A simple method for estimation of permeability of concrete from the compressive strength and pore size distribution based on literature survey

ABSTRACT This study aims to investigate the impacts of compressive strength and curing conditions of concrete on pore size distribution and air permeability. An empirical correlation was developed by analyzing the results from existing studies that measured both pore size distribution and air permeability. Air permeability was calculated using an empirical correlation newly developed. However, the correlation was targeted only to dry concrete since the influence of water content is very large on air permeability of concrete. To calculate air permeability of concrete in various water content, another correlation was developed between the degree of pore saturation and air permeability. Using the measurement result of pore size distribution, this study examined the effect of compressive strength and curing conditions on changes in pore size distribution and air permeability. Pore size distribution was measured by mercury intrusion porosimetry (MIP) using samples taken from mortar fragments collected from concrete. The compressive strength of concrete ranged from 21 to 150 MPa. Curing conditions were either water curing or air curing or sealed curing.


Introduction
The phenomenon of the explosive spalling of highstrength concrete at high temperatures like fire is well known. The reasons for explosive spalling are said to be thermal stress (Saito 1966) or water vapor pressure (Harmathy 1966). Even though the causes are not clearly verified yet, it was pointed out that the higher the compressive strength and the higher the water content increases the likelihood of explosive spalling (Lee et al. 2009(Lee et al. , 2013. It is necessary to investigate the effect of compressive strength and water content on permeability of concrete is in order to investigate the effect of vapor pressure on the cause of spalling.
Meanwhile, pore size distribution of concrete is a means of indirect evaluation of durability against salt damage and fire damage, which has a correlation with fluid flow inside the porous material. Air permeability is a quantitative measure of ease of fluid permeation. However, in order to measure air permeability of concrete, a special measuring device is needed, and especially, a longterm measurement is needed especially for highstrength concrete. It is practically difficlut to maintain constant water content condition during a long period of measurement.
In the area of durabaility of concrete, the effect of waterproofing admixture on the compressive strength and permeability of recycled aggregate concrete is discussed (Matar and Barhoun 2020) but the diffusion of chloride ion is the main problem. As for the surface repair mortar, the effect of material replacement and curing condutions are investigated and the effect of these parameters on gas permeability is discussed (Shi et al. 2020a(Shi et al. , 2020b.
For the above reasons, this study aimed to examine the effect of compressive strength and curing condition of concrete on air permeability by way of pore size distribution. A method to calculate air permeability from the measured pore size distribution is proposed.
The proposed formula is used as a property value related to moisture transfer which is necessary for simultaneous thermal and moisture transfer analysis of concrete exposed to high temperature like fire. When simultaneous thermal and moisture movement analysis is carried out, temperature distribution, moisture content distribution and pore pressure distribution are obtained as analysis results. It is also possible to investigate the relationship between pore pressure and occurrence of explosive spalling of normal and high strength concrete (Terada et al. 2017).

Correlation between compressive strength of concrete and air permeability in dry condition
In this section, air permeability of concrete in dry condition at room temperature is investigated. The effect of water content is discussed in section 2.2 and section 2.3.

