# Numerical modelling of ultrasonic wave propagation in concrete using embedded sensors

Rouba Hariri, Vincent Garnier, Jean François Chaix, Odile Abraham

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Rouba Hariri, Vincent Garnier, Jean François Chaix, Odile Abraham. Numerical modelling of ultrasonic wave propagation in concrete using embedded sensors. NDE NucCon 2023 - International Conference on Non-destructive Evaluation of Concrete in Nuclear Applications, VTT Technical Research Centre of Finland; Aalto University; BAM - Bundesanstalt Für Materialforschung und -prüfung, Jan 2023, Espoo, Finland. pp.296-306. [https://hal-04913656v1](https://hal-04913656v1)

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# Numerical modelling of ultrasonic wave propagation in concrete using embedded sensors

Rouba Hariri<fnref n="1" />, Vincent Garnier<fnref n="2" />, Jean-François Chaix<fnref n="2" /> and Odile Abraham<fnref n="1" />

## ABSTRACT

Due to the importance of reinforced concrete for civil engineering structures, monitoring the state of concrete over time has become a special interest in non-destructive testing. Although many different studies have been performed on concrete monitoring, it is still a challenge to have valid information on the first centimeters of the structure. An ultrasonic measurement system design is proposed using pairs of ultrasound sensors that are embedded inside the concrete structures during the construction phase. This system is designed to contribute to the local evaluation of the most relevant indicators in the first 20 cm of the structure: compressive strength, Young's modulus, porosity, and water content. The design involves the development of transmitter and receiver probes embedded in concrete to measure, as a function of depth, the evolution of two parameters: ultrasonic velocity and attenuation. A frequency pulse is sent by the emitter, and the receiver records the wave after propagation. Based on the recorded signal at regular time intervals, information about the micro-structure of concrete can be monitored. A set of sensors with frequencies from 50 kHz to 200 kHz is currently being tested in small concrete samples to define a system suitable for large full-scale mock-ups. To have accurate information, the uncertainties generated by the heterogeneity on the observables must be quantified, evaluated and compared to perturbations due to indicator's properties change.

**Keywords:** Concrete, Non-Destructive Testing, Ultrasound, Durability indicators.

## 1. INTRODUCTION

Monitoring and non-destructive evaluation (NDE) have become the main focus in civil engineering as safety and environmental concerns have shown the necessity for methods to ensure structural reliability and increase the lifetime of concrete structures. NDE methods based on ultrasound propagation have shown the possibility of overcoming the limitations of traditional methods. Ultrasonic testing has some capacity to estimate the structure of different alloys of the components having different acoustic properties. The results obtained are immediate and, in several cases, only one surface requires to be accessed [1]. The velocity and attenuation of ultrasonic waves can be used to determine the elastic properties (Young Modulus, Poisson ratio) and also to characterize the micro-structural properties of concrete (porosity, grain size, micro-cracks, etc.). Although different studies have been conducted in concrete monitoring, it is still a challenge to have valid data on the first centimeters of the structure.

This work is part of a French ANR Project called "Scaning" whose main objective is to equip the structures with onboard sensors to evaluate the state of concrete as a function of time. The final purpose of this work is to design a multi-physics monitoring system (that combines ultrasonic, electromagnetic, resistivity, and capacitive methods) able to evaluate the changes in concrete property gradients over time, more specifically the saturation gradient. Recently, piezoelectric transducers have been embedded into concrete structures for non-destructive permanent monitoring of concrete ([1], [3]). Large volumes of concrete structures are monitored for changes in material properties by a limited number of transducers. Compared to usual ultrasonic transducers, using embedded transducers improves the proficiency and reproducibility of ultrasonic tests and allows measurements in unreachable places when the structure is in service ([4], [5]).

<footnote n="1">¹Univ Gustave Eiffel, GERS-GeoEND, F-44344 Bouguenais, France</footnote>

<footnote n="2">²LMA UMR7031, Aix Marseille University, F-13625 Aix en Provence, France</footnote>
This paper focuses more particularly on ultrasonic wave measurements such as the time propagation and the maximum amplitude of the wave in concrete. A 2D numerical model is built and validated to study the ability of variable parameters, such as heterogeneity distribution and saturation ratio, represented by changes in the material properties, to influence the quality of the measurements. The variations generated by each parameter are estimated using numerical modeling in SPECFEM software [6]. A validation approach is first established to optimize the input parameters in the numerical model. This approach is detailed in the first part of the study. Then, data processing is used to calculate time propagation and amplitude in various configurations. The influence of heterogeneity and saturation ratio in concrete on the studied observables is presented. To accurately quantify all the uncertainties related to the material, no noise is added to the signal in this numerical study.

