This paper describes solutions to the hydraulic equations that govern flow in permeable friction courses (PFC). PFC is a layer of porous asphalt approximately 50 mm thick that is placed as an overlay on top of an existing conventional concrete or asphalt road surface to help control splash and hydroplaning, reduce noise, and enhance quality of storm water runoff. The primary objective of this manuscript is to present an analytical system of equations that can be used in design and analysis of PFC systems. The primary assumptions used in this analysis are that the flow can be modeled as one-dimensional, steady state Darcy-type flow and that slopes are sufficiently small so that the Dupuit-Forchheimer assumptions apply. Solutions are derived for cases where storm water drainage is confined to the PFC bed and for conditions where the PFC drainage capacity is exceeded and ponded sheet flow occurs across the pavement surface. The mathematical solutions provide the drainage characteristics (depth and residence time) as a function of rainfall intensity, PFC hydraulic conductivity, pavement slope, and maximum drainage path length.
 Porous asphalt is an alternative to traditional hot mix asphalt and is produced by eliminating the fine aggregate from the asphalt mix. A layer of porous asphalt approximately 50 mm thick can be placed as a sacrificial overlay on top of an existing conventional concrete or asphalt road surface. The overlay typically is referred to as permeable friction courses (PFC), or open graded friction courses (OGFC). Over time the sacrificial overlay is expected to degrade and be replaced at greater frequency than the underlying pavement. The void space in a PFC overlay layer generally is about 20% by volume [Anderson et al., 1998]. Specifications for PFC are available from the Texas Department of Transportation (Standard specifications for construction and maintenance of highways, streets, and bridges, item 342, pp. 312–329, 2004, available at http://www.dot.state.tx.us/business/specifications.htm) and other sources. Rain that falls on the friction course drains vertically through the porous layer to the original impervious road surface at which point the water drains along the boundary between the pavement types until the runoff emerges at the edge of the pavement.
 Porous asphalt overlays are used increasingly in Europe (Netherlands, Denmark, France, Italy [Ranieri, 2007b]) and by state transportation agencies in the U.S., including those in Arizona, Georgia, Florida, Texas, California, Oregon, and perhaps others. Acknowledged benefits include reduced splash and spray, better visibility, better traction, reduced hydroplaning and less noise [Berbee et al., 1999; Stotz and Krauth, 1994]. These pavements also may reduce the peak runoff velocity, as well as increase the lag time between rainfall and runoff, especially for smaller storm events.
 In addition to the safety benefits, research is also indicating that PFC may offer water quality benefits as well. Substantial reductions in the concentrations of pollutants commonly observed in highway runoff have been documented by several researchers in the U.S. and abroad [Barrett and Shaw, 2007; Berbee et al., 1999; Pagotto et al., 2000]. PFC might be expected to reduce the generation of pollutants, retain a portion of generated pollutants within the porous matrix, and impede the transport of pollutants to the edge of the pavement. It has been reported that the concentrations of selected constituents in highway runoff were affected by the number of vehicles passing the site during a storm event [Irish et al., 1998]. These constituents included oil/grease, copper, and lead. Spray generated from tires was assumed to wash pollutants from the engine compartments and bottoms of vehicles. It is reasonable to expect that the amount of material washed off vehicles while driving in the rain will be reduced since PFC reduces splash and spray. This reduction in the amount of material washed from vehicles is expected to decrease the concentrations of these pollutants in the runoff generated from roads paved with PFC.
 The porous structure of PFC also may act as a filter of the storm water. Runoff enters the pores in the overlay surface and is diverted toward the shoulder by the underlying conventional pavement. Pollutants in the runoff can be filtered out as the water flows through the pores, especially suspended solids and other pollutants associated with particles. Filtering occurs when pollutants become attached to the PFC matrix by straining, collision, and other processes. Material that accumulates in the pore spaces of PFC is difficult to transport and may be trapped permanently. Generally, water velocities within the pore spaces of the PFC are low and likely could only transport the smallest material. An exception could be beneath the tire lanes under saturated PFC conditions, where the surge caused by vehicle tires could locally initiate movement of deposited material. In contrast, on the surface of a conventionally paved road, splashing created by tires moving through standing water easily can transport even larger particulate matter rapidly to the edge of pavement.
