Porosity and Hydraulic Conductivity of Mixed Sand-Gravel Sediment

Porosity and Hydraulic Conductivity of Mixed Sand-Gravel Sediment

K. She1, D. Horn

2 and P. Canning

3

Abstract

Recharged mixed sand-gravel beaches are a common means of sea defence in the UK. A

widespread problem associated with these replenished beaches is that of cliffing. Cliffs of up to

2 m height have been recorded. Due to their instability, these cliffs are potentially hazardous to the

general public and must be removed at the first opportunity. This study looks at the causes of the

cliffing problem by means of a theoretical analysis with the support of laboratory experiments. A

number of equations have been derived giving the relationship between the porosity and hydraulic

conductivity of a mixed sand-gravel sediment and the properties of the component sand and gravel.

These equations are validated by a series of laboratory experiments on sediments of varying sand

percentages. An additional series of tests examines the relationship between the sand percentage

and loading capacity of the sediment. The test results indicate that serious cliffing is likely to occur

when the sand percentage of the sediment mix exceeds a critical value.

Key words

Permeability, Hydraulic conductivity, Bimodal, Cliffing

Introduction

Many of the beaches on the UK coast that constitute the main defence against erosion and flooding

are composed of highly permeable sediments, usually a mixture of sand and gravel. Recharge

material dredged from offshore is increasingly used to replenish these mixed sand and gravel

beaches. Because beach recharge materials may contain a larger proportion of fine sediment than

the natural beach, sediment size distributions, sorting and hydraulic conductivity can be

significantly altered, as is beach profile response. Even when the size distributions of the natural

sediment and the recharge sediment are quite similar, the standard recovery technique produces an

increased proportion of sand on the upper foreshore, which is normally composed of coarse

1 Lecturer, the School of the Environment, University of Brighton, Brighton, BN2 4GJ, U.K

Email: [email protected] 2 Lecturer, School of Geography, Birkbeck College, University of London, London, WC1E 7HX, U.K

Email: [email protected] 3 Assistant Engineer, River and Coastal Engineering, Atkins Water, Bristol, U.K

Email: [email protected]

mailto:[email protected]

Porosity and Hydraulic Conductivity of Mixed Sand-Gravel Sediment 2

sediment. Cliffs approaching 2m height have been recorded as demonstrated by Figure 1. In

addition to being a public hazard, these cliffs have the effect of enhancing beach erosion as a result

of increased wave reflection.

Figure 1. Cliffing photographed at Hayling Island (left, Nov/05) and Pevensey Bay (right, Jun/02)

The problem of cliffing of mixed sand-gravel beaches has been suspected to do with the bimodal

nature of the sediment mix despite a general lack of systematic investigations of the problem.

Mason (1997) showed that as the sand content increases the hydraulic conductivity of the sand-

gravel mixture reduces very quickly and reaches a value that approximates that of pure sand at

30~40%. Román-Blanco (2003) further discussed Mason’s results and also provided experimental

data that showed a similar behaviour in terms of the porosity of a bi-modal sediment mix. The

results of Mason (1997) and Román-Blanco (2003) suggest that the presence of sufficient sand can

dramatically alter the behaviour of the sediment in contrast to that of pure gravel. In the current

study, a bimodal mixed sand-gravel sediment is examined by means of a theoretical analysis

coupled with supporting laboratory experiments. The objectives of the study are to

Investigate the influence of sand percentage on the porosity and hydraulic conductivity of a

bimodal sediment mix.

Establish the relationship between the presence of sand and occurrence of cliffing.

Porosity versus Sand Percentage

Román-Blanco (2003) hypothesised that as the sand content increases from 0 to 100%, the sediment

mix may be said to be “under-filled”, “fully-filled” and “over-filled”. The “under-filled” mix is

where the sand fraction is not enough to fill the pore space between the gravel particles while the

“over-filled” mix is where there is more sand than that required to fill up the gravel pore space. The

“fully-filled” stage is a “transitional zone” between the under-filled and fully-filled stages.


Theoretically speaking, the “fully-filled” stage is a point at which the bulk volume of the sand is

equal to that of the gravel pore space. For clarity, we call this point the critical point.

Let be the percentage of the sand by weight. The porosity of the pure gravel is ng and the pure

sand has a value of ns. For simplicity, the same density s is assumed for both sand and gravel.

