Research on the Critical Value of Sand Permeability Particle Size and Its Permeability Law after Mixing

Abstract

The permeability of sand is an important factor in determining the movement and occurrence of liquids and gases in sand. The current work studied the influence of soil particle size and gradation on permeability by testing the permeability of different sand samples that consist of defined sand particles in certain ratios. The results of a total of 640 sets of experiments were analyzed and compared with the corresponding theoretical calculations. We found that the sand can be divided into four groups based on particle size: high-permeability particles, medium-permeability particles, low-permeability particles and non-permeable particles; and the critical particle size of sand for permeability is 0.050 mm. Permeation will be lost when non-permeable particles account for >75% of the total in the mixture of high-permeability particles and non-permeable particles. Permeation will be very easily lost when medium- or low-permeability particles are mixed with non-permeable particles. The current findings are of importance for assessing the permeability of sand based on particle size.

1. Introduction

The permeability of sand is one of the key factors that determine the movement and occurrence of underground liquids and gases [1]. After Darcy’s law was introduced experimentally based on permeability [2], subsequent studies have shown that many factors affect the permeability of porous media, including sand composition [3,4,5], pore size [6,7,8,9], compaction [10,11,12,13], dry density [14,15,16,17,18,19], etc.

Previous studies demonstrated that grain grade is the most important factor affecting permeability [20,21,22,23,24,25]. Many researchers [26,27,28,29,30,31] have developed empirical formulas to predict permeability from grain size distribution indexes. Among them, the Hazan formula is the most commonly used empirical formula to determine permeability. In addition, Burmister [32] demonstrated experimentally that the permeability of sand declines with increasing relative density. Guo [33] experimentally assessed the influence of coarse-grained content on permeability and correlated the permeability of coarse-grained soil with P5, defined as the proportion of coarse materials > 5 mm in diameter, and argued that permeability depends mainly on the nature of the fine materials when P5 < 30%, on the nature of the coarse material when P5 > 70%, and on the joint impact of fine and coarse material when 30% < P5 < 70%. By addressing the inaccuracies in permeability tests and related assumptions, Chapuis [34] extended the Hazen formula to saturated non-plastic soil based on the experimental results from the permeability tests of clean saturated sand and crushed stone. Cui [35] evaluated the critical hydraulic gradient that triggers piping for soil bodies with different flow patterns and different particle sizes. Yang [36] employed permeability tests and particle gradation analysis to derive, for sandy soil particles and through Darcy’s law, the relationship between the fractal dimension of gradation and the inhomogeneity coefficient. Specifically, poorer gradation leads to smaller inhomogeneity coefficient and larger fractal dimension, thus giving a larger permeability.

There are many experimental studies on grain grade, and empirical formulas for estimating the permeability coefficient of coarse-grained soil have been proposed [5,37,38]. The variables in the empirical formulas are mainly indicators that characterize grain grade, and some of them also include porosity and void ratio [39]. However, when empirical formulas are used to estimate the permeability coefficient of coarse-grained soil, the calculation results of different formulas may vary greatly and not match the measured results [40,41]. The empirical formulas thus have limited scope of application because there are inadequate experimental data and because they do not fully reflect the influence of void ratio or particle size composition.

In engineering practice, the aquifer can be divided simply according to the size of the sand particles; that is, as the particle size of the sand decreases, the permeability goes from high to low to a certain degree [1,42,43,44]. Although many scholars have put forward empirical formulas for particle size and permeability, the results calculated based on the empirical formulas are far from agreeing with the measured results. Previous studies were conducted on natural sandy soil, and the influencing factors only focused on the two variables of pore ratio and characteristic particle size. No one has studied the permeability of soil particles of each particle size, and the permeability of each particle size and its influencing factors are not clear. In addition, there is still a lack of knowledge on the critical value of particle size for permeability. The critical value of particle size for permeability refers to the maximum particle size at which sand does not have permeation in a specific application. The critical value of sand particle size for permeability has different requirements in different applications. This study mainly investigated the critical value of sand particle size for permeability in the field of hydrogeology. Through the seepage test, the critical value of sand soil for permeability was determined. The relationship between particle size and permeability was quantitatively analyzed, which can be used to determine the permeability of rock and soil, understand the water resource characteristics of rock and soil with different particle sizes, identify aquifers and determine their transport capacity for pollutants directly based on the particle size of sand.

