Introduction

The HydraulicPipe consists in the one-dimensional (1D) discretization of the generalized Darcy's law for fluid flow problems as shown below, in terms of pore pressure:

\begin{eqnarray*}\frac{d}{d\ell}\left(\frac{K}{\gamma}\frac{dp}{d\ell}\right) + q = 0 \end{eqnarray*}

such that:

\( K \) is conduit conductivity;
\( \ell \) is longitudinal coordinates of the conduit element;
\( p \) is the pore pressure;
\( \gamma \) is the specific weight of the fluid;

The conduit conductivity is modeled by the Hagen-Pouseille law and Darcy-Weisbach equation for laminar ( Reynolds number \(Re <= 2200 \)) and for turbulent ( \(Re > 2200 \)) conditions, respectively. A simple expression for the flow rate in conduits with turbulent flow has been proposed by Swamee and Jain (1986) which is based on the Darcy-Weisbach and the Colebrook equations. Both conductivities for laminar an turbulent flow are expressed as follows:

\begin{eqnarray*}K_{laminar} = \frac{\gamma d^2}{32\mu}\end{eqnarray*}

\begin{eqnarray*}K_{turbulent} = \sqrt{\frac{8gd}{i_h}}\log{\left(\frac{w}{3.7d}+\frac{2.51\nu}{d\sqrt{2gdi_h}}\right)}\end{eqnarray*}

where:

\( d \) is the diameter of the conduit;
\( \mu \) is the dynamic viscosity;
\( \nu \) is the kinematic viscosity;
\( g \) is the gravitational acceleration;
\( w \) is the wall roughness of the conduit;
\( i_h \) is the hydraulic gradient in the conduit;

Properties

The hydraulic pipe requires the following properties:

Property	Description	Type	Def. Unit	required
Dp	Pipe diameter	Scalar	m	Yes
Ufw	Dynamic fluid viscosity	Scalar	kPa*s	Yes
gw	Fluid specific weight	Scalar	kN/m3	Yes
wr	Wall roughness	Scalar	m	Yes
rhow	Water density	Scalar	kg/m3	Yes

Example:

PropertySet
{
  id          = 'MatProp',
  typeName    = 'GemaPropertySet',
  description = 'Hydraulic properties',
  properties  = {    
        {id = 'Dp', description = 'Pipe diameter',  unit = 'm'},
        {id = 'Ufw', description = 'Dynamic fluid viscosity',  unit = 'kPa*s'},  
        {id = 'gw', description = 'Fluid specific weight',  unit = 'kN/m3'},  
        {id = 'wr', description = 'Wall roughness',  unit = 'm'},  
        {id = 'rhow', description = 'Water density',  unit = 'kg/m3'},  
        {id = 'material', description = 'Hydraulic material type', constMap = constants.HydroFemPhysics.materialModels},
        
  },
  values = {
        {Dp = 352.6e-3, Ufw = 1.005709706501052e-06, gw = 9.983, wr = 3.0e-6, rhow = 998.3,  material= 'pipeFlow'},
  }
}

In terms to use the pipe in the hydraulic physics, the typename in the configurations of PhysicalMethod have to be set as 'HydroFemPhysics.Pipe'.

Example:

PhysicalMethod {
  id       = 'PipeFlow',
  typeName = 'HydroFemPhysics.Pipe',
  type     = 'fem',
  mesh = 'mesh',
  materials = 'pipeFlow',
  boundaryConditions = {'bc1'}, 
  ruleSet = 1,
}

Material attributes

During the flow analysis using the pipe elements, the following internal Gauss attributes are stored:

Attribute	Description	Type	Required	Def. Unit	History
FR	Flow regime*	boolean	Yes	–	Yes
Qf	Flow in the pipe element	scalar	Yes	m3/s	Yes
Re	Reynolds number	scalar	Yes	–	Yes

Units

The Hydraulic Fem physics uses internally the international system of units for its calculations, as can be seen above by the default units for properties and attributes. The whole unit system can be replaced through the unitSystem physics attribute, as described at the gemaFemProcessCommonFemPhysicsOptions page.

When making the substitution, the units on the following table should be replaced by the desired units, forming a coherent unit system. The unit system of the material should be also replaced.

local newUnitSystem = {
  time     = 's',
  coord    = 'm',
  Dp       = 'm',
  Ufw      = 'kPa*s',
  wr       = 'm',
  gw       = 'kN/m3',
  rhow     = 'kg/m3'
  P        = 'kPa',
  v        = 'm/s',
  qw       = 'm3/s',
  vFm      = 'm3/s',
  gr       = 'm/s2',
  inflow   = 'm3/s/m2',
}

References

1.Swamee, P. K. and Jain, A. K. (1976). Explicit equations for pipe-flow problems, Journal of the Hydraulic Division, 102, pp. 657 \( -\) 664.

2.Huyakorn, P. S., Lester, B. H. and Faust, C. R. (1983). Finite element techniques for modeling groundwater flow in fractured aquifers, Water Resources Research, 19(4), pp. 1019 \( -\) 1035.

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