Figure 1 shows a cross-section picture from a PFC core from a roadway near Austin, Texas. The core was taken from the middle of the outside traveled lane and has a diameter of 15 cm. The core was sliced in half vertically and impregnated with fluorescent epoxy to highlight the pores. About 1.5 cm of the underlying pavement is visible at the base of the section. The large typical pore size is evident, as well as the lack of fine-grain material. Because of the nature of placement, large-scale lateral heterogeneity in initial porosity and hydraulic conductivity are not anticipated. However, over time, deposition of suspended solids and other constituents in storm water runoff are expected to decrease both hydraulic characteristics, and could result in pavement internal characteristics that exhibit large-scale heterogeneity associated with the natural drainage pattern. Such characteristics have not been documented and are one focus of ongoing research efforts.
 One issue with the use of PFC is that at higher rainfall intensities, the entire volume of runoff cannot be contained within the 50 mm thick porous layer, causing sheet flow on the surface of the road. The sheet flow generally occurs in the outer lanes or in longitudinal sags (regions of local depression) in the roadway. This can present an unexpected danger to drivers who abruptly enter standing water after driving on relatively dry pavement. In addition, the water quality benefits may also be reduced under these circumstances. Interestingly, there are few specifications in the U.S. regarding appropriate design to maintain water within the pavement structure for a given storm intensity. Maintaining the runoff within the pavement may be accomplished by either controlling the thickness of the PFC or providing under drains to reduce the water depth. Potential use and design of under drains is discussed later in this manuscript.
Jackson and Ragan  were among the first researchers to address the issue of flow in porous pavements. Their research was focused on flow within the pavement base course rather than in the pavement itself with the goal of analyzing the effect of subdrain spacing on discharge rates (a subdrain is a drain placed within or beneath the pavement system to enhance storm water recovery). This was accomplished using numerical solutions to the Boussinesq equation for the situation where the impermeable base had a zero slope.
Ranieri  was the first researcher who published predictions of water depths within PFC itself. He developed a runoff model for PFC that links the hydraulic conductivity of PFC with the geometric characteristics of the road section and rainfall intensity. He validated his theoretical model in the laboratory with a device that simulated rainfall on porous pavements and, at the same time, measured the depth of water over the impervious layer during the seepage motion. On the basis of the experimental data, a chart for porous pavement design was presented. Ranieri [2007a] expands on this earlier work and provides some minor corrections to his original nomographs. In both papers, Ranieri relies on a numerical solution to solve the governing differential equations for water flow within PFC.
Tan et al.  also used a numerical modeling approach to analyze water depths within PFC. They used a three-dimensional finite element program to study the effects of cross and longitudinal slopes on the drainage performance of the porous asphalt surface course. A family of thickness requirement graphs, based on design rainfall, thickness of surface course layer, width of pavement, and longitudinal and cross slopes, was prepared for use by designers.
 The objectives of this current paper are to improve on this earlier modeling by Ranieri [2002, 2007a] and Tan et al.  by providing analytical solutions for the governing equations under a variety of boundary and rainfall conditions.
2. Mathematical Analysis
 The primary objective of this manuscript is to develop a set of model equations that can be used for design and analysis of PFC systems. Design equations are based on simplifying assumptions that include steady state flow, constant and uniform rainfall intensity, and homogeneous system properties. For urban drainage, the design rainfall intensity is determined from the size of the drainage area (on the basis of its time of concentration, which for this application is the time to steady state drainage for constant rainfall intensity) and the design frequency or return period, using the intensity-duration-frequency relationship of the area [Brown et al., 2001; Chow et al., 1988]. Flow within PFC is spatially variable porous media flow down a slope with an impermeable base and recharge from infiltrating rainfall. Among the features that distinguish PFC flow from groundwater flow are the typically larger hydraulic conductivity and the small thickness of the seepage layer. The primary assumptions in drainage hydraulics of PFC are that the flow can be modeled as one-dimensional, steady state Darcy-type flow, and that slopes are sufficiently small so that the Dupuit-Forchheimer assumptions apply [Charbeneau, 2000]. Figure 2 shows a schematic cross section along a flow path that is determined by the roadway cross slope and longitudinal slope. The pavement has slope s = tan(α), length L from the crown to the end of the overlay, and PFC thickness bp. Slope s generally ranges from 0.02 to 0.06, length L on the order of 5–15 meters, and thickness bp 3–8 cm. The infiltration rate r [L/T] is constant and uniform, and the PFC is underlain by impermeable pavement. The hydraulic head is H(x) while the drainage layer thickness within the pavement is h(x). The flow per unit width is U(x) [L2/T]. If the capacity of the PFC is exceeded, then drainage occurs as a combination of PFC flow plus overland sheet flow.
