Mathematical modeling of the infiltration in a permeable pavement on the field scale

Permeable pavement (PP) is an alternative for the management of urban rainwater that allows the reduction of effective precipitation through the infiltration process. In this study was evaluated the infiltration capacity of a PP of hollow concrete blocks in a parking lot of the Federal University of Pernambuco. The hydraulic characterization and the infiltration capacity were analyzed in real scale, using a simple ring infiltrometer of 100 cm in diameter through the Beerkan method. Infiltration tests were carried out at twelve points of the PP. The BEST algorithm was applied in it Best-Intercept and Best-Slope version, to estimate the hydraulic parameters of the van Genutchen and Brooks and Corey equations for the retention and hydraulic conductivity of the PP surface. The values of saturated hydraulic conductivity determined by the BEST Intercept method were higher than those obtained by BEST Slope. The sorptivity values estimated by BEST Slope and Intercept were similar, with BEST Slope values slightly higher. Moderate infiltration variability was observed on the PP surface, as well as within the same type of texture. The Beerkan method proved to be adaptable to measure, in field scale, the three-dimensional infiltration in the PP covering layer.


INTRODUCTION
In Brazil, due to the expansion of the built areas, generally spatially disordered, mainly in the large metropolitan regions with high urbanization, it has caused problems for the management of the stormwater runoff. Population growth and waterproofing of surfaces decrease the natural recharge of groundwater and the quality of surface water. According to Hager et al. (2019), traditional urban rainwater infrastructure is becoming increasingly insufficient due to the rapid urbanization process. In addition, the increase in runoff volume makes it urgent to apply Best Management Practices (BMPs) that favor the natural phenomena of infiltration and storage of urban rainwater (Li et al., 2019).
The management of urban rainwater has become a major challenge in coastal cities with a tropical climate. In these cities, the high frequency of rainfall, with great intensity and volume, associated with hydrographic basins with high waterproofing, bring serious problems and social, environmental, patrimonial and economic damages (Tucci, 2007). In particular, the city of Recife, located in the Northeast of Brazil, has an average annual rainfall of 2400 mm, of which 70% occurs in a concentrated manner between the months of March to August, a period with high flood records (Coutinho, 2011). The typical rainfall intensity for a 25-year return period event with 60 min duration in Recife was estimated at 66,46 mm.h -1 (Coutinho et al., 2013).
The use of green technologies that value infiltration and storage phenomena has been reported in several Brazilian studies (Agostinho & Poleto, 2012;Souza et al., 2012;Benini, 2015;Maruyama & Franco, 2016;Nunes et al., 2017). The compensatory technical term for urban drainage has been used in Brazil, assuming the equivalent of Best Management Practices (BMPs) in the United States. Among the alternative techniques for the management of urban rainwater, such as infiltration trenches, infiltration ditches and rain gardens, permeable pavements (PP's) have been increasingly used in sidewalks and parking lots to improve rainwater infiltration in urban (Liu et al., 2019). For the popularization of the use of this technology in Brazil, it is essential to prove its effectiveness in reducing rainwater inundations by the precise estimate of the PP's hydraulic parameters.
PP's are formed by a layer of porous covering and a reservoir of stones. In this type of structure, the knowledge of the infiltration capacity of the wear layer is essential to verify its hydrological performance. For this, the performance of infiltration tests in field scale, in the pavement structure, can estimate the hydraulic properties with the water retention curve and the hydraulic conductivity curve, helping in the decision making regarding the maintenance of the device.
In this sense, several studies such as that by Jabur et al. (2015) and Coutinho et al. (2016), has characterized the infiltration of PP's using the methodology provided for in ASTM (American Society for Testing and Materials) C1701 -Standard Test Method for Infiltration Rate of in Place Pervious Concrete (American Society for Testing and Materials, 2009). However, this method does not allow the estimation of the parameters of the water retention curves and the hydraulic conductivity curve of the soil. The parameters of both curves are prerequisites for the evaluation of hydraulic operating scenarios of the PP's through unsaturated numerical modeling, based on the resolution of the Richards (1931)