Measured values of air permeability in literatures
Air permeability of concrete of dry condition was investigated. In this study, air permeability cited from the literature survey was limited to that measured by the direct method. Here, the direct method refers to a method of directly measuring the amount of air (gas) passing through a material according to a pressure difference. This is to secure the reliability of this study. Table 1 shows the conditions of concrete used in literatures. Dhir, Hewlett, and Chan (1989) produced four mixtures of water-cement ratio (hereafter, W/C ratio) and in four curing conditions. The compressive strength ranged from 25 to 62 MPa. The specimens were dried at 105 °C until the rate of weight loss got below 0.1 %/Day. Jacobs (1998) investigated the correlation between the degree of saturation and air permeability. They produced concrete with three differrent of W/C ratio. The compressive strength ranged from 28 to 57 MPa. The specimens were cured for two years at a relative humidity of 60 % R.H. (20 °C) and dried at 80 °C for two weeks. Khan and Lynsdale (2002) investigated the correlation between air permeability and carbonization of high-strength concrete. The concrete specimens were produced varying the W/C ratio or W/B ratio, replacement ratio between FA and SF and the curing periods (28, 90 and 180 days). The compressive strength of concrete was 27-114 MPa. The specimens were dried at 105 ± 5 °C for one day and at RT for one day. Shi, Xu, and Zhou (2009) investigated the correlation between air permeability and carbonization of high-strength concrete. W/B Ratio, replacement ratio between FA and BFS and curing period were varied. Compressive strength of the specimen was between 39 to 106 MPa. The specimens were dried at 105 °C for seven days. The scope of the studies by Khan and Lynsdale (2002) and Shi, Xu, and Zhou (2009) were similar. However, Shi, Xu, and Zhou (2009) took the broader range of replacement ratio of the binder rather than W/B ratio.
In addition, Khan and Lynsdale (2002) and Shi, Xu, and Zhou (2009) presented an empirical formula between compressive strength and air permeability. As shown in Table 2, Khan and Lynsdale (2002) proposed three kinds of experimental formula according to the kind of cement replacement. Shi, Xu, and Zhou (2009) proposed four kinds of approximate expressions according to W/B Ratio and the kind of the cement replacement.   Figure 1 shows the air permeability measured by Dhir, Hewlett, and Chan (1989), Jacobs (1998), Khan and Lynsdale (2002) and Shi, Xu, and Zhou (2009) shown in Table 1 and proposed experimental formulae shown in Table 2. It is shown that air permeability is decreased with compressive strength. However, it should be noted that each proposed formula is in good agreement with each group of experimental data, but the difference between studies are large. For example, the results of Khan and Lynsdale (2002) and Shi, Xu, and Zhou (2009) differ one order of magnitude even though both studies use FA as cement replacement.

Proposal of an estimation formula for the global correlation between compressive strength and air permeability of over-dried concrete
The data in Figure 2 is re-plotted in Figure 2 by classifing concrete (C) and concrete using cement replacement such as FA and SF (C + P). The difference between two groups is not clear. Thus the correlation was developed using data in both groups as in Equation (1). The formula goes through the center of measured data. The value greatly decreases in the range of compressive strength between 15 and 60 MPa. Over 60 MPa, the value slowly decreases. The range of scatter is considerable. The range is one order of magnitude above and below the formula, if we estimate the permeability from compressive strength.

Correlation between the degree of saturation and air permeability
The degree of saturation (water content) has a large effect on the air permeability. Literature survey was made on the air permeability measurements on samples with known degree of pore saturation. A correlation between the degree of saturation and air permeability was developed. Table 3 shows the conditions of measurement in existing studies on permeability measurements on samples with known degree of saturation or relative humidity in Appendix. Figure 3 shows the correlation between the degree of saturation and air permeability by using the data in existing studies (Ujike and Nagataki (1988), Thomas and Matthews (1992) & Jacobs and Wittmann (1992), Jacobs (1998). Here, the values measured by Ujike and Nagataki (1988) and by Thomas and Matthews (1992) were originally shown against relative humidity. The degree of saturation was estimated from relative humidity by the method shown in Appendix.

Correlation between the degree of saturation and air permeability in existing researches
Air permeability gradually decreases with the increase of the degree of saturation in the range of 0 to 0.7. Over 0.7, permeability decreases sharply. Yet, there is no sharp decrease in air permeability in measured data by Thomas and Matthews (1992). The reason could be because the drying period is short for the specimen in each relative humidity condition. Jacobs (1998) showed an empirical formula for various W/C ratios as shown in Table 4. The results show that air permeability decreases with the increase of the degree of saturation. The range of application of the empirical formula was limited to degree of saturation smaller than 0.7 for concrete with W/C ratios at 45, 60 and 80 %. The reason for limitation was because of sharp decrease of air permeability over 0.7.

