A Prediction Model for the Unconfined Compressive Strength of Pervious Concrete Based on Mix Design and Compaction Energy Variables Using the Response Surface Methodology

Abstract

Pervious concrete is desirable for water drainage in building systems, but achieving both high strength and good permeability can be challenging. Also, the importance of compaction energy is significant in determining the efficiency of pervious concrete. However, research on the development of unconfined compressive strength (UCS) prediction models for pervious concrete materials that incorporate compaction energy parameters remains unexplored. Therefore, this study aimed to balance strength and permeability while optimizing the compaction energy required for concrete production. A Central Composite Design (CCD) was used to design experiments within the response surface methodology (RSM) and evaluate the UCS, the porosity and permeability of pervious concrete specimens produced with varying cement content (280.00–340.00 kg/m³), the water-to-cement ratio (0.27–0.33), the aggregate-to-cement ratio (4:1–4.5:1), and compaction energy (represented by VeBe compaction time, 13–82 s). A regression model with goodness of fit (R²_adjusted > 0.87) was calibrated to estimate the UCS of pervious concrete as a function of mix design parameters and VeBe compaction time (T_vc). This model can potentially guide field practices by recommending compaction strategies and mix designs for pervious concrete, achieving a desirable balance between mechanical strength and hydraulic permeability for building construction applications.

1. Introduction

When designing a building, it is essential to take into account the possibility of water and moisture getting trapped under different types of building elements [1,2,3,4]. This issue can arise over time, resulting in the problem of pumping and premature failures of structural members [5]. To mitigate this risk, it is suggested to utilize permeable layers [6]. The construction of a long-lasting permeable structure depends greatly on the utilization of suitable layers that have adequate strength and proper permeability [7]. To achieve this objective, various circumstances and requirements can be accommodated by utilizing different types of permeable material, such as pervious concrete [8].

As illustrated in Figure 1a,b, pervious concrete is typically implemented by closing a fixed mold at the desired location and discharging the concrete at the mold location. The concrete is then leveled with a vibratory screed and compacted using a steel roller. Similar to the roller compacted concrete pavement (RCCP) construction process, an asphalt paver is typically employed to expedite the construction process of pervious concrete pavements. Subsequently, a light roller is used to compact the surface (Figure 1c) [9,10]. Just like the RCC, pervious concrete has zero slumps [11,12]. The Gyratory [13] and VeBe compaction methods [14] are appropriate standards for compacting zero-slump mixes [15]. These experiments aim to simulate the movement of a roller during the compaction process of the pervious concrete layer [16,17,18].

It should be noted that excessive compaction efforts may lead to decreased permeability [19,20,21,22,23]. Thus, the level of compaction energy plays a vital role in influencing the effectiveness of the permeable layer [24,25]. However, compaction energy in the research conducted on the development of pervious concrete is largely overlooked. Essentially, the focus of researchers has been on optimizing the correlation between factors like permeability and compressive strength concerning the water-to-cement ratio, aggregate-to-cement ratio, and cement content. The models they presented lack any consideration for compaction energy [26,27,28,29,30,31]. Nevertheless, the significant impact of compaction energy on the permeability and compressive strength of pervious concrete must not be underestimated. Hence, the novelty of this study lies in presenting a model for the unconfined compressive strength (UCS) of pervious concrete that not only ensures suitable permeability but also harnesses the compaction energy. In simple terms, the model developed in this study enables the calculation of the necessary compaction energy based on structural requirements and desired strength, facilitating the determination of the compaction method and the selection of the required compaction machinery. For this purpose, in the current study, the VeBe test was utilized to analyze the influence of various compaction energy levels on the strength and permeability properties of pervious concrete.

Pervious concrete typically exhibits a permeability range of 0.14–1.22 cm/s [32,33,34], a compressive strength range of 2.80–28.00 MPa [35,36,37,38,39,40], a porosity range of 15–35% [41,42,43,44], and a density range of 1.60–2.00 g/cm³ [45,46,47]. Many studies have been conducted to investigate the characteristics, advantages, and limitations of pervious concrete pavements, resulting in diverse findings. The mix designs of different researchers for building the pervious concrete are outlined in

Table 1. Different types of mix designs for the construction of pervious concrete.

Author	Aggregate Content (kg/m3)	Cement Content (kg/m3)	Water Content (kg/m3)	A/C (by Mass) 1	W/C (by Mass) 2	Main Results
Zhang et al. (2024) [28].	1513.2 1522.7 1522.7	317.4 305.6 330.1	95.20 107.00 82.50	4.77:1 4.98:1 4.61:1	0.30 0.35 0.25	The performance of the pervious concrete pavement under extreme temperatures was improved by adjusting the mix of the pervious concrete and using microencapsulated phase change material (MPCM) and carbon nanotubes (CNTs).
Kanghao Tan et al. (2024) [9]	1715.2–1716.5	236.6–513.7	28.39–113.014	3.3:1–7.2:1	0.12–0.22	Incorporating hydroxypropyl methyl cellulose caused metallic sheen and filamentous features in cement pastes. The control mixture showed minor dry and hardened characteristics after 80 min of rest.
Kelly Brasileiro et al. (2024) [29]	1499.20	374.80	112.44	4:1	0.30	Samples with varying percentages of recycled aggregates (40%, 50%, and 60%) exhibited reductions in compressive strength (48%, 56%, and 52%, respectively) compared to samples with natural aggregates at 28 days.
Muthu & Sadowski (2023) [30]	1350	300	105	4.5:1	0.35	Geopolymer-based permeable pavements have ecological benefits. They harden quickly in moist air, gain strength early, resist acid, and aid in groundwater contamination remediation.
Hilal El-Hassan et al. (2023) [31]	1400	190–380	115	3.68:1–7.3:1	0.30–0.60	The addition of ground-granulated blast furnace slag improved the performance of the pervious concrete.
Behlul Furkan Ozel et al. (2022) [48]	519–1502	300	105	1.73:1–5:1	0.35	The addition of steel fiber positively impacted all the mechanical properties. Nevertheless, they experienced a significant decrease when polypropylene fiber was included.
Fan Wu et al. (2022) [49]	1684–1903.32	301	111.4	5.59:1–6.32:1	0.37	When 50% steel slag (1–2 mm) was added to pervious concrete, its compressive strength and flexural strength increased by 150.8% and 130.7%, respectively. Additionally, the mass loss from freeze–thaw cycles decreased by 73.7% compared to the control concrete.
Akkaya and Ismail Hakki (2021) [50]	1219.66–1868.31	448.72	95.00–115.00	2.7:1–4:1	0.21–0.25	The size of aggregates in pervious concrete positively affects its mechanical properties when reduced while maintaining a constant aggregate-to-cement ratio.

