Hydrogeology GEOL 611
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 highlevel, 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 = 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

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 (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 (h_{1}h_{2})/L or q = Q/A = K dh/dl, h: hydraulic
head, h = p/rg + 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*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 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
f
=
V_{void}/V_{total}

the mean pore water velocity is then: v = q/f (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 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
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 (watertable) 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 [L^{2 }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 (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.