Comparative analysis of models for design of infiltration basins in unpaved roads

Unpaved roads are of great importance for the economic and social development of a country. Water erosion provoked by the concentration of runoff along the road is a principle cause of its degradation. The construction of infiltration basins for runoff retention on unpaved roads is a usually alternative for the problem solution. The use of more coherent methodologies for designing infiltration basins is fundamental for the appropriate road degradation processes control. Therefore, the objective of this work was to perform a comparative analysis of a methodology for the design of infiltration basins that consider an intense rainfall associated to a determined return period, with a methodology that use all events of a determined precipitation series, and overlapping effects of their respective runoff volumes. According to the obtained results, it was found that the volume calculated for the infiltration basin by the model which considers all events of the precipitation series is greater when the rate of water infiltration into the soil at the basin bottom is lowest.


INTRODUCTION
Unpaved roads, also known as unsealed roads or rural roads, are of great importance for the economic and social development of a country (Lugo & Gucinski, 2000;Crossley et al., 2001;Carvalho et al., 2010).Their existence enable the access of rural population to health services, education, leisure (Gallego et al., 2008a), and transport of agricultural, stock-raising and products of forestry activities (Gallego et al., 2008b).
Due to the role of runoff as the main agent of unpaved roads erosion, its drainage before flow concentration and erosive energy acquirement is of fundamental importance for preventing accentuated damages to the roads (Griebeler et al., 2005).However, the simple act of removing water from roads is not sufficient in the absence of appropriate destination for the flow, which would resume in transferring the problem to road marginal areas.Therefore, the construction of infiltration basins for runoff retention on unpaved roads is alternative of great importance for the control of degradation processes on these roads.
For designing infiltration basins, normally only a single intense rainfall is considered, such as in methodology developed by Pruski et al. (1997).In order to develop appropriate methodology for designing infiltration basin, Silva (2009) proposed a model based not only on one rainfall event, but all events of historical series at the place of interest and consequently, the cumulative effect of its respective runoff events throughout the series.
Based on hypothesis that calculation of required volume to infiltration basins, the adoption of criteria that takes into account the analysis event to event of precipitation is more consistent than one based at a single event associated with a return period of project, this work aims to conduct a comparative analysis of the model developed by Silva (2009) (based on event to event analysis) with the one developed by Pruski et al. (1997) (based on a single event).

MATERIAL AND METHODS
The model proposed by Pruski et al. (1997), called the Soil Surface Water Balance Method (Pruski, 2009), obtains the maximum runoff volume to be retained by the basin considering the maximum rainfall for a given project return period, such as several other methods developed for designing infiltration basins.The model proposed by Silva (2009) is based on a new concept for designing infiltration basins, which takes into account not only one rainfall event, but all historical series events at the place of interest, as well as the overlapping effects of their respective runoff volumes.

Soil surface water balance method (Pruski et al., 1997)
The Soil Surface Water Balance Method (SSWB) (2005) was developed in order to obtain the maximum surface runoff depth (MRD, in mm).In this model, the volume required for the infiltration basin (V B ) is derived from the product of MRD (converted to m) by the road area contributing (m 2 ) to the infiltration basin.
To obtain MRD a physical model of water balance on soil surface water was used as described by: MRD = R -Ia -F where: R -total rainfall depth, mm Ia -initial abstraction, mm F -accumulation infiltration, mm The total rainfall depth corresponding to a time interval t (min) was obtained by: R = i m t 60 where: i m -average maximum rainfall intensity, mm h -1 (which is constant for a given rainfall and time duration) t -rainfall duration (min) To obtain i m , the classical rainfall intensity-durationfrequency relationship was used: where: a, b, c, K -parameters of the precipitation profile equation, dimensionless Substituting Eq. 3 into Eq. 2 and taking the derivative with respect to time, the instantaneous rainfall intensity (i int ), in mm h -1 , is obtained: Both i m and i int decrease as t increases.The MSR ends at the instant at which i int is equal water infiltration into the soil of the roadbed (INF R ).Under this condition: The t value corresponding to MSR was obtained using the Newton-Raphson method.
The values of initial abstraction were determined by the curve number method, using the equation (SCS, 1972): where CN is the curve number.
The time corresponding to the initial abstraction was obtained using: where t Ia is the time interval from the beginning of the rain to the beginning of the of runoff (min) where: INF R -rate of water infiltration into the soil of the roadbed, mm h -1 t inf -infiltration period, min (obtained from the equation t inf = t -t Ia )