equation.
Different methods can be used to determine the hydraulic parameters. The method must be chosen according to its complexity, robustness and related costs, among the various methods available the Beerkan method has been widely used (Coutinho et al., 2016). The Beerkan method is a semi-physical method that allows the estimation of seven hydraulic parameters, allowing to appreciate the properties of the water retention curve and the hydraulic conductivity curve of the soil (Lassabatère et al., 2006). Coutinho et al., (2016) used the Beerkan method with 7.5 cm diameter infiltrometers for the hydraulic characterization of fifty-two infiltration tests on a permeable pavement in the city of Recife. Obtaining the hydraulic parameters allowed the authors to simulate the water transfer processes on the permeable pavement.
This study aims to the estimation of hydraulic parameters of the PP wear layer using an analytical inverse technique (i.e. BEST) and Beerkan infiltration test., using the BEST algorithm -Beerkan Estimation of Soil Transfer Parameters through Infiltration Experiments in their Best-Slope and Best-Intercept versions.

Study site and soil charachterization
The study was carried out on a permeable pavement (PP) implanted in the parking lot of the Center for Legal Science (CCJ) of the Federal University of Pernambuco in Recife at coordinates 8°03'31"S and 34°52'57"W ( Figure 1). The climate of the study area is hot and humid with an average annual temperature of 25.8 °C (Köppen As').
Rainfall data from APAC (Pernambuco and Water and Climate Agency) for the period between the years 2010 and 2017 indicate annual average rainfall of 2,167.35 mm. June is the month with the highest rainfall, on average 379.7 mm; and November the driest month, with an average of 37.9 mm (Agência Pernambucana de Águas e Clima, 2018). Winter corresponds to the rainy season and lasts from March to August with about 70% of the total annual rainfall ( Figure 2).
The PP investigated in this study was built in 2012 ( Figure 3), has been in use for 6 years and has not yet undergone any maintenance. It consists in a layer of concrete hollow blocks, seated on a 7 cm thick layer of sand. The blocks are filled with grass and measure 39 cm × 21 cm and thickness of 10 cm. In this structure, it is considered that the global infiltration capacity of the PP depends mainly on the infiltration capacity of the soil that fills the concrete blocks during rain events. The effects of the lower layers on the infiltration of water were neglected. As a main approach, the focus is placed on the hydraulic characterization of the soil elements of the PP during a typical hydrological year in the region. The soil characterization was carried out in the samples taken from the hollow part of the concrete blocks according to the ABNT NBR 7181/2016 (Associação Brasileira de Normas Técnicas, 2016), the granulometric classification by United States Department of Agriculture (1993) and Empresa Brasileira de Pesquisa Agropecuária (1999).

Infiltration experiments
In order to verify the hydraulic and hydrological behavior of the PP, 12 points were chosen to perform the infiltration tests. The infiltration tests were carried out following the Beerkan methodology (until the time required for the infiltration of each water volume reaches steady state (permanent regime) (Mubarak et al., 2010;Coutinho et al., 2016). At each point the soil was sampled to determine the initial soil moisture condition (cm 3 .cm -3 ). After infiltrating the last applied volume, samples were taken to determine the final moisture (cm 3 .cm -3 ), particle size distribution and apparent density of the soil (g.cm -3 ) in the 12 points. In the Figure 4c is presented the schematic location of the points for experimental tests in the field, the black narrows indicate the car direction flow in the parking lot.

