Darcy’s Law

2.5.1 Darcy’s Law

The three main quantities that govern the flow of groundwater are as follows: hydraulic gradient, which is the driving force, hydraulic conductivity, which describes both the trans- missive properties of the porous media and the hydraulic properties of the flowing fluid (water), and the cross-sectional area of flow. Their relationship is described by Darcy’s law (Darcy was a French civil engineer who was first to quantitatively analyze the flow of water through sands as part of his design of water filters for the city of Dijon; his findings, published in 1856, are the foundation of all modern studies of fluid flow through porous media):

L This linear law states that the rate of fluid flow (Q) through porous medium is directly

proportional to the cross-sectional area of flow ( A) and the loss of the hydraulic head between these two points of measurement. K is the proportionality constant of the law

called hydraulic conductivity and has units of velocity. This constant is arguably the most important quantitative parameter characterizing the flow of groundwater. Following are the other common forms of Darcy’s equation:

v=K

[m/s]

L v = Ki [m/s]

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Well #1

Well #2

Distance (L)

Top of

Land surface

casing

Depth to water

Head loss(∆h)

Water table

Groundwater velocity

flow direction

ation

Well Screen

Elev

Hydrauliuc head (

Elevation

Bottom of aquifer

head

(z) Datum elevation

F IGURE 2.35 Schematic presentation of key elements for determining the hydraulic head and the hydraulic gradient in an unconfined aquifer. (Kresic, 2007a; copyright Taylor & Francis Group, LLC, printed with permission.)

Hydraulic Head and Hydraulic Gradient The principle of the hydraulic head and the hydraulic gradient is illustrated in Fig. 2.35. At the bottom of monitoring well #1, where the well screen is open to the saturated zone, the total energy (H) or the driving force for water flow at that point in the aquifer is

H=z+h p +

2g

where z = elevation above datum (datum is usually mean sea level, but it could be

any reference level)

h p = pressure head due to the pressure of fluid (groundwater) above that point v = groundwater velocity

g = acceleration of gravity Since the groundwater velocity in most cases is very low, the third factor on the

right-hand side may be ignored for practical purposes and the Eq. (2.10) becomes

(2.11) where h = hydraulic head, also called piezometric level. The pressure head represents the

H=h=z+h p

pressure of fluid ( p) of constant density (ρ) at that point in aquifer:

GroundwaterSystem

In practice, the hydraulic head is determined in monitoring wells or piezometers by subtracting the measured depth to the water level from the surveyed elevation of the top of the casing:

h = elevation of top of casing − depth to water in the well (2.13) As the groundwater flows from well #1 to well #2 (Fig. 2.35), it loses energy due to friction

between groundwater particles and the porous media. This loss equates to a decrease in the hydraulic head measured at the two wells:

h=h 1 − h 2 (2.14) The hydraulic gradient (i) between the two wells is obtained when this decrease in the

hydraulic head is divided by the distance (L) between the wells:

i=

[without dimension]

Groundwater flow always takes place from the higher hydraulic head toward the lower hydraulic head (just as in the case of surface water: “water cannot flow uphill”). It is also important to understand that, except in case of a very limited portion of an aquifer, there is no such thing as strictly horizontal groundwater flow. In an area where aquifer recharge is dominant, the flow is vertically downward and laterally toward the discharge area; in a discharge area, such as surface stream, this flow has an upward component (Fig. 2.36).

Downward gradient

sur face

90 Arbitrary scale

F IGURE 2.36 Movement of groundwater in an unconfined aquifer showing the importance of both vertical and horizontal hydraulic gradients. (Modified from Winter et al., 1998.)