Global correlation between the degree of saturation and the ratio of air permeability
As was shown in Figure 3 and Table 4 above, there is an empirical formula of the degree of saturation and air permeability in existing researches. However, the permeability could be calculated from compressive strength. Moreover, the formula might be applied to the degree of saturation over about 0.4 as well. Figure 4 shows the correlation between the degree of saturation and the ratio of air permeability k Øw /k dry . k Øw is the permeability at saturation Ø w ,k dry is that of dry condition. Here, in Figure 4, only the data of Ujike and Nagataki (1988) and Jacobs (1998) are shown, and only those which can confirm the ompressive strength for each sample were used. Here, the permeability in dry condition is the measured value in the case of Jacobs result, and the estimated value in the case of Ujike result. The estimated value was obtained by exponential approximate expression using the measurement results in the saturation condition. As a result, it is found the ratio of air permeability exponentially decreases with the degree of saturation in the range lower than 0.6. It seems that the rate of decrease is small in the range higher than 0.6. In case of 57.0 MPa strength, the ratio of permeability is small compared with others. High strength concrete has a small volume of large pores. Thus the fraction of blocked pore is larger compared with low strength concrete in the same value of degree of saturation.
From the above result, the ratio of air permeability can be expressed by an exponential function as Equation (2) in the range of the degree of saturation lower than 0.7. As shown in Figure 5, the coefficient of the exponential function m is a function of compressive strength as in Equation (3).
Jacobs proposes the values in Table 4 for m. When expressing this as compressive strength m, m ¼ 0:1178Fc c þ 2:8482; ð27:6 � Fc � 57:2Þ (3) No functional form was given for the degree of saturation higher than 0.7. However, it can be said that air permeability decreases to zero toward saturated condition and that air permeability is quite small at the degree of saturation of 0.7. Thus the relashipnship was approximated by linear correlatiuons as in Equation (4).

Estimation of air permeability in moisture condition by compressive strength
From investigations and discussions in section 2.1, Equation (1) is applicable to calculate air permeability in dry condition κ dr from compressive strength.
As to degree of saturation, Equation (4) is applicable to calculate the ratio of air permeability k Øw /k dry depending on compressive strength. Thus the air permeability k Øw can be calculated by the correlation of Equation (5).
In order to confirm the accuracy of the formula of air permeability considering the degree of saturation, calculations were carried out for the conditions in literature data with compressive strength and pore saturation. Air permeability was calculated by Equation (1) and Equation (5). Figure 6 shows the comparison of measured and calculated values. The estimated value shows a scatter in ± one order of magnitude reflecting the scatter in original correlation in Figure 6. The same dataset is plotted in Figure 7 as functions of compressive stress. Calculated values are plotted by lines with respective degree of pore saturation. In case of high strength concrete, estimated air permeability is considerably decreased by the degree of saturation. In case of compressive strength smaller than 60 MPa, estimated air permeability is higher than the measured values in Jacobs and Ujike. The difference came from the Jacobs's measured value and estimated values in dry condition. The error seems to be propagated to Figure 7. However, the range of difference by pore saturation could be explained.

Method of Estimating Air Permeability from Pore Size Distribution
As the estimation of air permeability only from compressive strength contains large scatter. To improve the accuracy, use of pore size distribution should be considered. If the flow in a cylindrical tube is laminar, the flow rate Q 1 (m 3 /s) is correlated with pressure gradient ∆P/L by Hagen-Poisuille equation (Pfitzner (1976)).
where d is a diameter of tube [m], η is the kinematic viscosity of air (m 2 /s), L is the length of tube (m), ∆P is the pressure difference over the tube length L (Pa).
Suppose that concrete be porous material and consider a cube with L (m) as shown in Figure 8. The volume of the pore using porosity ε (m 3 /m 3 ) is εL 3 (m 3 ). Assumeing that all pores are cylindrical tubes with a same diameter d (m), the total length of the pores ∑Øl is expressed as follows: In addition, assuming that all the pores are arranged parallel in one direction, the number of tubes n is Since n tubes are in the cross sectional area L 2 (m 2 ), flow rate per unit cross sectional area (m 3 /s) is On the other hands, flow of fluid in a porous material is described by Darcy's law (Darcy (1856)) where, k is specific air permeability (m 2 ). Comparing of Equations (9) and (10), specific air permeability k is given by However, coefficient C is introduced to account for variatuons of pore diameters and tortusity (Ujike and Nagataki 1988).
Coefficient C is determined experimentally. Coefficient C in Equation (12) was estimated from the measurement results in literature that measured both pore size distribution and air permeability in dry condition. Here, pore size distribution is measured from hardened cement paste taken from concrete. Figure 9 shows an empirical correlation by plotting the measured value of air permeability k against pore size distribution parameter ε 20 d 20 2 (measurement range: 0.1-20 µm). The empirical correlation between pore size distribution parameter ε 20 d 20 2 and air permeability in dry condition k dry follows the correlation of Equation (13) Here, porosity ε 20 is the total pore volume, mean diameter d 20 was defined as the diameter of the pore that correspond with half of the total pore volume (Ujike  denotes the maximum pore size of measurements in literature data. In contrast, the range of the measurement of pore size distribution in this study is 0.003-350 µm. As for the method for applying the pore size distribution measured in this study, the correlation between porosity ε 20 and porosity ε 350 and the correlation between mean diameter d 20 and mean diameter d 350 were established through the measured pore size distribution. Figure 10 shows the definitions of porosity ε 20 and porosity ε 350 mean diameter d 20 and mean diameter d 350 in cumulative pore size distribution. Figure 11 and Figure 12 show the correlation between porosity ε 20 ,ε 350 and mean diameter d 20 , d 350 , respectively. The plots included pore size distribution of existing study with same measurement range 0.003-350 µm (Ko 2006). As a result, porosity ε 20 is slightly smaller than porosity ε 350 .Mean diameters d 20 , d 350 are almost the same.
Substituting Equation (14) and Equation (15) Table 5 shows the mix proportion of concrete used for measurement. There are six kinds of mixtures different by water/binder ratio (hereafter, W/B ratio). In the condition where the W/B ratio is 35 or less, a small amount of a high range water reducing admixtures (Polycarboxylic acid) was added. As the curing condition, material age and aggregate type were different, the symbols were designated by K1, K2 and J. K1 is with a curing period of 28 days and two curing conditions of air curing and sealed-curing. K2 is with a curing period of 2 years or 4 years in air. J is with water curing for 28 days followed by air curing for 10 years. Table 6 shows the property values of the materials used in the production of concrete. Table 7 shows the physical property of concrete immediately before the measurement of pore size distribution. Compressive strength of concrete produced by these conditions was 21.5-150 MPa, and the sample used in the measurement of pore size distribution is mortar fragment collected from the concrete. The samples weight were between 2.05 g and 3.32 g.

