Darcy's Law Basics and More

by

Glenn Brown

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

    _____[1]

    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 [1] yields,

    [3]

    Equation [3] 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 [1] and [3] yields,

    _____[4]

    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_____[5]

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

    The Darcy flux is the volumetric flow per unit area. Substitution of equation [5] into [4] yields,

    _____[6]

    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_____[7]

    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)_____[8]
    dl = dx / cos(a)_____[9]
    dl = dz / sin(a)_____[10]

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

    Substitution of equation [8] and [9] into [4] produces a relation relative to the x direction.

    _____[11]

    Simplifying produces,

    _____[12]

    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 [13]

    Substitution of equation [13] into [12] produces

    _____[14]

    By similar methods the flow may be expressed relative to the vertical direction by substitution of equation [10] into [4]

    _____[15]

    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 [14] reduces to

    _____[16]

    Vertical Flow

    In vertical flow up, sin(a) = 1 and equation [15] reduces to

    _____[17]

    Unit Gradient Flow

    In vertical downward flow, if dp/dz = 0, equation [15] reduces to the unit gradient form.

    Q = AzK (down)_____ [18]

    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_____ [19]

    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_____ [20]

    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 [20]. However in practice, the solid matrix may swell or sink with different fluids and produce different values of k. Substitution of equation [20] into [4] yields,

    _____[21]

    Likewise, substitution into equation [6] produces,

    _____[22]