FLOW IN POROUS MEDIA

LUÍS RIBEIRO

INSTITUTO SUPERIOR TECNICO

UNIVERSITY OF LISBON

Where does groundwater flow?

How water flows?

Groundwater moves from areas of high hydraulic head to

areas of low hydraulic head

© United States Environmental Protection Agency

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.

Groundwater residence times

Water in the ground

Source: © United States Geological Survey

Ψ < 0

K e θ are function of ψ

Type of aquifers

Unconfined aquifer

Ground surface

Limit of the saturated zone

=

Phreatic level

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

Type of aquifers

Perched aquifer

PERCHED AQUIFERS are aquifers that have a confining layer below the

groundwater, and sits above the main water table.

Effects of groundwater pumping

Source: © United States Geological Survey

Artesian wells

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

DARCY LAW

Q/A = - K Δh / Δ l

q = - K i

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.

Macroscopic Law

Validity of Darcy law

Re – number of Reynolds

q = K im

Hydraulic conductivity and

permeability

• K – Hydraulic Conductivity– [ LT-1]

k - Permeability [L2]

K = k ( ρ g / μ )

k = C d2

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

POROSITY

θ

Heterogeneity

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

K is a tensor

9 components

3 components,

Main directions

Heterogeneity vs Anisotropy

i

ii

xm

mKK

ii

i

zKm

mK

/

Kx

Kz

mi – thickness of layer i

Head as Energy

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

Pressure work. Energy required to raise fluid packet

pressure from P1 to P2.

assuming a unit mass of incompressible fluid

Eq 3

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).

Hydraulic head

h = ψ + z [ L ]

datum

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.

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

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.

• 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.

GROUNDWATER FLOW THEORY

• 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

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

• 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.

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

Compressibility of water

The compressibility of water β is defined as:

eq2

Compressibility of aquifer

material

The compressibility of aquifer materiasl α is defined as:

eq3

eq4

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

• 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

eq9

• 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.

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.

• T > 0.015 m2/s

• S entre 0.005 e 0.00005

VALUES OF GOOD AQUIFER

PRODUCTIVITY

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)

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.

Steady-state saturated flow

Equation of continuity

If the fluid is incompressible, then density, ρ(x, y, z), is

constant and previous equation becomes :

From Darcy’s law, each of the specific discharge terms can be

expressed as:

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:

• 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.

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:

LEAKAGE

LEAKAGE MEASURES

LEAKAGE RATE

h1

h2

h1

h2

h1 > h2 h1 < h2

1 – unconfined aquifer ; 2 – confined aquifer 3 – aquitard

2

1

2

1

3 3

0m 5000m 10000m

Arade

Alcantarilha

Quarteira

AQUIFER SYSTEM OF QUERENÇA- SILVES

Influent and Efluent river sector streams