# Monitoring of drying and imbibition of concrete using embedded resistivity sensors for the estimation of permeability

Marie-Ange Eid, Nicolas Reuge, Géraldine Villain, Stéphanie Bonnet, Sérgio Palma Lopes

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## ► To cite this version:

Marie-Ange Eid, Nicolas Reuge, Géraldine Villain, Stéphanie Bonnet, Sérgio Palma Lopes. Monitoring of drying and imbibition of concrete using embedded resistivity sensors for the estimation of permeability. Cement and Concrete Composites, 2025, 157, pp.105900. 10.1016/j.cemconcomp.2024.105900. hal-05076591

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Submitted on 21 May 2025

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# Monitoring of drying and imbibition of concrete using embedded resistivity sensors for the estimation of permeability

Marie-Ange Eid<sup>a,b,*</sup>, Nicolas Reuge<sup>b</sup>, Géraldine Villain<sup>a</sup>, Stéphanie Bonnet<sup>b, ID</sup>, Sérgio Palma Lopes<sup>c</sup>

## ABSTRACT

Assessing the permeability of concrete is crucial as it governs the transport of aggressive agents, such as chlorides and carbon dioxide, which are key factors in the degradation mechanisms. Moreover, concrete's permeability constitutes an essential input parameter for durability models. Concrete's permeability can be measured directly (by experimental methods) or indirectly by fitting a transport model to saturation degree profiles. In this paper, we introduce a novel indirect method for estimating the permeability by monitoring the saturation degree profiles with embedded resistivity sensors. These embedded resistivity sensors are used for the evaluation of the saturation degree profiles over time during two experiments: drying and imbibition with tap water. Firstly, measured resistivity profiles are converted to saturation degree profiles, using a calibration curve established on concrete cores of the same formulation. Concrete's permeability is then estimated by fitting a hydric transport model to the experimental saturation degree profiles. Permeability values estimated using the embedded sensor are compared to those obtained by two reference methods: assessing the mass loss of a non-monitored specimen subjected to drying and saturation degree profiles obtained by gamma-densimetry measurements. The permeability values obtained with the monitoring method are consistent for drying and imbibition experiments and fall within the range of values found in the literature. This is very promising for the continuous monitoring of concrete by embedded resistivity sensors.

## 1. Introduction

The hydric transfer in a porous material is of major concern regarding its durability. Quantifying transport properties (mainly concrete permeability) is thus an important issue in civil engineering to estimate the service life of reinforced concrete structures. Thus, it is crucial to determine and monitor concrete's permeability, because it is one of the most important parameters governing the transport of aggressive agents. Several methods can be used in order to measure the water permeability of concrete. Some techniques are based on a direct measurement of water or gas permeability and are commonly used in laboratory and on-site conditions [1–6]. Moreover, an indirect measurement of concrete's permeability is also possible, by fitting a hydric transport model to saturation degree profiles obtained experimentally [7,8,9]. Literature studies have shown that non-destructive (ND) techniques are increasingly used for the evaluation of saturation degree profiles [10], particularly electrical resistivity methods due to their high sensitivity to the concrete's hydric state.

In laboratory conditions, a multi-ring electrode device is used for characterizing concrete cores conditioned at different saturation levels

[11,12]. Moreover, electrical resistivity tomography (ERT) is widely used for on-site and laboratory measurements, in order to evaluate the electrical resistivity and therefore the saturation degree of the concrete structure [11,13–15]. This technique combines several measurements for different electrode spacings and therefore allows to assess different investigation volumes of the structure. However, the aforementioned techniques do not allow a continuous monitoring of the concrete over time and the resolution they provide intrinsically diminishes with depth. Moreover, surface measurements are constrained by accessibility issues to certain areas of the structures. That is why, several types of embedded sensors were developed in order to evaluate resistivity profiles in concrete cover [16–19]. Yet, these embedded sensors present several limitations such as electrode polarization and wire intrusiveness. In order to overcome these constraints, [20] developed a printed circuit board (PCB) sensor that enables the evaluation of concrete electrical resistivity over time and over the entire thickness of the structure. The authors validated the capacity of the embedded sensor to evaluate the saturation degree of concrete specimens subjected to drying, by comparing saturation degree profiles obtained with the embedded sensor to those obtained by reference methods (gammadensimetry and THR probes). The

<footnote><sup>a</sup> Univ Gustave Eiffel, MAST-LAMES, Campus de Nantes, F-44344, Bouguenais, France</footnote>

<footnote><sup>b</sup> Nantes Université, Ecole Centrale Nantes, CNRS, GeM UMR 6183, F-44600, Saint Nazaire, France</footnote>

<footnote><sup>c</sup> Univ Gustave Eiffel, GERS-GeoEND, Campus de Nantes, F-44344, Bouguenais, France</footnote>

<footnote>* Corresponding author. Univ Gustave Eiffel, MAST-LAMES, Campus de Nantes, F-44344, Bouguenais, France.</footnote>

<footnote>E-mail address: marie-ange.eid@univ-eiffel.fr (M.-A. Eid).</footnote>
embedded sensor also showed its ability to measure the electrical resistivity of highly resistive concretes (such as slag concrete), which is not easily measured with surface techniques [21].

Several authors highlight the relationship between concrete permeability and electrical resistivity. [22] assessed carbonation depth and saturation degree using two ND techniques: resistivity measurement using a Wenner probe and a direct permeability measurement using a surface permeameter. The authors developed empirical models expressing resistivity or permeability versus saturation degree and carbonation depth. Although these laws are limited to saturation degrees ranging between 40 and 84 % due to the limitation of the measurement techniques and to the properties of the studied concrete, results showed that both techniques can be used for the combined assessment of saturation degree and carbonation depth in laboratory conditions. [23] used three different ND techniques to evaluate air permeability on laboratory samples and on-site structures. The first method used a surface Torrent permeameter whereas the two other techniques evaluated the saturation degree profiles obtained by capacitive or resistive probes. The authors showed that the capacitive and resistivity measurements provided a good evaluation of the saturation degree profile of concrete. Furthermore, these techniques, when combined with the Van Genuchten model, enabled to improve the evaluation of permeability using the surface Torrent device, making it more comparable to permeability values obtained by the Cembureau method [2]. However, the prediction of permeability using only the capacitive and resistive techniques presented some discrepancies, that were mainly explained by the high sensitivity of the methods to saturation degree. [24,25] showed that the concrete permeability is an exponential function of the degree of saturation. However, the authors did not estimate concrete permeability solely based on the data obtained from surface resistivity probes, as this measurement represents an average apparent resistivity over the investigated depth. To determine the transport properties, it is necessary to examine profiles, such as the variation of saturation degree with depth over time.

Concerning the indirect assessment of concrete permeability, [7,9] developed a hydric transport model describing transport phenomena in the concrete porous medium, then determined the concrete permeability by fitting this model either on the relative mass loss of a sample obtained during a controlled drying experiment or on saturation degree profiles obtained by gammadensimetry. [8] developed an improvement of the hydric transfer model, using a porous network model instead of semi-empirical Van Genuchten and Millington laws. The authors used the porous network model to determine the transport properties such as permeability and adsorption/desorption parameters. The authors validated their model by experimental results: relative mass loss of a sample as well as saturation degree profiles measured by a multi-ring resistivity device during a controlled drying of cylindrical samples. This device enables surface measurements on cylindrical laboratory samples, but are not suitable for monitoring reinforced concrete structures in the field.