¹ A/C = aggregate-to-cement ratio. ² W/C = water-to-cement ratio.

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Upon a careful examination of Table 1, as per ACI 522R [33], it is evident that the primary goal in all the proposed mix designs and guidelines has been to enhance the permeability and mechanical characteristics of pervious concrete. As a result, a specific range of mixture design components has been established. The outlined range of components for pervious concrete comprises a cement content varying from 280.00 to 340.00 kg per cubic meter [12,33,43], aggregate content from 1440 to 1800 kg per cubic meter [51,52,53], a water-to-cement (W/C) ratio from 0.26 to 0.45 [54], and aggregate-to-cement (A/C) ratio from 4:1 to 4.5:1 [55]. It should be stressed that these values may not be universally applicable. Finding the perfect blend depends on comprehending the characteristics of individual elements while also taking into account the structural requirements and environmental circumstances. Therefore, it is crucial to establish a harmonious equilibrium and encourage the interplay between permeability features and mechanical strength [56].

Although previous works have explored the effects of various factors (e.g., mix proportions, aggregate characteristics, types of admixtures, etc.) on the mechanical performance and permeability of pervious concrete, there is a notable gap in studies focusing on the development of strength prediction models that integrate compaction energy parameters. Therefore, this research aimed to develop a UCS model that addresses the question of how much pervious concrete pavements in the field must be compacted to attain the desired UCS while ensuring that the appropriate permeability is preserved. To address this issue, the novelty of this research lies in three key aspects: (1) A model was proposed for predicting the strength of pervious concrete. This model not only ensures an appropriate balance between mechanical and permeability properties but also utilizes the compaction energy effectively. It allows for the estimation of the required compaction energy based on structural needs and desired strength. This, in turn, helps in determining the suitable compaction method and selecting the necessary compaction equipment. This model considers the VeBe compaction time (T_vc) to simulate field compaction efforts using a roller in the process of a pervious concrete pavement construction. (2) An extensive experimental program comprising two different phases enabled the systematic evaluation of the effects of numerous factors (e.g., cement content, water-to-cement ratio, aggregate-to-cement ratio, and compaction energy) on the porosity, permeability, and UCS of pervious concrete. These findings were compared with existing research to highlight advancements in the current state of the art. (3) This work introduces the promising use of the Central Composite Design (CCD) to develop pervious concrete mixtures with appropriate compaction times using the response surface methodology (RSM). Response surfaces, regression models, and sensitive analyses provided a robust framework for the optimization of pervious concrete properties. In addition, minimum UCS thresholds were used to propose the best mix design for the pervious concrete pavement top layer on low-volume roads and permeable stabilized base.

2. Materials and Methods

The general experimental plan of the present work comprises tests for material characterization (Section 2.1) and two different phases of mixture experiments (Section 2.2).

2.1. Materials

2.1.1. Aggregates

To create long-lasting and high-quality pervious concrete, it is advisable to utilize a tough, pristine, chemical-free, and clay-free stone material [33]. Since the moisture content of the aggregates greatly affects the properties of the mix design, it is necessary to saturate the aggregates with a dry surface before producing the mixture. This will prevent the absorption of water by aggregates [33]. When considering permeability in the case of grading, it is advisable to use coarser aggregates with similar diameters and decrease the amount of fine parts [57]. However, to achieve an effective mix design that balances the required strength and hydraulic properties, approximately 5% to 7% of fine aggregates should be incorporated [58].

The aggregates used in this research were crashed aggregates obtained from the Shahriar mine (near Tehran, Iran). Their characteristics can be found in

Table 2. Characteristics of the crashed aggregate employed in this study.

Properties	Standards	Unit	Ranges	Reference
Specific gravity in a saturated surface dry state	ASTM C127-128	-	2.61–2.73	[59,60]
Bulk density in a saturated surface dry state	ASTM C29/C29M	g/cm3	1.35–1.44	[61]
Water absorption	ASTM C127-128	٪	1.72–4.95	[59,60]
Moisture content	ASTM C566	٪	0.25–1.67	[62]
Los Angeles aggregate abrasion loss	ASTM C131	٪	14.20–18.6	[63]

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ASTM C33/C33M was employed to grade and determine the characteristics of the aggregates [33,64]. The corresponding curves are plotted in Figure 2. As depicted in Figure 2, No. 89’s grading curve comprises a lower amount of fine and filler aggregates in contrast to the others that are utilized in a broader spectrum of research for pervious concrete. The reduced quantity of fine aggregate results in an augmentation of aggregate void content and creates sufficient space for the inclusion of cement paste as a stabilizing agent [55]. Hence, gradation No. 89 was employed in this research to meet the project’s needs in terms of permeability and mechanical strength.

To minimize the influence of changes in the distribution of aggregate sizes on the results of this study, the aggregates were divided into five distinct groups with varying desired size ranges, as illustrated in Figure 3. The number of aggregates needed, based on the average of the highest and lowest limits of ASTM No. 89, was determined by weighing each specific size range.

As shown in Figure 3, Group 2 (grain size ranging between 4.75 and 9.50 mm) presented the highest usage percentage. In addition, filler aggregates were incorporated at approximately 5% to enhance mechanical properties [58]. This practice follows the regulations specified in ACI 522R [33], wherein a small quantity of fine aggregate can be utilized to enhance the durability of the concrete.

2.1.2. Cementitious Materials

Type II Portland cement was used in this work, per ASTM C150 [65].

Table 3. Characterization of Type II Portland cement used in this study.

