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CHAPTER 3
SELECTION AND DEVELOPMENT OF MODELS FOR PAVDRN
In order to develop guidelines for Improving the drainage of water from pavement
surfaces, it was necessary to select a mode! for predicting the depth of sheet flow on pavement
surfaces. Since the maximum allowable water fun thickness is the min~murn flow depth at
which hydropl~nmg is Initiated, it was also necessary to select a model that can be used to
predict the onset of hydroplaning as a function of water fun thickness. A literature search was
conducted to identify and evaluate water film thickness and hydroplaning models ~ the
existing literature. From the literature search, a one~mensional, steadystate form of the
kinematic wave equation was selected for the prediction of the depth of sheet flow. Additional
development of the mode! was also undertaken as cart of this protect.
, . ~
The models arid predictive equations that are contained ~ PAVDRN are described In
this section. A more comprehensive discussion of surface flow models and the development of
values for Manmng's n can be found In Appendices B and C and the references cited therein.
WATER lILM THICKNESS MODEL
Before discussing the water film thickness model, the terms water fihn thickness
(WFT), mean texture depth (MTD), and water depth, y, must be clearly be defined (see figure
I). Water depth is defined as the total thickness of the water film on the surface of the
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pavement. It is the sum of the MTD and Me WFT. The water below the MTD is trapped in
the macrotexture and is considered immobile and does not contribute to the flow depth, y. The
WFT is the thickness of Me water fihn above the tops of the asperities on the pavement
surface. The total depth of flow, y, is the thickness of the WFT plus the MTD.
There are two general types of models that can be used to detenn~ne water film
thickness: (~) an analytical mode} and (2) an empirical model. Bow types have been developed
and used for this purpose.
Empirical Mode!
A one~mensional, steadystate model was developed by Gallaway f2) as presented in
equations 3 and 4:
WFT = 0.003726 L0519 i0.s62 MTD0.l2s
s0364
and
where
 MTD
y = WET + MTD
(3)
(4)
WFT = Water film thickness (in) (1 In 25.4mm)
L Plane length of Towpath (ft) (1 It = 305 mm)
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i = Rainfall intensity (in/h)
MID = Mean texture depth (in)
Pavement slope (mlm)
Depth of water contributing to flow (in)
(1 inlh = 25.4 mmJh)
(1 in = 25.4 mm)
(1 in = 25.4 mm)
(1 in = 25.4 mm)
In equations 3 and 4, the flow path, L, is presumed to be over a simple, planar surface.
Gallaway's equation is based on an extensive set of water depth data for a variety of pavement
types. The equation, however, does not contain a variable, such as M~nning's n, to describe
the hydraulic resistance of the pavement surface. The equation was developed by combining
data from pavements with different types of surfaces in a single regression analysis. Since
Gallaway's equation does not include a term for the hydraulic resistance of the pavement
surface, does not differentiate between different surfaces, and is empirical, the equation was
not considered for use ~ this project.
OneDimensional Analytical Models
Two analytical models were considered during the project. The first is a fully dynamic
model based on the principles of conservation of mass and momentum. The model has been
used by many investigators to represent the equations of state for shallow wave motion,
equations 5 and 6. Equations 5 and 6 represent spatially varied, unsteady flow in one
· ~
dlmenslon.
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dh flu Sh
+ h + u = ~  f = ~`5'
at fix fix
where
Depth of flow
u  Spatially averaged velocities (x  directions
.
1
I
Rainfall rate over the domain
Infiltration rate
Rainfall rate adjusted for infiltration
em + u ~ + g ~ = g(Sox Sfx) h h (6,
where u, h, i, and f are the same as described In equation 5 and
g  Acceleration due to gravity (32.2 ft/s 2 or 9.~1 m/s 2)
Sax
Sex
vr
Slope of the flow path In the xdirection
Slope of the energy grade Ime In the Erection
Terminal rainfall velocity
Ox  Angle of rainfall input with respect to the xaxis
The last term on the righthand side of equation 6 represents the momentum due to the
angle of incidence of rainfall velocity In the Indirection. The term By is often taken to be
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negligible; Pus costly can be assumed equal to zero, and the rightmost tenn is dropped from
