water

Article
Research on the Critical Value of Sand Permeability Particle Size
and Its Permeability Law after Mixing
Cunjin Lu , Longfei Li, Jinpeng Xu *, Hui Zhao and Mingyue Chen

School of Resources and Geosciences, China University of Mining and Technology, Xuzhou 221116, China;
tshjlcj@126.com (C.L.)
* Correspondence: xjp319@126.com

Abstract: The permeability of sand is an important factor in determining the movement and oc-
currence 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.

Keywords: sand; permeability; critical value; particle size; ratio

1. Introduction
The permeability of sand is one of the key factors that determine the movement
Citation: Lu, C.; Li, L.; Xu, J.; Zhao, and occurrence of underground liquids and gases [1]. After Darcy’s law was introduced
H.; Chen, M. Research on the Critical experimentally based on permeability [2], subsequent studies have shown that many factors
Value of Sand Permeability Particle affect the permeability of porous media, including sand composition [3–5], pore size [6–9],
Size and Its Permeability Law after compaction [10–13], dry density [14–19], etc.
Mixing. Water 2024, 16, 393. https:// Previous studies demonstrated that grain grade is the most important factor affecting
doi.org/10.3390/w16030393 permeability [20–25]. Many researchers [26–31] have developed empirical formulas to
Academic Editors: Chin H Wu and predict permeability from grain size distribution indexes. Among them, the Hazan formula
Bommanna Krishnappan is the most commonly used empirical formula to determine permeability. In addition,
Burmister [32] demonstrated experimentally that the permeability of sand declines with
Received: 25 December 2023 increasing relative density. Guo [33] experimentally assessed the influence of coarse-
Revised: 19 January 2024
grained content on permeability and correlated the permeability of coarse-grained soil
Accepted: 20 January 2024
with P5, defined as the proportion of coarse materials > 5 mm in diameter, and argued that
Published: 24 January 2024
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
Copyright: © 2024 by the authors.
related assumptions, Chapuis [34] extended the Hazen formula to saturated non-plastic
Licensee MDPI, Basel, Switzerland. soil based on the experimental results from the permeability tests of clean saturated sand
This article is an open access article and crushed stone. Cui [35] evaluated the critical hydraulic gradient that triggers piping
distributed under the terms and for soil bodies with different flow patterns and different particle sizes. Yang [36] employed
conditions of the Creative Commons permeability tests and particle gradation analysis to derive, for sandy soil particles and
Attribution (CC BY) license (https:// through Darcy’s law, the relationship between the fractal dimension of gradation and the
creativecommons.org/licenses/by/ inhomogeneity coefficient. Specifically, poorer gradation leads to smaller inhomogeneity
4.0/). coefficient and larger fractal dimension, thus giving a larger permeability.

Water 2024, 16, 393. https://doi.org/10.3390/w16030393 https://www.mdpi.com/journal/water

Water 2024, 16, 393 2 of 16

There are many experimental studies on grain grade, and empirical formulas for esti-
mating 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 [1,42–44]. 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.
WaterWater
2024, 16,
Water2024,x16,
2024, FOR PEER
16,x393
FOR REVIEW
PEER REVIEW 3ofof1616
3 of 316