 The flow per unit width is the integral of the Darcy velocity over the saturated thickness and is proportional to the slope of the drainage water surface (water table)
In equation (1)K is the PFC hydraulic conductivity. Figure 2 shows that the hydraulic head and saturated thickness are related through
Equation (3) implies that the flow per unit width may be written
 For one-dimensional steady state flow conditions with recharge at a rate r, the principle of continuity gives
From Figure 2, the origin of the x axis (x = xo = 0) is the roadway crown and acts as a no-flow boundary. Thus equation (5) may be written
Equation (7) describes the change in drainage flow thickness through PFC. A couple of points are of interest.
 1. The origin is automatically a no-flow boundary since for x = 0, this equation requires dh/dx = s, and from equation (3), this implies dH/dx = 0.
 2. Since s > 0, the flow thickness initially increases (dh/dx > 0).
 3. At the location of maximum depth within the flow profile dh/dx = 0, and the following relationship must hold among the variables
The location of maximum depth changes with flow conditions.
 4. Methods for estimating the upstream thickness h(0) are not well defined by problem conditions, and specification of a downstream boundary condition h(L) = hL is more useful.
Equation (7) is a homogeneous ordinary differential equation [Boyce and DiPrima, 1969, p. 43] that may be solved using variable transformation followed by separation of variables. Use the following variable transformation
Equation (10) states that the variable η (=h/x) depends only on distance x with parameters s and relative recharge rate R = r/K. The relative magnitude of these parameters is important. The variables in equation (10) may be separated to give
Formally the problem now becomes one of quadrature. Integration gives
The last term in equation (12) determines the form of the solution, and it depends on the sign of the quantity
 First consider the case for Φ < 0 which corresponds to low rainfall intensity. For this case equation (12) gives
The constant is evaluated by applying a boundary condition with known thickness at a specified location. One may use h(L) = hL to find (with ηL = hL/L)
 Next consider the special case for Φ = 0 so that R = s2/4. For this case equation (11) becomes
With h(L) = hL the result is
 The last case is for Φ > 0. For this case equation (12) gives
With h(L) = hL the result is
3. Analysis and Discussion
 The three solutions for cases with (1) Φ < 0, (2) Φ = 0, and (3) Φ > 0 are given by equations (15), (17), and (19) for boundary condition h(L) = hL. These cases are discussed separately.
The notation used in equation (20) is meant to specify that h and x are variables for the function F1, while s, R, L, and hL are function parameters. The function F1 defines the relationship h(x) implicitly, and in order to extract this relationship one must invert the function using a selected numerical technique. If the appropriate root of F1 = 0 can be bracketed, then the method of bisection is easy to implement [Press et al., 1992]. Investigation of function behavior shows that the critical parameters are the roots of the quadratic equation appearing in the numerator of the first term on the left side. The symbol h has been used to denote the flow depth within the PFC; the special case for roots of the quadratic equation is denoted by h′, so that values of h′ are found as solutions of the quadratic equation
In particular, for the roots evaluated at x = L one finds
The magnitude of hL compared with hL′(−) and hL′(+) determines the range that must be searched for the appropriate root of F1 = 0. These are given explicitly as follows.
In the special case, of minor interest, where hL is equal to one of the roots given by equation (22), then the drainage profile is a straight line.
Figure 3 shows the solution from equation (20) for three different downstream depth values corresponding to the three different ranges specified by equation (23). The dashed lines correspond to the “root” equations. For case 1 (Φ < 0), the water level profiles do not cross the root equations. Also, if hL ≤ (L/2) (s + ), then h(0) = 0.
3.2. Case 2: Φ = 0
 Case 2 (Φ = 0) represents a special (limited) condition with r = K s2/4. In essence, the solution corresponds to the limit Φ → 0 from case 1, so that the roots corresponding to equation (21) or (22) collapse to a double root with hL = sL/2. The middle range in equation (23) does not exist. The solution gives a drainage profile that is similar to either the lower or upper profiles shown in Figure 3. The root equation gives h′(x) = sx/2. The drainage profile is found from roots of the function
3.3. Case 3: Φ > 0
 Case 3 (Φ > 0) corresponds to higher rainfall rates, is of greater interest, and appears to be fundamentally different from cases 1 and 2 in terms of mathematical behavior of the solution. The flow profile is found from solution of equation (19), written as follows.
