Surface water and groundwater flow
Global water budget
 Earth's water inventory (Fig)
 groundwater in the hydrologic cycle (Fig)
 hydrological cyclereservoirs and fluxes (Fig)
 there is significant uncertainty in some of these numbers (Fig)
 residence times
 atmosphere: 2 weeks
 surface waters: 4 years
 groundwater: 20,000 years
Surface water flow
Forces on fluids
 the basic forces that make fluids move are gravity, pressure
differences,
and surface stresses
 pressure is a normal stress
Bernoulli equation
 let us look at the movement of a fluid in
a
pipe
(Fig3.5)
 the statement of the conservation of
energy
for a
frictionless fluid along a flowline is the following:
 p*V + m*g*z + 1/2*mu^{2} =
constant,
means:
 work due to pressure + potential energy +
kinetic
energy
 by dividing this equation by m = v*r*g
we get the Bernoulli equation:
 u^{2}/(2*g) + z +p/(r*g)
= constant, means:
 velocity head + elevation head + pressure
head =
total head
 this head can be measured by looking at
the
level
to which the water rises in a vertical tube stuck into the pipe
 assumptions we are making:
 no friction (viscosity = 0)
 incompressible fluid
 homogeneous fluid
 flow steady with time
 example:
expansion joint in a hose (Fig3.7)
 apply continuity equation Q = u*A
=
const,
then Bernoulli
equation
 decrease of mean velocity results
in
increase of
pressure
 experiment: blow on sheet of paper
Friction
 in reality, if we do an experiment, we do
see
a loss
in head along a pipe due to friction (Fig3.8)
 instead of having a uniform value in the
pipe, there
is a velocity profile (Fig3.9)
 u^{2}/(2*g) + z +p/(r*g)
= constant needs to be modified to: u^{2}/(2*g) + z +p/(r*g)
+h_{L}= constant
 the head loss can be described as: h_{L}
= f*L*u^{2}/(D*2*g), f being the friction factor
 the friction factor has been measured
under a
range
of circumstances (Fig3.10
for smooth pipes)
 The key parameter to describe a flow
situation is
the Reynolds number: R = r*U*D/m
 at R>2000 something happens, the flow
becomes turbulent
 Reynolds experiment, laminar
("layered") flow,
turbulent ("restless") flow (Fig3.11)
 is flow in a straw laminar or turbulent?
 let us calculate R for an example:
 U=2 m/s
 D = 0.03 m
 r = 10^{3}
kg/m^{3}
 m =
1.139
10^{3}
Pa s
 how can we make the flow laminar?
 reducing the velocity;
 reducing the diameter;
 reducing the density of the fluid; or
 increasing the viscosity. reduce
Flow in channels
 in most cases, friction does play a big
role
in stream
flow, so we are actually loosing head in the flow (Roughness
of a stream)
 flow in a channel is described by
Manning's
equation (Fig4.8):
R_{H}=wh/(2h+w): Hydraulic radius
k=1 m^{1/3}s^{1}
S: slope
n: roughness coefficient
Measuring flow in natural channels
 weirs can be used to measure discharge
rates

 we can look impirically at the discharrge
rate vs.
height relationship for a stream without a welldefined weir and can
can
use this to estimate the discharge rate (Fig4.13,
for this particular creek the exponent in the empirical relationship
between
Q and h is 2.85).
 schematic diagram of a stream gaging
station (Fig5.2)
 velocity distribution as a function of
depth
(Fig4.10),
most flow is turbulent, average flow velocity is at a depth of
0.6*total
depth
 we can measure the discharge rate by
taking
measurements
at 0.6* depth (from surface) wading through the river (Fig4.11)
 Q = U*A = S w_{i}*h_{i}*U_{i}
 Small
stream gaging
 Gaging
larger streams and rivers
 stream gaging on the Colorado River at Lee's Ferry
Groundwater
 groundwater is the water in the saturated zone (Fig)
(Fig)
 recharge is the water entering the saturated zone
 in many parts of the world, groundwater is the only source of
fresh
water
 in the US about 10% of the rainfall becomes groundwater
eventually.
