Method to quantify the notional permeability

1

Henk Jan Verhagen1 _{, Daan Jumelet}2_{, Ana Vilaplana Domingo and Patrick van}

Broekhoven

In de Van der Meer formulas for armour stability the Notional Permeability is used as a parameter. Unfortunately the physical basis of this parameter is weak. It is therefore suggested to use a relation between the Notional Permeability P and the reduction of wave run-up due to infiltration into the breakwater. The advantage is that the latter can be computed with VOF models. This makes is possible to estimate the value of P from mathematical models. Also the run-up reduction can be measured in a physical model, which has the advantage that physical tests for run-up are much faster to execute than models for armour stability.

INTRODUCTION

The notional permeability factor, as used in the generally applied armour stability formula of Van der Meer can only be determined from experience and from a rough estimate using the figure as published in the original thesis of VAN DER MEER [1988]. A better way to quantify this parameter before starting

exten-sive physical tests would be beneficial for designers. This paper describes an approach to determine the P-value using parameters which can be determined without physical modelling, but for example using VOF-modelling.

THE VOLUME EXCHANGE MODEL General

To give this factor a physical description a volume-exchange-model is intro-duced to express the effect of core permeability on the external wave run-up process. This volume-exchange-model couples the external process with the internal process. The external process is described by a wave run-up model. In this model the wave run-up wedge approach of HUGHES [2004] is linked to the wave kinematics in front of the structure. The internal process is described by the ‘Forchheimer’ equation for the water flow through a porous medium. In this study it is assumed that the notional permeability factor P is highly related to the volume exchange model. By coupling the volume-exchange-model with the notional permeability factor this relation is investigated. This coupling is realized by elaboration of the volume-exchange-model for the four defined ‘notional permeability structures’. In case of a vertical structure transition the elaboration of the volume-exchange-model works well. For a sloped structure transition the volume-exchange-model is subject to a phase difference between the separate layers. This phenomenon should be studied more extensively.

1 _{Section Hydraulic Engineering, Delft University of Technology, PO Box 5048, NL2600DA Delft, Netherlands,} H.J.Verhagen@tudelft.nl

However, in both cases (sloped and vertical structure transition) the correlation between the P-factor and the so-called run-up reduction coefficient cr (followed

from the volume-exchange model) is clearly visible. With this correlation it is possible to choose a value of the notional permeability factor P that is based on a physical description. Besides this, the study also shows that the permeability of the structure not only depends on structural properties, as stated by Van der Meer, but also on the hydraulic parameters. With this consideration the dual permeability notation in the stability formula (for surging waves) is explained. This paper gives a proposal to separate the P-factor from the stability formulae by incorporating the influence of the permeability in the stability formulae in the run-up reduction factor. However, without an additional test program this is in the present form of the volume-exchange-model not possible.

The stability formulae

VAN DER MEER [1988] presented two stability formulas for breakwater armour:

0.2 0.18 0.5 50 s pl m n

H

S

c P

for plunging waves

d

N









_

_



_

_

cot

H

S

c P

for surging waves

d

N

 









_

_



_

_

For notation see the Appendix 1, Notations. In these formulas Van der Meer has introduced the Notional Permeability P. This parameter describes the effect of the permeability of the sublayers, however there is no direct relation with the magnitude of the permeability. Van der Meer defined the value of P in a figure, which he based on a regression analysis, see Fig. 1.

Figure 1. Notional permeability coefficient P according to VAN DER MEER [1988]

The values for P=0.1, P=0.5 and P=0.6 were based on curve fitting of data, while P=0.4 was based on interpolation.

Because of the fact that the definition of P is not directly based on a physical process, it is difficult to determine the exact value of P on beforehand. In this research it is tried to link P to a physical process.

