# Darcy's Law Calculator

Created by Joanna Śmietańska, PhD
Reviewed by Dominik Czernia, PhD and Steven Wooding
Last updated: Jun 05, 2023

Use our Darcy's law calculator to simply calculate the flow rate through a porous medium. Darcy's law formula is one of the principles in hydrogeology describing water flow through a porous medium.

According to Darcy's law equation, the amount of fluid flowing through a porous medium per unit of time is directly proportional to the hydraulic gradient, the cross-sectional area conducting the fluid, and the permeability of the material. If you want to know more about flow rate, check out our flow rate calculator.

In our calculator, you can easily calculate the permeability of a specific medium coefficient based on knowledge of flow duration and geometric parameters.

• What is Darcy's law?
• How to calculate fluid flow through a porous material.
• How to calculate the permeability of different materials.

## Filtration and fluid flow

Water can flow through a porous medium under the influence of a pressure difference. It depends mainly on the physical properties of the water being filtered and on the physical properties of the soil – the ability of the soil to allow water to pass through a system of interconnected pores (hydraulic permeability, water permeability).

There are three types of filtration:

• Laminar filtration – a very slow seepage of water through a system of pores and tubules, when water molecules in the individual layers move in parallel with each other and filtration is linearly dependent on the hydraulic gradient (as we described in the hydraulic gradient calculator, it's the ratio of the difference in groundwater height to the length of its flow);
• Turbulent filtration – water movement taking place at a higher velocity (when water molecules move in a vortex motion) so that they often collide and mix; and
• Mixed filtration – water movement characterized by the occurrence of laminar filtration and turbulent filtration movements in individual layers

We can encounter many examples of laminar flows in real life, e.g., blood circulation throughout the body or the flow of air over an aircraft. To predict whether the flow of a liquid will be laminar or turbulent, you can check out Reynolds number calculator.

On the other hand, viscosity $\mu$ is the resistance of fluid against flowing. If there is friction between different molecules in a fluid, it will need some energy to make a fluid flow. You can check the dynamic viscosity of water at different temperatures using our water viscosity calculator.

## What is Darcy's law? - Darcy's law formula

Darcy's law equation describes the laminar flow of water through a porous medium. It is named after the French engineer and hydrologist Henry Philibert Gaspard Darcy (1803-1858). He passed water through sand samples placed in a metal cylinder in order to measure the difference in water levels in the cylinder and the outflow vessel and the amount of water flowing in a given unit of time.

According to Darcy's law equation, the flow rate $Q\ [\text{m}^3/\text{s}]$, i.e., the amount of water flowing through a porous medium per unit time, is proportional to the hydraulic gradient i, the cross-sectional area conducting the water $A\ [\text{m}^2]$ and the permeability coefficient $k\ [\text{m}/\text{s}]$:

$Q = k \times I \times A$

We can calculate hydraulic gradient $i$ using the pressure difference on either side of the material $\Delta p$, the flow distance $L\ [\text{m}]$ and the dynamic viscosity of the water $\mu$ at a given temperature:

$i=\frac{\Delta p}{\mu \times L}$

## Definition of permeability – how to calculate permeability?

The permeability $k$ in Darcy's law calculator determines how many water molecules moving in a laminar motion can percolate through a system of pores and tubules. It depends both on the properties of the liquid (water) being filtered (temperature, chemical composition, density, viscosity) and the physical properties of the soil (grain size, porosity, mineral composition, structure, and texture).

We will now show you how to calculate permeability using Darcy's law calculator:

$k=\frac{Q}{A×i}$

Flow rate $Q$ can be described as flow volume per unit time $Q=V/t$, therefore:

$k=\frac{V}{t×A×i}$

💡 We can see from the above relationship that the hydraulic gradient $i$ is inversely proportional to the permeability $k$. This means that in materials characterized by good permeability, e.g., coarse-grained sands, hydraulic gradients will be lower compared to formations that are difficult to permeate (e.g., fine-grained sands) at the same filtration velocity value.

## How to use Darcy's law calculator

You only need a few steps to calculate the flow rate using our Darcy's law calculator. First, we can determine the pressure gradient by specifying the pressures on both sides of the material.

Then you can enter further data:

1. The viscosity of a fluid.
2. The distance the fluid flows through the material.

At this stage, you should obtain the hydraulic gradient that the calculator will use in further calculations. Now enter the following terms:

1. The cross-sectional area of the material.
2. The time taken for the fluid to move through the material.
3. The volume of fluid leaving the material.

The permeability coefficient $k$ will be obtained from the definition of permeability and used directly in Darcy's law equation to calculate the flow rate through the material.

Joanna Śmietańska, PhD
Pressure in
psi
Pressure out
psi
Pressure difference (Δp)
psi
Viscosity (μ)
P
Distance (L)
ft
Flow rate
Area (A)
ft²
Time (t)
sec
Volume (V)
US gal
Permeability (k)
ft/s
Flow rate (Q)
ft³/s
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