9. (Darcy's law, Henry Darcy found that when ground water flow is laminar and the aquifer is fully saturated, the flow rate of water depends on the pressure gradient and the permeability (or conductivity) of the groundwater zone. There are two forms of Darcy's law: (a) Q = -KA (P2-P₁) μL (b) Q = KA22 where Q is the flow rate (m³ s¹), K is the permeability (m²) of a porous medium, A is the surface are (m²), p2 and p₁ are the pressures (Pa) of heights h₂ and h₁ respectively, u is the viscosity (Pa s) of fluid, L is the length over which the pressure drop (p2-P₁) takes place, K is the hydraulic conductivity (m s¹), and Ah is the difference of heights h₂ and h₁. Given two additional relationships: K = K (i) Pg Ap = -pgAh (ii) where p is the density (kg m-³), g is gravitational acceleration (m s-2). 9.1. Prove that the two forms of Darcy's law are equivalent. 9.2. For both forms of Darcy's law, perform unit check on the right hand side of the equation to confirm that the resulted unit is the same as that of the left hand side.

9. (Darcy's law, Henry Darcy found that when ground water flow is laminar and the aquifer is fully saturated, the flow rate of water depends on the pressure gradient and the permeability (or conductivity) of the groundwater zone. There are two forms of Darcy's law: (a) Q = -KA (P2-P₁) μL (b) Q = KA22 where Q is the flow rate (m³ s¹), K is the permeability (m²) of a porous medium, A is the surface are (m²), p2 and p₁ are the pressures (Pa) of heights h₂ and h₁ respectively, u is the viscosity (Pa s) of fluid, L is the length over which the pressure drop (p2-P₁) takes place, K is the hydraulic conductivity (m s¹), and Ah is the difference of heights h₂ and h₁. Given two additional relationships: K = K (i) Pg Ap = -pgAh (ii) where p is the density (kg m-³), g is gravitational acceleration (m s-2). 9.1. Prove that the two forms of Darcy's law are equivalent. 9.2. For both forms of Darcy's law, perform unit check on the right hand side of the equation to confirm that the resulted unit is the same as that of the left hand side.

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Question

9. (Darcy's law,
Henry Darcy found that when ground water flow is
laminar and the aquifer is fully saturated, the flow rate of water depends on the
pressure gradient and the permeability (or conductivity) of the groundwater zone.
There are two forms of Darcy's law:
(a) Q = -KA (P2-P₁)
μL
(b) Q = KA22
where Q is the flow rate (m³ s¹), K is the permeability (m²) of a porous medium, A
is the surface are (m²), p2 and p₁ are the pressures (Pa) of heights h₂ and h₁
respectively, u is the viscosity (Pa s) of fluid, L is the length over which the pressure
drop (p2-P₁) takes place, K is the hydraulic conductivity (m s¹), and Ah is the
difference of heights h₂ and h₁.
Given two additional relationships:
K = K (i)
Pg
Ap = -pgAh (ii)
where p is the density (kg m-³), g is gravitational acceleration (m s-2).
9.1.
Prove that the two forms of Darcy's law are equivalent.
9.2. For both forms of Darcy's law, perform unit check on the right hand side
of the equation to confirm that the resulted unit is the same as that of the left
hand side.

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