Parameters	Value
Loss on ignition (%)	1.30
Insoluble residue (%)	0.60
Specific gravity	3.15
SiO2 (%)	21.74
Al2O3 (%)	5.00
Fe2O3 (%)	4.00
CaO (%)	63.04
MgO (%)	2.00
SO3 (%)	2.30
Alkalis [Na2O (%) + 0.658K2O (%)]	1.00
Free Cao (%)	1.40
C3S (%)	45.50
C2S (%)	28.00
C3A (%)	6.50
C4AF (%)	12.20

provides the characteristics of the cement used in this study.

2.1.3. Preparation of Test Conditions and VeBe Apparatus

Due to the significance of the T_vc in simulating field compaction efforts, modifications were implemented to uphold the effectiveness of the apparatus and meet the demands of this work. The VeBe apparatus utilized in this study is illustrated in Figure 4.

The VeBe apparatus is equipped with a motor for vibration with a power of 250 W (Figure 4a). A metal overhead with a mass of 22.25 kg and dimensions of 232 mm in length and 72 mm in height was placed on the sample molded in the vessel with an inner diameter of approximately 243 mm and a depth of 202 mm. According to the ASTM C1170 standard [66], only 13.4 ± 0.7 kg of fresh mixture should be poured into the vessel. Finally, the test continued until the material formed a ring around the overhead. The time required to reach this state is referred to as the VeBe consistency time. In this research, the apparatus mold was replaced by 3 cylindrical molds (Figure 4c), with an inner diameter of 104 mm, and a height of 208 mm as per the ASTM C470 standard [67]. A cylindrical metal overhead with a height of 85 mm, diameter of 100 mm, and weight of 5.66 kg was used for each of the 3 molds (Figure 4b). The weight for the overhead of the cylindrical sample was determined by taking into account the ratio between the standard overhead weight and the volume of the VeBe apparatus mold compared to the volume of a cylindrical mold.

2.2. Methods

2.2.1. Phases of Experiments

In this study, different experiments were carried out. Figure 5 shows the two phases of experiments, including the number of samples and their objectives.

In the initial phase, a total of 27 samples were constructed using VeBe compaction energies corresponding to compaction time intervals ranging from 0 to 150 s, followed by evaluations of porosity, permeability, and UCS. For this purpose, molds with dimensions of 20.80 × 10.40 cm were employed, according to the ASTM C470 standard [67]. After pouring, samples were compacted using the VeBe apparatus following ASTM C1170 [66]. Due to the varying options for sample density, the specimens were compacted with different T_vc to identify the most suitable time duration for compaction. After a curing period of 28 days, porosity, permeability, and UCS experiments were carried out [68]. Ultimately, Phase I employed SPSS v.28 software for statistical analysis of the experimental data. This analysis aimed to establish a model that predicts the VeBe compaction time as a function of the mix design parameters. The model ensured that all concrete mixtures achieved the target permeability of 0.4 cm/s while simultaneously optimizing the compaction time for each specific mix design. Some of the pervious concrete samples constructed in this phase are shown in Figure 6.

After defining T_vc for each of the mix designs, in Phase II of the present experiments program, 3 samples for each of the 24 mix designs were constructed (a total of 72 samples). Following this, experiments were conducted to evaluate the strength and permeability properties of the samples. Ultimately, this research culminated in the development of a regression model for predicting the UCS of pervious concrete. The model incorporates the VeBe compaction time alongside other relevant mix design characteristics as influencing factors. Also, under the appropriate standards, the recommended mix design for pervious concrete pavement was determined based on the pervious concrete limitation (2.80 to 28.00 MPa) and the target permeability of 0.4 cm/s.

2.2.2. Response Surface Methodology (RSM) for Determining Mix Designs

There are various strategies for designing experiments, such as the Taguchi method, factorial design, and RSM, among others [69,70,71]. All these strategies focus on determining the number of experimental designs by considering the variables that affect the analysis [72]. In this research, the RSM was applied to reduce the number of experiments, and therefore, it was employed to perform relevant analyses such as model prediction and sensitivity analysis [73]. It is important to recognize that machine learning (ML) methods such as random forest regression, artificial neural networks, and other techniques [74,75], which are used for data analysis, involve computational models intended to forecast output variables when the relationships between inputs and outputs are so intricate that mathematical models struggle to accurately calibrate with the data. ML algorithms can yield more accurate predictions if optimization processes across multiple parameter grids are finely adjusted, and an adequate volume of data is available, which was not the case in the present research [72,76,77,78,79].

The RSM is one of the most traditional approaches derived from Taguchi’s concept and used to optimize modeling experiments using static data [80]. Originally developed in 1951 by Box and Wilson [81], this method aims to optimize the output variable based on the input factors while minimizing the number of experiments required to obtain optimal conditions [82]. In this study, the CCD was employed to design experiments within the RSM. This is a partial factorial experiment design method [83,84] that allows for the establishment of a suitable relationship between nonindependent factors (response) and independent factors involved in the experiment under different scenarios [85]. One of the key advantages of this approach is its ability to provide valuable information about the relationship between factors, making it superior to independent factor analysis methods that only test one factor at a time [86]. The factors in the CCD are defined based on their distance from the range of changes, namely central, factorial, and axial distances. The different points in a CCD are illustrated in Figure 7.

The points located in the middle of the range of changes are referred to as central points. These points are the key focus as they play a significant role in the analysis. On the other hand, the intersections known as factorial points are situated at the furthest distance from the central points. Lastly, the axial points lie between these two types of points [87]. By utilizing the data obtained from the designed experiments and mathematical equations, a static relationship is established between independent and nonindependent variables. This relationship can be visually represented through various software tools, such as curves, surfaces, or contours [88]. The quadratic or reduced quadratic model was employed in the RSM for this research, and it is shown in Equation (1) [72].

$$Y={\beta }_0+\sum _{i=1}^n{\beta }_i{X}_i+\sum _{i=1}^n{\beta }_{ii}{X_i}^2+\sum _{i\ne j=1}^n{\beta }_{ij}{X}_i{X}_j+\epsilon$$

(1)

In this relationship, one can observe the response variable (Y), regression coefficients (β), independent variables (X_i, X_j), the number of factors examined (n), and the experiment’s error rate (ε). Subsequently, following the utilization of the least-square method, the coefficients for the quadratic equation were determined, and the predicted answers were calculated. Then, to assess the quality of the predicted model and the interactions between different variables, the results of the experiments and the model’s responses were evaluated through analysis of variance (ANOVA) [72,89,90,91]. In this study, the RSM was employed to initially construct a design based on the limits and requirements identified in prior research. This was accomplished using the CCD through the utilization of DESIGN EXPERT software [92]. To facilitate this process, the necessary input information is provided in

Table 4. Information about the factors investigated in this study.