Model proposed by Silva (2009)
This model uses synthetic precipitation data obtained from the ClimaBr model, developed by Oliveira et al. (2005a;2005b) and refined by Zanetti et al. (2005) and Baena et al. (2005).From a historical series of rainfall data the ClimaBr model generates the following variables: total precipitated depth, total rainfall duration, time of maximum intensity rainfall and parameters related to the precipitation profile.
The precipitation profile considered in ClimaBr is described by a double exponential function (Figure 1) that represents the variation of the instantaneous precipitation intensity (i i ) as a function of time (t).
A double exponential function that represents the precipitation profile is given as: where: a', b', c', d' -parameters of the double exponential function in relation to the precipitation event, dimensionless ti max -time of occurrence of the maximum instantaneous precipitation intensity, dimensionless From the daily series of rainfall data, calculations are performed to obtain the daily runoff volume.The runoff volume (RV) is derived from the product of runoff depth (RD) by the road area contributing to the infiltration basin.
In the determination of RD it is assumed that runoff starts when i i is greater than the rate of water infiltration into the soil of the roadbed (INF R ) at time t b, and ends when i i returns to be equal to INF R , at time t e (Figure 1).
The precipitation depth occurring in the interval between the times t b and t e is obtained from the sum of the value resulting from the integration of the area corresponding to the ascending branch of the precipitation profile, t b to ti max, and the area corresponding to the descending branch, ti max to t e (Figure 1).The value of RD is derived from the difference between the precipitation depth occurring during the time interval from t b and t e , and the infiltration depth occurring in the roadbed over the same time interval.The infiltration depth is obtained from the product of INF R by the time interval in which surface runoff occurs (t e -t b ).
The volume stored in the infiltration basin on a given day of the series (V S i ) is obtained from the total volume of runoff reaching the basin that day (RV i ) and the stored volume remaining from the previous day (V Ri -1 ).These volumes are obtained by the equations: In the first rainfall of the series, which corresponds to the beginning of the simulations, the value of V S is equal to the volume of runoff that occurs on this day.If this volume does not completely infiltrate to the bottom of the basin that day, part of this volume will remain in the basin, corresponding to the value of V R i-1 for the next day.
The values of V R i-1 are directly related to the geometrical characteristics of the basin (width and slopes) and the water infiltration rate into the soil at the bottom of the basin (INF B ). Basins that have lower embankment slopes and greater width tend to have a greater surface area of liquid when full, a fact which implies a greater infiltrated volume for the same INF B .
The V S values calculations are repeated for all days in the series, in order to obtain a daily series of volumes stored.In each year the highest value of V S from the series is identified , which provides an annual series of maximum volumes stored in the infiltration basin (V max ).
The volume required for the infiltration basin (V B ) is obtained by applying the Gumbel distribution to the series of maximum stored volumes.According to the methodology proposed by Kite (1988), to calculate the magnitude of an event for finite series the value of V B is calculated from the following equation: where: V max -average of the maximum storage volume, m 3 K' -frequency factor, dimensionless  -standard deviation of the maximum quantity stored in each year of the series, dimensionless The frequency factor for the finite series is calculated as follows: K' = -0,45 + 0,7797 ln -ln 1-1 T where T is the return period, years.

Comparative analysis of models
The analyses were made based on comparison of VB values obtained by two models.The VB values were calculated considering five return periods (5, 10, 15, 18 and 20 years) and five values of water infiltration rate through the basin bottom soil -INFB (2, 5, 10, 15 and 20 mm h -1 ).The comparison of the VB values was separately made for each INF B value, since this variable is not considered in the model proposed by Pruski et al. (1997).
For both models it was considered that runoff is generated only by the drainage area comprised of the roadbed and corresponding to 300 m 2 , allowing for a water infiltration rate in the roadbed (INF R ) of 1 mm h -1 .The considered infiltration basin was rectangular shaped with a width (L B ) of 6 m and downstream (s 1 ) and upstream (s 2 ) slope declivities of 1 and 0.5 m m -1 , respectively.In the comparative analysis of the models, were considered precipitation data from the João Pinheiro and São Gonçalo do Abaeté towns, both from Minas Gerais State, Brazil, and localized into of the Paracatu watershed.In the S M model, the effect of overlapping runoff events on volumes stored during each day of the series is taken into account and, consequently, it influences the volume required for the infiltration basin.The SSWB model only takes into consideration one event associated with the determined return period without taking into account the fact that there are many events throughout the series where the runoff volume does not completely infiltrate the soil during a given day, and that the infiltration of volumes over a rainy season may represent a more critical condition for the design of the infiltration basin than the volume resulting from a single rainfall, as considered by the SSWB model.

