Effect of coarse particle volume fraction on the hydraulic conductivity of fresh cement based material

Abstract

Hydraulic conductivity or permeability is a key parameter that affects the stability of fresh cement-based materials. For example, bleeding and water retention are among the most important issues governed by concrete permeability. Whereas this parameter is relatively easy to measure at the cement paste scale, it is difficult to evaluate with the addition of sand or gravel due to the sample size needed to obtain a representative sample. For ordinary permeability measurement, the test duration is linearly linked to the sample height. The aim of this study is to relate the concrete permeability to the cement paste permeability for various aggregate volume fractions in mortar or concrete. The proposed model uses an adaptation of Darcy’s law to compute concrete permeability from measured cement paste permeability and mix-design parameters. This model has been successfully tested with kaolin and cement pastes with spherical aggregates (glass beads) and sand.

Appendix

1.1 Elementary volume definition

The inclusion volume fraction φ is defined as the ratio of the inclusions volume V _b to the total volume V _t:

$$\varphi = \frac{{V_{\text{b}} }}{{V_{\text{t}} }}.$$

(11)

The number of inclusions included in the sample can be expressed as the ratio of the inclusions volume to the volume of one spherical inclusion of radius R:

$$N = \frac{{\varphi V_{\text{t}} }}{{\frac{4}{3}\pi R^{3} }}.$$

(12)

A cubic elementary volume can then be expressed as the ratio of the volume sample to the number of inclusions:

$$V_{\text{e}} = \frac{4}{3}\frac{{\pi R^{3} }}{\varphi }.$$

(13)

Then, the edge L of the cubic elementary yields:

$$L = R\root{3}\of{{\frac{4}{3}}{\frac{\pi}{\varphi}}}.$$

(14)

1.2 Average length of the flow path in the elementary volume

If the flow doesn’t intercept the inclusion, the length of the flow path is equal to L (direct path).

If the flow intercepts the inclusion, the length of the flow path is a function of the distance of the symmetrical axis (Fig. 5) and noted h(r). It should be noted that this computation underestimate the average length of the flow as we take into account only the lowest possible flow path length, neglecting the interaction between flow lines.

Fig. 5

Flow path considered when inclusion is intercepted

Regarding to the notation adopted in Fig. 5, h(r) can be expressed as:

$$h(r) = L - 2R\sin \beta + 2R\beta ;\,\,\beta = \cos^{ - 1} \left( \frac{r}{R} \right).$$

(15)

Then, it follows:

$$h(r) = L - 2R\sin \cos^{ - 1} \left( \frac{r}{R} \right) + 2R\cos^{ - 1} \left( \frac{r}{R} \right).$$

(16)

The length of the flow path can then be averaged on the elementary volume:

$$h_{\text{m}} = \frac{1}{{L^{2} }}\left( {L(L^{2} - \pi R^{2} ) + \int\limits_{0}^{2\pi } {\int\limits_{0}^{R} {h(r)r{\text{d}}\theta {\text{d}}r} } } \right),$$

(17)

where h _m is the average length of the flow path. The integration of Eq (7) leads to:

$$h_{\text{m}} = R\pi \left( {\frac{4\pi }{3\varphi }} \right)^{{ - {\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} \left( {\frac{{4\left( {1 - \varphi } \right)}}{3\varphi } + \frac{\pi }{2}} \right).$$

(18)

The tortuosity, defined as the ratio of the real average length of the flow path to sample length is then:

$$\tau = \pi \left( {\frac{4\pi }{3\varphi }} \right)^{ - 1} \left( {\frac{{4\left( {1 - \varphi } \right)}}{3\varphi } + \frac{\pi }{2}} \right).$$

(19)