Continuous monitoring of concrete saturation and permeability is critical for durability modeling, as it provides essential data on the material's evolution over time, under varying environmental conditions [26,27]. To address this need, the present study introduces a novel approach for the estimation of intrinsic water permeability of concrete, by means of saturation degree profiles obtained with the resistivity PCB sensor above-mentioned [20]. The adopted methodology consists in using the embedded sensor to monitor resistivity profiles through the depth of the specimen ($\rho(z)$) during two experimental campaigns: drying and imbibition. These resistivity profiles (the measured observable) are

subsequently converted to saturation degree profiles ($S_i(z)$) using a calibration curve ($\rho(S_i)$). Finally, the saturation degree profiles are used to estimate water permeability (the durability indicator) by fitting a transport model. The adopted methodology is schematized in Fig. 1.

The paper begins with a presentation of the materials and methods used in this work, including the concrete formulation and characteristics, instrumentation and measurement principle, calibration curve and experimental conditions of the drying and imbibition campaigns. Subsequently, the resistivity and saturation degree profiles obtained during the drying and imbibition campaigns are presented. Finally, the results of water permeability, estimated by fitting the transport model to the experimental saturation degree profiles, are discussed. The results obtained by the embedded sensor are compared to those obtained by two reference methods: assessing the mass loss of a non-instrumented specimen subjected to drying and saturation degree profiles obtained by gammadensimetry measurements.

## 2. Materials and methods

### 2.1. Concrete formulation and characteristics

The concrete mix used in this work is representative of numerous concretes studied in the literature. It is composed of a normal CEM I 52.5 Portland cement with a water/cement ratio of 0.5. The concrete has a compressive strength of 50 MPa at 28 days and a porosity of 14.1 % measured at 90 days according to [28]. The formulation and characteristics of the studied concrete (referred to as concrete OPCC50) are presented in Table 1.

### 2.2. Description of the embedded sensor and measurement principle

The embedded resistivity sensor used in this work was developed and validated by [20]. The PCB sensor has a ladder shape (with a total length of 144 mm) and is composed of 15 electrodes distributed on both sides of the PCB. The metallic electrodes (of $5 \times 3 \text{ mm}^2$ each) are made of copper and coated with a layer of nickel-gold. The spacing between two consecutive electrodes on the same side of the PCB is of 20 mm (as shown in Fig. 2).

The sensor was placed in the center of a cylindrical mold ($\varnothing11 \times 18 \text{ cm}$) along its z axis (the direction in which resistivity profile is to be studied), between two metal grids (each with a diameter of 100 mm) well anchored in concrete and serving for electrical current injection (Fig. 3(a)). Resistivity measurements were done according to the “Transmission” configuration: an electric current is injected between two metal grids (C1 and C2) and the potential drop is measured between

<table><caption>**Table 1**

Formulation and characteristics of concrete OPCC50.</caption><thead><tr><th>Components and characteristics</th><th>[kg/m<sup>3</sup>]</th></tr></thead><tbody><tr><td>Gravel, 4–12 mm (Boulonnais)</td><td>1041.7</td></tr><tr><td>Sand, 0–4 mm (Boulonnais)</td><td>910.3</td></tr><tr><td>CEM I 52,5 N - SR 5 CE PM-CP2 NF HTS (Lafarge)</td><td>302.5</td></tr><tr><td>Superplasticizer MasterGlenium 201 (BASF)</td><td>2.94</td></tr><tr><td>Efficient water (W<sub>eff</sub>)</td><td>151.7</td></tr><tr><td>Water-to-binder ratio (W<sub>eff/B</sub>)</td><td>0.5</td></tr><tr><td>Compressive strength at 28 days (MPa)</td><td>52.03 ± 2.7</td></tr><tr><td>Young's modulus at 28 days (GPa)</td><td>38.75 ± 0.4</td></tr><tr><td>Water porosity at 90 days (%)</td><td>14.10 ± 1.0</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Adopted methodology for the estimation of water permeability.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 2. Photo of the PCB resistivity sensor.</figcaption></figure>

two consecutive electrodes (P1 and P2) on the same side of the PCB sensor (Fig. 3(b)). Thus, a total of 13 measurements is obtained through the height of the PCB sensor and the depth of each measurement is assumed to be in the middle of the electrodes where the potential drop is measured [20].

It is important to note that Fig. 3(b) presents the intended layout of the sensor and injection grids, as well as the intended measurement positions along the depth of the specimen. However, when casting the instrumented specimens, the embedded sensors slightly moved from this position. Therefore, the experimental resistivity profiles obtained for each specimen (Figs. 7 and 8) will be presented according to the actual positions of the electrodes.

For each pair of electrodes on the sensor, an apparent resistivity is obtained by multiplying the transfer resistance (Ω), which is the ratio of

the measured potential drop (mV) to the transmitted current intensity (mA), by a geometric factor (m) that accounts for the geometric conditions of the measurement [29,30]. During the monitoring, the sensor embedded in the concrete specimen is connected to a commercial resistivity meter: a Syscal Junior Switch 72 (IRIS Instruments) with a resolution of 1 µV and a precision of 0.2 %. The resistivity meter allows to inject a square-wave current of frequency 2 Hz. Therefore, measurements are considered to be carried out in a static regime, usually called DC-resistivity.

Each measurement along the sensor, corresponding to a specific pair of electrodes, is mainly sensitive to the volume of concrete located between these electrodes [20]. Thus, each apparent resistivity value represents an integration of concrete's resistivity over a volume of about 2 cm in height. Thus, the 13 measured apparent resistivities can be considered as a close estimation of the concrete resistivity profile with respect to depth [20]. Consequently, the sensitivity of the sensor is nearly uniform over all measuring pairs, with a value close to 1 Ωm/Ωm. Since the electrodes pairs are spaced 1 cm apart on either side of the sensor (Fig. 2), the sensor yields a spatial resolution of about 1 cm along the direction of the PCB sensor.

### 2.3. Concrete specimens and curing conditions

The experimental campaign involved different types of concrete specimens that underwent different curing conditions. Table 2 summarizes the details corresponding to each concrete sample: name, dimensions, type of cure and corresponding experimental campaign.

First of all, several non-instrumented (Ø11 × 22 cm) specimens were used for mechanical and durability characterization and only one specimen (referred to as specimen D1) was used for monitoring the unidirectional drying by assessing its mass loss. Specimen D1 was cured under water for 90 days and it was then sawn in order to remove the upper and lower surfaces and keep only the core of Ø11 × 10 cm. Moreover, a calibration campaign was conducted on 3 cylindrical cores of Ø70 × 75 mm cored from a slab, which was made with the same concrete and kept under water for 90 days as well. These cores are referred to as specimens C1 to C3.

Two monitored specimens of Ø11 × 18 cm were used for monitoring the drying (of specimen M1) and the imbibition (of specimen M2) by means of embedded sensors. Note that due to experimental issues, drying and imbibition campaigns were carried out on two different samples, yet having similar formulation and instrumentation. Due to instrumentation constraints, instrumented specimens were not cured under water. They were rather kept in a tray containing a small quantity of water for 90 days, at a temperature of 21 ± 0.5 °C and a relative humidity of 95 ± 2 %. These curing conditions were regularly controlled

<figure><img src="image_6.png" /><figcaption>Fig. 3. Instrumentation and measurement method: (a) instrumented mold before casting, (b) measurements principle according to “transmission” configuration.</figcaption></figure>
<table><caption>**Table 2**
Details of concrete samples involved in the experimental campaign.</caption><thead><tr><th>Name</th><th>Dimensions (mm)</th><th>Embedded sensor</th><th>Type and period of cure</th><th>Corresponding experimental campaign</th></tr></thead><tbody><tr><td>D1</td><td>Ø110 × 100</td><td>No</td><td>Cured under water (90 days)</td><td>Assessing mass loss during unidirectional drying</td></tr><tr><td>C1 to C3</td><td>Ø70 × 75</td><td>No</td><td>Cured under water (90 days)</td><td>Calibration campaign</td></tr><tr><td>M1</td><td>Ø110 × 180</td><td>Yes</td><td>Kept in a humid environment (90 days)</td><td>Monitoring of drying by the embedded sensor</td></tr><tr><td>M2</td><td>Ø110 × 180</td><td>Yes</td><td>Kept in a humid environment (90 days)</td><td>Monitoring of imbibition by the embedded sensor</td></tr></tbody></table>

using a thermo-hygrometric probe (THR).