(a) ST-70A permeameter (b) Instrument diagram (c) Experimental diagram

(a) ST-70A permeameter (b) Instrument diagram (c) Experimental diagram
Figure 1. Schematic drawing and photograph of the customized permeameter.
Figure 1. Schematic
Figure 1. Schematicdrawing
drawingand
and photograph
photograph ofofthe
the customized
customized permeameter.
permeameter.
2.2. Test Materials
2.2.2.2.
TestTest Materials
Materials
2.2.1.
2.2.1. Composition ofofTest
Composition TestMaterials
Materials
2.2.1. Composition of Test Materials
To avoid the influence of mineral salts on the permeability of sand, river sand from
To avoid the influence of mineral salts on the permeability of sand, river sand from
theTo
the avoid
same
same sitethe
site wasinfluence
was used.X-ray
used. ofdiffraction
X-ray mineral salts
diffraction on the permeability
measurement
measurement revealed
revealed of
thesand,
thatthat the
main river
main sand from
component
component
theofsame
the site
samplewas is used.
SiO , X-ray
which diffraction
does not measurement
contain hydrophilicrevealed
or that the
water-swellable
the sample is SiO2 , which does not contain hydrophilic or water-swellable components.
2 main component
components.
of the
Composition
Composition SiO2, which
sample isanalysis
analysis ofoftestdoes
test not
sand
sand bybycontain
X-ray
X-ray hydrophilic
diffraction
diffraction or water-swellable
is shown
is shown 2. 2. components.
in Figure
in Figure
Composition analysis of test sand by X-ray diffraction is shown in Figure 2.

2. Composition
Figure 2.
Figure Compositionanalysis
analysisofoftest sand
test byby
sand X-ray diffraction.
X-ray diffraction.
2.2.2. Sieving of Sand Particles
2.2.2.
Figure 2. Sieving of Sand
Composition Particles
analysis of test sand by X-ray diffraction.
Vibrating screens from 20 to 300 mesh (a total of 12 sizes) were used to sieve the river
sand according to particle size20into
Vibrating screens from to 300 mesh with
segments (a total of 12 sizes)
different were used
granularity to1).
(Table sieve the river
2.2.2.
sandSieving of Sand
according Particles
to particle size into segments with different granularity (Table 1).
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).
Water 2024, 16, 393 4 of 16

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 thor-
oughly 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:

H1 − H2
I= (1)
L
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

Q
k= (2)
IA

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.
 (6) The procedures were carried out for samples with different formulations to derive
the corresponding permeability k.
Water 2024, 16, 393 5 of 16
3. Results
3.1. Single-Particle-Size Samples
3. Results
3.1. A total of 45 sets ofSamples
Single-Particle-Size data were obtained for analysis. The permeability of certain sand
types was taken as the average of the calculated permeability of the corresponding sam-
A total of 45 sets of data were obtained for analysis. The permeability of certain
ples (see Table 2 and the corresponding scatter plot in Figure 3). Figure 3 shows that the
sand types was taken as the average of the calculated permeability of the corresponding
permeability gradually decreases with particle size. Permeation was extremely slow for
samples (see Table 2 and the corresponding scatter plot in Figure 3). Figure 3 shows that
the sand sample of <0.050 mm particles: the water level in the column moved only about
the permeability gradually decreases with particle size. Permeation was extremely slow for
10 cm after 8 h. The type of tested sand has very low permeability. In hydrogeology, it can
the sand sample of <0.050 mm particles: the water level in the column moved only about
be considered as having no permeability. For convenience, the permeability coefficient
10 cm after 8 h. The type of tested sand has very low permeability. In hydrogeology, it
was treated as zero in this study. As shown in Table 2, the sand samples can be divided
can be considered as having no permeability. For convenience, the permeability coefficient
into four groups according to the permeability coefficient: high permeability, medium
was treated as zero in this study. As shown in Table 2, the sand samples can be divided
permeability, low permeability and no permeability. The classification results refer to the
into four groups according to the permeability coefficient: high permeability, medium
classification
permeability,criteria of ‘strong permeability
low permeability (8.64~864The
and no permeability. m/d)’ and ‘medium
classification permeability
results refer to the
(0.0864~8.64 m/d)’ in Appendix F of the Code for Engineering Geological Survey
classification criteria of ‘strong permeability (8.64~864 m/d)’ and ‘medium permeability of Water
Resources and Hydropower (GB 50487-2008, 2022 Edition), but the
(0.0864~8.64 m/d)’ in Appendix F of the Code for Engineering Geological Survey of classification criteria
Water
ofResources
‘weak permeability and below’ are applicable to ‘silt~clay’. Combined with
and Hydropower (GB 50487-2008, 2022 Edition), but the classification criteria the change
boundary points of permeability
of ‘weak permeability and below’coefficient and to
are applicable sand particleCombined
‘silt~clay’. size in Figure
with 3,
thethe low
change
permeability
boundary points of permeability coefficient and sand particle size in Figure 3, the the
and non-permeability range of sand samples are determined. In Figure 3, low
relationship
permeabilitybetween sand sample particle
and non-permeability range ofsize
sandand permeability
samples coefficientInwas
are determined. fitted
Figure to
3, the
three curves. The three curves correspond to the sand groups of high permeability,
relationship between sand sample particle size and permeability coefficient was fitted to me-
dium
three permeability and curves
curves. The three low permeability.
correspond The
to the critical value of
sand groups ofparticle size based medium
high permeability, on per-
meability is 0.050
permeability and mm.
low permeability. The critical value of particle size based on permeability
is 0.050 mm.