## 2. PROBLEM STATEMENT

An ultrasonic measurement system design is suggested using one pair of ultrasound sensors embedded inside the concrete structure. The Spectral Element Method (SEM) in the elastic domain is used to test several configurations of a concrete model in order to study the influence of physical and ultrasonic parameters on wave travel time and amplitude [8]. Significant effort will also be made to clean up information collected from the unwanted influential parameters in order to estimate the precision that can be expected in the field for travel time and attenuation measurements. First, the material properties and the frequency domain applied in the simulations are presented. The physical parameters for the simulations are similar to those used in [9]. Second, a validation of the numerical approach is presented to optimize the input parameters in the 2D model and therefore reduce the numerical uncertainties of the problem. Then, other configurations are tested to analyze the effect of the influencing factors (heterogeneity distribution, material properties change) on the observables.

## 3. MATERIALS AND 2D NUMERICAL MODELLING

### 3.1 Materials properties

The material properties used in the following numerical simulations are given (Table.1). The concrete properties in this paper are set to correspond to the values given in [8].

<table><caption>Table 1 Material properties for the matrix and the circular aggregates ($\rho$ is the material density, E is the Young's modulus, v is the Poisson's ratio, and Vp and Vs are respectively the longitudinal (pressure) and transverse (shear) wave velocities) [8]</caption><thead><tr><th>Material properties</th><th>Mortar</th><th>Aggregate</th></tr></thead><tbody><tr><td>ρ (kg/m³)</td><td>2050</td><td>2610</td></tr><tr><td>E (GPa)</td><td>26</td><td>40</td></tr><tr><td>v</td><td>0.26</td><td>0.25</td></tr><tr><td>Vp (m/s)</td><td>3950</td><td>4300</td></tr><tr><td>Vs (m/s)</td><td>2250</td><td>2475</td></tr></tbody></table>

To be able to generate the structure, the distribution of the aggregates should be representative of real concrete. An example of the cumulative volume fraction of aggregates as a function of the size taken in [8] is presented (Fig.1). In this study, assuming that aggregates smaller than 4 mm are homogenized in the mortar, the volume fraction of aggregates is taken at 40%. To define a random distribution of these aggregates, a Gaussian distribution is used in which the mean and variance values are respectively $\mu = 7$ mm and $\sigma = 2$ mm (Fig.2). The aggregate radius varies between 4 mm and 10 mm.
<figure><img src="image_2.png" /><figcaption>Figure 1. Cumulative volume fraction of aggregates as a function of size for VeRCoRs concrete</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2. Gaussian distribution for random circular radii between 4 mm and 10 mm</figcaption></figure>

**3.2 Numerical modelling**

Numerical modeling will be used to conduct a sensitivity study to evaluate the variations of the studied observables and to define the uncertainties in the experimental model. It would therefore be considered an "experiment" with digital tools that allow us to optimize the conditions of the real experiment set up. SPECFEM software is a free code based on the Spectral Element Method (SEM), allowing the simulation of wave propagation in acoustic, elastic, and viscoelastic conditions [7]. It is used in this work to simulate elastic wave propagation in both homogeneous and heterogeneous domains with randomly distributed circular aggregates [6]. We simulate a bounded computational domain by creating artificial boundaries on which absorbing conditions are imposed. The role of the absorbing boundaries is to avoid spurious waves from the reflection on the edges.

**3.2.1 Geometry of a 2D numerical model.**

The 2D numerical geometry is illustrated (Fig.3). An incident longitudinal wavelet propagates across the study domain [X,Y] = [18 cm, 8 cm] from the left to the right. In this paper, the emitter is a point source that generates a wave through the specimen. The emission signal which imposes the displacement corresponds to a Ricker signal that has a maximum amplitude and two minimums (Fig.4). In the snapshot, the color scale represents the displacement amplitude, with red color for a positive amplitude displacement and blue color for a negative amplitude displacement (Fig.3). The receiver is punctual and is placed in the study domain to record the displacements of the wave in both the X and Y directions of the wave field. The distance between the receiver and the emitter is d = 10 cm. This numerical geometry is meshed in the GMSH software [10]. To increase the precision
of measurements in the simulations, a function is used in SPECFEM software to optimize the mesh quality, including mesh size and element skewness.

