Evaluation modelling : single hole sonic logging : draft report

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Evaluation modelling

Single hole sonic logging

1202249-003

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1202249-003-GEO-0003, 14 September 2011, draft

Inhoud

1 Introduction 1

1.1 Background and aim of this report 1

1.2 On the content of this report 1

2 Surface wave on the interface 3

2.1 Plane surface wave 3

2.2 Tube wave 5

2.2.1 General solution 5

2.2.2 Influences 8

2.2.3 Comparison with plane strain solution 8

2.2.4 Amplitude 9

2.2.5 Final remarks 11

3 General overview of the waves in a pile during single hole sonic logging 13

3.1 Basics of the model 13

3.2 A perfect pile 14

3.3 A pile with a neck 15

3.4 A pile with an inclusion 15

3.5 A pile with an eccentric hole 15

3.6 Alternative interpretation of the SHSL test 16

4 Modelling a neck in a pile 17

4.1 Introduction 17

4.2 Symmetric neck 17

4.3 Non-symmetric neck 20

4.3.1 Principle 20

4.3.2 Influences 20

4.3.3 Determining the depth of a neck 25

4.4 Borehole is not symmetric in the pile 26

4.5 Conclusion 28

5 An inclusion in the pile 29

5.1 General 29

5.2 Influence of a soft layer in a pile 29

5.2.1 Stoneley and refracted wave 30

5.2.2 Reflections of body waves 31

5.2.3 Water solid interaction 33

5.3 Scattering of waves around a spherical or cylindrical inclusion 34

5.3.1 Considerations 34

5.3.2 Worked example 35

5.4 Conclusions 36

6 Analysis of measurements 37

6.1 Preliminary analysis 37

6.2 Cross correlation for one signal 37

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Evaluation modelling ii

7 Conclusions and recommendations 43

8 References 45

Appendices

A Selected measurements A-1

B Cross correlation results B-1

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1 Introduction

1.1 Background and aim of this report

The Dutch ’Geo-impuls’ program aims to reduce the costs of geotechnical failure during construction activities. A major problem is the quality assurance of in-situ constructed piles. Both the construction and the inspection of the created piles are difficult.

A method that can be used for medium diameter piles (diameter about 0.7 m) is the method of single hole sonic logging (SHSL). Here, one hole is prepared at the moment the steel reinforcement is placed. This hole can be used to lower the equipment consisting of a source and a receiver in one instrument.

This method is comparable with the cross-hole sonic logging (CHSL) method. However, for a CHSL at least two, but standard three or more holes are needed, for a SHSL only one. This makes the method cheaper and more equipped for medium sized piles. CHSL is more suited for large diameter piles (diameter more than 1 m).

However, the interpretation of SHSL is complicated. For a homogeneous pile, the signal in the receiver is quite constant. Flaws can be seen easily by changes in the signal. However, the judgement of the severity of a flaw requires more insight in the properties (dimensions, stiffness) of the flaw.

This report tries to derive the relation between the possible flaws and the expected observations during a SHSL. This helps the interpretation of test results.

The result of the full project might be e.g. a catalogue of common flaws and related measurement results, which can be used by a practical engineer to support his judgement of the quality of a pile using SHSL.

1.2 On the content of this report

In a preliminary study, the outline of the research is given [Hölscher, 2010]. This report is a logical follow up and elaboration of the preliminary study.

The method of ray tracing is chosen as a suitable method to create this catalogue. The advantage of this method is that the several wave types are distinguished before hand. This advantage is at the same moment also its’ disadvantage: the relative importance of each wave is not known before hand.

The preliminary report suggests creating a model that is able to simulate a full synthetic measurement for a pile with given flaws. This idea is abandoned. The report focuses on the theoretical resuls that are calculated for several flaws by using ray-tracing theory. This seems a more comprehensive method, fitting better to the current status of the research. The flaws considered in this report are the neck and an inclusion. Both symmetrical cases as well as a-symmetrical cases are considered.

Chapter 2 the existence of a surface wave along the edge of the borehole is evaluated. It is important to understand the possible wave paths for the non-body waves. In the reciever, the information of all waves is observed. To understand their meaning, the basic properties of all possible waves must be known.

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Evaluation modelling 2 of 52

The following chapters concentrate on the influence of several flaws on the body waves. Due to the reflections to the edge of the pile, these will influence the measured signals. Typical observations are derived.

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2 Surface wave on the interface

Firstly, the phenomenon is studied for plane waves. Based on the result, a tube wave is considered.

2.1 Plane surface wave

[Brekhovskikh and Godin, 1998] derive the equation to find the wave velocity of the surface wave on a fluid-solid interface (see their equation 4.4.20). This equation reads

2 2

( )

4 (1

) (1

)

(

2)

s

((1

)(1

))

0 f s

s

qs

s

qs

rs

m

(1.1) With: 2 2 2 2 2 2 ss ps ss f s f ss

c

q

c

r

c

m

v

s

c

The first subscript refers to the materials f: fluid, s: solid; the second subscript to the wave type s: secondary wave and p: primary wave

c is the wave velocity in the material, the volumetric mass of the material, and v is the wave speed of the surface wave.

If s is a solution of the equation above, a surface wave exists.

For an interface air-concrete the function f(s) can be drawn. Figure 2.1 shows the result. The material properties are shown in Table 2.1.

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Figure 2.1 Function f(s) for air-concrete interface

A zero is found for s=0.83, which means that the wave speed is 0.911 of the shear wave speed in the solid, which coincides the Rayleigh-wave speed in the solid. It must be noted that the function is real for s<1.

Figure 2.2 shows the function for a water-concrete interface. Now, the function is complex for much smaller values of s. In order to find a zero, both the real and imaginary part should be zero, which is not the case for any s > 0. This means that the equation has no solution thus no surface wave exists.

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It is noted that the function shows a limitation for the possible occurrence of a surface wave on a fluid-solid interface.