2. Methodology

This research mainly studied two topics: the critical value of sand particle size for permeability and the impact of the ratio of particles under the critical value of size on permeability. To determine the critical value of sand particle size for permeability, the natural river sand was divided into 12 different particle sizes below 0.375 mm, and seepage measurement was conducted. A total of 45 sets of data were obtained, and each type of sand permeation was selected. The average value of the permeability is analyzed to obtain the critical value of permeability. Based on the two types of sand classified, a total of 595 sets of seepage experiments were carried out after mixing with other types of sand to investigate the influence of particle size and mass ratio on the permeability of mixed sand under the critical value.

2.1. Instrument

A customized instrument based on an ST-70A permeameter (Shanghai Civil & Road Instrument Co., Ltd., Shanghai, China) was used for the experiments (Figure 1). Because the body of the instrument is made of acrylic, sand sample can be easily loaded into and emptied from the instrument, cleaning is easy, and the instrument is robust against damage. Moreover, the water flow through the sand particles can be directly observed through the transparent acrylic wall to determine the path of water flow and check if water exits from the wall of the instrument.

2.2. Test Materials

2.2.1. Composition of Test Materials

To avoid the influence of mineral salts on the permeability of sand, river sand from the same site was used. X-ray diffraction measurement revealed that the main component of the sample is SiO₂, which does not contain hydrophilic or water-swellable components. Composition analysis of test sand by X-ray diffraction is shown in Figure 2.

2.2.2. Sieving of Sand Particles

Vibrating screens from 20 to 300 mesh (a total of 12 sizes) were used to sieve the river sand according to particle size into segments with different granularity (

Table 1. Material properties of sand samples.

Mesh Size	Particle Granularity (mm)
20~40	0.375~0.750
40~60	0.250~0.375
60~80	0.188~0.250
80~100	0.150~0.188
100~120	0.125~0.150
60~120	0.125~0.250
120~140	0.107~0.125
140~160	0.094~0.107
160~180	0.083~0.094
180~200	0.075~0.083
200~250	0.060~0.075
250~300	0.050~0.060
<300	<0.050

).

2.2.3. Gradation

The sieved particles are divided into four categories: high-permeability particles, (40~60 mesh, 60~120 mesh, 120~140 mesh), medium-permeability particles: (140~160 mesh, 160~180 mesh, 180~200 mesh), low-permeability particles: (200~250 mesh, 250~300 mesh), and non-permeable particles (>300 mesh). The four types of sand samples were mixed in pairs. The first type of sand samples was mixed with the second, third, and fourth types of sand samples with different particle sizes, and then the second type was mixed with the third and fourth types of sand samples. Four types of sand samples were mixed at the ratios of 4:1, 3:1, 2:1, 1:1, 1:2, 1:3, 1:4, and each ratio was tested four times. Final data of 595 groups were obtained.

2.3. Test Method

The impact of sand particle sizes on permeation was studied as follows:

(1)

The sand sample of a single particle size or a defined gradation scheme (the sand samples with different particle sizes are mixed fully and evenly in proportion) was loaded into the instrument, and the cross-sectional area (A) and the permeation path of water (L) were measured;

(2)

Water was slowly charged into the instrument to fully saturate the sand and thoroughly drain the trapped air (i.e., until the position of the head in the piezometric tubes became level with the water surface in the instrument);

(3)

Water was slowly discharged from the instrument at a rate that maintains the steady water level in the piezometric tubes. The hydraulic gradient I was then calculated from the recorded head of the piezometric tubes as follows:

$$\mathrm{I}=\frac{{\mathrm{H}}_1{-\mathrm{H}}_2}{\mathrm{L}}$$

(1)

where I is the hydraulic gradient, H₁ − H₂ (m) is the head loss, and L (m) is the length of seepage path.

(4)

The volume of water (V1) leaving the instrument during step (3) was measured with a graduated cylinder, and the corresponding time was recorded with a stopwatch to determine the flow rate Q;

(5)

The hydraulic conductivity k was calculated as

$$\mathrm{k}=\frac{\mathrm{Q}}{\mathrm{IA}}$$

(2)

where A (m²) is the cross-sectional area.

(6)

The procedures were carried out for samples with different formulations to derive the corresponding permeability k.

