# Experimental Study on Measurement of Drag Coefficient of Sandy Soil Nonlinear Vadose

Authors: Yu-ping Xu; Jing-yuan Liu; Xiu-wei Chai; Sha-sha Shi
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

The seepage characteristics of sand affect its strength, mechanical deformation characteristics, and find safety and stability. The drag coefficient is a dimensionless physical quantity expressing the interaction between fluid and particles, it is also an important parameter that characterizes the permeability of granular soils. Through the seepage test of standard sand with single particle sizes and different grades, analyze the seepage laws of standard sand with different particle sizes in the Darcy and non-Darcy seepage. The relationship between parametersa and b in the seepage equation and porosity (n), average particle size (d50), nonuniformity coefficient (CU), and curvature coefficient (CC), and their influence degree were studied. An empirical formula for determining the permeability coefficient of standard sand is established. By adding a correction factor and verifying its rationality, the empirical formula of standard sand permeability parameters is finally obtained.


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## Body

## 1. Introduction

Sandy soil is the particles with a particle size greater than 0.075 mm accounting for more than 50% of the total mass. Coarse-grained soil refers to the soil with a particle size of 2 to 20 mm gravel content less than or equal to 50%. It is widely used in construction materials, dam filter material, foundation treatment, and other engineering materials. The seepage of water in the sand will exert seepage force on the soil particles, which will cause changes in the internal stress state, structure, and strength of the soil.When the seepage force is too large, relative movement between soil particles and even overall movement of the soil body will occur, and seepage deformation or even destruction will occur [1, 2], causing damage to dams and foundation pits [3, 4], slope instability [5, 6], ground uplift, and other issues [7, 8]. Thaisiam et al. [9] believed that after seepage erosion, with the continuous movement of fine particles, some pores in the sand may be blocked or enlarged, which will result in the change of the soil permeability coefficient. Moffat and Fannin [10] proposed that seepage erosion causes the permeability coefficient of soil to increase from upstream to downstream. However, Benamar and Correia dos Santos [11] found that under the action of seepage erosion, the permeability coefficient of the downstream soil mass decreases, while the permeability coefficient of the upstream soil mass does not change much. In order to explore the factors affecting the permeability of sandy soil, Su et al. [12] carried out a single particle size and different gradation sand soil permeability tests and found that the factors affecting permeability are sand gradation, porosity, and particle size. Finally, the relationship between the permeability coefficient and various factors is fitted to provide guidance for the actual project. Liu et al. [13] concluded that the thin elastic layer assumption is significantly robust to the elastic layer assumption for simulating fracture opening and closing under coupled conditions. Due to the complex fracture geometry, the evolution of fracture permeability shows some heterogeneity. Qi et al. [14] collected a large number of seepage deformation test data of noncohesive soil (gravel soil and sand) and conducted comparative analysis and found that when the sample unevenness coefficient is less than or equal to 5, there are two types of seepage deformation of noncohesive soil: Crushed gravel soil is piping and sandy soil is flowing soil. Ju et al. [15] used the head control tube to control the average hydraulic gradient of the entire soil column when the steady seepage flow was 0.8 and conducted permeability tests on medium sand, coarse sand, and gravel sand samples. In the experiment, it is concluded that the lower part of the coarse sand and gravel sand samples is easier to be carried away by seepage, and the increase in permeability coefficient is larger. The research results of scholars at home and abroad show that the seepage curve of low-permeability porous media (that is, the relationship between flow velocity and hydraulic gradient) is no longer a simple linear feature but presents a non-Darcy seepage feature. Researchers such as Liu [16], Zhu and Ling [17], Zhang [18], and Qian et al. [19] conducted a large number of indoor permeability experiments in terms of particle size, porosity, gradation, etc. Get the respective permeability coefficient formula. Each researcher has a different emphasis on sand selection and analysis factors, and the obtained formulas are also different in form, and none of them can be directly used for the determination of the permeability coefficient of standard sand and sand.Through the single particle size and different grade standard sand particle seepage test, analyze the seepage law of different standard sand particles in the Darcy-non-Darcy flow zone. Study the relationship between the parameters in the seepage equation and the porosity, average particle size, nonuniformity coefficient and curvature coefficient, and the degree of influence, establish an empirical formula suitable for determining the permeability coefficient of standard sand, and verify its rationality by adding correction factors, and finally obtain the empirical formula of standard sand permeability parameters to guide the safety and stability analysis and evaluation of actual sand foundation engineering.


## 2. Materials and Methods

### 2.1. Physical Properties of Test Materials

The test material is Fujian standard sand. After the sand is fully dried, it is subjected to a screening test to obtain five particle size ranges of 0.1 to 0.25 mm, 0.25 to 0.5 mm, 0.5 to 1 mm, 1 to 2 mm, and 2 to 5 mm. The original grading curve is shown in Figure1. The specific gravity of standard sand measured by the pycnometer method is about 2.65. Using PartAn 3D scanning equipment to measure the average particle size of standard sand, the results are shown in Table 1.Figure 1
Cumulative curve of standard sand particle size gradation.Table 1
Values of average particle size in different particle size intervals.
Particle size (mm)0.1–0.250.25–0.50.5–11–22–5The average grain diameter (mm)0.1750.3750.751.5992.057


### 2.2. Experimental Scheme

First, the permeability tests of standard sand with different dry densities in a single particle size range were carried out. Dry density values of sand samples in the test are shown in Table2. Second, the influence of different particle size ranges on the permeability of standard sand under the same dry density was studied. Finally, the data obtained from the test were processed, and the permeability results of different samples were compared.Table 2
Maximum and minimum dry densities in each particle size range.
Particle size (mm)0.1–0.250.25–0.50.5–11–22–5Maximum dry density (g/cm3)1.6421.6531.7451.8021.822Minimum dry density (g/cm3)1.3801.3901.5441.6121.632The permeability of soil is affected by particle distribution. The permeability was studied by adjusting the mass percentage of each particle-size interval to prepare different grades. The quality of soil particle gradation can be judged by the two parameters of nonuniformity coefficient (CU) and curvature coefficient (CC). When considering the influence of grading changes, there are two cases: Keep the curvature coefficient (CC) as 1 and change the value of the nonuniformity coefficient (CU), respectively: 5, 5.25, 5.5, 6, and 6.25. Keep the nonuniformity coefficient (CU) as 6 and change the value of the curvature coefficient (CC), respectively: 0.5, 0.75, 1, 1.25, and 1.43. Take the average particle size d50 as 1 mm, the effective particle size d10 as 0.25 mm, d60 and d30 are obtained by formulas. Finally, the mass percentage of each particle size range of samples with different grades was determined as shown in Tables 3 and 4. The cumulative curve of particle size of each group can be drawn, as shown in Figures 2 and 3.Table 3
Mass percentage of each particle size interval of samples with different uneven coefficient.
CCCUMass percentage of each particle size interval (%)d50 (mm)0.1–0.250.25–0.50.5–11.0–2.02.0–5.0151017.322.7401015.251016.523.5321815.51015.624.426.723.316101426203016.251013.326.717.832.21Table 4
Mass percentage of each particle size interval of samples with different curvature coefficients.
CUCCMass percentage of each particle size interval (%)d50 (mm)0.1–0.250.25–0.50.5–11.0–2.02.0–5.060.51027.312.7203010.751018.721.32030111014.026.0203011.25108.331.7203011.43102.537.520301Figure 2
Cumulative curve diagram of particle size of different nonuniformity coefficientCU samples.Figure 3
Cumulative curve diagram of particle size ofCC samples with different curvature coefficients.


