# Two-Dimensional Microstructure-Based Model for Evaluating the Permeability Coefficient of Heterogeneous Construction Materials

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Objectives

## 3. Model Development

#### 3.1. Modelling of Aggregates and Mortar

#### 3.2. Modelling of ITZ

#### 3.3. Calculation of the Permeability Coefficient

^{3}in this paper. Therefore, Darcy’s law can be applied to calculate the seepage. The model was discretized into a finite number of cells with meshing, and the flow rate of each node on the water boundary was calculated. The overall flow rate (Q) of the concrete specimen can then be determined by summing the flow rate (Q

_{i}) at each node, as illustrated in Figure 4. i.e., $Q=\sum {Q}_{i}$, then Darcy’s law can be written as Equation (1).

_{i}represents the flow rate at each node, m

^{3}/s; ∆h is the hydraulic slope, m; L is the seepage path, m; A is the section area, m

^{2}. In this paper, A = 1 × the width of the specimen W.

#### 3.4. FE Model Parameters

^{−4}m. Some models were partially encrypted (Figure 5) to ensure normal calculation. In the numerical simulation, the same hydraulic gradient used was applied to the microstructure of cement concrete. The flow rate of the specimen per unit of time was obtained through numerical simulation. This value was substituted into Equation (1) to calculate the permeability coefficient K

_{eff}of cement concrete.

## 4. Experimental Measurements

#### 4.1. Materials

^{3}), and limestone aggregate (apparent density of 2720 kg/m

^{3}). The fine aggregate used was river sand, and the coarse aggregate was limestone with a particle size greater than 4.75 mm. Tap water was added to the mixing process. The round table samples with a base diameter of 175 mm, top diameter of 185 mm, and height of 150 mm were cast. The samples were demolded and subjected to a 28-day curing period under controlled conditions. The curing environment maintained a relative humidity of greater than 95% and a room temperature of 20 °C. Table 1 and Table 2 provide a summary of the concrete mix proportions and coarse aggregate gradation used in the experiment.

#### 4.2. Test Method and Results

^{2}, and d is the diameter of the specimen, m.

^{2}m.

## 5. Modeling Results and Discussion

#### 5.1. Model Validation

_{m}of cement mortar was set to the average value measured in the experiment at 5.724 × 10

^{−12}m/s, and the permeability coefficient of ITZ K

_{i}was equal to 30 K

_{m}[28]. According to previous studies, the permeability coefficient of limestone is relatively small, generally ranging from 10

^{−10}to 10

^{−14}[37,38,39]. In this model, the permeability coefficient of aggregate K

_{a}was set at 1.0 × 10

^{−12}m/s.

^{−12}m/s, 3.411 × 10

^{−12}m/s, 3.417 × 10

^{−12}m/s, and 3.471 × 10

^{−12}m/s, respectively. The mean value of the measured data was 3.359 × 10

^{−12}m/s. The relative errors between the predicted and measured values were 4.61%, 1.55%, 1.73%, and 3.33%, respectively. In general, the accuracy of the permeability calculation model based on microstructure is acceptable. Besides, the computation time was calculated for each model. As shown in Figure 9, the time spent increases exponentially with the increase in model size, while the differences in calculation results are not significant. Therefore, the trade-off between calculation accuracy and computational time was taken into consideration, and the model size was determined to be 100 mm × 100 mm for subsequent analysis.

#### 5.2. Effects of Coarse Aggregate Content

#### 5.3. Effects of Aggregate Mesoscopic Parameters

_{a}/K

_{m}was set from 0.001 to 100,000, and the permeability coefficients of 36 models were calculated. According to previous studies, the w/c ratio of concrete had a great influence on ITZ. Therefore, a fixed water−cement ratio of 0.5 was used when calculating the permeability coefficient of aggregates. The logarithmic coordinate system was drawn according to the above simulation results, as shown in Figure 11. It can be seen that when K

_{a}/K

_{m}is less than or greater than a certain threshold, K

_{eff}/K

_{m}basically converges to a constant value. However, within this range, the value of K

_{eff}/K

_{m}exhibits significant changes with varying K

_{a}/K

_{m}values, which also reflects that the influence of the aggregate permeability coefficient on the effective permeability coefficient of concrete cannot be ignored within a certain range. The lower and upper limits of the permeability coefficients of aggregates in the sensitive region were determined by interpolating the fitted logarithmic curve, resulting in values of 1.1 × 10

^{−13}m/s and 4.5 × 10

^{−9}m/s, respectively. The selected parameter for the model in this paper was 1.0 × 10

^{−12}m/s, which was located in the sensitive region, indicating the need to consider the permeability of aggregates.

