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Unformatted text preview: 5 5.1 \ Principles of Ground
Water Flow just as above the earth, small drops form and these join others, till finally water descends
in a body as rain, so too we must suppose that in the earth the water at first trickles
together little by little and that the sources of rivers drip, as it were, out of the earth and
then unite. Meteorologica, Aristotle (384—322 B.C.) INTRODUCTION Ground water possesses energy in mechanical, thermal, and chemical forms.
Because the amounts of energy vary spatially, ground water is forced to move
from one region to another in nature’s attempt to eliminate these energy differ
entials. The ﬂow of ground water is thus controlled by the laws of physics and
thermodynamics. To enable a separate examination of mechanical energy, we will
make the simplifying assumption that the water is of nearly constant temperature.
Thermal energy must be considered, however, in such applications as geothermal
ﬂow systems and burial of radioactive heat sources. There are three outside forces acting on the water contained in the
ground. The most obvious of these is gravity, which pulls water downward. The
second force is external pressure. Above the zone of saturation, atmospheric
pressure is acting. The combination of atmospheric pressure and the weight of
overlying water creates pressures in the zone of saturation. The third force is
molecular attraction, which causes water to adhere to solid surfaces. It also
creates surface tension in water when the water is exposed to air. The combina
tion of these two processes is responsible for the phenomenon of capillarity. When water in the ground is ﬂowing through a porous medium, there are
forces resisting the ﬂuid movement. These consist of the shear stresses acting 5.2 MECHANICAL —————ﬁ——— ENERGY physics. Of these, we will consider kinetic energy, gravitational potential energy,
and energy of ﬂuid pressures. A moving body or ﬂuid tends to remain i n motion, according to Newto
nian physics. This is because it possesses ene rgy due to its motion—kinetic
energy. This energy is equal to onehalf the product 0 l f l f its mass and the square of l the magnitude of the velocity: ii I
Ek = 1/2mv2 (5—1) t.
where Ek is the kinetic energy (MLZ/TZ; slugftZ/s2 or kgmz/sz)
v is the velocity (L/T; ft/s or m/s) If m is in kilograms and v in meters per second, then Ek has the units of kg'mZ/s2 or newtonmeters. The unit of energy is the joule, which is one newton
meter. The joule is also the unit of work. Imagine that a weightless container ﬁlled with water of mass m is moved
vertically upward a distance, z, from some reference surface (a datum). Work is w
done in moving the mass of water upward. This work is equal to g e(
W = Fz = (mg)z (5—2) of
i ha
where i
Wis work (MLz/Tz; slugftz/s2 or kg~m2/s2) i
z is the elevation of the center of gravity of the ﬂuid above the i
reference elevation (L; ft or m) Ste
m is the mass (M; slugs or kg) ani
g is the acceleration of gravity (L/TZ; ft/s2 or m/sz) ﬂu“
F is a force (ML/T2; slugft/s2 or kgm/sz) g?
The mass of water has now acquired energy equal to the work done in usef
lifting the mass. This is a potential energy, due to the position of the ﬂuid mass
with respect to the datum. Eg is gravitational potential energy: resu,
W = Eg = mgz (5—3)
A ﬂuid mass has another source of potential energy owmg to the pressure
‘ of the surrounding ﬂuid acting upon it Pressure 18 the force per unit area acting on This
a body f are J/
len th
P = F/A (5—4) ten‘i MECHANICAL ENERGY 133 where P is the pressure (M/LTZ; slug/fts2 or (kg‘m/s2)/m2) A is the crosssectional area perpendicular to the direction of the force
(L2; ft2 or m2) The units of pressure are pascals (Pa), or N/mz. A N/m2 is equal to a N~m/m3, or
J/m3. Pressure may thus be thought of as potential energy per unit volume of ﬂuid. For a unit volume of ﬂuid, the mass, m, is numerically equal to the
density, p, since density is deﬁned as mass per unit volume. The total energy of
the unit volume of ﬂuid is the sum of the three components—kinetic, gravita
tional, and ﬂuidpressure energy: EN = l/2pv2 + pgz + P (5—5) where EW is the total energy per unit volume. If Equation 5—5 is divided by p, the result is total energy per unit mass,
E tm' 2
v P
E,m=§+gz+3 (56)
which is known as the Bernoulli equation. The derivation of the Bernoulli
equation may be found in textbooks on ﬂuid mechanics (Streeter 1962). For steady ﬂow of a frictionless, incompressible ﬂuid along a smooth line P
—— + g2 + E = constant (5—7) Steady ﬂow indicates that the conditions do not change with time. The density of
an incompressible ﬂuid would not change with changes in pressure. A frictionless
ﬂuid would not require energy to overcome resistance to ﬂow. An ideal ﬂuid
would have both of these characteristics; real ﬂuids have neither one. Real ﬂuids
are compressible and do suffer frictional ﬂow losses; however, Equation 5—7 is
useful for purposes of comparing the components of mechanical energy. If each term of Equation 5—7 is divided by g, the following expression
results: 2
P v— + z + — = constant (5—8) 2g pg This equation expresses all terms in units of energy per unit weight. These
are J/N, or m. Thus, Equation 5—8 has the advantage of having all units in
length dimensions (L). The ﬁrst term of v2/2g is (m/s)2/(m/s2), or m; the second
term, z, is already in m; and the third term, P/pg, is Pa/(kg/m3)(m/s2), or 134 5.3 PRINCIPLES OF GROUNDWATER FLOW FIGURE 5.1 Piezometer measuring fluid pressure and the elevation of water. \ (N/m2)/(kg/m3)(m/sz), which reduces to m. The sum of these three factors is the
total mechanical energy per unit weight, known as the hydraulic head, h. This is
usually measured in the ﬁeld or laboratory in units of length. HYDRAULIC HEAD A piezometer* is used to measure the total energy of the ﬂuid ﬂowing through a
pipe packed with sand, as shown in Figure 5 .1. The piezometer is open at the top
and bottom, and water rises in it in direct proportion to the total ﬂuid energy at the
point at which the bottom of the piezometer is open in the sand. At point A, which
is at an elevation, z, above a datum, there is a ﬂuid pressure, P. The ﬂuid is ﬂowing
at a velocity, v. The total energy per unit mass can be found from Equation 5—6. i ...
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 Fall '10
 ROOT
 Hydrogeology, Energy, Meteorologica

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