The first term in the function F3 is the same as in the function F1, and thus the roots of the quadratic are also given by equation (21). The fundamental difference is that in the function F3, Φ > 0 so the roots are complex conjugates. There are no roots corresponding to singularities of the function on the real axis. A simple method is used for bracketing the root of F3 = 0. Start with hi = Rx/s, which is the solution corresponding to dh/dx = 0 in equation (7). The lower limit is then found by the sequence hi+1 = 0.9 * hi until the condition F3(hi+1,…) < 0 is met. Similarly, the sequence hi+1 = hi + hL is evaluated until the condition F3(hi+1,…) > 0 is met. The method of bisection is then used to find the solution to F3 = 0 → h(x).
Figure 4 shows drainage profiles corresponding to rainfall rates r = 0.25, 0.5, 1.0, and 2.5 cm/h with other conditions specified in the caption. For these parameters, the critical rainfall intensity corresponding to Φ = 0 is r = Ks2/4 = 0.0001 cm/s (0.36 cm/h), so that the lower profile corresponds to Φ < 0 while the upper three profiles have Φ > 0. Summary results from this example are shown in Table 1. As anticipated, the maximum drainage depth increases with rainfall intensity, as does the specific storage (volume per unit width) which is calculated from the integral of the drainage depth over the profile and an assumed porosity n = 0.2. What may not be expected is that the mean residence time (specific volume divided by rainfall recharge rate to the pavement) decreases with increasing rainfall rate. Discussion of the equilibrium time (time of concentration) is presented below.
 One cannot directly evaluate the functions F1, F2, or F3 at x = 0. For F1 and F2 one knows that if hL ≤ hL′(+), then h(0) = 0. For other cases the h(0) value is found by extrapolating from the two adjacent points x1 and x2, and fitting a quadratic using the following algorithm. The general quadratic is
This is consistent with the upper boundary being a no-flow boundary (dH/dx = 0).
3.5. Upslope Drainage
 Depending on conditions near the upstream edge of a PFC, the boundary may not act as a no-flow boundary, but rather as a drainage boundary as shown in Figure 5. In this case there must be a drainage divide located at some station along the pavement surface. The coordinate system is located at this station, so that the boundary condition dH/dx(0) = 0 applies for both upstream and downstream drainage. The flow depth at the divide is h(0) = h0.
 The solution for upstream drainage is essentially the same as that already considered. To simplify notation let ξ = −x. Then the flow per unit width in the upstream direction is given by U = −K h dH/dξ. Also, H(ξ) = h(ξ) + ξs. Combining these with continuity (U = rξ) gives
Comparing equation (30) with equation (7) shows that the only change is in the sign of the slope (s is still considered a positive number). The solution development remains the same, and the critical parameter for solution behavior is still the magnitude of Φ. Practically, only the solution for Φ > 0 is of interest, and this solution is still given by equation (25) with the sign of s changed. The interesting question is how to match the upstream and downstream solutions. One must have h(0) the same from both sides, where x = 0 is the location of the divide. For a given slope, rainfall intensity, etc., one solves both problems with selected L1 and L2 = L − L1 values, and compare the depth h(0) from the two solutions. The value of L1 is adjusted until the solutions match. An example profile with r = 1.0 cm/h, hL1 = 1 cm, and hL2 = 0.5 cm (other conditions are the same as in Figure 4) is shown in Figure 6. For this example, L1 = 469.4 cm and L2 = 30.6 cm. The mean residence time for downstream drainage is 0.64 h while that for upstream drainage is 0.09 h. The maximum flow depth within the PFC is 3.94 cm.
3.6. Combining Overland (Sheet) and PFC Flow
 The discussion so far has considered PFC as if it had an infinite thickness. However, because the thickness is finite (and generally small), one will often find conditions where storm water drains from the roadway through a combination of PFC drainage plus overland sheet flow upon the roadway surface. Because the depth of overland flow is very small (even compared with the PFC thickness bp), the maximum hydraulic gradient for the saturated PFC is limited to
Once overland flow occurs, continuity gives
In equation (32) the PFC transmissivity is defined by T = Kbp. For the sheet flow per unit width one may use either the Darcy-Weisbach equation or Manning's equation. The sheet flow Reynolds number is generally very small, and the friction factor can be expressed as f = KDW/Re = KDWν/Usheet, where KDW is the Darcy-Weisbach coefficient, Re = Usheet/ν is the Reynolds number, and ν is the kinematic viscosity. With laminar flow on a smooth surface, KDW = 24 [Chow, 1959]. However, because the roadway surface is rough, and because of the impacts from rainfall, KDW is much larger for PFC applications. With the Darcy-Weisbach equation the thickness of sheet flow hs(x) is calculated using
Care must be taken in applying equation (33) because there are sections of roadway that are saturated but equation (33) does not apply because the hydraulic gradient in equation (31) is less than s (for these sections, rx < Ts).