Factors	Name	Min	Max	Mean	Lower Limit (−1)	Upper Limit (+1)
A	Water-to-cement ratio	0.27	0.33	0.30	−1 ↔ 0.27	+1 ↔ 0.33
B	Aggregate-to-cement ratio	4.00	4.50	4.25	−1 ↔ 4.00	+1 ↔ 4.50
C	Cement content (kg/m3)	280.00	340.00	310.00	−1 ↔ 280.00	+1 ↔ 340.00

.

The input factors of experiments and their boundaries are specified in Table 4. The chosen ranges for W/C (A: 0.27, 0.30, and 0.33), A/C (B: 4.00, 4.25, and 4.50), and cement content (C: 280.00, 310.00, and 340.00) were carefully selected based on previous studies, structural needs, and the standards outlined in ACI 522R [33,43,56]. Subsequently, utilizing the RSM and CCD, the mix designs were constructed for the reduced quadratic model. In

Table 5. Mix designs developed using the DESIGN EXPERT software.

Run	Water-to-Cement Ratio	Aggregate-to-Cement Ratio	Cement Content (kg/m3)
1	0.33	4.50	280.00
2	0.27	4.00	280.00
3	0.27	4.50	340.00
4	0.33	4.00	280.00
5	0.30	4.00	310.00
6	0.30	4.25	310.00
7	0.30	4.25	280.00
8	0.30	4.50	310.00
9	0.30	4.25	340.00
10	0.30	4.25	310.00
11	0.27	4.00	340.00
12	0.30	4.25	310.00
13	0.30	4.25	310.00
14	0.27	4.50	280.00
15	0.30	4.25	310.00
16	0.33	4.25	310.00
17	0.33	4.50	340.00
18	0.27	4.25	310.00
19	0.3	4.25	310.00
20	0.33	4.00	340.00
21	0.33	4.00	300.00
22	0.33	4.00	320.00
23	0.30	4.00	300.00
24	0.30	4.00	320.00

, all mix designs prepared by DESIGN EXPERT software are shown. The present study evaluated the variables A, B, and C, in addition to the T_vc variable. Consequently, two separate scenarios were analyzed, concerning the experimental mixtures’ design. In the first scenario, T_vc was treated as a set of variables. For each mix design, three different T_vc values were utilized for compacting the pervious concrete. Within this scenario, there were four variables, each with three distinct levels for creating mix designs. If T_vc, which affects UCS and permeability, was considered as a variable, the contrasting behaviors of UCS and permeability factors would lead to significant variations in the results, making it difficult to identify the optimized UCS in relation to sufficient permeability. In the second scenario investigated in this study, T_vc was excluded from the primary variables aimed at developing mix designs and was instead regarded as a response. Subsequently, efforts were made to identify the T_vc for each mix design while maintaining the permeability at approximately 0.4 cm/s. The results indicated that the permeability value stayed within the expected range. Through this approach, the number of design experiments was notably minimized. In the concluding phase, the UCS model of the pervious concrete was calibrated.

Table 5 outlines that the values of input factors in the six experiments concentrated on central points, while the remaining experiments were designed based on factorial and axial points.

2.2.3. Analyzing the Results

Statistical analysis was performed using the SPSS v.28 and the DESIGN EXPERT software [92] to examine the relationship between independent and dependent variables, due to the extensive range of data. Sensitivity and prediction analyses for UCS and permeability factors were conducted using the DESIGN EXPERT software [92]. The SPSS v.28 software was employed to ascertain the optimal T_vc for each mix design following the specified permeability. Additionally, MATLAB Ver. R2014a [93] was utilized for image processing.

2.2.4. Porosity, Permeability, and UCS Experiments

The porosity of hardened samples was determined with the methods presented in ASTM B962-17 [94]. According to this guideline, porosity can be calculated using Equation (2) [94] as follows:

$$P=\left(1-{\displaystyle \frac{M_d-M_u}{{\rho }_W\times V}}\right)\times 100$$

(2)

where P is the porosity; ρ_w is the density of water; V is the initial volume; and M_d and M_u are the mass of the dry and drowned samples, respectively.

Permeability was determined using the falling head method. For this purpose, an apparatus was designed and constructed, similar to the one developed by Neithalath et al. [95]. Figure 8 depicts the schematic of this apparatus. Figure 8 shows that it consists of a valve, a drain pipe, a movable pipe, a sample, and metal fasteners. After running the test, permeability was determined using Equation (3) [96] as follows:

$$K={\displaystyle \frac{aL}{At}}ln{\displaystyle \frac{h_1}{h_2}}$$

(3)

where K is the coefficient of permeability, A is the cross-section area of the sample, L is the sample length, a is the pipe cross-section area, h₁ is the water depth in the pipe at the beginning of the experiment, and h₂ is the water depth at time t.

Following the 28-day curing period, the UCS of the samples was determined according to the methodology outlined in ASTM C39 [97,98]. To accommodate variations in sample heights, a universal testing machine (UTM) equipped with an adjustable loading jack was employed to ensure precise alignment with the upper surface of the pervious concrete.

3. Results and Discussion

3.1. Phase I

In this phase, the UCS and permeability of mix designs in Run 1 to 3 were examined and modeled in the SPSS v.28 software to determine the T_vc for each of the mix designs in a manner in which permeability remains in the same predicted range for all mixes. Accordingly, a total of nine samples were constructed for each mix design, and tests for the determination of porosity, permeability, and UCS were conducted per the specified standards. Figure 9 shows the relationship between the UCS, permeability, and T_vc.