First consider the under-filled sediment mix. Let VB be the bulk volume of the sand-gravel mixture

and n be its porosity. Note that the bulk volume of the gavel is also VB. The weight of gravel is

thus

Wg = sVB(1 ng) (1)

And the weight of sand is

Ws = Wg

=

sVB(1 ng)

(2)

The net volume of sand is therefore

Vs = VB(1 ng)

(3)

The pore space of the sand-gravel mixture is the gravel pore space taken away the volume of sand,

i.e.,

VP = ngVB Vs = VB(ng )

(4)

By definition, the porosity of the sediment mix in an under-filled state may be expressed as

nunder-filled = VP

VB =

ng

(5)

The bulk volume occupied by the sand fraction is given by

VBs = VB(1 ng)

(1 ns) (6)

At the critical point, the bulk volume occupied by the sand is equal to the pore space of the gravel,

thus

ngVB = VB(1 ng)c

c (1 ns) (7)

The critical value is therefore


c = ng(1 ns)

ngns (8)

Substituting (8) into (5) we have a critical porosity of the sand-gravel mixture of

nc = ngns (9)

Now consider the situation where the sand fraction exceeds the critical percentage. The bulk

volume of the sediment mix is the sum of the bulk volume of the sand (VBs) and the net volume of

gravel (Vg) while the pore space is that of the sand (nsVBs). The porosity of the sediment mix is

thus

nover-filled = nsVBs

(VBs Vg) (10)

Vg is related to VBs through :

Vg = nsVBs

(11)

Substituting (11) into (10)

nover-filled = ns

1 ns(1 ) (12)

It can be shown that substituting c into (12) also leads to equation (9).

To summarise, the porosity of a bimodal mixed sand-gravel sediment may be represented by

n =

ng

≤ c)

n = ns

1 ns(1 ) ≥ c)

(13)

Permeability versus Sand Percentage

Let kg and ks be the hydraulic conductivity of pure gravel and pure sand, respectively. According to

Darcy’s law, the discharge velocity v through the sediment media is

v = ki (14)

where i is the hydraulic gradient dh

dl. The seepage velocity vseepage is related to v by


vseepage = v/n = ki/n (15)

In order to assess the effect of the addition of sand to gravel, we assume that the flow through the

sediment pores may be approximated by that through parallel pipes each having a length L and an

effective diameter d. It may be assumed that the diameter d is proportional to the sediment size and

porosity. We also assume laminar flow as required by Darcy’s law. Given a laminar pipe flow, the

energy loss may be expressed in terms of the Hagen-Poiseuille equation (Chadwick and Morfett,

1993):

h = 32Lvseepage

gd2 =

Lvseepage

Cd2 (16)

where and are the viscosity and density of the water, respectively, and g is the acceleration due

to gravity. By definition, the hydraulic gradient may be written as

i = h

L =

vseepage

Cd2 (17)

For a pure gravel system, we denote the effective pore diameter with d0. Comparing (15) and (17)

leads to

kg = Cd02ng (18)

Now consider the case of an under-filled sand-gravel mixture. By nature, the fluid finds the path of

least resistance. The effect of this is the creation of “two” flow pathways within the gravel pore

space, one through the sand and the other through the pore space that is free from the sand

occupation. The two flow pathways run in parallel just like two parallel pipes of different

diameters. Let A0 be the cross-sectional area of the pathway completely free from sand. A system

partially filled with sand has a cross-sectional area A and diameter d in relation to:

A

A0 =

d2

d02 =

VBng VBs

VBng = 1

VBs

VBng

Substituting (6) into above, we have:

A

A0 =

d2

d02 = 1 (19)

where

(1 ng)

ng (1 ns)() (20)

From (17) we have the seepage velocity through the reduced gravel pore space as


vseepage,g = iCd2 (21)

The corresponding discharge is then

qg = vseepage,g (ATng

A

A0)

AT is the cross-sectional area of the mixed sediment. Substituting (19) and (21) into above:

qg = iCd02ngAT

d2

d02

A

A0 = ikgAT(1)

2 (22)

The discharge through the sand may be given by

qs = iks[(ATng)(1 A

A0)] = iksATng (23)

The total discharge through the mixed sediment media is thus

q = qg + qs = iAT( )kg(1)2

+ ksng (24)

By definition, the hydraulic conductivity of the sand-gravel mixture is given by

kunder-filled = kg(1)2

+ ksng (25)

For an over-filled sediment mix, the gravel particles simply act to reduce the cross-sectional area of

the sand, leading to reduced cross-sectional area of the flow.