Now we assume that the solid is stiffer than the fluid, so r > 1 (the shear wave speed in the solid is higher than the pressure wave speed in the fluid). The first possibility is that the function f(s) is real. Since q is always smaller then 1, the function changes from real to complex for increasing s when:

2 2 2 2

1

0

1

f ss ss f

rs

s

r

c

v

c

v

c

(1.2)

The function is real for 0<s<1/r. In this part of the domain, no zeros are found. The physical background of this observation is not studied further.

For the case 1/r < s < 1, the function f(s) is complex:

2 2

Re[ ( )]

4 (1

) (1

)

(

2)

0 Im[ ( )]

((1

)(

1))

0 f s

s

qs

s

f s

qs rs

m

(1.3)

Starting with the imaginary part, we see that s = 1/q or s=1/r. Substitution of the first solution in the first relation of Eq. (1.3) leads to a very special solid, since Poison ratio should be zero. Substitution of the second solution in the first relation of Eq. (1.3) leads to a fifth order in r (with parameter q), which leads to a special requirement in material properties for the shear wave in the solid and the wave speed in the fluid. It seems not to have real solutions.

Therefore, it is concluded that no surface waves will occur along the plane fluid-solid interface from water-concrete. The existance of a surface wave along the interface requires a continuous comptability of the displacement and stresses in the interface. The outcome of this study shows that for the concrete and water this is not possible. Therefore, a wave with an amplitude that decreases with distance to the interface is not possible.

2.2 Tube wave

[Brekhovskikh and Godin, 1998] remark that the decay of the surface waves with distance in the fluid (gas) is very low. The problem at hand is defined in an axial symmetric geometry. Reflections, which vanish in plain geometry, will be reflected by the edge of the borehole ‘at the other side’. These considerations suggest that a special type of wave should be evaluated. The amplitude of this wave is (almost) constant in the water, and decays rapidly in the concrete. The existence of this type of waves will be evaluated in this Section.

2.2.1 General solution

The analytical solution for waves propagating along a circular fluid filled hole in a solid is presented by [Henry, 2005]. Henry presents in the Eq. 2-93 the characteristic equation of the coupled wave propagation along the borehole:

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1202249-003-GEO-0003, 14 September 2011, draft Evaluation modelling 6 of 52 1 1 2 2 2 2 0 2 2 1 0 1 2 2 2 2 0 2 1

(

)

(

)

(

)

2 (

)

(

)

(

)

0 (

)

2

4 (

)

P S P S r r r r P P z r P S r r z r P P r r f s s f f r r r s f _S S r z r P s r

K k a K k a k k

k

_{K k a}

k

K k a

V

I k a

I k a k

V

K k a

k k

aV

K k a

(1.4)

with the wave numbers defined by:

1 2 1 2 2 2 2 2 2 2 P r z p S r z s z

k

V

k

V

k

c

(1.5)

K and I are modified Besselfunctions.

For each frequency, a zero can be searched. If found, the wave speed and other properties can be found for that wave. Since the first four terms of the equation give no or trivial zeros, only the part between brackets is considered.

Eq. 2-50 and 2-51 of Henry then deliver the decay of the displacements with distance to the borehole: 1 1 0 0

( , )

(

)

( , )

(

)

( , )

(

)

( , )

(

)

P z S r z r S z r r P S z r P z r z r r

ik

u

B k

K k r

C k

K k r

k

ik

u

B k

K k r

C k

K k r

k

(1.6)

Where B and C are constants for the frequency considered.

The solution depends on two radial wave numbers, which is comparable with Rayleigh wave solutions.

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Property Value Unit

frequency 50 kHz

compression modulus fluid 2000 MPa

volumetric mass fluid 1000 kg/m3

Youngs’ modulus concrete 30 GPa

Poisson ratio concrete 0.2 [-]

volumetric mass solid 2400 kg/m3

radius borehole 30 mm

wave speed water 1414 m/s

P-wave speed concrete 3727 m/s

S-wave speed concrete 2282 m/s

Table 2.1 Properties for Stoneley wave study

Figure 2.3 shows the characteristic equation as a function of the wave number (kz = /c) for

frequency f = 50 kHz ( = 314E3 rad/s). Table 2.1 shows the material properties.

Figure 2.3 Characteristic equation for single hole sonic logging case

In the upper part of the Figure (with the absolute value), the wave numbers related with the compression wave and shear wave in the solid and the wave in the fluid are presented by markers. For low wave numbers, the imaginary part is non-zero, so no solution is found. For higher wavenumbers, the imaginary part is zero. Obviously, a zero is found at kz = 315, which

shows a wave speed of 997 m/s. This wave is slower than all waves in the water and the concrete.

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2.2.2 Influences

Two variations are evaluated:

the stiffness of the water might be lower than the value used, due to gas bubbles. For a compressibility of 200 MPa, the resulting wave speed is 226 m/s, which is indeed much lower. For 100 MPa a wave speed of 136 m/s is found.

the frequency of the waves is evaluated more in detail. Figure 2.4 shows that the wave speed decreases with increasing frequency. At low frequency, the value is just a little bit below the wave speed in the water. At the working frequency (50 kHz) only about 2/3 of this value is found.

•

Wave speed Stoneley wave

0 200 400 600 800 1000 1200 1400 0 20 40 60 80 100 120 Frequency [kHz] W a v e s p e e d [ m /s ]

Figure 2.4 Influence of frequency on Stoneley wave speed (radius hole is 30 mm)

2.2.3 Comparison with plane strain solution

Analysis of a water-filled borehole in concrete shows that in the radial case a Stoneley wave exists. This obviously differs strongly from the plane strain case discussed before. This aspect will be discussed now, by studying the influence of the borehole radius. The material properties of Table 2.1 are used, with frequency also 50 kHz.

Radius [mm] Wave speed [m/s]

30 997 60 731 90 584 120 490 200 351 300 264 400 215 450 no answer

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Table 2.2 shows that the radius of the borehole has a tremendous influence on the solution and for a large radius, the solution cannot be found anymore. This suggests that the solution only exists for small boreholes.

