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Darcy's law: The French hydrologist Henri Darcy discovered that the velocity \(V\) of underground water is proportional to the magnitude of the slope \(S\) of the water table (see Figure 1.71). The constant of proportionality in this case is the permeability of the medium through which the water is flowing. This proportionality relationship is known as Darcy's law, and it is important in modern hydrology. a. Using \(K\) as the constant of proportionality, express Darcy's law as an equation. b. Sandstone has a permeability of about \(0.041 \mathrm{me}-\) ter per day. If an underground aquifer is seeping through sandstone, and if the water table drops \(0.03\) vertical meter for each horizontal meter, what is the velocity of the water flow? Be sure to use appropriate units for velocity and keep all digits. c. Sand has a permeability of about 41 meters per day. If the aquifer from part a were flowing through sand, what would be its velocity?

Step by step solution

01

Expressing Darcy's Law as an Equation

The velocity \( V \) of underground water flow according to Darcy's Law is related to the slope \( S \) by the following proportionality equation: \( V = K \cdot S \), where \( K \) is the constant of proportionality, also known as the permeability of the medium.

02

Calculating Velocity in Sandstone

Given that the permeability of sandstone \( K = 0.041 \) meters per day and the slope \( S = 0.03 \), we substitute these into Darcy's Law equation to get the velocity. Thus, \( V = 0.041 \times 0.03 = 0.00123 \). Therefore, the velocity of the water flow is \( 0.00123 \) meters per day.

03

Calculating Velocity in Sand

For sand, given that the permeability \( K = 41 \) meters per day and keeping the slope \( S = 0.03 \), substitute into Darcy's Law: \( V = 41 \times 0.03 = 1.23 \). Hence, the velocity of the water flow in sand is \( 1.23 \) meters per day.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hydrology

Hydrology is the scientific study of water in its various forms on and beneath the Earth's surface. It focuses on the distribution, movement, and properties of water within environmental systems. Hydrologists examine how water interacts with the surrounding landscape, including soil, vegetation, and geological formations like rocks and sediments. Their research is crucial for managing water resources, predicting flooding, and understanding the environmental impacts of human activities.

Understanding hydrology is essential for applying concepts like Darcy's Law, which governs the flow of underground water. By analyzing factors such as slope and permeability, hydrologists can predict water movement and assess groundwater availability. These insights are vital for various applications, including water resource management and environmental conservation.

Permeability

Permeability is a measure of how easily fluids can pass through a material. In the context of hydrogeology, it specifically refers to the ease with which water can move through pores in sediments or rock layers.

• High permeability means water can flow through a material quickly.
• Low permeability indicates obstacles to water flow, causing it to move slowly.

Materials like sand and sandstone vary greatly in their permeability values. Sand, with its larger pores, allows water to pass through efficiently (41 meters per day), whereas sandstone, being more compact, has a lower permeability (0.041 meters per day).

This concept is central to applying Darcy's Law, where permeability ( K ) acts as a constant of proportionality. By understanding permeability, one can infer how different materials affect water movement underground, helping inform decisions in water management and engineering projects.

Water Flow Velocity

Water flow velocity is the speed at which water travels through a medium, such as soil or rock. Determining this velocity is critical for understanding groundwater movement, which impacts both natural ecosystems and human activities.

According to Darcy's Law, water flow velocity ( V ) is calculated by multiplying the permeability ( K ) of the medium by the slope ( S ) of the water table.

• For sandstone, with low permeability, water flows at a slower pace (0.00123 meters per day).
• In sand, due to higher permeability, the flow is significantly faster (1.23 meters per day).

Understanding these velocities aids hydrogeologists in predicting how quickly water will transport through an area, which is crucial for managing aquifers and ensuring sustainable water supplies.

Proportionality

Proportionality in the context of Darcy's Law refers to the relationship between two quantities that change at the same rate. In groundwater hydrology, this concept is used to describe the relationship between water flow velocity and the slope of the water table, moderated by the permeability of the material through which the water is flowing.

This relationship is mathematicallagin expressed with the formula:
\[ V = K \cdot S \]
where - V  represents velocity,  K  is permeability, and  S  is the slope.

• If the slope increases, the velocity will increase proportionally, assuming a constant permeability.
• Similarly, changing the permeability will directly affect the velocity when the slope remains unchanged.

Proportionality enables accurate predictions of water flow under varying conditions. By understanding this concept, one can apply Darcy's Law to real-world scenarios, such as predicting the effects of changes in terrain or material properties on groundwater movement.