Measurement Method of Pore Size Distribution by MIP
An outline of the measurement method is as follows: After installing a porosimeter by putting a mortar sample, mercury is injected into the glass container. Next, measure the volume of infused mercury in the pore of the sample by adding pressure. Measurement is taken by increasing the pressure by stages. Since the pore diameter is correlated with pressure as in Equation (17), the relevant pore size is calculated from each pressure in the measurement. Measurment range was 0.003-350 μm. (Webb, 1993) where d is pore diameter (m), γ is surface tension of mercury (0.485 N/m), θ is contact angle 130° (cosθ = −0.643) between mercury and the sample; and P is the pressure (MPa) at the point of mercury injection.

Result of the measurement of pore size distribution 3.2.3.1. Cumulative pore size distribution.
The effects of compressive strength and curing condition on changes in pore size distribution were examined using pore size distribution. Total cumulative pore volume (=porosity) was defined by the pore volume over 0.003 μm. The volume ratio over 0.1 μm diameter was selected according to P. K. Mehta's report (Mehta and Manmohan 1980) that the pore with a diameter larger than 0.1 μm has a close correlation with permeability while the pore with a diameter under 0.1 μm corresponds with durability. Figure 13 shows the pore size distribution of a specimen in air curing condition during first 28 days. The result of J-25, which is water cured, is also shown in Figure 13. Compressive strength ranged 21.5-150 MPa. As the W/B ratio is small, i.e. compressive strength is high, the porosity were small.
The pore size distribution is classified roughly into three strengths, 21.5 MPa, 56.3 MPa and over 77.7 MPa. The pore diameter where there is a sharp increase in cumulative pore volume differs. It turns out that the diameter is smaller as the compressive strength is larger. The volume ratio of a pore with a diameter larger than 0.1 μm was about 0.07 for compressive strength under 56.3 MPa, 0.035 for compressive strength larger than 77.7 MPa. In the air curing condition, the volume ratio of pore over 0.1 μm considerably decreased in the range of compressive strength between 21.5 MPa and 56.3 MPa, 56.3 MPa and 77.7 MPa. Figure 14 shows the pore size distribution of sealed curing specimens. The range of compressive strength is 23. 8-85.4 MPa. Similar to air curing case, the porosity    tends to be small as W/B ratio is small. Comparing the form over 0.1 μm, the pore size distribution is classified into two types, 23.8 MPa and over 58.9 MPa. The volume ratio of a pore with a diameter larger than 0.1 μm was about 0.55 in case of 23.8 MPa compressive strength, 0.38 in case of over 58.9 MPa.