2.2.4 Amplitude

Now we focus on the importance of the wave with distance from the borehole. The penetration depth of the Stoneley waves can be derived from Eq. 2-50 and 2-51 from [Henry 2005]. First, the solution for water with compressibility of 1000 MPa is shown: Figure 2.5 shows the characteristic function and Figure 2.6 the dependancy of the amtlitude of the tube wave with distance from the pile center. It turns out that the penetration depth of the wave is very limited, up to one or two cm.

Figure 2.5 Characteristic function for borehole 30 mm and compressibility of water 1000 MPa

Althought Figure 2.6 shows that the absolute value, the real part and the imaginary part, only one line is seen. This is due to the fact that the imaginary part is zero. This will be explained: The zero of the characteristic equation is found at kz = 473.5. The wave number for the shear

component is 2 2 2 2 S r z z s s

k

V

(1.7)

This is a real number for kz > ks. For the case of a fluid filled borehole in concrete, ks =

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real. For real arguments, the modified Bessel functions are real and have continuously decreasing behaviour.

Since the wave number kz increases with decreasing compressibility of the water in the

borehole, this behaviour is expected for all compressibility’s of the water.

The penetration depth depends strongly on frequency. For lower frequency, a higher penetrationdepth is found. For frequency 5 kHz (10 times lower), Figure 2.7 shows the amplitude with distance to the pile axis. Now the penetration depth increases to more than 10 cm. This is understandable. For frequency 50 kHz and wave number 474, the wavelength is 13 mm. For frequency 5 kHz, the wave number is 32.6 and the wavelength is 193 mm. These numbers are observed in Figure 2.6 and Figure 2.7 respectively.

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Figure 2.7 Penetration of Stoneley wave for 5 kHz wave (compressibility water 1000 MPa, radius hole 30 mm)

2.2.5 Final remarks

The reflection on fractures is studied by [Henry 2005]. This study shows that thin fractures (order mm’s) at high frequencies (10-100 kHz) generate almost no reflections of Stoneley waves.

The Stoneley wave might be influenced by fractures in the pile (which are open and in the centre) but that the (high frequency) pulses do not give any information about the pile properties deeper in the pile. This aspect is studied by [Henry 2005]. Henry’s study shows that thin fractures (order mm’s) at high frequencies (10-100 kHz) generate almost no reflections of Stoneley waves. This means that all information should be derived from the arrivals of the (reflected) body waves.

This is a support for the application of the ray-tracing technique for simulation, since the incorporation of surface waves is a weak point of ray tracing. It should however be noted that the surface waves will be observed in the measurements, but flaws in the pile farther away from the centre (borehole) will have minor influence on the signals from the surface waves. The conclusions are based on the processing of the equation derived by Henry. The theoretical background of the equation is sound, but the equation itself is not checked within this study. The existance of this type of waves might be checked from field measurements.

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3 General overview of the waves in a pile during single hole

sonic logging

In this Chapter a short discussion on the ray tracing method will be given. The following cases are considered:

1. A perfect pile. 2. A pile with a neck. 3. A pile with an inclusion. 4. A pile with an eccentric hole. 3.1 Basics of the model

The source sends waves in all directions. The angle between the direction of propagation at the source and the line from source to receiver, determines what will happen with the wave. The angle is /2 – is the angle of incident against the edge of the borehole. Two critical angles exist: One for the P-wave and one for the S-wave. According the graphs in [Ewing et al, 1957] the critical angles are about 20o and 30-40o respectively. In the preliminary study [Hölscher, 2010] values of 6-22 and 10-36 respectively are found, depending on the air-saturation of the water.

1. Angle = 0: direct wave from source to receiver through the water.

2. Angle is small, but above both critical angles: the Stoneley wave along the borehole will be generated. Additionally, the waves that are partly reflected against the edge of the borehole, might reach the receiver.

3. The angle is between the two critical angles: a refracted wave will be generated, which might also lead to a Stoneley wave (see the final remark in this Section).

4. The angel is above both critical angles: a compression wave and a shear wave are generated. These will reflect against the edge of the pile. Some of these waves will reach the receiver. This might happen after one or more reflections against the edge of the pile. Since the reflected waves are focussed towards the borehole, The decrease of this type of waves is mainly determined by the loss due to material damping and the reflections. The preliminary study [Hölscher, 2010] the case for a water filled borehole in a concrete pile was treated. This can be worked out here in more detail. Based on the study in Section 2.2 the Stoneley wave will be much slower then the compression wave in the water, say half of the speed. These waves will reach the reciever after the body waves (compression an dshear waves) in the concrete have reached the reciever, if the distance between te source and the reciever is suficient. The strength of the arrivals depends on the strength of the waves with direction of emission at the source, the material properties and the source reciever distance. In this report, we will follow the waves along the rays from the source to the receiver. Final remark in area 3 ‘the angle is between the two critical angles’:

The situation between two critical angles is at this moment not clear to me. The solution exists of a first potential that decreases exponentially with distance to the edge, and a second potential that is a propagating wave in the pile. At this moment is seems reasonable that this leads to a kind of surface wave with compression wave speed in the concrete and a shear wave which might reflect against the edge of the pile and finally reach the receiver.

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3.2 A perfect pile

In a perfect pile, the hole is in the centre of the pile, the diameter of both the borehole and pile are constant and the material properties are constant all over the pile.

For a pile with diameter 50 cm and source-receiver distance 50 cm and the properties in Table 3.1, we find the travel times for the available paths in Table 3.2. For the Stoneley wave 700 m/s is used. Figure 3.1 shows the paths which are mentioned in

Material Compression wave [m/s] Shear wave [m/s] Volumetric mass [kg/m3] water 1400 - 1000 concrete 3500 2200 2400 sand 300 123 2000

Table 3.1 Material properties of pile and water

Type of wave Path length [m] Arrival time [ms]

direct wave in water 0.5 0.36

direct pressure wave in concrete 0.5 0.16

surface wave concrete interface 0.5 0.72

single reflected pressure wave in concrete

0.7 0.21

single reflected shear wave in concrete 0.7 0.32 double reflected pressure wave in

concrete

1.12 0.32

double reflected shear wave in concrete 1.12 0.51

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Figure 3.1 Possible paths from source to reciever

3.3 A pile with a neck

Here, the work is limited to a symmetric neck. Two things happen:

The reflected waves which should reflect on the place where the neck is, cannot reflect. On the neck other waves are reflected.