## 2.4. Calibration campaign

Apparent resistivity profiles obtained by the embedded sensors were converted to saturation degree profiles by means of a calibration campaign. The latter was conducted on 3 concrete cores of Ø70 × 75 mm (C1 to C3) using the resistivity cell developed by [11]. The measurement protocol assumes that moisture content is homogeneous within the volume of the Ø70 × 75 mm concrete core, thus the resistivity cell enables to measure the “real” resistivity of the material at different saturation rates [12]. A calibration curve is then obtained, linking the observable (electrical resistivity) to the durability indicator (saturation degree) (Fig. 4). The data is available in [31].

The calibration curve shown in Fig. 4 shows that for a saturation degree varying between 40 % and 100 %, the electrical resistivity varies between 50 Ω m and 1500 Ω m. The experimental points shown in the graph correspond to the average resistivity obtained throughout the resistivity cell [11] for each concrete core and the error bars represent the ±standard deviation. The calibration curve (represented by a dashed line) shows a power relationship between the electrical resistivity and the saturation degree (Eq. (1)), in agreement with Archie’s law [12,32].

$$S_l = \left( \frac{\rho}{41.717} \right)^{-\frac{1}{3.37}} \qquad \text{Eq. (1)}$$

It is important to note that the parameters in Eq. (1) are specific to the concrete mix studied in this work. These parameters vary depending on the concrete formulation, such as the type and size of aggregates

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Calibration curve obtained for concrete OPCC50.</figcaption></figure>

which influence the concrete’s resistivity [33].

## 2.5. Experimental conditions of the drying campaigns

As previously mentioned, two different drying campaigns were conducted on instrumented (M1) and non-instrumented (D1) specimens. Assessing the unidirectional drying of the non-instrumented cylinder by its mass loss serves as a reference method in order to validate the ability of our approach based on the use of the embedded sensor to estimate the intrinsic water permeability.

### Drying of non-instrumented specimen D1

This method is widely used in the literature in order to evaluate the intrinsic water permeability of concrete [7,9]. The method consists of monitoring the drying of a cylindrical specimen by evaluating the kinetics of its mass loss. Specimen D1 was placed in a desiccator in which a constant temperature of 21 ± 0.5 °C and a constant ambient relative humidity ($RH_{amb}$) of 55 ± 3 % are maintained by means of a concentrated saline solution (magnesium nitrate). In order to ensure a unidirectional drying within the sample, the latter was covered on its lateral surface with two layers of adhesive aluminum foil (Fig. 5(a)). The relative mass loss $\Delta M$ was calculated using Eq. (2) at different times of the drying phase and the corresponding curve is shown in Fig. 5(b). The latter shows a rapid increase in the mass loss during the first two months, followed by a slow increase for the rest of the drying phase.

$$\Delta M = \frac{M_{sat} - M_t}{M_{sat}} \qquad \text{Eq. (2)}$$

Moreover, gammadensimetry measurements were conducted on concrete sample D1 in order to determine the saturation rate during drying, as described in [34] but by using a horizontal (instead of vertical) gamma device, equipped with a similar radioactive cesium source and a similar scintillator that receives and counts the incidental photons. In summary, the method enables to monitor the density according to the depth of the concrete sample using the Lambert law (Eq. (3)).

$$d = -\frac{1}{\mu l} \ln \left( \frac{N}{N_0} \right) \qquad \text{Eq. (3)}$$

where $d$ (kg/m$^3$) is the density of concrete, $\mu$ (m$^2$/kg) is the mass absorption coefficient of the material, $l$ (m) is the thickness of the sample and $N_0$ and $N$ are the number of the incidental photons emitted by the radioactive source and the number of photons that crossed the concrete sample respectively. Gammadensimetry measurements were carried out at the saturated state (during which $N_0^{sat}$ and $N^{sat}$ represent respectively the number of the incidental photons emitted by the radioactive source and the number of photons that crossed the saturated concrete sample), the dry state at 105 °C (during which $N_0^{105}$ and $N^{105}$ represent respectively the number of the incidental photons emitted by the radioactive source and the number of photons that crossed the dried concrete sample), and at $t = 6$ months of drying (during which $N_0(t)$ and $N(t)$ represent respectively the number of the incidental photons emitted by the radioactive source and the number of photons that crossed the concrete sample at time $t$). These gammadensimetry measurements allow to obtain the corresponding densities $d_{sat}$, $d_{105}$ and $d_t = 6$ months, and therefore calculate the saturation degree $S_l(t)$ at time $t$ for each depth along the concrete sample using Eq. (4).

$$S_l(t) = \ln \left( \frac{N_0(t) N^{105}}{N(t) N_0^{105}} \right) / \ln \left( \frac{N_0^{sat} N^{105}}{N^{sat} N_0^{105}} \right) \qquad \text{Eq. (4)}$$

Fig. 6(a) and (b) respectively show the density of the concrete specimen at different states and its saturation degree profile obtained at 6 months of drying. Fig. 6(a) shows that the density of specimen D1 is relatively homogeneous with depth, despite the small heterogeneities probably due to the presence of voids or aggregates. Whereas, the
<figure><img src="image_8.png" /><figcaption>Fig. 5. Drying of sample D1: (a) specimen placed in a desiccator, (b) evolution of the mass loss (%) along the drying period.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Experimental results of gammadensimetry measurements conducted on specimen D1: (a) concrete density at three saturation states, (b) saturation degree calculated at 6 months of drying.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. Experimental set-up for the imbibition of monitored specimen M2 with tap water.</figcaption></figure>

saturation degree profile obtained at $t = 6$ months shows that specimen D1 dried only from both sides and stayed relatively wet in its core, highlighting the unidirectional drying of the sample (Fig. 6(b)).

Nevertheless, Fig. 6(b) shows that the saturation degree profile is asymmetric, which can be explained by some uncertainties brought by a slight concrete segregation [34]. Moreover, it is important to note that $d_{sat}$ was obtained after a second saturation of specimen D1 carried out after 6 months of drying, which can also present a source of error in experimental measurements.

