What is Darcy’s Law??

Petroleum Engineering / By

Darcy’s law is the MOST important fundamental equation in petroleum engineering. If you don’t know it, well you have come to the right place.

The Origin of Darcy’s Law

Way back in 1856, a French engineer by the name of Henry Darcy set up an experiment to investigate the flow of water through sand filter beds with the intent of understanding the relationship between ground water flow and pressure loss. He conducted experiments with different packings of river sand and varied the inlet and outlet pressures. Remarkably, he was able to come up with a single expression that accurately depicted his experimental results for multiple test cases. Since then, his equation has become one of the cornerstone equations in petroleum engineering. The equation is as follows:

    \[ \frac{q}{A_c} = -\frac{k}{u} \frac{dP}{dL} \]

The Meaning of Darcy’s Equation

The Darcy equation is basis for describing flow in a porous rock system in the oil and gas industry. The differential form of the equation can be integrated to describe all types of flow conditions in porous geological materials, including linear flow, radial flow, pseudosteady-state flow, ect. In addition, the equation is often modified to describe multiphase flow and more intricate reservoir systems. We will look at some of these later, but first its important to understand the terms that make up the equation and the assumptions used in deriving the equation. The terms that make up the equation in oilfield units include the following:

  • q  = flow rate of fluid, bbl/day
  • k = permeability of the porous rock, Darcy
  • A_c =  cross sectional area of the rock, ft^3
  • u = viscosity of the fluid, cp
  • l = length of the rock sample, ft
  • \frac{dP}{dL} = pressure gradient in the direction of flow, psi/ft

A term you may not be familiar with is permeability. Permeability is defined as the capacity of a rock to transmit a fluid or the ease at which the fluid flows through the rock. The higher the permeability is, the more efficient a rock is able to transfer the fluid. It has units of area, i.e. Darcy is a unit describing area. The experiment Henry Darcy developed the equation from occurred under simple circumstances. Therefore his equation assumes the following:

  1. Single phase flow
  2. No rock fluid interactions
  3. Incompressible fluid
  4. Laminar flow

The equation above describes the simplest form of fluid flow encountered in porous medium. Although this is not what actually occurs in real life, it is a building block that helps describe more complicated fluid flow systems. Just remember that.

Whys is Darcy’s Law Negative??

Darcy’s experimental setup was more or less shown in the figure above where P_0 > P_L. This is analogous to flow through a pipe for you mechanical and chemical engineers out there. For Darcy’s equation to make since, the equation has to include a negative sign. I will show you why: 

Applying Darcy’s law WITHOUT the negative sign we arrive at the following expression:

    \[ \frac{q}{A_c} =\frac{k}{u} \frac{dP}{dL} \]

Plug in the inlet and outlet pressures and there corresponding spacial coordinate in the derivative/slope term, we end up with:

    \[ \frac{q}{A_c} =\frac{k}{u}\frac{P_0- P_L}{0-L} \]

The final expression is the following:

    \[ \frac{q}{A_c} =-\frac{k}{u}\frac{P_0- P_L}{L} \]

By inspection we know that the equation needs a negative sign. Because P_0 -P_L > 0 the expression ends up giving us a negative flow rate which does not make sense because we know that fluid flow goes from high pressure to low pressure. Therefore, a negative sign is included in the Darcy equation to ensure that the flow rate is positive, i.e flowing in the positive x-direction. Applying the negative sign, we end up with the following correct expression:

    \[ \frac{q}{A_c} =\frac{k}{u}\frac{P_0- P_L}{L} \]