Question

A porous medium (shown schematically in the figure) has the following properties. 
\(\text{Length } L = 600 \, \text{m}, \text{ Width } W = 8 \, \text{m}, \text{ Height } h = 0.5 \, \text{m}, \text{ Permeability } k = 100 \, \text{mD}, \text{ Porosity } \phi = 15%.} \)

An incompressible fluid having a viscosity of 2 cP is flowing through a porous medium at the inlet and exit pressures of \(7 \times 10^6 \, \text{Pa and } 6 \times 10^6 \, \text{Pa, respectively.} \) The actual fluid velocity through the porous medium is \(\underline{\hspace{2cm}} \times 10^{-7} \, \text{m/s.}\)

Hint:

When calculating fluid velocity through porous media, use Darcy's law and ensure proper conversion of units.

Answer 1

Correct Answer: 5

We can use Darcy's law to calculate the fluid velocity: \[ v = \frac{k \cdot (P_{\text{inlet}} - P_{\text{exit}})}{\mu \cdot L} \] where:
- \( k = 100 \times 10^{-3} \, \text{m}^2 \),
- \( P_{\text{inlet}} = 7 \times 10^6 \, \text{Pa} \),
- \( P_{\text{exit}} = 6 \times 10^6 \, \text{Pa} \), - \( \mu = 2 \, \text{cP} = 2 \times 10^{-3} \, \text{Pa.s} \),
- \( L = 600 \, \text{m}. \)
Substitute the values into the equation: \[ v = \frac{(100 \times 10^{-3}) \cdot (7 \times 10^6 - 6 \times 10^6)}{2 \times 10^{-3} \cdot 600} = \frac{100 \times 10^{-3} \cdot 10^6}{1.2} = 8.33 \times 10^{-7} \, \text{m/s}. \] Thus, the fluid velocity is approximately \( 8.33 \times 10^{-7} \, \text{m/s} \).