Estimation of the soil hydraulic properties
The characterization of the hydraulic properties of PP was carried out using the BEST method (Beerkan Estimation of Soil Transfer Parameters through Infiltration Experiments) (Lassabatère et al., 2006;Souza et al., 2008). This method has the advantage of providing a complete characterization of the hydraulic conductivity curve and the soil water retention curve. The parameters of the hydraulic conductivity curve of the soil K(θ) were described by the Brooks & Corey (1964) model (Equation 1). The parameters of the water retention curve θ(h) were described by the Van Genuchten (1980) with Burdine (1953) The θ is the volumetric humidity (L 3 .L -3 ); θ r and θ s the residual and saturated volumetric humidity (L 3 .L -3 ), respectively; h the matric potential (L); h g (L) a scale value of h considered the potential for air intake; m and n are shape parameters; K s the saturated hydraulic conductivity of the soil (L.T -1 ) and η the shape parameter for the hydraulic conductivity curve. The parameter p is a tortuosity, that depends on the capillary model, that is, zero (Childs & Collis-George, 1950), 0.5 (Mualem, 1976), 1 (Burdine, 1953), or 1.33 (Millington & Quirk, 1961).
To estimate all hydraulic parameters, the BEST method requires two sets of data: (1) the particle size distribution and the apparent density of the soil and (2) the infiltration accumulated throughout the infiltration experiment and the initial and final   humidity. In the BEST θ r is considered zero. The humidity in saturation θ s it is derived from the value of the apparent density of the soil assuming that it is equal to porosity. The shape parameter n was estimated from a pedotransfer function developed from the particle size distribution of the smaller fractions than 2 mm (Lassabatère et al., 2006). The scale parameters K s and h g are derived from the analysis of accumulated infiltration.
The shape parameters are linked to the texture and have similar shape between the particle size distribution F(D) and θ(h). Haverkamp & Parlange (1986) presented the Equation 3 to express F(D).
where 2 m 1 n = − Being D the particle diameter (mm); Dg the particle size scale parameter (mm); m and n the shape parameters of the particle size distribution curve.
In turn, the normalization parameters depend on the soil structure. The parameters h g and K s are obtained by minimizing I(S, K s ), that is, of the squares of the differences between the infiltrated water observed and calculated. The infiltrated water depth is calculated using the equation proposed by Haverkamp et al. (1994), valid for short and medium times (Equation 4).
where a and b 2 are obtained by Brooks & Corey (1964): where S is the sorptivity; r is the cylinder radius; γ and β are coefficients that are commonly set at 0.75 and 0.60 . ∆θ = θs -θ 0 , which apply for most soils when θ 0 < 0.25 θs Haverkamp et al., 1994).
To minimize I(S, K s ) is used the algorithm of Levenberg-Marquardt, iteration technique used to find the minimum of a function expressed as the sum of squares of non-linear functions (Marquardt, 1963). The Levenberg-Marquardt method is the determination of the vertex of a parabola. The performance of the adjustments analyzed by the values that correspond to the mean square error. Obtained the values of θ s and K s , the scale parameter for water pressure is determined h g , by Equation 7 (Lassabatère et al., 2006).
Being p c a parameter that depends only on the parameters n, m and η (Condappa et al., 2002;Haverkamp et al., 1998), determined in Equation 8.
Γ is the classic Gamma function, which is an extension of the factorial function to numbers. During the three-dimensional infiltration process the factors that can affect the flow of water into the soil, that are: the geometry of the water source, capillarity and gravity (Lassabatère et al., 2019). One of the ways to characterize these factors is from the capillary length scales (λ c ) (White & Sully, 1987) and the characteristic radius of the hydraulically active pores (λ m ) (Philip, 1987), determined by Equations 9 and 10, respectively: Being σ the surface tension of the water (0.0719 N.m -1 ); ρ a the specific gravity of water (10 3 Kg.m -3 ); g the gravity acceleration; δ a diffusivity shape parameter [1⁄(2≤δ≤π⁄4)], which was considered 0.55 (White & Sully, 1987).
According Souza et al. (2008), the capillary length scale represents the relative importance of capillary forces in relation to gravity, when water is transmitted from a source through the soil, with initial humidity θ 0 . The characteristic pore radius defines the average size of the pores that participate in the infiltration process subjected to applied pressure h; the larger the characteristic radius, λ m , greater is the effect of gravity compared to that of capillarity.
The number of pores C λm is estimated using the Poiseuille law for flow in a capillary tube (Equation 11). μ is the dynamic viscosity of the water (0.00089 kg·m -1· s -1 ).
The degree of variability of parameters was analyzed based on the classification proposed by Warrick & Nielsen (1980), who suggest CV limits <12%, 12<CV <52% and CV> 52% for low, medium properties and high variability, respectively.