Figure 7 shows an example of a roadway section with L = 10 m and s = 0.03. The PFC has thickness bp = 5 cm and K = 1 cm/s. The rainfall intensity r = 2.0 cm/h. The flow divide occurs approximately 70 cm from the upstream end of the roadway section. Three different regions are noted on the flow profile. In region 1 the PFC remains unsaturated (h < bp). The hydraulic gradient in this region is small and the PFC does not become saturated until a station 130 cm from the upstream end. In region 2 the PFC is saturated, but there is no flow across the roadway surface. Within this region the hydraulic gradient I increases in the downstream direction according to
In equation (34), x is measured from the flow divide. At station 340 cm from the upstream end the hydraulic gradient reaches s, and sheet flow begins. Equation (33) applies within this region 3. For this example the flow per unit width at the downstream end is U = 0.518 cm2/s, while the flow per unit width from the PFC is Upfc = 0.150 cm2/s. Most of the drainage is associated with sheet flow. The sheet flow depth at the downstream end of the pavement is hs = 0.24 cm with Re = 37.
3.7. Estimation of Drainage Equilibrium Time (Time of Concentration)
 The analyses presented in this manuscript have focused on steady state flow, which is conventional for engineering design. A constant rainfall rate is estimated on the basis of the time of concentration and the intensity-duration-frequency curve for the location [American Society of Civil Engineers, 1992; Brown et al., 2001]. The time of concentration is often estimated using the kinematic wave method, where the estimate depends on the rainfall intensity, so that the procedure can be iterative. There are no methods for estimating the time of concentration of drainage segments including porous pavement.
 An estimate of the minimum possible equilibrium drainage time corresponds to the mean storm water residence time that is calculated as the ratio of the steady state storage volume (from the drainage profile) to the discharge (rainfall intensity multiplied by the drainage path length). Izzard  assumes that the equilibrium time for surface drainage is twice the residence time, as calculated above. A more fundamental approach is based on analysis of the continuity equation for transient drainage in PFC.
 Combining equation (5) with the one-dimensional transient continuity equation gives
In equation (35) the leading term uses the porosity n in place of the specific yield since estimates of the specific retention of PFC are not available, and it is expected that r ≪ K so that the specific retention is small. The initial condition is
Formulation of boundary conditions is not considered since no attempt at solving equation (35) directly is made. For early time and within the interior of the domain, the depth gradient vanishes and equation (35) simplifies to the following form:
Equation (37) is a linear, first-order hyperbolic equation which may be solved using the method of characteristics. The wave celerity (c) is constant and equal to
Along each characteristic within this region the depth is equal to
An approximate estimate for the equilibrium drainage time (time of concentration) may be taken as the time duration required for the hyperbolic wave depth to reach the maximum drainage depth within the profile. With equations (38) and (39) this gives
In equation (40), Tc = time of concentration, hmax = maximum drainage depth within the PFC profile, and xmax = location of the maximum depth. Equation (40) is consistent with equation (8). Estimates of the equilibrium drainage time using equation (40) are presented in Table 1 for comparison with the mean residence time values.
4. Example Application
 Storm water quality and runoff hydrology and hydraulics from PFC are monitored at a station along Loop 360 in Austin, Texas. Loop 360 is a four lane divided highway. At the monitoring station the two-lane roadway width is 10.4 m and the longitudinal and cross slope are 0.023 and 0.02, respectively. The PFC has bp = 5 cm. It is of interest to evaluate the reduction in direct runoff from pavement surfaces due to placement of PFC, and the required spacing between drains beneath the PFC that could be placed across the roadway to meet certain objectives. These issues are discussed in this example.