ATs

AT =

VTs

VT =

1

1 Vg/VTs

+ (1 ns)() (26)

The discharge through the sand-gravel mixture is equal to the discharge through the sand:

q = ATkover-filledi = ksiATs

Therefore

kover-filled = ks

ATs

AT =

ks

+ (1 ns)() (27)

In summary, the hydraulic conductivity of a mixed sand-gravel media may be estimated by

k = kg(1)

2 + ksng ≤ c)

k = ks

+ (1 ns)() ≥ c)

(28)

Corresponding to the critical sand percentage, the hydraulic conductivity is given by


kc = ksng (29)

Bulk Density versus Sand Percentage

The bulk density of the sand-gravel mixture can be easily worked out from the porosity

(Equation 13):

bulk

s =

1 ng

≤ c)

bulk

s =

1 ns

1 ns(1 ) ≥ c)

(30)

Permeability Experiment

In order to validate equations (13) and (28), a series of permeability tests were carried out at the

University of Brighton. Although Mason (1997) performed permeability tests for sand-gravel

mixtures of varying sand content, the data could not be used due to lack of details on porosity.

Similarly, the data of Román-Blanco (2003) contained porosity but permeability was not measured.

Having taken into consideration of the theoretical predictions and the results from Mason (1997),

we decided to test sand-gravel mixtures with sand percentages of 0%, 10%, 20%, 30%, 40%, 60%,

80% and 100% by weight. The gravel has a D50 of 4 mm while three sand sizes were used,

representing the fine, medium and coarse range. The size distribution of the gravel and sands is

shown in Figure 2. The test procedure follows that of British Standard 1377 (BSI, 1990). A

constant head permeameter is used, which has a cell chamber of 75 mm diameter and 600 mm

length with three manometer tappings spaced at 225 mm apart.

Comparison between Theory and Experimental Data

Figure 3 compares the porosity measured from the current permeability tests with the prediction

given by Equation (13). The measured data followed the trend as predicted by the equation and the

minimum porosity also appears to occur in the vicinity of the predicted critical sand percentage.

The experimental data of Román-Blanco (2003) did not include the porosity of pure sand, which

may assume the value measured for the fine sand used in the present study. No gross error is

anticipated due to the fact that fine sand was used in Blanco’s experiment. It can be seen from

Figure 4 that the analytical solution gives very good prediction against the Blanco’s experimental

data.


0

20

40

60

80

100

0.01 0.1 1 10

Sieve Size (mm)

%P

assin

g

Gravel

Coarse sand

Medium sand

Fine sand

Figure 2. Sediment Grading of sand and gravel used for permeability tests

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0% 20% 40% 60% 80% 100%

Sand Percentage

Poro

sity n

Coarse sand (Exp) Coarse sand (Analytical)

Medium sand (Exp) Medium (Analytical)

Fine sand (Expt) Fine sand (Analytical)

Figure 3. Comparison between analytical prediction and measured porosity (present study)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0% 20% 40% 60% 80% 100%

Sand Percentage

Poro

sity n

Román-Blanco (2003) Analytical prediction

Figure 4. Comparison between analytical prediction and measured porosity of Román-Blanco (2003)


Figures 5&6 compare the theoretically predicted hydraulic conductivity with the experimental

results of the current study and those of Mason (1997). Again, the sand porosity in Mason’s

experiment was estimated based on the present experimental data. It can be seen that there is very

good agreement between the experimental data and analytical predictions across the whole range of

sand percentages for all three sand grain sizes.

0.0001

0.001

0.01

0.1

1

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Sand Percentage

Hyd

rau

lic C

on

du

ctivity k

(m

/s)

Laboratory (Coarse sand) Analytical (Coarse sand)Laboratory (Medium Sand) Analytical (Medium sand)Laboratory (Fine Sand) Analytical (Fine sand)

Figure 5. Comparison between analytical prediction and measured permeability (present study)

0.00001

0.0001

0.001

0.01

0.1

1

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Sand Percentage

Hyd

rau

lic C

on

du

ctivity k

(m

/s)

Mason, 1997 (Coarse Sand) Analytical (Coarse)

Mason, 1997 (Medium Sand) Analytical (Medium)

Mason, 1997 (Fine Sand) Analytical (Fine)

Figure 6. Comparison between analytical prediction & measured permeability of Mason (1997

Discussions on Porosity and Hydraulic Conductivity

Having shown the validity of the analytical solution, we can now look at the significance of the

derived equations. The first important point is that as the sand content increases from 0% to the

critical value c, the hydraulic conductivity reduces rapidly. This effect can be better seen from

Figure 7 which shows the ratio between the hydraulic conductivity of sand-gravel mixtures and that


of pure gravel. A major advantage of a gravel beach is its ability to efficiently absorb wave energy

over a short distance as a result of large percolation flow allowed in the beach. As the sand

percentage increases in a mixed sand-gravel sediment, hydraulic conductivity reduces very quickly.