### Drying of monitored specimen M1

Another drying campaign was conducted on monitored specimen M1, in order to evaluate the capacity of the embedded sensor in estimating the intrinsic water permeability of concrete. The drying campaign was divided into 3 phases described in Table 3 ($t_0$ being the beginning of the drying campaign). The specimen first underwent a three-dimensional drying in the oven. It was then removed from the oven and kept in a room at $T = 21 \pm 0.5$ °C for almost one month. In the meantime, two layers of epoxy resin were applied on the lateral surface of specimen M1. The latter was then packaged in a plastic bag in order to homogenize its internal hydric state. In the last phase, the concrete specimen was placed in the oven again for one-dimensional drying. Resistivity profiles were recorded throughout the drying period ($t_0 < t < t_0+166$ days) and converted into saturation degree profiles by means of the calibration curve. Nevertheless, only saturation degree profiles obtained during the third phase ($t_0+105$ days $< t < t_0+166$ days) will be used for the estimation of the water permeability by fitting the hydric transport model, the latter being only applicable to one-dimensional drying.
<figure><img src="image_11.png" /><figcaption>Fig. 8. Experimental results obtained during the drying of specimen M1: (a) temperature corrected apparent resistivity profiles, (b) saturation degree profiles.</figcaption></figure>

<table><caption>**Table 3**
Drying conditions at each phase of the drying campaign conducted on specimen M1.</caption><thead><tr><td>Time interval</td><td>Drying conditions</td></tr></thead><tbody><tr><td>t<sub>0</sub> &lt; t &lt; t<sub>0</sub> +71 days</td><td>Three-dimensional drying in the oven (T = 45 °C and RH<sub>amb</sub> = 50 %)</td></tr><tr><td>t<sub>0</sub> +71 days &lt; t &lt; t<sub>0</sub> +105 days</td><td>Concrete sample kept in a room at T = 21 °C while applying two layers of epoxy resin</td></tr><tr><td>t<sub>0</sub> +105 days &lt; t &lt; t<sub>0</sub> +166 days</td><td>Unidirectional drying in the oven (T = 45 °C and RH<sub>amb</sub> = 50 %)</td></tr></tbody></table>

## 2.6. Experimental conditions of the imbibition campaign

The imbibition campaign with tap water was conducted on monitored specimen M2. After 90 days of cure, sample M2 underwent a three-dimensional drying in the oven (at T = 45 °C and RH<sub>amb</sub> = 75 %) for 28 days, until reaching an average saturation rate of 57 % ± 1 %. The specimen was then covered with two layers of adhesive aluminum foil on its lateral surface, in order to ensure one-dimensional water penetration. The imbibition campaign lasted for 21 days and was carried out at T = 21 ± 0.5 °C and RH<sub>amb</sub> = 66 ± 2 %. Resistivity profiles were obtained at different times of the imbibition campaign and were converted to saturation degree profiles by means of the calibration curve.

## 3. Monitoring results and discussion

This section presents the resistivity profiles and the saturation degree profiles obtained by the embedded sensors during drying and imbibition campaigns.

It is important to note that the embedded sensor used in this work enables the measurement of “apparent” resistivity values. The concrete being a heterogeneous material (presenting water content gradients), the transformation from measured apparent resistivity ($ρ_{app}$) data to the distribution of the real resistivity ($ρ$) of the material usually requires an inversion process [35]. Nonetheless, given the ladder shape of the sensor and the distribution of electrodes along the depth of the concrete, the measured apparent resistivities provide a direct estimation of the real resistivity profile [20]. Thus, we assume that $ρ_{app}(z)$ is nearly equal to $ρ(z)$, which allows us to convert resistivity profiles obtained by the embedded sensors to saturation degree profiles by means of the calibration curve (Fig. 4).

Moreover, the calibration campaign described in section 2.4 was conducted at a controlled ambient temperature of 20 ± 1 °C, therefore resistivity profiles measured at 45 °C during the drying campaign must be corrected to the reference temperature (20 °C). The correction of resistivity profiles is carried out using Arrhenius' law, commonly used in the literature (Eq. (5)) [36–39].

$$ρ(T_i) = ρ(T_0)e^{\frac{E_a}{R}\left(\frac{1}{T_i} - \frac{1}{T_0}\right)} \qquad \text{Eq. (5)}$$

where $ρ(T_i)$ (Ω·m) is the resistivity of the material measured at temperature $T_i$ (K), $ρ(T_0)$ (Ω·m) is the resistivity calculated for the reference temperature $T_0$ (K), R is the gas constant (equal to 8.314 J/mol·K) and $E_a$ (J/mol) is the activation energy that depends on the type of the studied concrete and its saturation rate. In this work, we used a value of $E_a$ calculated by [39] for a CEM I based concrete with a water-to-cement ratio of 0.59 and a porosity of 15.3 % ± 0.3 %, therefore similar to the studied OPCC50 concrete [21]. The value of $E_a$ used if of 20 kJ/mol [39].

### 3.1. Resistivity and saturation degree profiles obtained during the drying campaign

In this section, experimental results obtained during the drying of specimen M1 are presented. Fig. 8(a) and (b) respectively show the evolution of temperature corrected apparent resistivity profiles and the evolution of saturation degree profiles throughout the drying period. Results show that the evaporation of the liquid phase initially present in the porosity of the concrete leads to the decrease of its saturation degree and thus, the increase of its electrical resistivity [22,40–42].

An average saturation degree of 78 % ± 2 % is measured at the end of the curing period (beginning of the drying campaign $t_0$), showing a non-saturated state of specimen M1 (Fig. 8(b)). This can be explained by the curing conditions described in section 2.3. In fact, due to instrumentation constraints, the concrete sample was not kept under water during the curing period. It was rather stored in a relatively humid atmosphere, which could have led to some drying of the sample. The saturation degree of specimen M1 decreases progressively until it reaches an average value of 22 % ± 2 % at the end of the drying period ($t_0+166d$). This saturation degree observed at the end of the drying of specimen M1 (Fig. 8(b)) is lower than that observed after 6 months of drying of specimen D1 (Fig. 6(b)). This can be explained by the differences in the drying conditions and in the measurement techniques.

Note that a saturation degree of 22 % ± 2 % is not within the range of values studied during the calibration campaign (Fig. 4) and presented in red dashed lines on Fig. 8(a) and (b). This value of saturation degree is obtained using Eq. (1), considering that the latter is applicable even for resistivity values exceeding 1500 Ω m. This assumption may not be valid for such high resistivity values and thus, the exponential form of the calibration curve may present a source of error in the experimental results. In fact, the saturation degree profile observed at $t_0+166d$ is almost homogeneous in the depth of the concrete sample (Fig. 8(b)), whereas the apparent resistivity profile shows a sharp gradient at $t_0+166d$ of drying (Fig. 8(a)). Finally, the excessive drying of specimen M1
prevented the latter from being used for the imbibition campaign. That is why the imbibition was carried out on specimen M2, whose prior drying was better controlled.

## 3.2. Resistivity and saturation degree profiles obtained during the imbibition campaign

Fig. 9(a) and (b) respectively show apparent resistivity profiles and saturation degree profiles obtained at different times of the imbibition campaign. Note that $t_0(imb)$ is the initial state and $z = 0$ cm is the lower surface of specimen M2 through which the tap water penetrates. Results highlight the effect of water penetration increasing the saturation degree of concrete and thus leading to the decrease of its electrical resistivity.

Fig. 9(a) shows that apparent resistivity values vary between 120 Ω m and 300 Ω m during the imbibition period. These values are within the range of values studied in the calibration campaign (Fig. 4). Moreover, Fig. 9(b) shows that the initial saturation degree profile measured at $t_0(imb)$ is nearly flat throughout the depth of the specimen (despite the presence of local heterogeneities of small magnitude). The saturation degree starts with the mean value (of 57 %) at both sides of the specimen (at $z = 5.55$ cm and $z = 16.55$ cm). As water penetrates into the poral space at $z = 0$ cm of the concrete, a gradient of saturation degree appears and continues to increase gradually. After 3 weeks of imbibition ($t_0(imb)+21d$), the saturation degree measured at $z = 5.55$ cm reaches 71 %, while maintaining approximately the same initial value (58 %) at $z = 16.55$ cm. The fact that the upper surface of the specimen maintains the same dry state even after 21 days of imbibition can be explained either by the initial state of the specimen or by a possible evaporation of water at the upper surface which was not covered during the imbibition campaign.

## 4. Estimation of the intrinsic permeability

Saturation degree profiles obtained experimentally with the embedded sensors were used to determine the intrinsic water permeability of concrete OPCC50, by adjusting a hydric transport model to experimental data.

