Darcy's Law

The law of Darcy's is an equation that describes the fluids flowing through a porous medium. The law which we are discussing here was formulated by Henry Darcy based on the results of experiments on the flow of water through sand beds that are forming the basis of hydrogeology under the branch of earth sciences.

The law of Darcy's was first determined experimentally by Darcy but we can say that we have since been derived from the Navier–Stokes equations via homogenization methods. It is the law analogous to Fourier's in the field of conduction of heat that is Ohm's law in the field of electrical networks. 

What is Darcy's Law?

The derivation of the law of Darcy's is used extensively in petroleum engineering. It determines the flow through permeable media, the most simple of which is for a one-dimensional homogeneous rock formation with a fluid that is single phase and constant fluid viscosity.

Almost all reserves of oil have a water zone below the oil leg and some have also a cap of gas which is above the oil leg. When the reservoir pressure drops due to oil production the flow of water is into the oil zone from below and flow of gas into the oil zone from above if the gas cap exists, and we get a flow of simultaneous and immiscible mixing of all fluid phases in the zone of oil. 

The operator which we are talking about here is about the oil field which may also inject water and/or gas in order to improve oil production. The flow of Multiphase which is in oil and gas reservoirs is a comprehensive topic and one of many articles that are about this topic is the law of Darcy's for multiphase flow.

Darcy's Law Further Explained

This is the law in geology that describes the rate at which a fluid flows through a permeable medium. The law of Darcy's states that this rate is directly proportional to the drop which is in vertical elevation which is between two places in the medium and indirectly proportional to the distance that is between them. The law is used to describe the water flow from one part of an aquifer to another and the flow of petroleum that is through the gravel and the sandstone.

The law of Darcy's is valid for laminar flow through sediments. In fine-grained sediment, the dimensions which are of interstices are small, and thus flow is laminar. The Coarse-grained sediments also behave similarly but in very coarse-grained sediments the flow may be turbulent. Hence the law of Darcy's is not always valid in such sediments. 

For flow-through commercial pipes which are circular, the flow is laminar when the number of Reynolds is less than 2000 and turbulent when it is more than 4000 but we can say that in some sediments it has been found that flow is laminar when the value of a number of Reynolds is less than 1.

For a very short period of time, the scales at a time which is derivative of flux may be added to the law of Darcy's which results from in valid solutions at very small times that is in heat transfer this is known as the modified form of Fourier's law.

The main reason which we are discussing for doing this is that the regular groundwater flow equation is the diffusion equation which leads to singularities that are at constant head boundaries at very small times. This is a form that is more mathematically rigorous but this leads to groundwater which is a hyperbolic flow equation that is more difficult to solve and is only useful at very small times that is typically out of the realm of practical use.

Application of Darcy’s Law

One of the applications of laws of Darcy's is in the analysis of water flow through an aquifer that is the law of  Darcy's which, along with the equation of conservation of that mass, simplifies to the groundwater equation flow. Which is one of the basic relationships of hydrogeology.

Muskat Morris was the first refined equation of Darcy's which is for a single-phase flow by including viscosity in the single that is the fluid phase equation of Darcy. This change which we notice is made by it suitable for researchers in the petroleum industry. The generalized flow which is multiphase equations by Muskat and others provides the analytical foundation for engineering which is a reservoir that exists to this day.

Properties of Darcy's Law

The law of Darcy's is a simple mathematical statement that neatly summarizes many or we can say that several familiar properties that flowing groundwater in aquifers exhibits which is including:

• That if there is no pressure gradient over a distance which is  no flow occurs as these are hydrostatic conditions,

• Another if there is a pressure gradient the flow will occur from high pressure that is towards low pressure that is opposite the direction of increasing gradient hence the negative sign in the law of Darcy's.

• Another the greater the pressure which is a gradient that is through the same formation material the greater the discharge rate.

Darcy's law is a conservation law that states the flow of a fluid through a pipe or other closed conduit is proportional to the transverse force acting on the fluid and inversely proportional to the viscosity of the fluid. It was introduced by the Scottish engineer Lord Kelvin in 1890. Its first experimental demonstration was performed by Lord Rayleigh in 1900.