### 2.3. Experimental Procedure

Standard sand is a noncohesive soil. In order to achieve a breakthrough in the seepage test under the state of Darcy flow to the turbulent zone, the permeability test device adopts an improved device for measuring the permeability coefficient and drag force coefficient of sandy soil at a higher flow rate. The sample section is equally divided into five layers, each layer is 4.9 cm, and the sample is loaded by the subloading method. Five equal parts of the standard sand quality are loaded into each layer, and each layer is compacted with a compactor. Ensure that all samples just fill the sample section. The criterion for determining the saturation of the sample is that within 5 minutes, the data fluctuation of the flow meter and the differential pressure transmitter digital display instrument should not exceed 0.5 L/h and 10, respectively. During the test, all valves must be adjusted slowly to prevent excessively fast adjustment from causing greater disturbance to the sample.


## 2.1. Physical Properties of Test Materials

The test material is Fujian standard sand. After the sand is fully dried, it is subjected to a screening test to obtain five particle size ranges of 0.1 to 0.25 mm, 0.25 to 0.5 mm, 0.5 to 1 mm, 1 to 2 mm, and 2 to 5 mm. The original grading curve is shown in Figure1. The specific gravity of standard sand measured by the pycnometer method is about 2.65. Using PartAn 3D scanning equipment to measure the average particle size of standard sand, the results are shown in Table 1.Figure 1
Cumulative curve of standard sand particle size gradation.Table 1
Values of average particle size in different particle size intervals.
Particle size (mm)0.1–0.250.25–0.50.5–11–22–5The average grain diameter (mm)0.1750.3750.751.5992.057


## 2.2. Experimental Scheme

First, the permeability tests of standard sand with different dry densities in a single particle size range were carried out. Dry density values of sand samples in the test are shown in Table2. Second, the influence of different particle size ranges on the permeability of standard sand under the same dry density was studied. Finally, the data obtained from the test were processed, and the permeability results of different samples were compared.Table 2
Maximum and minimum dry densities in each particle size range.
Particle size (mm)0.1–0.250.25–0.50.5–11–22–5Maximum dry density (g/cm3)1.6421.6531.7451.8021.822Minimum dry density (g/cm3)1.3801.3901.5441.6121.632The permeability of soil is affected by particle distribution. The permeability was studied by adjusting the mass percentage of each particle-size interval to prepare different grades. The quality of soil particle gradation can be judged by the two parameters of nonuniformity coefficient (CU) and curvature coefficient (CC). When considering the influence of grading changes, there are two cases: Keep the curvature coefficient (CC) as 1 and change the value of the nonuniformity coefficient (CU), respectively: 5, 5.25, 5.5, 6, and 6.25. Keep the nonuniformity coefficient (CU) as 6 and change the value of the curvature coefficient (CC), respectively: 0.5, 0.75, 1, 1.25, and 1.43. Take the average particle size d50 as 1 mm, the effective particle size d10 as 0.25 mm, d60 and d30 are obtained by formulas. Finally, the mass percentage of each particle size range of samples with different grades was determined as shown in Tables 3 and 4. The cumulative curve of particle size of each group can be drawn, as shown in Figures 2 and 3.Table 3
Mass percentage of each particle size interval of samples with different uneven coefficient.
CCCUMass percentage of each particle size interval (%)d50 (mm)0.1–0.250.25–0.50.5–11.0–2.02.0–5.0151017.322.7401015.251016.523.5321815.51015.624.426.723.316101426203016.251013.326.717.832.21Table 4
Mass percentage of each particle size interval of samples with different curvature coefficients.
CUCCMass percentage of each particle size interval (%)d50 (mm)0.1–0.250.25–0.50.5–11.0–2.02.0–5.060.51027.312.7203010.751018.721.32030111014.026.0203011.25108.331.7203011.43102.537.520301Figure 2
Cumulative curve diagram of particle size of different nonuniformity coefficientCU samples.Figure 3
Cumulative curve diagram of particle size ofCC samples with different curvature coefficients.


## 2.3. Experimental Procedure

Standard sand is a noncohesive soil. In order to achieve a breakthrough in the seepage test under the state of Darcy flow to the turbulent zone, the permeability test device adopts an improved device for measuring the permeability coefficient and drag force coefficient of sandy soil at a higher flow rate. The sample section is equally divided into five layers, each layer is 4.9 cm, and the sample is loaded by the subloading method. Five equal parts of the standard sand quality are loaded into each layer, and each layer is compacted with a compactor. Ensure that all samples just fill the sample section. The criterion for determining the saturation of the sample is that within 5 minutes, the data fluctuation of the flow meter and the differential pressure transmitter digital display instrument should not exceed 0.5 L/h and 10, respectively. During the test, all valves must be adjusted slowly to prevent excessively fast adjustment from causing greater disturbance to the sample.


## 3. Experimental Results and Analysis

### 3.1. Comparison of Seepage Experiments with Different Porosities in Different Particle Size Intervals

Conduct penetration tests on standard sands with different porosities in the five groups of single particle size intervals of 0.1 to 0.25 mm, 0.25 to 0.5 mm, 0.5 to 1 mm, 1 to 2 mm, and 2 to 5 mm. The relationship between the standard sand seepage velocityU and hydraulic gradient I in a single particle size range is shown in Figure 4.Figure 4
U−I relationship diagram under different porosity in the 0.1–0.25 mm particle size range.It can be seen from Figure4 that under the same hydraulic slope, as the porosity decreases, the seepage velocity in the sample gradually slows down. Analyzing the range of coefficients a and b in the simplified seepage formula, it can be seen that the value of b is one order of magnitude higher than the value of a, and the values of a and b increase with the decrease of porosity.


#### 3.1.1. Relationship between Drag Coefficient (a) and Porosity

In the permeability test, the control variablesCC, CU, and d50 are fixed values, and the relationship between the drag coefficient and the porosity in each particle size range is explored. Refer to Ergun's formula and initially draw up α∝λ11−nλ2nλ3. As n ∈ (0, 10), according to the boundary conditions: When n ⟶ 0, a ⟶ +∞, we have λ3<0 and λ1>0; when n ⟶ 1, a ⟶ 0, we have λ2>0.By determining the value ranges of the parametersλ1, λ2, and λ3 and defining the type of the fitting function, the relationship between the drag force coefficient and the porosity in each particle size interval is shown in Figure 5.Figure 5
The relationship between the drag coefficienta and the porosity n of each particle size class: (a) single particle size 0.1–0.25 mm, (b) single particle size 0.25–0.5 mm, (c) single particle size 0.5–1 mm, (d) single particle size 1–2 mm, and (e) single particle size 2–5 mm.
(a)(b)(c)(d)(e)It can be seen from Figure5 that the drag coefficient a is positively correlated with 1−n2n−3. The correlation between a and porosity n in each single particle size range is as follows: 0.84252, 0.80163, 0.95891, 0.85015, and 0.85994. The degree of correlation is above 0.8, with a good linear correlation. In the same particle size class, the increase of the dry density of the sample leads to the decrease of the internal porosity of the sample, which is manifested by the decrease of the seepage path or the weakening of the hydraulic path of the water cross section, resulting in the decrease of the overall permeability of the sample and the increase of the drag force coefficient.


