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Darcy's Law (Specific Discharge)

Flow of fluid through a porous medium.

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Core idea

Overview

Darcy's Law defines the relationship between the flow rate of a fluid through a porous medium and the hydraulic gradient. In hydrogeology, it is primarily used to calculate the specific discharge, also known as the Darcy flux, which represents the volume of water flowing through a unit cross-sectional area per unit time.

When to use: Apply this equation when analyzing laminar flow through saturated materials like sand, gravel, or fractured rock. It assumes steady-state conditions and is most accurate for low-velocity groundwater systems where the Reynolds number is less than 1 to 10.

Why it matters: This principle is fundamental for predicting groundwater movement, managing water supply wells, and tracking the spread of underground pollutants. It allows engineers to design effective drainage systems and assess the stability of earth-fill dams or embankments.

Symbols

Variables

v = Velocity (v), K = Hydraulic Cond., i = Gradient (i)

Velocity (v)
Hydraulic Cond.
Gradient (i)

Walkthrough

Derivation

Formula: Darcy's Law (Specific Discharge)

Describes the flow of a fluid through a porous medium.

  • Laminar flow (low Reynolds number).
  • Homogeneous and isotropic medium.
1

Relate velocity to gradient:

The velocity of flow is proportional to the hydraulic conductivity (K) and the hydraulic gradient (head loss over distance).

Result

Source: University Hydrogeology — Porous Flow

Free formulas

Rearrangements

Solve for

Darcy's Law (Specific Discharge): Make K the subject

Start from Darcy's Law (Specific Discharge). Substitute the hydraulic gradient `i` and then divide to make K the subject.

Difficulty: 2/5

Solve for

Darcy's Law (Specific Discharge): Make i the subject

Start from Darcy's Law (Specific Discharge). Substitute the definition of the hydraulic gradient , then divide by to solve for .

Difficulty: 2/5

Solve for

Make v the subject

Rearrange Darcy's Law to express specific discharge `v` in terms of hydraulic conductivity `K` and hydraulic gradient `i`.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a straight line passing through the origin, showing that discharge velocity increases linearly as the gradient increases. For a geology student, this means that a small gradient results in slow fluid movement through the porous medium, while a large gradient drives a much faster flow. The most important feature is that the relationship is directly proportional, meaning that doubling the gradient will always result in a doubling of the discharge velocity.

Graph type: linear

Why it behaves this way

Intuition

Imagine water seeping through a sponge-like material, always moving 'downhill' along the internal water surface, driven by the difference in water levels from one point to another.

v
Specific discharge (Darcy flux)
Represents the effective volume of water flowing through a unit cross-sectional area of the porous medium per unit time. It's an average flow rate, not the actual velocity of individual water particles.
K
Hydraulic conductivity
A measure of how easily water can flow through a porous material. High K means water flows readily (e.g., sand), while low K means it flows with difficulty (e.g., clay).
dh/dl
Hydraulic gradient
The 'slope' of the hydraulic head. A steeper gradient (larger value) indicates a stronger driving force for water to flow, similar to how a steeper hill causes water to run faster.

Signs and relationships

  • -: The negative sign indicates that the flow (v) occurs in the direction of decreasing hydraulic head. Water naturally moves from areas of higher hydraulic head to areas of lower hydraulic head.

Free study cues

Insight

Canonical usage

This equation is used to calculate the specific discharge (or Darcy flux) by ensuring consistent units for hydraulic conductivity and the dimensionless hydraulic gradient.

Common confusion

A common mistake is using inconsistent units for hydraulic conductivity (K) and specific discharge (v), or for the length terms in the hydraulic gradient (dh and dl).

Dimension note

The hydraulic gradient (dh/dl) is a dimensionless quantity. It is crucial that the units for 'dh' (change in hydraulic head) and 'dl' (change in length) are consistent before calculating the gradient.

Unit systems

m/s (SI), ft/day (Imperial) · Specific discharge, also known as Darcy flux. Represents the volume of water flowing through a unit cross-sectional area per unit time, not the actual average velocity of water particles.
m/s (SI), ft/day (Imperial) · Hydraulic conductivity, an intrinsic property of the porous medium and the fluid. Its units must be consistent with the desired units for specific discharge.
dimensionless · Hydraulic gradient, which is the change in hydraulic head (dh) over a change in length (dl). The units for 'dh' and 'dl' must be identical, making the ratio dimensionless.

One free problem

Practice Problem

A sandy aquifer has a hydraulic conductivity of 12 meters per day. If the measured hydraulic gradient between two observation wells is 0.005, calculate the specific discharge in meters per day.

Hydraulic Cond.12 m/s
Gradient (i)0.005

Solve for:

Hint: Multiply the hydraulic conductivity by the hydraulic gradient to find the specific discharge.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Water moving through a sandy aquifer with K = 0.01 m/s and gradient 0.05. v = 0.0005 m/s.

Study smarter

Tips

  • Ensure units for specific discharge (v) and hydraulic conductivity (K) are identical (e.g., m/day).
  • The hydraulic gradient (i) is a dimensionless ratio calculated as the change in head over the flow distance.
  • The negative sign in the theoretical formula indicates that flow occurs in the direction of decreasing hydraulic head.
  • Remember that specific discharge is a bulk velocity and does not represent the actual speed of a water molecule through pores.

Avoid these traps

Common Mistakes

  • Ignoring the negative sign when calculating direction.

Common questions

Frequently Asked Questions

Describes the flow of a fluid through a porous medium.

Apply this equation when analyzing laminar flow through saturated materials like sand, gravel, or fractured rock. It assumes steady-state conditions and is most accurate for low-velocity groundwater systems where the Reynolds number is less than 1 to 10.

This principle is fundamental for predicting groundwater movement, managing water supply wells, and tracking the spread of underground pollutants. It allows engineers to design effective drainage systems and assess the stability of earth-fill dams or embankments.

Ignoring the negative sign when calculating direction.

Water moving through a sandy aquifer with K = 0.01 m/s and gradient 0.05. v = 0.0005 m/s.

Ensure units for specific discharge (v) and hydraulic conductivity (K) are identical (e.g., m/day). The hydraulic gradient (i) is a dimensionless ratio calculated as the change in head over the flow distance. The negative sign in the theoretical formula indicates that flow occurs in the direction of decreasing hydraulic head. Remember that specific discharge is a bulk velocity and does not represent the actual speed of a water molecule through pores.

References

Sources

  1. Applied Hydrogeology, C.W. Fetter
  2. Groundwater, R.A. Freeze and J.A. Cherry
  3. Wikipedia: Darcy's law
  4. Freeze, R. Allan, and Cherry, John A. Groundwater. Prentice-Hall, 1979.
  5. Fetter, C.W. Applied Hydrogeology. 4th ed. Prentice Hall, 2001.
  6. University Hydrogeology — Porous Flow