This leads that there are source-receiver positions in which two single reflected waves arrive (from the perfect edge and the neck), but also positions where no single reflected waves arrive.

3.4 A pile with an inclusion

An inclusion is much harder to model, since it is a priori not symmetric. Moreover, the waves are partly reflected from the inclusion, but also partly transmitted. One reflected wave will surely give an additional ray that reaches the receiver. The transmitted waves will be dispersed by the inclusion acting like a lens.

3.5 A pile with an eccentric hole

In this case the angle of horizontal direction must be considered. The theory of the standard neck is applicable, but integration over all horizontal angles must be done. For each angle the actual pile radius differs. This leads to shorter and longer rays, which leads to dispersion in the signals: at the thin side the signals arrive earlier, at the thick side the signals arrive later. For the other waves (in the borehole or the refraction wave and Stoneley waves, the edge of the pile is far away and these waves are not considered.

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A non-symmetric neck can be solved using the same algorithm. 3.6 Alternative interpretation of the SHSL test

At the moment, the amplitudes of all types of waves are not known. The project proposal points at the simulation of the full measurements for some theoretical flaws (circular neck, circular inclusion). This might be interesting from the theoretical point of view and for the creation of a library with examples for interpretation of measurements. Nevertheless, it is time consuming.

It might be a suggestion to analyse measured signals beforehand. Using cross-correlations between the source and the receiver signals, it should be possible to distinguish the waves in the measurement and the relative strength of each wave.

Assuming the source signal is not known, a test in water can be carried out. The device is placed in a big basin filled with water. Using several source receiver distances, the wave speed in the water is measured and the time base of the source can be determined.

Then a signal in a good pile is interpreted, again for several distances between source and receiver (preferably similar to those used in the water basin. This will give the required answer: the importance of the waves.

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4 Modelling a neck in a pile

4.1 Introduction

A pile might have a neck in the shaft. This Chapter develops the ray-tracing model for this case.

A neck can be observed from the reflections of the body waves to the edge of the pile. Here, we limit the considerations to the most important waves: the compression waves. In the shear waves, similar phenomena should be observed.

4.2 Symmetric neck

Here is assumed that the problem is axial symmetry. A neck is defined by a circular ring, defined by a centre M and a radius Rn. See Figure 4.1. (The neck has the form of a donut

filled with soil.)

Figure 4.1 Sketch of a neck with main variables

The waves follow the rays. The source emits compression waves in the fluid filled borehole. These hit in the borehole edge and generate a compression wave and a shear wave. These waves reflect on the pile edge and will hit the borehole edge close to the receiver. The wave enters the water and will excite the receiver. Moreover, rays which reflects on the borehole and once again against the pile edge may also play a role. these are called multiple reflected rays.

The following two rays are possible in the pile with a neck:

M: center

R

_n

: radius of neck

R

_p

: radius of pile

Source

Reciever

+ z-axis

M: center

R

_n

: radius of neck

R

_p

: radius of pile

Source

Reciever

+ z-axis

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1. A reflection on the straight part of the pile edge. 2. A reflection on the edge in the neck.

It should be taken into account that sometimes both rays are possible.

Figure 4.2 elucidates this point. A pile with radius 30 cm is considered. At depth z = 0, a neck with depth 3 cm and radius 30 cm is assumed. A measurement device with 60 cm between source and receiver is simulated. Calculations are done for the source position from 30 cm above the centre of the neck to the level of the neck, at intervals of 3 cm, resulting in 11 calculations. The problem is symmetric in the plane z = 0.

For (the lower part of) a circular neck the reflection points are calculated, these are shown by dark blue circles. Green asterisks show the sources, the receiver positions are marked by red circles. Light blue diamonds shows the reflection points on the ordinary edge of the pile. Each source point relates to one receiver point (60 cm below) and one reflection point at the neck and one reflection point at the pile edge. The points are in each set ordered from top to bottom. Of course, points at the pile edge in the neck do not contribute to a physical reflection.

From Figure 4.2 it is concluded that in this case the first 4 or 5 points give a reflection of the neck only. The remaining points give a reflection on both the neck and the edge.

When we pass this neck, we first observe changes in the signal, since two reflections may occur. While approaching the neck, the path along the neck is a bit longer, so we observe a longer signal. Going along the neck, a point were the two paths have the same length is passed; there, a double signal is observed. Shortly after that situation, the normal pile edge reflection is not observed anymore, only the one from the neck. When the source and receiver are symmetric around the neck, the path reflected on the neck is a bit shorter, so that reflection arrives a bit earlier than normal. This pattern will repeat inversely on the symmetric part of the neck. It will occur for both the compression wave as well the shear wave.

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Figure 4.2 Source points, reflection points and receiver points close to a neck

Figure 4.3 Signals for reflected waves close to neck

For the specified case, the signals are calculated. Figure 4.3 shows the result. Each signal relates to a 3 cm displacement of the equipment. The center of the neck is at zero. When the first observation of the neck arrives at the transducer (at -33 cm) two separate signals are

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observed. Then, moving the equipment towards the neck, the two signals merge. The start of the signal is defined by the pile radius and this initially does not change. The peak is getting higher, due to the summation of the signals. Then suddenly (at this case close to level -10 cm) the reflections of the edge of the pile vanish, and only the reflection of the neck is observed. Since the neck is thinner than the pile itself, the signals arrive earlier then in the perfect pile. This scheme is made for a compression wave, a similar scheme will be observed for the shear wave, with different amplitude.

For a bulb with similar properties (thickness 3 cm, radius 30 cm), a comparable pattern will be found, but for the bulb the additional wave will be a mirror view compared with the neck. The reflection at the pile edge is assumed constant for all cases. The real value of the reflection coefficient depends on the properties of the material around the pile and the angle of incident of the incoming wave. These two aspects are neglected in the discussion in this chapter.