RESULTS AND DISCUSSION
The comparison between the models for different INF B values shows that lower INF B values resulted in greater differences in the V B calculated by the two models.For a INF B of 2 mm h -1 , it was noted that the V B values obtained by the S M model estimated higher V B values, the magnitude of which was 2.65 times higher on average than the values obtained by the SSWB.For INF B of 5, 10, 15 and 20 mm h -1 , differences of 1.48, 1.21, 1.14 and 1.11 times greater were noted on average, respectively.
This behavior is the result of a more accentuated effect of overlapping runoff events when INF B is lower, making the volumes stored in the basin and, consequently, the value of V B calculated by the S M model larger.Because The SSWB model does not consider this characteristic, the difference between the V B values calculated by the models increases.
Studies relating the water infiltration rate into the soil at the bottom of the infiltration basin are still very scarce.However, it is anticipated that consideration of smaller values of this variable are more representative for the majority of the cases, even when dealing with infiltration basins constructed of soils with good infiltration capacity.This affirmation is based on the fact that there are processes which expressively interfere in the reduction of basin infiltration rates, such as: i) compaction of the soil at the base by tractor tires during construction such as illustrated by Miranda et al. (2009) and from the weight of the basin water (Ma & Spalding, 1997); and ii) soil surface sealing by the suspended solids accumulation (Lassabatere et  2010), silt and clay particles transported from the road to the infiltration basin, biofilm development by microbial growth on bottom of the basin and precipitation of calcium carbonate due to increase of pH caused by algal activity (Ma & Spalding, 1997).
When comparing the models for different values of T, it was noted that for different values of INF B, there was a linear trend between the V B values obtained by the models (Figure 2).precipitation depth of 93.3 mm.Thus, the precipition depth in São Gonçalo do Abaeté for the critical condition, i.e. that which results in maximum runoff, is about 64% greater than observed in João Pinheiro.
With regard to the differences between V B values obtained by the S M model, the fact that João Pinheiro presented higher V B values is due to the expressive effect of events overlapping throughout the series that was noted in this locality.An analysis of the historical precipitation series from the two sites showed that the average annual rainfall and the maximum rainfall event associated with a return period of 10 years are more critical to the design of the basin for the conditions of João Pinheiro than for São Gonçalo do Abaeté.
The average annual precipitation calculated for the municipality of João Pinheiro was 1164.5 mm and corresponds to a value of roughly 29% greater than the value calculated for São Gonçalo do Abaeté, which was only 900.9 mm.With respect to an isolated rainfall event associated with a return period of 10 years (allowing for the use of the precipitation series used in the S M model), the value obtained for the conditions in Joao Pinheiro was about 10% higher, i.e., while for this location the value was 113.9 mm and the corresponding value in São Gonçalo do Abaeté was 103.1 mm.This data indicates that from the V B values obtained (Figures 2 and 3), the effect of overlapping events is more pronounced when the rain volume and magnitude of these are greater, especially for low values of INF B .
The effect of overlapping runoff events can be better understood when analyzing Figure 5.In this figure are show on the average number (considering a data series of 50 years) of rain days per month and average number of days per month when it rainy while the basin still had rainwater stored from previous days.According to Figure 5A the average number of rainy days in João Pinheiro is more than the average number of rainy days in São Gonçalo do Abaeté (Figure 5B).The Figure 5 also shows that the average number of rainy days occurring when the basin still contained rainwater remaining from previous events is greater for João Pinheiro, for all INF B values.model proposed by Pruski et al. (1997), when the events overlap effect is higher.
3. For soils with high INF B , the return period effect in the V B values is higher in the model proposed by Pruski et al. (1997) than the model proposed by Silva (2009).