3. Results

3.1. Single-Particle-Size Samples

A total of 45 sets of data were obtained for analysis. The permeability of certain sand types was taken as the average of the calculated permeability of the corresponding samples (see

Table 2. Table of particle size distribution and permeability.

Granularity	Mesh Number	Particle Size (mm)	Permeability k (m/d)
High-permeability particles	20~40	0.375~0.750	178.396
	40~60	0.250~0.375	124.003
	60~80	0.188~0.250	78.464
	80~100	0.150~0.188	55.554
	100~120	0.125~0.150	30.344
	120~140	0.107~0.125	18.396
Medium-permeability particles	140~160	0.094~0.107	10.909
	160~180	0.083~0.094	8.966
	180~200	0.075~0.083	8.141
Low-permeability particles	200~250	0.060~0.075	0.361
	250~300	0.050~0.060	0.232
Non-permeable particles	<300	<0.050	0.000

and the corresponding scatter plot in Figure 3). Figure 3 shows that the permeability gradually decreases with particle size. Permeation was extremely slow for the sand sample of <0.050 mm particles: the water level in the column moved only about 10 cm after 8 h. The type of tested sand has very low permeability. In hydrogeology, it can be considered as having no permeability. For convenience, the permeability coefficient was treated as zero in this study. As shown in Table 2, the sand samples can be divided into four groups according to the permeability coefficient: high permeability, medium permeability, low permeability and no permeability. The classification results refer to the classification criteria of ‘strong permeability (8.64~864 m/d)’ and ‘medium permeability (0.0864~8.64 m/d)’ in Appendix F of the Code for Engineering Geological Survey of Water Resources and Hydropower (GB 50487-2008, 2022 Edition), but the classification criteria of ‘weak permeability and below’ are applicable to ‘silt~clay’. Combined with the change boundary points of permeability coefficient and sand particle size in Figure 3, the low permeability and non-permeability range of sand samples are determined. In Figure 3, the relationship between sand sample particle size and permeability coefficient was fitted to three curves. The three curves correspond to the sand groups of high permeability, medium permeability and low permeability. The critical value of particle size based on permeability is 0.050 mm.

3.2. Mixed-Particle-Size Samples

A total of 595 sets of data were obtained for the mixed-particle-size samples (Table 2), from which the permeability was calculated to assess the relationship between particle size and permeability, and in particular the impact of the content of particles below the particle size threshold.

(1)

From Figure 4, we can see that: When high-permeability particles are mixed with medium-permeability particles, the permeability of the mixed sample is always lower than that of the high-permeability particles. When the medium-permeability particles account for >50% of the total, the permeability of the mixed sample is lower than that of the medium-permeability particles.

(2)

From Figure 5, we can see that: When high-permeability particles are mixed with low-permeability particles, the permeability of the mixed sample is always lower than that of the high-permeability particles. When the high-permeability particles account for >50% of the total, the permeability of the mixed sample is higher than that of the low-permeability particles.

(3)

From Figure 6, we can see that: For the mixture of high-permeability particles and non-permeable particles, the mixed sample loses permeability (i.e., k = 0 or k < 0.05 m/d) when the content of non-permeable particles is no less than 80%.

(4)

From Figure 7, we can see that: When the medium-permeability particles are mixed with low-permeability particles, the permeability of the mixed sample is always lower than that of the medium-permeability particles. When the low-permeability particles account for >50% of the total, the permeability of the mixed sample is lower than that of the low-permeability particles.

(5)

From Figure 8, we can see that: The mixed sample loses permeability (i.e., k = 0 or k < 0.05 m/d) entirely when medium-permeability particles are mixed with non-permeable particles.

According to the results, when both particles are larger than the critical value of water conductivity, they still have water conductivity after mixing. The size of permeability after mixing is related to the proportion of different particles, and the larger proportion of particles has a greater decisive effect. The reason for this result is that the pores of the sand sample become smaller after the small particles are mixed into the large-particle-size sand sample, and the effective cross section of the seepage decreases, so the permeability coefficient of the mixed sand sample decreases. Permeability is not lost when high-permeability particles are mixed with medium- or low-permeability particles because the particle size of the sand sample is higher than the critical permeability value, but will be lost in a mixture of >80% non-permeable particles with high-permeability particles. Permeability is easily lost when medium-permeability particles are mixed with non-permeable particles.

