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According to Darcy’s law, the flow of liquid in a particular medium is governed by certain principles. Using an aquifer as an example, one can better understand Darcy’s law. In geology, an aquifer is a body of permeable rock, incoherent materials, or deposits. With aquifers, groundwater is filtered. Groundwater is forced through pores and deposits in the aquifers which helps to filter out contaminants. As we explore further what the Darcy law is, it is essential to understand what permeability means. To put it simply, permeability is the property that allows fluid to flow through a porous substance.
What is Darcy’s Law?
Darcy’s law states that the rate at which a fluid moves through a permeable medium (like groundwater moving through an aquifer) is directly related to the difference between vertical elevations at different points in the medium, and indirectly directly related to the distance between those points in the medium. Henry Darcy first tested this law by passing water through a column filled with sand. Water flowed through the sand beds as it passed. Based on the results of these experiments Darcy’s law equation was formulated.
According to Darcy’s law, the flow of fluid between two points is directly related to pressure differences, distances, and interconnectedness of flows within rocks/sediments.
Darcy’s law equation is expressed as follows:
Q = -kA (Pb – Pa)/μL
Q is the rate at which the liquid flows
k is the instantaneous discharge rate
A is the cross-sectional area of a column
(Pb – Pa) is the pressure drop over a given distance
μ is the fluid’s viscosity
L is the length over which the pressure tends to drop
Where is Darcy’s Law applied?
Darcy’s law equation is primarily used in hydrogeology, and it forms the basis of many hydrogeological equations. It is the formula that states how water flows through a porous (permeable) substance. By applying Darcy’s law formula, groundwater quantities can also be determined. In aquifers under the ground, Darcy’s law is applied to study the groundwater movement.
Other applications of Darcy’s law are:
- Equations for topographic groundwater flow in saturated confined and unconfined aquifers are derived from Darcy’s law. Now let’s look at the confined and unconfined flow formulas.
Confined Flow:
The formula for confined flow in a constant thickness aquifer can be expressed as:
∂2h/∂x² + ∂²h/∂y² + ∂²h/∂z² = S/T * ∂h/∂t or ∇²h = S/T * ∂h/∂t
Unconfined flow:
The formula for unconfined flow in a constant thickness aquifer can be expressed as:
∂²h/∂x² + ∂²h/∂y² + ∂²h/∂z² = 0 or ∇²h = 0
- Aquifers that are confined and unconfined both use the Darcy law to determine the steady-state flow.
Confined aquifer:
A confined aquifer can be characterized by the quantity of groundwater flow per unit area (q) as follows:
q = Kb dh/dx
K is the hydraulic conductivity
b is the thickness
dh/dx is the hydraulic gradient
X is the direction of the hydraulic gradient
Unconfined aquifer:
Groundwater discharge per unit of aquifer width (q) can be determined by Darcy’s law when the aquifer is unconfined:
q = -Kh dh/dx
K is the hydraulic conductivity
h is the saturated thickness
dh/dx is the hydraulic gradient
x is the direction of the hydraulic gradient
Drawbacks of Darcy’s law
While Darcy’s law formula is a basic formula for calculating fluid flow through porous bodies, it does have some drawbacks. Some drawbacks of Darcy’ law are:
- Darcy’s law does not apply to all types of soil, but rather to some types only. These include clay, sandy soil, and silt soil. Darcy’s law does not apply to permeable surfaces, such as gravel and cobbles.
- Neither the saturated nor the unsaturated flow of fluid follows Darcy’s law.
- It is only applicable when the flow of fluid is laminar. Laminar flow is denoted by Re which stands for Reynold’s number and it is expressed as:
Re = puL/μ
p is the density of the fluid
u flow speed
L is the characteristics linear dimension
μ is the dynamic viscosity of the fluid
Having discussed the Reynolds number, we can now understand that Darcy’s law is only valid when the Reynolds number is less than or equal to 1.
Re ≤ 1
Conclusions
The basis for the calculation of groundwater quantity is Darcy’s law. It is also a fundamental element of many hydrogeological equations, as we have seen in this article. Consequently, these equations are used to determine the rate at which fluids flow through a permeable substance and are very useful for a wide variety of operations that involve the flow of water. Even though Darcy’s law is only applicable when a fluid is laminar, its applications can’t be neglected in other fields. Additionally, this law can be used to determine the amount of linear and radiating groundwater flow.
