## Darcy's law

Page content By Kimberly Mullen, CPG

**Objective**

Students will learn about Darcy’s law and use it to predict different scenarios pertaining to groundwater.

**General introduction**

Henry Philibert Gaspard Darcy was born June 10, 1803, in Dijon, France. In 1821, at the age 18, Darcy entered the Polytechnic School in Paris. Two years later he was admitted to the School of Bridges and Roads in Paris. After graduation, he was eventually assigned to a position in Dijon.

In 1828, Darcy was assigned to a deep-well drilling project that found water but could not provide an adequate supply for the town. However, soon after the disappointment of the well, and under his own initiative, he set out to provide a clean, dependable water supply to the city from more conventional surface water sources. That effort eventually produced a system that delivered 8 m3/min. The entire system was enclosed and gravity-driven, thus it required no pumps or filters.

By 1848 Darcy was chief engineer, but due to political pressures, he was forced to leave Dijon. He was, however, soon appointed to the higher position of chief director for Water and Pavements, Paris. In Paris, he carried out significant research on the flow and friction losses in pipes, and created an improved design for the Pitot tube.

In 1855, due to failing health, he resigned from normal duties, but was allowed to return to Dijon to carry out research of his own interest. In 1855 and 1856, he conducted column experiments that established Darcy's law for flow in sands. Since his time, Darcy's law been generalized by many writers to allow for differential solutions, vector analysis, unsaturated flow, and multiphase flow. Likewise, the equation's theoretical basis and applicability in several fields has been well defined.

Unexpectedly, on January 3, 1858, Darcy died of pneumonia while on a trip to Paris.

Darcy’s law describes how rate of flow of groundwater, is related to the variables described below. The rate at which groundwater flows is equal to the product of the hydraulic conductivity multiplied by the hydraulic gradient.

V = K(G)V = groundwater velocity

K = hydraulic conductivity, a number that describes how easily groundwater flows through a particular type of rock or soil; undergroundwater flow is related to the porosity and permeability of the aquifer

G = hydraulic gradient, the slope of the water table; groundwater flows downhill and the gradient determines the rate of flow and is computed as H/L (the difference in head or H, the static water level, divided by the distance between wells or L).

NOTE:Students will need some examples to allow them to truly understand the use of Darcy’s law. Draw examples on the board to clearly illustrate what each variable is and how it is used to calculate velocity. Plan to work through a few different examples prior to setting up a worksheet for the students to try on their own. Be sure students understand that all units of measure are in the same system (for example feet vs. meters). This is also a great opportunity to work on converting this information.

**Student questions**

- What would Darcy’s law predict would happen to the rate of flow of groundwater as the hydraulic gradient, (i.e. the slope of the water table) increased?
- What would Darcy’s law predict would happen to the rate of flow of groundwater if the distance between the wells increased?
- If the hydraulic conductivity doubled and the hydraulic gradient decreased fourfold, what would happen to the rate of flow?
- Pumping out water from an aquifer affects which variable?
- If contamination is flowing into groundwater, how many years would it take for it to flow from its current location to the drinking water well located 10 miles away, using an average flow rate of 1 foot per day?

**Answer key**

- The rate of flow would increase.
- The rate of flow would increase.
- The rate of flow would be halved.
- The hydraulic gradient.
- 10 miles x 5,280 ft/mile ÷ 1 foot/day ÷ 365 days/year = 145 years.

**Note:** The biographical material presented here on Darcy has been excerpted in large part from Henry Darcy and His Law by Glenn Brown, Ph.D., Oklahoma State University, http://biosystems.okstate.edu/darcy/1pagebio.htm (last accessed 8/25/2011).

This page was last updated on 09-05-2012