We explain what Darcy’s law is, its equations, applications, limitations and we propose an exercise to solve
What is Darcy’s law?
The Darcy ‘s law is applicable mathematical relationship to the flow of fluid permeable or porous media, for example, water draining in the sand.
As the fluid advances through the porous medium, its hydrostatic pressure varies, specifically it is higher at the points closest to the source and lower at the points closest to the drain. In this way, the concept of hydraulic gradient appears, a physical quantity that will be denoted by the letter I.
Moreover, the porous medium is characterized by a quantity called hydraulic conductivity K . There is clearly a relationship between porosity, determined by K, the hydraulic gradient I, and the flow rate per unit cross-sectional area q.
The relationship between them was discovered by the French hydraulic engineer Henry Darcy (1803-1858), who was in charge of the water supply of his hometown: Dijon.
Darcy’s law equations
Darcy’s law shows the relationship between various physical quantities that describe flow through a porous medium. It indicates that the flow rate of water Q that moves through a certain porous medium is directly proportional to the cross section through which A and the hydraulic gradient I :
Q ∝ A ∙ I
The constant of proportionality is the permeability K of the porous medium, also called hydraulic conductivity . In this way, Darcy’s law is presented as:
Q = K ∙ A ∙ I
Differential form of Darcy’s law
Darcy’s equation can be expressed as a differential relationship between the flow velocity at each point and the local hydraulic gradient:
Since the hydraulic gradient is a negative quantity when calculated in the flow direction, it is then necessary to multiply by the negative of the hydraulic conductivity to obtain the average flow velocity q, in each cross section.
Flow rate, hydraulic gradient and permeability
1.- Flow Q
Flow is defined as the volume of water that circulates through a certain cross-sectional area to the direction of flow, per unit of time:
Q = ΔV / Δt
In the International System of SI Units, flow is measured in cubic meters per second, but it is often also expressed in liters per minute or liters per second.
The flow rate per unit area q, which is the ratio of the flow rate Q to the cross-sectional area, is often required:
q = Q / A
In SI, q is expressed in m / s, which is why q represents the average velocity of the fluid in the cross section of the pipe.
It is important to note that, while the flow rate Q is the same in all sections of the pipe, the flow rate per unit area q or simply the flow velocity is higher in the narrower sections and lower in the wider ones .
2.- Hydraulic gradient I
When a fluid circulates through a porous medium, the hydrostatic pressure decreases in the same direction as the flow.
It is known that the hydrostatic pressure, at a certain point in the pipeline, is proportional to the height h marked by an open tube manometer at that place. The constant of proportionality is the product of the density of the fluid and the acceleration of gravity.
In this way, the hydraulic gradient I is defined as the quotient between the height difference Δh of the columns of two manometers and ΔL, the latter quantity being the distance that separates the manometers (see the figure below):
I = Δh / ΔL
This is the mean hydraulic gradient in the length ΔL, a dimensionless and negative quantity.
If we want to find the hydraulic gradient at each point in the pipeline, we take the limit for ΔL tending to zero, resulting in the derivative of the hydraulic gradient function with respect to the position L, along the flow:
3.- Permeability K
The permeability of a porous medium or hydraulic conductivity K is the quotient between the flow rate Q and the product of the cross section of area A and the hydraulic gradient I :
K = Q / A ∙ I
Hydraulic conductivity has units of velocity, meters over second in SI.
A unit has been defined for K, called darcy , in honor of Henry Darcy and defined as follows:
A darcy is the permeability of a milliliter of fluid, with viscosity of one centipoise, moving along one centimeter at a differential pressure of one atmosphere , through a cross section of one square centimeter.
Darcy’s law applications
The main application of Darcy’s law is to predict the flow of water through an aquifer, before drilling wells.
Likewise, Darcy’s law is used regularly in agricultural and hydrological engineering. It can also be used in the oil industry to describe the flow of oil and gas in porous media. However, in that case K may vary, depending on whether the flow is gas or oil and it may not depend solely and exclusively on the permeable substrate.
Limitations
Darcy’s law assumes that the hydraulic conductivity K is a proper quantity of the medium, which is true in many cases. However, sometimes K depends on the dynamic viscosity of the fluid, which in turn can depend on the flow rate and temperature gradients.
Darcy’s assumption is plausible when considering groundwater flow, where the viscosity is practically constant, since its value is almost unaffected in view of the few temperature differences across the aquifer.
In the cases of oil flow through porous media, the Darcy equation cannot be applied as presented here, but rather certain modifications are incorporated that go beyond the purpose of this article.
Exercise resolved
Determine the hydraulic conductivity of a beach sand, by using a laboratory permeameter.
Suppose that the permeameter tube has a diameter of 20 cm and that the distance between the two manometers is 50 cm. It is also known that water flows at a rate of 300 cubic decimeters per minute and the difference in level between the two gauges is 25 cm.
Solution
The flow Q is 300 cubic decimeters per minute, but expressing it in units of the international system it would look like this:
Q = 300 x 10 -3 m 3 /60 s = 5 x 10 -3 m 3 / s
The cross section of area A is calculated by:
A = π ∙ R 2 = π ∙ (10 cm) 2 = π ∙ (0.1 m) 2 = 0.314 m 2
The hydraulic gradient I is the quotient between the difference in manometric height and the separation of the pressure gauges:
I = 25 cm / 50 cm = 0.5
According to the definition of hydraulic conductivity K given above:
K = Q / A ∙ I = (5 x 10 -3 m 3 / s) / (0.314 m 2 ∙ 0.5) = 3.185 x 10 -2 m / s ≈ 2 m / min.