Darcy's Law in physics describes an asymmetry in the permeability of porous materials. It was discovered by Lord Kelvin in 1856.

One way to describe Darcy's Law is to view it as a description of how flow velocity in the pores of a porous medium is determined by the applied pressure and the fluid properties.  Another way is to view it as a statement about how the permeability of the material is determined by the size and geometry of the pore structure.  The latter is the usual and more familiar way to view Darcy's Law.

The equation is often described as a generalization of Darcy's law.  As the flow increases, so does the pressure drop through the medium. The permeability of the material determines the amount of fluid that is able to flow in a given area and the effect of this flow is a drop in the pressure of the fluid. The formula is where Q is flow rate, Δp is pressure drop, and L is the length of the pipe.  Here K is the permeability,  is the Darcy velocity, and ν is the fluid viscosity.  The equation is true regardless of whether the system is static (no flow) or if flow occurs.

Equation of Continuity

The equation of continuity for mass flow (mass/volume per unit time) may be written as

where ρ is density,  is velocity and v is the volume per unit time.

The equation may be expressed in terms of Darcy's Law by using and is valid for Newtonian fluids as well as for non-Newtonian fluids like gelatin or suspensions of particles.

Formulation of Darcy's Law 

Darcy's law may be considered in terms of two components, the flow rate, and the pressure drop. The flow rate is generally determined by the hydraulic conductivity, but this also affects the viscosity. Therefore, the resistance to flow in a porous medium with variable permeability (e.g. a porous rock, porous sand, a porous concrete, ...) is generally not constant and may depend on the porous media structure or the fluid viscosity (e.g. water in a rock).  The pressure drop is given. where ΔP is the pressure drop,  is the fluid velocity,  is the permeability, and is the length of the pipe.

This equation relates Darcy's law to the fluid viscosity and length. It is also possible to generalize the equation by including the porous media resistance, as expressed by Darcy's law. This means that the porous medium resistance may vary in time and space, for example as a result of flow erosion.

History 

Lord Kelvin had been studying the pressure distribution in a fluid flowing in a tube.  In a lecture delivered to the British Association for the Advancement of Science in Edinburgh on 29 June 1890 he introduced the law:

This equation was introduced to Kelvin without mathematical justification, but he had earlier applied similar equations to other areas of science, such as sound and water waves. Lord Rayleigh provided the first experimental proof of this law in 1900.

Lord Rayleigh published a book, Fluid mechanics, containing the first experimental proofs.

The equation has since been applied in various other areas of science, including fluid dynamics, seismology, geophysics, nanoscale physics, biology, and materials science.

The mathematical formulation of Darcy's law was given by J.D. Fardin, R. C. Desai, and S. M. Wise in 1975.  More recently, T. M. Truskett has provided a rigorous mathematical derivation of the equation.

Derivation of Darcy's Law 

Darcy's Law is an approximation to a more complicated Navier–Stokes equation. It is an alternative method to the Reynolds Stress Decomposition, and it was first proposed by Lord Kelvin in 1890. Darcy's law has been derived using a number of methods and the first example of a derivation is a paper published by J.D. Fardin, R. C. Desai, and S. M. Wise in 1975. This paper discusses the derivation of an approximate Navier–Stokes equation for a simple pipe flow.

Darcy's law states that the flow of liquid is proportional to the pressure difference between two points in the pipe. The following are the two points:

• A point A which is located at the inlet of the pipe and

• A point B which is located in the center of the pipe and is also at half the length of the pipe from A.

The flow of liquid through the pipe can be represented by two parameters. They are the cross-section area of the pipe and the kinematic viscosity. The cross-section area of the pipe is constant throughout the pipe length. The kinematic viscosity varies with the flow rate. This is known as Darcy's law since Lord Kelvin derived it using the fluid mechanic's principle.

The exact formulation of Darcy's law  where:

• The flow rate of liquid (liters per second)

• The velocity (m/s) of the liquid

• The pressure of the liquid (N/m²)

• The kinematic viscosity (m²/s)

and the Darcy velocity is the actual flow velocity (m/s).

The above formulation is valid for the case of steady, laminar flow.  The Darcy velocity is obtained by plugging Darcy's law into the general expression for steady laminar flow in a pipe. The expressions for the terms are obtained from the equation.