<figure><img src="image_4.png" /><figcaption>Figure 3. Numerical geometry of the study domain: Example of a heterogeneous medium with cylindrical diffusers of various radii</figcaption></figure>

### 3.2.2 Frequency domain

In concrete, high-frequency (up to 500 kHz) is suggested [4] to carry out a faster-rising edge of the wave and therefore a reduced error when pointing its arrival time. Nevertheless, the large attenuation in concrete (due to scattering and viscous damping) controls the effective range of frequencies over which waves can be coherently propagated to a given distance. In this study, the wave frequency domain is limited to a maximum frequency of 200 kHz, which corresponds to a wavelength of around 24mm when compared with the larger aggregate size of 10mm. A Ricker wavelet with a peak frequency equal to 50kHz, 100 kHz, or 200kHz is taken as an incident longitudinal wave.

## 4. STUDIED OBSERVABLES AND MEASUREMENT METHODS

In this paragraph, the methodology to measure the propagation time ($t_p$) and the maximum amplitude ($A_{max}$) of coherent P-waves in a homogeneous medium is presented. Studies show that these two parameters can provide complementary information on concrete properties.

<figure><img src="image_5.png" /><figcaption>Figure 4. Displacement measurement simulation for the case of homogenized concrete</figcaption></figure>

### 4.1 Propagation time of the longitudinal wave ($t_p$) and wave velocity

Wave propagation time ($t_p$) is the travel time from the start point of the source wavelet ($t_{0\_ES}$) to the arrival time of the received signal ($t_{0\_RS}$) (Fig.4).

$$t_p = t_{0\_RS} - t_{0\_ES} \qquad (1)$$

Many approaches have been used to accurately pick the arrival time of the wave: The tangential line approach [11] requires drawing a tangential line from the point of the greatest gradient in front of the first peak and finding the intersection of that line with the time axis. The advantage of this approach is that only data near the rising part of the signal is necessary. However, when using this approach, the arrival time value is sometimes slightly earlier than the experimental value. Another
method is to choose the arrival time of the signal as the instant when the amplitude of the signal exceeds the noise level by a factor of 4, which makes it possible to detect the signal even if the noise is very large [8]. Because noise is not considered, a certain time range is selected between the beginning of the signal and the time corresponding to its first peak ($t_{max}$) (Fig.4). In this range, the arrival time ($t_{0_RS}$) is determined as the instant when the amplitude is higher than the peak amplitude by a factor of 0.01. This method, hereafter called the threshold method, was found to give the best match between the measured value of $t_p$ and the actual value calculated in the homogeneous medium (Fig.4).

In the following the velocity ($V_{P-wave}$) of the wave is determined as:

$$V_{P-wave} = \frac{d}{t_p} \qquad (2)$$

In the case of a homogeneous material with the properties of the aggregates (table 1), the propagation time and the velocity of the wave are found to be in the same order as the values in the case of a homogeneous material without aggregates. When comparing the numerical values to the experimental results, the errors in time and velocities are found to be 1.3% for the tangential method and 0.6% for the threshold method. Therefore, the latter method is chosen in the following to calculate the propagation time of the wave.

## 4.2 Maximum Amplitude ($A_{max}$)

The amplitude of a wave is a very precise tool to evaluate material changes. It is not frequently used as it requires reproducible coupling of the ultrasonic transducers, which is not always feasible. Using embedded sensors eliminates coupling complications and permits the evaluation of the ultrasonic wave amplitude. For this paper, the change in amplitude is monitored at the first peak of the received signal (Fig.4). The maximum amplitude is considered to be an ultrasonic wave propagation index that reflects variations in apparent viscosity related to changes in physical properties during the evolution of concrete [12]. Furthermore, the variation of the maximum amplitude is related to the heterogeneity of the concrete and gives information about the propagation paths of waves: a wave that propagates in the same direction as the incident wave after diffraction by heterogeneity has a lower amplitude than the one that does not interact with it. In this paper, we are interested in studying the variation of amplitude as a function of heterogeneity and material changes in concrete. The observable that is used in the following is the maximum amplitude at the first peak of the received signal.