#### 3.1.2. The Relationship between Drag Coefficient (b) and Porosity

Compared with the aforementioned methods, the relationship between drag force coefficient and porosity was analyzed. Refer to Ergun's formula and initially draw upa∝λ11−nλ2nλ3. Since n ∈ (0, 1), according to the boundary conditions, when n ⟶ 0 and b ⟶ +∞, we have λ3<0 and λ1>0, and when n ⟶ 1 and b ⟶ 0, we have. λ2>0.Define the type of fitting function by determining the value range of parametersλ1, λ2, and λ3. The correlation between drag coefficient b and porosity n in different particle size ranges is shown in Figure 6.Figure 6
The relationship between drag coefficientb and porosity n in each single particle size interval: (a) single particle size 0.1–0.25 mm, (b) single particle size 0.25–0.5 mm, (c) single particle size 0.5–1 mm, (d) single particle size 1-2 mm, and (e) single particle size 2–5 mm.
(a)(b)(c)(d)(e)From Figures6(a) and 6(b), it can be seen that the drag coefficient b and 1−n2n−3 are positively correlated, and the correlation is very high. In Figure 6(c), the relationship between b and n is completely inconsistent with 1−n2n−3. Figures 6(d) and 6(e) show a positive correlation between fit b and 1−n2n−3, but it is not very relevant. In the same particle size class, the decrease of the dry density of the sample leads to an increase in the internal porosity of the sample, which is manifested as an increase in the seepage path of the cross section of the water, which leads to an increase in the overall permeability of the sample and a decrease in the drag coefficient b value.


### 3.2. Comparison of Seepage Experiments in Different Particle Size Intervals under the Same Porosity

When the porosity is the same, the particle size of the sample accumulation is the only factor that affects its permeability. Multiple sets of samples are tested separately for seepage, and the test results are fitted and analyzed. The correlation between the drag coefficientsa, b and the average particle size d50 is shown in Figure 7.Figure 7
The relationship between drag coefficienta, b and average particle size d50: (a) the correlation between a and d50 and (b) the correlation between b and d50.
(a)(b)It can be seen from Figure7 that the overall linear regression correlation between drag coefficient a and d50−2 reaches 0.913, the overall linear regression correlation between drag coefficient b and d50−2 reaches 0.861. Both have a good linear correlation. When the porosity is constant, as the average particle size d50 of the sample increases, the more pores formed between the particles, the larger the pore volume, and the large internal pores will not be filled by small particles. It shows that the water seepage channel is more open, the seepage resistance of the water flow is weakened, and it shows better permeability.


### 3.3. Seepage Experiment of Mixed Particle Size under Different Gradations

The particle size of the soil has a great influence on its permeability, the inhomogeneity coefficientCU indicates the degree of unevenness of the particles, and the curvature coefficient CC indicates the degree of continuity of the soil particles. In order to explore the influence of the inhomogeneity coefficient CU and the curvature coefficient CC on the permeability of the standard sand, seepage tests were carried out and the test results were compared and analyzed.


#### 3.3.1. Effect of Uneven Coefficient on Drag Coefficient

In order to explore the influence of theCU on the permeability of the standard sand, the value of the CC of the sample is all 1. The CU takes the values 5, 5.25, 5.5, 6, and 6.25 in sequence. The preloaded sample method is used to estimate the mass of the required standard sand samples of different grades to be about 3.5 kg. Control five groups of samples to have the same dry density (1.819 g/cm3). The relationship between the drag coefficient a, b and the uneven coefficient in the five groups of different gradation tests is shown in Figure 8.Figure 8
The relationship between drag coefficienta, b and uneven coefficient: (a) the relationship between a and CU and (b) the relationship between b and CU.
(a)(b)It can be seen from Figure8 that the drag coefficients a, b and CU−2 are all positively correlated. The correlation degree distribution is 0.98802 and 0.88682. When the curvature coefficient is a fixed value and the inhomogeneity coefficient increases, the permeability coefficient of the sample also increases. Control the average particle size to 1 mm and the effective particle size to 0.25 mm, from the calculation formula of the grading parameter, it can be known that when the CU is 5.25, the corresponding value of d60 is greater than the value of d60 when the CU is 5. That is, the proportion of coarse particles in the sample is greater than that of fine particles. And so on, when CU is 6.25, the content of coarse particles is the largest. The arrangement and combination of coarse particles will form larger voids, while the content of fine particles is reduced, and these voids cannot be completely filled, resulting in a more developed internal hydraulic path, making it easier for water to pass through, that is, better permeability.


#### 3.3.2. Effect of Curvature Coefficient on Drag Coefficient

In order to study the influence ofCC on the permeability of standard sand, the value of CU is all 6, and the value of CC is 0.50, 0.75, 1.00, 1.25, and 1.50. Control five groups of samples to have the same dry density (1.819 g/cm3). The relationship between the drag coefficient a, b and the curvature coefficient in the five groups of different gradation tests is shown in Figure 9.Figure 9
The relationship between drag coefficienta, b and curvature coefficient: (a) the relationship between a and CC and (b) the relationship between b and CC.
(a)(b)It can be seen from Figure9 that the drag coefficients a, b and CC−0.5 are all positively correlated. The correlation degree distribution is 0.82742 and 0.80447. With the increase of CC, the content in the 0.5 to 1 mm particle size range increases, and the content in the 0.25 to 0.5 mm particle size range decreases accordingly. As a result, the pores formed by larger particles cannot be fully filled by fine particles, resulting in the easier flow of water inside the sample, better permeability, and the drag coefficient a and b values are getting smaller and smaller.


### 3.4. Empirical Formula Fitting

#### 3.4.1. Preliminary Empirical Formula Fitting of Drag Coefficient

According to the experimental data, the correlation between the average particle size, porosity, curvature coefficient and uneven coefficient, and the drag coefficient is studied by controlling a single variable and analyzing the boundary conditions. The values of drag force coefficientsa and b were fitted by empirical formula with four parameters, d50, n, CU, and CC, respectively, such as formulas (1) and (2). The fitting results of the empirical formulas of the drag coefficients a and b are shown in Figure 10.(1)a′=40.95⋅d50−2⋅1−n2⋅n−3⋅CC−0.5⋅CU−2,(2)b′=e6.7⋅a0.06.Figure 10
The preliminary empirical formula fitting of drag coefficienta.The results show that the correlation between test dataaTest and fitting formula a′ can reach 0.75. a′ and aTest are highly correlated. However, it can be seen that part of the data is more discrete when the test data are about aTest1000. This part of the data mainly comes from the gradation test. It shows that compared with the two, the single particle size test data are more reliable. The main reason is that the composition of the gradation test particles is more complex, which leads to a large difference in the pore structure. Even if the same group of samples is affected by the preloading and postloading, the test results are also very different.It can be seen from the aforementioned comparative analysis that the correlation between theb value and the d50, n, CU, and CC parameters is weak. Therefore, you can choose to explore the correlation between b value and a value, and the fitting result is shown in Figure 11.Figure 11
The relationship between drag coefficienta and b.By comparing the drag coefficienta value representing the coefficient of the U−I linear segment and the drag force coefficient b representing the coefficient of the U−I nonlinear segment, and the sample data bTest and aTest, it is found that the drag force coefficient b has a relatively stable magnitude and a higher drag coefficient a is an order of magnitude. The correlation between the two is 0.85, so formula (2) can be used as a preliminary empirical formula for drag coefficient b.