Soil texture and infiltration curves
The particle size distribution curves are shown in Figure 5. The classification of soil texture was determined according to the fractions of clay, silt and sand (United States Department of Agriculture, 1987) and can be viewed on the (Appendix A). As a result, soil samples were divided into three main texture: Loamy sand (9 samples) ( Figure 5a); Sandy loam (2 samples) ( Figure 5b) and sand (1 sample) (Figure 5c). There is a predominance of the sand fraction with percentages above 80% in the 12 samples analyzed.

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• Parameter m C λ , m λ and c λ Table 1 shows the values of m C λ , m λ and c λ , for the 12 sample points. The values of the capillary length scales ( c λ ), characteristic radius of hydraulically active pores ( m λ ) and the number of pores ( ) m C λ ) varied from 784.85 (Point 11) to 82.73 mm (Point 6); 0.088592 (Point 6) to 0.009338 mm (Point 11); 9.02x10 10 (Point 11) to 5.27x10 6 pores/m 2 (Point 6), respectively. Point 7 was discarded, as it did not show numerical convergence with the Beerkan method, not meeting the criteria of the method.

Water infiltration experiments and hydraulic characterization
The Figure 6  According to Figure 6, the curves show that there is a significant spatial variability in the infiltration behavior on the surface of the PP. Despite the variability, the response curves of Loamy Sand and Sandy Loam are in the same order of magnitude and resemble the curve of the sand. The different infiltration behaviors distributed in the PP can be attributed to the interference of other factors such as degree of compaction, dimension and connectivity between the pores (Lassabatère et al., 2010;Aiello et al., 2014), content of organic matter, in addition to the textural class (Coutinho et al., 2016).
Analyzing the average infiltration rates, it is noted that the average response is similar for all types of soil of the PP. The average infiltration rate was 4.56 mm·min -1 , the texture of Loamy Sand presented an average rate of 4.44 mm·min -1 , the Sandy Loam 5.03 mm·min -1 and the sand presented 4.24 mm·min -1 . It is important to highlight that the small number of samples does not allow to characterize the dispersion in detail for textural classes Sandy Loam and Sand. The observed differences in the infiltration rates considering all the experiments also indicate  moderate variability of the infiltration on the surface of the PP, as well as within the same type of texture. Based on the classification proposed by Warrick & Nielsen (1980), the statistical parameters of the infiltration rate in the permeable pavement indicated average variability, with a CV of 35.37% in the data with intermediates and coefficients of variation (12 < CV <52%) In the same way, Coutinho et al. (2016) obtained an average infiltration rate of 14.5 mm·min -1 for a PP with interlocking concrete blocks and high spatial variability in the infiltration capacity through the Beerkan method. Jabur et al. (2015) obtained infiltration rate between 1.67 mm·min -1 and 2.84 mm·min -1 for a permeable pavement of interlocked concrete blocks. Bruno et al. (2013) obtained an infiltration rate of 1.4 mm·min -1 in an experimental plot of interlocked concrete blocks. Bean et al. (2007) obtained an infiltration rate of 0.84 mm.min -1 for permeable pavement applied to sidewalks and parking lots without routine maintenance and 1.44 mm.min -1 for permeable pavement applied to sidewalks and maintenance parking.
In general, the infiltration rates of all curves fall to a value that tends to the permanent regime. Only at Point 1 or Point 7 (Loamy Sand), the infiltration rate starts from 6.67 mm.min-1 and increases up to 13.3 mm.min -1 According to the infiltration theory, this is a lagged process, so that the appearance of phenomena such as the acceleration of the infiltration rate, is reported in the literature as hydrophobicity (Bastos et al., 2005;Maia et al., 2010;Vogelmann, 2014). According Vogelmann (2014), this is due to changes in the contact angle between water and the solid phase. By covering the soil particles by organic hydrophobic substances (such as leaves and roots) that cause water repellency.
It can be seen that P9, P10, P6 and P3 presented the lowest actual infiltration rates, due to the intense and constant traffic of the vehicles in the places of access to the spaces and maneuver, which cause the compaction. That is, reduction of the void volume of the soil (reduce its porosity) that generates the increase of the impermeability index (Bean et al., 2004). And the presence of leafy trees that protect the soil from direct irradiation, which also have scattered roots with a large radius of influence on the soil (which may be observed in the Figure 1b), can also interfere with water infiltration into the pavement (Fini et al., 2016). According to the tested points, the average of the infiltration rate was 355 mm·h -1 , being the lowest found in P6 (149.02 mm·h -1 ) and the highest for P1 (527.80 mm·h -1 ).
In addition, in this work it was not possible to make a correlation between the decrease of the infiltration rate with the classification by the textural triangle, since soils with the same textural classification presented very different values. This difference in hydraulic properties may be caused by factors that directly interfere with hydraulic conductivity how soil texture and structure, pore size and connectivity (Rahmati et al., 2018).