4.3 Non-symmetric neck

4.3.1 Principle

For an a-symmetric neck both the center (rM) and the radius (rn) might be a function of the

angle of horizontal direction:

0 1 0 1

sin(

)

sin(

)

n M M Mj rj j n n n nj Rj j

r

j

r

j

(1.8)

More general, also the height of the center might be a function of the angle

0 1

sin(

)

n M M Mj zj j

z

j

(1.9)

The non-symmetric neck means that in principle, each angle must be considered separately, and to obtain the required signals, integration over the angle must be carried out. This means that a numerical integration must be done.

In practical sense, the routine developed for the previous section (4.2) has been reordered, so that the summation over the angle is done. The implementation is worked out for the first order only (n=1), the phase angles are not implemented. The methodology is elaborated more strictly; the graph with reflection points cannot be made anymore.

4.3.2 Influences

To show the influences of a neck and the properties of the neck three cases are considered. Table 4.1 shows the properties

Property Symbol Mean value First term

radius of the neck rn 0.30 0.05

distance center to pile axis rM 0.55 0.05

depth center zM 0.00 0.10

distance source receiver d 0.6

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The pile diameter is 0.6. The integration over the angle is carried out using 120 steps. Eight sine waves with a frequency of 50 kHz are assumed. The amplitudes of all eight sine waves are equal.

First, the symmetric neck is presented. Figure 4.4 shows the reflection points. Physical impossible points are included, but of course, these are not taken into account for the calculation of the amplitudes

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Figure 4.5 Seismogram for symmetric neck

Figure 4.5 shows the calculated seismogram. The result is as expected. The neck distorts the waves. The lines with zero signals are noticeable. Here the reflections from the neck extinguish the reflections from the pile edge.

Figure 4.6 shows the seismogram for a comparable case, but the neck radius is not constant (first line of Table 4.1). The properties of the neck and distance of the centre is chosen so, that at one side the maximum neck is reached and at the other side the neck is just not observed. The cross-sectional area of the two cases in the neck is the same.

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Figure 4.6 Seismogram for non-constant neck radius [ Rn=0.3+0.05*sin( ) ]

The results are very comparable with those in Figure 4.5, but the signals in the center of the neck are more extinguished. The noticeable lines in Figure 4.5 are less pronounced, or almost invisible.

Figure 4.7 shows the seismogram for a comparable case, but the neck centre distance is not constant (second line of Table 4.1). This case is very similar to the case of non-constant neck radius, see Figure 4.6.

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Figure 4.7 Seismogram for non-constant center radius [ RM =0.55+0.05*sin( ) ]

Finally, Figure 4.8 shows the seismogram for a third case, where the neck centre height is not constant (third line of Table 4.1). This figure can be compared with Figure 4.5. The pattern is more or less similar, but for the non-constant height, the sharp changed are not visible.

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Figure 4.8 Seismogram for non-constant center height

4.3.3 Determining the depth of a neck

The test should be able to define a depth of a neck, since this is a flaw in a pile that might have important consequences. The length of the neck is constant: 0.5 m. The pile diameter is 1.2 m. The depth increases from 5 to 10 and 15% of pile radius (2.5-7.5% of diameter). These refer to cases where the cross-sectional area of the pile is reduced by 5, 10 or 15%.

Figure 4.9 shows the properties of the necks studied in this Section. The deeper the neck, the earlier the reflected signals in the middle of the neck arrive. Therefore, the arrival time is a measure for the depth of the neck. The length of the neck does not change, while the length of the disturbance shows a minor increase with depth. This means that the length of the neck can be reasonably estimated from the length of the disturbance. Half the length is an upper limit.

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form of the neck calculated signal

depth neck 2.5 cm

depth neck 5.0 cm

depth neck 7.5 cm

Figure 4.9 Influence depth of the neck on reflections

4.4 Borehole is not symmetric in the pile

The consequences of a borehole that is not in the centre of the pile can be studied using the model for a non-symmetric neck. The radius of the neck must be very large, and the neck centre position far away. The numerical pile radius must be large enough to avoid additional non-physical reflections.

This trick has been worked out with for a pile with radius 0.3 m and the borehole 0.05 m out of the centre: a neck radius of 30±0.05 m and centre distance 30.25 m has been used. Figure 4.10 shows the result. For this case, we observe that the first wave is a bit stronger and a

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ninth shadow wave. The strength of the shadow wave increases with the eccentricity of the borehole.

Figure 4.10 Result for a non-symmetric borehole

Finally, one might wonder whether it is possible to connect the tube for the device to the reinforcement. Assuming a pile with radius 0.3 m (diameter 0.6 m), the centre of the hole will be at e.g. 0.2 m from the pile axis. This is simulated by using a neck radius of 30±0.2 m and centre distance 30.30 m.

Figure 4.11 shows the results. The number of shadow waves increases to three plus a minor one in between. That means that such a measurement is possible, but the post processing should be adjusted to this case.

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Figure 4.11 Seismogram for measurement connected to reinforcement

4.5 Conclusion

The body waves that reflect on the edge of the pile and the neck provide a clear pattern. It seems possible to find these patterns in the measurements and it seems possible to derive the properties of the neck from a measurement: the smallest diameter from the pile can be estimated from the time shift in the center, the length can be estimated from the position where the reflections of the pile edge vanish.

For an asymmetric neck, the patterns are similar, but some other phenomena are visible. The asymmetry of the neck tends to mask the phenomena.

An asymmetric borehole influences the measurements, but seems not to be able to disturb a measurement completely. Even a measurement tube connected to the reinforcement seems possible, but requires a more advanced signal processing.

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5 An inclusion in the pile

5.1 General

An inclusion will create a complicated wave pattern in the pile.