#### 3.4.2. Comparison of Empirical Formula Correlation

Use goodness of fit (R2) to compare the fit of different empirical formulas to standard sand permeability data, the formula of R2 is as follows:(3)R2=1−∑ITest−IEmpirical−formula2∑ITest−I¯Test2.Among them,ITest is the I value obtained from the test data, I¯Test is the average value of the test data, IEmpirical−formula is the I value calculated by the empirical formula. The preliminary empirical formula calculation value I′, the Ergun formula calculation value, the Van Gent formula calculation value, the Liu formula calculation value, and the correlation between the Jingsui Wu formula calculation value, and the experimental data ITest are listed in Table 5.Table 5
Comparison of fitting results between common seepage formulas and standard sand seepage experiment data.
Particle size range (mm)NCUCCCorrelation comparison (1−SSres/SStot)The formula of this paperI′ErgunVan GentLiuJingsui WuThe formula of this paperI″0.1–0.250.381110.943−2.047−9.1060.583−1.7150.9430.455110.982−3.361−7.231−0.056−2.5440.9820.449110.978−2.028−5.3980.320−1.4780.9780.25–0.50.381110.404−2.9690.479−0.827−1.9420.0530.381110.219−2.5700.799−0.724−1.9050.0050.402110.590−1.9210.585−0.368−1.1400.3330.5–10.36111−4.672−1.380−30.4100.6940.0900.1890.37411−0.157−1.677−10.8530.7370.8970.9660.3811−2.131−2.126−18.7720.8250.7830.6291–20.348110.030−2.112−6.4350.595−0.1950.8940.361110.154−1.824−3.9760.5290.0630.9020.377110.334−2.083−3.6300.4840.5560.8382–50.34411−0.213−2.158−5.0670.551−2.0590.8530.35311−0.011−2.137−4.5560.527−1.1360.8200.38111−0.134−2.335−5.3940.509−0.1400.8190.1–50.31460.75−1.907−2.0720.536−0.2400.1870.9290.3145.51−2.093−2.2810.490−0.3210.1810.9630.31461−1.573−1.7040.323−0.0160.3770.9090.31461−2.169−2.3030.838−0.530−0.0660.9520.3146.251−2.179−2.2890.887−0.585−0.1940.951It can be seen from Table 5 that the preliminary fitting formula is only suitable for solving the permeability parameters of standard sand with a single particle size of 0.1 to 0.25 mm. The formula of Van Gent is suitable for the solution of standard sand permeability parameters in grading experiments. The formula of Liu is suitable for the solution of permeability parameters of standard sand in the interval of 0.5 and 1 mm. In general, the aforementioned five formulas are lacking in the process of solving the standard sand permeability parameters, and the correlation is low.


#### 3.4.3. Improve Empirical Formula

There is a certain deviation between the empirical formulaI′ and the standard sand drag coefficient in all particle sizes. However, the analysis of the empirical formula I′ and the experimental data show that there is a linear correlation between the two. The comparison is shown in Figure 12.Figure 12
Comparison ofU−I relationship between fitting formula I′ and experimental data: (a) the relationship between I′ and U and (b) the relationship between I′ and I.
(a)(b)It can be seen from Figure12(a) that there is a certain correlation between the data points obtained in the interval of 0.1 to 0.25 mm for a single particle size and the fitting formula I′. In order to make the empirical formula reflect the experimental data more accurately, a correction factor is introduced to improve the empirical formula. That is, add correction C. Assuming ITestC⋅I′, exploring the correlation coefficient C between the two is shown in Figure 12(b). The improved empirical formula is in good agreement with the experimental data points, and the correlation is high, which reflects that the improved empirical formula can be closer to the experimental results. The correlation coefficient between the single particle size interval and the gradation experimental data is not a fixed value, and the closer the coefficient is to 1, the better the empirical formula fits. The relationship between the coefficient C and each parameter is shown in Figure 13.Figure 13
The correlation between the coefficientC and various parameters.It can be seen from the figure that the coefficientC is related to d50, CC, and CU, and the correlation of the fitting formula is as high as 0.96, so the empirical formula for the coefficient C is as follows:(4)C=0.18⋅d500.3⋅CC⋅CU0.8⋅CC+CU1.8.The improved empirical formula is as follows:(5)I′C⋅I′.The correlation between the modified empirical formulaI″ and the empirical formula I′ is shown in Table 6.Table 6
Comparison of the fitting accuracy between the modified empirical formulaI″ and the empirical formula I′
Particle size range (mm)Dry density (g/cm3)CUCCFitting accuracy comparison (1−SSres/SStot)I′I″0.1–0.251.4461.001.000.98180.98181.4611.001.000.97810.97811.6401.001.000.94340.94340.25–0.51.6401.001.000.40360.0531.5851.001.000.58950.3331.5991.001.000.21940.0050.5–11.6941.001.00−4.67150.1891.6601.001.00−0.15730.9661.6401.001.00−2.13100.6291–21.6931.001.000.15360.9021.7291.001.000.02960.8941.7671.001.00−0.01110.9042–51.6401.001.00−0.13400.8191.7151.001.00−0.01130.8201.7381.001.00−0.21290.8530.1–51.81906.000.75−1.90650.9291.81906.001.00−1.57270.9091.81906.001.00−2.16850.9521.81906.251.00−2.17860.9511.81905.501.00−2.09250.963Comparative analysis shows that the calculation results of the revised empirical formula are more accurate. Finally, the empirical formula for the permeability parameters of the standard sand is obtained:(6)I=Ca∪b∪2,where C=0.18⋅d50−0.3⋅CC⋅CU0.8⋅CC⋅CU1.8 and a′=40.95⋅d50−2⋅1−n2⋅n−3⋅CC−0.5⋅CU−2.


## 3.1. Comparison of Seepage Experiments with Different Porosities in Different Particle Size Intervals

Conduct penetration tests on standard sands with different porosities in the five groups of single particle size intervals of 0.1 to 0.25 mm, 0.25 to 0.5 mm, 0.5 to 1 mm, 1 to 2 mm, and 2 to 5 mm. The relationship between the standard sand seepage velocityU and hydraulic gradient I in a single particle size range is shown in Figure 4.Figure 4
U−I relationship diagram under different porosity in the 0.1–0.25 mm particle size range.It can be seen from Figure4 that under the same hydraulic slope, as the porosity decreases, the seepage velocity in the sample gradually slows down. Analyzing the range of coefficients a and b in the simplified seepage formula, it can be seen that the value of b is one order of magnitude higher than the value of a, and the values of a and b increase with the decrease of porosity.