the equation.
Under steadystate conditions where the friction slope, Sf,,
, is equal to the slope of the
flow plane, SOx, equations 5 and 6 simplify to a form known as the kinematic wave equations as
presented in equations 7 through 9:
hen = ~ teq
teq =
where
(7)
( I) ( ~ I)
u = ahm~1
(8)
(9)
he = Equilibrium water flow depth (ft)(1 It = 305 man)
i = Excess rainfall rate (ft3/s/ft2)(1 ft3/s/ft2 = 305 mm3/s/mTn2)
to = Time to equilibrium (s)
L = Plane length (ft)(1 It = 305 mm)
a = Friction loss coefficient
m = Friction loss exponent
u = Velocity of flow (ft/s)(1 ft/s 305 mm/s)
Equations 7 through 9 represent the kinematic wave solution for steadystate, one~'mensional
flow.
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TwoDimensional Analytical Models
An analytical two~mensional flow mode! written by Zhang and Cundy (33) was
evaluated as part of this study. Their mode] is representative of two~mensional models and
illustrates the difficulties that are typical of twodimensional models. The primary limitation of
their mode} is the lack of stability and convergence In the solutions. Stability and convergence
are relatively easy to obtain with sets of linear equations, but difficult to acquire win ache non
linear sets of equations implicit in the Zhang and Cundy model. The computational stability
and convergence for nonlinear sets of equations are usually obtained by assuming linearity and
using the linear solution as a conservative first approximation of the nonlinear case. As a
consequence, extremely small tune steps are used that result In dramatically Increasing
execution times as ache geometric complexity of the flow surfaces augments. For a multilane
pavement, this may result in hundreds of thousands of iterations to reach equilibrium
conditions. Therefore, no further use was made of two~mensional models, which are
discussed in Appendix B.
Mode} of Choice
The one~imensional kinematic model, represented by equations 7 through 9, was
chosen as the preferred model for predicting water film thickness. This mode} is based on
theories and includes a variable, M~nmng's n, that accounts for the hydraulic effect of surface
roughness on water depths. The one~mensional, steadystate form of me model was used
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developing the surface drainage guidelines and the PAVDRN program. The selection of the
one~mensional flow equation was determined by its computational stability and efficiency of
solution.
The flow domain, i.e. flow paths and channels, for a one~unensional model must be
defined by the user or must be established by an analysis of the topography of the surface prior
to applying equations that determine flow, depth, and/or velocity. For example, PAVDRN
analyzes the topography of a section and determines the longest flow path length before
applying the kinematic wave equation to determine water film thickness at points along the
flow path. The longest flow path is determined from geometric conditions as the path from the
point where Me water falls on the pavement to the pout where it exits the surface of the
pavement.
where
n
L
1
WFr =
The following equation is used In PAVDRN to calculate the water fiDn thickness:
 MID
36.1 Sash
Mann~ng's roughness coefficient
Drainage path length (in)
Rainfall rate (in/h)
51
(10)
(1 in = 25.4 mm)
(1 in/in = 25.4 mm/in)
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S = Slope of drainage path (mm/mm)
MID = Mean texture depth (in) (1 ~ = 25.4 mm)
Values of water film thickness calculated according to equation 10 are presented In figure 8.
Subsurface Flow Mode}
For porous asphalt surfaces, flow within the porous asphalt layer parallel to the
pavement surface must be considered. Two options were explored for the subsurface flow
model. The first was based upon ache consideration of threedunensional, fully saturated flow,
represented by equation ~ ~ . The second, a one~unensional model, is discussed In the
following.
where
_ (Kxx She
fix fix
h _
Kss =
_ (Kyy ah) + ~ (Kzz ah) WS ah (11)
BY Liz Liz s St
Piezometric head or potential
Hydraulic conductivity of the porous material in the direction of the
. · .
prmc~pa1 axes
W = Sources arid sinks of water
Ss = Storage coefficient or specific storage of the porous material
52
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 Portland cement concrete,
MTD = 0.91 mm, Manning's n = 0.031,
 ~ Dense graded asphalt concrete,
MTD = 0.91 mm, Manning's n  0.0327,
Open graded asphalt concrete,
MTD = 1.5 mm, Manning~s n = 0.0355
100
90
80
E