Effects of compressive strength and curing conditions on pore size distribution. The correlation
between compressive strength and porosity (up to 350 μm) is shown in Figure 15. The porosity decreases linearly with compressive strength. The range of porosity was 0.19-0.06 m 3 /m 3 . Figure 16 shows the position and value at the maximum differential pore volume obtained in this study. In the range of compressive strength between 20 and 77.7 MPa, the position of the maximum differential pore volume moved from 1.48 μm toward 0.045 μm according to the increase of the compressive strength. The maximum differential pore volume increased as compressive strength. This means that the volume of the relatively large pore decreased with compressive strength in the range of 20 to 77.7 MPa. In contrast, in the range of compressive strength between 77.7 and 150 MPa, the position of the maximum differential pore volume did not change with compresive strength. The value of the maximum differential volume decreased with the increase of compressive strength, which is consistent with the result of pore size distribution. Figure 16. Pore size at the maximum differential volume and value of the maximum differential volume.

Estimation result by pore size distribution
Using porosity ε 350 and mean diameter d 350 of the pore size distribution in Figure 13 and Figure 14, air permeability in dry condition k dry was estimated by Equation (16). As for porosity ε 350 , the values shown in Figure 15 was used. Figure 17 shows mean pore diameter d 350 . Mean pore diameter decreased with compressive strength in the range of compressive strength from 20 to 80 MPa, while it hardly changed in the range of compressive strength from 80 to 150 MPa. In addition, in the range of compressive strength from 20 to 60 MPa, mean diameter of the sealed-cured specimen was large. Thus there was almost no difference by curing condition at compressive strength at 80 MPa. Figure 18 is the estimation result of air permeability in dry condition. As the compressive strength is increased, the air permeability is decreased. The degree of decrease is large in the range of compressive strength 20 to 80 MPa. At the compressive strength from 80 to 150 MPa, change is fairly small. Moreover, it was found that air-curing concrete had larger air permeability than sealed-curing concrete. The estimation results by the pore size distribution are in agreement on average as compared with the estimation result from the compressive strength (equation (1)). However, considering the pore size distribution, it is considered that the influence of curing condition and others could also be considered.

Conclusions
In order to examine the differences in pore size distribution and air permeability according to compressive strength (21.5-150 MPa) and curing conditions (air curing and sealing-curing), air permeability was estimated through measurement of pore size distribution and analysis of literature data. The results are summarized as follows.
• Changes in pore size distribution according to compressive strength and curing conditions 1) From the result of the measurement of pore size distribution, the increase of compressive strength decreased not only porosity but also the volume fraction of the pore larger than 0.1 μm, which corresponds with air permeability.
2) There were little effects of air curing and sealingcuring on the porosity of concrete. However, in concrete with less than 60 MPa of strength, the volume fraction of the pore larger than 0.1 μm, is decreased in the sealed-cured specimen.
• Method of estimating air permeability of concrete 1) An empirical correlation was developed between compressive strength and air permeability in dry condition k dry . The range of compressive strength was 15-114 MPa.
2) An empirical correlation was developd between pore size distribution parameters and air permeability in dry condition k dry . The range of compressive strength was 27.6-150 MPa.
3) An empirical correlation was developed to calculate the correlation between the degree of saturation and the ratio of air permeability from compressive strength. 4) By summarizing the above items 1) to 3), a simple estimation equation to calculate air permeability of concrete was developed.
• Change in air permeability according to compressive strength and curing condition of concrete 1) Air permeability was estimated in dry condition using the measured result of pore size distribution. Air permeability decreases as compressive strength increases. The decrease was large at a compressive strength range of 21-80 MPa, while decrease was little at a compressive strength range of 80-150 MPa.
2) Air permeability of air-cured concrete was greater than that of the sealed-cured concrete. In case of small compressive strength concrete, the effect of initial curing conditions is significant.

Appendix Conversion from relative humidity to degree of pore saturation
Relative humidity during curing is related to degree of saturation using equilibrium moisture content of concrete. Examples of equilibrium moisture content are shown in Figure A1. The compressive strength range of the used concrete is from 25.1 MPa to 150 MPa (Véronique (2007)). Desorotion isotherm curve was used to correlate the relative humidity in main text.