Different time intervals for sample compaction, ranging from 0 to 150 s, were investigated, as shown in Figure 9. It is evident that the concentration of data, in terms of both mechanical and hydraulic properties, was within the range of 0 to 80 s. Consequently, by utilizing this specific time duration, it is feasible to enhance the research accuracy and determine the appropriate compaction time for each mix design. Furthermore, it can be inferred that there is a strong correlation between T_vc, UCS, and permeability. As the permeability increased, the porosity also increased, resulting in a reduction in the sample’s strength. Additionally, the results showed that as T_vc increased, the permeability decreased, while the strength increased. This can be attributed to the same reasons previously mentioned, namely the increase in density and decrease in porosity.

By leveraging the in-depth characterization of the samples using the SPSS v.28 software, the VeBe compaction time was predicted for each specific pervious concrete mix design, ensuring that all mixes achieved a target permeability of approximately 0.4 cm/s. Permeability, cement content, and water-to-cement ratio were taken into account as independent variables, while T_vc was considered the response variable. Regression analysis details and the obtained regression model are presented in

Table 6. Summary of the regression analysis for T_vc.

Independent Variables	Dependent Variable	R	R2	${\mathbf{R}}_{\mathbf{A}\mathbf{d}\mathbf{j}\mathbf{u}\mathbf{s}\mathbf{t}\mathbf{e}\mathbf{d}}^2$	p-Value	Standard Error of the Estimate
Permeability, cement content, and water-to-cement ratio	Tvc	0.749	0.63	0.582	<0.0001	35.3134

and Equation (4), respectively.

$$T_{vc}=578.016-0.789C-630.398\left({\displaystyle \frac{W}{C}}\right)-807.92K$$

(4)

where C, W/C, K, and T_vc represent cement content, water/cement ratio, permeability, and VeBe compaction time, respectively.

This model was used to estimate the T_vc for each specific mix design of Phase II, ensuring that the values of permeability remained approximately 0.4 cm/s and adhered to the established standard. The VeBe compaction times obtained for each mix design can be observed in

Table 7. Value of T_vc obtained for each mix design of Phase II.

Run	Tvc (s)	Run	Tvc (s)	Run	Tvc (s)	Run	Tvc (s)
1	80	7	47	13	47	19	47
2	13	8	47	14	13	20	82
3	14	9	48	15	47	21	80
4	80	10	47	16	81	22	80
5	47	11	14	17	82	23	80
6	47	12	47	18	13	24	80

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Table 7 shows that the developed model was able to provide the T_vc value for each mix design of Phase II, assuming that the permeability of each design was approximately 0.4 cm/s.

3.2. Phase II

In this Phase, the UCS, permeability, and porosity of all mix designs (Run 1 to 24) were determined. The test results of all mixes are displayed in

Table 8. Results obtained from the experiments conducted in Phase II.

Run	Density (g/cm3)	Porosity (%)	Permeability (cm/s)	UCS (MPa)
1	1.909	23.241	0.368	4.955
2	1.656	28.683	0.444	1.851
3	2.313	16.155	0.418	5.202
4	2.279	17.407	0.302	7.679
5	1.592	29.542	0.444	2.560
6	2.223	17.639	0.461	3.623
7	1.774	26.404	0.430	3.103
8	1.741	27.287	0.471	2.964
9	1.824	25.572	0.438	3.230
10	2.182	17.717	0.466	3.738
11	1.462	31.048	0.442	2.379
12	2.133	19.413	0.462	3.503
13	2.167	17.753	0.465	3.627
14	1.797	25.898	0.450	5.283
15	2.359	15.952	0.470	3.831
16	1.888	23.596	0.395	3.931
17	1.855	24.205	0.449	4.497
18	1.741	26.941	0.507	4.247
19	2.260	17.632	0.466	3.719
20	2.451	12.243	0.032	10.380
21	2.288	16.393	0.037	8.164
22	2.323	12.289	0.032	9.512
23	2.306	13.513	0.032	8.584
24	2.473	10.707	0.029	9.704

.

A comparison was made between the obtained test results and the findings of previous researchers (Figure 10) [27,41,50,96,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115]. Figure 10 serves as a reliable confirmation of the selected ranges for the formulation of pervious concrete mix designs in this study, with most findings from this research, as well as those from earlier studies, falling within the minimum acceptable thresholds concerning UCS, permeability, porosity, and density. Furthermore, Figure 10 indicates that exceeding the specified limits, derived from prior research and ACI 522R guidelines [32,33,34,35,36,37,38,39,40], in the quantity and ratio of materials for pervious concrete construction can lead to significant errors, potentially compromising the mechanical or permeability characteristics of pervious concrete such as reduced strength or lower drainage capacity. This implies that the typical range employed in this research for the design and casting of pervious concrete is sufficiently effective for its construction, eliminating the necessity to explore ranges that are either lower or higher than the chosen range in this investigation. Furthermore, Figure 10a illustrates the correlation between porosity and density. It can be concluded that as the porosity increases, the density of pervious concrete decreases. This can be attributed to the increase in voids within the sample, resulting in a reduction in density. Similarly, the results showed that as porosity increased, the strength decreased (Figure 10b). The existence of voids resulted in a decrease in the level of interdependence between aggregates, as well as a reduction in the pathways for internal force transmission, which ultimately led to a decrease in compressive strength. Additionally, the presence of angular voids increases the chances of stress concentration, thereby increasing the likelihood of crack initiation and propagation. This study further indicates that once the porosity exceeds the 29% threshold, the strength of pervious concrete diminishes beyond the permissible limits (below 2.8 MPa) defined by the ACI 522R guidelines and prior studies [35,36,37,38,39,40,41,42,43,44], making it structurally and mechanically ineffective (Figure 10b). Also, the relationship between porosity and strength in pervious concrete is exponential rather than linear [33,56]. In this research, our findings confirmed this essential principle. Furthermore, the correlation between permeability and porosity can also be observed (Figure 10c) [116]. Since this study aimed to determine the optimal amount of compaction energy for different mix designs assuming a specific value of permeability (K = 0.4 cm/s), the value of K remained constant in the range of 0.4 cm/s (Figure 10c). As a result, the increase in empty spaces had a minimal impact on permeability.