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Hydraulic Conductivity and Permeability In addition to hydraulic conductivity, another quantitative parameter called intrinsic per- meability (or simply permeability) is also used in studies of fluid flow through porous media. It is defined as the ease with which a fluid can flow through a porous medium. In other words, permeability characterizes the ability of a porous medium to trans- mit a fluid (water, oil, gas, etc.). It is dependent only on the physical properties of the porous medium: grain size, grain shape and arrangement, or pore size and intercon- nections in general. On the other hand, hydraulic conductivity is dependent on the properties of both the porous medium and the fluid. The relationship between the permeability (K i ) and the hydraulic conductivity (K ) is expressed by the following formula:

[m ρ 2 ]

where μ = absolute viscosity of the fluid (also called dynamic viscosity or

simply viscosity) ρ= density of the fluid

g = acceleration of gravity The viscosity and the density of the fluid are related through the property called kinematic

Inserting the kinematic viscosity into Eq. (2.16) somewhat simplifies the calculation of the permeability since only one value (that of υ) has to be obtained from tables or graphs (note that, for most practical purposes, the value of the acceleration of gravity (g) is 9.81

m/s 2 , and is often rounded to 10 m/s 2 ):

K i = K [m 2 ]

Although it is better to express permeability in units of area (m 2 or cm 2 ) for reasons of consistency and easier use in other formulas, it is more commonly given in darcys (which is a tribute to Darcy):

1 darcy = 9.87 × 10 − 9 cm 2 = 9.87 × 10 13 m 2

When laboratory results of permeability measurements are reported in darcys (or meters squared), the following two equations can be used to find the hydraulic conduc- tivity:

K=K i υ or K=K i μ [m/s]

GroundwaterSystem

Water temperature influences both water density and viscosity and, consequently, the hydraulic conductivity is strongly dependent on groundwater temperature. Kinematic viscosity of water at temperature of 20 ◦

C is approximately 1 × 10 − 6 m 2 /s, and rounding gravity acceleration to 10 m/s 2 , gives the following conversion between permeability (given in m 2 ) and hydraulic conductivity (given in m/s):

K [m/s] = K i [m 2 ] × 10 7 (2.20) Since effective porosity, as the main factor influencing the permeability of a porous

medium, varies widely by rock types, the hydraulic conductivity and permeability also have wide ranges as shown in Fig. 2.37. As is the case with porosity, limestones have the widest range of hydraulic conductivity of all rocks. Vesicular basalts can have very high hydraulic conductivity, but they are on average less permeable than medium to coarse sand and gravel, which are rock types with the highest average hydraulic conductivity. Pure clays and fresh igneous rocks generally have the lowest permeability, although some field-scale bedded salt bodies were determined to have permeability of zero (Wolff, 1982). This is one of the reasons why salt domes are considered as potential depositories of high radioactivity nuclear wastes in some countries.

Except in rare cases of uniform and nonstratified, homogeneous unconsolidated sed- iments, hydraulic conductivity and permeability vary in space and in different directions

Feet per day(ft/d)

Feet per minute (ft/min)

gal/ft 2 /d

Meters per day (m/d)

Relative Permeability

Very High High

Moderate

Low

Very Low

Clean gravel Clean sand and

Massive clay sand and gravel

Fine sand

Silt, clay, and mixtures

of sand, silt, and clay

Vesicular and scoriaceous

Massive igneous basalt and cavernous

Clean sandstone

Laminated sandstone

and metamorphic limestone and dolomite

and fractured

shale, mudstone

igneous and

rocks

metamorphic rocks

F IGURE 2.37 Range of hydraulic conductivity for different rock types (USBR, 1977).

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within the same rock mass due to its heterogeneity and anisotropy. Most practitioners tend to simplify these inherent characteristics of porous media by dividing the com- plex three-dimensional hydraulic conductivity tensor into just two main components: horizontal and vertical hydraulic conductivity. Unfortunately, it seems common prac- tice to apply some “rules of thumb” indiscriminately, such as vertical conductivity is ten times lower than the horizontal conductivity, without trying to better character- ize the underlying hydrogeology. This difference in the two hydraulic conductivities can vary many orders of magnitude in highly anisotropic rocks and, in many cases, it may be completely inappropriate to apply the concept altogether: a highly trans- missive fracture or a karst conduit may have any shape and spatial extent, at any depth.