Infiltration of water

The reason why the permeability of sublayers is relevant is the fact that in case of impermeable sublayers the water pressure under the armour layer increases and consequently the stability decreases. Therefore one could try to link the magnitude of P to the amount of water which is infiltrating into the breakwater. Infiltration into the breakwater leads to a reduction in the wave run-up. This reduction is defined as:

, , u r r u f

R

c

R



In this equation Ru,f is the run-up along an impermeable slope, but including

friction. The value of Ru,f is therefore smaller than the run-up along a smooth

slope. Ru,f = γf Ru,s, in which Ru,s is the run-up on a smooth, impermeable slope.

For the determination of Ru,s quite reliable prediction formulas exist (e.g.

PULLEN ET.AL [2007])

One may conclude that there has to be a relation between the magnitude of P and the run-up reduction due to infiltration. The value of cr can be determined in

practice from simple physical lab tests (these tests are more simple than stability tests), but it is nowadays also possible to determine this value with VOF models. Apart from that, one may also approximate the value of cr using the Volume

Exchange Model, as will be explained in the next section. The Volume Exchange Model

The Volume Exchange Model has been worked out in detail by JUMELET

[2010]. As a first approximation a vertical, permeable wall is considered, see Fig 2.

Figure 2. Volume exchange model for a homogeneous structure with a vertical transition.

In this figure Ru,f is the run-up on a rough, impermeable slope. Because of

infiltration, this run-up reduces to Ru,r. From a water balance follows:





V

V

R

R

V





The volume VRu,f is determined using a simple linear approach [HUGHES, 2004] :

0 , 2 ,

2

3

L

V

R





The internal volume Vb can be determined when the gradient of the water (I) is

known.

Application on a multi-layered slope

In case of a slope and a non-homogeneous structure the equations (3)-(5) remains in principle identical. However a slope will be included.

Figure 3. The iterated gradient (red line) and the instantaneous water level (blue line)

The gradient inside the structure can be calculated by solving eq. (6) with iteration:













1 1

1

1 cos

2

sin

R

I

n

R

_b

T

I





 









It is assumed that the water is infiltrating only during a part of the wave period. As a first assumption it is assumed that γinf = 0.25 (so infiltration only during a

quarter of the wave period). Unknown values in this equation are the internal gradient I and the reduction of the run-up internally Ru (the run-up at the core is

less as the run-up outside the armour layer). Based on experiments of MUTTRAY

[2000] for Ru a value of 0.5 is suggested. When the iteration leads to values of I

larger than 1, a value I=1 should be used. The internal volume is defined by:









1

1 cos

sin

R

I

V

_b



 

T









In eq. (6) b is the Forchheimer parameter for turbulent flow, defined as: 3 50,

1

1

n

b

n

gd







In order to investigate the relations four sample structures have been defined, each with given armour units, filter dimensions and core dimensions. The values are chosen in such a way that they coincide with a breakwater with P-values of 0.1, 0.4, 0.5 and 0.6. For all these structures the run-up reduction coefficient is calculated for different values of wave period and wave steepness. The results are presented in Table 1.

Table 1. values of cr for various values of P and various wave conditions.

Hs T0 steepness P=0.6 P=0.5 P=0.4 P=0.1 2.91 12 1.30% 0.967 0.977 0.984 0.997 2.91 13 1.10% 0.970 0.979 0.986 0.998 2.91 14 0.95% 0.972 0.981 0.987 0.998 2.91 15 0.83% 0.975 0.983 0.988 0.998 2.00 14 0.65% 0.976 0.984 0.988 0.998 3.00 14 0.98% 0.972 0.981 0.987 0.998 4.00 14 1.31% 0.969 0.979 0.986 0.998 5.00 14 1.64% 0.966 0.977 0.985 0.998

Regression analysis of the data, including the Iribarren number, the correction factor cr and the ratio of grain sizes of the core and armour lead to the following

equation:





0.72

1

2.5

d

P

c

d











_

_





In VILAPLANA [2010] is shown that a value of  = 4.2 leads on average to a

good correlation between the suggested values of VAN DER MEER [1988] and

equation (9). See also Fig. 4.