Figure 3. Permeability of single-particle-size samples.

Figure 3. Permeability of single-particle-size samples.
Table 2. Table of particle size distribution and permeability.

Granularity Mesh Number Particle Size (mm) Permeability k (m/d)

20~40 0.375~0.750 178.396
40~60 0.250~0.375 124.003
60~80 0.188~0.250 78.464
High-permeability particles
80~100 0.150~0.188 55.554
100~120 0.125~0.150 30.344
120~140 0.107~0.125 18.396
 Granularity Mesh Number Particle Size (mm) Permeability k (m/d)
20~40 0.375~0.750 178.396
40~60 0.250~0.375 124.003
60~80 0.188~0.250 78.464
High-permeability
Water 2024, 16, 393 particles 6 of 16
80~100 0.150~0.188 55.554
100~120 0.125~0.150 30.344
Table 2. Cont.
120~140 0.107~0.125 18.396
140~160 0.094~0.107 10.909
Granularity
Medium-permeability particles Mesh Number
160~180 Particle Size (mm)
0.083~0.094 Permeability k (m/d)
8.966
180~200
140~160 0.075~0.083
0.094~0.107 10.909 8.141
Medium-permeability particles 160~180 0.083~0.094 8.966
200~250
180~200
0.060~0.075
0.075~0.083 8.141
0.361
Low-permeability particles
250~300 0.050~0.060 0.232
200~250 0.060~0.075 0.361
Low-permeability
Non-permeable particles
particles <300 <0.050
250~300 0.050~0.060 0.232 0.000
Non-permeable particles <300 <0.050 0.000
3.2. Mixed-Particle-Size Samples
A total of 595 sets
3.2. Mixed-Particle-Size of data were obtained for the mixed-particle-size samples (Table
Samples
2), from which
A total the
of 595 permeability
sets of data werewas calculated
obtained for the to assess the relationship
mixed-particle-size samplesbetween
(Table 2), particle
size
fromand
whichpermeability, andwas
the permeability in calculated
particulartothe impact
assess of the content
the relationship of particles
between below the
particle size
particle size threshold.
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
(1) medium-permeability
From Figure 4, we can see that: When
particles, thehigh-permeability
permeability of particles are mixed
the mixed samplewith
is always
medium-permeability
lower than that of theparticles, the permeability
high-permeability of the mixed
particles. Whensample is always lower
the medium-permeability
than that of
particles the high-permeability
account for >50% of the particles. When
total, the the medium-permeability
permeability particlesis lower
of the mixed sample
account for >50% of the total, the permeability of the mixed sample is lower than that
than that of the medium-permeability particles.
of the medium-permeability particles.

Figure
Figure 4.4. The
The relationship
relationshipbetween
between permeability
permeability and and the content
the content ratio ofratio of 0.250~0.375
0.250~0.375 mm andmm and
0.094~0.107 mmsand
0.094~0.107 mm sandparticles.
particles.

