (Figure 7b). For a specific shale fracture surface, the values of λ₁ and λ₂ would be roughly determined by the function relationship, and then the basic change in seepage discharge via the shale fracture would be described during the application of normal stress.

4.1.4. Relationships between K and e_L

Assuming that the shale fracture seepage followed the cubic law, the equivalent hydraulic aperture could be inversely calculated [36]. The relationship between seepage discharge q(m³/s) and equivalent hydraulic aperture e_h(m) was listed as follows:

$$q=K\Delta H=\frac{e_h^3}{12\mu }J=\frac{e_h^3}{12\mu }\frac{\Delta H}{L}$$

(15)

$$K=\frac{q}{\Delta H}=\frac{e_h^3}{12\mu L}=C{e}_h^3$$

(16)

$$e_h=\sqrt[3]{12K\mu L}$$

(17)

where J is the hydraulic gradient and ∆H is the water head difference between both ends of the fracture with the unit of meter. In this case, the fracture could be approximated as a smooth parallel plate, where e_h is the equivalent hydraulic aperture with the unit of mm; μ is the dynamic viscosity coefficient of water with μ = 0.001 Pa∙s; and L is the seepage length with L = 0.2 m.

After e_h was inversely calculated from Formula (17), the relationship between Δb and e_h can be obtained as shown in Figure 8. The fitting coefficients α and β and the fitting degree R² of the fitting curve are shown in

Table 4. The fitting coefficients α and β of S1–S5.

Sample Number	α	β	R2
S1	−0.476	3.099	0.995
S2	−0.504	2.870	0.999
S3	−0.484	2.581	0.992
S4	−0.489	2.958	0.996
S5	−0.491	2.638	0.999

. Δb decreased linearly with the increase in e_h, as follows:

$$\Delta b=\alpha e_h+\beta$$

(18)

where ∆b is the fracture closure with the unit of mm and α and β are the fitting coefficients with α < 0 and β > 0.

As the main factor affecting fracture seepage, the actual value of fracture aperture was usually not easy to measure, and the fracture closure was a relatively easy parameter to measure. Therefore, it could be assumed that when e_h was zero, the fracture closure reached the maximum value, which also corresponded to the maximum mechanical aperture e_m. e_m minus the fracture closure under a normal stress state was the mechanical aperture e_L under the normal stress, and the relationship was as follows:

$$e_L=e_m-\Delta b=\beta -\Delta b$$

(19)

Figure 9 shows the relationship between K and e_L. And the fitting coefficients λ₃ and λ₄ and the fitting degree R² of the fitting curve are shown in

Table 5. The fitting coefficients λ₃ and λ₄ of S1–S5.

Sample Number	λ3	λ4	R2
S1	0.412	2.871	0.984
S2	0.312	2.965	0.969
S3	0.349	2.914	0.859
S4	0.325	3.176	0.970
S5	0.334	3.063	0.993

. K decreased with the decrease in e_L, and the decreasing rate decreased gradually. This showed that with the continuous contact and closure of the upper and lower fracture surfaces, the seepage channel was continuously compressed, and the compressibility of the fracture gradually decreased. The fitting relationship was as follows:

$$K=\frac{q}{\Delta H}={\lambda }_3{e}_L^{{\lambda }_4}$$

(20)

where λ₃ and λ₄ are the fitting coefficients with λ₃ > 0 and λ₄ > 0. About the index λ₄, the maximum value was 3.176, the minimum value was 2.871, and the average value was 2.998. Therefore, it could be considered that the relationship between K and e_L was approximately cubic relationship. In Formula (20), λ₄ was assumed to be 3, and the following formula was obtained:

$$K=\frac{q}{\Delta H}={\lambda }_3^{\prime }{e}_L^3$$

(21)

where ${\lambda }_3^{\prime }$ is the fitting coefficient with ${\lambda }_3^{\prime }$>0. The cubic relationship between K and e_L can be obtained, as shown in Figure 10. And the fitting coefficients ${\lambda }_3^{\prime }$ and the fitting degree R² of the fitting curve are shown in

Table 6. The fitting coefficients ${\lambda }_3^{\prime }$ of S1–S5.

Sample Number	$${\mathit{\lambda }}_3^{\prime }$$	R2
S1	0.381	0.980
S2	0.306	0.967
S3	0.338	0.850
S4	0.359	0.966
S5	0.344	0.992

. The fitting degree R² of the fitting curve was above 0.96, except for sample S3, which was 0.85.

It can be seen from Figure 10 that the parameter ${\lambda }_3^{\prime }$ is not a constant like $C=\frac{1}{12 \mu L}$ in the cubic law. This indicated that the seepage law of rough shale fractures did not fully conform to the cubic law. Therefore, it was necessary to add a coefficient θ related to the fracture surface topography characteristics to describe the law as follows:

$$K=\frac{q}{\Delta H}=\theta C{e}_L^3$$

(22)

The correction coefficient θ has an obvious functional relationship with K_a, that θ increases exponentially with K_a (Figure 11). This was because the shale had the structural characteristics of thin layers. In this test, the shale fracture was made by splitting along the direction of its layered structure, which made the fracture surface relatively flat and the upper and lower fracture surfaces well matched. The roughness distribution of the fracture surface directly affected the flow conductivity of the fracture. The larger the K_a, the more uneven the roughness distribution, the more flow channels generated, and the greater the flow conductivity. When the difference in roughness distribution was small, K_a had little effect on the flow conductivity; when the difference was large, there might have been more flow channels, and the influence of K_a on the flow conductivity was more obvious. For a specific shale fracture, the correction coefficient θ would be roughly determined via the function relationship so as to describe the relationship between the seepage discharge of shale fractures and the mechanical aperture. However, it should be noted that due to the limitation of the number of samples, the exact relationship between the correction coefficient θ and the fracture surface topography would need to be further explored.