_{r}represents roundness, A represents the aggregate area, and L represents aggregate contour circumference.

_{r}decreases, the aggregate becomes flatter and longer, while aggregates with larger K

_{r}values tend to be closer to circular in shape.

#### 5.4. Effects of Aggregate Segregation

#### 5.5. Effects of ITZ

_{i}= λK

_{m}[29], where λ = 10, 20, 30, 40, 50. The calculation results are presented in Figure 19.

## 6. Conclusions

- (1)
- The water−cement ratio of concrete greatly impacts permeability. The experimental results show that the lower w/c ratio leads to more impermeability. The proposed prediction method in this study shows acceptable accuracy, with a relative error of 1.73%.
- (2)
- The permeability coefficient of cement concrete gradually decreases with the increase in coarse aggregate content.
- (3)
- The permeability of aggregate significantly influences the effective permeability coefficient of concrete. The roundness of aggregates has little influence on the permeability coefficient of concrete. However, when the proportion of smaller-sized aggregates is higher, the ITZ increases, leading to a corresponding increase in the permeability coefficient.
- (4)
- The separation of cement concrete leads to the decrease of permeability coefficient. And the decrease is gradual in the 0–20% range, while it becomes more pronounced between 20% and 30%.
- (5)
- With the increase of ITZ thickness, the effective permeability coefficient of concrete increases.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 12.**Aggregate models with different roundness (

**a**) K

_{r}= 0.5805, (

**b**) K

_{r}= 0.6434, (

**c**) K

_{r}= 0.7173, (

**d**) K

_{r}= 0.7870, (

**e**) K

_{r}= 0.8540.

**Figure 13.**Concrete models with different roundness aggregates (

**a**) K

_{r}= 0.5805, (

**b**) K

_{r}= 0.6434, (

**c**) K

_{r}= 0.7173, (

**d**) K

_{r}= 0.7870, (

**e**) K

_{r}= 0.8540.

**Figure 15.**Concrete models with different particle size aggregates (

**a**) Smaller particle size, (

**b**) Normally distributed particle size, (

**c**) Larger particle size.

Sieve size (mm) | 25.0 | 19.0 | 16.0 | 9.5 | 4.75 |

Passing percentage (%) | 100 | 68.0 | 48 | 30 | 0 |

No. | W/C | Cement | Water | Sand | Gravel |
---|---|---|---|---|---|

1 | 0.3 | 450 | 150 | 777 | 1073 |

2 | 0.4 | 450 | 180 | 766 | 1058 |

3 | 0.5 | 400 | 200 | 777 | 1073 |

4 | 0.5 | 525 | 263 | 1477 | 0 |

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**MDPI and ACS Style**

Chen, J.; Yu, S.; Huang, W.; Wang, H.
Two-Dimensional Microstructure-Based Model for Evaluating the Permeability Coefficient of Heterogeneous Construction Materials. *Materials* **2023**, *16*, 5892.
https://doi.org/10.3390/ma16175892

**AMA Style**

Chen J, Yu S, Huang W, Wang H.
Two-Dimensional Microstructure-Based Model for Evaluating the Permeability Coefficient of Heterogeneous Construction Materials. *Materials*. 2023; 16(17):5892.
https://doi.org/10.3390/ma16175892

**Chicago/Turabian Style**

Chen, Jiaqi, Shujun Yu, Wei Huang, and Hao Wang.
2023. "Two-Dimensional Microstructure-Based Model for Evaluating the Permeability Coefficient of Heterogeneous Construction Materials" *Materials* 16, no. 17: 5892.
https://doi.org/10.3390/ma16175892