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Porosity and Flow Drag Parameters
The option to specify a Porosity and Flow Drag Layer was added in HEC-RAS 6.6. This option allows the user to specify a spatially variable porosity and flow drag parameters (see section on Creating a Porosity and Flow Drag Layer). The parameters are assumed to be constant over a thickness from the bed elevation. Porosity is defined as the volume of voids over the total volume of the medium. Therefore, porosity is a physical property that is relatively simple to measure. Typical values are available for various materials are available from literature and have a relatively narrow range. The flow drag parameters are much more difficult to estimate. For one, the linear and quadratic flow drag coefficients used in HEC-RAS are not variables commonly found in literature and must be computed form other variables, as described below, or calibrated. The linear and quadratic flow drag coefficients represent the viscous and inertial energy losses, respectively. The linear flow drag coefficient is useful for simulating losses through various bed materials, rocks, and rubble mound structures, while the quadratic or inertial flow drag coefficient is most useful for simulating energy losses through flow obstructions, coarse bed materials, and rubble mound structures. The sections below provide some guidance on typical values for the porosity and flow drag parameters for simulating flow through porous media, flow obstructions such as buildings, and vegetation.
Porous Media
The coefficients for flow through porous media can be calculated as (Nield and Bejan 2013)
\begin{equation} a = \dfrac{\nu \phi}{k_v} = \dfrac{g \phi}{K_v} \\ b = \dfrac{c_F \phi^2} {\sqrt{k_v} } = \dfrac{\phi^2} {k_i } \end{equation} |
where
a : linear (viscous) flow drag coefficient [1/T]
b : quadratic (inertial) flow drag coefficient [1/L]
\nu : water kinematic viscosity [L2/T]
g : gravity [L/T2]
k_v : viscous permeability [L2]
k_i : inertial permeability [L]
c_F : dimensionless form-drag coefficient or Forchheimer coefficient [-]
In which the following relationship between the viscous permeability and conductivity is utilized: k_v =\frac{\nu K_v}{g}.
Zhou et al. (2019) found an empirical formula to relate the viscous and inertial permeability coefficients
k_i = 10^{10} k_v^{3/2} |
Hazen (1911) developed the following empirical formula for the viscous hydraulic conductivity
\begin{equation} K_v = C d_{10}^2 \end{equation} |
where C is an empirical coefficient between 0 and 1.5, d_{10} is the 10th percentile diameter in mm, and K_v is in cm/s.
The viscous permeability and Forcheimer coefficient can be estimated with the empirical formulas of Ghimire (2009)
\begin{equation} k_v = \dfrac{\phi^2 d^2}{150(1 - \phi)^2} \end{equation} |
\begin{equation} c_{F} = \dfrac{1.75}{\sqrt{150 \phi^3}} \end{equation} |
The Forcheimer coefficient ranges between 0.55 and 10.
The following table below shows common porous material porosities and viscous permeabilities.
Table 1. Common porous material properties (data from Scheidegger 1974 and Bejan and Lage 1991).
Material | Porosity | Viscous Permeability (cm2) |
---|---|---|
Black Slate Powder | 0.57 - 0.66 | 4.9 x 10-10 - 1.2 x 10-9 |
Brick | 0.12 - 0.34 | 4.8 x 10-11 - 2.2 x 10-9 |
Limestone | 0.04 - 0.10 | 2 x 10-11 - 4.5 x 10-10 |
Sand | 0.37 - 0.50 | 2 x 10-7 - 1.8 x 10-6 |
Soil | 0.43 - 0.54 | 2.9 x 10-9 - 1.4 x 10-7 |
Buildings
The momentum loss due to subgrid building can be represented with the following expression (Guinot et al. 2017)
\begin{equation} a = 0 \\ b = C_{D,B} a_{F,B} \end{equation} |
where
C_{D,B} : building drag coefficient [-]
a_{F,B} : building frontal area by unit volume [1/L]
Rubble Mound Structures
Permeable rubble mound structures, such as spur dikes and longitudinal groins, are commonly used in rivers as river training structures. Three options for computing the flow drag coefficients for rubble mound structures are:
Sidiropoulou et al. (2006)
\begin{equation} a = 0.003 g D^{-1.5} \phi^{0.06} \\ b = 0.194 g D^{-1.265} \phi^{-1.14} \end{equation} |
Kadlec and Night (1998)
\begin{equation} a = \frac{255 \nu (1 - \phi)}{\phi^{3.7} D^2} \\ b = \frac{2(1 - \phi)}{\phi^{3} D} \end{equation} |
Ward (1964)
\begin{equation} a = \frac{ 360 \nu }{ D^2 } \\ b = \frac{ 10.44 }{ D } \end{equation} |
where
D : Rock diameter [L]
\phi_{rm} : Porosity [-]
Vegetation
In reality, vegetation flow drag is a complex function of vegetation density, stem diameter, vegetation flexibility, frontal area, etc. As a simple example here emergent vegetation may be treated as an array of rigid circular stems with a constant diameter. In this case, the vegetation flow drag coefficients may be modeled as (Wu 2007)
1) | \begin{equation} a = 0 \\ b = \dfrac{1}{2} C_{D,v} N_v \alpha_v D_v \end{equation} |
where
C_{D,v}: vegetation drag coefficient [-]
N_v: vegetation density defined as the number of vegetation elements per unit area in the horizontal plane [1/L2]
\alpha_v: vegetation shape factor [-]
D_v: vegetation diameter [L]
The vegetation shape factor, \alpha_v, is equal to 1 for rigid cylinders but can vary as a function of the vegetation shape including the resulting shape from bending.