### 3.1.1. Relationship between Drag Coefficient (a) and Porosity

In the permeability test, the control variablesCC, CU, and d50 are fixed values, and the relationship between the drag coefficient and the porosity in each particle size range is explored. Refer to Ergun's formula and initially draw up α∝λ11−nλ2nλ3. As n ∈ (0, 10), according to the boundary conditions: When n ⟶ 0, a ⟶ +∞, we have λ3<0 and λ1>0; when n ⟶ 1, a ⟶ 0, we have λ2>0.By determining the value ranges of the parametersλ1, λ2, and λ3 and defining the type of the fitting function, the relationship between the drag force coefficient and the porosity in each particle size interval is shown in Figure 5.Figure 5
The relationship between the drag coefficienta and the porosity n of each particle size class: (a) single particle size 0.1–0.25 mm, (b) single particle size 0.25–0.5 mm, (c) single particle size 0.5–1 mm, (d) single particle size 1–2 mm, and (e) single particle size 2–5 mm.
(a)(b)(c)(d)(e)It can be seen from Figure5 that the drag coefficient a is positively correlated with 1−n2n−3. The correlation between a and porosity n in each single particle size range is as follows: 0.84252, 0.80163, 0.95891, 0.85015, and 0.85994. The degree of correlation is above 0.8, with a good linear correlation. In the same particle size class, the increase of the dry density of the sample leads to the decrease of the internal porosity of the sample, which is manifested by the decrease of the seepage path or the weakening of the hydraulic path of the water cross section, resulting in the decrease of the overall permeability of the sample and the increase of the drag force coefficient.


### 3.1.2. The Relationship between Drag Coefficient (b) and Porosity

Compared with the aforementioned methods, the relationship between drag force coefficient and porosity was analyzed. Refer to Ergun's formula and initially draw upa∝λ11−nλ2nλ3. Since n ∈ (0, 1), according to the boundary conditions, when n ⟶ 0 and b ⟶ +∞, we have λ3<0 and λ1>0, and when n ⟶ 1 and b ⟶ 0, we have. λ2>0.Define the type of fitting function by determining the value range of parametersλ1, λ2, and λ3. The correlation between drag coefficient b and porosity n in different particle size ranges is shown in Figure 6.Figure 6
The relationship between drag coefficientb and porosity n in each single particle size interval: (a) single particle size 0.1–0.25 mm, (b) single particle size 0.25–0.5 mm, (c) single particle size 0.5–1 mm, (d) single particle size 1-2 mm, and (e) single particle size 2–5 mm.
(a)(b)(c)(d)(e)From Figures6(a) and 6(b), it can be seen that the drag coefficient b and 1−n2n−3 are positively correlated, and the correlation is very high. In Figure 6(c), the relationship between b and n is completely inconsistent with 1−n2n−3. Figures 6(d) and 6(e) show a positive correlation between fit b and 1−n2n−3, but it is not very relevant. In the same particle size class, the decrease of the dry density of the sample leads to an increase in the internal porosity of the sample, which is manifested as an increase in the seepage path of the cross section of the water, which leads to an increase in the overall permeability of the sample and a decrease in the drag coefficient b value.


## 3.1.1. Relationship between Drag Coefficient (a) and Porosity

In the permeability test, the control variablesCC, CU, and d50 are fixed values, and the relationship between the drag coefficient and the porosity in each particle size range is explored. Refer to Ergun's formula and initially draw up α∝λ11−nλ2nλ3. As n ∈ (0, 10), according to the boundary conditions: When n ⟶ 0, a ⟶ +∞, we have λ3<0 and λ1>0; when n ⟶ 1, a ⟶ 0, we have λ2>0.By determining the value ranges of the parametersλ1, λ2, and λ3 and defining the type of the fitting function, the relationship between the drag force coefficient and the porosity in each particle size interval is shown in Figure 5.Figure 5
The relationship between the drag coefficienta and the porosity n of each particle size class: (a) single particle size 0.1–0.25 mm, (b) single particle size 0.25–0.5 mm, (c) single particle size 0.5–1 mm, (d) single particle size 1–2 mm, and (e) single particle size 2–5 mm.
(a)(b)(c)(d)(e)It can be seen from Figure5 that the drag coefficient a is positively correlated with 1−n2n−3. The correlation between a and porosity n in each single particle size range is as follows: 0.84252, 0.80163, 0.95891, 0.85015, and 0.85994. The degree of correlation is above 0.8, with a good linear correlation. In the same particle size class, the increase of the dry density of the sample leads to the decrease of the internal porosity of the sample, which is manifested by the decrease of the seepage path or the weakening of the hydraulic path of the water cross section, resulting in the decrease of the overall permeability of the sample and the increase of the drag force coefficient.


## 3.1.2. The Relationship between Drag Coefficient (b) and Porosity

Compared with the aforementioned methods, the relationship between drag force coefficient and porosity was analyzed. Refer to Ergun's formula and initially draw upa∝λ11−nλ2nλ3. Since n ∈ (0, 1), according to the boundary conditions, when n ⟶ 0 and b ⟶ +∞, we have λ3<0 and λ1>0, and when n ⟶ 1 and b ⟶ 0, we have. λ2>0.Define the type of fitting function by determining the value range of parametersλ1, λ2, and λ3. The correlation between drag coefficient b and porosity n in different particle size ranges is shown in Figure 6.Figure 6
The relationship between drag coefficientb and porosity n in each single particle size interval: (a) single particle size 0.1–0.25 mm, (b) single particle size 0.25–0.5 mm, (c) single particle size 0.5–1 mm, (d) single particle size 1-2 mm, and (e) single particle size 2–5 mm.
(a)(b)(c)(d)(e)From Figures6(a) and 6(b), it can be seen that the drag coefficient b and 1−n2n−3 are positively correlated, and the correlation is very high. In Figure 6(c), the relationship between b and n is completely inconsistent with 1−n2n−3. Figures 6(d) and 6(e) show a positive correlation between fit b and 1−n2n−3, but it is not very relevant. In the same particle size class, the decrease of the dry density of the sample leads to an increase in the internal porosity of the sample, which is manifested as an increase in the seepage path of the cross section of the water, which leads to an increase in the overall permeability of the sample and a decrease in the drag coefficient b value.


## 3.2. Comparison of Seepage Experiments in Different Particle Size Intervals under the Same Porosity

When the porosity is the same, the particle size of the sample accumulation is the only factor that affects its permeability. Multiple sets of samples are tested separately for seepage, and the test results are fitted and analyzed. The correlation between the drag coefficientsa, b and the average particle size d50 is shown in Figure 7.Figure 7
The relationship between drag coefficienta, b and average particle size d50: (a) the correlation between a and d50 and (b) the correlation between b and d50.
(a)(b)It can be seen from Figure7 that the overall linear regression correlation between drag coefficient a and d50−2 reaches 0.913, the overall linear regression correlation between drag coefficient b and d50−2 reaches 0.861. Both have a good linear correlation. When the porosity is constant, as the average particle size d50 of the sample increases, the more pores formed between the particles, the larger the pore volume, and the large internal pores will not be filled by small particles. It shows that the water seepage channel is more open, the seepage resistance of the water flow is weakened, and it shows better permeability.


## 3.3. Seepage Experiment of Mixed Particle Size under Different Gradations

The particle size of the soil has a great influence on its permeability, the inhomogeneity coefficientCU indicates the degree of unevenness of the particles, and the curvature coefficient CC indicates the degree of continuity of the soil particles. In order to explore the influence of the inhomogeneity coefficient CU and the curvature coefficient CC on the permeability of the standard sand, seepage tests were carried out and the test results were compared and analyzed.