4. Discussion

4.1. Theoretical Analysis of Critical Value

Water existing in the pores of sand includes gravity water, bound water, and capillary water. Gravity water can move freely under the action of gravity, whereas bound water cannot due to the adsorption of sand particles. When liquid contacts a solid surface, thermodynamic equilibrium is reached when enough molecules of the liquid accumulate on the solid surface. An “adsorption layer” of water molecules is formed as a result of the adsorption effect. At the equilibrium state, the chemical potential of the adsorption layer (U_s) is equal to the chemical potential of the adsorbate (U_a), and dn moles of adsorbate are transferred to the adsorption layer. Physical adsorption can quickly achieve an equilibrium that is reversible, except when mass transfer is limited in the gas phase or in a porous adsorbent. Reversibility means that an equilibrium exists between adsorption and desorption. In the case of physical adsorption, more than a single molecular layer will be formed on the adsorbent surface. When water is adsorbed on a solid surface, multiple layers that arise from molecular positioning and cooperative adsorption will eventually develop in the three-dimensional space from the two-dimensional water layer. Three liquids with distinct properties exist within immersed stacked solid particles, i.e., (1) free water in the pores of stacking particles, (2) capillary water retained between particles due to capillary force, and (3) adsorbed water on the solid surface due to the interaction between the liquid and the particle surface. The thickness of the layer of adsorbed water mainly depends on the properties of the solid surface and the liquid. It has been reported that in the absence of driving pressure, the thickness of the adsorbed water layer is 0.7 μm [45].

Theoretically, when the pores are less than twice the thickness of adsorbed water, the pores should be filled with bound water, in which case the groundwater becomes largely immobile and the soil essentially is not permeable. Thus, the critical value of water permeation through the pores can be considered as twice the thickness of the bound water layer. Because sand particles in reality are irregular instead of compactly stacked tetrahedrons or cubes, the actual critical value is generally 0.155D = d₂ < d < d₁ = 0.414D. The relationship between pore throat diameter and particle diameter can be expressed as d = kD (0.155 ≤ k ≤ 0.414).

When the thickness of the layer of bound water is W, the minimum diameter (i.e., critical value) of the pore throat between particles is then d = 2 W, and the minimum particle size (i.e., critical value) to allow any permeation is D_l = 2 W/k (0.155 ≤ k ≤ 0.414).

It can then be derived from the prior report of W = 0.7 μm that 3.4 μm ≤ D ≤ 9.1 μm. That is, the maximum particle diameter is 9.1 μm. The measured critical value from the current experiment is 50 μm, which deviates from the theoretical calculation value possibly due to the following reasons:

(1)

In the previous literature, the thickness of the water absorption layer of the glass beads was measured; there are no micropores on the surface of glass beads, which is smaller than the surface. But the geotechnical particles were used in this test, and there was a big difference between the glass beads and the geotechnical particles;

(2)

Aside from bound water, the capillary water also has certain influence on the permeation;

(3)

The materials used in this test are not perfectly spherical and thus deviate from the ideal. The shape of sand has a certain influence on the test results;

(4)

The conclusion of previous studies is based on the dynamic centrifugal test, which will throw out most of the retained water in the pores, resulting in a small theoretical value;

(5)

Combined with the close relationship between the thickness of water and the physical properties of the surface of the object, the seepage device used in this experiment may have boundary effects, resulting in larger test results.

4.2. Comparison between Experimental and Calculated Permeability

Zhang [46] calculated the permeability according to the following:

$$\mathrm{K}={\mathrm{r}}^2\mathsf{\gamma }/8\mathsf{\epsilon }$$

(3)

where K is the permeability (m/d), γ is the gravity density of water (10⁴ N/m³), and ε is the viscosity coefficient of the liquid (Pas).

When the diameter of the pore throat has 0.155D = d₂ < d < d₁ = 0.414D, Equation (3) can be transformed into the following:

$$\mathrm{K}=(0.00075\text{~}0.0054){\mathrm{D}}^2\mathsf{\gamma }/\mathsf{\epsilon }$$

(4)

The theoretical permeability calculated with Equation (4) is compared with the experimentally derived permeability for all particle sizes (

Table 3. Comparison table between the calculated permeability and the test result of each particle- size sand sample.