First of all, a brief description of the model is presented. The intrinsic water permeability is then estimated in adsorption and desorption conditions, by fitting the hydric transport model on saturation degree profiles obtained during the imbibition of specimen M2 and the drying of specimen M1. In order to validate the capacity of the embedded sensor in estimating the water permeability, the latter is compared to that estimated during the drying of specimen D1 (serving as a reference method).

### 4.1. Hydric model applied in unsaturated conditions

In porous materials, two different phases are considered: the gas phase (g) and the liquid phase (l). The simplified hydric model considers that the transport of gas phase can be neglected, as long as the ambient relative humidity remains greater than about 50 % [9,43]. In this case, the transport is ruled only by the flow of the liquid phase. The velocity of the liquid phase $v_l$ (m/s) in the concrete depth is then described by Darcy's law (Eq. (6)), where $\eta_l$ (Pa.s) is the dynamic viscosity of phase l, $P_l$ (Pa) is the total pressure of phase l, $K_l$ (m$^2$) and $k_{r,l}$ (–) are the intrinsic and relative permeabilities of the liquid phase respectively:

$$v_l = - \frac{K_l k_{r,l}}{\eta_l} \frac{\partial P_l}{\partial x} \qquad \text{Eq. (6)}$$

The water sorption phenomenon in concrete depends on the local relative humidity RH (which by definition is equal to $P_v/P_{sat}$ at a specific temperature, in this case at $T = 20$ °C) and is described by the Van Genuchten (VG) model (Eq. (7)) [44] commonly used in the literature [22,9,45,46].

$$S_l(RH) = \left[ 1 + \left( - \frac{d_w RT}{M_w} \frac{1}{a} \ln(RH) \right)^{\frac{1}{1-m}} \right]^{-m} \qquad \text{Eq. (7)}$$

where $S_l(RH)$ is the saturation degree representing the volumetric ratio of liquid water in the concrete's porosity $\varphi$ (%), $d_w$ (kg/m$^3$) and $M_w$ (kg/mol) are respectively the density and the molar mass of the liquid water and $a$ (Pa) and $m$ (–) are VG parameters adjustable to experimental measurements.

The capillary pressure $P_c$ (Pa) existing in unsaturated pores is the difference between gas and liquid total pressures (Eq. (8)) and is expressed as a function of $S_l$ and VG parameters $a$ and $m$ (Eq. (9)). Note that in the simplified model, $P_g$ is considered constant and equal to the atmospheric pressure $P_{atm}$.

$$P_c(S_l) = P_g - P_l \qquad \text{Eq. (8)}$$

$$P_c(S_l) = a \left( S_l^m - 1 \right)^{1-m} \qquad \text{Eq. (9)}$$

According to [44], liquid relative permeability can be estimated by Eq. (10) with $q = 2$. In fact, Eq. (10) represents a semi-empirical expression which can be adjusted to experimental measurements and is only valid in adsorption conditions. In desorption conditions, there is no simple model under the form of an analytical expression. From [47], $k_{r,l(\text{des})}$ could be determined by a complex method that involves an integration of capillary pressure expressions. Eq. (11) represents an empirical expression of the liquid relative permeability under desorption conditions, where coefficients $\alpha$ and $\beta$ vary according to the

<figure><img src="image_12.png" /><figcaption>Fig. 9. Experimental results obtained during the imbibition of specimen M2: (a) apparent resistivity profiles, (b) saturation degree profiles.</figcaption></figure>
variation of $q$ in Eq. (10) [43,47]. Note that coefficient $\alpha$ is adjusted to ensure similar slopes between the $k_{rl,(ads)}$ and $k_{rl,(des)}$ expressions for saturation degrees ranging between 30 % and 90 %, while coefficient $\beta$ is adjusted to optimize the fit of numerical simulations to experimental measurements.

$$k_{rl,(ads)} = S_l^{\frac{1}{2}} \left[ 1 - \left( 1 - S_l^{\frac{1}{m}} \right)^{m-q} \right] \quad \text{Eq. (10)}$$

$$k_{rl,(des)} = \exp \left[ \ln(10) \times \beta \times \frac{S_l - 1}{1 + 2.34 \times S_l^{0.05} - 3.32 \times S_l^q} \right] \left( 2 - \frac{0.7}{S_l^{0.315}} \right) / 1.3 \quad \text{Eq. (11)}$$

In this study, the hydric transport model is firstly adjusted to experimental saturation degree profiles obtained during the imbibition of specimen M2 by varying the value of $q$ in Eq. (10). The corresponding $k_{rl,(des)}$ expressions are then used in order to simulate the drying of specimens D1 and M1. Table 4 presents the three values of $q$ considered in this study and the corresponding values of coefficients $\alpha$ and $\beta$. Fig. 10 shows relative permeabilities corresponding to the coefficients appearing in Table 4 as a function of the saturation degree.

Given that the transport of the gas phase is neglected, the transfer model is then reduced to the mass balance equation of the liquid phase given by Eq. (12).

$$\frac{\partial}{\partial t}(\varphi S_l) - \frac{\partial}{\partial x} \left[ \frac{K_l k_{rl}(S_l)}{\eta_l} \frac{\partial P_l}{\partial x} \right] = 0 \quad \text{Eq. (12)}$$

It should be noted that the input parameters of the model are material properties (porosity, intrinsic permeability and adsorption and desorption parameters) and the output parameter is the saturation degree profile $S_l(RH)$. Numerical simulations are carried out using a finite element solver software called FlexPDE® based on an iterative procedure, an adaptative time step and an adaptive mesh.

## 4.2. Estimation of the intrinsic water permeability in adsorption conditions

As mentioned above, the hydric transport model is firstly adjusted to experimental saturation degree profiles obtained in adsorption conditions (imbibition of specimen M2) using Eq. (10). Different values of $q$ are considered (Table 4) in order to find the expression of $k_{rl,(ads)}$ that best fits the experimental data.

It is important to note that there are 3 unknown input parameters: the adjustable VG parameters $\alpha$ and $m$ and the intrinsic water permeability $K_l$. This being, the input parameters of concrete OPCC50 are estimated based on a literature review summarized in Table 5. The latter presents input data of different concretes similar to OPCC50 (in terms of formulation and porosity). Note that VG parameters ($\alpha$ and $m$) presented in Table 5 were obtained experimentally by assessing the water vapor sorption isotherms of the different concretes under adsorption and/or desorption conditions and intrinsic permeability values were obtained by adjustment of the Van Genuchten model to drying and/or imbibition experiments. Table 5 illustrates a disparity in the values of parameter $\alpha$ under adsorption and desorption conditions, whereas the value of parameter $m$ remains nearly identical in both scenarios.

Table 6 summarizes the input parameters considered for the simulation of the imbibition of specimen M2. The adopted VG parameters $\alpha$

<caption>**Table 4**</caption>

Considered values of coefficient $q$ in the expression of $k_{rl,(ads)}$ and corresponding values of coefficients $\alpha$ and $\beta$ in the expression of $k_{rl,(des)}$.

<table><thead><tr><th>q (k<sub>rl,(ads)</sub>)</th><th>α (k<sub>rl,(des)</sub>)</th><th>β (k<sub>rl,(des)</sub>)</th></tr></thead><tbody><tr><td>0.25</td><td>1.2</td><td>12.15</td></tr><tr><td>0.7</td><td>1.9</td><td>17</td></tr><tr><td>2</td><td>2.7</td><td>21</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Considered relative permeabilities in adsorption and desorption conditions as a function of the saturation degree.</figcaption></figure>

<caption>**Table 5**</caption>

Input data of different concretes comparable to the studied concrete OPCC50.