 Lateral drains for PFC drainage enhancement are slots cut into the base pavement that can be filled with coarse material or metal drains before placement of the PFC. A schematic layout is shown in Figure 8. The placement of the drains is shown perpendicular to the roadway alignment, though this is not necessary. The drainage path is “down slope” on the basis of the longitudinal and cross slope, sL and sc. For this location s = 0.0305, and the drainage path length is L = 15.85 m. It is assumed that K = 1 cm/s (no measurements have been made for this location; the estimated value is based on literature values for porous media gravel material [0.1–10 cm/s] [Charbeneau, 2000], and on estimates for PFC from European experience [1–2 cm/s] [Ranieri, 2007b]).
Figure 9 shows the cumulative rainfall intensity curve for Austin, Texas area based on 15-min rainfall data from 1987 to 2004 collected as part of the City of Austin's Flood Early Warning System. The curve gives the fraction of the total rainfall duration over that period that occurred at intensities less than the given value.
 The model equations were used to estimate the maximum rainfall intensity that would not result in overland flow from the pavement surface under steady state conditions. For a drainage path length of 15.85 m with slope s = 0.0305, rainfall of intensity less than 0.425 cm/h will not cause overland flow. With this rainfall intensity and the data shown in Figure 9, it is estimated that just over 75 percent (0.758) of the rainfall duration is at intensities that will not cause direct pavement surface runoff. Such estimates are approximate because they are based on a steady state flow model, and natural rainfall events exhibit significant variability and drainage flow is transient.
 One option for design of drain spacing is to control water quality (and/or splash from traffic caused by sheet flow on roadways) for ninety percent of the annual rainfall events, which for this case, would give a design rainfall intensity of 0.88 cm/h. The model equations were used to investigate the drainage profile for rainfall intensity r = 0.88 cm/h and slope s = 0.0305. The drainage path length was adjusted to allow for both upstream and downstream drainage with hL1 = 1 cm and hL2 = 0.5 cm, and the maximum drainage depth was limited to h < bp = 5 cm. The results are shown in Figure 10. The maximum drainage path depth is L = 850 cm and results in only downstream drainage, which corresponds to a longitudinal distance between drains Ld = 850 (0.023/0.0305) = 641 cm. The Reynolds number (based on Darcy velocity and effective grain diameter from Hazen's formula [Fetter, 2001]) ranges from 0.15 to 0.76. Also shown in Figure 10 is the flow profile for conditions where the drainage path length is increased to L = 900 cm (with the same rainfall intensity). Sheet flow occurs on the pavement surface with a maximum depth hs = 1.36 mm at the downstream pavement edge. However, nearly seventy percent (0.693) of the drainage occurs through the PFC.
5. Summary and Conclusions
 This paper describes solutions to the hydraulic equations that govern flow in permeable friction courses (PFC). The primary assumptions used in this analysis are that the flow can be modeled as one-dimensional, steady state Darcy-type flow, and that slopes are sufficiently small so that the Dupuit-Forchheimer assumptions apply. Solutions are derived for three distinct cases that represent low rainfall intensity, high intensity, and a point of singularity that divides the two previous cases. At low rainfall intensities, three different water surface profiles are predicted on the basis of the selected downstream boundary condition (water depth at the edge of pavement). At higher rainfall rates the mathematical solution is substantially different. The maximum drainage depth increases with rainfall, as does the specific storage; however, the mean residence time (specific volume divided by rainfall recharge rate to the pavement) decreases with increasing rainfall rate. Depending on conditions near the upstream end of a PFC, this boundary may not act as a no-flow boundary, but rather as a drainage boundary. In this case there must be a drainage divide located at some station along the pavement surface, and a search algorithm may be used to locate the drainage divide and apportion down slope and upslope drainage.
 These solutions have considered PFC as if it had an infinite thickness; however, because the thickness is finite, one will often find conditions where storm water drains from the roadway through a combination of PFC drainage plus overland sheet flow upon the roadway surface. A solution is also presented for this case. Finally, a method for estimating the time of concentration from the steady state flow profiles is presented.
 For design rainfall intensity, the mathematical solutions provide estimates of the maximum drainage depth within PFC, and the maximum ponded depth of sheet flow on the roadway surface when the PFC drainage capacity is exceeded. These variables depend on the rainfall intensity, PFC hydraulic conductivity, roadway slope, and maximum drainage path length. Necessary spacing between under drains for controlling surface flow on pavement surfaces can also be calculated using the model equations.
 This work is supported by the Texas Department of Transportation under Project 0-5220. The authors would particularly like to thank the TxDOT Project Director, Gary Lantrip, for his interest and support for research related to the use of porous pavements on highways.