Particle Size (mm)	Calculated Permeability (m/d)	Experimental Permeability (m/d)
0.375~0.750	50.625~1445.625	167.212~191.963
0.250~0.375	22.5~361.406	116.119~140.076
0.188~0.250	12.724~160.625	68.083~84.584
0.150~0.188	8.1~90.834	46.203~66.921
0.125~0.150	5.625~57.825	30.069~30.802
0.107~0.125	4.122~40.156	17.723~19.557
0.094~0.107	3.181~29.424	10.267~11.367
0.083~0.094	2.48~22.71	8.031~9.901
0.075~0.083	2.025~17.705	7.334~9.314
0.060~0.075	1.296~14.456	0.342~0.379
0.050~0.060	0.9~9.252	0.208~0.257
<0.050	<0.9	0

). There is a certain difference between the permeability coefficient obtained from the test and the corresponding calculated permeability coefficient for particles larger than 0.075 mm in diameter. The pore channel is generalized into an equivalent circular tube of equal diameter in the calculation process, but the actual pore channel is not equivalent and of equal diameter and not a straight line, so it is normal that there is a certain difference between the calculated results and the experimental results. For smaller particles (<0.075 mm), the experimental value is less than the theoretical value, and a critical value exists for the experimental permeability but not the theoretical permeability.

The observed discrepancy arises because bound water is not considered in the theoretical calculation. The influence of bound water on permeability becomes greater when the particle size is smaller. When the pore size is less than twice the thickness of the layer of adsorbed water, the pores are entirely filled with bound water, at which time it becomes very difficult for groundwater to migrate. In other words, the soil sample consisting of small particles loses permeability, hence giving rise to the observed critical value of permeability. According to the test results, when the sand particle size is less than 0.075 mm, there is a critical value of permeability coefficient, and when the sand particle size is less than 0.05 mm, the permeability coefficient is zero, causing pore water to stop flowing and become bound water.

4.3. Discussion of Permeation in Mixed-Particle-Size Soil

It can be seen for ideal isogranular soil that the soil has no permeability when the particle size is less than a certain threshold, and the permeability of soil of mixed particle size depends heavily on the content of non-permeable soil particles.

Mixed soil will still be permeable if the particles of individual soil are all larger than the critical value regardless of the mixing ratio, as the size of the smallest pores is larger than the pore size of the smaller soil particles.

On the other hand, the permeability of mixed soil will have to depend on the mixing ratio when one soil component has a particle size that is smaller than the threshold and is thus non-permeable. For this case, assume that the finer soil component has a particle diameter of D₁ that is lower than the threshold, and the permeable soil component has a particle diameter of D₂ that is higher than the threshold; the following will apply.

(1)

When D₁ > 0.414D₂, only a minor amount of the finer soil will reduce the permeability of the mixed soil to zero because it is larger than the pores of the permeable soil particles and cannot readily enter those pores (Figure 9). Even in an ideal state of particle arrangement shown in Figure 10, only a minimum amount of non-permeable soil particles are needed to annihilate the permeability of the mixed soil. This analysis is supported by experimental data: when the mixing ratio of medium-permeable and non-permeable soil is 4:1, the mixed soil has a permeability of k < 0.05 m/d and thus loses permeability.

(2)

When D₁ < 0.414D₂, the finer soil can fill in the pores of permeable soil to affect the permeability of mixed soil. According to geometry, when the permeable soil is assumed to have spherical particles of equal size, the compaction is the minimum (pores occupying 47.64% of the space) when they are ordered in a cubic array and maximum (pores occupying 25.95% of the space) when they are ordered in a tetrahedral array [1]. That is, the pore size of the permeable soil can fall within 25.95%~47.64%. Accordingly, the finer soil can completely fill the pores of the permeable soil completely when its content in the mixed soil is greater than 47.64%, which will then lose permeability entirely. The analysis here is supported by the measured data: when the high permeability soil is mixed with non-permeable soil, the permeability of the mixed soil drops below 0.05 m/d only when the mixing ratio reaches 1:2.

4.4. Comparison between Test Data and General Laws of Sand Permeability

The four kinds of soil particles with different permeability can be associated with soil type (

Table 4. Soil type and particle size.

Particle Size (mm)	Soil Type
0.250~0.375	Medium sand (mSa)
0.125~0.250
0.107~0.125	Fine sand
0.094~0.107
0.083~0.094
0.075~0.083
0.060~0.075	Coarse silt (cSi)
0.050~0.060
<0.050

), according to the above-mentioned code (EN ISO 14688-1:2017) [47].