In conclusion, the cubic law was not suitable to directly describe the seepage characteristics of rough shale fractures, but the cubic relationship between seepage discharge and fracture aperture was still approximately satisfied after adding the correction coefficient related to fracture surface topography. However, the test results might be different with different rock types and sizes. Firstly, the rock type used in this test was shale, and the artificial fracture surface was relatively flat with small roughness. If other rocks were used, JRC and S_dr of the fracture surface and other morphological parameters would be much larger, which would also have a greater influence on the seepage law. Secondly, the seepage length in this test was 20 cm. If a larger fracture sample was used, a longer seepage length could be provided, including more factors such as fracture surface topography and contact distribution [37], and more reasonable results might be obtained. However, there was also the possibility that the accuracy of stress and displacement measurement might be reduced.

4.2. Sand-Filled Shale Fracture Seepage

4.2.1. Relationship between Seepage Discharge and Normal Stress

Figure 12 shows the relationship between sand-filled fracture seepage discharge and normal stress during the normal stress loading phase when the water head difference is 5 m. When the water head difference was 1 and 3 m, the above-mentioned relationship was similar to that when the water head difference was 5 m, so it would not be given again.

As shown in Figure 12, the normal stress is loaded from 0.5 MPa to 2.5 MPa step by step, and the seepage discharge decreases linearly, which is different from the seepage law without filling. When the fracture was not filled, with the increase in fracture closure, the increasing rate of contact rate decreased, the rate of seepage area decreased, and the seepage discharge decreased exponentially with the normal stress. When the filled fracture seepage mainly occurred in sand pores, and the seepage area decreased proportionally with the fracture closure, that is, with the increase in the fracture closure, the seepage discharge showed a linear decreasing trend (Figure 13). This can be seen in the example of sample S6 (Figure 14), during the process of normal loading, where, when the position of the filling sand particles was adjusted, the pores between the sand particles and the pores between the sand particles and the fracture surface were compressed, and the fracture closure increased linearly, so the seepage discharge decreased linearly with the increase in normal stress.

The sand-filled fracture seepage discharge decreased with the increase in normal stress, and its change curve was almost straight line. And the normal stress had a certain influence on the seepage discharge of sand-filled smooth fracture, but it was less than that of unfilled fracture (Yu, B., 1993) [38]. This verified the linear law between the seepage discharge and the normal stress under the filling condition in this paper.

4.2.2. Relationship between Seepage Discharge and Thickness and Particle Size of Filling Sands

Figure 15, respectively, shows the relationship between fracture seepage and normal stress in the normal loading phase under different filling-sand thicknesses and particle sizes when the water head difference is 5 m. When the water head difference was 1 and 3 m, the above-mentioned relationship was similar to that when the water head difference was 5 m, so it would not be given here.

It can be seen from Figure 15a that under the initial normal stress state, the sandy soil is relatively loose. The greater the thickness of the filling sand, the more the pores of the sand, and the stronger the flow conductivity. With the increase in the normal stress, the filling sand particles were compressed via the position adjustment, and the seepage channels were reduced. The seepage discharge of the fractures with the three filling thicknesses was reduced and tended to be close to each other.

According to Figure 15b, when other conditions are constant, larger sand particle size can produce a larger fracture aperture, larger pores between sand particles, and stronger flow conductivity. With the increase in normal stress, the seepage discharge of the three groups of samples all decreased, but compared to the case of different thicknesses, the difference values of the three groups of samples changed little. This indicated that the sand laid in the single layer had no obvious position adjustment, but it was embedded in the fracture surface continuously with the increase in normal stress, and the flow conductivity decreased proportionally with the increase in normal stress. Combined with the test results, it could be predicted that when the normal stress is large enough, the filling sand would be crushed, and the seepage discharge results would be similar under different sand particle sizes.

5. Conclusions

Aiming at the coupling characteristics of shale fracture seepage–stress, the seepage tests under different normal stress and water head differences were carried out on the unfilled and sand-filled rough shale fractures by using the self-developed shale fracture seepage testing equipment. The main conclusions were as follows:

(1) The normal stress loading caused the increase in the normal stiffness, which made the seepage discharge of unfilled fractures decrease in a negative exponential function with the increase in the normal stress, and the variation trend of the seepage discharge of different fractures was different, which was related to the roughness and relief degree of the fracture surface.

(2) The relationship between the seepage discharge of unfilled fracture and the mechanical aperture was the exponential function, which basically conformed to the cubic law. After introducing K_a, the modified cubic law was obtained.

(3) Different from the seepage law of unfilled shale fractures, the seepage discharge of filled fractures decreased linearly in the process of normal stress loading because the seepage of filled fractures mainly occurred in sand pores and was not affected by the fracture surface topography.

(4) The seepage discharge of filling fractures increased with the increase in the thickness of the filling sand. With the increase in normal stress, the seepage discharge of different filling thicknesses decreased linearly and tended to approach. The larger the filling sand particle size, the larger the seepage discharge, and the seepage discharge also decreased linearly with the increase of normal stress. However, the variation in the difference in seepage discharge under the different filling particle sizes was smaller than that under the different thicknesses.

On the basis of considering the fracture surface topography and filling condition, this paper can more accurately describe the seepage–stress law of natural shale fracture via the shale fracture seepage test. In the future, this will have broad prospects in the important links of shale gas exploitation, such as hydraulic fracturing, reservoir filling reconstruction, and so on.