Introduction

It is of increasing importance to predict the long-term chloride ingress in concrete as ever longer design service life requirements are prescribed. Roughly, two types of models for chloride ingress prediction exists: empirical and physical. Empirical models are widely used by engineers in practical applications, which is probably due to their simplicity relative to physical models. Physical models seek to model chloride ingress by e.g. including free chloride as driving force, chloride binding, additional transport processes besides diffusion, multi-ionic characteristics, moisture and dissolution/precipitation reactions [1,2,3].

Empirical chloride ingress models generally require fewer input parameters. Commonly chloride ingress is modeled by the error function solution to Fick’s 2nd law of diffusion for a semi-infinite medium. The total chloride content is often considered as the driving force, although only free chloride is available for diffusion [4,5,6,7]. To account for the observed decrease of the diffusion coefficient over time, often ingress prediction models include an ageing function [1, 4,5,6,7,8,9,10,11]. A drawback of the ageing function is that it causes the chloride ingress to stop at infinite age (the diffusion coefficient goes towards zero). To avoid this issue, the service life model Life 365 keeps the apparent chloride diffusion coefficient (Da) constant after 25 years exposure [12]. The surface concentration (Cs) is constant in many models, although it has been observed to increase over time. The HETEK model takes this into account by a time dependent surface concentration [8] based on work by Mejlbro [11]. Finally, the maximum concentration is often observed at a certain depth; either due to convection (as accounted for in e.g. [4]) or due to elemental zonation and leaching [13, 14], which is currently only accounted for by omitting the outer measuring point(s) when fitting data [4].

Another empirical approach to predict chloride ingress could be to follow the ingress of a chloride concentration of interest (e.g. the critical chloride content) rather than the entire chloride profile. Poulsen and Sørensen [15] observed a linear relationship between a reference concentration of 0.05% chloride by mass of concrete and the square root of time and termed this the “square root observation”. Poulsen et al. [16] showed an excellent correlation with experimental data of up to 20 years for submerged exposure in Träslövsläge Field Exposure Site in Kattegat and up to five years for submerged exposure at Fehmarn Belt Exposure Site.

According to Poulsen et al. [16] the ingress depth of a reference concentration of 0.05% chloride by mass of concrete (x0.05) can for their data be described by Eq. (1).

$$x_{0.05} (t) = a_{0.05} \times \sqrt t + b_{0.05}$$
(1)

where a0.05 is the slope of a straight line, when x0.05 is plotted against the square root of exposure time, and b0.05 is the intercept. Poulsen et al. [16] interpreted a0.05 as an indicator for the long-term chloride penetration rate of the reference concentration (Cr = 0.05% chloride by mass of concrete), while b0.05 is interpreted as an indicator for the ingress relatively faster than a0.05 occurring at early age. Hereafter the slope of the straight line for a Cr is mentioned as aCr (ingress parameter), the intercept as bCr (early ingress depth), and the ingress depth as xCr.

In the square root observation, the ingress parameter is constant with square root of exposure time and the challenge of the chloride ingress approaching zero exponentially for long term exposure is thereby overcome. However, the analysis by Poulsen et al. [16] only applied the square root method on data from submerged exposure at Fehmarn Belt and Träslövsläge Exposure Sites and only for a reference concentration of 0.05% chloride by mass of concrete.

Covering other types of exposure (wetting and drying), Thomas and Matthews [17] plotted the depth of penetration of 0.4% chloride by mass of binder versus the square root of exposure time for concrete with water-to-cement ratio (by mass) (w/c) 0.37–0.57 and 0–50% fly ash (FA) substitution for tidal exposure. They found that the ingress depth of 0.4% chloride by mass of binder was 18–21 mm already after 28 days, but only increased to 28–31 mm after ten years for concrete with 30% FA substitution [17]. Thomas and Matthews [17] plotted the ingress depth of 0.4% chloride by mass of binder versus the square root of time, but did not mention a linear relationship, although the graph indicated it.

