Introduction

Darcy's law is a fundamental principle in fluid mechanics that describes the flow of fluids through porous media. It states that the rate of flow is directly proportional to the hydraulic gradient and the cross-sectional area, and inversely proportional to the coefficient of permeability.

In this problem, we are given a bed of sands consisting of three horizontal layers of equal thickness. The permeability of the upper and lower layers is given as 1×10⁻² cm/sec, while the permeability of the middle layer is given as 1×10⁻¹⁰ cm/sec. We are required to find the ratio of the permeability of the bed in the horizontal direction to that in the vertical direction.

Understanding the Problem

To solve this problem, we need to understand the concept of permeability and its relationship with Darcy's law. Permeability is a measure of how easily a fluid can flow through a porous medium, such as sand. It depends on the properties of the medium, such as the size and shape of the particles, as well as the connectivity of the pores.

Darcy's law states that the rate of flow (Q) through a porous medium is given by the equation:

Q = -kA(dh/dl)

Where:
- Q is the flow rate
- k is the coefficient of permeability
- A is the cross-sectional area
- dh/dl is the hydraulic gradient

From this equation, we can see that the flow rate is directly proportional to the coefficient of permeability. Therefore, a higher value of k indicates a higher flow rate.

Solution

To find the ratio of the permeability in the horizontal direction to that in the vertical direction, we need to compare the flow rates in these two directions.

Let's assume the hydraulic gradient is the same for both directions. Since the layers have equal thickness, the cross-sectional area for both directions will also be the same. Therefore, the only factor that affects the flow rate is the coefficient of permeability (k).

Comparing Horizontal and Vertical Flow Rates

Let's consider the horizontal flow rate first. According to Darcy's law, the flow rate (Q) in the horizontal direction is given by:

Q_horizontal = -k_horizontal * A * dh/dl

Similarly, the flow rate in the vertical direction is given by:

Q_vertical = -k_vertical * A * dh/dl

Since the cross-sectional area and hydraulic gradient are the same for both directions, we can cancel them out and compare the flow rates directly:

Q_horizontal / Q_vertical = -k_horizontal / -k_vertical

Simplifying the equation, we get:

Q_horizontal / Q_vertical = k_horizontal / k_vertical

Calculating the Ratio

Now, let's substitute the given values of k_horizontal and k_vertical into the equation:

Q_horizontal / Q_vertical = (1×10⁻² cm/sec) / (1×10⁻¹⁰ cm/sec)

Dividing these values, we get:

Q_horizontal / Q_vertical = 1×10⁸

Therefore, the ratio of the permeability of the bed in the horizontal direction to that in the vertical direction is 1×10⁸.

Conclusion

In this problem, we used Darcy's law to compare the flow rates in the horizontal and vertical directions through a bed

A bed of sands consists of three horizontal
layers of equal thickness the value of Darcy’s
k for the upper and the lower layers is 1102
cm/sec and that for the middle layers is
1 1 10  cm/sec. The ratio of the permeability
of the bed in the horizontal direction to that
in vertical direction is