### 3.3.1. Effect of Uneven Coefficient on Drag Coefficient

In order to explore the influence of theCU on the permeability of the standard sand, the value of the CC of the sample is all 1. The CU takes the values 5, 5.25, 5.5, 6, and 6.25 in sequence. The preloaded sample method is used to estimate the mass of the required standard sand samples of different grades to be about 3.5 kg. Control five groups of samples to have the same dry density (1.819 g/cm3). The relationship between the drag coefficient a, b and the uneven coefficient in the five groups of different gradation tests is shown in Figure 8.Figure 8
The relationship between drag coefficienta, b and uneven coefficient: (a) the relationship between a and CU and (b) the relationship between b and CU.
(a)(b)It can be seen from Figure8 that the drag coefficients a, b and CU−2 are all positively correlated. The correlation degree distribution is 0.98802 and 0.88682. When the curvature coefficient is a fixed value and the inhomogeneity coefficient increases, the permeability coefficient of the sample also increases. Control the average particle size to 1 mm and the effective particle size to 0.25 mm, from the calculation formula of the grading parameter, it can be known that when the CU is 5.25, the corresponding value of d60 is greater than the value of d60 when the CU is 5. That is, the proportion of coarse particles in the sample is greater than that of fine particles. And so on, when CU is 6.25, the content of coarse particles is the largest. The arrangement and combination of coarse particles will form larger voids, while the content of fine particles is reduced, and these voids cannot be completely filled, resulting in a more developed internal hydraulic path, making it easier for water to pass through, that is, better permeability.


### 3.3.2. Effect of Curvature Coefficient on Drag Coefficient

In order to study the influence ofCC on the permeability of standard sand, the value of CU is all 6, and the value of CC is 0.50, 0.75, 1.00, 1.25, and 1.50. Control five groups of samples to have the same dry density (1.819 g/cm3). The relationship between the drag coefficient a, b and the curvature coefficient in the five groups of different gradation tests is shown in Figure 9.Figure 9
The relationship between drag coefficienta, b and curvature coefficient: (a) the relationship between a and CC and (b) the relationship between b and CC.
(a)(b)It can be seen from Figure9 that the drag coefficients a, b and CC−0.5 are all positively correlated. The correlation degree distribution is 0.82742 and 0.80447. With the increase of CC, the content in the 0.5 to 1 mm particle size range increases, and the content in the 0.25 to 0.5 mm particle size range decreases accordingly. As a result, the pores formed by larger particles cannot be fully filled by fine particles, resulting in the easier flow of water inside the sample, better permeability, and the drag coefficient a and b values are getting smaller and smaller.


## 3.3.1. Effect of Uneven Coefficient on Drag Coefficient

In order to explore the influence of theCU on the permeability of the standard sand, the value of the CC of the sample is all 1. The CU takes the values 5, 5.25, 5.5, 6, and 6.25 in sequence. The preloaded sample method is used to estimate the mass of the required standard sand samples of different grades to be about 3.5 kg. Control five groups of samples to have the same dry density (1.819 g/cm3). The relationship between the drag coefficient a, b and the uneven coefficient in the five groups of different gradation tests is shown in Figure 8.Figure 8
The relationship between drag coefficienta, b and uneven coefficient: (a) the relationship between a and CU and (b) the relationship between b and CU.
(a)(b)It can be seen from Figure8 that the drag coefficients a, b and CU−2 are all positively correlated. The correlation degree distribution is 0.98802 and 0.88682. When the curvature coefficient is a fixed value and the inhomogeneity coefficient increases, the permeability coefficient of the sample also increases. Control the average particle size to 1 mm and the effective particle size to 0.25 mm, from the calculation formula of the grading parameter, it can be known that when the CU is 5.25, the corresponding value of d60 is greater than the value of d60 when the CU is 5. That is, the proportion of coarse particles in the sample is greater than that of fine particles. And so on, when CU is 6.25, the content of coarse particles is the largest. The arrangement and combination of coarse particles will form larger voids, while the content of fine particles is reduced, and these voids cannot be completely filled, resulting in a more developed internal hydraulic path, making it easier for water to pass through, that is, better permeability.


## 3.3.2. Effect of Curvature Coefficient on Drag Coefficient

In order to study the influence ofCC on the permeability of standard sand, the value of CU is all 6, and the value of CC is 0.50, 0.75, 1.00, 1.25, and 1.50. Control five groups of samples to have the same dry density (1.819 g/cm3). The relationship between the drag coefficient a, b and the curvature coefficient in the five groups of different gradation tests is shown in Figure 9.Figure 9
The relationship between drag coefficienta, b and curvature coefficient: (a) the relationship between a and CC and (b) the relationship between b and CC.
(a)(b)It can be seen from Figure9 that the drag coefficients a, b and CC−0.5 are all positively correlated. The correlation degree distribution is 0.82742 and 0.80447. With the increase of CC, the content in the 0.5 to 1 mm particle size range increases, and the content in the 0.25 to 0.5 mm particle size range decreases accordingly. As a result, the pores formed by larger particles cannot be fully filled by fine particles, resulting in the easier flow of water inside the sample, better permeability, and the drag coefficient a and b values are getting smaller and smaller.


## 3.4. Empirical Formula Fitting

### 3.4.1. Preliminary Empirical Formula Fitting of Drag Coefficient

According to the experimental data, the correlation between the average particle size, porosity, curvature coefficient and uneven coefficient, and the drag coefficient is studied by controlling a single variable and analyzing the boundary conditions. The values of drag force coefficientsa and b were fitted by empirical formula with four parameters, d50, n, CU, and CC, respectively, such as formulas (1) and (2). The fitting results of the empirical formulas of the drag coefficients a and b are shown in Figure 10.(1)a′=40.95⋅d50−2⋅1−n2⋅n−3⋅CC−0.5⋅CU−2,(2)b′=e6.7⋅a0.06.Figure 10
The preliminary empirical formula fitting of drag coefficienta.The results show that the correlation between test dataaTest and fitting formula a′ can reach 0.75. a′ and aTest are highly correlated. However, it can be seen that part of the data is more discrete when the test data are about aTest1000. This part of the data mainly comes from the gradation test. It shows that compared with the two, the single particle size test data are more reliable. The main reason is that the composition of the gradation test particles is more complex, which leads to a large difference in the pore structure. Even if the same group of samples is affected by the preloading and postloading, the test results are also very different.It can be seen from the aforementioned comparative analysis that the correlation between theb value and the d50, n, CU, and CC parameters is weak. Therefore, you can choose to explore the correlation between b value and a value, and the fitting result is shown in Figure 11.Figure 11
The relationship between drag coefficienta and b.By comparing the drag coefficienta value representing the coefficient of the U−I linear segment and the drag force coefficient b representing the coefficient of the U−I nonlinear segment, and the sample data bTest and aTest, it is found that the drag force coefficient b has a relatively stable magnitude and a higher drag coefficient a is an order of magnitude. The correlation between the two is 0.85, so formula (2) can be used as a preliminary empirical formula for drag coefficient b.