Baroghel-Bouny et al. [18] found a linear relationship with a R2 of > 0.95 between the average free chloride penetration depth measured by 0.1 N AgNO3 spray test (assumed equal to 0.15% chloride by mass of binder) and the square root of the number of wetting–drying cycles after 10 years at marine tidal field exposure.

Based on the observations by Thomas and Matthews [17], Baroghel-Bouny et al. [18], Poulsen and Sørensen [15] and Poulsen et al. [16] it is likely that the observation of an ingress depth versus square root of time dependency can be generalized. The square root observation is based on a very simple linear relationship between the ingress depth of a single reference concentration and the square root of exposure time. This linear relationship is consistent with the error function solution to Fick’s 2nd law of diffusion for a semi-infinite medium with constant Da and Cs [5]. However, the square root observation adds an intercept value to account for early ingress.

This paper formulates the square root observation as a method and analyzes the applicability to describe chloride ingress by including field data from submerged, tidal, atmospheric and inland deicing salt exposure at various geographical locations. Five different reference concentrations were analyzed for each exposure condition and geographical location. By applying the square root method on additional exposure conditions, it can be tested under which conditions the square root method is applicable.

Data

This paper analyzes data from a number of studies of in-situ exposed concretes from marine field exposure sites including exposure sites at Fehmarn, Denmark [16]; Träslövsläge, Sweden [19]; Østmarksneset, Norway [20] and Dornoch, Scotland [21]. Analyzed marine exposure conditions include submerged, tidal, splash and atmospheric. Further analyzed are data from concretes exposed in deicing environment at the inland field exposure site RV40, Sweden [22]. For each concrete composition and exposure condition data from three to eight exposure times, ranging from six months and up to maximum 31 years, were reported.

An overview of exposure conditions, curing and sampling is given in Table 1. An overview of mixture proportions of the analyzed concretes is given in Table 2. All cement notations are according to EN 197-1 [23]. The nomenclature (IDs) used in this study reflects the exposure site and the original ID used in the reference paper. When results obtained from the concretes are displayed in figures, the label additionally includes the content of supplementary cementitious materials (SCMs) and information on the exposure type, e.g. F-G12FA4SF_sub stands for Fehmarn Belt Exposure Site, concrete G, which include 12% fly ash (FA) and 4% silica fume (SF) and is exposed at submerged exposure. Binder types are described in groups comprising plain Portland cement (PC) and blends with FA, SF, ground granulated blast furnace slag (GGBS) and combinations of fly ash and silica fume (FA + SF).

Table 1 Overview of exposure conditions, curing and sampling at included exposure sites
Table 2 Mixture proportions of concretes used in this study

The square root method for chloride ingress prediction

The square root dependency of a constant (reference) chloride concentration, \({\text{C}}_{\text{r}}\), proposed for chloride ingress prediction by Poulsen and Sørensen, was used to analyze the data [15, 32]:

$$x_{{C_{{\text{r}}} }} \left( t \right) \, = a_{{C_{{\text{r}}} }} \times \surd t + b_{{C_{{\text{r}}} }}$$
(2)

where t is the exposure time, xCr(t) is the depth of Cr at t, and aCr and bCr are constants.

The method was applied using the steps described in Fig. 1.

Fig. 1
figure 1

Application of the square root method for determination of the chloride ingress constants aCr and bCr (Eq. 3)

Selection of input data for the square root method

In this study, only data sets with minimum three exposure times were used, and for which the chloride fronts penetrating from opposing sides of the specimens did not overlap. The entire data set (data from all exposure times) was omitted when lower chloride ingress was measured at a given exposure time relative to an earlier exposure time. This was the case for the combinations of concrete and exposure listed in Table 3.

Table 3 Excluded combinations of concrete and exposure conditions due to measured lower chloride ingress at a given exposure time relative to an earlier exposure time

A minimum exposure time of 200 days was applied in this study and the applicability of the data from the first exposure time was checked (see Sect. 5.1). The minimum exposure time was selected based on studies by Tang [33]. Investigating a range of binders (PC, SF) and w/b (0.32–0.70) Tang [27] found six months as sufficient time for laboratory samples to obtain an insignificant change of properties [e.g. water assessable porosity and diffusivity determined by the rapid chloride migration test (RCMT)] for prediction of free and total chloride ingress in submerged marine concrete using the so-called engineering expression of the ClinConc model [1].

