MOHID Land Equations
Porous media
Is not much different from the flow in a pipe….
Flow in a pipe:
1
 p  z 
U 2 f  
2
x
Laminar flow (Poiseuille flow):
f 
16
1
16
 16

UD UD
Re

1
1
16
 p  z 
U 2 f  U 2

2
2
UD
x
 D   p  z 
U   
 8  x
 p  z 
U  k
x
Hydrology
Water Flux:
Φ   κθ px z
Η2Ο i
i
Why does k depend on the soil moisture? Is it because the higher is the soil moisture the larger
is the diameter of the porous filled with water, i.e. the diameter or the larger porous where
water is flowing?
Rate of accumulation (Richards Equation):


 
p  z  
 κθ


ΦΗ2Ο i 
t
xi
xi 
xi 


Pressure in unsaturated flow:
Is a function of the water content (pF curve):
Curves calculated using the van Genuchten Formula:
http://en.wikipedia.org/wiki/Water_retention_curve.
Other models: Mualem; Brooks and Corey.
Hydraulic conductivity (http://en.wikipedia.org/wiki/Hydraulic_conductivity):
It is proportional to the inverse of the friction coefficient introduced in Fluid Mechanics/
Hydraulics. It increases with the pore diameter and consequently it is higher on sandy soils and
lower in clay soils.
http://www.fao.org/docrep/w4367e/w4367e0m.htm
Saturated flow:
The pressure is given by:
p
  g 1  w / k s 
z
And the saturated hydraulic conductivity (ks) depends and oil properties only and not of the
water content, because all pores are saturated.
Solutes in the water
c j
 c  s

c




 s0  si 
t
t
x j
x j x j
  D   
   
vj 
v

7
 3
2
j

Where:
c is the concentration in the dissolved phase
s is the concentration in the adsorbed phase
 is the soil density
 is soil water moisture,
D is the molecular diffusivity,
 is the pore tortuosity,
 is the porosity,
 is the dispersivity.
Adsorption/desorption depends on the soil ion exchange capacity and can be described using
several models
River Network
Q
Q
z  s  b
v
 gA


t
x
x H H
A
z
Q
L

 ql
t
t
x
Where is L the channel width, A is the cross section, qlis the lateral flow per unit of length, Q is the river
discharge, v is the river velocity τ is the shear stress (at the surface and bottom) and z is the
free surface level.