Page 71 - Water Engineering Hydraulics, Distribution and Treatment

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3.7 Darcy’s Law
water held by capillary attraction at less than atmospheric
tends to be concentrated in zones of higher permeability, that
pressure may fully saturate the interstices to levels above
is, where the interstices are larger in size and have a better
those observed in wells. Thus the upper limit of the zone of
saturation and water table are not coincident. The capillary
interconnection.
In aquifers of high yield, velocities of 5–60 ft/d
fringe may be significant for sediments with small interstices
(1.5–18.3 m/d) are associated with hydraulic gradients of
and low permeability, such as clay.
10–20 ft/mi (1.89 to 3.78 m/km). Underflow through gravel
More than one zone of saturation occurs when an imper-
deposits may travel several hundred feet per day (m/d).
vious or semipervious layer or lens in the zone of aeration
supports a less extensive zone of saturation above the main
Depending on requirements, flows as low as a few feet per
water table, giving rise to the so-called perched water table.
year (m/year) may also be economically useful.
In homogeneous, isotropic aquifers, the dominant move-
If a porous stratum in the zone of saturation dips beneath
ment is in the direction of greatest slope of the water table
an impervious layer, the flow is confined in much the same vary considerably within the same geologic formation. Flow
way as in a pipe that drops below the hydraulic grade line. or piezometric surface. Where there are marked nonhomo-
There is no free surface in contact with the atmosphere in the geneities and anisotropies in permeability, the direction of
area of confinement. The water level in a well tapping this groundwater movement can be highly variable.
confined or artesian aquifer will rise, under pressure, above
the base of the confining layer to an elevation that defines the
piezometric level. If the recharge areas are at a sufficiently 3.7 DARCY’S LAW
high elevation, the pressure may be great enough to result in
Although other scientists were the first to propose that the
free-flowing wells or springs. An imaginary surface connect-
velocity of flow of water and other liquids through capillary
ing the piezometric levels at all points in an artesian aquifer
tubes is proportional to the first power of the hydraulic gra-
is called the piezometric surface (Fig. 3.1 depicts some of
dient, credit for verification of this observation and for its
these terms). The rise and fall of water levels in artesian
application to the flow of water through natural materials or,
wells result primarily from changes in pressure rather than
more specifically, its filtration through sand must go to Darcy.
from changes in storage volume. The seasonal fluctuations
The relationship known as Darcy’s law may be written as
are usually small compared with unconfined conditions.
Aquifers that are overlain or underlain by aquitards are v = K(dh∕dl) = KI (3.1)
called leaky aquifers. In natural materials, confining layers
seldom form an absolute barrier to groundwater movement. where v is the hypothetical, superficial or face velocity (darcy;
The magnitude of flow through the semipervious layer is not the actual velocity through the interstices of the soil)
called leakage. Although the vertical permeability of the through the gross cross-sectional area of the porous medium;
aquitard is very low and the movement of water through I = dh∕dl is the hydraulic gradient, or the loss of head per
it extremely slow, leakage can be significant because of the unit length in the direction of flow; and K is a constant of
large horizontal areas involved. proportionality known as hydraulic conductivity,orthe coef-
ficient of permeability. The actual velocity, known as effec-
tive velocity, varies from point to point. The average velocity
3.6 GROUNDWATER MOVEMENT
through pore space is given by
Groundwater in the natural state is constantly in motion. Its
v = KI∕ (3.2)
rate of movement under the force of gravity is governed e
by the frictional resistance to flow offered by the porous
where is the effective porosity. Because I is a dimensionless
medium. The difference in head between any two points pro-
ratio, K has the dimensions of velocity and is in fact the
vides the driving force. Water moves from levels of higher
velocity of flow associated with a hydraulic gradient of unity.
energy potential (or head) to levels of lower energy potential,
The proportionality coefficient in Darcy’s law, K, refers
the loss in head being dissipated as heat. Because the magni-
to the characteristics of both the porous medium and the fluid.
tudes of discharge, recharge, and storage fluctuate with time,
By dimensional analysis:
the head distribution at various locations is not stationary.
Groundwater flow is both unsteady and nonuniform. Com- K = Cd ∕ (3.3)
2
pared with surface water, the rate of groundwater movement
is generally very slow. Low velocities and the small size of where C is a dimensionless constant summarizing the geo-
passageways give rise to very low Reynolds numbers and metric properties of the medium affecting flow, d is a repre-
consequently the flow is almost always laminar. Turbulent sentative pore diameter, is the viscosity, and is the specific
2
flow may occur in cavernous limestones and volcanic rocks, weight of fluid. The product Cd depends on the properties
where the passageways may be large, or in coarse gravels, of the medium alone and is called the intrinsic or specific
2
particularly in the vicinity of a discharging well. Depend- permeability of a water-bearing medium, k = Cd .Ithas the
ing on the intrinsic permeability, the rate of movement can dimensions of area.