Darcy's law

Darcy's law

Darcy's law. Equation and notations

Darcy's law is the fundamental law of fluid motion in porous media published by Henry Darcy in 1856 . French engineer Henry Darcy has earned himself a special place in history as the first experimental reservoir engineer .

Darcy's law has been successfully applied to determine the flow through permeable media since the early days of Petroleum Engineering.

The basic form of Darcy's law is very similar to in form to other physical laws. For example Fourier's law for heat conduction and Ohm's law for flow of electricity .

Darcy's law formula:

$q=\frac{kA}{\mu} \frac{\Delta P}{L}$

where

$A$ = cross-sectional area, cm²

$k$ = permeability, Darcy

$L$ = length, cm

$P$ = pressure, atm

$q$ = flow rate, cm³/sec

$\mu$ = fluid viscosity, cp

The permeability of 1 Darcy defined as permeability which allows fluid with viscosity of 1 centipoise flow a distance of 1 cm with velocity of 1 cc/sec through the crossectional area of 1 cm2 with the pressure gradient of 1 atm.

Example

Determine the permeability given the core lab test data: A=2.5cm2, L=3cm, qw=0.6 cm3/sec, dP=2 atm, water viscosity 1 cP.

$k=\frac{q \mu L}{A \Delta P} = \frac{0.6 *1 *3}{2*2.5} * 1000=360$ mD

History

Darcy's experimental equipment

Henry Darcy worked on the design of a filter large enough to process the Dijon towns daily water requirement .

By flowing water through the sand pack Darcy established that, for any flow rate, the velocity of the flow was directly proportional to the difference in manometric heights:

$u=K\frac{h1-h2}{L}$

All the experiments were carried out with water changing the type of sand pack. The effects of fluid density and viscosity on the flow was not investigated and therefore accounted for in the constant K.

Subsequently, others experiments performed with a variety of different liquids revealed the dependence of fluid flow on fluid density and viscosity.

The new constant k has therefore been isolated as being solely dependent on the nature of sand and is described as the permeability.

Equation

Differential form

If distance is measured positive in the direction of flow, then the pressure gradient must be negative in the same direction since fluids move from high to low pressure. Therefore, Darcy's law is:

$q = -\frac{kA}{\mu} \frac{dP}{dL}$

Linear form

$q = \frac{k}{\mu} \frac{A}{L} (P_1 - P_2)$

Radial form

$q = \frac{2 \pi kh (P_e - P_w)}{\mu ln(r_e/r_w)}$

Conditions

• Single fluid
• Steady stay flow
• Constant fluid compressibility
• Constant temperature

Inflow Equations Derivation

Derivation of the Linear and Radial Inflow Equations

See Also

• Converting from the Darcy's law units to the field units in the well's inflow equations 141.2 derivation
• Calculating Production Potential with the Darcy's law
• Petroleum Engineering