GATE-2018 (24) :Wyllie – Time average equation

1. The hydraulic conductivity measure of how easily water moves through the porous media. It depends on the permeability of the material, the degree of saturation, and the density and viscosity of the fluid.

According to Darcy’s law, the specific discharge can be defined as volume flux per unit area

 

$\frac{Q}{A}=K \frac{dh}{dl}$------------------(1)

 

Q- Discharge (m³/s)

 

dh / dl – Hydraulic head gradient 

 

K – Hydraulic conductivity (m/s)

 

A- cross-sectional area (m²)

 

The Hydraulic gradient is the change in total head divided the distance over which the change occurs (dh / dl).

2.Hydraulic conductivity (K):

It is a second-order tensor and depends on properties of both the fluid (density ρ_f and viscosity μ [Pa s] ) and porous material (permeability k [m^2]).

$K= \frac{ρ_f g k}{μ}---(2)$

 

3. Discharge velocity

$v=K \frac{dh}{dl}---(3)$

Darcy’s law is found to be a good approximation at Reynolds number up to ~10, where the Reynolds number is defined as $\frac {ρ_f v d}{μ}$, where d is the pore dimension and v is the speed of the fluid itself (variably called the interstitial, pore, linear, or seepage velocity),

4. Seepage velocity is

$v_s=\frac{v}{ϕ}---(4)$

Where v is the discharge velocity, ϕ is porosity. 

5. Permeability in fluid mechanics and the earth sciences is a measure of the ability of a porous material to allow fluids to pass through it, SI unit for permeability is m². The permeability unit darcy $(~10^{-12} m^2 )$ is named in Henry Darcy honour.

 

6. The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress. For liquids, it corresponds to the informal concept of "thickness"; for example, honey has a much higher viscosity than water.

 

Both the physical unit of dynamic viscosity in SI units, the poiseuille (Pl), and cgs units, the poise (P), are named after Jean Léonard Marie Poiseuille. The poiseuille, which is rarely used, is equivalent to the pascal second (Pa·s), or (N·s)/m², or kg/(m·s).

 

7. Hydraulic head or piezometric head is a specific measurement of liquid pressure above a geodetic datum. It is usually measured as a liquid surface elevation, expressed in units of length, at the entrance (or bottom) of a piezometer.

 

8. Transmissivity is a measure of how much water can be transmitted horizontally, such as to a pumping well.

An aquifer may consist of n soil layers. The transmissivity for the horizontal flow of T_i in the i-th soil layer with a saturated thickness di and horizontal hydraulic conductivity K_i is:

$T_i= K_i d_i ---(5)$

Transmissivity is directly proportional to horizontal hydraulic conductivity K_i and thickness d_i. Expressing K_i in m/day and d_i in m, the transmissivity Ti is found in units m²/day.]

Reference:

1. http://geology.wwu.edu/rjmitch/L21_groundwater2.pdf

2. http://qu.edu.iq/el/pluginfile.php/59583/mod_resource/content/1/Lect.15_Viscosity.pdf

3.https://www.brighthubengineering.com/hydraulics-civil-engineering/58490-darcys-law-for-modeling-groundwater-flow/

4. Water and Earthquakes by Chi-Yuen Wang and Michael Manga