Experimental Study onMeasurement of Drag Coefficient of Sandy Soil Nonlinear Vadose

*e seepage characteristics of sand affect its strength, mechanical deformation characteristics, and find safety and stability. *e 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. *rough 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. *e relationship between parameters a 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.


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. e 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]. aisiam 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. e 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.
rough 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.

Physical Properties of Test Materials.
e 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. e original grading curve is shown in Figure 1. e 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.

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 Table 2. 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. e permeability of soil is affected by particle distribution. e permeability was studied by adjusting the mass percentage of each particle-size interval to prepare different grades. e quality of soil particle gradation can be judged by the two parameters of nonuniformity coefficient (C U ) and curvature coefficient (C C ). When considering the influence of grading changes, there are two cases: Keep the curvature coefficient (C C ) as 1 and change the value of the nonuniformity coefficient (C U ), respectively: 5, 5.25, 5.5, 6, and 6.25. Keep the nonuniformity coefficient (C U ) as 6 and change the value of the curvature coefficient (C C ), respectively: 0.5, 0.75, 1, 1.25, and 1.43. Take the average particle size d 50 as 1 mm, the effective particle size d 10 as 0.25 mm, d 60 and d 30 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. e cumulative curve of particle size of each group can be drawn, as shown in Figures 2 and 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. e 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. e 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.

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. e relationship between the standard sand seepage velocity U and hydraulic gradient I in a single particle size range is shown in Figure 4.
It can be seen from Figure 4 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.

Relationship between Drag Coefficient (a) and Porosity.
In the permeability test, the control variables C C , C U , and d 50 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 ∝ λ 1 (1 − n) λ 2 n λ 3 . As n ∈ (0, 10), according to the boundary Table 4: Mass percentage of each particle size interval of samples with different curvature coefficients.    conditions: When n ⟶ 0, a ⟶ +∞, we have λ 3 < 0 and λ 1 > 0; when n ⟶ 1, a ⟶ 0, we have λ 2 > 0.
By determining the value ranges of the parameters λ 1 , λ 2 , and λ 3 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.
It can be seen from Figure 5 that the drag coefficient a is positively correlated with (1 − n) 2 n − 3 . e 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. e 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.
Define the type of fitting function by determining the value range of parameters λ 1 , λ 2 , and λ 3 . e correlation between drag coefficient b and porosity n in different particle size ranges is shown in Figure 6.
From Figures 6(a) and 6(b), it can be seen that the drag coefficient b and (1 − n) 2 n − 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 − n) 2 n − 3 . Figures 6(d) and 6(e) show a positive correlation between fit b and (1 − n) 2 n − 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.

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. e correlation between the drag coefficients a, b and the average particle size d 50 is shown in Figure 7.
It can be seen from Figure 7 that the overall linear regression correlation between drag coefficient a and d − 2 50 reaches 0.913, the overall linear regression correlation between drag coefficient b and d − 2 50 reaches 0.861. Both have a good linear correlation. When the porosity is constant, as the average particle size d 50 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.

Seepage Experiment of Mixed Particle Size under Different
Gradations.
e particle size of the soil has a great influence on its permeability, the inhomogeneity coefficient C U indicates the degree of unevenness of the particles, and the curvature coefficient C C indicates the degree of continuity of the soil particles. In order to explore the influence of the inhomogeneity coefficient C U and the curvature coefficient C C on the permeability of the standard sand, seepage tests were carried out and the test results were compared and analyzed.

Effect of Uneven Coefficient on Drag Coefficient.
In order to explore the influence of the C U on the permeability of the standard sand, the value of the C C of the sample is all 1.
e C U takes the values 5, 5.25, 5.5, 6, and 6.25 in sequence. e 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/cm 3 ). e relationship between the drag coefficient a, b and the uneven coefficient in the five groups of different gradation tests is shown in Figure 8.
It can be seen from Figure 8 Figure 5: e relationship between the drag coefficient a 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.  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 C U is 5.25, the corresponding value of d 60 is greater than the value of d 60 when the C U is 5. at is, the proportion of coarse particles in the sample is greater than that of fine particles. And so on, when C U is 6.25, the content of coarse particles is the largest. e 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.

Effect of Curvature Coefficient on Drag Coefficient.
In order to study the influence of C C on the permeability of standard sand, the value of C U is all 6, and the value of C C 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/cm 3 ). e relationship between the drag coefficient a, b and the curvature coefficient in the five groups of different gradation tests is shown in Figure 9.
It can be seen from Figure 9 that the drag coefficients a, b and C − 0.5 C are all positively correlated. e correlation degree distribution is 0.82742 and 0.80447. With the increase of C C , 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  Figure 6: e relationship between drag coefficient b 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. the easier flow of water inside the sample, better permeability, and the drag coefficient a and b values are getting smaller and smaller.

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. e values of drag force coefficients a and b were fitted by empirical formula with four parameters, d 50 , n, C U , and C C , respectively, such as formulas (1) and (2). e fitting results of the empirical formulas of the drag coefficients a and b are shown in Figure 10.    Advances in Civil Engineering b ′ � e 6.7·a 0.06 . (2) e results show that the correlation between test data a Test and fitting formula a ′ can reach 0.75. a ′ and a Test are highly correlated. However, it can be seen that part of the data is more discrete when the test data are about a Test 1000.
is 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.
e 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 the b value and the d 50 , n, C U , and C C parameters is weak. erefore, you can choose to explore the correlation between b value and a value, and the fitting result is shown in Figure 11.
By comparing the drag coefficient a 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 b Test and a Test , 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.
e correlation between the two is 0.85, so formula (2) can be used as a preliminary empirical formula for drag coefficient b.

Comparison of Empirical Formula Correlation.
Use goodness of fit (R 2 ) to compare the fit of different empirical formulas to standard sand permeability data, the formula of R 2 is as follows: Among them, I Test is the I value obtained from the test data, I Test is the average value of the test data, I Empirical− formula is the I value calculated by the empirical formula. e 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 I Test are listed in Table 5. It 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. e formula of Van Gent is suitable for the solution of standard sand permeability parameters in grading experiments. e 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.

Improve Empirical Formula.
ere is a certain deviation between the empirical formula I ′ 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. e comparison is shown in Figure 12.  It can be seen from Figure 12(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.
at is, add correction C. Assuming I Test C · I ′ , exploring the correlation coefficient C between the two is shown in Figure 12(b). e 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. e 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. e relationship between the coefficient C and each parameter is shown in Figure 13.  It can be seen from the figure that the coefficient C is related to d 50 , C C , and C U , and the correlation of the fitting formula is as high as 0.96, so the empirical formula for the coefficient C is as follows: e improved empirical formula is as follows: e correlation between the modified empirical formula I ″ and the empirical formula I ′ is shown in Table 6.
Comparative 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: where

Conclusion
rough 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. e relationship between the parameters a 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. e diameter gradation curve to obtain the average particle size of standard sand is 1.599 mm and 2.057 mm, respectively. (2) e 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 coefficient a and b in the seepage equation. e 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, e content of coarse particles increases while the content of fine particles decreases. e arrangement and combination of coarse particles will form larger voids and cannot be effectively filled. e hydraulic path inside the soil is wider, and the soil exhibits better permeability.
(3) e drag coefficient b 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 e 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. e empirical formula can be applied to different Darcynon-Darcy flow zones. Prediction of the relationship between flow velocity and hydraulic gradient under the condition of particle size standard sand.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.