# On the Tortuosity-Connectivity of Cement-Based Porous Materials
## 5 Determining the Tortuosity-Connectivity Parameter
### Calculation of the tortuosity-Connectivity Parameter  
Experimental measurements of the tortuosity-connectivity would require determining the tortuosity-connectivity for various values of the pore water content, which should be distributed uniformly within the material. Methods to obtain a uniformly distributed pore water content in concrete can be found in research work [28; 29; 30]. According to Equation (68), the tortuosity-connectivity parameter of the porous medium can be calculated if its permeability is known. The permeability is often modelled by means of the van Genuchten model:  
\[\hat{k}_{l}(\phi_{l})=\hat{K}\sqrt{\frac{\phi_{l}}{\phi}}\left(1-\left[1-\left( \frac{\phi_{l}}{\phi}\right)^{\frac{1}{2}}\right]^{e}\right) \tag{69}\]  
where \(e\) is a material constant. The expression for the tortuosity-connectivity parameter then becomes:  
\[\tau(\phi_{l})=\tau(\phi)\sqrt{\frac{\phi}{\phi_{l}}}\left(1-\left[1-\left( \frac{\phi_{l}}{\phi}\right)^{\frac{1}{2}}\right]^{e}\right) \tag{70}\]  
In order to calibrate this model with experimental results, Equation (70) was used in this study, where the material constant \(e\) and the tortuosity-connectivity in saturated conditions \(\tau(\phi)\) were measured for four different concretes, by fitting Darcy's law to experimentally obtained data.The procedure consists of solving the equation of mass flux of the pore solution, based on the Darcy flow, which calculates a profile of water in the pores (see [5] for more details). Then this profile has been integrated over the spatial domain to obtain the total mass of water penetrated in thin concrete samples of 2 mm thickness. As the Darcy flow depends on the tortuosity-connectivity model Equation (70), this mass comparison has allowed us to determine the values of \(e\) and \(\tau(\phi)\). In order to determine the hygroscopic properties of concrete, it is necessary to measure the intrinsic permeability coefficient of the material, as well as the relation between relative humidity and water content within the material. The permeability coefficients of the materials were indirectly obtained by absorption tests. The relation between relative humidity and water content was obtained for each material by means of absorption and desorption tests at different relative humidities. The absorption and desorption tests were performed according to the instructions of the standard test method EN ISO 12571 [31]. The test consists basically in placing a concrete sample in some constant environment (constant temperature and constant relative humidity) and weigh the sample when it is entirely in equilibrium with its surroundings. The greater the concrete samples, the longer it takes them to reach equilibrium. Therefore, very thin samples (approximately 2 mm) were used. The four concretes were designed in accordance with usual mixes used in agressive environments to improve the impermeability and resistance against penetration of ionic species. Concrete dosages are referred to as _mat1_, _mat2_, _mat3_ and _mat_. The cement type used for materials _mat1_ (without additions), _mat2_ (with addition of 20% of fly ash) and _mat4_ (with addition of of 10% of silica fume) is CEM I 42.5 R/SR according to the EN 197-1 Standard [32]. The cement employed for material _mat3_ contains 66% blast furnace slag and its type is known as CEM III/B 42.5 L/SR, in agreement with the EN 197-1 Standard [32]. The properties of the materials are described in detail in [33]. The dosages of the materials are given in Table 1.  
The tortuosity-connectivity function of the materials, obtained from the experimental results, is plotted in Figure 4. It is shown that the tortuosity-connectivity parameter decreases significantly for lower pore water contents, meaning that the mass fluxes are significantly reduced in the direction of the net mass flux. In saturated conditions, the tortuosity-connectivity functions reach their maxima.