Selection of input parameters for the square root method

Based on the range of observed Cmax-t1 values (see Online Resource 1), two sets of Cr values were chosen. One set for submerged and tidal exposure and another set for splash, atmospheric, and deicing exposure, see Table 4. When choosing the maximum Cr value a compromise was made between (1) ensuring that all Cr values could be applied for all concretes (maximum Cr ≤ minimum Cmax-t1) and (2) including the widest possible span of Cr values. The compromise was made slightly below the 20th percentile of the Cmax-t1 values, thus ensuring that all Cr values were tested for at least 80% of the concretes. The remaining up to 20% of the concretes (see Table 4) were still tested at lower Cr values present in those concretes. Additionally, four reference concentrations were chosen as the pentiles. All chloride profiles are given in Online Resource 1 with the utilized Cr values marked as horizontal lines.

Table 4 Range, average and 20th percentile of observed Cmax-t1 and chosen range of Cr for all analyzed concretes and exposure types

Results

Before applying the square root method on all the data (Sect. 2), it was checked, whether the assumption made in the methods section (Sect. 3), that aCr is constant, when using data after minimum 200 days exposure (t1), is correct.

Applicability of the chosen t1

Initially, it was verified whether the t1 applied in this study i.e. 203–370 days, is sufficient to have reached insignificant changes of properties as mentioned in Sect. 3.1. The result of this verification is seen at Fig. 2, where the ingress depth of Cr = 0.75% chloride by mass of binder (x0.75) at t1 predicted by the square root method using data from t2 and t3 is compared to the ingress depth found by linear interpolation on the measured chloride profiles for t1. Data from submerged exposure for Fehmarn Belt and Träslövsläge Field Exposure Sites (Fig. 2a) as well as data from tidal exposure from Fehmarn Belt Exposure Site (Fig. 2b) was used for this comparison.

Fig. 2
figure 2

The ingress depth of a concentration of 0.75% chloride by mass of binder (x0.75) at t1 predicted by the square root method using data from t2 and t3 compared to the ingress depth found by linear interpolation on measured data for t1. t1 varied between 203 and 217 days. a Data from submerged exposure for Fehmarn and Träslövsläge was used. b Data from tidal exposure from Fehmarn was used. A dotted straight line corresponding to x = y is drawn as guidance

The data points are closely distributed around the line x = y with a single outlier on either side, which indicate, that the chloride profiles from t1, can be used in the square root method without any systematic change of aCr and bCr. The remaining data series show similar behavior. This implies that for the investigated concretes and exposures, the quality of the datapoint (xCr-t1, t1) is as good as the remaining data points.

Application of the square root method

Data from 59 combinations of exposure site, exposure condition and concrete were treated by the method described in Fig. 1 for five different Cr values. The resulting aCr, bCr and R2 values (237 sets in total) are tabulated in Online Resource 2.

Figure 3 illustrates the results of the application of the square root method at five different Cr values on concrete T-H8 submerged exposed at Träslövsläge Field Exposure Site.

Fig. 3
figure 3

Application of the square root method at five different Cr values (in % chloride by mass of binder) on concrete T-H8 submerged exposure at Träslövsläge Field Exposure Site. a Example of application of step 3 of the square root method (see Fig. 1). Concrete T-H8 exposed submerged at Träslövsläge Field Exposure Site (see Table 3). Interpolated ingress depths of Cr = 0.25, 0.50, 0.75, 1.00 and 1.25% chloride by mass of binder are marked with symbols. b Example of application of step 4 of the square root method (Eq. 3; see Fig. 1). Concrete T-H8 submerged exposed at Träslövsläge Field Exposure Site (see Table 3). xCr = penetration depth of Cr

Discussion

This section includes discussions on the impact of the choice of Cr on aCr, bCr, R2 and xCr. Further the impact of exposure condition and binder is treated and hypotheses for the observed linearity of xCr versus √t are proposed. Finally, the ability of the square root method to predict further ingress is evaluated and suggestions are given on how to use the method.

