flow in porous media - autenticação · flow in porous media luÍs ribeiro instituto ... saturated...
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FLOW IN POROUS MEDIA
LUÍS RIBEIRO
INSTITUTO SUPERIOR TECNICO
UNIVERSITY OF LISBON
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Where does groundwater flow?
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How water flows?
Groundwater moves from areas of high hydraulic head to
areas of low hydraulic head
© United States Environmental Protection Agency
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Type of hydrogeological formations
Aquifer A formation, group of formations, or part of
a formation that contains sufficient saturated
permeable material to yield significant quantities of
water to wells and springs (after Lohman and others,
1972).
Aquitard - A confining bed that retards but does not
prevent the flow of water to or from an adjacent
aquifer; a leaky confining bed. It does not readily
yield water to wells or springs, but may serve as a
storage unit for ground water (AGI, 1980)
Aquiclude - A hydrogeologic unit which, although
porous and capable of storing water, does not
transmit it at rates sufficient to furnish an appreciable
supply for a well or spring (after WMO, 1974).
Aquifuge A hydrogeologic unit which has no
interconnected openings and, hence cannot store or
transmit water.
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Groundwater residence times
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Water in the ground
Source: © United States Geological Survey
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Ψ < 0
K e θ are function of ψ
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Type of aquifers
Unconfined aquifer
Ground surface
Limit of the saturated zone
=
Phreatic level
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Type of aquifers
Confined aquifer
What does it happen when the saturated zone reaches
the top of the aquifer?
Note that:
The pressure that water places on
top of the aquifer can be measured
by measuring the height h that
water rises in a piezometric tube
that taps the confined aquifer
Piezometric
level
Limit of the
saturated zone
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Type of aquifers
Perched aquifer
PERCHED AQUIFERS are aquifers that have a confining layer below the
groundwater, and sits above the main water table.
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Effects of groundwater pumping
Source: © United States Geological Survey
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Artesian wells
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DARCY LAW
• The French municipal hydraulic engineer Henry Darcy
(1803–1858) studied the movement of water through
sand and from empirical observations defined the basic
equation, universally known as Darcy’s law, that
governs groundwater flow in most alluvial and
sedimentary formations.
• Darcy’s law is the foundation of the theoretical aspects
of groundwater flow
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DARCY LAW
Q/A = - K Δh / Δ l
q = - K i
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Darcy velocity
• The term q, referred to as the specific discharge,
has the dimensions of velocity and is also known
as the darcy velocity or darcy flux.
• It is important to remember that the darcy velocity
is not the true, microscopic velocity of the water
moving along winding flow paths within the soil
or rock.
• Instead, by dividing the specific discharge by the
fraction of open space (in other words, effective
porosity) through which groundwater flows across
a given sectional area, this provides an average
measure of groundwater velocity.
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Macroscopic Law
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Validity of Darcy law
Re – number of Reynolds
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q = K im
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Hydraulic conductivity and
permeability
• K – Hydraulic Conductivity– [ LT-1]
k - Permeability [L2]
K = k ( ρ g / μ )
k = C d2
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DENSITY AND VISCOSITY
• The density and viscosity of water are
functions of temperature and pressure but
these effects are not great for the ranges of
temperature and pressure encountered in
most groundwater situations
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POROSITY
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θ
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Heterogeneity
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Histogram: ce
K-S d=.18063, p<.20 ; Lilliefors p<.01
Expected Normal
-1 0 1 1 2 2 3 3 4 4
X <= Category Boundary
0
2
4
6
8
10
12
14
16N
o.
of
ob
s.
K is lognormal distributed
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K is a tensor
9 components
3 components,
Main directions
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Heterogeneity vs Anisotropy
i
ii
xm
mKK
ii
i
zKm
mK
/
Kx
Kz
mi – thickness of layer i
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Head as Energy
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Kinetic Energy . Energy required to accelerate fluid packet
from velocity v1 to velocity v2.
Gravitational work. Energy required to raise fluid packet from
elevation z1 to elevation z2.
Eq 1
Eq 2
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Pressure work. Energy required to raise fluid packet
pressure from P1 to P2.
assuming a unit mass of incompressible fluid
Eq 3
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the sum of eq1, eq2 and eq3 is the total mechanical energy for the
unit mass (i.e. m = 1)
Assuming that v is small (true for flow in porous
media).
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Hydraulic head
h = ψ + z [ L ]
datum
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HYDRAULIC HEAD
• Equation confirms that the hydraulic head at
a point within a saturated porous material is
the sum of the elevation head, z, and
pressure head, ψ, thus providing a
relationship that is basic to an
understanding of groundwater flow.
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EQUIPOTENTIAL LINES
• Observation boreholes and piezometers located
within a district provide a picture of the three-
dimensional distribution of hydraulic head
throughout an aquifer system.
• Lines drawn joining points of equal groundwater
head, or groundwater potential, are termed
equipotential lines.
• Lines perpendicular to the equipotential lines are
flow lines and can be used in the construction of a
flow net
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EQUIPOTENTIAL CONTOURS
• In plan view, the construction of equipotential contours
results in a map of the potentiometric surface. In an
unconfined aquifer, the potentiometric surface is a map of
the water table, where the groundwater is by definition at
atmospheric pressure.
• In a confined aquifer the potentiometric surface predicts
the position that the water level would rise to in a borehole
that penetrates the buried aquifer.
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• The areas of high hydraulic head may be
interpreted as groundwater recharge zones
while areas of low hydraulic head are
typically in groundwater discharge zones.