Before discussing inclusions, an idea of what type of inclusions will be considered. In the preliminary study, a spherical inclusion was assumed. This can be symmetric (center of the inclusion is on the axis of symmetry) or a-symmetric. Another type of inclusion is a ‘layer’ in the pile, where the stiffness of the layer (inclusion) is much lower the stiffness of the pile. The fact that a spherical inclusion can influence the wave pattern by multiples should be mentioned, but it seems allowed to neglect that phenomenon. The first step is to find the influence of an inclusion. An inclusion can be considered as a spherical body in the pile, with properties that differs from the pile properties. In general, the stiffness and volumetric mass will be lower, due to e.g. low densification of the concrete or a gravel inclusion without cement.

5.2 Influence of a soft layer in a pile

Figure 5.1 shows the situation when the device enters a soft layer.

Figure 5.1 Sketch of situation while approaching a soft layer

stiff

soft

source

receiver

stiff

soft

source

receiver

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Several influences are to be considered:

1. the effect of the source changes (other entrance for Stoneley wave, strength body wave might change)

2. the Stoneley wave and the refraction wave reflect on the surface between the pile and the inclusion

3. the reflected waves will partly reflect on the surface

4. The lower wave speed will lead to a later arrival of the body waves 5. the transition from the body waves back to the receiver changes 5.2.1 Stoneley and refracted wave

First, effect number 2 is discussed. In fact, these two waves are ordinary one-dimensional waves, which will be reflected and transmitted against the boundary. Two effects are expected:

Due to the lower wave speed in the soft layer, the arrival time of the refraction wave will be retarded. This retardation increases with penetration of the equipment in the soft layer. Due to the transmission, the amplitude changes in a relative short interval.

The retardation is a linear function, which changes linearly from the wave speed in the concrete (when the receiver just enters the soft layer) to the wave speed in the inclusion (when the source enters the soft layer). The retardation can be calculated from the arrival times in the two materials (concrete and inclusion):

1 2 S r c S r i

z

t

c

z

t

c

(1.10)

The transmission depends on the impedance of the two materials for compression waves. The transmission coefficient reads:

1 1 2

2 (1

)

(1

) * (1 2 )

transmitted incoming

A

Z

T

A

Z

E

(1.11)

Assuming that Poison ratio is equal for the pile and the inclusion, the transmission coefficient reads 1 1 1 1 2 2

2 E

T

E

(1.12)

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Material Youngs modulus

[MPa] Poison ratio [-] Volumetric mass [kg/m3] concrete 30 0.2 2400 inclusion 6 0.2 2400

Table 5.1 Material properties for “soft layer” in the pile

Using these properties, the retardation of arrival time is 0.36 – 0.16 = 0.20 ms and the transmission coefficient is 1.36. This means that the signal that is transmitted has larger amplitude that the original signal.

5.2.2 Reflections of body waves

Secondly, effect number 3 (influence of reflections against the soft layer) is studied. The results are based on the properties mentioned in Table 5.1 and the same configuring as in the previous discussion: source at top, 60 cm above the receiver.

Figure 5.2 shows the resulting wave rays. The effect of the boundary on the propagation paths is clearly visible. The type conversions (at the edge of the pile and at the layer boundary are neglected)

Figure 5.2 The rays from source (at top) to receiver for a layer at z = 0 m

For both the compression wave (Figure 5.3) and the shear wave (Figure 5.4) the results are calculated. The arrival time in the concrete pile are 0.23 ms and 0.37 ms for the compression and shear wave respectively. These numbers can be recognized at entrance of the soft layer, so at vertical coordinate is 0.3 m.

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Figure 5.3 Compression wave arrivals while passing into a ‘soft layer’ in a pile

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For the transmission of the shear waves, it turns out that the transmission coefficients can be a complex number. This means that the transmission boundary generates an additional phase shift of the signal. This phase shift is introduced in the result for the frequency of 50 kHz. Here, it is assumed that this phase shift does not lead to significant deformations of the signal, but this must be tested more in detail.

It is concluded that the arrivals of both compression and shear waves generate a linear change in arrival time, together with a decrease of the amplitude. Since the source and the receiver are always in the same material (source in concrete, receiver in inclusion) this aspect does not play a role. However, since the impedance of the borehole edge material changes when the source (or receiver) enters the softer material, all signals are changing suddenly. If the soft layer has a finite depth (thickness) in axial direction, leaving the soft layer will lead to a similar but opposite behaviour. If the layer is thick, in the middle an area is found, where the arrival times of the waves does not change. If the soft layer is thinner than the distance between the source and the receiver, this area is not found.

5.2.3 Water solid interaction

Since the angle (with the horizontal plane) of the ray changes, the angle of incidence from the water to the concrete also changes. This leads to a change in transmission coefficient from the water to the concrete (at the source) and from the concrete to the water (at the receiver). Figure 5.5 shows the transmission of a compression wave from water to concrete as function of the angle of the transmitted compression wave, based on [Ewing et al, 1957].

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If the angles of the ray in the concrete are about 40-60 the efficiency of the source is not very strong dependant on the angle. However, for a high angle (thus large source receiver distance) the efficiency of the p-wave is extremely high. For low angles, the efficiency of S-waves is low.

5.3 Scattering of waves around a spherical or cylindrical inclusion

5.3.1 Considerations

The references in literature general studies complete analytical solutions, e.g. [Korneev & Johnson, 1993]. In general, it possible but very complicated to implement these in this problem. Therefore, a more practical approach is followed. This approach holds for inclusions that are large compared with the wavelength of the measurements.

Large inclusions can be considered as a body in the pile. On the boundary between the pile and the inclusion, reflection and transmission will occur. The transmitted wave might be reflected against the ‘rear side’ of the inclusion, of (after transmission through the rear side) against the pile edge, and finally reaches the receiver. However, due to the multiple passes of boundary-separations the relevance of these waves is expected to be low. These will be ignored.

The reflected wave will spread over a large area. Always one ray will reach the receiver. Some other rays might reach the receiver after reflection against the edge of the pile somewhere. In fact, many of such reflections can reach the receiver, but the amplitude will be low and the delay may also be (too) large. This means that these waves do not lead to reflections. This phenomenon will be called the shadow of the inclusion.