Generally, when porosity increases, permeability also increases. This can be explained by the probability of more interconnected pores, which has a substantial impact on the rate of permeability. However, as illustrated in Figure 10c, permeability and porosity do not always have a direct relationship. There may be instances where permeability does not increase with increasing porosity. This is because an increase in permeability is contingent upon the formation of new interconnected pores, which may not occur with an increase in voids. The permeability laboratory results, displayed in Figure 10c, also indicate that they correspond with the modeling and are approximately 0.4 cm/s, thus showcasing a high degree of modeling accuracy.

To investigate further, a 3D diagram was plotted in Phase II to establish the correlation among various characteristics of pervious concrete. By utilizing this method, it became possible to visually represent the connection between porosity, UCS, and permeability, as shown in Figure 11.

As illustrated in Figure 11, there exists an evident correlation between the UCS and both porosity and permeability. As the porosity and permeability increase, it is noticeable that the UCS, on the other hand, undergoes a significant decrease. Moreover, it is worth noting that Figure 11 conveniently displays the distribution of permeability, which is accurately confined within the range of 0.35 to 0.45 cm/s. In simpler terms, this graphical representation undeniably affirms and validates the precise accuracy of the modeling that was performed during Phase I.

3.2.1. Developing a Regression Model for the UCS of Pervious Concrete

Examinations were carried out in this part of the study to analyze the sensitivity of the key variables that significantly influence both the UCS and permeability factors. Furthermore, an evaluation of the accuracy of the UCS regression model and the specified permeability was carried out using prediction analysis. Since the different mix designs in this study were established based on the assumption of a permeability of approximately 0.4 cm/s, it was not possible to calibrate the comprehensive permeability regression model. Consequently, only the UCS regression model was generated. The regression modeling was conducted using the DESIGN EXPERT software [92]. A summary of the created model is shown in

Table 9. Summary of the UCS regression model calibrated in the current study.

Name of the Model	Type of the Model	${\mathbf{R}}^2$	${\mathbf{R}}_{\mathbf{A}\mathbf{d}\mathbf{j}\mathbf{u}\mathbf{s}\mathbf{t}\mathbf{e}\mathbf{d}}^2$	${\mathbf{R}}_{\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{d}\mathbf{i}\mathbf{c}\mathbf{t}\mathbf{e}\mathbf{d}}^2$	p-Value	Standard Deviation
UCS	2FI	0.9049	0.8785	0.8379	<0.0001	0.8836

. In addition, the relationship of the aforementioned model is illustrated in Equation (5). Since the T_vc was determined for each of the mix designs, it was then utilized as an independent variable in the model, which helps to estimate the UCS of pervious concrete in real fields based on some mix design parameters such as the aggregate-to-cement ratio (A/C), water-to-cement ratio (W/C), and compaction energy levels (T_vc). Based on the DESIGN EXPERT software [92], Equation (5) can be utilized to predict the response for specific factor levels. High levels of the factors are presented as +1 and low levels as −1.

$$UCS=3.70-2.22\left({\displaystyle \frac{W}{C}}\right)-0.4318\left({\displaystyle \frac{A}{C}}\right)+5.46T_{VC}+2.55\left({\displaystyle \frac{W}{C}}\right)\times T_{VC}-2.90\left({\displaystyle \frac{A}{C}}\right)\times T_{VC}$$

(5)

This model facilitates the estimation of the UCS of pervious concrete. Moreover, it can serve as a tool for the determination of the resilience modulus, a critical parameter for designing pavement thickness.

Statistical analysis using a 5% error level and a 95% confidence interval indicated that none of the variables exhibited a strong correlation with each other. This finding suggests that multicollinearity, a potential issue in regression models, was not a concern. However, the subsequent regression analysis revealed that the p-values for all independent variable coefficients were close to zero. While statistically significant (p-value < 0.05), these coefficients may not have had a meaningful practical impact on the model’s outcome.

3.2.2. Sensitivity, Prediction Analyses, and Pore Dispersion

After UCS model calibration and a better understanding of the specified assumption for permeability, it was possible to determine the key variables that impact the UCS and permeability through the implementation of sensitivity analysis. Figure 12 shows the sensitivity analysis of the key independent variables and their impact on permeability and UCS.

In sensitivity analysis related to the permeability factor (Figure 12a), the correlation between the quantity of cement content (C) utilized in constructing the pervious concrete and its permeability is feeble and inverse. This implies that the permeability marginally diminishes as the cement content increases. Concerning the compaction energy level (D), this connection is strong and inverse, albeit in a nonlinear manner. This denotes that as the compaction energy increases, the permeability decreases significantly and in a nonlinear fashion. Another influential variable is the aggregate-to-cement ratio (B), which exhibits a robust and direct association with the permeability value. With an increase in this variable, it becomes evident that the amount of free cement paste between the aggregates diminishes, thereby resulting in an increase in the channels for water movement and subsequently leading to enhanced permeability. Additionally, it is anticipated that the UCS will decline as the bond between the paste and aggregates weakens. In this study, the aggregate-to-cement ratio and compaction energy level were identified as the key variables influencing the permeability factor. Furthermore, the sensitivity analysis of the UCS model shown in Figure 12b demonstrates that the variables of the compaction energy level (D) and water-to-cement ratio (A) have the greatest impact on the output of the UCS factor. The compaction energy level and the amount of cement content (C) have a positive influence, while the aggregate-to-cement ratio (B) and water-to-cement ratio have a negative influence. The compaction energy level increases condensation, while a higher amount of cement enhances adhesion, resulting in greater strength. Conversely, a higher aggregate-to-cement ratio increases the specific surface area of the aggregate covered by cement paste and reduces the thickness of the cement paste between the aggregates, leading to a decrease in adhesion and strength. Additionally, a higher water-to-cement ratio leads to a substantial reduction in the UCS because the paste becomes too loose and moves toward the end of the sample, thereby increasing the void content of the cementitious material.