Impact of reference concentration, C r

As mentioned in Sect. 3, to apply the square root method a reference concentration, Cr, must be chosen. In this section, the influence of the chosen Cr is discussed. The datasets for submerged and tidal exposure are used, as they represent the widest variety of concretes and identical Cr values, see Online Resource 2.

The influence of Cr on aCr, bCr and R2 is separated by binder type and exposure to highlight potential differences. The influence of binder is further treated in Sect. 5.4.

Impact of C r on a Cr

Figure 4 shows the calculated ingress parameter, aCr, as a function of the reference chloride concentration, Cr, for submerged conditions (Fig. 4a) and tidal conditions (Fig. 4b). For each combination of concrete and exposure, aCr is normalized to a0.75. Thereafter the results are grouped and averaged per binder type (PC, FA, SF, FA + SF, GGBS); see Online Resource 3 for an example of how data is normalized and averaged. From Fig. 4 it is seen, that there is in general a slightly decreasing trend of aCr/a0.75 with increasing Cr. This trend is most pronounced for PC, SF and GGBS containing concretes, whereas aCr/a0.75 for concretes with FA (FA and FA + SF) is almost independent of the Cr. The varying dependency between binders of aCr/a0.75 on Cr implies that aCr/a0.75 could be higher for a concrete based on binder i than binder ii at one Cr, while the opposite is the case at another Cr (See Online Resource 4).

Fig. 4
figure 4

Influence of chosen Cr on average aCr value for concretes based on different binders in a submerged and b tidal exposure. Note that aCr is normalized to a0.75. Error bars show ± 1 standard deviation

Figure 4 shows decreasing aCr/a0.75 with increasing Cr. This means, that the calculated ingress parameter is lower for higher Cr. Physically this observed trend coincide with a reduction in the driving force for diffusion, as the concentration gradient decreases with increasing Cr (all other things being equal).

When comparing the results from submerged exposure (Fig. 4a) with the ones from tidal exposure (Fig. 4b), there is a trend that a0.25/a0.75 is higher for tidal exposure than for submerged exposure irrespective of the binder type.

Impact of C r on \({b}_{\text{Cr}}\)

The impact of the selected Cr on bCr is illustrated in Fig. 5. It is seen that bCr depends on both Cr and the binder composition. A clear decreasing trend of bCr with increasing Cr is seen for FA, SF and SF + FA containing concretes independent of exposure condition (submerged and tidal). The analyzed PC and GGBS containing concretes have bCr values near 0 irrespective of Cr in the range 0.25–1.25. This implies that aCr governs the chloride ingress for the analyzed PC and GGBS concretes.

Fig. 5
figure 5

Influence of chosen Cr on average bCr value for concretes based on different binders in a submerged and b tidal exposure. Error bars show ± 1 standard deviation. Note that the values of bCr are not normalized as some b0.75 are near zero and the corresponding concretes thus affect the average too much

Figure 5 shows a decreasing bCr with increasing Cr. This might be explained by two competing sets of mechanisms:

  1. (i)

    Capillary suction and high diffusivity at early age

    Poulsen et al. [16] interpreted bCr as partly due to initial capillary suction due to self-desiccation or partial drying and partly due to limited degree of hydration resulting in a concrete, which is initially more permeable than in a more mature state. Thus, it is expected that chloride ingress proceeds faster initially, than what can be described by aCr alone. This leads to a positive contribution to bCr.

  2. (ii)

    Gradual phase changes and increasing Cmax.

    Selecting a high Cr which is only present in the concrete after e.g. 4 years, the intercept at the x-axis of the straight line constituting xCr versus the square root of time (step 4, Fig. 1) would be 2 (√4). As the ingress parameter aCr is positive, this implies that bCr is negative. An increasing Cmax could e.g. be caused by phase changes as ions from the sea water such as chloride are bound in the system and others leached [13, 34]. The higher the chosen Cr, the longer time it will take for the reference concentration to be present in the concrete and the larger the negative contribution on bCr will be.