(2) From
(2) From Figure
Figure5,5,we
wecan seesee
can that: When
that: high-permeability
When particles
high-permeability are mixed
particles are with
mixed with
low-permeability particles, the permeability of the mixed sample is always lower than
low-permeability particles, the permeability of the mixed sample is always lower
that of the high-permeability particles. When the high-permeability particles account
than thatof
for >50% ofthe
thetotal,
high-permeability
the permeabilityparticles. When
of the mixed the high-permeability
sample is higher than that ofparticles
the ac-
count for >50% of
low-permeability the total, the permeability of the mixed sample is higher than that
particles.
of the low-permeability particles.
Water 2024, 16, x FOR PEER REVIEW 7 of 16
Water
Water2024, 16,16,
2024, x FOR
393 PEER REVIEW 7 of7 of
1616

Figure 5. The relationship between permeability and the content ratio of 0.250~0.375 mm and
The sand
Figure 5. mm
0.050~0.060 relationship between permeability and the content ratio of 0.250~0.375 mm and
particles.
Figure 5. The relationship between permeability and the content ratio of 0.250~0.375 mm and
0.050~0.060 mm sand particles.
0.050~0.060 mm sand particles.
(3) From Figure 6, we can see that :For the mixture of high-permeability particles and
(3) From Figure 6, we can see that: For the mixture of high-permeability particles and non-
non-permeable
(3) From particles,
Figure 6, we can seethe mixed
that sample
:For the losesofpermeability
mixture (i.e., k =particles
high-permeability 0 or k <and
0.05
permeable particles, the mixed sample loses permeability (i.e., k = 0 or k < 0.05 m/d)
m/d) when the particles,
non-permeable content ofthe
non-permeable
mixed sampleparticles
loses is no less than
permeability 80%.
(i.e., k = 0 or k < 0.05
when the content of non-permeable particles is no less than 80%.
m/d) when the content of non-permeable particles is no less than 80%.

Therelationship
Figure6.6.The
Figure relationship between
between permeability
permeabilityand
andthe content
the ratio
content of 0.250~0.375
ratio mmmm
of 0.250~0.375 andand
<0.050 mm
<0.050
sand
mm particles.
sand particles.
Figure 6. The relationship between permeability and the content ratio of 0.250~0.375 mm and <0.050
mm sand particles.
(4) From
(4) FromFigure
Figure7,7,we
wecan
cansee
seethat:
that:When
Whenthethemedium-permeability
medium-permeabilityparticles
particlesare
aremixed
mixed
with low-permeability particles, the permeability of the mixed sample is always lower
with Figure
(4) From low-permeability
7, we can see particles,
that: Whenthe the
permeability of the mixedparticles
medium-permeability sampleareis mixed
always
than that of the medium-permeability particles. When the low-permeability particles
lowerlow-permeability
with than that of the particles,
medium-permeability
the particles.
permeability of theWhen
mixedthesample
low-permeability
is always
account for >50% of the total, the permeability of the mixed sample is lower than that
particles
lower thanaccount
that offor
the>50% of the total, the permeability
medium-permeability of the mixed
particles. When sample is lower
the low-permeability
of the low-permeability particles.
than that of the low-permeability particles.
particles account for >50% of the total, the permeability of the mixed sample is lower
than that of the low-permeability particles.
Water 2024, 16, x FOR PEER REVIEW 8 of 16
Water 2024, 16, x FOR PEER REVIEW 8 of 16
Water 2024, 16, 393 8 of 16

Figure 7. The relationship between permeability and the content ratio of 0.094~0.107 mm and
0.060~0.075
Figure 7.
Figure mmrelationship
7. The
The sand particles.
relationship between permeability
between and and
permeability the content ratio of
the content 0.094~0.107
ratio mm andmm and
of 0.094~0.107
0.060~0.075 mm sand particles.
0.060~0.075 mm sand particles.
(5) From Figure 8, we can see that :The mixed sample loses permeability (i.e., k = 0 or k
(5) From Figure 8, we can see that: The mixed sample loses permeability (i.e., k = 0 or
(5) <From
0.05 Figure
m/d) entirely when
8, we can medium-permeability
see that :The mixed sampleparticles are mixed with
loses permeability (i.e., knon-per-
k < 0.05 m/d) entirely when medium-permeability particles are mixed with non-
= 0 or k
meable particles.
<permeable
0.05 m/d) entirely when medium-permeability particles are mixed with non-per-
particles.
meable particles.