Table 4 shows that the non-permeable particles correspond to coarse silt (cSi), which is in agreement with the common understanding that coarse silt (cSi) is a water barrier. The medium-permeability particles correspond to medium sand. The high-permeability particles correspond to medium sand and fine sand.

Naturally occurring sand contains particles of various sizes. According to the results of the preceding experiments, it can be argued that only the following types of sand are not permeable (

Table 5. Types of non-permeable soil.

Content of Particle 1 (Clay)	Type of Particle 2	Soil Type
>80%	High-permeability particles	Clay with (medium) coarse sand
>25%	Medium-permeability particles	Clay with fine sand

).

4.5. Limitations

This study mainly investigated the critical value of sand particle size for permeability in the field of hydrogeology. Due to the complexity of sand particle permeability, factors other than particle size need to be studied. This study only conducted tests of two kinds of sand. The experiments of sand with different particle sizes need to be performed. Other research [48] reported that the shape of the particles also has an impact on permeability, which is not considered in this article. However, through the test, we can also see that only when 0.094~0.107 mm-particle-size sand is mixed with 0.107~0.125 mm-particle-size sand, the permeability coefficient of the sand sample is relatively small, which may be caused by the irregular shape of the 0.094~0.107 mm-particle-size sand sample. There are more irregular sand particles that can fill the pores of 0.107~0.125 mm particle size, so its permeability is relatively small. In future studies, the effects of other factors on permeability will be investigated.

5. Conclusions

The current work tested the permeability characteristics of river sand to examine the critical particle size of permeable sand and its hydrological and geological significance. The following conclusions can be drawn:

(1)

For a sand sample with a defined particle size, the permeability declines with decreasing particle size, and the extent of the decline becomes lower with decreasing particle size. The particle size of sand can be segmented according to permeability as follows:

(a)

High-permeability particles: particle size >0.107 mm, hydraulic conductivity >10 m/d;

(b)

Medium-permeability particles: particle size 0.075~0.107 mm, permeability 1~10 m/d;

(c)

Low-permeability particles: particle size 0.050~0.075 mm, permeability 0.1~1 m/d;

(d)

Non-permeable particles: particle size <0.050 mm, permeability <0.05 m/d. The permeability should theoretically be zero, but because of the compactness of sample and other reasons in the laboratory, sand particles giving <0.05 m/d permeability are considered non-permeable. Therefore, 0.050 mm is the critical value of sand particle size for permeation.

(2)

Permeation test of binary sand mixtures of different particle size gave a total of 595 sets of data, from which the following conclusions could be drawn:

(a)

When larger particles are mixed with small particles, the permeability of the mixed sand sample is always smaller than that of the larger sand sample. When the content of small particles is 50%~75%, the permeability of the mixed sand sample becomes no greater than the permeability of the small sand sample.

(b)

At the same ratio, when two different particles are mixed, the permeability of the mixed sand sample decreases as the particle size of one sand component decreases. However, when the two kinds of particles are similar in size, the permeability becomes smaller after mixing.

(c)

When two particles of different size are mixed, the permeability of the mixed sand sample always decreases as the proportion of small particles increases. Permeability will not be lost when high-permeability particles are mixed with only medium- or low-permeability particles. Permeability will be lost when high-permeability particles are mixed with non-permeable particles and the proportion of non-permeable particles exceeds 75%. Permeability is easily lost when medium- or low-permeability particles are mixed with non-permeable particles.

(3)

Comparison between test results and theoretical calculation leads to the following:

(a)

Theoretical analysis and calculation based on prior research shows that the critical particle size for soil permeability is 3.4~9.1 μm, whereas the measured critical particle size in this work is 50 μm. The difference can be accounted for by the fact that the theoretical analysis considers glass beads whereas the experiments measure river sand.

(b)

The relationship between particle size and permeability can be calculated as k = (0.00075~0.0054) D²γ based on the generalization of channel. Some discrepancy exists between the calculated and measured permeability for low-permeability particles and non-permeable particle, because bound water is considered in the theoretical calculation.

(c)

When soil particles of different size are mixed, the mixed sand sample can lose permeability as long as a small proportion of the smaller sand exists when the smaller sand cannot readily enter the pores of the larger sand. In contrast, the mixed sand sample loses permeability only when the proportion of non-permeable particles exceeds 1:2.