Reinforced concrete (RC) is a material widely used in the construction of infrastructure and buildings because of its relative low cost and large durability. However, there are some environmental conditions where physical, chemical, and biological deterioration processes reduce significantly its durability and safety (Bastidas-Arteaga et al., 2009; de Larrard et al., 2014; Marquez-Peñaranda et al., 2016). Among these deterioration processes, atmospheric carbonation of RC structures is one of the major causes of depassivation and then corrosion of steel reinforcing rebars (Ann et al., 2010). Carbonation-induced corrosion damage could certainly increase in the future years by the rise of environmental CO₂ concentration inducing additional maintenance costs (IPCC, 2013; de Larrard et al., 2014; Peng and Stewart, 2016; Stewart et al., 2014).

Maintenance strategies of corroding RC structures aim at predicting corrosion and planning repair operations (coating, replacement of concrete cover, cathodic protection, etc.) in order to maintain acceptable serviceability and safety levels. For new or non-corroded structures, inspection, and data collection are crucial to characterize parameters of carbonation models. The inherent spatial variability of concrete properties and cover depth is of prime importance and must be properly characterized and modeled (Li, 2004; Stewart and Mullard, 2007; Peng and Stewart, 2014); it implies that models must be selected, on the one hand, for representing the carbonation process and predicting corrosion initiation. On the other hand, models must be capable of integrating the spatial variability of input and propagating it onto the output. A convenient way to characterize the spatial variability of stationary random fields is to assess the spatial correlation of data (Schoefs et al., 2009, 2016; O’Connor and Kenshel, 2013; Pasqualini et al., 2013). Knowing the spatial correlation before inspection helps to define an optimal inspection by reducing inspection cost and increasing the predictions accuracy (Bastidas-Arteaga and Schoefs, 2012; O’Connor et al., 2013; Gomez-Cardenas et al., 2015). Inspection or repair decision-making can be efficiently conducted in a probabilistic context especially when statistical and spatial variability of data have been characterized (Stewart, 2004, 2006; Papakonstantinou and Shinozuka, 2013). Data collected from real structures can be perturbed by: spatial variability, measurement error, inaccuracy of experimental devices, complexity of experimental process, etc. Therefore, a dedicated treatment is often applied to data in order to discard gross outliers. In addition, it is not possible to generalize outcomes regarding spatial variability between structures even for the same material property. For instance, concrete mix, execution, and environmental conditions have an important impact on the concrete porosity, and then, the spatial variability of porosity between two components supposedly casted with the same concrete is not necessarily the same. This issue was addressed within the framework of the ANR-EVADEOS project (funded by the French National Research Agency) where a wide experimental investigation was undertaken on several RC structures. Destructive and non-destructive evaluations (NDE) were performed to estimate durability properties of concrete as well as carbonation depth. Only results of destructive tests are considered in this paper. Measurements were taken over a representative part of a concrete wall.

The main objective of the present study is to the estimate the ability of analytical carbonation models to propagate the spatial variability of measured inputs (porosity and saturation degree). The method relies on a comparison between simulated and measured outputs (carbonation depth). A peculiar attention is paid to the quality of data. The effect of gross outliers on the correlation profile of concrete properties is hence studied as well as the influence of unintended deviations in experimental measurements. Moreover, a statistical approach is proposed to study the capability of analytical models to deal with the spatial variability. Section “Investigated Structure” presents the studied structure and the data collected in one experimental investigation of the ANR-EVADEOS Project. Section “Simulation of Random Fields” introduces the computational tools used to assess the correlation profiles and to simulate stationary random fields. We evaluate in Section “Uncertainties from Data and Computation” the uncertainties related to experimental measures. Finally, we propose in Section “Metrics for Estimating the Quality of the Spatial Variability Predictions” various metrics used to evaluate and compare the capability of analytical models to deal with spatial variability.