The set of mechanisms (1) is relatively independent of Cr and therefore relatively more pronounced at low Cr, whereas the set of mechanisms (2) is concentration dependent and more pronounced at high Cr. As bCr is the sum of positive and negative contributions, the result is a decreasing bCr with increasing Cr.

Discussion on limits of reference concentration

For service life modeling the reference concentration of interest is typically the critical chloride content causing corrosion initiation. However, the critical chloride content can vary considerably [35, 36]. It is therefore important to know the range of reference concentrations with which the square root method can be applied.

Choosing a Cr value higher than the maximum chloride content at the earliest recorded exposure times naturally rules out the use of these chloride profiles for the square root method. Additionally, for a fixed exposure time, it forces the use of a point closer to the concrete surface, which could potentially lead to disturbances caused by leaching and elemental zonation [13, 34].

The lower limit of the Cr value must be related to the method of obtaining the chloride profile, as well as the detection limit and accuracy. The lowest Cr value must be distinguishable from the initial chloride content (Ci). Further, to enable interpolation, the profile grinding must be conducted deep enough to detect a concentration lower than the lowest Cr value.

To test the chloride concentration range in which the method is applicable; the square root method was applied on concretes exposed in submerged and tidal zones for Cr values of 0.125 and 1.8% chloride by mass of binder. (Note that for some of the combinations, the highest concentration was not reached at the shortest analyzed exposure time.) From Fig. 6 it can be seen that R2 for both Cr values was comparable to R2 for the Cr values generally used in this paper (see Table 4), except when using Cr = 1.8% chloride by mass of binder for T-12, F-F and F-G and Cr = 0.125% chloride by mass of binder for T-H2 and Dor_sub, where a decreasing tendency of R2 is observed. It can be concluded that the square root method is valid for the investigated concretes in submerged and tidal exposure for the whole range of tested Cr values, i.e. 0.125–1.80% chloride by mass of binder. Similarly, the method was found applicable for concretes in the atmospheric zone using Cr values in the range 0.1–0.5% chloride by mass of binder (data not shown here).

Fig. 6
figure 6

R2 value of linear regression on xCr versus square root of time for five concretes from submerged exposure at Träslövsläge Field Exposure Site, when using Cr values of 0.125 and 1.8% chloride by mass of binder in comparison to the five Cr values used throughout this study for submerged and tidal exposure

Selection of Cr for remaining discussion

When investigating the influence of exposure conditions and binders in further detail, it was chosen to proceed with a Cr of 0.75% chloride by mass of binder for tidal and submerged exposure. For atmospheric and deicing exposure, it was chosen to proceed with a Cr of 0.3% chloride by mass of binder. These reference concentrations were chosen from a perspective of benefitting from using as much of the raw data as possible, i.e. avoiding the challenge of using a Cr which is yet not obtained at t1 or which is not measured due to lack of grinding depth. In theory, if the chloride threshold value is known and within the limits of reference concentrations discussed in Sect. 5.1.3, it could be chosen as Cr.

Accuracy of interpolation

Figure 7 shows the ratio between the ingress depth of Cr = 0.75% chloride by mass of binder calculated by the square root method by using data from all available exposure times, and the ingress depth found by linear interpolation between the nearest data points directly below and above the Cr in the measured chloride profiles (x0.75(predicted)/x0.75(measured)).

Fig. 7
figure 7

Ratio between x0.75 predicted by Eq. (1) (xCr(t) = aCrx√t + bCr) and measured by linear interpolation between the nearest data points directly below and above the reference concentration (0.75% chloride by mass of concrete). white fill = Fehmarn, black fill = Träslövsläge, diamond = PC, triangle = FA, circle = SF, square = FA + SF, line = GGBS. The dashed lines show a 90% confidence interval

For the analyzed concretes, x0.75(predicted)/x0.75(measured) varied by ± 11% (90% confidence interval). The largest variation is observed for concretes containing SF, whereas the least variation is observed for concretes with PC or FA. Considering the usual variation in service life estimates, a prediction error of 11% (90% confidence interval) appears acceptable [21, 37, 38].