Above a sand fraction of approximately 30%, the hydraulic conductivity of the sand-gravel

sediment becomes comparable to that of pure sand. Consequently, the advantage of a gravel beach

diminishes.

Ratio of hydraulic conductivity of mixed sediment to that of gravel

0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Sand Percentage

k/k

g

ng = 0.45, ns = 0.42, kg = 0.075 m/s, ks = 0.00001 m/s

Figure 7. Relative hydraulic conductivity as a function of sand percentage

It is clear that the hydraulic conductivity of a mixed sand-gravel sediment can be smaller than that

of pure sand. The hydraulic conductivity reaches a minimum at the critical sand percentage c,

which corresponds to the point when gravel pore space is just fully occupied by sand. It can be seen

from Equation (29) that the hydraulic conductivity at the critical sand percentage is around 50% of

that of pure sand. The percolation in the beach is thus significantly reduced, leading to less efficient

dissipation of wave energy. It is thus desirable to avoid the sand percentage in the region of

30%~40%.

Loading Experiment and Implications on Cliffing

The particular nature of porosity, bulk density and permeability of the sand-gravel mixture as a

function of sand percentage seems to indicate that serious cliffing starts to occur when sand fraction

is equal or more than the critical value. The critical point is also likely to be the worst point for

cliffing. To verify this hypothesis, we carried out some simple but effective laboratory tests. The

experiment was intended to be indicative rather than definitive. The experimental procedure is as

follows:


a) Weigh the amount of dry sand and gravel according to the required sand percentage, and mix

the two by hand.

b) Place the mixture in a (63m) sieve and add water into the sand-gravel mix. Allow the excess

water to drain and then mix content again.

c) Place the sand-gravel mixture into a plastic container in layers of ~5cm, hand-tamping each

layer as happened in the permeability experiment. The container is 185 mm diameter and

120 mm height with a wall slope of 1:15.

d) Place a laminated chipboard on the top of the container, turn the board and container upside

down and lay them on a level concrete base.

e) Slowly remove the container to form a “sand castle”.

f) If the sand castle does not collapse, place an empty container on top of the “sand castle”.

Slowly add gravel into the container until the “sand castle” collapses.

g) Record the load at which the “sand castle” collapses.

We chose to experiment with the medium sand and gravel mixture, with the sand content of 20%,

30%, 36%, 40%, 50% and 100%. The 36% represents the critical point. Each mixture was tested

twice. At 20% and 30%, the “sand castle” collapsed on removing the container. In both cases, the

collapse was symmetric, as shown in Figures 8, but the central portion of the “sand castle” remained

upright. As can be expected, the size of the free standing part of the “sand castle” became smaller

as the sand percentage was reduced. In addition, this remaining part completely collapsed when

given a gentle touch with a finger.

Figure 8. Collapsed “sand castles” without loading


At 36 and higher percentages, the “sand castles” stood firm on removal of the container. The

recorded failure load varied from sample to sample for a given sand percentage, and from one sand

percentage to another. Figure 9 shows the 40% “sand castle” at different stages of the test.

Figure 10 shows the collapsed “sand castles” at 36% and 100% of sand. In all cases tested, the

collapses were asymmetric, which may have been caused by the imperfections (non-symmetric) in

the “castle” itself and/or in loading. Table 1 summarises the collapsing loads for all tests. It

seemed that the “sand castle” at the critical percentage was able to sustain a similar or greater load

than at higher sand percentages. The reason for this may be due to the fact that the bulk density of

the mixed sediment is at its highest (Equation 30), and its behaviour is closest to that of a solid

body.

The implication of the “sand castle” test results is that cliffing is likely a phenomenon when the

sand percentage of the sediment exceeds the critical value. No serious cliffing should be expected

below the critical percentage.