Figure 4. Computed values of P with eq. 9 compared with the suggested value of P for this condition by Van der Meer (P=0.5) for a variation coefficient  = 4.2.

A similar analysis for the values lead to a value of f = 0.75. For the run-up

reduction factor a dependency was found on the type of structure. For impermeable structures Ru=0.3, for permeable structures Ru=0.48 and for

homogeneous structures Ru=0.42

COMPARISON WITH THE ORIGINAL VAN DER MEER DATA Comparison

Using the original data of Van der Meer, the above mentioned model has been applied [VILAPLANA, 2010]. Using default values the model performed as good

as the estimate of P directly derived from the Van der Meer tests. See also Fig. 5.

Figure 5. Comparison of measured damage level with computed damage level; left: using P-values from this study, right: using fixed P-values (0.1, 0.5 and 0.6) (based on the original experimental data of Van der Meer).

Next step

The comparison has been carried out with default values. No real run-up data from the tests were included, neither real information regarding the porosity. A weak point in the analysis is the assumptions regarding the value of f and Ru.

Therefore further research has been carried out in order to separate the influence of friction and infiltration on the total run-up on a breakwater.

EFFECT OF RUN-UP PREDICTION General

One of the unknown parameters in the model was the relation between the run-up reduction due to roughness and the run-run-up reduction due to infiltration of water into the filter layer and the core. In order to make this clear, additional model tests have been performed to determine the run-up reduction due to roughness only (so for an impermeable slope). See VAN BROEKHOVEN [2011]. Given this value, a much more reliable estimate of the P-value can be determined. In order to investigate this, a series of tests were performed measuring the run-up for:

 a rough, impermeable slope

 armour on a rough, impermeable core  armour on a rough, permeable core

The tests were performed both with regular and irregular waves in one of the flumes of the WaterLab of Delft University of Technology. Wave boards were equipped with active reflection compensation. The results of the tests of run-up on smooth slopes were completely in line with other researchers. For the results with rough, impermeable slopes results were as expected. See Fig. 6.

Figure 6. Run-up on smooth slopes, on rough impermeable slopes and on rough permeable slopes on an impermeable core.

The tests with a rough, permeable slope on an impermeable core showed a much lower value of run-up (as expected, due to infiltration in the armour layer), but there was no dependency any more on the Iribarren number. The variation of the results was quite large. See also Fig. 6. In the next test series the run-up on permeable armour was compared for the case “permeable core” with the case “impermeable core”. As can be seen in Fig. 7, there is no significant difference in both cases. This is a remarkable observation. For the stability of the armour it makes quite some difference whether the core is permeable or not, but for the run-up at the surface (Ru,r) is does not make difference.

The large spread in data makes predictions of run-up rather difficult, and consequently this parameter is not very practical for design calculations.

Figure 7. Run-up on rough permeable armour, comparison of an impermeable core with a permeable core.

However, one may consider that the run-up does depend on de roughness itself, but more on the relative roughness dn50/H. When this factor is included Fig.7

reduces to Fig. 8, which has a considerably smaller spread and is a good basis for further design calculations.

Figure 8. Relative run-up scaled with dn50 as function of the Iribarren number for

From the tests followed also that the reduction factor Ru was a function of the Iribarren number:





1.0 tanh 0.31







This is illustrated in Fig. 8.

Figure 8. Relative run-up scaled with dn50 as function of the Iribarren number for

armour on both permeable and impermeable core.

Also a difference was observed between the calculated run-up at the core and the observed run-up at the core. For this an empirical correction factor γcr has

been introduced. This factor depends on the porosity of the core and the Iribarren number:





1.60

1

cr

n



_

_



The tests also showed that the infiltration period is somewhat shorter than assumed in eq. (6). The experiments showed that γinf ≈ 0.15.

Summary

With this method the various values of run-up are calculated with the equations given. : , , , , Ru c b u c cr u imp Ru c

V

V

R

R

V









 

_



_





For the calculation of VRu,c eq (5) can be used using Ru,c instead of Ru,f.