Impact of exposure condition

Considering aCr describes the ingress of chloride ions over the square root of time, a given concrete should have a higher aCr value, the higher the environmental load it is exposed to. The environmental load can be described by chloride, moisture and temperature load. When tidal, submerged and atmospheric exposure are compared, aCr is expected to be lowest for atmospheric exposure due to the relatively low chloride and moisture load present there. The relative ingress rate in the tidal and submerged exposure is expected to be depth dependent; in the tidal zone convection will cause rapid ingress and accumulation of chlorides in the surface near region giving rise to a high chloride gradient, whereas an expected higher moisture state of submerged concrete will facilitate a higher diffusion coefficient [39].

As shown in Table 1, in this study data has been analyzed from three exposure sites, where elements of the same concrete composition are exposed under different exposure conditions. A comparison between aCr at Cr = 0.75 (submerged, tidal, low splash) and Cr = 0.3 (atmospheric, high splash) for identical concretes in different exposure conditions is shown in Fig. 8a–c.

Fig. 8
figure 8

a Ratio between a0.75 for concretes in tidal vs. submerged exposure at Fehmarn Belt Exposure Site. The dashed line indicates the average ratio. b Ratio between a0.30 for concretes in atmospheric versus a0.75 for concretes in submerged exposure at Träslövsläge Field Exposure Site. The dashed line indicates the average ratio. c Ratio between aCr for the concrete Dor exposed at XS3HT (high tide, Cr = 0.75), XS3LS (low splash, Cr = 0.75) and XS3HS (high splash, Cr = 0.30) versus XS2 (below mid tide, Cr = 0.75) at Dornoch Field Exposure Site. The dashed line indicates the average ratio

For Dornoch Field Exposure Site no data is available for submerged exposure. However, one concrete composition is exposed in almost saturated below mid tide level and thus used as reference [21].

Figure 8a–c indicates that for the compared exposure conditions, aCr depends not only on the material, but also on the exposure. Large variations are observed between concrete compositions. However, for the investigated combination of concretes, exposure conditions and Cr, aCr typically descended in the order: tidal > submerged > atmospheric, high tidal and splash. The mentioned order of descension did not change by varying Cr within the investigated span (not reported here). Note that lower Cr values were used for atmospheric and high splash exposure than for submerged. However, due to the increasing trend of aCr with decreasing Cr (Fig. 4), the ratio is expected to be lower, if aCr for atmospheric and splash exposure was compared to aCr for submerged exposure at the same Cr. Thus, this effect would increase the influence of environment as compared to what is shown in Fig. 8a–c.

The square root method was applied on data for deicing exposure from the highway RV40 field exposure site in Sweden [22, 24, 25]. However, only a single concrete complied to the criteria set up in Sect. 3.1. The other concrete compositions failed on the criterium of having an increasing chloride concentration with time. This could partly be due to seasonal changes as de-icing salts are spread during the cold seasons and the chloride profiles were measured at different times of the year (January, May, June, July and September). Further the de-icing procedure changed, while the data collection was ongoing. When only summer sampling (May–July) from vertical surfaces is utilized for concrete 206, R2 values of 0.98–1.0 are obtained—see Online Resource 2. However, as data from only a single concrete in de-icing exposure complied to the criteria set up in Sect. 3.1 and the data due to seasonal variations is expected less reliable, de-icing is not treated further.

Influence of binder

Based on the findings in Sects. 5.1 and 5.3, the analysis on the influence of binders is conducted for a fixed Cr of 0.75 and submerged exposure conditions. Submerged exposure is chosen as the widest variety of binder compositions is available for this exposure condition. Figure 9 shows the experimental and modelled chloride ingress data for concretes with different binder types (PC, FA, GGBFS, FA + SF and SF). Table 5 presents the corresponding parameters of the square root fits.