This reasoning leads to three phenomena, which must be considered (see Figure 5.6): 1. in the center of the inclusion a strong and fast reflection is generated

2. outside this area a shadow area is observed

3. the edge of the pile which is outside the shadow area will act as a normal reflector for the waves.

Figure 5.6 Zones in cross-section of a pile with inclusion

point reflection shadow zone inclusion point reflection shadow zone shadow zone inclusion

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This idea will be work out for a spherical inclusion, defined by a center and a radius. Without loss of generality, the center of the inclusion is placed on position = 0. First, it is assumed that the center of the pile is outside the inclusion. Considering the plane = 0, the ray will behave similar as for the neck. The strength of the reflection depends on the distance between the source, receiver and inclusion edge, thus the path length. Outside the plane, we reach first the shadow area, where no reflections that reach the receiver are generated. In addition, for larger values of the angle standard pile edge reflections are found. This means that the calculation method for the neck can be applied, using appropriate factors for the waves. These factors depend on the properties of the neck.

Until now, the reflection coefficient of the pile edge is not discussed. No need for, it is everywhere the same. Now however, the reflection coefficient of the inclusion may differ from the reflection coefficient of the edge of the pile. This will be included by a factor for the reflection coefficient on the inclusion. If the impedance of the inclusion is higher than the one of the soil, this factor is lower than 1, is the impedance of the inclusion is higher, the factor will be lower than 1.

5.3.2 Worked example

A pile with a diameter of 1.2 m is considered. The inclusion has a radius of 0.6 m; the center of the inclusion is 0.8 m from the axis of the pile.

Figure 5.7 Simulated wave pattern at a inclusion of radius 0.6 m with centre 0.8 m from axis

Figure 5.7 shows the result. At depth z = 0 m, the front wave from the inclusion is just visible. The amplitude of the main wave is visually almost constant. However, if the amplitude of the waves that are reflected outside the shadow area is drawn as a function of depth, the

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differences are more visible, see Figure 5.8. Since the amplitude decreases gradually, this effect is not easy visible.

Figure 5.8 Influence of shadow zone on amplitude of reflected waves

5.4 Conclusions

The results in this chapter show that all types of inclusions in the pile all have their own typical wave pattern. This makes it in principle possible to distinguish inclusions. However, it should be taken into account that the effects are rather small. To get these effects from a field measurement, will need a high level of signal processing

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6 Analysis of measurements

6.1 Preliminary analysis

Brem Funderingsexpertise BV in Reeuwijk delivered three measurements. These measurements are on the same pile with three source receiver distances: 80, 60 and 40 cm. Based on the assumption that a wave with frequency 50 kHz is generated and the available date the following additional data are derived:

Sample time: Dt = 2 s.

The source generates eight sine waves during a test.

Furthermore, it is assumed that the source signal is modulated by a perfect sine wave. Figure 6.1 shows the resulting assumed source signal.

Figure 6.1 Synthetic assumed source window

6.2 Cross correlation for one signal

This source signal is cross-correlated by the measurement with depth number 10. Figure 6.2 shows the result for source-receiver distance 80 cm.

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Figure 6.2 Result cross correlation for distance 80 cm

The peaks in Figure 6.2 are related to durations of wave propagation. The higher the peak, the stronger the wave is. The expected length of a bump is twice the duration of the original wave, thus 2*0.16 = 0.32 ms (cf Figure 6.1). Four clear peaks are visible at 0.18, 0.28, 0.42 and 0.65 ms. Maybe, additional peaks are observed at 0.50(?) and 0.77 ms.

The first arrival wave is at 0.18 ms. This must be the head wave in the concrete, showing a wave speed of 0.8 m/0.18 ms = 4.44 m/ms. This is quite high for concrete. It must be taken into account that this value is a high estimate, since the wave partly travels through the water. Now it is assumed that the peak at 0.42 ms is the reflection of the compression wave (PP). With wave speed 3.5 m/ms the length of this path is 1.47 m. The radius of the pile should then be 0.62 m (diameter = 1.24 m), which is quite reasonable. Using the value from the head wave, the path length is 1.86 m (pile diameter = 1.68 m).

The reflected shear wave (SS) should arrive at 0.42*3.5/2.2 = 0.68 ms (The relative wave speed depends on Poison ratio only, so the theoretical wave speeds can be used). This value is quite close to the observed value 0.65 ms.

The other peak between these two waves might be related to the PS and SP reflections, which arrive almost at the same moment. The arrival time should be between the PP and SS waves. The wave speed in the water is 1.5 m/ms (or lower). This wave takes 0.53 ms to reach the receiver. This is another explanation for the peak at 0.50 ms.

The wave arrival at 0.77 ms is presumably a multiple reflection of the P-wave (the PPPP ray). The second wave at 0.28 ms is at this moment not recognised. Maybe the peak at 0.28 ms is related to the head wave. This leads to a compression wave speed of 2.9 m/ms for the

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compression wave in concrete. The arrival at 0.18 ms is than related to the equipment. It seems reasonable to assume that the source and the receiver have a structural connection. The related wave speed is 4.44 m/ms

To be complete: the wave speed of 2.9 m/ms leads to a path length for the PP ray of 1.22 m, and a pile diameter 0.90 m.

This last option is very realistic. Therefore, the other two measurements will be evaluated using the compression wave speed of 2.9 m/ms. The shear wave speed is then estimated to be 1.8 m/ms.

6.3 Generalisation for the measurements

Using these wave speeds and pile diameter 0.9 m, Table 6.1 shows the estimated arrival times for other pile diameters. The path length for the PS and SP waves are not corrected for the exact reflection point.

wave wave speed 80 cm 60 cm 40 cm

equipment 4440 0.36 0.27 0.18 head wave 3500 0.55 0.41 0.28 PP 3500 0.42 0.37 0.34 PP/SS 2850 0.50 0.45 0.41 SS 2200 0.63 0.57 0.52 PPPP 2850 0.50 0.45 0.41

Table 6.1 Arrival times as function of pile diameter for several waves

For the interpretation of the table, it should be taken into account that the duration of the source is 8*0.02 ms = 0.16 ms. Thus, arrivals less than this duration starts before the end of the previous reflection. However, since the strength of the generated waves is not constant, it seems reasonable to use half of this duration.

