Calculating Permeability (K) Using Darcy's Law
This problem involves calculating the permeability ($K$) of a soil formation using Darcy's Law and the concept of seepage velocity. We are given the porosity ($n$), the hydraulic gradient ($i$), and the seepage velocity ($v_s$).
Key Formulas and Relationships
- Darcy's Law relates discharge velocity ($v_d$) to hydraulic gradient ($i$) and permeability ($K$): $v_d = K * i$.
- Seepage velocity ($v_s$) is related to discharge velocity ($v_d$) and porosity ($n$) by: $v_s = v_d / n$.
Step-by-Step Calculation
- Relate Seepage Velocity to Permeability: Substitute the expression for $v_d$ from Darcy's Law into the seepage velocity equation:
$v_s = (K * i) / n$
- Rearrange for Permeability (K): Solve the equation for $K$:
$K = (v_s * n) / i$
- Input Given Values: Plug in the provided values:
- Seepage velocity, $v_s$ = 0.0025 m/s
- Porosity, $n$ = 20% = 0.20
- Hydraulic gradient, $i$ = 2% = 0.02
- Calculate K:
$K = (0.0025 m/s * 0.20) / 0.02$
$K = 0.0005 m/s / 0.02$
$K = 0.025 m/s$
The calculated permeability is 0.025 m/s. This value falls within the range specified in the answer choice (between 0.024 and 0.026 m/s).
Important Questions from Darcy Law
- A homogeneous isotropic confined aquifer of uniform thickness $30 \text{ m}$ has hydraulic conductivity of $5 \text{ m/day}$ and porosity of $0.3$. There are two observation wells X and Y along a radial line from a fully penetrating pumping well at $100 \text{ m}$ and $200 \text{ m}$ distance, respectively. The well is pumped at a uniform rate to produce steady drawdown of $5 \text{ m}$ at X and $3 \text{ m}$ at Y. If a non-reactive pollutant enters at the observation well Y, then the time taken by the pollutant (under advection) to reach the observation well X is ________ days. (rounded off to two decimal places)
- The kinematic viscosity of glycerin and kerosene are 1.2 times and 0.95 times of that of water, respectively. Glycerin and kerosene flow through two identical porous media having same hydraulic gradient. Assuming Darcy's law is valid for the porous media, the ratio of flow rate of kerosene to that of glycerin is
Two reservoirs having a difference of 20 m in their water surface elevations are connected through a confined aquifer (thickness = 5 m, length = 2 km, hydraulic conductivity $K = 3 \times 10^{-3}$ m/s, and porosity $\eta = 0.3$) as shown in the following figure (drawn not to scale).
If the Reservoir 1 is contaminated by a pollutant then the time (in days, rounded off to one decimal place) taken by the pollutant to reach to the Reservoir 2 under advection will be ________Two fully penetrating wells are dug 1.4 km apart in a homogenous confined aquifer. The difference in their piezometric levels is 4.0 m. The groundwater flow is steady and unidirectional. If the aquifer has a hydraulic conductivity of $3.5\ m\cdot day^{-1}$ and effective porosity of 40%, the time taken for water to move from one well to the other in days is _____________. [in integer]
