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3.7 Darcy’s Law 67
3.6 GROUNDWATER MOVEMENT

Groundwater in the natural state is constantly in motion. Its rate of movement under the force
of gravity is governed by the frictional resistance to flow offered by the porous medium. The
difference in head between any two points provides the driving force. Water moves from
levels of higher energy potential (or head) to levels of lower energy potential, the loss in head
being dissipated as heat. Because the magnitudes of discharge, recharge, and storage fluctuate
with time, the head distribution at various locations is not stationary. Groundwater flow is both
unsteady and nonuniform. Compared with surface water, the rate of groundwater movement is
generally very slow. Low velocities and the small size of passageways give rise to very low
Reynolds numbers and consequently the flow is almost always laminar. Turbulent flow may
occur in cavernous limestones and volcanic rocks, where the passageways may be large, or in
coarse gravels, particularly in the vicinity of a discharging well. Depending on the intrinsic
permeability, the rate of movement can vary considerably within the same geologic formation.
Flow tends to be concentrated in zones of higher permeability, that is, where the interstices are
larger in size and have a better interconnection.
In aquifers of high yield, velocities of 5 to 60 ft/d (1.5 to 18.3 m/d) are associated with
hydraulic gradients of 10 to 20 ft/mi (1.89 to 3.78 m/km). Underflow through gravel de-
posits may travel several hundred ft/d (m/d). Depending on requirements, flows as low as
a few ft/year (m/year) may also be economically useful.
In homogeneous, isotropic aquifers, the dominant movement is in the direction of
greatest slope of the water table or piezometric surface. Where there are marked nonhomo-
geneities and anisotropies in permeability, the direction of groundwater movement can be
highly variable.

3.7 DARCY’S LAW

Although other scientists were the first to propose that the velocity of flow of water and
other liquids through capillary tubes is proportional to the first power of the hydraulic gra-
dient, credit for verification of this observation and for its application to the flow of water
through natural materials or, more specifically, its filtration through sand, must go to
Darcy. The relationship known as Darcy’s law may be written:
v = K(dh>dl) = KI (3.1)
where v is the hypothetical, superficial or face velocity (darcy; not the actual velocity
through the interstices of the soil) through the gross cross-sectional area of the porous
medium; I = dh>dl is the hydraulic gradient, or the loss of head per unit length in the di-
rection of flow; and K is a constant of proportionality known as hydraulic conductivity, or
the coefficient of permeability. The actual velocity, known as effective velocity, varies from
point to point. The average velocity through pore space is given by:
v = KI>u (3.2)
e
u
where is the effective porosity. Because I is a dimensionless ratio, K has the dimensions
of velocity and is in fact the velocity of flow associated with a hydraulic gradient of unity.
The proportionality coefficient in Darcy’s law, K, refers to the characteristics of both
the porous medium and the fluid. By dimensional analysis:
2
K = Cd g>m (3.3)
where C is a dimensionless constant summarizing the geometric properties of the medium
g
affecting flow, d is a representative pore diameter, is the viscosity, and is the specific
m