Further:

, ,

u imp Ru u f

and , , , , u c Ru c b r cr u imp Ru c

R

V

V

c

R

V









Elaborating this for the tests of Van der Meer resulted in: 1 3.4

0.5

P

_



c

Eq. (15) is comparable to eq. (9), however eq.(15) is based on infiltration and eq. (9) on the size of particles. The advantage of eq. (15) is that a relation between P and infiltration has more physical basis than using a curve fit with a

dn50/da-ratio.

Figure 9. Overview of the different values for run-up for breakwaters with permeable and impermeable core. Note that Ru,r is identical in both cases (see also Fig. 7)

FUTURE APPLICATION

Modern VOF programs are able to calculate wave run-up along a slope and water penetration into a filter layer rather exact from any shape of construction. By combining the results of this work with the data from VOF computations, a more reliable stability analysis of coastal structures can be made.

It is not expected that a VOF model in the near future will be able to calculate stone stability. There this approach may be a good solution to enter more physical processes in the stability calculations for breakwaters and use the benefits of VOF models.

ACKNOWLEDGMENTS

This work is partly sponsored by DEME Group, Zwijndrecht, Belgium. Their contribution is greatly appreciated. The contribution of mr. Jonas Maertens of DEME is especially acknowledged.

APPENDIX: NOTATIONS

b Turbulent friction coefficient in the Forchheimer equation (s/m)

cr Wave run-up reduction coefficient (Ru,r / Ru,f) (-) crγ Wave run-up reduction coefficient (Ru,c / Ru,imp) (-)

cs Constant in the Van der Meer formula for surging waves (-) da Thickness of the armour layer (m)

dn50 Median grain size (m) Hs Significant wave height (m) I Hydraulic gradient (-)

N Number of waves in a wave field (-)

n Porosity (-)

P Notional Permeability (-)

Ru,c Run-up at the breakwater core (m)

Ru,f Run-up on an impermeable slope, but with friction (m) Ru,imp Imposed run-up at the breakwater core, identical to Ru,c for a

impermeable core (m)

Ru,r Run-up on a permeable slope (m) Ru,s Run-up on a smooth slope (m) S Damage level (-)

s Wave steepness (-)

 Slope of the breakwater (degrees)  Fit factor for the Forchheimer factor b (-)

γcr correction factor for observed/calculated run-up at the core (-)

f Run-up reduction coefficient (Ru,f / Ru,s) (-)

γinf Reduction for the time the water is infiltrating into the core

Ru Run-up reduction coefficient for infiltration (Ru,c / Ru,f ) (-)

 Relative material density (-)  Iribarren number (-) ω Angular frequency (2/T) REFERENCES

BRUUN, P., GÜNBAK, A.R., 1977 Stability of sloping structures in relation to = tan /(H/L0) risk criteria in design, Coastal Engineering 1:287-322 doi:10.1016/0378-3839(78)90007-8

HUGHES, S.A., 2004. Estimation of wave run-up on smooth, impermeable slopes using the wave momentum flux parameter, Coastal Engineering 51:1085-1104

doi:10.1016/j.coastaleng.2004.07.026

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http://repository.tudelft.nl/view/ir/uuid:38e5cf5c-bcf5-4433-a7cd-80334d79bb88

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VAN BROEKHOVEN, P.J.M., 2011. The influence of armour layer and core permeability on the wave runup. MSc thesis, Delft University of Technology

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VILAPLANA DOMINGO, A.M., 2010. Evaluation of the volume-exchange model with Van der Meer laboratory test results. Additonal MSc thesis, Delft University of Technology

1 Paper P0197

Method to quantify the notional permeability

1st _{Verhagen, Henk Jan}

2nd _{Jumelet, Daan}

3rd _{Vilaplana Domingo, Ana}

4th _{Van Broekhoven, Patrick} keywords: Breakwaters Coastal structures Notional permeability Armour stability Core permeability