# 5. RESULTS AND DISCUSSIONS

## 5.1 Validation of the numerical modelling at f=100 kHz

In order to validate the numerical approach used in this study, a comparison was made between the results of the simulation on a homogeneous matrix composed of mortar and a homogeneous matrix composed of mortar and aggregates with the same mechanical properties as the mortar. This computation makes possible the quantification of the errors related to the mesh only. First, aggregates are added to the homogeneous medium. The source wave frequency is 100 kHz and the wave traveling distance is 10 cm. The wave field recorded for the mortar and aggregates matrix is compared to the wave field recorded for the mortar matrix and is presented in addition to the source wave (Fig.4). Wavefronts are overlaid. The variation of the amplitude of wave displacement in one direction and the variation of the arrival time of the wave are determined as:

$$\Delta A_{max} = A_{max\_homo} - A_{max\_homo\_with\_aggregates} \qquad (3)$$

$$\Delta t_{0\_RS} = t_{0\_RS\_homo} - t_{0\_RS\_homo\_with\_aggregates} \qquad (4)$$

These variations are equal to $\Delta A_{max} = 0.02\%$ and $\Delta t_{0\_RS} = 0.007\%$. The errors related to mesh generation are below 0.02% both for time travel time and amplitude, so the numerical uncertainties are negligible compared to other variations that will be detailed in this paper, such as variations related to the heterogeneity and to property changes in concrete.
## 5.2 Influence of the heterogeneity of concrete on the studied observables

During their propagation time, ultrasonic waves will interact strongly with the multiple heterogeneities of the concrete. The resulting effects due to these interactions should be quantified in order to separate them from the ones related to the evolution of the saturation ratio in concrete. Aggregate distribution was found to have a significant effect on measured travel time and ultrasonic wave velocity [9]. In several studies, ultrasonic measurements were performed on concrete samples and the wave travel time was recorded and used to calculate the wave velocity, which was used as a parameter in the evaluation of concrete properties [13].

Concrete and aggregate properties are presented in Table 1, with spherical aggregates whose surface ratio is 40% and whose size distribution varies from 4 mm to 10 mm (Fig.1). Numerical simulations are performed to study wave propagation in 10 different disorder realizations using an emission frequency of 100 kHz. The aim of using multiple configurations is to quantify the uncertainty of velocity measurement related to the distribution of aggregates. For each realization, the number of aggregates between the source and the receiver is different. Therefore, in each simulation, the source-receiver distance is the same (d=10 cm) but the waves propagate along different paths. Once these variations are quantified, they will be compared to the ones related to property change in concrete in part 5.3. An example of a wave propagation snapshot taken during the simulation of the first realization is illustrated (Fig 3). Emitted and received signals are illustrated for two different disorder realizations in Fig.5 and Fig.6.

<figure><img src="image_6.png" /><figcaption>Figure 5. Displacement measurement simulation for aggregates distribution n°1</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6. Displacement measurement simulation for aggregates distribution n°2</figcaption></figure>

The propagation time ($t_p$) and maximum amplitude ($A_{max}$) of P-waves are calculated and $\Delta t$ is the variation between the propagation time of the wave for each disorder realization ($t_p$) and the average travel time of all realizations ($t_{ave}$) (Table2).
<table><caption>Table 2 Variation of observable values related to the heterogeneity of concrete for 10 realizations of disorder</caption><thead><tr><th>disorder realization number</th><th>Maximum amplitude of the received signal (A<sub>max</sub>) (10<sup>-12</sup> m)</th><th>Propagation time of the wave (tp) (10<sup>-6</sup> s)</th><th>Δtp (%) variation of tp compared to the average travel time of the wave</th></tr></thead><tbody><tr><td>1</td><td>0.88</td><td>25.07</td><td>0.83%</td></tr><tr><td>2</td><td>0.76</td><td>25.19</td><td>0.36%</td></tr><tr><td>3</td><td>0.78</td><td>25.49</td><td>0.82%</td></tr><tr><td>4</td><td>0.86</td><td>25.22</td><td>0.23%</td></tr><tr><td>5</td><td>0.87</td><td>25.27</td><td>0.03%</td></tr><tr><td>6</td><td>0.69</td><td>25.43</td><td>0.58</td></tr><tr><td>7</td><td>0.92</td><td>24.97</td><td>1.22%</td></tr><tr><td>8</td><td>0.67</td><td>25.37</td><td>0.35%</td></tr><tr><td>9</td><td>0.71</td><td>25.48</td><td>0.78%</td></tr><tr><td>10</td><td>0.77</td><td>25.33</td><td>0.19%</td></tr><tr><td>Average value</td><td>0.791</td><td>25.282</td><td>0.539%</td></tr><tr><td>Standard deviation</td><td>+/- 0.08</td><td>+/- 0.164</td><td></td></tr></tbody></table>

Results show that the variation in the propagation time of the P-wave related to the heterogeneity of concrete can be up to 1.22%. Therefore, given that these variations are quantified, the ultrasonic observables are tested to verify their ability to accurately measure other variations, related to the evolution of the water gradient in concrete, despite the observed influence of heterogeneity.