### 3.4.2. Comparison of Empirical Formula Correlation

Use goodness of fit (R2) to compare the fit of different empirical formulas to standard sand permeability data, the formula of R2 is as follows:(3)R2=1−∑ITest−IEmpirical−formula2∑ITest−I¯Test2.Among them,ITest is the I value obtained from the test data, I¯Test is the average value of the test data, IEmpirical−formula is the I value calculated by the empirical formula. The preliminary empirical formula calculation value I′, the Ergun formula calculation value, the Van Gent formula calculation value, the Liu formula calculation value, and the correlation between the Jingsui Wu formula calculation value, and the experimental data ITest are listed in Table 5.Table 5
Comparison of fitting results between common seepage formulas and standard sand seepage experiment data.
Particle size range (mm)NCUCCCorrelation comparison (1−SSres/SStot)The formula of this paperI′ErgunVan GentLiuJingsui WuThe formula of this paperI″0.1–0.250.381110.943−2.047−9.1060.583−1.7150.9430.455110.982−3.361−7.231−0.056−2.5440.9820.449110.978−2.028−5.3980.320−1.4780.9780.25–0.50.381110.404−2.9690.479−0.827−1.9420.0530.381110.219−2.5700.799−0.724−1.9050.0050.402110.590−1.9210.585−0.368−1.1400.3330.5–10.36111−4.672−1.380−30.4100.6940.0900.1890.37411−0.157−1.677−10.8530.7370.8970.9660.3811−2.131−2.126−18.7720.8250.7830.6291–20.348110.030−2.112−6.4350.595−0.1950.8940.361110.154−1.824−3.9760.5290.0630.9020.377110.334−2.083−3.6300.4840.5560.8382–50.34411−0.213−2.158−5.0670.551−2.0590.8530.35311−0.011−2.137−4.5560.527−1.1360.8200.38111−0.134−2.335−5.3940.509−0.1400.8190.1–50.31460.75−1.907−2.0720.536−0.2400.1870.9290.3145.51−2.093−2.2810.490−0.3210.1810.9630.31461−1.573−1.7040.323−0.0160.3770.9090.31461−2.169−2.3030.838−0.530−0.0660.9520.3146.251−2.179−2.2890.887−0.585−0.1940.951It can be seen from Table 5 that the preliminary fitting formula is only suitable for solving the permeability parameters of standard sand with a single particle size of 0.1 to 0.25 mm. The formula of Van Gent is suitable for the solution of standard sand permeability parameters in grading experiments. The formula of Liu is suitable for the solution of permeability parameters of standard sand in the interval of 0.5 and 1 mm. In general, the aforementioned five formulas are lacking in the process of solving the standard sand permeability parameters, and the correlation is low.


### 3.4.3. Improve Empirical Formula

There is a certain deviation between the empirical formulaI′ and the standard sand drag coefficient in all particle sizes. However, the analysis of the empirical formula I′ and the experimental data show that there is a linear correlation between the two. The comparison is shown in Figure 12.Figure 12
Comparison ofU−I relationship between fitting formula I′ and experimental data: (a) the relationship between I′ and U and (b) the relationship between I′ and I.
(a)(b)It can be seen from Figure12(a) that there is a certain correlation between the data points obtained in the interval of 0.1 to 0.25 mm for a single particle size and the fitting formula I′. In order to make the empirical formula reflect the experimental data more accurately, a correction factor is introduced to improve the empirical formula. That is, add correction C. Assuming ITestC⋅I′, exploring the correlation coefficient C between the two is shown in Figure 12(b). The improved empirical formula is in good agreement with the experimental data points, and the correlation is high, which reflects that the improved empirical formula can be closer to the experimental results. The correlation coefficient between the single particle size interval and the gradation experimental data is not a fixed value, and the closer the coefficient is to 1, the better the empirical formula fits. The relationship between the coefficient C and each parameter is shown in Figure 13.Figure 13
The correlation between the coefficientC and various parameters.It can be seen from the figure that the coefficientC is related to d50, CC, and CU, and the correlation of the fitting formula is as high as 0.96, so the empirical formula for the coefficient C is as follows:(4)C=0.18⋅d500.3⋅CC⋅CU0.8⋅CC+CU1.8.The improved empirical formula is as follows:(5)I′C⋅I′.The correlation between the modified empirical formulaI″ and the empirical formula I′ is shown in Table 6.Table 6
Comparison of the fitting accuracy between the modified empirical formulaI″ and the empirical formula I′
Particle size range (mm)Dry density (g/cm3)CUCCFitting accuracy comparison (1−SSres/SStot)I′I″0.1–0.251.4461.001.000.98180.98181.4611.001.000.97810.97811.6401.001.000.94340.94340.25–0.51.6401.001.000.40360.0531.5851.001.000.58950.3331.5991.001.000.21940.0050.5–11.6941.001.00−4.67150.1891.6601.001.00−0.15730.9661.6401.001.00−2.13100.6291–21.6931.001.000.15360.9021.7291.001.000.02960.8941.7671.001.00−0.01110.9042–51.6401.001.00−0.13400.8191.7151.001.00−0.01130.8201.7381.001.00−0.21290.8530.1–51.81906.000.75−1.90650.9291.81906.001.00−1.57270.9091.81906.001.00−2.16850.9521.81906.251.00−2.17860.9511.81905.501.00−2.09250.963Comparative analysis shows that the calculation results of the revised empirical formula are more accurate. Finally, the empirical formula for the permeability parameters of the standard sand is obtained:(6)I=Ca∪b∪2,where C=0.18⋅d50−0.3⋅CC⋅CU0.8⋅CC⋅CU1.8 and a′=40.95⋅d50−2⋅1−n2⋅n−3⋅CC−0.5⋅CU−2.


## 3.4.1. Preliminary Empirical Formula Fitting of Drag Coefficient

According to the experimental data, the correlation between the average particle size, porosity, curvature coefficient and uneven coefficient, and the drag coefficient is studied by controlling a single variable and analyzing the boundary conditions. The values of drag force coefficientsa and b were fitted by empirical formula with four parameters, d50, n, CU, and CC, respectively, such as formulas (1) and (2). The fitting results of the empirical formulas of the drag coefficients a and b are shown in Figure 10.(1)a′=40.95⋅d50−2⋅1−n2⋅n−3⋅CC−0.5⋅CU−2,(2)b′=e6.7⋅a0.06.Figure 10
The preliminary empirical formula fitting of drag coefficienta.The results show that the correlation between test dataaTest and fitting formula a′ can reach 0.75. a′ and aTest are highly correlated. However, it can be seen that part of the data is more discrete when the test data are about aTest1000. This part of the data mainly comes from the gradation test. It shows that compared with the two, the single particle size test data are more reliable. The main reason is that the composition of the gradation test particles is more complex, which leads to a large difference in the pore structure. Even if the same group of samples is affected by the preloading and postloading, the test results are also very different.It can be seen from the aforementioned comparative analysis that the correlation between theb value and the d50, n, CU, and CC parameters is weak. Therefore, you can choose to explore the correlation between b value and a value, and the fitting result is shown in Figure 11.Figure 11
The relationship between drag coefficienta and b.By comparing the drag coefficienta value representing the coefficient of the U−I linear segment and the drag force coefficient b representing the coefficient of the U−I nonlinear segment, and the sample data bTest and aTest, it is found that the drag force coefficient b has a relatively stable magnitude and a higher drag coefficient a is an order of magnitude. The correlation between the two is 0.85, so formula (2) can be used as a preliminary empirical formula for drag coefficient b.