~n
to
t' 50

e~
~ 40
a)

~ 30
70
60
20
10
O i
'.~k
MY ,
ad/
it,'
a. ~
Slope of flow plane: 2%,
Rainfall intensity: 40 mm/in
1

0 5 10 15 20 25
Drainage path length, m
Figure 8. Water film thickness versus distance along flow path for several pavement surfaces
as calculated using PAVDRN.
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The solution of equation ~ ~ for the quantity of flow can Include the full dynamic nature
of the effects of hydroplaning In the porous asphalt drainage layer. However, the
computational effort required to arrive at a solution for either a two or three~mensional
mode! is unjustified when the desired solution is the surface water film thickness. In this case
a one~vmensional mode] is sufficient. In some cases, the multidimensional models do not
converge and require operator intervention In selecting different boundary conditions or, in the
case of transient analysis, varying time steps and computational grids.
The other option for flow through porous asphalt, a steadystate, one~mensional
model, has several advantages. Like the one~mensional surface How model, it is
unconditionally stable. As a consequence, little or no operator intervention is necessary to
arrive at a solution. Appropriate material properties can be quantified with a reasonable level
of effort. Therefore, the one~nnensional surface flow model, which was modified as
described in the following, was chosen to compute water film thicknesses on porous
pavements.
For dete~n~ng water film thickness on porous pavement sections it is necessary to
modify equation 10 to account for the infiltration rate. The addition of a term to account for
infiltration rate results In equation 12. This form of the equation was used In PAVDRN to
estimate the water film thickness for porous surfaces:
DIVEST . n L T
L
36.1
3.6
 MTD
54
(12)
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where
and
I
.
1
f
f
=
MANNING'S N
n = Mann~ng's roughness coefficient
(i f) Excess rainfall rate (in/h)
Rainfall rate
fiItracion rate or permeability of pavement
(iffy)
(1 in/in = 25.4 mm/in)
(1 inA1 = 25.4 mm/in)
(1 inlh = 25.4 mm/in)
In order to predict the water fihn thickness that occurs on a pavement surface during
sheet flow, as presented In equations 10 and 12, Maurung's n must be known. The hydraulic
roughness of a surface, Mann~g's n, can be expressed In terms of an evalue as used above
and defined by Manning (34J. ManT~ng's evalue is surfacespecific, requiring different
expressions for different surfaces. During the course of this project, the hydraulic resistance
of three different types of pavement surfaces was determined:
.
Portland cement concrete
· Porous asphalt
.
Densegraded asphalt concrete
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experunental data and the regression of the data with various forms of equations 13 through
15, were developed and used In PAVDRN to determine Manning's n:
I. Portland cement concrete surfaces:
n 0 319 (NR ~ 1000) (16)
NOSO2 (NR ~ 50O) (17)
2. Densegraded asphalt concrete :
n = 0.0823 N~0~74 (~)
3. Porous asphalt concrete:
1.490 S 0 306
N 0.424 (19)
where
NR = q
58
(20)
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and
NR  Reynold's number
q = Quantity of flow per unit width (m31slm)
v = Kinematic viscosity of water
Equations 16 through 19 are presented graphically In ISgures 9 through 12.
HYDROPLANING SPEED MODEL
The hydroplaning model selected for the study is based upon the work of Gallaway and
his colleagues (4) and as further developed by others (37, 38). On the basis of the work
reported by these authors, for water hum thicknesses less than 2.4 mm (0.095 ill), the
hydroplaning speed is determined by:
HPS = 26.04 WFI0259
where
(21)
HPS = Hydroplaning speed (mi/h)
WFr  Water film thickness (in)
(1 mi/h = 1.61 1an/h)
(1 in 25.4 mm)
For water film thicknesses greater than or equal to 2.4 mm (0.095 in), the hydroplaning speed
IS:
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.
0.12
0.10
0.08
to
·= 0.06
0.04
0.02
0.00
500 ~ NR ~ 1 000,
Water temperature = ~ SAC.
Ago.
lb

~ .,

_,
~ .,
R ain fall intensity = ~ mcn/h
R ainfall intensity = ~ O mm/in
 R ainfall intensity = 20 mcn/h
\
\

b
R ainfall intesity = 40 mm/in
R ainfall intensity = 60 mm/in

10 30 50 70 90 110
Length of flo w path, m
Figure 9. Manning's n versus length of flow path for various rainfall rates, Portland cement
concrete, 500 < NR < 19000.
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0.12
0.10
0.08
0.06
w
0.04
0.02
0.00
10
l
NR ~ 50O,
Water temperature = ~ SAC.
\
\ ~
\
\ ~

·


Rainfall intensity = 5 mm~h
Rainfall intensity = 10 mm~h
      Rainfall intensity = 20 mm~h
Rainfall intensity = 40 mm~h
 Rainfall intensity = 60 mm~h

30
50
70
Length of Towpath, m
90
110
Figure 10. Mannir g's n versus length of flow path for various rainfall rates, Portland cement
concrete, NR < 500
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0.06
0.05
0.04
a)
·' 0.03
ce
0.02
0.01
0.00

.


Water temperature = 1 0°C.