This
amount equals the annual use of water in the US, about 3 inch per year
 Water use in the US (Fig)
 water may stay in the groundwater reservoir between several days
and
thousands
of years. We will discuss tracer techniques that may be used to derive
residence times later in the class
 management of catchment areas requires understanding of
groundwater
flow
 many environmental issues involve groundwater
Conceptual model of groundwater flow
the flow of water through a porous medium (Fig
6.1)
water flows tortuous paths
geometry of channels is very complex
frictionles flow is totally meaningless!
conceptual model of flow through a porous medium is flow through
a
bundle
of very small (capillary) tubes of different diameters (Fig
6.2)
the flow (Q) through a horizontal tube can be described as: Q = p*D^{4}/(128*m)*dp/dx
(Poiseuille's law)
=> size of the capillary tubes is important!
Darcy's law
 what drives groundwater flow?
 water flows from high elevation to low elevation and from high
pressure
to low pressure, gradients in potential energy drive groundwater flow
 two ways how to derive the potential energy:
 1: Bernoulli
 Bernoulli equation said: u^{2}/(2*g)
+ z
+p/(r*g) = constant, means: velocity head +
elevation head + pressure head = total head
 in groundwater flow, we cannot make
the
assumption
that there is no friction, therefore the head is not constant
 also u is so small that that term can
be
typically
neglected
 2: direct determination of potential
energy
 lifting up a parcel of water: W_{1}=m*g*z
 creating space for it in the
groundwater
system:
W_{2}=p*V
 total work/m = g*z + p/r or
h = z + p/rg
 groundwater flows from high to
low
head
 how do you measure the head or potential? => drill an
observation
well,
the elevation of the water level in the well is a measure of the
potential
energy at the opening of the well: h=z + p/rg
= z + F/(Arg) ......
 in 1856, a French hydraulic engineer named Henry Darcy published
an
equation
for flow through a porous medium that today bears his name (Fig.
6.3)
 Q = KA (h_{1}h_{2})/L or q = Q/A = K dh/dl, h: hydraulic
head, h = p/rg + z
 q = Q/A is the specific discharge [L/T], dh/dl
is the hydraulic
gradient
 K is the hydraulic conductivity
[L/T]
 the law is very similar to Ohm's law for electrical
curcuits I =
1/R * U (current = voltage divided by resistance)
 the orginal Darcy experiment yielded
these
data (Fig
6.4)
 the analogy between Darcy's law and
Poiseulle's law
suggests that K = (const*d^{2})*rg/m
 the first term (const*d^{2}) is
k,
the intrinsic permeability [L^{2}], summarized the
properties
of the porous medium, while rg/m describe
the fluid
 hydraulic conductivities and permeabilities vary over many orders
of
magnitude
(Fig 6.5)
Example: calculation of a typical hydraulic gradient of 1/100 in a
salt formation with a hydraulic conductivity of 10^{10 }m s^{1}
will produce a specific discharge of 10^{12} m s1, or less
than
1 mm per 30 years!  specific discharge has the dimension of a
velocity, but it is not
the velocity
at which the water flows in the porous medium, the water has to squeeze
through the pores
 tagged parcels that are averaged together, will appear to
move
through
a porous medium at a rate that is faster than the specific discharge
 porosity is the fraction of a porous material which is
void
space f =
V_{void}/V_{total}
 the mean pore water velocity is then: v = q/f
(Fig)
 Darcy's law has been found to be invalid for high values of
Reynolds
number
and at very low values of hydraulic gradient in some very
lowpermeability
materials, such as clays.
 example :
 K= 10^{5} m/s, h_{2}h_{1}
= 100m, L = 10km, A = 1m^{2} > Q = 3.15 m^{3}/y; the
K
value above is typical for a sandstone aquifer
 the actual flow velocity v may be
calculated with
the following formula: v=Q/(A*f)=q/f, f
is the porosity, and q the specific discharge
 if the porosity n is 30%, the flow
velocity
in the
example above is 10.5 m/y
Resources
Freeze, R.A. and Cherry, J.A.
(1979)
Groundwater.
Prentice Hall, 604p.
Hornberger, G.M., Raffensberger, J.P.,
Wiberg, P.L.,
and Eshleman, K.N. (1998) Elements
of physical hydrology. Johns Hopkins University Press,
Baltimore,
302p.
see also BC 3025
Hydrology
class