<table><thead><tr><th>Concrete</th><th>W/ C</th><th>φ (%)</th><th>a(des) (MPa)</th><th>a(ads) (MPa)</th><th>m (–)</th><th>K<sub>l</sub> (10<sup>-21</sup> m<sup>2</sup>)</th></tr></thead><tbody><tr><td>OPCC60 [48]</td><td>0.48</td><td>14.0</td><td></td><td>4.75</td><td>0.40</td><td></td></tr><tr><td>B1 [8]</td><td>0.43</td><td>13.2</td><td></td><td>8.48</td><td>0.41</td><td>250</td></tr><tr><td>B1 [8]</td><td>0.43</td><td>13.2</td><td>35.29</td><td></td><td>0.43</td><td>2.5</td></tr><tr><td>BO [46]</td><td>0.48</td><td>12.0</td><td>18.62</td><td></td><td>0.44</td><td>3</td></tr><tr><td>B1* [49]</td><td>0.45</td><td>12.3</td><td>17.98</td><td></td><td>0.31</td><td>3</td></tr></tbody></table>

($ads$) and $m$ were obtained by adjustment of the Van Genuchten model to experimental data and are respectively equal to 18 MPa and 0.4. The initial value $S_{l0}$ (%) corresponds to the saturation degree profile measured by the embedded sensor at $t_{0(imb)}$ (see Fig. 9(b)). The boundary conditions $S_{lext1}$ (%) and $S_{lext2}$ (%) represent respectively the saturation degrees at the lower face and the upper face of specimen M2, as shown in Fig. 11. $S_{lext1}$ is set at 100 % given that the lower face of specimen M2 is in contact with water, while $S_{lext2}$ depends on the ambient relative humidity and is calculated using Eq. (7). Finally, Eq. (10) is used to express the liquid relative permeability under adsorption conditions, considering the three different values of $q$ appearing in Table 4. The intrinsic permeability is estimated by adjusting the hydric transport model to the saturation degree profile obtained by the embedded sensor after 3 weeks of imbibition (at $t_{0(imb)}+21d$), as shown in Fig. 12.

Fig. 12 shows that, for the considered input parameters shown in Table 6, numerical simulations fit well the experimental saturation degree profile obtained by the embedded sensor at 21 days of imbibition. In order to properly estimate the values of $q$ and $K_l$ that best fit the

<caption>**Table 6**</caption>

Input parameters considered for the simulation of the imbibition of specimen M2.

<table><tbody><tr><td>Porosity</td><td>φ (%)</td><td>14.1</td></tr><tr><td>Adsorption parameters</td><td>a(ads) (MPa)</td><td>18</td></tr><tr><td></td><td>m (–)</td><td>0.4</td></tr><tr><td>Intrinsic permeability</td><td>K<sub>l</sub> (m<sup>2</sup>)</td><td></td></tr><tr><td>Initial values</td><td>S<sub>l0</sub> (%)</td><td>S<sub>l</sub> (%) profile measured by the embedded sensor at t<sub>0(imb)</sub> Fig. 9(b))</td></tr><tr><td>Ambient relative humidity</td><td>RH<sub>amb</sub> (%)</td><td>66 ± 2 %</td></tr><tr><td>Boundary conditions</td><td>S<sub>lext1</sub> (%)<br>S<sub>lext2</sub> (%)</td><td>100 %<br>44.17 % (Eq. (7))</td></tr><tr><td>Liquid relative permeability</td><td>k<sub>rl,(ads)</sub> (–)</td><td>Eq. (10)</td></tr></tbody></table>
<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Boundary conditions considered for the simulation of the imbibition of specimen M2.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Estimation of the intrinsic permeability by adjustment of the hydric transport model to the saturation degree profile obtained at $t_0(imb)+21d$ using different expressions of $k_{rl,(ads)}$.</figcaption></figure>

experimental data, the Root Mean Square Error (RMSE (%)) is calculated between experimental and numerical saturation degree profiles, as shown in Table 7. The latter shows that the calculated RMSE (%) are close when comparing between the considered values of $q$ and $K_l$, with a minimal value obtained for $q = 2$ and $K_l = 15 \times 10^{-21} m^2$.

The variation of coefficient $q$ appearing in the expression of $k_{rl,(ads)}$ highly influences the shape of the saturation degree profile. Fig. 12 shows that, for $q = 0.25$ and $q = 0.7$, numerical profiles exhibit a gradual decrease in the saturation degree near the penetration face and a slight divergence from the experimental profile in the upper part of the specimen. For these two values of $q$, the estimated intrinsic permeability is of $\sim 10^{-21} m^2$. However, for $q = 2$, the numerical profile shows a sharp drop in the saturation degree with a relatively good fit deeper into the specimen. Achieving this close fit between the model and the

<table><caption>**Table 7**

RMSE (%) calculated between experimental and numerical saturation degree profiles for the different values of $q$ and $K_l$.</caption><thead><tr><th>q (-)/K<sub>l</sub> (m<sup>2</sup>)</th><th>0.25/2 × 10<sup>-21</sup></th><th>0.7/5 × 10<sup>-21</sup></th><th>2/15 × 10<sup>-21</sup></th></tr></thead><tbody><tr><th>RMSE (%)</th><td>7.38</td><td>6.07</td><td>4.95</td></tr></tbody></table>

experimental data required a higher value of $K_l$ (around $10^{-20} m^2$).

To conclude, the intrinsic permeability estimated by the simulation of the imbibition experiment ranges between $10^{-21} m^2$ and $10^{-20} m^2$, depending on the value of the exponent $q$. This order of magnitude of permeability is comparable to findings in existing literature (Table 5) and will be further compared to that estimated by the drying experiments.

### 4.3. Estimation of the intrinsic water permeability in desorption conditions

#### 4.3.1. Drying of specimen D1

In this section, the transport model is adjusted to experimental results obtained by the reference methods: mass loss curve and saturation degree profile obtained by gammadensimetry at $t = 6$ months of drying. The input parameters considered for the simulation of the unidirectional drying of sample D1 are presented in Table 8.

As previously mentioned, the values of parameters $a$ and $m$ vary between adsorption and desorption conditions (Table 5), as adsorption and desorption isotherms typically differ. However, given the number of unknown parameters, we assume that parameters $a$ and $m$ do not vary and maintain the previously adopted values in the imbibition simulation ($a = 18$ MPa and $m = 0.4$). The initial value $S_{l0}$ is set at 100 % (given the prior saturation of specimen D1), while the boundary conditions $S_{lext1}$ and $S_{lext2}$ (representing the saturation degrees at both ends of specimen D1) depend on the ambient relative humidity and are both calculated using Eq. (7). Finally, Eq. (11) is used to express the liquid relative permeability under desorption conditions, considering the three pairs of values ($a, \beta$) appearing in Table 4.

Fig. 13(a) and (b) show that an intrinsic permeability of $80 \times 10^{-21} m^2$ allows a close fit between the model and the experimental measurements obtained during the unidirectional drying of specimen D1. Moreover, Table 9 shows that the RMSE (%) computed between numerical and experimental profiles are relatively low and notably close, especially when calculated between mass loss curves.

Results indicate better fitting between the saturation degree profiles (Fig. 13(b)) compared to the mass loss curves (Fig. 13(a)). Indeed, Fig. 13(a) shows that, for the different considered expressions of $k_{rl,(des)}$, the model underestimates the mass loss when compared to experimental data. This discrepancy may be attributed to experimental uncertainties associated with the drying protocol, particularly in controlling the amount of saline solution used to maintain a constant ambient relative humidity. Conversely, notable alignments are observed between the model and the saturation degree profile obtained by gammadensimetry at $t = 6$ months of drying, regardless of the expression of $k_{rl,(des)}$ (Fig. 13(b)). Minor discrepancies are observed (especially at both ends of the specimen) and can be explained by the slight segregation initially present within the material, as well as by the uncertainty introduced by the gammadensimetry measurement (which is of 5 % for the 1 cm ray beam used in this work) [34].