Figure 9. Different stages of “sand castle” test ( = 40%)


Figure 10. Collapsing at critical percentage (36%, left) and at 100% (right)

Sand percentage Test 1 Load (N) Test 2 Load (N) Mean (N)

36% 7.2 10.8 9.0

40% 5.9 7.3 6.6

50% 5.4 9.6 7.5

100% 7.7 8.0 7.8

Table 1. Collapsing load of “sand castles”

Effects of Compaction

When recharging a beach, heavy machinery is normally used to move the material around the

beach. As a result, the sediment is greatly consolidated and compacted by the end of the recharge

operations. This will significantly reduce the porosity of the sediment media, which in turn reduces

the hydraulic conductivity and makes the beach more “solid” due to increased bulk density. At the

same time, the critical sand percentage becomes smaller. This can be best demonstrated by Figures

11, 12 & 13 showing the porosity, hydraulic conductivity and bulk density in response to 10% and

20% compactions. A basic assumption has been made here with regard to the effect of porosity on

the permeability. According to Kozeny (1927), the hydraulic conductivity may be approximated by

k = Cd

2n

3

(1 n)2

where C is a constant and d is the sediment size. Given a percentage reduction in the porosity of

gravel and sand, the reduction in the hydraulic conductivity of the sand-gravel mixture may be

worked out accordingly. It can seen that with a 20% compaction, the critical point is moved from

36% to 30%. The greater the compaction, the more reduction in the critical percentage. It is likely

that cliffing starts to occur at a sand percentage of around 30% in practice.


In addition to the shift of the critical point, the compaction also has the effect of increasing the

loading capacity of the sediment mix. This point can be demonstrated by the result of two

additional tests at 50% and 100% sand where a greater tamping force was applied in making these

“sand castles”. The failure load was increased by 250% for the 50% “sand castle” and 350% for the

100% “sand castle”. The implication is that the more compaction, the more likely and more severe

the cliffing.

Effects of compaction on porosity (medium sand)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0% 20% 40% 60% 80% 100%

Sand Percentage

Poro

sity n

Without compaction

With 10% compaction)

With 20% compaction

Reduced porosity and reduced critical sand percentage

Figure 11. Effects of compaction on porosity.

Effects of compaction on permeability (medium sand)

0.0001

0.001

0.01

0.1

0% 20% 40% 60% 80% 100%

Sand Percentage

Hyd

rau

lic C

on

du

ctivity k

(m/s

)

Without compaction

With 10% compaction

With 20% compaction

Reduced hydraulic conductivity

Figure 12. Effects of compaction on hydraulic conductivity.


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0% 20% 40% 60% 80% 100%

Sand Percentage

Ratio o

f B

ulk

Density to

Sedim

ent D

ensity

Bulk/Sediment Density (without compaction)

Bulk/Sediment Density (with 10% compaction)

Bulk/Sediment Density (with 20% compaction)

Increased bulk density

Figure 13. Effects of compaction on bulk density.

Conclusions

Two analytical equations have been derived expressing the porosity and hydraulic conductivity of a

bimodal mixed sand-gravel sediment in relation to the sand percentage and the porosity and

hydraulic conductivity of the component gravel and sand fractions. A critical sand percentage or

critical point can be identified given the sediment properties of the component sand and gravel. At

the critical point, the hydraulic conductivity and porosity of the mixed sediment reach their

respective minimums while the bulk density is at a maximum. There are two important

implications in terms of engineering applications. The first is that cliffing becomes a problem when

the sand percentage exceeds the critical value. The second is that compaction due to heavy plant

operations will greatly reduce the critical sand percentage, making cliffing more likely to occur and

more severe.

References

British Standards Institution, 1990, Methods of Test for soild for civil engineering purposes,

BS1377, British Standards Institution, London

Chadwick A and Morfett J, Hydraulics in Civil and Environmental Engineering, 1993, E & FN

Spon, London

Kozeny J. 1927, Uber kapillare Leitung des Wassers in Boden. S. B. Akad. Wiss. Wien Math.

Naturwiss 136, 271-306.


Lopez de San Román-Blanco, B. 2003. Dynamics of gravel and mixed sand and gravel beaches.

Unpublished PhD thesis, Imperial College, University of London.

Mason, T. 1997. Hydrodynamics and sediment transport on composite (mixed sand/shingle) and

sand beaches. Unpublished PhD thesis, University of Southampton.

Acknowledgement

This work forms part of a Defra funded project (FD1923). Contributions and assistance have been

kindly provided by Bill Symmons (Defra), Roger Spencer (Arun DC), Ian Thomas (PCD), Jonathan

Clark (Canterbury CC) and Clive Moon (Havant BC).

Documents

Porosity and Hydraulic Conductivity of Mixed Sand-Gravel Sediment