### 5.3 Impact of the variation of the saturation ratio on the studied observables for one disorder realization

Water saturation profiles in concrete are essential to evaluate its durability and can be determined using non-destructive measurement methods. Ultrasonic methods are applied for strength assessment through elastic modulus measurement. Several studies have been performed to establish relations between the saturation ratio and the strength of concrete [14].

The aim of this study is to examine the possibility of the NDE of concrete from a knowledge of the relationship between the saturation ratio and the ultrasonic propagation in concrete. The concrete saturation rate represents the volumetric content of free water in the material. Its value is between 0% (completely “dry” material) and 100% (completely saturated), although these extreme values are never reached in practice [15].

In the current investigation, the same configuration of concrete (disorder realization N°1 in table 2) with changes in the material properties, representing different saturation ratios, is conducted to collect reliable data for estimating the propagation time of the wave. To accurately describe the variation of the saturation ratio in concrete, the properties (ρ, Vp, Vs) are increased from 1 to 5% and are compared to the properties (ρ, Vp, Vs) of the concrete at the initial state 0. Results are presented in table 3.
<table><caption>Table 3 Variation of the studied observables related to variation (%) of the mortar properties (disorder realization n°1 of table 2)</caption><thead><tr><th>Percentage of mortar properties change</th><th colspan="3">Aggregates properties</th><th colspan="3">Mortar properties</th><th>Maximum Amplitude of the received signal</th><th>Propagation time of the wave</th><th>Δtp =<br>t<sub>pi</sub> - t<sub>p0</sub><br>t<sub>pi</sub><br>With i=[1,5]</th></tr><tr><th>(%)</th><th>ρ<br>(kg/m<sup>3</sup>)</th><th>Vp<br>(m/s)</th><th>Vs<br>(m/s)</th><th>ρ<br>(kg/m<sup>3</sup>)</th><th>Vp<br>(m/s)</th><th>Vs<br>(m/s)</th><th>Amax<br>(10<sup>-12</sup>m)</th><th>tp<br>(10<sup>-6</sup>s)</th><th>(%)</th></tr></thead><tbody><tr><td>0</td><td rowspan="5">2610</td><td rowspan="5">4300</td><td rowspan="5">2475</td><td>2050</td><td>3950</td><td>2250</td><td>0.88</td><td>25.07</td><td></td></tr><tr><td>1</td><td>2070.5</td><td>3989.5</td><td>2272.5</td><td>0.869</td><td>24.97</td><td>0.39</td></tr><tr><td>2</td><td>2091</td><td>4029</td><td>2295</td><td>0.86</td><td>24.94</td><td>0.51</td></tr><tr><td>3</td><td>2111.5</td><td>4068.5</td><td>2317.5</td><td>0.844</td><td>24.92</td><td>0.56</td></tr><tr><td>4</td><td>2132</td><td>4108</td><td>2340</td><td>0.829</td><td>24.77</td><td>1.19</td></tr><tr><td>5</td><td></td><td></td><td></td><td>2152.5</td><td>4147.5</td><td>2362.5</td><td>0.816</td><td>24.61</td><td>1.83</td></tr></tbody></table>

Results show that when the change of the mortar properties increased from 1% to 5%, both the maximum amplitude and the propagation time of the wave decreased.

The variation of the studied observables related to the change of mortar properties is found to be comparable to that related to the heterogeneity of concrete. Only mortar variation superior to 5% gives a variation of arrival time $\Delta t_{p\_mortar\_changes} < 1.83\%$ superior to $\Delta t_{p\_heterogeneity}$ than can take values as high as 1.22%. Therefore, the measurements taken at a certain depth below the concrete surface should be compared to a reference state to be able to measure the effect of the water gradient in concrete despite the variations due to the heterogeneity of concrete.

## 5.4 Influence of sensors emission frequency on the studied observables

The sensor's emission frequency is an important parameter that can present sources of errors and uncertainties. The influence of this parameter on the performance of observables has been acknowledged in [13]. When changing the emission frequency, the wavefront changes and therefore the terms defining "tp" and "Amax" become frequency dependent.