Fig. 9
figure 9

Experimental data (markers) and graphical result of the square root method (lines) for concretes in submerged exposure. a PC, b FA, c GGBS, d FA + SF, e SF. Cr = 0.75. Although concretes with similar binders are compared for submerged exposure and identical Cr, some differences are seen between the concretes due to differences in w/ceq, exposure temperature, salinity etc

Table 5 aCr, bCr and R2 values for concretes in submerged exposure. Binder groups are divided by thick borders (FA = Fly ash, GGBS = Ground granulated blast furnace slag, SF = Silica fume). Cr = 0.75

Note that the choice of Cr has an impact on the predicted chloride ingress as described in Sect. 4.2 i.e. if a lower Cr was chosen, the predicted chloride ingress in a PC concrete would be even higher relative to a FA containing concrete. An example of the influence of Cr on the ingress depth is given in Online Resource 4, where the graphical result of the square root method applied on concretes exposed at submerged exposure for Cr = 0.75 (as shown at Fig. 9) is compared to the graphical result for Cr = 0.25.

As expected, some differences are seen between the concretes in Fig. 9. Although concretes with similar binders are compared for submerged exposure and identical Cr, differences exist in w/c, exposure temperature, salinity etc. (see Table 1). aCr is e.g. expected to increase with increasing w/c.

From Fig. 9, Table 5, Online Resource 2 and Online Resource 3 it is seen that the regression coefficients are very high. Further it is noted that similarities exist between similar compositions. e.g. aCr of the analyzed PC concretes are all in the range 15–20 mm/√year, which is higher than for any other concretes. Contrary, bCr is low for the PC concretes. A low bCr combined with a high aCr is interpreted as the PC concretes quickly obtain their maximal potential chloride resistance in the given environment they are exposed to, but that this maximal potential chloride resistance is relatively poor (assuming similar moisture level).

The values of aCr obtained for FA, FA + SF and GGBS concretes are low compared to the PC concretes and they are all in the same range (4–8 mm/√year). The long-term chloride resistance is high for both FA, FA + SF and GGBS, but due to an additional high early resistance, reflected in the low bCr, the GGBS concretes have the lowest extrapolated ingress depths after 100 years as shown in Fig. 9c.

Hypotheses for linearity of x Cr versus √t

The ingress of ions (1) in concrete might be explained by the Nernst–Planck equation (Eq. 1). The left hand term of the equation describes the change of concentration over time. The three right hand terms describe diffusion, migration and convection respectively [40]:

$$\frac{{\partial c_{i} }}{\partial t} = \nabla \left( {D_{i} \nabla c_{i} + z_{i} u_{m,i} Fc_{i} \nabla V - c_{i} v} \right)$$
(1)

where ci = ionic concentration, Di = ionic diffusion coefficient, zi = electrical charge, um,I = the ionic mobility, F = Faradays constant, V = electrostatic potential, v = velocity of the solvent.

Assuming one-dimensional diffusion in a non-reactive and homogeneous media, the diffusion coefficient is constant, and the diffusion is square root of time dependent.

At initial exposure to chloride ions, concrete is in general neither fully saturated nor fully hydrated. Therefore, initial ingress might be enhanced by capillary suction (convection) and a higher initial diffusion due to an initial higher and more interconnected porosity. The latter is especially expected for concretes containing slowly reacting SCMs as e.g. fly ash.

Over time it is expected that convection ceases or becomes insignificant compared to diffusion. However, other factors such as elemental zonation [13], leaching [41], moisture [42] and temperature [43] are also known to influence the ingress, binding and resulting chloride profiles. Predicting the ingress depth of a total fixed (reference) chloride concentration, the concentration dependency of chloride binding (illustrated with a non-linear binding isotherm), can be neglected. In addition, the potential impact of leaching and elemental zonation on chloride binding is reduced. Finally, focusing on deeper ingress depths, limited impact of moisture changes on the transport coefficient is expected.

In summary, when the dominant chloride ingress mechanism is diffusion and the diffusion coefficient and chloride binding capacity are constant, linearity is expected of xCr versus √t, as the square root of the diffusion coefficient is length/√t.