The structure investigated is a concrete wall (Figure 1) built in 1979 enclosing a yard where inert wastes are stored. From carbonation point of view, the exposed surface of the wall represents perfectly a vertical surface of a bridge girder or a column and can be investigated with a lower cost. However, the wall is not cyclically loaded. If the structure was subjected to external mechanical loading, two cases may occur: (i) the loading causes significant mechanical degradation (excessive cracking on given zones for instance): the concrete is hence subjected to substantial supplementary heterogeneity and, therefore, the methodology could not be applied due to the non-stationarity of random fields; (ii) the loading causes negligible or uniform degradation: the results presented in the following would not be affected. The wall is 2.3 m high, several tens of meters long, and 20 cm width. The portion of wall considered is East–West oriented and 3.5 m length. Non-destructive and destructive measurements were carried out on the North face while only destructive measurements were carried out on the South face. The 21 successive measurements along a single horizontal line situated at 1.5 m above the ground were located at center of the reinforcement meshes with a constant distance of 16 cm between measurements. These are common operational conditions: limited number of semi-destructive tests or small distance between tests to ensure the condition of similar exposure zone. It was shown that second order statistical properties of the random filed could be characterized from a single sample function (or trajectory) when using NDEs on similar exposure zones (Schoefs et al., 2016). Concrete saturation degree and porosity were estimated by both destructive and non-destructive techniques. For the destructive tests, cores were extracted according to EN-13791 (2007), porosity and density were determined following the procedure described by NF 18-459 (2010). The distance of the measurement line to the ground (1.5 m) and to the top (0.8 m) was selected to avoid edge effects both from environmental conditions and material variability due to concreting. Compressive strength was estimated by non-destructive techniques (rebound hammer). After inspection, carbonation depth was estimated from extracted cores immediately placed into sealed plastic bags and measurements were conducted in lab.

Non-destructive evaluation of the same parameters results from a procedure combining different non-destructive technics (ultrasonic waves, radar, impact echo, surface waves, and capacitive sound) by data fusion. This technique was adapted because any of these NDEs can provide a direct measurement of the quantities of interest. Data fusion is based on possibility and fuzzy set theories (Dubois and Prade, 2001). Nevertheless, the assessment of uncertainties affecting the output of the NDEs cannot be easily and directly determined from the observations (frequency, wave velocity, etc.), and European standards do not address this topic yet. Since we want to analyze the capacity of models to propagate uncertainties and spatial variability, NDEs are, therefore, not considered in this paper.

Exposure conditions after 35 years of each wall face are rather different and, consequently, their effect on measured quantities (carbonation depth or saturation degree) is not negligible: on the South side, the drying is faster and the carbonation is supposed to be accelerated. The results indicate (Tables 1 and 2) that the mean value of carbonation depth is 1.96 cm for North side (Side A) and 2.42 cm for South side (Side C). It was, therefore, decided to analyze separately the measurements obtained on each face.

This paper will focus on modeling spatial variability of three properties: concrete saturation degree, porosity and carbonation depth. Given that many studies have been devoted to numerical simulation of random fields—e.g., Kenshel (2009) and Schoefs et al. (2009), this work employs well-known numerical methods toward this aim.

A random field X(x, ω) is a set of random variables X(ω) (where ω denotes hazard) indexed by a parameter x (continuous or discrete) whose values belong to R_n. In this case, x represents the space in the horizontal direction. For a given realization ω₀, X(x, ω₀) represents a sample function (or trajectory) of the random field. This study assumes that the field is ergodic (stationary) to be able to estimate all properties of X(x, ω) (mean, variance, correlation length, statistical moments) from a unique sample function X(x, ω_i). We thus consider only one sample function for each field in the following and ω is not mentioned anymore. For each x₀, X(x₀) is a random variable whose probability density function is fXx0h. The n-order spatial moment mDn and the global statistical moment mx0n are respectively defined as: (1)mDn=limD→∞1D∫DXnx0dx (2)mx0n=EXnx0=∫RXnx0fXx0hdh where D describes the geometry of the field and R is the space of real numbers. In the case where mX0n and fXx0h do not depend on x₀, the random field is stationary (Property P1). On the other hand, when mDn and mX0n are equal, the random field is ergodic (Property P2): an ergodic field is thus stationary. Moreover, if the spatial variance mD2 is also equal to the global statistical variance mX02, the random field is second order stationary (Property P3). In this case, the autocorrelation function ρ(x − x′) [correlation between random variables X(x, ω) and X(x′, ω)] depends only on the lag distance Δx = x − x′between two positions x and x′.

Due to the fact that only one realization of each random field is available and for the sake of simplicity, we assume that the random fields are ergodic (P2) and Gaussian. However, additional experimental observations are required to confirm this assumption. Moreover considering more complex type of random fields (non-stationarity or piecewise stationarity) (Schoefs et al., 2009) is beyond the scope of this study and would complicate the comparison between models. That is the case for excessive cracking on given zones. For many practical cases in civil engineering, the amount of data is insufficient to justify or use P1 and we assume P3 (second-order stationarity).

The autocorrelation function of a stationary random field describes the decay of the correlation with respect to the distance between points. Many autocorrelation functions were proposed in the literature [see, for instance, Sudret and Der Kiureghian (2000) and Kenshel (2009) for an overview]. These functions are characterized by the scale fluctuation θ.