8. The
Figure 8.
Figure Therelationship between
relationship permeability
between and theand
permeability content
the ratio of 0.094~0.107
content mm and <0.050
ratio of 0.094~0.107 mmmm
and <0.050
sandsand
mm particles.
particles.
Figure 8. The relationship between permeability and the content ratio of 0.094~0.107 mm and <0.050
mm sand particles.
According to the results, when both particles are larger than the critical value of water
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
conductivity,
mixing isthey
after According still
to the
related have water
toresults, when
the proportion conductivity
both after
particles
of different aremixing.
larger
particles, Thelarger
andthan
the sizecritical
the of permeability after
valueofof water
proportion
mixing is
conductivity,related to the
they still
particles has a greater proportion
have water
decisive of different
effect. conductivity particles,
The reason for after and
mixing.
this result the
The
is that larger
size
the proportion
of permeability
pores of
of the sand afterpar-
ticles
sample
mixing hasis arelated
becomegreater todecisive
smallertheafter effect.
the smallThe
proportion reason
particles
of are for
different thisinto
mixed
particles,result is
thethat the proportion
pores sand
the large-particle-size
and larger of theofsand
par-
sample
sample, become
and the smaller
effective after
cross the small
section of particles
the seepage are mixed
decreases, into
so the
the large-particle-size
permeability
ticles has a greater decisive effect. The reason for this result is that the pores of the coeffi- sand
cient of the
sample,
sample and mixed
become sand sample
the smaller
effective cross
after decreases.
thesection Permeability
areismixed
of the seepage
small particles not lost when
decreases,
into the high-permeability
so the permeability coef-
large-particle-size sand
particles
ficient of are
the mixed
mixed with
sandmedium-
sample or low-permeability
decreases. particles
Permeability isbecause
not the
lost particle
when size of
high-permeabil-
sample, and the effective cross section of the seepage decreases, so the permeability coef-
the sand sample is higher than the critical permeability value, but will be lost in a mixture
ity particles
ficient of the are mixed
mixed sandwith medium-
sample or low-permeability
decreases. particles
Permeability is not because
lost when the particle
high-permeabil-
size of the sand sample is higher than the critical permeability value, but
ity particles are mixed with medium- or low-permeability particles because the will be lost in a
particle
mixture of sand
size of the >80%sample
non-permeable
is higher particles
than the with high-permeability
critical particles.
permeability value, but willPermeability
be lost in a
mixture of >80% non-permeable particles with high-permeability particles. Permeability
Water 2024, 16, 393 9 of 16

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, ther-
modynamic 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
(Us ) is equal to the chemical potential of the adsorbate (Ua ), 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 ad-
sorbent. 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 Dl = 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.
Water 2024, 16, 393 10 of 16

4.2. Comparison between Experimental and Calculated Permeability

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

K = r2 γ/8ε (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:

K = (0 .00075~0 .0054)D2 γ/ε (4)

The theoretical permeability calculated with Equation (4) is compared with the experi-
mentally derived permeability for all particle sizes (Table 3). There is a certain difference
between the permeability coefficient obtained from the test and the corresponding calcu-
lated 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.