Shape parameters
The BEST algorithm estimates the shape parameters based on the size of the particles and calculates the infiltration capacity by adjusting the infiltration rate as a function of time, using the theoretical model of Haverkamp et al. (1994). As a prerequisite for the application of the BEST method, the infiltration curve has to be described satisfactorily by the model, both in the permanent and in the transient regime (Lassabatère et al., 2010;Di Prima et al., 2016). Most of the accumulated infiltration curves from this work were well adjusted by the BEST method.
In Figure 7 box-plot graphs of shape parameters are presented, separated by soil type. The values of n varied between 2.23 e 2.24 to the soil Sandy Loam, 2.23 a 2.37 to the Loamy Sand and 2.43 to the sand. It is important to highlight that the values of n are higher on coarse soils than on thin soils. The n values are reported to the soil Loamy Sand between 2.42 and 2.56, Santos et al.  (Coutinho et al., 2016;Santos et al., 2012;Souza et al., 2008). Santos et al. (2012) obtained n of 2.19 and 2.42, to a Sandy Loam soil, and Souza et al. (2008) obtained n between 2.13 and 2.16 for that same type of soil.
Small changes in the n values provide substantial differences in η. The values of η varied between 7.67 to 11.63 in this study considering all the soils studied. Some works with predominantly sandy soils such as Souza et al. (2008) obtained values η between 6.70 and 18.16, Silva et al. (2009) between 6.23 and 8.09 and Santos et al. (2012) between 1.34 and 7.83. Comparatively, the values of m, n, η and c p obtained by Souza et al. (2008), Silva et al. (2009), Santos et al. (2012 and Coutinho et al. (2016), for the same textural classes they are compatible with those obtained in this study.
The shape parameters in general depend on the type of soil. The parameter values n and m for coarser soils (predominantly sandy soils) they are usually high. This can illustrate the piston effect of the conductivity and water retention curves on coarse materials. The piston effect is related to the condition of an abrupt increase in hydraulic conductivity and soil moisture in potential matrixes close to zero. In contrast, the shape parameter values for thin materials are lower and result in smoother retention and conductivity curves (Coutinho et al., 2016). The values presented in this work were higher in soils that have higher sand content and inversely the values of η and c p were smaller for the same points.

Scale parameters
Primarily, normalization parameters do not depend on the type of soil. Among the normalization parameters adjusted by BEST, the average pore length (h g ) it depends directly on the structure of the soil. Saturated hydraulic conductivity (K s ) is the parameter that indicates how easily a fluid is transported through the soil. Sorptivity (S) corresponds to the capacity of the soil to absorb water by capillarity in the absence of gravitational effects (Borges et al., 1999). In Figure 8 box-plot graphs of normalization parameters (h g , K s and S) are shown adjusted by BEST Slope and Intercept separated by soil type.
The average pore length ( g h ) varied between -229.952 to -365.303 mm in the BEST   In the Sandy Loam soil, the BEST Intercept estimated Ks between 0.0323 and 0.01057 mm.s -1 and BEST Slope estimated the Ks values between 0.0434 and 0.0293 mm.s -1 . The Ks values estimated by the BEST Intercept were also higher than the values of Ks estimated by BEST Slope for Sand soil. These results for Loamy Sand and Sandy Loam are in accordance with several studies in the literature (Souza et al., 2008;Di Prima et al., 2017;Sousa et al., 2019;Lassabatère et al., 2019). The Sorptivity values estimated by BEST slope and intercept were similar, with BEST slope values slightly higher. The greatest value of Sorptivity was 4.416 mm.s -0.5 of Loamy Sand soil by Slope and the lowest value was 1.7984 mm.s -0.5 to Loamy Sand soil by Intercept. For some cases where the portion of Sorptivity is very large, BEST Slope can estimate null or negative values of saturated hydraulic conductivity, in this case BEST Intercept corrects the error (Yilmaz et al., 2010).
Analyzing the results according to the texture, it is expected that the highest values of s K and S they should have occurred on Sand-type soil. However, the extreme values occurred within the texture Loamy Sand. This is due to the s K and S be parameters that depend on factors other than texture, such as the apparent density of the soil and connectivity between the hydraulically active pores (Castellini et al., 2019). These results highlight the importance of the soil structure for its hydraulic behavior (Lassabatère et al., 2019). In general, the saturated hydraulic conductivity is of the same order of magnitude for all types of soil in this study. There were significant differences between the two BEST methods in estimating the scale parameter for water pressure. Noting that there are only two Sandy Loam soil samples and a sample of sand the focus should be placed on the most frequent type of soil (Loamy Sand).