In conclusion, the intrinsic permeability estimated by the simulation of the unidirectional drying of specimen D1 is of $80 \times 10^{-21} m^2$. This value allows a good alignment with the saturation degree profile

<table><caption>**Table 8**

Input parameters considered for the simulation of the unidirectional drying of specimen D1.</caption><thead><tr><th>Porosity</th><th>φ (%)</th><th>14.1</th></tr></thead><tbody><tr><td rowspan="2">Desorption parameters</td><td>a(des) (MPa)</td><td>18</td></tr><tr><td>m (-)</td><td>0.4</td></tr><tr><td>Intrinsic permeability</td><td>K<sub>l</sub> (m<sup>2</sup>)</td><td></td></tr><tr><td>Initial values</td><td>S<sub>l0</sub> (%)</td><td>100 %</td></tr><tr><td>Ambient relative humidity</td><td>RH<sub>amb</sub> (%)</td><td>55 ± 3 %</td></tr><tr><td rowspan="2">Boundary conditions</td><td>S<sub>lext1</sub> (%)</td><td>35.52 % (Eq. (7))</td></tr><tr><td>S<sub>lext2</sub> (%)</td><td>35.52 % (Eq. (7))</td></tr><tr><td>Liquid relative permeability</td><td>k<sub>rl,(des)</sub> (-)</td><td>Eq. (11)</td></tr></tbody></table>
<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Estimation of the intrinsic permeability by adjustment of the hydric transport model to experimental data obtained during the unidirectional drying of specimen D1: (a) adjustment to the mass loss curve, (b) adjustment to the saturation degree profile obtained by gammadensimetry at *t* = 6 months of drying.</figcaption></figure>

**Table 9**

RMSE (%) calculated between numerical simulation and experimental measurements obtained during the drying of specimen D1.

<table><thead><tr><th>(α, β)</th><th>(1.2, 12.15)</th><th>(1.9, 17)</th><th>(2.7, 21)</th></tr></thead><tbody><tr><td>K<sub>l</sub> (m<sup>2</sup>)</td><td>80 × 10<sup>-21</sup></td><td>80 × 10<sup>-21</sup></td><td>80 × 10<sup>-21</sup></td></tr><tr><td>RMSE (%) on mass loss curves (Fig. 11 (a))</td><td>10.7</td><td>15.7</td><td>17.3</td></tr><tr><td>RMSE (%) on saturation degree profiles (Fig. 11(b))</td><td>13.2</td><td>8.32</td><td>6.18</td></tr></tbody></table>

obtained by gammadensimetry. According to Table 9, the minimum RMSE value is obtained when considering $α = 2.7$ and $β = 21$. The discrepancies observed when fitting the model to the mass loss curve can be attributed to experimental uncertainties. These findings will serve as a reference to validate the capacity of the embedded sensor in estimating the intrinsic permeability of concrete.

### 4.3.2. Drying of specimen M1

In this section, the intrinsic permeability obtained by the reference methods (drying of specimen D1) is compared to that estimated by the embedded sensor during the drying of specimen M1. The input parameters considered for the simulation of the drying of instrumented specimen M1 are summarized in Table 10. Adjusting the transport model to the experimental saturation degree profiles was somehow complicated, given the particular drying conditions experienced by specimen M1.

First of all, it is important to note that specimen M1 underwent different conditions during the drying campaign (as described in Table 3). Yet, only the last phase (corresponding to unidirectional drying) is considered for numerical simulations. Therefore, we consider that the unidirectional drying starts at $t_0+105d$ and that the initial saturation state of sample M1 ($S_{l0}$) corresponds to the saturation degree profile obtained at $t_0+105d$. The intrinsic permeability is therefore estimated by fitting the transport model to the available saturation degree profiles, obtained at $t_0+126d$ and $t_0+166d$. Table 11 presents each time limit considered in the unidirectional drying phase and its equivalent in the actual drying campaign.

Since the initial state of sample M1 represents a sufficiently dry state, the resulting boundary conditions required an adjustment of the previously adopted parameters $α$ and $m$, setting them to 12 MPa and 0.47 respectively. The boundary conditions ($S_{lext1}$ and $S_{lext2}$) are then computed using Eq. (7).

Finally, the liquid relative permeability is expressed using Eq. (11). Nonetheless, the parameter set previously conducted on coefficients $α$ and $β$ appearing in Eq. (11) was not feasible for simulating the drying of specimen M1. The only possible adjustment was observed when setting $α$ and $β$ to 1.2 and 12.15 respectively.

The intrinsic permeability estimated by the simulation of the drying of specimen D1 ($K_l = 80 \times 10^{-21} m^2$) appears to be slightly low. That is why, the transport model is fitted to the experimental saturation degree profiles using a higher value of $K_l$, equal to $800 \times 10^{-21} m^2$. Fig. 14 shows that, for $K_l = 800 \times 10^{-21} m^2$, numerical simulations fit well the experimental profiles at different times of the unidirectional drying phase. A slight overestimation of the saturation degree is observed in the upper part of the specimen (from $z = 8$ cm to $z = 16$ cm) at $t_0'+60d$, due to the asymmetric saturation degree profile obtained by the embedded sensor.

The complexity of the simulation of the unidirectional drying of specimen M1 may have several explanations. First of all, as indicated in Table 3, specimen M1 was previously subjected to a three-dimensional drying phase for about two months. Thus, it was already dry enough at the beginning of the unidirectional drying phase (at $t_0'=t_0+105d$) considered in the numerical simulations. This being, it may be difficult to simulate the drying of a previously dried sample, even while considering a high value of intrinsic permeability. Besides, this high permeability value can be explained by the possible presence of an Interfacial Transition Zone (ITZ) between the PCB sensor and the

**Table 10**

Input parameters considered for the simulation of the unidirectional drying phase of specimen M1.

<table><thead><tr><th>Porosity</th><th>φ (%)</th><th>14.1</th></tr></thead><tbody><tr><td rowspan="2">Desorption parameters</td><td>a(des) (MPa)</td><td>12</td></tr><tr><td>m (–)</td><td>0.47</td></tr><tr><td>Intrinsic permeability</td><td>K<sub>l</sub> (m<sup>2</sup>)</td><td></td></tr><tr><td>Initial values</td><td>S<sub>l0</sub> (%)</td><td>S<sub>l</sub> (%) profile measured by the embedded sensor at t<sub>0</sub>+105d (Fig. 8(b))</td></tr><tr><td>Ambient relative humidity</td><td>RH<sub>amb</sub> (%)</td><td>50 ± 1 %</td></tr><tr><td rowspan="2">Boundary conditions</td><td>S<sub>lext1</sub> (%)</td><td>15.95 % (Eq. (7))</td></tr><tr><td>S<sub>lext2</sub> (%)</td><td>15.95 % (Eq. (7))</td></tr><tr><td rowspan="2">Liquid relative permeability</td><td>k<sub>rl(des)</sub> (–)</td><td>Eq. (11) only with α=1.2 and β = 12.15</td></tr></tbody></table>

**Table 11**

Adjustment of the exact times of the actual drying campaign to the considered times in the unidirectional drying phase.