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

The observed discrepancy arises because bound water is not considered in the theoret-
ical 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 per-
meability. 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.
 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
Water 2024, 16, 393 particle size is less than a certain threshold, and the permeability of soil of mixed particle 11 of 16
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
Mixed soil regardless
will of the mixing
still be permeable if theratio, as the
particles of size of the smallest
individual pores
soil are all is larger
larger than
than the pore size of the smaller soil particles.
the critical value regardless of the mixing ratio, as the size of the smallest pores is larger
Onpore
than the the other hand,
size of the permeability
the smaller of mixed soil will have to depend on the mixing
soil particles.
ratioOn
when one soil
the other component
hand, has a particle
the permeability size that
of mixed is smaller
soil will have tothan the threshold
depend and is
on the mixing
thus when
ratio non-permeable. For this case,
one soil component has aassume
particlethatsizethe
thatfiner soil component
is smaller has a particle
than the threshold and
isdiameter of D1 that is lower
thus non-permeable. than
For this theassume
case, threshold,thatand
thethe permeable
finer soil component
soil component has a
has a particle
particle diameter
diameter of D1 thatofisDlower
2 that is higher
than than the threshold;
the threshold, the following
and the permeable soilwill apply. has a
component
particle
(1) Whendiameter of D2 that
D1 > 0.414D is higher than the threshold; the following will apply.
2, only a minor amount of the finer soil will reduce the permeability

(1) of the D
When mixed soil to
1 > 0.414D zero abecause
2 , only it is larger
minor amount than
of the thesoil
finer pores
willofreduce
the permeable soil par-
the permeability
ticles
of theand
mixed cannot
soilreadily
to zeroenter thoseit pores
because is larger(Figure
than9).the
Even in an
pores of ideal state of particle
the permeable soil
arrangement
particles shown readily
and cannot in Figure 10, those
enter only aporesminimum (Figure amount
9). Even of in
non-permeable
an ideal statesoil of
particlesarrangement
particle are needed shown to annihilate the 10,
in Figure permeability
only a minimum of the mixed
amountsoil. This analysis is
of non-permeable
supported
soil particlesby areexperimental data: when
needed to annihilate the mixing ratio
the permeability of mixed
of the medium-permeable
soil. This analysis and
non-permeable
is soil is 4:1, thedata:
supported by experimental mixed soil the
when has mixing
a permeability of k < 0.05 m/d and and
ratio of medium-permeable thus
loses permeability.
non-permeable soil is 4:1, the mixed soil has a permeability of k < 0.05 m/d and thus
(2) loses
Whenpermeability.
D1 < 0.414D2, the finer soil can fill in the pores of permeable soil to affect the
(2) When D1 < 0.414D
permeability 2 , thesoil.
of mixed finerAccording
soil can filltoingeometry,
the pores whenof permeable soil to affect
the permeable soil isthe as-
permeability of mixed soil. According to geometry, when
sumed to have spherical particles of equal size, the compaction is the minimum the permeable soil is as-
sumed
(pores to have spherical
occupying 47.64% particles
of the of equalwhen
space) size, the
theycompaction
are ordered is the
in aminimum
cubic array (pores
and
occupying
maximum47.64% of the space)
(pores occupying when of
25.95% they
theare ordered
space) when inthey
a cubicarearray andinmaximum
ordered a tetrahe-
(pores occupying
dral array [1]. 25.95%
That is,of the space)
pore when
size of theythe are permeable
ordered in asoil tetrahedral
can fall array [1].
within
That is, the pore size
25.95%~47.64%. of the permeable
Accordingly, the finersoil
soilcan
can fall within 25.95%~47.64%.
completely fill the pores of Accordingly,
the perme-
the
ablefiner
soil soil can completely
completely when itsfill the pores
content in theof mixed
the permeable soil completely
soil is greater than 47.64%, when whichits
content in the mixed soil is greater than 47.64%, which will
will then lose permeability entirely. The analysis here is supported by the measured then lose permeability
entirely.
data: when Thetheanalysis here is supported
high permeability soil isbymixed
the measured data: when the
with non-permeable soil,high
the perme-
perme-
ability soil is mixed with non-permeable soil, the permeability
ability of the mixed soil drops below 0.05 m/d only when the mixing ratio of the mixed soilreaches
drops
below
1:2. 0.05 m/d only when the mixing ratio reaches 1:2.

Figure 9. Schematic diagram of particle mixing for D1 > 0.414D2 .

