Hydrology BC ENV 3025
Darcy's law
(Fig)(Fig
6.1)(Fig 6.2)(Fig.
6.3)(Fig
6.4)(Fig 6.5)(Fig)(Fig
6.6)(Fig 6.7)(Fig
6.8)(Fig 6.9)(Fig
6.10)(Fig 6.11)
Introduction
- groundwater is the water in the saturated zone (Fig)
- recharge is the water entering the saturated zone
- 30% of freshwater on Earth trapped below the surface
- 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
- residence time = reservoir/flux = ~1000 m / 3 inch/year = 10,000
y!
This
is a very rough estimate.
- 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
- WIPP site case study (A Tour of
the
WIPP Site)
- underground repository built for certain radioactive wastes
- not high-level, but they contain isotopes that remain
radioactive for
very
long periods of time (tens of thousands of years)
- storage of waste in salt formations
- quantitative description of groundwater flow necessary to
evaluate risk
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 = -ρ*D4/(128*μ)*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
- Bernoulli equation said: u2/(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 (example!)
- 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
- 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 (h1-h2)/L or q = Q/A = -K dh/dl, h: hydraulic
head, h = p/ρg + z
- thought experiment: hydraulic head
distribution in
a lake
- 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*d2)*ρg/μ
- the first term (const*d2) is
k,
the
intrinsic permeability [L2], summarized the
properties
of the porous medium, while ρg/μ 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 s-1,
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 θ
=
Vvoid/Vtotal
- the mean pore water velocity is then: v = q/θ
(Fig)
(experiment)
- 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
low-permeability
materials, such as clays.
- example :
- K= 10-5 m/s, h2-h1
= 100m, L = 10km, A = 1m2 > Q = 3.15 m3/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*θ
)=q/θ
, θ
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
Water in natural formations
- an aquifer is a saturated geological formation that
contains
and
transmits "significant" quantities of water under normal field
conditions
(=> gravel, sand, volcanic and igneous rocks, limestone) (Fig
6.6)
- an aquitard is a formation with relatively low
permeability
- an aquiclude is a formation that may contain water but
does not
transmit significant quantities (clays and shales)
- an aquifuge is a
foamtion that does not contain or transmit significant amounts of water
| formation |
contains
water |
permeability |
| aquifer |
Y |
high |
| aquitard |
Y |
low |
| aquiclude |
Y |
very low |
| aquifuge |
N |
negligible |
- confined and unconfined (water-table) aquifers
- an unconfined aquifer has a water table (water table aquifer)
- a confined aquifer does not have a water table. If you drill a
well,
water
will rise (in the well) above the top of the aquifer
- perched groundwater is groundwater sitting on top of a poorly
permeable
layer with an unconfined aquifer underneath
- the height to which water rises in a well defines the piezometric
or potentiometric surface
- geology of aquifers (show examples)
- unconsolidated sediments: loose granular deposit, particles are
not
cemented
together (e.g.: Long Island)
- consolidated sediments, most important: sandstone, porosity
varies
depending
on the degree of compaction (e.g. Zion, Bryce, and Grand Canyon
National
Parks)
- limestone: composed mainly of calcium carbonate, CO2 rich water
dissolves
limestone, e.g.: limestone caves, karst (e.g. Floridan aquifer)
- volcanic rock
- basalt lava, fractures (e.g.: Hawaii, Palisades)
- crystalline rocks: igneous and metamorphic rocks, e.g. Granite,
have
often
very low porosity, flow through fractures
- porosities and hydraulic conductivities of different aquifer
rocks (Fig
6.5)
Steady groundwater flow
- flow in a horizontal confined aquifer (Fig
6.7)
- lines of equal hydraulic head are called equipotentials
- flow occurs perpendicularly to those, lines indicating those are
called flowlines
- together, the equipotentials and the streamlines constitute a flow
net
(Fig
6.8)
- generally, groundwater flow follows topography, in detail the
situation
can be more complicated though
- groundwater flow not only occurs near the water table, but does
penetrate
deep into the aquifer (Fig 6.9)
- flownets provide a lot of information about groundwater flow,
they are
generated by computer models these days
Quantifying groundwater flow using flownets
- T = Kb [L2 T-1] is called the
transmissivity
of the aquifer, this term is often the more useful parameter for
estimating
the yield of an aquifer, it is relevant when we want to estimate the
discharge
per unit length of stream, for example
- the area between a pair of streamlines is referred to as a streamtube
- in more complicated flow nets, these squares might become
"curvilinear
squares," as can be seen in (Fig
6.9)
- if we isolate one of these squares (ds=dm) (Fig
6.10) and make use of Darcy's law, we can calculate the discharge
through
the streamtube: Q = K*b*dh
- you can imagine each streamtube as a "pipe," because water cannot
cross
a streamline
- the specific discharge will be greatest where the streamtube is
narrowest,
analogue to the laminar flow table
- the total discharge through the streamtube must be the same at
any
cross
section
- by counting the number of stream tubes, we can determine the
total flow
- another examle is steady flow under a dam (Fig
6.11)
- the dam is 100m wide (direction into the page), and the
hydraulic
conductivity
beneath the dam is 10-10 m s-1
- we use the length of the dam (100 m) in place of the aquifer
thickness
(b)
Heterogeneity and anisotropy
- so far we have considered only homogeneous aquifer (the same K
everywhere)
- virtually all natural materials through which groundwater flows
display
variations in intrinsic permeability from point to point, this is
referred
to as heterogeneity (Example: Fig)
- permeable zones tend to focus groundwater flow, while,
conversely, flow
tends to avoid less permeable zones (Exploring
further,
explore6.doc)
- in anisotropic media the permeability depends on the direction of
measurement,
in isotropic media, it does not
Resources
Manning, J.C. (1997) Applied Principles of
Hydrology.
Prentice Hall, third edition, 276p.
Freeze, R.A. and Cherry, J.A. (1979)
Groundwater.
Prentice Hall, 604p.
WIPP
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