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GROUNDWATER FLOW THEORY
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• At the beginning of the last century, Meinzer and Hard (1925) observed in a
study of the Dakota sandstone
• that more water was pumped from the region than could be accounted
for (as water was pumped, a cone of depression developed and the rate
of abstraction decreased, but with no apparent effect on groundwater
levels in the recharge zone), such that the water-bearing formation was
demonstrating elastic behaviour in releasing water from storage.
• Later, in deriving the general partial differential equation describing
transient groundwater flow, Jacob (1940) formally described the elastic
behaviour of porous rocks.
• There are two mechanisms that explain how water is produced by confined
aquifers: the porosity of the aquifer is reduced by compaction and
groundwater is released; and the water itself expands since water is
slightly compressible
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The total downward stress, σT, applied at the top of
a confined aquifer is supported by an upward
effective stress, σe, on the aquifer material, and the
water pressure contained in the pore space Pw.
eq1
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• If the pore water pressure is decreased by
groundwater pumping or by natural
groundwater outflow, the stress on the
aquifer material will increase causing it to
undergo compression.
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Specific storage
• Specific storage represents the volume
of water that an aquifer releases from
storage per unit surface area of aquifer
per unit decline in the component of
hydraulic head normal to that surface
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Compressibility of water
The compressibility of water β is defined as:
eq2
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Compressibility of aquifer
material
The compressibility of aquifer materiasl α is defined as:
eq3
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eq4
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dσe = 0 − ρgdψ = −ρgdh
For a unit decline in head, dh = −1, and if unit volume
is assumed (VT = 1), then eq4 becomes:
dVw = α(1)(−ρg)(−1) = αρg
The water produced by the expansion of water is
found from eq2 thus:
dVw = −βVwdPw
eq5
eq6
eq7
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• Recognizing that the volume of water, Vw, in the
total unit volume of aquifer material, VT, is nVT
where n is porosity, and that dP = ρgdψ or −ρg for
a unit decline in hydraulic head (where ψ = h − z
with z remaining constant), then for unit volume,
VT = 1 :
dVw = −βn(1)(−ρg) = βnρg eq8
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eq9
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• In other words, groundwater pumped from a confined
aquifer does not represent a dewatering of the physical
pore space in the aquifer but, instead, results from the
secondary effects of aquifer compaction and water
expansion. As a consequence, for an equivalent unit
decline in hydraulic head, yields from confined aquifers
are much less than from unconfined aquifers.
• Hence, storage coefficient values of confined aquifers are
much smaller than for unconfined aquifers.
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Transmissivity
• T = K x b [L2T-1]
It represents the rate at which water of a given density and
viscosity is transmitted through a unit width of aquifer or aquitard
under a unit hydraulic gradient.
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• T > 0.015 m2/s
• S entre 0.005 e 0.00005
VALUES OF GOOD AQUIFER
PRODUCTIVITY
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Transmissivity and specific yield of
unconfined aquifers
• For an unconfined aquifer, the transmissivity is not as well defined as
in a confined aquifer, but the equation can be applied with b now
representing the saturated thickness of the aquifer or the height of the
water table above the top of a lower aquitard boundary.
• The transmissivity will, therefore, vary if there are large seasonal
fluctuations in the elevation of the water table or if the saturated
thickness of the aquifer shows lateral variation as a result of an
irregular lower aquitard boundary or differences between recharge and
discharge areas in the same aquifer.
• The storage term for an unconfined aquifer is known as the specific
yield, Sy, (or the unconfined storativity)
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Equations of groundwater flow
• Equations of groundwater flow are derived from a
consideration of the basic flow law, Darcy’s law , and an
equation of continuity that describes the conservation of
fluid mass during flow through a porous material
• Under steady-state conditions, the magnitude and
direction of the flow velocity at any point are constant with
time.
• For transient conditions, either the magnitude or direction
of the flow velocity at any point may change with time, or
the potentiometric conditions may change as groundwater
either enters into or is released from storage.
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Steady-state saturated flow
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Equation of continuity
If the fluid is incompressible, then density, ρ(x, y, z), is
constant and previous equation becomes :
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From Darcy’s law, each of the specific discharge terms can be
expressed as:
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Transient saturated flow
The law of conservation of mass for transient flow in a
saturated porous material requires that the net rate of fluid
mass flow into the control volume is equal to the time rate of
change of fluid mass storage within the control volume.
The equation of continuity is now:
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• The first term on the right-hand side of equation describes
the mass rate of water produced by expansion of the water
under a change in its density, ρ, and is controlled by the
compressibility of the fluid, β. The second term is the mass
rate of water produced by the compaction of the porous
material as influenced by the change in its porosity, n, and
is determined by the compressibility of the aquifer, α.
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By expanding the terms on the left-hand side of equation using
the chain rule (eliminating the smaller density gradient terms compared
with the larger specific discharge gradient terms) and, at the same time,
inserting Darcy’s law to define the specific discharge terms, then:
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LEAKAGE
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LEAKAGE MEASURES
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LEAKAGE RATE
h1
h2
h1
h2
h1 > h2 h1 < h2
1 – unconfined aquifer ; 2 – confined aquifer 3 – aquitard
2
1
2
1
3 3
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0m 5000m 10000m
Arade
Alcantarilha
Quarteira
AQUIFER SYSTEM OF QUERENÇA- SILVES
Influent and Efluent river sector streams