Water 2024, 16, x FOR PEER REVIEW 12 of 16

Water 2024, 16, 393 12 of 16

Figure 9. Schematic diagram of particle mixing for D1 > 0.414D2.

Figure 10.
10. Schematic
Schematic diagram
diagram of
of particle
particle mixing
mixing for
for D
D1 <
< 0.414D2.
Figure 1 0.414D2 .

4.4.
4.4. Comparison
Comparison between
between Test
Test Data
Data and
and General
General Laws
LawsofofSand
SandPermeability
Permeability
The
The four kinds of soil particles with different permeability can
four kinds of soil particles with different permeability can be
be associated
associated with
with soil
soil
type
type (Table
(Table 4),
4), according
accordingtotothe
theabove-mentioned
above-mentionedcodecode(EN
(ENISO
ISO14688-1:2017)
14688-1:2017)[47].
[47].

Table4.4. Soil
Table Soil type
typeand
andparticle
particlesize.
size.

Particle Size(mm)
Particle Size (mm) Soil Type
Soil Type
0.250~0.375
0.250~0.375 Medium
Medium sandsand(mSa)
(mSa)
0.125~0.250
0.125~0.250
0.107~0.125
0.107~0.125
0.094~0.107
0.094~0.107 Fine sand
0.083~0.094 Fine sand
0.083~0.094
0.075~0.083
0.075~0.083
0.060~0.075
0.050~0.060
0.060~0.075 Coarse silt (cSi)
<0.050
0.050~0.060 Coarse silt (cSi)
<0.050
Table 4 shows that the non-permeable particles correspond to coarse silt (cSi), which
is in agreement with
Table 4 shows thethe
that common understanding
non-permeable that
particles coarse siltto(cSi)
correspond is asilt
coarse water
(cSi),barrier.
which
The medium-permeability particles correspond to medium sand. The high-permeability
is in agreement with the common understanding that coarse silt (cSi) is a water barrier.
particles correspond to medium sand and fine sand.
The medium-permeability particles correspond to medium sand. The high-permeability
Naturally occurring sand contains particles of various sizes. According to the results
particles correspond to medium sand and fine sand.
of the preceding experiments, it can be argued that only the following types of sand are not
Naturally occurring sand contains particles of various sizes. According to the results
permeable (Table 5).
of the preceding experiments, it can be argued that only the following types of sand are
not permeable (Table 5).
Table 5. Types of non-permeable soil.
Table 5. Types of non-permeable soil.
Content of Particle 1 (Clay) Type of Particle 2 Soil Type
Content of>80%
Particle 1 (Clay) Type of Particleparticles
High-permeability 2 Clay withSoil Type coarse sand
(medium)
>25%
>80% Medium-permeability particles
High-permeability particles Clay with
Clay with (medium) finecoarse
sand sand
>25% Medium-permeability particles Clay with fine sand
Water 2024, 16, 393 13 of 16

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 decreas-
ing 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
Water 2024, 16, 393 14 of 16

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) D2 γ 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.

Author Contributions: Conceptualization, J.X.; methodology, C.L.; software, H.Z.; validation, L.L.;
formal analysis, M.C.; investigation, C.L.; data curation, M.C.; writing—original draft preparation,
L.L.; writing—review and editing, C.L.; visualization, H.Z.; supervision, J.X.; project administra-
tion, L.L.; funding acquisition, J.X. All authors have read and agreed to the published version of
the manuscript.
Funding: This project is funded by the General Program of National Natural Science Foundation
of China (No. 52274243), the Postdoctoral Fellowship Program of CPSF (No. GZC20233005), the
Fundamental Research Funds for the Central Universities (No. 2024QN11025), the National Key
Research and Development Program of China (No. 2019YFC1805400) and the National Natural
Science Foundation of China (No. 42174165).
Data Availability Statement: The data used to support the findings of this study are included within
the manuscript.
Conflicts of Interest: The authors declare no conflict of interest.

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