<table><thead><tr><th>Exact time of the drying campaign</th><th>Corresponding time in the unidirectional drying phase</th></tr></thead><tbody><tr><td>t<sub>0</sub> +105d</td><td>t<sub>0</sub>'</td></tr><tr><td>t<sub>0</sub> +126d</td><td>t<sub>0</sub>' +21d</td></tr><tr><td>t<sub>0</sub> +166d</td><td>t<sub>0</sub>' +60d</td></tr></tbody></table>
<figure><img src="image_17.png" /><figcaption>**Fig. 14.** Estimation of the intrinsic permeability by adjustment of the hydric transport model to experimental saturation degree profiles obtained by the embedded sensor during the unidirectional drying phase of specimen M1.</figcaption></figure>

surrounding concrete, which leads to an increase in the porosity and consequently an increase in permeability [50,51]. Moreover, the conversion of resistivity profiles measured by the embedded sensor to saturation degree profiles may present some uncertainties. First of all, resistivity profiles measured at 45 °C were corrected using Arrhenius' law as described in section 3. The application of Arrhenius' law is based on a study conducted by [39] on a concrete similar to OPCC50 and thus, some parameters may not be well adapted to our studied concrete. Additionally, the calibration curve shown in Fig. 4 was used to convert the apparent resistivity profiles to saturation degree profiles. Nevertheless, most of resistivity values measured during the drying of specimen M1 were not in the range of values studied in the calibration campaign. This experimental protocol may account for the complexity encountered when simulating the drying the specimen M1.

In order to compare between the different simulated experiences, Table 12 summarizes the parameters used for the simulation of the drying and the imbibition phenomena (sorption parameters, liquid relative permeability expression and intrinsic permeability), that allowed the optimal alignment of the model with experimental data.

First of all, Table 12 shows that the intrinsic permeability estimated when simulating the unidirectional drying of specimen M1 (~$10^{-19}$ m$^2$) is 10 times higher than that estimated by the simulation of the drying of specimen D1 and the imbibition of specimen M2 (~$10^{-20}$ m$^2$). This can be attributed to the excessive prior drying of specimen M1, likely resulting in the formation of new percolation pathways near the electrodes and thus, an increase in the permeability. Despite the disparity observed in the results, the intrinsic permeability values estimated in this study align with the magnitudes commonly found in the literature [8,9,43,46].

In general, the chosen parameters *a* and *m* showed a good agreement between the model and the experimental data. An adjustment of these parameters was necessary specifically for simulating the unidirectional drying of sample M1, which experienced particular drying conditions. Moreover, it is important to note the variation of parameters *a* and *m* between adsorption and desorption isotherms, although they were treated as invariant. The hysteresis phenomenon during adsorption/

desorption results from variations in pore accessibility and water retention within the pores, which may explain the differences observed between the results of the drying and imbibition simulations [52]. Therefore, the attempt to fit the model to the saturation degree profile obtained during the imbibition of specimen M2 was challenging using literature-based values of *a* under desorption conditions (see Table 5). This difficulty stemmed also from measurement limitations starting 5.5 cm from the penetration face, affecting accuracy near the water front. Additionally, the conversion of “apparent” resistivity to approximate “real” resistivity in a homogeneous material might be inadequate in the presence of strong water gradients during imbibition. Consequently, interpreting these results might require an inversion process to derive more accurate resistivity profiles. These experimental uncertainties likely contributed to the requirement for a relatively high value of *a* during the imbibition simulation.

Furthermore, Table 12 illustrates some discrepancies in the results of relative permeability expression and the corresponding coefficients among the simulated experiments. These differences arise from variations among the samples and their experimental conditions, as well as uncertainties inherent in the model itself. It is important to note that the expressions of relative permeabilities (Eq. (10) and Eq. (11)) are empirical laws adjusted to fit experimental data. Despite these variations, the simulation results indicate that the variation in the expressions of $k_{RL(ads)}$ and $k_{RL(des)}$, as well as the parameter set performed on coefficients $q$, $a$ and $\beta$ primarily affects the shape of numerical profiles and does not have a significant effect on the estimated intrinsic permeability values.

Finally, it is important to emphasize that the results presented in Table 12 were obtained for a concrete formulated with CEM I. If the study is conducted on other mixtures using different binders, it is essential to consider the changes in the composition of the pore solution, which affects the concrete's resistivity [53,54].

## 5. Conclusions and outlook

The objective of the present paper is to estimate the intrinsic permeability of concrete using saturation degree profiles, obtained by monitoring concrete specimens using embedded resistivity sensors. From this study, the following conclusions can be drawn:

- The embedded resistivity sensors make it possible to evaluate the saturation degree profiles in the core of the concrete sample, and to monitor these profiles over time in an accurate and trustable way. A calibration curve is needed to convert the resistivity to saturation degree.

- The hydric transport model chosen to fit the saturation degree profiles is well adapted to assess the concrete permeability $K_l$. However, this model depends also on other parameters that are difficult to adjust and that can reduce the accuracy of the estimation of $K_l$. In particular, parameters *a* and *m* were chosen based on previous studies and adjusted to achieve the best fit between the model and the experimental data. Using similar values for both adsorption and desorption conditions may need further investigation, as adsorption and desorption isotherms differ.

- The relative permeability expressions used in this study successfully aligned the model to experimental data. The discrepancies in coefficients between the simulated experiments stem from differences

<table><caption>**Table 12**

Input parameters used for the adjustment of the transport model to experimental data obtained during the different experimental campaigns.</caption><thead><tr><th>Sample</th><th>Experimental campaign</th><th>a(ads) (MPa)</th><th>a(des) (MPa)</th><th>m (-)</th><th>q (Eq. (10))</th><th>(a, β)<br>(Eq. (11))</th><th>K<sub>l</sub> (10<sup>-21</sup> m<sup>2</sup>)</th></tr></thead><tbody><tr><td>M2</td><td>Monitoring of the imbibition by the embedded sensor</td><td>18</td><td></td><td>0.4</td><td>2</td><td></td><td>15</td></tr><tr><td>D1</td><td>Assessing mass loss during unidirectional drying</td><td></td><td>18</td><td>0.4</td><td></td><td>(2.7, 21)</td><td>80</td></tr><tr><td>M1</td><td>Monitoring of the drying by the embedded sensor</td><td></td><td>12</td><td>0.47</td><td></td><td>(1.2, 12.15)</td><td>800</td></tr></tbody></table>
in experimental conditions and uncertainties in the model regarding the relative permeability expressions.

- The permeability values obtained align closely with those reported in the literature, indicating the reliability of our approach.

These results thus are very promising for the development of concrete monitoring by embedded resistivity probes. Indeed, the concrete permeability found by analyzing the resistivity profiles, could be used in durability models for estimating the service life of concrete.

The perspectives that can be drawn from this work are the following:

- The same procedure could be applied to other concrete formulations with different binders, in order to study the effect of changes in pore solution on concrete's resistivity.

- The embedded sensor could be used also in saturated conditions, for example during a diffusion campaign, in order to estimate the chloride diffusion coefficient (which is also a major transport property of concrete).

- Finally, the embedded sensor could be tested on structures exposed to real-world drying/wetting conditions, in order to account for the coupling of phenomena and therefore include the hysteresis effects.

## CRediT authorship contribution statement

**Marie-Ange Eid:** Writing – original draft, Visualization, Validation, Methodology, Investigation. **Nicolas Reuge:** Validation, Methodology, Conceptualization. **Géraldine Villain:** Writing – review & editing, Validation, Supervision, Project administration. **Stéphanie Bonnet:** Writing – review & editing, Validation, Supervision, Project administration. **Sérgio Palma Lopes:** Writing – review & editing, Validation, Supervision.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

Authors would like to thank the "Agence Nationale de la Recherche" for funding the ANR-20-CE22-0008 DEMCOM project. We are also grateful to Jean-Luc Geffard from Université Gustave Eiffel, Denis Fournol from Nantes Université and Laurence Guiheneuf from Capacités for their contribution to the experimental program.

## Data availability

Data will be made available on request.

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