Ability to predict further ingress

To test the ability of the square root method to predict further ingress, the dataset from Träslövsläge Exposure Site comprising data from five exposure times up to 20 years was used. The square root method was applied to the first four data points (up to 10 years or 3.3 √year) and the predicted ingress after 20 years was compared to the measured ingress. The deviation between the predicted and measured ingress depth was similar for the measured ingress depths of t1−t4 to which the prediction was fitted (9% deviation) as to the measured ingress depth at t5, which was not used for fitting the straight line (12% deviation)—see Fig. 10 and Online Resource 5. It can therefore be concluded that the square root method is able to predict chloride ingress with an acceptable accuracy over 20 years.

Fig. 10
figure 10

Predicted versus measured chloride ingress (Cr = 0.75) at 20 years for four submerged concretes exposed at Träslövsläge Field Exposure Site. Prediction was made from data up to 10 years exposure (shown with filled markers). The diameter of the measured data point for T-H2-10SF_sub is doubled to distinguish it from that of T-H1-5SF_sub

Suggestions of how to use the square root method

The square root method is suggested to be used for (1) estimating remaining service life of existing structures. Depending on the availability of data, it might also be used for service life design of new structures.

When used for estimating the remaining service life of existing structures, the following information is required:

  • Chloride threshold value. The reference chloride concentrations, Cr, should reflect the expected range of chloride threshold values for corrosion initiation.

  • Cover depth.

  • Chloride profiles to establish a straight line by the square root method considering the following aspects:

  • Chloride profiles must be measured with a sufficient period in time between them to allow the difference in chloride content to exceed the measurement uncertainties.

  • The exposure time of the first data point must be sufficiently long, so potential initial effects of e.g. capillary suction and prolonged hydration have ceased. The present investigation indicates the applicability of data from the earliest recorded exposure times (200 days for concrete initially cured for 43–49 maturity days and 217–370 days for concrete initially cured for 12–19 maturity days).

  • Although two exposure times are sufficient to establish a straight line by the square root method, it is suggested to determine three profiles at minimum three exposure times. If any of the points deviate from the long-term trend (e.g. with R2 less than 0.95), additional data points should be obtained.

When estimating the ingress depth for a long design service life (e.g. 100 years) by the square root method, the impact of aCr heavily exceeds that of bCr (see Table 5). Based on the findings in Sects. 5.15.4 multiple parameters including binder composition and exposure conditions do have an influence on the parameters aCr and bCr. There exists a potential of improving design values for new structures by increasing the number of investigated datasets and parameters to include e.g.:

  • The influence of binder compositions.

  • The influence of production and execution.

  • The influence of exposure time and conditions.

  • The influence of maintenance.

Conclusion

Based on analysis by the square root method of 34 concretes containing various amounts of PC, FA, SF and GGBS field exposed in up to five environments, it is concluded that:

  • The ingress depth (xCr) of a fixed (reference) chloride concentration (Cr) in the investigated concretes followed a square root of time dependency: xCr(t) = aCrxt + bCr, where aCr and bCr are constants.

  • The square root method appeared valid for marine exposed concrete in submerged, tidal, splash and atmospheric zone. Too little data was available for assessing the applicability for concrete in de-icing exposure.

  • The square root method was found to be valid for the whole range of tested Cr values (0.125–1.80% chloride by mass of binder for submerged, tidal and low splash; 0.1–0.5% chloride by mass of binder for high splash and atmospheric exposure).

  • The square root method appeared valid for the investigated PC, SF, FA, FA + SF and GGBS containing concrete.

  • For the analyzed concretes, x0.75(predicted)/x0.75(measured) varied by ± 11% (90% confidence interval).

  • For the investigated concretes and conditions, aCr was found to be constant after 200 days of exposure.

  • A decreasing trend of aCr with increasing Cr was observed for submerged and tidal exposure. It was most pronounced for PC, SF and GGBS containing concretes.

  • A clear decreasing trend of bCr with increasing Cr was observed for FA, SF and SF + FA containing concretes in submerged and tidal exposure.

  • Based on a dataset from 1 to 10 years of exposure we were able to predict the chloride ingress after 20 years with a deviation of 12%.

In summary, the square root method seems to offer a reliable possibility to predict chloride ingress.