Figure 13 shows the prediction analysis for the UCS model, which has a prediction value of 0.83, and confirms the accuracy of the specified permeability compared to the laboratory data.

The ratio between the specified permeability value and the laboratory results is illustrated in Figure 13a. It is evident that the test results align with the assumed permeability and validate its accuracy. Additionally, the UCS regression model ratios compared to the actual values obtained from the conducted experiments are presented in Figure 13b. The model’s prediction is satisfactory and applicable to almost all mix designs.

To depict the behavior of the permeability and UCS criteria concerning the two significant and sensitive variables within each factor, a three-dimensional diagram is presented in Figure 14.

Figure 14a provides additional evidence supporting the validity of the assumption made regarding permeability and the VeBe compaction time for each specific mix design. This is evident from the fact that the results of the permeability tests predominantly revolve around the value of 0.4 cm/s, which is consistent with the assumption. Moreover, it can be observed that when the T_vc value increases, it has a significant effect on decreasing permeability. Also, if the ratio of aggregate to cement is increased, it leads to a corresponding elevation in the K coefficient. The previous discussions have provided explanations for the aforementioned conditions. In addition, the 3D model of UCS demonstrates a noticeable relationship between an increase in T_vc and an increase in strength (Figure 14b). A prime example of this can be seen when the T_vc is extended from 13 s to 82 s, resulting in a significant increase in UCS from 1.8 MPa to 10.3 MPa, along with a decrease in K coefficient from 0.44 to 0.032 cm/s. Furthermore, exceeding the necessary water quantity for hydration, along with the resulting increase in shrinkage and cracks, brings about an alteration in the characteristics of the hydration components. This alteration weakens the cement paste and diminishes the strength of the cement mixture.

Since the VeBe compaction method was employed in the current study, the formation of porosity dispersion did not occur uniformly across the entire sample. Figure 15 shows the pore dispersion in a permeable sample of 10 by 15 cm. The dispersion of pores in the sample was analyzed through the utilization of image processing techniques in MATLAB [93]. Figure 15 demonstrates the compaction of the permeable sample during the VeBe test. This test effectively replicates the movement of the roller on the pavement. Consequently, the compaction energy beneath the overhead is the highest, and it progressively decreases with an increase in the layer depth. This variation in energy distribution leads to nonuniform porosity dispersion throughout the sample.

Using the Digital Image Correlation and Tracking toolbox in MATLAB [93], we found that the top of the sample (Figure 15a) demonstrated a higher density and a lower porosity compared to its lower counterpart (Figure 15b). In addition, Figure 15c also shows a clear change in porosity from the top to the bottom of the sample. It is revealed that the upper 2 cm section of the sample has a higher density and lower porosity compared to the rest of the sample. As one moves from the top to the bottom, there is a rise in porosity and interconnected pores, which is caused by a reduction in compaction energy in the same direction. As a result, the upper layer is more prone to clogging due to the fewer and narrower connected pores [44,117,118,119,120].

3.2.3. Comparison of UCS Prediction Models Reported in Various Studies

In the past few decades, the prediction of UCS has made significant progress with the implementation of different programs and software [121]. This study introduces a regression model for the UCS, allowing for a direct comparison between the model derived from this research and previous studies. This idea is further explained and expanded upon in

Table 10. Comparison of UCS prediction models reported in various studies.

No.	Model	Reference
Study 1	$\mathrm{U}\mathrm{C}\mathrm{S}=3.70-2.22\left({\displaystyle \frac{\mathrm{W}}{\mathrm{C}}}\right)-0.4318\left({\displaystyle \frac{\mathrm{A}}{\mathrm{C}}}\right)+5.46{\mathrm{T}}_{\mathrm{V}\mathrm{C}}+2.55\left({\displaystyle \frac{\mathrm{W}}{\mathrm{C}}}\right)\times {\mathrm{T}}_{\mathrm{V}\mathrm{C}}-2.90\left({\displaystyle \frac{\mathrm{A}}{\mathrm{C}}}\right)\times {\mathrm{T}}_{\mathrm{V}\mathrm{C}}$ $({\mathrm{R}}_{\mathrm{A}\mathrm{d}\mathrm{j}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{d}}^2=0.878)$ The unconfined compressive strength (UCS) varied from 1.85 to 10.38 MPa. The water-to-cement ratio (W/C) was between 0.27 and 0.33, the aggregate-to-cement ratio (A/C) ranged from 4.00 to 4.50, and Tvc varied between 13 and 82 s. Based on the DESIGN EXPERT software, this model can be utilized to predict the response for specific factor levels. High levels of the factors are presented as +1 and low levels as −1.	Current study
Study 2	$f_{c-28}=25.38-0.539P-13.12\left({\displaystyle \frac{W}{C}}\right)+0.0045C$ $\left(R^2=0.92\right)$ $\mathrm{The}\mathrm{compressive}\mathrm{strength}\left(f_{c-28}\right)$ varied from 0.90 to 23.70 MPa, with porosity (P) ranging between 13% and 24%. The water-to-cement ratio (W/C) ranged from 0.14 to 0.40, while the cement content (C) ranged from 150 to 412 kg/m3.	[26]
Study 3	$f_{c-28}=0.8967+0.7289C+8.163\left({\displaystyle \frac{W}{C}}\right)+1.4418W-0.1824A_1-0.1637A_2-0.1693A_3$ $\left(R^2=0.85\right)$ $\mathrm{The}\mathrm{compressive}\mathrm{strength}\left(f_{c-28}\right)$ ranged from 1 to 6.95 MPa, while the water-to-cement ratio (W/C) varied between 0.30 and 0.40. The quantity of cement (C) and water (W) utilized ranged from 150 to 250 kg/m3 and 45 to 100 kg/m3, respectively. The mixes A1 to A3 were designed with aggregate sizes of 4.50 mm, 9.50 mm, and 19.50 mm.	[27]

. Since the design of the pervious concrete in this study was derived from the restrictions imposed by the pervious concrete standards, the model calibrated in this study was compared to the models developed by various researchers in this domain.