## 3.4.2. Comparison of Empirical Formula Correlation

Use goodness of fit (R2) to compare the fit of different empirical formulas to standard sand permeability data, the formula of R2 is as follows:(3)R2=1−∑ITest−IEmpirical−formula2∑ITest−I¯Test2.Among them,ITest is the I value obtained from the test data, I¯Test is the average value of the test data, IEmpirical−formula is the I value calculated by the empirical formula. The preliminary empirical formula calculation value I′, the Ergun formula calculation value, the Van Gent formula calculation value, the Liu formula calculation value, and the correlation between the Jingsui Wu formula calculation value, and the experimental data ITest are listed in Table 5.Table 5
Comparison of fitting results between common seepage formulas and standard sand seepage experiment data.
Particle size range (mm)NCUCCCorrelation comparison (1−SSres/SStot)The formula of this paperI′ErgunVan GentLiuJingsui WuThe formula of this paperI″0.1–0.250.381110.943−2.047−9.1060.583−1.7150.9430.455110.982−3.361−7.231−0.056−2.5440.9820.449110.978−2.028−5.3980.320−1.4780.9780.25–0.50.381110.404−2.9690.479−0.827−1.9420.0530.381110.219−2.5700.799−0.724−1.9050.0050.402110.590−1.9210.585−0.368−1.1400.3330.5–10.36111−4.672−1.380−30.4100.6940.0900.1890.37411−0.157−1.677−10.8530.7370.8970.9660.3811−2.131−2.126−18.7720.8250.7830.6291–20.348110.030−2.112−6.4350.595−0.1950.8940.361110.154−1.824−3.9760.5290.0630.9020.377110.334−2.083−3.6300.4840.5560.8382–50.34411−0.213−2.158−5.0670.551−2.0590.8530.35311−0.011−2.137−4.5560.527−1.1360.8200.38111−0.134−2.335−5.3940.509−0.1400.8190.1–50.31460.75−1.907−2.0720.536−0.2400.1870.9290.3145.51−2.093−2.2810.490−0.3210.1810.9630.31461−1.573−1.7040.323−0.0160.3770.9090.31461−2.169−2.3030.838−0.530−0.0660.9520.3146.251−2.179−2.2890.887−0.585−0.1940.951It can be seen from Table 5 that the preliminary fitting formula is only suitable for solving the permeability parameters of standard sand with a single particle size of 0.1 to 0.25 mm. The formula of Van Gent is suitable for the solution of standard sand permeability parameters in grading experiments. The formula of Liu is suitable for the solution of permeability parameters of standard sand in the interval of 0.5 and 1 mm. In general, the aforementioned five formulas are lacking in the process of solving the standard sand permeability parameters, and the correlation is low.


## 3.4.3. Improve Empirical Formula

There is a certain deviation between the empirical formulaI′ and the standard sand drag coefficient in all particle sizes. However, the analysis of the empirical formula I′ and the experimental data show that there is a linear correlation between the two. The comparison is shown in Figure 12.Figure 12
Comparison ofU−I relationship between fitting formula I′ and experimental data: (a) the relationship between I′ and U and (b) the relationship between I′ and I.
(a)(b)It can be seen from Figure12(a) that there is a certain correlation between the data points obtained in the interval of 0.1 to 0.25 mm for a single particle size and the fitting formula I′. In order to make the empirical formula reflect the experimental data more accurately, a correction factor is introduced to improve the empirical formula. That is, add correction C. Assuming ITestC⋅I′, exploring the correlation coefficient C between the two is shown in Figure 12(b). The improved empirical formula is in good agreement with the experimental data points, and the correlation is high, which reflects that the improved empirical formula can be closer to the experimental results. The correlation coefficient between the single particle size interval and the gradation experimental data is not a fixed value, and the closer the coefficient is to 1, the better the empirical formula fits. The relationship between the coefficient C and each parameter is shown in Figure 13.Figure 13
The correlation between the coefficientC and various parameters.It can be seen from the figure that the coefficientC is related to d50, CC, and CU, and the correlation of the fitting formula is as high as 0.96, so the empirical formula for the coefficient C is as follows:(4)C=0.18⋅d500.3⋅CC⋅CU0.8⋅CC+CU1.8.The improved empirical formula is as follows:(5)I′C⋅I′.The correlation between the modified empirical formulaI″ and the empirical formula I′ is shown in Table 6.Table 6
Comparison of the fitting accuracy between the modified empirical formulaI″ and the empirical formula I′
Particle size range (mm)Dry density (g/cm3)CUCCFitting accuracy comparison (1−SSres/SStot)I′I″0.1–0.251.4461.001.000.98180.98181.4611.001.000.97810.97811.6401.001.000.94340.94340.25–0.51.6401.001.000.40360.0531.5851.001.000.58950.3331.5991.001.000.21940.0050.5–11.6941.001.00−4.67150.1891.6601.001.00−0.15730.9661.6401.001.00−2.13100.6291–21.6931.001.000.15360.9021.7291.001.000.02960.8941.7671.001.00−0.01110.9042–51.6401.001.00−0.13400.8191.7151.001.00−0.01130.8201.7381.001.00−0.21290.8530.1–51.81906.000.75−1.90650.9291.81906.001.00−1.57270.9091.81906.001.00−2.16850.9521.81906.251.00−2.17860.9511.81905.501.00−2.09250.963Comparative analysis shows that the calculation results of the revised empirical formula are more accurate. Finally, the empirical formula for the permeability parameters of the standard sand is obtained:(6)I=Ca∪b∪2,where C=0.18⋅d50−0.3⋅CC⋅CU0.8⋅CC⋅CU1.8 and a′=40.95⋅d50−2⋅1−n2⋅n−3⋅CC−0.5⋅CU−2.


## 4. Conclusion

Through the single particle size and different grade standard sand particle seepage test, analyze the seepage law of different standard sand particles in the Darcy-non-Darcy flow zone. The relationship between the parametersa and b in the seepage equation and the porosity, average particle size, inhomogeneity coefficient and curvature coefficient and their influence degree are studied, and the following research results have been obtained.(1)
Using electron microscope scanning and image processing technology, the data of the equal-area circle diameter of standard sand particles in the range of 1 to 2 mm and 2 to 5 mm in single particle size is measured, and the equal-area circle diameter is analyzed to show a right skewed distribution, and the equal-area circle is drawn. The diameter gradation curve to obtain the average particle size of standard sand is 1.599 mm and 2.057 mm, respectively.(2)
The results of the constant head seepage test show that porosity, average particle size, uneven coefficient, and curvature coefficient are the main factors affecting the drag coefficienta and b in the seepage equation. The drag coefficient a is related to the average particle size, porosity, and uneven coefficient. Both have a negative correlation with the curvature coefficient and have a high correlation. It is found that the permeability of standard sand increases with the increase of average particle size and porosity. When the uneven coefficient of the sample increases, the greater the curvature coefficient, The content of coarse particles increases while the content of fine particles decreases. The arrangement and combination of coarse particles will form larger voids and cannot be effectively filled. The hydraulic path inside the soil is wider, and the soil exhibits better permeability.(3)
The drag coefficientb has a negative correlation with the uneven coefficient and the curvature coefficient, and the correlation is high, but the correlation between the b value and the average particle size and porosity is relatively poor. By analyzing the relationship between the drag coefficient a and the drag coefficient b. It is concluded that the two have a good functional relationship, and the correlation between the two is very high. The empirical formula of standard sand is obtained by fitting, and the empirical formula is improved by adding correction items and its rationality is verified, and finally, the empirical formula for predicting the drag force coefficient of standard sand is determined. The empirical formula can be applied to different Darcy-non-Darcy flow zones. Prediction of the relationship between flow velocity and hydraulic gradient under the condition of particle size standard sand.



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*Source: 8104842-2022-03-08.xml*