This leads to a rule of thumb that the distance between the transducers must be about the pile diameter from each other.

The method developed above is generalised for all measurements.

Appendix C shows some measured signals for all three distances, Appendix B shows the cross correlation functions for some selected singles, see Appendix A. Figure 6.3, Figure 6.4 and Figure 6.5 show the result of the cross correlation with depth for source receiver distances of 80, 60 and 40 cm respectively. The left figure give the highest value of the integral; the right figure the delay for which this highest value was found. The depth between two successive measurements is assumed 1 cm, so the depth -1 refers to measurement number 100.

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Figure 6.3 Result cross correlation for distance 80 cm

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7 Conclusions and recommendations

This report shows that the ray-tracing theory is an extremely strong tool to study the reflection patterns for flaws in piles for the application of single hole sonic logging.

The source of a SHSL device generates several waves. All of them can be evaluated using ray tracing theory:

The amplitude and arrival times of the head wave, which propagates along the edge of the borehole, delivers information on inclusions that are in the centre of the pile.

Stoneley waves along the borehole may be observed, but are in general very slow. However, if observed, these give similar information as the head waves.

The body waves (single and eventually multiple reflected waves) give information on necks and inclusions in the pile. It seems possible to derive the dimensions of the flaws from the measurements.

This report offers many possibilities for continuation. E.g.

Develop a method for back analysing measured signals. A distinction between wave types in measured signals can be developed. The distinguished waves can be interpreted more easily.

Development of signal processing tools for the case that the tube is connected to the reinforcement.

Analyse the optimal distance between source and receiver as a function of pile diameter, taking into account the dimension of the flaw. This is useful for practical engineers in order to determine flaw properties, if a first measurement indicates a flaw.

Given a measurement set-up, a catalogue of flaws and related observations can be created.

Develop a method that generates a complete signal given a flaw and a measurement set-up.

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8 References

[Brekhovskikh and Godin, 1998] Brekhovskikh, L.M., Godin, O.A. Acoustics of layered media I

Springer-Verlag Berlin, ISBN 3-540-64724-4 [de Bruin, 2011a]

de Bruin, H.

private communications per e-mail [Ewing et. al, 1957]

Ewing, W.M., Jardetzky, W.S. and Press, F. Elastic waves in layered media

McGraw-Hill Book Company, New York 1957 [Henry, 2005]

Henry, F.

Characterization of borehole fractures by the body and interface waves

PhD-thesis, Delft University of Technology, Delft, the Netherlands, January 2005 [Hölscher, 2010]

Hölscher, P.

Single hole sonic logging, vooronderzoek (in Dutch)

Deltares report 1202240-0003-geo-001, draft October 2010 [Korneev & Johnson, 1993]

Korneev, V.A. & Johnson, L.R.

Scattering of elastic waves by a sperical inclusion, part 1 and 2

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Evaluation modelling A-1

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Stieltjesweg 2, NL 2628 CK DELFT P.O. Box 177, NL 2600 MH DELFT

Telephone Telefax 31 (0) 88 335 72 00 31 (0) 15 261 08 21 Homepage: www.deltares.nl date drw. ctr. 194_6_s_80cm 2011-08-01 1202249.003 Hols none 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 0.1 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 0.4 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 0.7 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 1 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 1.3 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 1.6 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 1.9 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 2.2 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 2.5 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 2.8 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 3.1 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 3.4 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 3.7 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 4 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 4.3 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 4.6 m

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Telephone Telefax 31 (0) 88 335 72 00 31 (0) 15 261 08 21 Homepage: www.deltares.nl date drw. ctr. form. A3

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0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 0.1 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 0.4 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 0.7 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 1 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 1.3 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 1.6 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 1.9 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 2.2 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 2.5 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 2.8 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 3.1 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 3.4 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 3.7 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 4 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 4.3 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 4.6 m

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Telephone Telefax 31 (0) 88 335 72 00 31 (0) 15 261 08 21 Homepage: www.deltares.nl date drw. ctr. 194_6_s_40cm 2011-08-01 1202249.003 Hols none 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 0.1 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 0.4 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 0.7 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 1 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 1.3 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 1.6 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 1.9 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 2.2 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 2.5 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 2.8 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 3.1 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 3.4 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 3.7 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 4 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 4.3 m 0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 −500 0 500 1000 4.6 m

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Evaluation modelling B-1

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Telephone Telefax 31 (0) 88 335 72 00 31 (0) 15 261 08 21 Homepage: www.deltares.nl date drw. ctr. 194_6_s_80cm 2011-08-01 1202249.003 Hols none 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{0.1 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{0.4 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{0.7 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{1 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{1.3 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{1.6 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{1.9 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{2.2 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{2.5 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{2.8 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{3.1 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{3.4 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{3.7 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{4 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{4.3 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{4.6 m}

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Telephone Telefax 31 (0) 88 335 72 00 31 (0) 15 261 08 21 Homepage: www.deltares.nl date drw. ctr. form. A3

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cross correlations each 30 cm

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0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{0.1 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{0.4 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{0.7 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{1 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{1.3 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{1.6 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{1.9 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{2.2 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{2.5 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{2.8 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{3.1 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{3.4 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{3.7 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{4 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{4.3 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{4.6 m}

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Telephone Telefax 31 (0) 88 335 72 00 31 (0) 15 261 08 21 Homepage: www.deltares.nl date drw. ctr. 194_6_s_40cm 2011-08-01 1202249.003 Hols none 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{0.1 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{0.4 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{0.7 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{1 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{1.3 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{1.6 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{1.9 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{2.2 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{2.5 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{2.8 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{3.1 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{3.4 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{3.7 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{4 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{4.3 m} 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3x 10 4 _{4.6 m}

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Evaluation modelling C-2

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