Figures 7, 8, and 9 show waveforms of the source (Ricker signal) and the received signal of numerical simulations that are conducted for a heterogeneous medium at various excitation frequencies of 50 kHz, 100 kHz, and 200 kHz. Results are illustrated in tables 4 and 5 respectively for the homogeneous medium and the heterogeneous medium of disorder realization n°1.

<figure><img src="image_8.png" /><figcaption>Figure 7. Displacement measurement simulation with a transmission frequency f=50 kHz</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 8. Displacement measurement simulation with a transmission frequency f=100 kHz</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 9. Displacement measurement simulation with a transmission frequency f=200 kHz</figcaption></figure>

<table><caption>Table 4 Variation of observables related to the emission frequency in the homogeneous medium having the properties of the mortar (Table 1)</caption><thead><tr><th>Sensors emission frequency</th><th>Maximum amplitude of the received signal (Amax) (10<sup>-12</sup>m)</th><th>P-Wave propagation time (tp) (10<sup>-6</sup>s)</th><th>Δtp = <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">d / V<sub>p</sub> - t<sub>p</sub></div><div style="padding: 0 5px;">d / V<sub>p</sub></div></div></th></tr></thead><tbody><tr><td>50 kHz</td><td>1.56</td><td>25.69</td><td>1.5%</td></tr><tr><td>100 kHz</td><td>1.03</td><td>25.47</td><td>0.63%</td></tr><tr><td>200 kHz</td><td>0.69</td><td>25.29</td><td>0.079%</td></tr></tbody></table>

<table><caption>Table 5 Variation of observables related to the emission frequency in the heterogeneous medium (disorder realization n°1 of table 2)</caption><thead><tr><th>Sensors emission frequency</th><th>Maximum amplitude of the received signal (Amax) (10<sup>-12</sup>m)</th><th>P-Wave propagation time (tp) (10<sup>-6</sup>s)</th></tr></thead><tbody><tr><td>50 kHz</td><td>3.96</td><td>27.76</td></tr><tr><td>100 kHz</td><td>3.10</td><td>26.55</td></tr><tr><td>200 kHz</td><td>1.42</td><td>24.79</td></tr></tbody></table>

The variation of $t_p$ compared to the true value $t=0.1/3950=25.31$ µs is calculated for the homogeneous medium (Table 4). Results show that the use of a high-frequency domain is suggested
to carry out a faster-rising edge of the wave (Fig.9) and therefore a more accurate measurement when defining t<sub>p</sub> (Δtp = 0.12%). Results in Table 5 show that a clear correlation exists between the propagation time of the wave, hence the wave velocity, and the emission frequency. When the emission frequency increases, the maximum amplitude and the propagation time of the wave decrease. The sensor's emission frequency is shown to have a significant influence on the accuracy and reliability of picked arrival times: the use of a high frequency makes it possible to have a faster rise of the wavefront and therefore a higher precision in the detection of the arrival point.

# 6. CONCLUSION

Ultrasonic measurements are a commonly used and reliable form of non destructive evaluation of materials. The nature of concrete as a heterogeneous material makes it challenging to have a direct interpretation of ultrasonic data. A 2D numerical model of wave propagation in a heterogeneous medium is built and validated by the numerical comparisons in the case of two homogeneous mediums, first composed of mortar properties and the second composed of mortar and aggregates with the same properties. The simulation of numerical heterogeneous mediums, representing real concrete samples, shows that wave propagation time and amplitude evolve with two important parameters: first, the heterogeneity of concrete; and second, the variation of water gradient, which is represented in this paper by the change of mortar properties. The variation of the studied observables related to the change of mortar properties up to 1.83% is found to be comparable to that related to the heterogeneity of concrete. A comparison is also made to illustrate the influence of the emission frequency on the precision of measurements. However, in order to accurately quantify all the uncertainties related to the material, no noise is added to the signal in this numerical study. This numerical model will be extended in further studies by taking into account the size of the sensors, the aggregate distribution, and the evolution of the water gradient in the concrete and by introducing noise. Knowing that the distribution of the aggregates does not change with time, it will be possible to neglect the uncertainties due to the distribution for an identical realization. For this reason, it will be possible to follow the evolution of the saturation rate or properties of the concrete with time by making differential measurements between a reference state and a subsequent measurement.

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