It is important to note that utilizing porosity as a modeling factor is not ideal because it is directly linked to the dependent variable. To obtain porosity, a sample must be created, which renders the prediction model ineffective (Study 2). Moreover, the independent variables must not exhibit any correlation with one another. In the current study, as previously noted, the statistical analysis did not reveal any significant correlations between the investigated variables (the water-to-cement ratio, aggregate-to-cement ratio, and compaction energy levels). Additionally, by comparing the goodness of fit of all the models in Table 10, it is evident that the effectiveness of this particular study model surpasses that of Study 3. This can likely be attributed to the inclusion of the compaction energy parameter, which is a significant factor in determining the compressive strength of the samples. Also, Study 3 examined the maximum size of different aggregates, which was assumed to remain constant in this study and should be also investigated in further research.

3.2.4. Failure Modes Observed in UCS Tests

The samples subjected to UCS tests can fail in various states and shapes. ASTM C39 [97] has considered some of these patterns. Examples of these failures concerning the UCS tests are displayed in

Table 11. Observed failure types in cylinder samples in this study.

Failure Modes	Photos	Failure Modes in ASTM C39	Explanation
Columnar cracks			The cracks are evenly distributed vertically around the sample, following the Type III model [97]. Columnar crack failure was observed in the samples tested for compressive strength, within an estimated UCS range of 1.00 to 4.00 MPa.
Cone			Two cones were formed at both ends of the sample with intact corners, as described by the Type I model [97]. Cone failure occurred in the samples examined for compressive strength, falling within a projected UCS range of 8.50 to 10.50 MPa.
Shear			The fracture extended throughout the entire sample without affecting the starting and ending points, as indicated by the Type IV model [97]. Shear failure was detected in the samples examined for compressive strength, falling within an estimated UCS range of 4.50 to 6.00 MPa.
Side fractures			The side fractures occurred either above or below the sample, in alignment with the Type V model [97]. Side fracture failure was observed in the samples tested for compressive strength, within an estimated UCS range of 6.00 to 8.00 MPa.

.

As shown in Table 11, the failure pattern of the samples with low strength was characterized by the formation of columnar cracks, while the samples with high strength demonstrated a cone-shaped rupture. Nassiri et al. [43] reported similar findings in their experimental study.

3.2.5. Determining the Recommended Mix Designs

To identify the most effective mix design, it is necessary to take into account two fundamental factors. These two factors encompass the highest strength and desirable permeability, which should be chosen based on the specifications and guidelines outlined in ACI 522R [33]. For this purpose, the minimum permeability and strength are 0.14 cm/s and 2.80 MPa, respectively [33]. Figure 16 illustrates these specific ranges, showcasing the mixture designs that exist within this range. It should be noted that the respective diagram is sorted by permeability from low to high. Designs 20 to 24, in particular, have permeability outside the range due to the extreme sensitivity of the samples to the amount of cement used. As a result, these designs have the highest UCS and the lowest permeability, which can be attributed to the utilization of the highest amount of cement.

Based on the acceptable mix design specifications and standards set by ACI 522R [33], it is possible to recommend the best mix design by considering the highest permeability and UCS, as well as the volume of cement and the VeBe compaction time.

Table 12. Recommended mix design in this study for low-volume road pavement.

Run	UCS (MPa)	Permeability (cm/s)	Cement Content (kg/m3)	Tvc (s)	Standard
4	7.679	0.302	280.00	80	ACI 522R [33]

indicates the recommended mix design.

Based on Table 12, it is feasible to choose the recommended mix design by taking into account the acceptable range and guidelines. When considering the maximum UCS level alongside the suitable permeability, it is possible to make use of Run 4, which possesses a strength of 7.679 MPa and can be implemented on low-volume roads and surfaces such as pathways. Also, this specific mix design requires a reduced amount of cement.

3.2.6. Preliminary Schematic of a Pervious Concrete Structure

The recommended mix design (Run 4) can be used to construct the permeable structural layer. Figure 17 illustrates a cross-section of a preliminary schematic of the pervious concrete structure.

Figure 17 shows that the proposed structure comprises various components. The thickness of every layer can be determined according to specific standards and project specifications. Furthermore, a geogrid layer can be utilized between the base layer and the subgrade to improve the soil’s strength while also acting as a barrier against the entry of fine aggregates into the permeable layer [52].

4. Conclusions

The novelty of this research lies in the calibration of a model for predicting the UCS of pervious concrete by taking into account different mixture characteristics and levels of compaction energy. The mix design procedure aimed at achieving a balance among the UCS, permeability, and compaction levels. The model established in this study allows for estimating the necessary compaction energy according to structural needs and the desired UCS. Consequently, it enables the selection of suitable compaction methods and machinery. In summary, the key findings of this research are as follows:

• Based on the ACI 522R criteria, Run 4 emerged as the recommended mixture for the pervious concrete layer. This mix comprised a water-to-cement ratio of 0.33, an aggregate-to-cement ratio of 4.00, and a cement content of 280.00 kg/m³. It achieved a UCS of 7.68 MPa, a permeability of 0.302 cm/s, a porosity of 17.4%, and a density of 2.28 g/cm³, making it suitable for low-traffic applications.
• The established UCS model demonstrated a good fit (R² = 0.878). Sensitivity analysis revealed that the compaction energy had the most significant influence on the UCS, followed by the water-to-cement ratio. This suggests that compaction can potentially enhance strength while reducing cement content.
• The compaction energy level significantly influenced the mechanical and permeability properties. An increase in the compaction time from 13 to 82 s resulted in an increase in the UCS from 1.851 to 10.38 MPa, while the permeability decreased from approximately 0.4 cm/s to nearly 0.03 cm/s.
• The results of this study revealed a nonuniform distribution of porosity and permeability within the pervious concrete layer. These properties increased from the upper two centimeters of the layer (compaction zone) toward the bottom.

Based on the findings of the present study, several recommendations for future research can be provided. For example, future studies should explore the effects of aggregate grading and properties such as size, shape, and origin on the proposed UCS prediction model, as these parameters were assumed constant in the present research. It is also recommended to examine the effects of various types of admixtures commonly added to pervious concrete, as they can also influence both strength and permeability. Lastly, practical trials should be conducted under real-world construction conditions to complement the results of the present study.