# Darcy's Law Basics and More

### Oklahoma State University

Back to Groundwater

• Introduction
• One-Dimensional Flow
Simple Discrete Form
Differential Form
Flow Variables
Darcy Flux
Seepage Velocity
• One Dimensional Flow at an angle to the coordinate axis
• Special 1-D Flows
Horizontal flow
Vertical Flow
Unit Gradient Flow
• Other Measures of the Flow Proportionality
Transmissivity
Permeability
• ### Introduction

Darcy's Law is a generalized relationship for flow in porous media. It shows the volumetric flow rate is a function of the flow area, elevation, fluid pressure and a proportionality constant. It may be stated in several different forms depending on the flow conditions. Since its discovery, it has been found valid for any Newtonian fluid. Likewise, while it was established under saturated flow conditions, it may be adjusted to account for unsaturated and multiphase flow. The following outlines its common forms and assumes water is the working fluid unless otherwise stated.

### One-Dimensional Flow

Simple Discrete Form

A one-dimensional flow column is shown in Figure 1. Figure 1. Simple column.

For a finite 1-D flow, it may be stated as _____

where,
Q = volumetric flow rate (m3/s or ft3/s),
A = flow area perpendicular to L (m2 or ft2),
K = hydraulic conductivity (m/s or ft/s),
l = flow path length (m or ft),
h = hydraulic head (m or ft), and
D = denotes the change in h over the path L.

The hydraulic head at a specific point, h is the sum of the pressure head and the elevation, or

h = (p/r g + z)_____[2a]
h = (p/g + z)_____[2b]

where,
p = water pressure (N/m2, lb/ft2),
r = water density (kg/m3),
g = water specific weight (lb/ft3),
g = acceleration of gravity (m/s2 or ft/s2), and
z = elevation (m or ft).

Equation [2a] is the normal SI form of the equation, while [2b] is the usual form used with English units. The hydraulic head is the height that water would rise in a peizometer. Thus, Dh is simply the difference in height of water in peizometers placed at the inlet and the outlet (Dh = hin-hout). Substituting [2a] into  yields, Equation  is approximately the form Darcy used to analyze his experimental data. Note that the flow is not a function of the absolute pressure or the elevation. It is only a function of the change in hydraulic head.

Differential Form

A more general form of the equation results when the limit of Dh with respect to the flow direction l, as the flow path L goes to zero. Applying that step to equations  and  yields, _____

The minus signs on the right hand terms reflects that the hydraulic head always decreases in the direction of flow.

### Flow Variables

Darcy Flux

The Darcy flux is defined as,

q = Q /A_____

where q = Darcy flux (m/s or ft/s).

The Darcy flux is the volumetric flow per unit area. Substitution of equation  into  yields, _____

Seepage Velocity

While the Darcy flux has the units of velocity, it is not the velocity of the water in the pores. The solid matrix takes up some of the flow area. The average pore water velocity is termed the seepage velocity, v, and is given by

v = Q/Af = q/f_____

where f is the porosity of the porous media. The maximum pore velocity is a function of the pore geometry and cannot be easily predicted except for simple shaped. In circular tubes the maximum velocity is twice v.

### One Dimensional Flow at an Angle to the Coordinate Axis

Darcy's Law is not a function of the flow direction in a homogeneous material. However, the gradient of h is calculated along the flow path, l, and the flow area, A is measured normal to l. Therefore, the geometry of flow must be accounted for if the flow is measured relative to a different direction. Figure 2 shows the simple column tilted up. Figure 2. Flow at an angle to the horizontal.

Assuming a 2-D space,

z = x tan(a)_____
dl = dx / cos(a)_____
dl = dz / sin(a)_____

where,
a = angle to horizontal, and
x = horizontal distance (m or ft).

Substitution of equation  and  into  produces a relation relative to the x direction. _____

Simplifying produces, _____

If the area of flow is measured normal to the x axis, Ax will be larger than the area normal to l. The two areas are related by,

A = cos(a)Ax 

Substitution of equation  into  produces _____

By similar methods the flow may be expressed relative to the vertical direction by substitution of equation  into _____

where Az is the area of flow normal to the vertical axis.

### Special 1-D Flows

Horizontal flow

In horizontal flow, a = 0 and equation  reduces to _____

Vertical Flow

In vertical flow up, sin(a) = 1 and equation  reduces to _____

Unit Gradient Flow

In vertical downward flow, if dp/dz = 0, equation  reduces to the unit gradient form.

Q = AzK (down)_____ 

### Other Measures of the Flow Proportionality

Transmissivity

In saturated groundwater analysis with nearly horizontal flow, it is common practice to combine the hydraulic conductivity and the thickness of the aquifer, b into a single variable,

T = bK_____ 

where T = transmissivity (m2/s, ft2/s).

Permeability

When the fluid is other than water at standard conditions, the conductivity is replaced by the permeability of the media. The two properties are related by,

K = krg / m = kg / n_____ 

where,
k = permeability, (m2 or ft2),
m = fluid absolute viscosity, (N s/m2 or lb s/ft2) and
n = fluid kinematic viscosity, (m2/s or ft2/s).

Ideally, the permeability of a porous media is the same to different fluids. Thus, you may predict the flow of one fluid, from the measurement of a second with equation . However in practice, the solid matrix may swell or sink with different fluids and produce different values of k. Substitution of equation  into  yields, _____

Likewise, substitution into equation  produces, _____