Question

For a system of air conditioning water is extracted from a layer of \(10 \mathrm{~m}\) thickness, having a hydraulic conductivity of \(1 \mathrm{~m} / \mathrm{d}\). The discharge is \(50 \mathrm{~m}^{3} / \mathrm{d}\). At a distance of \(100 \mathrm{~m}\) the water is being injected into the same layer by a recharge well. What is the influence on the groundwater head in the

Short answer

Answer: The influence on the groundwater head in the system is a decrease of 5 meters.

Step-by-step solution

Understand the given information and variables

In this exercise, we are given: - Thickness of the layer (h) = 10 m - Hydraulic conductivity (K) = 1 m/d - Discharge (Q) = 50 m³/d - Distance between wells (L) = 100 m

Apply Darcy's Law

Darcy's Law states that for a given porous media, the flow rate (Q) is proportional to the hydraulic gradient and the coefficient of hydraulic conductivity: $$Q = -K * A * \frac{\Delta h}{L}$$ Where: - Q is the flow rate (m³/d) - K is the hydraulic conductivity (m/d) - A is the cross-sectional area normal to the direction of flow (m²) - Δh is the change in hydraulic head between two points (m) - L is the distance between the two points (m)

Determine the cross-sectional area

We know the thickness of the layer (h) and the distance between the wells (L), so we can determine the cross-sectional area (A) as: $$A = h * L$$ Now, plugging in the values given in the problem: $$A = 10 \mathrm{~m} * 100 \mathrm{~m} = 1000 \mathrm{~m}^2$$

Solve for the change in hydraulic head

Now, using Darcy's Law, we can solve for the change in hydraulic head (Δh) as follows: $$\Delta h = - \frac{Q * L}{K * A}$$ Substituting the given values: $$\Delta h = - \frac{50 \mathrm{~m}^3 \mathrm{~/d} * 100 \mathrm{~m}}{1 \mathrm{~m/d} * 1000 \mathrm{~m}^2}$$ Calculating the result: $$\Delta h = - \frac{5000 \mathrm{~m}^3 \mathrm{~/d}}{1000 \mathrm{~m}^2} = - 5 \mathrm{~m}$$

Interpret the result

The influence on the groundwater head in the system is a decrease of 5 meters. This means that in the area between the extraction well and the recharge well, the groundwater head will drop by 5 meters as a result of the air conditioning system's operation.

Key concepts

Hydraulic Conductivity

Hydraulic conductivity is a fundamental property of soil and rock that describes the ease with which water can move through pore spaces or fractures. It's often denoted by the symbol 'K' and measured in units such as meters per day (m/d). In the context of Darcy's Law, hydraulic conductivity is critical because it factors into how fast water can flow through the ground. The higher the hydraulic conductivity, the more permeable the material is, meaning water can pass through it more quickly. For instance, gravel has a much higher conductivity compared to clay. Understanding 'K' helps in predicting the movement of groundwater and contaminants, designing drainage systems, and in our exercise, it was used to calculate the effect of pumping and injection wells on groundwater head.

Groundwater Head

Groundwater head essentially represents the elevation to which water will naturally rise in a well; it is a measure of the potential energy available to drive the flow of groundwater through porous materials. This height is influenced by several factors, including the hydraulic conductivity of the material and the water flow rate. In soil mechanics, groundwater head is a crucial concept as it helps understand and predict how water will behave under natural conditions and when subjected to external influences like pumping. It's also used to evaluate the condition and risks of water logging or drought in an area. In the exercise, the change in groundwater head was calculated to quantify the impact of water withdrawal and recharge in the system.

Flow Rate Calculation

Flow rate in the context of groundwater and soil mechanics is the volume of water passing through a cross-sectional area over a specific time period, typically expressed in cubic meters per day (m³/d). Darcy's Law allows us to calculate this flow rate by considering the hydraulic conductivity, the cross-sectional area perpendicular to the flow, and the hydraulic gradient. The calculation of flow rate is essential in various fields such as civil engineering, hydrology, and environmental science for the design of water supply systems, management of water resources, and assessment of pollution spread. Our exercise required calculating the change in groundwater head by using the flow rate, exemplifying the law's utility in practical scenarios.

Porous Media Flow

Porous media flow refers to the movement of fluids—like water—through a material that contains pores or void spaces. Soils and rocks are typical examples of porous media. Understanding the characteristics of the flow through such materials is indispensable in several fields like hydrogeology, petroleum engineering, and soil mechanics. Flow through porous media is governed by factors like pore size distribution, the viscosity of the fluid, and the material's hydraulic conductivity. The direction and rate of flow are significant when managing groundwater resources, predicting the transport of contaminants, or designing subsurface structures. As demonstrated in our exercise, Darcy's Law is a fundamental principle to describe and quantify the fluid flow within these porous systems.