Figure 1 .
Figure 1.Precipitation profile as a double exponential function

Figure 2
Figure 2 shows a comparison between the V B for different values of INF B calculated by the model developed by Silva (2009) (S M ) and those calculated by the model developed by Pruski et al. (1997) (SSWB).Relations considering different values of T were obtained for each INF B studies with regard to precipitation conditions in the municipality of João Pinheiro.The lines shown in the figure relate the identity function and the linear regression fitted to the V B points calculated by the S M model versus the V B calculated by the SSWB model.In the S M model, the effect of overlapping runoff events on volumes stored during each day of the series is taken into account and, consequently, it influences the volume required for the infiltration basin.The SSWB model only takes into consideration one event associated with the determined return period without taking into account the fact that there are many events throughout the series where the runoff volume does not completely infiltrate the soil during a given day, and that the infiltration of volumes over a rainy season may represent a more critical condition for the design of the infiltration basin than the volume resulting from a single rainfall, as considered by the SSWB model.The comparison between the models for different INF B values shows that lower INF B values resulted in greater differences in the V B calculated by the two models.For a INF B of 2 mm h -1 , it was noted that the V B values obtained by the S M model estimated higher V B values, the magnitude of which was 2.65 times higher on average than the values obtained by the SSWB.For INF B of 5, 10, 15 and 20 mm h -1 , differences of 1.48, 1.21, 1.14 and 1.11 times greater were noted on average, respectively.This behavior is the result of a more accentuated effect of overlapping runoff events when INF B is lower, making the volumes stored in the basin and, consequently, the value of V B calculated by the S M model larger.Because The SSWB model does not consider this characteristic, the difference between the V B values calculated by the models increases.Studies relating the water infiltration rate into the soil at the bottom of the infiltration basin are still very scarce.However, it is anticipated that consideration of smaller values of this variable are more representative for the majority of the cases, even when dealing with infiltration basins constructed of soils with good infiltration capacity.This affirmation is based on the fact that there are processes which expressively interfere in the reduction of basin infiltration rates, such as: i) compaction of the soil at the base by tractor tires during construction such as illustrated byMiranda et al. (2009) and from the weight of the basin water(Ma & Spalding, 1997); and ii) soil surface sealing by the suspended solids accumulation(Lassabatere et

Figure 2 .
Figure 2. Volumes required for the infiltration basin calculated by the S M model, considering the INF B of 2 (A), 5 (B), 10 (C), 15 (D) and 20 mm h -1 (E) and by the SSWB model considering the return periods of 5, 10, 15, 18 and 20 years with the precipitation conditions in the municipality of João Pinheiro, MG For a INF B of 2 mm h -1 , the V B values obtained by the S M showed an increased difference in relation to those obtained by the SSWB model with an increase in T, represented by the angular coefficient of the equation greater than 1.With the increase in INF B the angular coefficients of the regression equations decrease, also for INF B values of 10, 15 and 20 mm h -1 less than 1, indicating the reduction of differences between the models with increases in T.The trend highlighted how the difference between the obtained values increases with the return period for lower INF B values due to the differences in procedures used to estimate V B .In the SSWB model, the value of V B is estimated considering a single rainfall associated with the return period in question, estimated by the equation of intensity, duration and frequency (Eq.3), whereas in the S M model the value of V B is calculated based on the application of the Gumbel distribution for annual series of maximum stored volumes.The V B value calculated by SSWB model does not change with the variation of INF B at the bottom of the basin. .In contrast, the S M model presents an expressive sensitivity to variation in INF B, which is more pronounced for lower values of this variable V

Figure 4 .Figure 5 .
Figure 4. Precipitation profiles corresponding to the maximum expected rainfall for a return period of 10 years in the municipalities of São Gonçalo do Abaeté and João Pinheiro

CONCLUSIONS 1 .
For soils with low water infiltration rate through the basin bottom soil (INF B ) the method proposed by Silva (2009) leads to higher required volumes values of infiltration basins (V B ), independently of location and return period considered.2.For soils with high INF B , the V B values calculated by the model developed bySilva (2009)  tends to be higher than the Average number of rainy days Months of the years A. B.