Retention and hydraulic conductivity curves
Defined the shape parameters (m or n and η) and scale (θ s , K s and h g ), retention curves were obtained θ(h) and hydraulic conductivity K(θ). The points of origin of each curve (θ s ) were set in 0.46792 m 3 .m -3 , considering the same volumetric humidity for all experiments. According Di Prima et al. (2016), the estimates for the scale parameters depend on the BEST methods chosen. However, in this study, the use of the Slope and Intercept methods led to results comparable with similar trends in most parameters. Given the dependence of the method, it was decided to average the values of the scale parameters in both methods. Table 2 shows the averages of hydraulic parameters by type of soil. Based on these values, the hydraulic conductivity curve and the water retention curve were plotted (Figure 9).
In relation to hydraulic conductivity, the most sandy soil presented less hydraulic conductivity in saturation. Regarding the soil water retention curve θ(h), it was observed that all cases presented very similar curves. The higher the percentage of sand, the lower the water retention capacity in the soil. In general, soils with a high percentage of sand (as in the case of the study) have many hydraulically active pores. Fine Soils or clayey soils presents a greater proximity between the particles, results in higher adsorption and capillarity effects and most capacity of retention of water. The saturated hydraulic conductivity curves Ks contribute to a better understanding of hydraulic behavior at the points tested. For greater volumetric water content (θ), the K(θ) values tend to increase in soils with a higher percentage of Sand.

CONCLUSIONS
The present study evaluated the infiltration capacity of a permeable pavement (PP) of hollow concrete blocks. The hydraulic characterization and the infiltration capacity were analyzed in real scale, using a simple ring infiltrometer of 100 cm in diameter through the Beerkan method.
In this experimental work, 12 points of a permeable pavement consisting of concrete blocks filled with compacted soil were evaluated. From the results obtained, the following conclusions are presented: The dispersion of the observed infiltration rate set of values demonstrated intermediate variability of pavement infiltration capacity. This variance may be due to the degree of compaction that interferes with the arrangement of the particles and pore connectivity due to the pavement usage conditions. Besides, the presence of organic matter, boulders, and other contaminants in the floor void filling soil, as well as in the lower layers, modify the behavior and the rate of water transfer into the soil (i.e. hydrophobicity phenomenon, observed in P1).
The Beerkan method produced, quickly, simply and at a low cost, an important set of data that helped evaluating and characterize the capacity of PP of hollow blocks as a compensatory technique. The BEST algorithm obtained acceptable values for S and Ks, in 11 of 12 analyzed points, besides providing precise adjustments of the accumulated infiltrations.
The results obtained in this study through the estimation of the hydraulic parameters can demonstrate the hydraulic efficiency of the PP of hollow blocks for the Decrease of the effective precipitation through the use of simulation models that allow evaluating the dynamics of the water in these structures. Also, the monitoring of the hydraulic behavior of permeable pavements at field scale using infiltration tests allows the knowledge of the hydrological performance of these structures, allowing them to be used and integrated with urban environments knowing their limitations.  Figure 9. Hydraulic conductivity and water retention curves for the three types of soil.