Two main procedures are reported in the literature for the estimation of θ. The Maximum Likelihood Estimate method consists in searching for the value of θ that maximizes the joint probability density of the data, supposed to be the realizations of the same distribution function (Li, 2004). Initially correlated according to the ongoing value of θ, these realizations must be transformed into uncorrelated variables so as to compute the joint probability density as a simple product of independent standardized Gaussian variables. The fitting method aims at assessing θ that best fits the correlation profile ρ_D(Δx) obtained from the measured data (Vanmarcke, 1983). For a one dimensional and stationary random field, the correlation profile along the domain is determined as the successive values of the correlation coefficient with respect to the distance Δx between points: (3)ρDΔx=∑i=1mXxi−mXXxi+Δx−mX where m_X and s_X are, respectively, the mean and SD of X estimated from independent values of data and m is the number of points at a distance Δx from each other.

Among other possible techniques (Lévy, 1965; Vanmarcke, 1983), the Karhunen–Loève expansion (Karhunen, 1947; Lévy, 1965) was used in this study to simulate a Gaussian stationary random field: (4)Xx,θ=mX+sX∑i=1nλiξifix where n is number of terms in the truncated expansion, ξ_i is a standardized Gaussian random variable, λ_i and f_i are, respectively, the eigenvalues and eigen-functions of the autocorrelation function ρ_D(Δx).

Only few papers in the literature recommend the use of a given autocorrelation function. We propose, herein, to use an exponential autocorrelation function, generally used for representing the autocorrelation of concrete property or durability indicators (Kenshel, 2009; Schoefs et al., 2016). Figure 2B shows that it is well adapted in the present case also. In the case of an exponential autocorrelation function ρ(Δx) = exp(−|Δx|/b) that depends on the correlation parameter b, the eigenvalues λ_i, and eigen-functions f_i are expressed under the assumption that the field is second-order stationarity (Sudret and Der Kiureghian, 2000): (5)λi=2b1b2+ωi2 (6)fix=cosωixa0+sin2ωia02ωiforioddsinωixa0−sin2ωia02ωiforieven where a₀ is half the length of the domain and ω_i is the solution of the following transcendental equations: (7)1b−ωtanωa0=0forioddω−1btanωa0=0forieven

This section proposes quality indicators to analyze the ability of predictive models to represent the spatial correlation of degradation indicators (carbonation depth). The idea is to quantify how well the spatial correlation of the degradation estimated by the propagation of uncertain measured data through models fits the spatial correlation of the measured degradation. In the aforementioned on-site investigation, each extracted core was divided into three zones, namely, A, B, and C (Figure 1). Sides A and C comprised the edges of the concrete wall exposed to the atmosphere while B was the inner part of the wall. As the carbonation depth is the matter of interest and was not detected on the portion B, only the destructive measurements on sides A and C are considered in this study. The mean values and their related uncertainties are reported in Tables 1 and 2.

The spatial variability of degradation processes (chlorination, carbonation, depassivation of steel reinforcing bars, etc.) has to be considered for structural inspection, diagnosis, and maintenance. Indeed, when the maintenance strategy is risk-based, which is a global trend nowadays, assessing the probability of depassivation over a global surface of a concrete wall or structure can help to define a portion of that surface presenting a substantial risk and then prescribe appropriate maintenance measures. The probabilistic assessment must be based on a more or less refined knowledge of the spatial correlation existing in concrete cover depth, concrete properties (for instance porosity), and state indicators (for instance saturation degree). Theoretical considerations were developed from decades to represent this spatial variability within the rational framework of the stochastic finite element method. The probabilistic assessment combines these stochastic methods with deterioration (carbonation) models to estimate failure risks. Carbonation models could take more or less complex analytical or numerical forms.

This paper estimated the capability of one-dimension analytical carbonation models to deal with spatial variability based on in-field data. The database contains some input and output model parameters determined from destructive testing on cores extracted from an enclosure wall. The correlation profiles of the carbonation depth either obtained from the measurements or computed thanks to the experimental input used in the models were established. Two normalized so-called correspondence indices were proposed based on (i) the “distance” between profiles for perfect measurements or (ii) the “overlapping” of simulated and experimental data for uncertain measurements. The correspondence index determined without accounting for uncertainties is not relevant in practice because it does not allow the models to be properly discriminated. Contrarily, the correspondence index incorporating uncertainties reveals clearly the various capabilities of models to transfer the spatial variability from input to output, compared to the experimental one. It was found that some models are more or less appropriate to propagate spatial variability depending on the exposure conditions. However, it was not possible to establish a unique ranking from the existing database because carbonation processes were influenced by environmental conditions that differ for each side of the wall. It can be, therefore, concluded that the proposed methodology allows determining the capability of carbonation models to deal with uncertainties and spatial variability as a function of the exposure zone. More experimental data are required for generalization purposes.

FS: probabilistic modeling of spatial variability, FD: carbonation models, NR: NDT and destructive techniques, TL: numerical simulations, EB-A: propagation of uncertainties through models of degradation.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.