Rainfall intensity = ~ mm~h
Rainfall intensity = JO mm/in
Rainfall intensity = 20 mm/in
Rainfall intensity = 40 mm/in
Rainfall intensity = 60 mm/in
~ ~ ~ ~ ~a c I ~
_
_ __ ~ ~
1 0 30 50 70 90 1 1 0
Length of Towpath, m
Figure ~ ~ . Mar mr~g's n versus length of flow path for various rainfall rates, densegraded
asphalt concrete.
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0.25
0.20
0.15
v,

._
~5
0.10
0.05
0.00
Water temperature = 1 0°C
 Rainfall intensity= ~ mm/in 
\  Rainfall intensity = 10 mm/in
\ ~ Rainfall intensity = 20 mmfh
\  Rainfall intensity = 40 mm/in
\ \
\ ~ Rainfall inte laity = 60 mmlh
., ____~~
, ~ . , ,,,, _ _ __ _
_, ~  . _,    ~ ·      _ · 
._. _,   _ . _ _._ _._ _._
.
1 0 30 50 70 90 1 1 0
Length of flow path, m
Figure 12. Manning's n versus length of flow path for various rainfall rates, porous asphalt
concrete.
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HPS = 3.09 A
where A is the greater of the values calculated using equations 23 and 24:
10.409 ~ 3 507
FIT 0 06
or
(22)
28.952 7 817
WET 0.06

(23)
MID 0.14
(24)
The model predicts the onset of hydroplaning on the basis of the water film thickness
where water hen thickness is the thickness of the wafer film above the mean tenure depth, as
presented In figure 1. The results of equations 21 through 24 are shown graphically ~ figure
13. The mean texture depth can be determined from sand patch or micro profile measurements.
Suggested levels of macrotexture are presented In the guidelines for design situations where
acme measurements are not available. Although the hydroplaning prediction models
(equations 2 through 24) are empirical, they represent the state of the art: The development of
rational equations was considered too ambitious an undertaking given the complexity of We
problem, the resources required, and the extent of the data that would be needed for a rational
model.
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CD ~ O
· . .
to to
11 11 11
~ Q Q
C~e ~ C  e C~e
t I I I
\ee (  e c~et
\e C~] (~ (e
~ ~ ~ cr cr
t ~e lLi l]J
E ~ ~ ~ ~
· . . ·
C ~C~e C\e N
V ~ ~ ~ ~
r~
e I
1 1
e ~
I
I :
/
1
~ e
1 1
e
e
1: 1
a e
e e
1: 1
~ e
e e
~ : 1
4' · e
de. ~
1:1
e e
lli!
ee ~ ~
11: 1
lee ~
.1. e
11: 1
e.
.1e .
ICI
t. ~
.
Ir1
te
e. .
1~1
~e ~
Itl
1e
· ~ ~
lll
t ~
. ~ e
lll
~ e
e er .
1:11
C~te
~n
a
~
Y
C)
· 

· 
~ 
e_
C~
_
C~
O O O
~( ~C`~`e
O O O O O O
~ 0 ~~ ~ w)
4/~ Spasds Ou!ueldo!ep/H
Figure 13. Hydroplaning speed versus water film thickness.
65
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Unfortunately, experimental test data needed to verify the prediction of water depths on
the surface of porous asphalt pavement was not obtained during this study and such data are
not available In the literature. In order to obtain reliable experimental data, very high rainfall
rates are needed and the experimental section must be well isolated so that the entire flow is
contained within the boundaries of the test sections. In simple terms, very high, uniform
rainfall rates are needed on a "leakproof" test section. These conditions could not be satisfied
with the available equipment, neither In the laboratory nor In the Users. Capturing a natural
event was considered but the idea was discarded because of the coordination effort and costs
associated with capturing such an event. Therefore, it was not possible to validate equations
21 Trough 24 for porous asphalt surfaces.
Rainfall Intensity
The AASHTO highway design guides Include an equation for relating rainfall intensity,
vehicle speed, and maximum allowable sight distance as follows:
i  [80,000/(SVVi)] 147
where
i Rainfall intensity (in/h)
sv Sight distance (ft)
V; Vehicle velocity (mi/h)
66
(25)
(1 in/in25.4 mm/in)
(1 ft _ 0.305 mm)
(1 m~/h 1.61 l~n/h)
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This relationship is depicted In figure 14.
Although it appears In equation 25 that intensity is a function of sight distance, sight
distance is a function of Intensity. As Intensity increases, sight distance decreases. Likewise,
vehicle velocity decreases with increased intensity and decreasing sight distance.
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   V ehicIe speed = 60 knn/h
 V chicle speed = 80 krr`/h
V chicle speed = 100 k m/h
Vehicle speed = 120 krn/h
Vehicle speed =160 k m/h
80
s
~ 60

~n
0


ce
._
ce
40
20
100
200 300
Sight distance Sv, m
400 500
Figure 14. Rainfall ~ntemit~r versus sight distance for various vehicle speeds.
68